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 THE 
 
 ELEMENTS OF MECHANICS, 
 
 COMPREHENDING 
 
 STATICS AND DYNAMICS. 
 
 WITH 
 A COPIOCB COLLECTION OF ' 
 
 MECHANICAL PROBLEMS. 
 
 INTENDED FOR THE DSE OF 
 
 MATHEMATICAL STUDENTS IN SCHOOLS AND UNIVERSITIES. 
 WITH NUMEROUS PLATES. 
 
 BY J. R. YOUNG, 
 
 ▲CTROR OF "THE ELEMENTS OF ANALYTICAL GEOMETRY;" "ELEMENTS OF THE 
 DiFFERENTIAt AND INTEGRAL CALCULUS." 
 
 REVISED AND CORRECTED 
 
 BY JOHN D. WILLIAMS, 
 
 ACTHOR OF "KEY TO BUTTON'S M A TH E M ATI C S," &C. 
 
 PHILADELPHIA : 
 HOGAN AND THOMPSON, 
 
 No. 30, NORTH FOURTH STREET. 
 
 1839. 
 
 JOHM S. PRELL 
 
 Gvil & Mechanical Engineer. 
 
 SAN FRANCISCO, CAL.
 
 Entered, accordincr to the Act of Congress, in the year 1834, by 
 Carky, Lea & Blanchard, in the Clerk's Office of the District Court of 
 the Eastern District of Pennsylvania. 
 
 STEREOTYPED BY U JOHNSON, 
 miLADELFHIA. 
 
 •J3fiq .2 VIHOl
 
 Ulrary 
 
 PREFACE. i^3l 
 
 The following Treatise is an attempt to exhibit, in small 
 compass, the principles of Mechanical Science in its present 
 improved state, and to supply the English student with a 
 clear and comprehensive manual of instruction on this im- 
 portant branch of Natural Philosophy. 
 
 Our language already possesses some very valuable works 
 in this department of science, as for instance, the treatises of 
 Professors Gregory and Whewell ; works Avhich, for the abun- 
 dance of real information that they convey, are not, perhaps, 
 to be equalled by any similar performances of our continental 
 neighbours. 
 
 The bulk and consequent high price, however, of these 
 works must necessarily place them beyond the reach of many 
 students desirous to be informed on the subjects of which they 
 treat ; and there can, I think, be no doubt that at a time like 
 the present, when a taste for analytical science is so widely 
 extending itself, a treatise, of moderate price, on Analytical 
 Mechanics, if well executed, would prove acceptable both to 
 teachers and to students. 
 
 Under these impressions I have been led to undertake this 
 elementary Treatise, with the hope that by economizing the pa- 
 per, and adopting a small clear type. I might be able to compress 
 into one small volume a course of instruction on elementaiy 
 mechanics, of extent amply sufficient for all the purposes of 
 academical education. My desire, however, having been to 
 teach the elements of the science, not to write a book, I have 
 spared no pains to render the whole clear and intelligible ; to 
 develope the several theories with as much simplicity as I 
 could ; to explain fully the meaning and extent of the various 
 analytical expressions in which these theories are embodied ; 
 and finally, to illustrate each by a sufficient number of useful 
 and interesting practical examples. 
 
 To what extent these endeavours may have been successful, 
 it is for others now to determine; but, from the very flattering 
 reception which my former mathematical publications have 
 met with, both in this country and in America, I am encou- 
 
 3 
 
 7137S5 

 
 i PREFACE. 
 
 raged to hope tlial the present volume will be found, upon ex- 
 amination, not altogether undeserving of notice. 
 
 The work is divided into two principal divisions, Statics 
 or the theory of Equilibrium, and Dynamics or the theory of 
 Motion ; and these are again subdivided into sections and 
 chapters. A very short account of these will suffice here, as 
 a copious analysis is presented in the table of contents. 
 
 The first section of the Statics treats on the equilibrium of a 
 point, viewed under two aspects : first, as entirely free ; and 
 second, when constrained to rest on a given curve or surface. 
 Into this section too is introduced the theory of the funicular 
 polygon and catenary: this, I am aware, is not in strict accord- 
 ance with a scientific arrangement of the parts of the subject, 
 nor do I consider such arrangement to be absolutely essential in 
 an elementary treatise ; the principle which with me has all 
 along governed the arrangement is this, viz., so to dispose the 
 several topics that each may present itself to the student pre- 
 cisely at that place where he is best prepared to receive it, 
 and thus the acquisition of the whole be facilitated. 
 
 The second section is on the theory of the equilibrium of 
 a rigid body delivered in all its generality, and applied to a 
 variety of examples. The closing ciiapter of this section is 
 devoted to a subject of considerable practical importance, the 
 strength and stress of beams ; for the principal materials of 
 it I am indebted to Mr. Barlow's experimental inquiries on 
 this subject. 
 
 These two sections comprise the theory of Statics ; the se-~^ 
 cond part, or Dynamics, is divided into three sections ; the 
 first being on the rectilinear motion of a free point, the se- 
 cond on its curvilinear motion, and the third discussing the 
 general theory of the motion of a solid body. 
 
 In the opening chapter of tlie first section, the fundamental 
 equations of motion are deduced from simple and obvious con- 
 siderations, and pains are taken to give a clear and distinct 
 meaning to the several analytical expressions involved in 
 these fundamental equations. All these are then fully illus- 
 trated by interesting practical exercises. 
 
 In the second section a pretty comprehensive view is taken 
 of the theory of curvilinear motion ; and some attempts have 
 been made to simplify those p^irts of it which seemed most to 
 require simplification. 
 
 The last section, on the motion of a solid body, is the most 
 A 2
 
 PREFACE. 5 
 
 extensive, as well as the most difficult, and will be found to 
 embrace a great variety of important particulars, treated, I hope, 
 with sufficient clearness to be abundantly intelligible to an 
 ordinary student ; several interesting dynamical problems are 
 interspersed throughout this section, and to the end is append- 
 ed a miscellaneous collection, as further illustrative of the use 
 and application of the general theories before established, 
 especially o{ the Principle of D'Alemhert^ow account of the 
 importance of this principle in a great variety of dynamical 
 inquiries. 
 
 For the manner in which the subjects here briefly enume- 
 rated are discussed, I must now refer to the book itself ; and 
 shall be glad if it be thought calculated to promote, in any de- 
 gree, the study of the science, or to form a useful introduction 
 to works of higher pretensions and of acknowledged ability. 
 
 J. R. YOUNG. 
 April 10, 1832.
 
 CONTENTS. 
 
 PART I. 
 
 ELEMENTS OF STATICS. 
 
 Introduction. 
 
 Article Page 
 
 1. MECHASfics defined - - - - - -13 
 
 2. Force defined - - - - - - - ib. 
 
 3. Equilibrium of a point acted on by two equal and opposite forces - ib. 
 
 4. Linear measure of force - - - - - - 14 
 
 5. The point of application of a force may be anywhere in the line of its 
 
 direction - - - - - - - -ib. 
 
 On the Equilibrium of a Point, 
 
 6. The resultant of forces acting in the same straight line is equal to the 
 
 algebraical sum of those forces - - - - - 17 
 
 7. The resultant of several forces in one plane is in that plane - - 18 
 
 8. When the intensities of equilibrating forces vary, the resultant varies 
 
 in the same ratio, but retains its direction - - - - ib. 
 
 9. If three equal forces are inclined to one another in angles each of 
 
 120°, any one will balance the other two - - - - ib. 
 
 10. Two of these forces are equivalent to a third, represented in direction 
 
 and intensity by the diagonal of the rhombus constructed on the 
 lines which represent the forces - - - - - ib. 
 
 11. Any two equal forces, represented by the sides of a rhombus, are equiva- 
 
 lent to the force represented by the diagonal - - - 19 
 
 12. Any two forces represented by the sides of a rectangle are equivalent to 
 
 the single force represented by the diagonal - - - 20 
 
 13. Any two forces represented by the sides of a parallelogram are equiva- 
 
 lent to the single force represented by the diagonal - - - 21 
 
 14. Of three equilibrating forces the intensity of any one is proportional to 
 
 the sine of the angle between the other two - - - - 22 
 
 15. Composition of several concurring forces by geometrical construction - ib. 
 
 16. To resolve a single force into two concurring forces acting in given direc- 
 
 tions - - - - - - - - 23 
 
 17. Analytical determination of the intensity of the resultant of a system of 
 
 concurring forces situated in one plane - - - - ib. 
 
 18. Remarks on the signs of the forces - - - - - 24 
 
 19. Determination of the direction of the resultant of a system of concurring 
 
 forces - - - - - - - -ib. 
 
 20. Particular example of the composition of forces - - - 25 
 
 21. On the parallelopiped of forces - - - - - ib. 
 
 22. Determination of the resultant of any system of concurring forces situ- 
 
 ated in space - - - - - - -27 
 
 — Equations of equilibrium of a system of concurring forces - \b 
 
 n
 
 8 CONTENTS. 
 
 .IrlicU Pag' 
 
 23. When any number of forces are in equilibrium, their projections upon 
 
 any plane or line will also be in equilibrium - - - - 28 
 
 — Equation of the line representing the resultant of a system of forces - ib 
 
 24. Problems respecting the action of forces through the intervention of 
 
 flexible cords ... . . • - ib. 
 
 2.5. On the funicular polygon - - - - - - 34 
 
 26. The polygon formed by the vertical action of weights - - 36 
 
 27. On the catenary curve - - - - - - 37 
 
 28. Equations of the cur%'e - - - - - - 38 
 
 29. Determination of the tension and direction of the catenary at any point 
 
 when the points of suspension are in a horizontal line - - 40 
 
 30. Forms of the equations of the curve, and of the tension, when the 
 
 origin is at lowest point of the catenary - - - - ib 
 
 31. Problems on suspended chains - - - - - 41 
 
 32. General theory of the equilibrium of a point on a surface - - 45 
 
 33. Conditions of equilibrium when the acting force is gravity - - 46 
 
 34. Problems on the equilibrium of bodies of surfaces - - - 48 
 
 SECTION II. 
 
 On the Eqtdlibrinm of a Solid Body. 
 
 '■\b. Preliminary remarks - - - - - - 53 
 
 36. Determination of the resultant of two parallel forces - - - ib 
 
 37. Determination of the resultant of a system of parallel forces - - 55 
 
 35. Determination of the centre of parallel forces - - - ib. 
 
 — Conditions of equilibrium of a system of parallel forces - 57 
 
 39. The same conditions more concisely expressed - - ib. 
 
 40. On the centre of gravity - - - - - - 58 
 
 41. Analytical expressions for the co-ordinates of the centre of gravity - 60 
 
 42. Determination of the centre of gravity of a plane line - - 62 
 
 43. Centre of gravity of a plane area - - - - - 64 
 
 44. Centre of gra^^ty of a surface of revolution - - - - 66 
 
 45. Centre of gravity of a solid of revolution - - - - 67 
 
 46. Centre of gravity of any volume generated by the motion of a varying 
 
 surface along a fixed axis pcri>endicular to its plane, and passing 
 
 through its centre of gravity - - - - - 68 
 
 47. Expressions for the co-ordinates of the centre of gravity of a surface or 
 
 solid which is not symmetrical - - - - - 69 
 
 48. The theorem of GtiUUn - - - - - - 70 
 
 49. On the equilibrium of a solid body acted on by any system of forces 
 
 whatever - - - - - - - -73 
 
 50. Determination of the resultant of a system of forces situated in one 
 
 plane, and applied to different points of the body - - - ib. 
 
 51. The moment of the resultant of this system of forces is equal to the 
 
 sum of the moments of the components - - - - 74 
 
 52. Equations necessary to determine the resultant in intensity and direc- 
 
 tion - - - - - - - -75 
 
 53. Equations of equilibrium of a system of forces in one plane, acting on 
 
 different points of a body - - - - - - 76 
 
 .54. Examination of the import of the several equations of condition - 77 
 
 55. Determination of the equations of equilibrium when the acting forces 
 
 are situated in different planes .... ib 
 
 56. Examination of the import of the several equations of condition - 79
 
 CONTENTS' 9 
 
 Article _ _ ■Pag'C 
 
 57. Conditions of equilibrium when there is an immoveable point in the 
 
 body or system - - - - - - -81 
 
 — Conditions of equilibrium when there is a fixed axis in the system - ib. 
 
 58. Analogy between the theory of moments and the theory of projections 
 
 in geometry - - - - - - -82 
 
 59. Problems on the equilibrium of a solid body - - - - 84 
 
 — A bent lever ACB (fig. 47,) is suspended at C, about which point it is 
 
 free to move in a vertical plane, and weights are attached to its ex- 
 tremities : to find the position in which it will rest - - - ib. 
 
 60. An obhque cylinder stands on a horizontal plane : to determine the great- 
 
 est weight that will hang suspended from its upper edge without over- 
 turning it- - - - - - - -85 
 
 61. One extremity of a heavy rod is moveable about a fiLxed point in a ver- 
 
 tical plane, and to the other extremity is fastened a cord which goes 
 over a pulley in the same horizontal line with a fixed end, and sup- 
 ports a weight equal to half the weight of the rod : to determine the 
 position in which the rod will rest - - - - - ib. 
 
 62. Two heavy bars, moveable each in the same vertical plane about one 
 
 extremity, mutually support each other : to determine their position 86 
 
 63. A beam of variable thickness has one end suspended by a cord of given 
 
 lengtli, fixed at a given point above an incUned plane of given incli- 
 nation, and the other end of the beam is sustained by the inclined 
 plane : to determine the position of the beam, weight sustained by 
 the cord, and pressure against the plane, when the beam is at rest - 87 
 
 64. A given beam is supported by strings which go over pulleys, and have 
 
 given weights attached to them : to find the position of equilibrium 88 
 
 65. A given beam hangs, by two strings of given lengths, from two fixed 
 
 points : to find the position of equilibrium - - - 89 
 
 66. A given solid hemisphere, with its convex part upon a smooth inclined 
 
 plane of given inclination, is kept from sliding by a string of given 
 length, fastened at one end to a given point, and at the other to the 
 edge of the hemisphere : to determine the point where the hemisphere, 
 when at rest, touches the plane, the pressure on the plane, and the 
 tension of the string - - - - - - 89 
 
 67. To determine the position in which a paraboloid will rest upon a hori- 
 
 zontal plane - - - - - - -90 
 
 68. To determine the pressures exerted by a door on its hinges - - 91 
 
 69. 73. Additional problems for exercise - - - - 92 
 
 74. On the mechanical powers - - - - - 93 
 
 75. Theory of the lever - - - - - - - ib. 
 
 76. Advantage of a combination of levers - - - - 94 
 
 77. A body being weighed successively in the two scales of a false balance ; 
 
 to determine the true weight - - - - - ib. 
 
 78. Construction of the common steelyard - - - - ib. 
 
 79. To determine what must be the length of one arm of a lever of the 
 
 second kind, in order that a weight at the extremity of the other may 
 
 be supported by the least power possible - - - - 95 
 
 80. Theory of the wheel and axle - - - - - ib. 
 
 81. Of toothed wheels - - - - . - - 97 
 
 82. On the figure of the teeth of wheels - - - - - ib. 
 
 83. Theory of the pulley " - - - - - - 98 
 
 84. Advantage of a system of pulleys - - - - - 99 
 85,86. On difierent systems of pulleys - - - - - 100 
 
 2
 
 10 CONTENTS. 
 
 Arliele Pog' 
 
 87. Thcon' of the inclined plane - - - - - 100 
 
 88. Thfory of the screw - - - - - - 101 
 
 89. On die wedge - - - - - • - 103 
 
 90. Theory of the wedge abstracting from the influence of friction - 104 
 
 91. On the strength and stress of l)caras - - - - - 106 
 
 92. Galileo's hypothesis of the resistance of the fibres - - - ib. 
 
 93. The hypothesis of Leibnitz ... - 107 
 
 — Results of ,1/r. Barloivs experiments - - - - 1 08 
 
 94. Problems on the strength of beams in various positions - - ib. 
 
 95. Useful deiluctions from the preceding problems, as to the most econo- 
 
 mical forms of beams - - - - - -111 
 
 96. Method of determining the actual weight which any given beam is 
 
 competent to sustain in given circumstances - - - ib 
 
 97. Remarks on the erroneous notions of Emerson and others, respecting 
 
 the comparative strengths of beams when firmly fixed at each end, 
 and when but loosely supported - - - - - 114 
 
 — To determine the strongest rectangular beam that can be cut out of a 
 
 given cylindrical tree - - - - - -115 
 
 PART II. 
 
 ELEMENTS OF DYNAMICS. 
 
 SECTION 1. 
 
 On the Rectilinear Motion of a Free Point. 
 
 98. Preliminary remarks - - - - - -117 
 
 99. Of inertia - - - - - - - - ib. 
 
 100. Equations of uniform motion - - - - - 118 
 
 101. Expression for the velocity in variable motion - - - 119 
 
 102. Expression for the force when constant - - - - ib. 
 
 103. Expression for the force when variable .... 120 
 
 104. Remarks on the methods of representing force - . - ib. 
 
 105. Explanation of the true meaning of the foregoing differential expres- 
 
 sions for velocity and force - - - - -121 
 
 106. Table of the equations of uniform and accelerated motion - - 122 
 
 107. On the force of gravity ...... 124 
 
 108. Examples on the vertical motion of heavy bodies - - - 125 
 
 109. On the motion of bodies along inclined planes ... 127 
 
 110. Examples of this motion .-..-. 128 
 
 111. Additional problems ...... 129 
 
 112. On the motion of projectiles ...... 131 
 
 113. Problems on projectiles - - - - - -132 
 
 114. Table of the equations containing the theory of projectiles - - 135 
 
 115. On the rectilinear motion produced by a variable force - - 136 
 
 116. To determine the vertical motion of a heavy body towards the earth; 
 
 the force of gravity varying inversely as the square of the distance 
 
 from the centre - - . . . - - ib. 
 
 117. To determine the motion of a body attracted towards a fixed centre, 
 
 the force varying directly as the distance - - . . 138 
 
 118. To determine the motion of a heavy body near the earth's surface, con- 
 
 sidering the resistance of the air to vary as the square of the velocity 140
 
 CONTENTS. 11 
 
 SECTION II. 
 
 On the Theory of Curvilinear JMution. 
 Article Page 
 
 119. Preliminary remarks - - - - - -143 
 
 120. Determination of the general equations of the curvilinear motion of a 
 
 material particle - - - - - - -144 
 
 121. The dilTerential expression for the velocity in the curve is always exact 
 
 when the motion is due to any number of central forces - - 146 
 
 122. The trilineal spaces described by the radius vector about a single centre 
 
 of force, are to each other as the times of describing them - - 148 
 
 123. Expression for the tangential force, and useful deductions from it - 149 
 
 124. Deduction of the several equations of motion employed in last section 
 
 from the more general theory established in this - - - ib. 
 
 125. On the motion of a body constrained to move on a given curve - 150 
 
 126. General expression for the resistance or normal force at any point of 
 
 the constraining curve ...... 152 
 
 — To determine the centrifugal force at different places on the earth's 
 
 surface, from knowing the time of one rotation on its axis - - 154 
 
 127. On the simple pendulum . - - . . 157 
 
 128. Formulas concerned in the theory of the pendulum - - - ih. 
 
 129. To determine the time of oscillation in a circular pendulum - - 158 
 
 130. To determine the compression or ellipticity of the earth by means of 
 
 seconds' pendulums - - - - - -162 
 
 131. To determine the time of oscillation in a cycloidal pendulum - 163 
 
 1 32. When a body vibrates in a circular arc, to determine the tension of the 
 
 string at any point - - - - - -164 
 
 133. To determine the centrifugal force and the tension of the string in the 
 
 cycloidal pendulum - - - - - - 165 
 
 134. To determine the time of gyration of a conical pendulum - - 166 
 
 135. On the theory of central forces - - - - - ib. 
 
 136. Expression for the central force in terms of the perpendicular on the 
 
 tangent - - - - - - - - 168 
 
 137. Expression for the force in terms of the chord of curvature - - 169 
 
 138. Expressions for the centrifugal force, centripetal force, paracentric 
 
 force, &c. - - - - - - - -ib. 
 
 139. Motions of the planets — Kepler's laws - - - - 170 
 
 140. The force which retains the planets in their orbits must necessarily be 
 
 directed towards the sun --.... 171 
 
 141. The law of the attractive force determined - - - - ib. 
 
 142. Expression for the velocity at any point of the orbit - . - 172 
 
 143. Another expression for the velocity - - - - - 173 
 
 144. The attractive force like terrestrial gravity influences all bodies alike - ib. 
 
 145. If a body move in a conic section in virtue of a central force, that force 
 
 must be in the focus, and must vary inversely as the square of the 
 distance at which it acts - - - - - -174 
 
 146. Conversely ; if the central force vary inversely as the square of the 
 
 distance, the body must describe a conic section having that force in 
 
 its focus - - - - ... . -ib. 
 
 147. Determination of the particular form of the orbit from knowing the ini- 
 
 tial velocity of the planet and its initial distance from the sun - 175 
 
 — Expressions for the intensities of the solar and planetary attractions - 1 76
 
 12 CONTENTS. 
 
 SECTION III. 
 
 On the Motion of a Solid Body. 
 
 JirticU Pogt 
 
 148. Preliminary remarks - - - . - - - 177 
 
 149. Expressions for the moving force, velocity, &c. of a solid body - 178 
 160. On the collision of bodies - - - - - -179 
 
 151. Direct impact of inelastic bodies . . . - . 180 
 
 152. Direct impact of elastic bodies ..... \%\ 
 
 153. On oblique impact - - - - - - -183 
 
 154. The principle of D'Alembert - - - - - 184 
 
 155. On the moments of inertia ------ 187 
 
 156. To determine the moment of inertia with respect to an axis by means 
 
 of the known moment with respect to some other axis parallel to it - 190 
 
 157. Rotation of a solid body, acted upon by impulse about a I'lxcd axis - 192 
 
 1 58. Rotation of the body when every particle of it is actuated by an accele- 
 
 ralive force - - - - - - -193 
 
 159. Formulas for the determination of the centre of oscillation of a vibra- 
 
 ting body on copper ...... jh. 
 
 160. Examples of the determination of the centre of oscillation - - 196 
 
 161. Determination of the centre of percussion .... igg 
 
 162. On the centre of spontaneous rotation - - - - 199 
 
 163. Determination of the progressive and angular velocities of a body 
 
 moving in consequence of an impulsion which does not pass through 
 the centre of gravity ...... 202 
 
 1 64. To determine the distance from the centre of gravity at which the im- 
 
 pulsion must have been given to produce the progressive and rotatory 
 motion observed in any body ..... 204 
 
 — Application of this to the double motion of the earth - - - ib. 
 
 165. A variety of dynamical problems illustrative of the preceding theories - 205 
 
 166. On the general theory of the motions of a system of bodies acted on by 
 
 any accelerative forces whatever - - - - -215 
 
 167. Motion of the system when the only forces acting are the mutual at- 
 
 tractions of the bodies - - - - - -217 
 
 168. The general principle of the conservation of areas - . . 218 
 
 169. The general principle of the conservation of vis viva - - - ib. 
 
 170. On the composition of rotatory motions .... 220 
 
 171. On the principal axes of rotation ..... 223 
 — Through every point of space three principal axes may always be 
 
 drawn --.-.... 224 
 
 172. The same shown otherwise .--... 226 
 
 173. If a body revolve about a principal axis through the centre of gra%'ity 
 
 this axis will suffer no pressure ..... 229 
 
 174. Further particulars respecting the rotation about a principal axis - 230 
 
 175. Equations of the rotatory motion of a body about its centre of gravity 
 
 when fixed - - - - - - - ib. 
 
 176. If the instantaneous axis of rotation does not coincide with a principal 
 
 axis at the commencement of motion it can never aftervyards coincide 
 with one - . . .... 23' 
 
 177. Determination of the circumstances of the rotation when the instanta- 
 
 neous axis is nearly coincident with a principal axis at the com- 
 mencement ....... 233 
 
 178. A collection of miscellaneous dynamical problems ... 234 
 
 179. Note, Poissoft'o proof of the parallelogram of forces - - - 236
 
 PART I. 
 
 ELEMENTS OF STATICS. 
 INTRODUCTION. 
 
 Artide (1.) Mechanics, taken in its most extensive acceptation, 
 is the science which embraces all inquiries respecting the equilibrium 
 and motion of bodies, whether solid or fluid, and, therefore, consti- 
 tutes a very large and important part of Natural Philosophy, or 
 that vast body of knowledge which explains the laws that govern 
 the various operations of nature. It is usual however to give a more 
 limited signification to the term Mechanics, and to treat under that 
 denomination only of the equilibrium and motion of solid bodies. 
 The theory of the equilibrium of solid bodies is called Statics ; 
 the theory of their motion Dynamics ; these, therefore, are the two 
 great branches of the science of Mechanics. 
 
 (2.) If a body be submitted to any influence which would, if not 
 opposed by an equal counteracting influence, move it, such influ- 
 ence, whatever it be, is called force. The term force, therefore, as 
 employed in Mechanics, applies not merely to what, in common 
 language, we understand by power or physical energy, but also to 
 every cause which either produces or tends to produce motion, 
 however hidden or inexplicable that cause may be. 
 
 (3.) A body subjected to the action of a force or moving influ- 
 ence, ought, necessarily, to move in the direction of that force ; 
 towards it, if the force draw or attract it, and from it, if the force 
 push or repel it. Hence, it is the tendency of a body influenced 
 by a single force applied to it at rest to move in a straight direction. 
 But if to the same point two equal and directly opposite forces are 
 applied, the tendency to motion in one direction being equal to the 
 tendency in the opposite direction, the point will necessarily remain 
 at rest, and will be as much prepared to obey the influence of any 
 third force as it was before the two counterbalancing forces, of which 
 we have just spoken, were applied. 
 
 It is obvious, that although two equal forces be applied to the 
 same point they will not keep that point at rest, unless they are di- 
 rectly opposite as well as equal. This indeed may be easily proved 
 thus : Suppose two equal forces P and Q (fig. 1 ,) tend to draw the 
 point M in their respective directions MP, MQ, which are not oppo- 
 site to each other ; if we suppose the point to remain at rest, let us 
 B 13
 
 14 INTRODUCTION. 
 
 introduce a third force P' equal and opposite to P, that is, tending 
 to draw M towards P' with the same energy as P tends to draw it 
 towards P. Now tlie point being acted upon by three forces, of 
 which two, viz. P and P', are in eiiuilibriuni, the point will tend to 
 move in the direction MQ of the third force Q. But, by hypothesis, 
 the two forces P, Q are in equilibrium ; hence the tendency to mo- 
 tion must be in the direction MP' of the third, w'hich is absurd. 
 
 (4.) As it is the business of Statics to investigate the laws of 
 forces in equilibrium, it will readily occur to the student that one of 
 the principal probleius of this brancli of Mechanics is to determine, 
 from knowing the magnitudes and directions of forces applied to a 
 point, what must be the magnitude and direction of that counterba- 
 lancing force, which would j)revent motion ensuing. 
 
 We may estimate the magnitudes of the forces of which we 
 speak by means of weights : for it is plain that whatever influence 
 at P (fig. 2,) solicits M, M may yet be kept in its place by the 
 counteraction of some weight W tending to draw M in the opposite 
 direction MP'. If for the force P were substituted another, which, 
 in order to keep M unmoved, would render it necessary to double 
 the weight W, we should then say that this new force was double 
 the former, and, in like manner, the ratio of any two forces acting 
 separately at P would be determined by the ratio of the separate 
 counteracting weights acting in the directing MP'. We see that 
 weight is a very suitable representative or measure of force ; but for 
 all the purposes of comparison it matters not by what we represent 
 a force, taking care only that the representing quantities shall have 
 the same ratio to one another as the forces represented, we are, 
 therefore, at liberty to choose that mode of representation most con- 
 ducive to the ends in view, viz. the investigation of the theory of 
 equilibrating forces. We accordingly represent a force by a line 
 drawn from the point on which it acts, called the point of application, 
 in the direction of the force. The length of this line for one of the 
 forces of the equilibrating system may be arbitrary, but for any other 
 of the forces the length of the representing line will be to the former 
 as the represented force is to the former force. Hence the theory 
 of statics is reduced to that of lines and angles. 
 
 (5.) We have just said, that a force is represented by a line drawn 
 from the point of application in the direction of that force ; but we 
 are at liberty to consider any point of this direction as the point of 
 application of the force, and not merely the material point on which it 
 acts : thus it matters not whether the force acting upon M (fig. 2,) 
 to pull it in the direction MP be applied to the point P or to any other 
 point in MP, provided we consider MP to be a perfectly inextensible 
 line connecting P with M. Or if P be a repulsive force tending to 
 push M in the direction MP' then, supposing MP to be a perfectly
 
 INTRODUCTION'. 15 
 
 rigid line, it matters not whereabouts in this line P may be. All this 
 is too obvious to require any laboured proof, and we shall now 
 proceed to the general theory of equilibrating forces acting on a free 
 point, viz., the point where the directions of these forces meet, and, 
 to avoid confusion, we shall generally consider the forces concerned 
 as pulling forces ; for a pushing force may obviously be always 
 supplied by a pulling force of the same intensity, and acting in an op- 
 posite direction.
 
 JOHW S. PRELL 
 
 \oil & Mechanical Engineen 
 
 8AN FRANCISCO, CAU 
 
 SECTION I. 
 
 ON THE EQUILIBRIUM OF A POINT. 
 
 CHAPTER I. 
 
 ON THE COMPOSITION AND RESOLUTION OF CONCURRING FORCES. 
 
 Article (6,) By concurring forces we are to understand forces 
 of which the directions all meet in a point, upon which point they 
 simultaneously act; and they are said to be situated in the same 
 plane when their directions are all in the same plane. To deter- 
 mine the resultant of two such forces, that is, a single force equally 
 effective with the two, is a problem of great importance, and to this 
 the present chapter will be chiefly devoted. The simplest case is 
 that in which the concurrent forces act not only in the same plane, 
 but even in the same straight line ; if in this case the forces should 
 both conspire or tend to move the point in the same direction, then 
 their resultant would be equal to their sum ; for it is plain that the 
 weight W (fig. 2,) has the same effect on M as two smaller weights, 
 together equal to W. If the two forces are opposite, then their re- 
 sultant must be equal to their difference, and the direction of it to- 
 wards the greater force, because so much of the greater force as is 
 equal to the less force which opposes it, is employed in keeping 
 the point in equilibrium, and the tendency to motion is the effect of 
 the remaining force. 
 
 If, instead of two conspiring forces, there were three, then, by 
 adding together two, we should obtain the resultant of those two, 
 and this, added to the third, would furnish the resultant of the three ; 
 and in like manner the resultant of four, or of any number of con- 
 spiring forces, is found by merely adding together the component 
 conspiring forces. If there be two systems of conspiring forces 
 directly opposed to each other, they may thus be reduced to a single 
 pair of opposing forces, and the difference of these two will be the 
 resultant of the whole system. The direction, as well as the mag- 
 nitude of this resultant, will be expressed algebraically, if we agree 
 to consider the forces which conspire in one direction as all positive, 
 and those which conspire in the opposite direction, as all negative j 
 for we may then say that the resultant of any number of forces act- 
 ing in the same straight line, is equal to the sum of those forces; 
 b2 3 17
 
 vi<n^ 
 
 18 ELEiMENTS OF STATICS. 
 
 the sign of tliis sum pointing "out the direction. This tlieorem, of 
 course, includes the case in which the forces all pull one way, that 
 is, where there is hut a single system of conspiring forces. 
 
 Having disposed of this simple case of the general problem, we 
 are now to determine the magnitude and direction of the resultant 
 of any two concurring t'orces situated in a plane ; we speak of only 
 two concurring forces, because, as we shall soon see here, as in the 
 case just considered, that when we know how to compound two, 
 the composition of any number can present no difficulty. 
 
 There are several ways of arriving at the solution of this impor- 
 tant problem, but the simplest process with which we are acquainted 
 is that given by Professor Gregory, in his valuable treatise on Me- 
 chanics, who conducts the investigation as follows : 
 
 (7.) Prop. The equivalent of several forces situated in one plane, 
 is in the same plane. 
 
 For if we suppose the equivalent to be out of the plane of the 
 forces, on either side, we may always find a line on the other side 
 of the plane situated in a perfectly similar manner ; and since there 
 can be no reason why the resultant should he in one of these direc- 
 tions, rather than in the other; it is therefore in neither of them, un- 
 less we admit the absurd consequence that it is in both, that is, unless 
 we admit that the same forces acting in like manner can produce two 
 distinct eflects. 
 
 Cor. The resultant of two equal forces must be in their plane ; 
 and it must be in the line which bisects the angle of their direction, 
 since there is no reason Avhy it should tend more to one side than to 
 another. 
 
 (8.) Prop. If to a material point, already kept in equilibrio by a 
 system of forces, another system is applied also in equilibrio, this 
 will not destroy the pre-existing equilibrium : this is manifest. 
 
 Cor. Hence, if the three forces, C, C, 0, (fig. 3,) are in a state 
 of equilibrium, and if each of the forces were doubled, or tripled, 
 or quadrupled, &c., or if they were halved, quartered. Sic, or changed 
 in any proportion, the equilibrium would remain so long as they con- 
 tinued to act in the same directions CP, C'P, OP. 
 
 Cor. 2. Hence, also, since the resultant R is always equal and op- 
 posite to one of the forces (as O), it follows, that when the magni- 
 tudes of equilibriated forces concurring in a point are made to vary 
 in any ratio, the resultant retains its position but changes its magni- 
 tude in the same ratio. 
 
 (9.) Prop. If three equal forces are inclined to one another in an- 
 gles each of 120 degrees, any one of them will balance the joint ac- 
 tion of the other two. This is likewise incontrovertible ; for neither 
 of the forces can prevail. 
 
 (10.) Prop. Two equal forces inclined in an angle of 120 degrees,
 
 CONCURRIXG FORCES IN ONE PLANE. 19 
 
 have for their equivalent a third, which has the direction and pro- 
 portion of the diagonal of the rhombus constructed on the lines which 
 represent the forces. 
 
 For if C, C are the forces (fig. 3,) acting on the point P, the force 
 O whose measure is OP=CP=C'P, and is situated so that the an- 
 gles CPO and C'PO are each equal to CPC, will (9) ensure the 
 equilibrium. But RP, the measure of the equivalent R, is equal and 
 opposite to OP (6); therefore CP=PR=C'P and because angle 
 CPR=60°=C'PR,CR=CP and C'R=C'P. Consequently CPC'R 
 is a rhombus, and RP, the representative of the equivalent of the 
 forces C, C, is its diagonal. 
 
 Cor. If half the angle CPC be denoted by a, v/e shall have 
 PD=PC cos. fl=C cos. a, whence the equivalent RP=2 C cos. a. 
 
 (11.) Prop. Any two equal forces have for their equivalent the 
 diagonal of the rhombus constructed on the right lines which repre- 
 sent them in magnitude and direction. 
 
 1. If this proposition be true with regard to any two equal forces 
 C, C acting in the directions CP, C'P (fig. 4), and forming Avith 
 their resultant R the angles CPR, C'PR each for example equal to 
 a, it is true, likewise, for two other equal forces c, c' acting accord- 
 ing to the directions c P, c'P, which bisect those angles. In this 
 case c may be considered (7 Cor.) as the resultant of two equal forces, 
 X and y, acting in the directions CP, RP ; and in like manner, c' 
 may be considered as the resultant of two other equal forces, x' and 
 ?/', acting in the directions C'P, RP : so that, in lieu of the two 
 equal forces c, c', we may consider four equal but unknown forces x 
 x', y, y' acting in the directions just assigned them. The two first 
 of these, x, x', acting in the directions CP, C'P, have, by hypothe- 
 sis, the diagonal of the rhombus for their resultant: that is, they 
 are equivalent to a force expressed by 2 x cos. a acting in RP, 
 therefore the resultant z of c and c' will be equal to 2y-\-2x cos. a; 
 but a?=y, therefore z=2x(l-t- cos. a). Now the angles CPR, 
 c P c' being each equal to half CPC, are equal to each other; and 
 c being the resultant of two equal components acting in CP and 
 RP, we have 
 
 . J^ ' " ' ~ z 
 
 Substituting this value of x for it in the preceding equation, we obtain 
 
 2^2 
 
 Z = (l+COS. «) .-. ;22_2c2(l -fcos. fl) .'. z = c v/2(l+cos. a). 
 
 But it is known that cos. i a=^ ^ — , (Gregory^ s Trigono- 
 
 7netry, p. 46,) whence, by substitution, z=2c cos. 5 a. Conse- 
 quently, the proposition if true for a is true for I a.
 
 20 ELEMENTS OF STATICS. 
 
 2. In exactly the same manner may the proposition be proved 
 true with respect to the half of 5 a or 4 a, and in succession for 
 i a, j\a, ^^a, 6ic. That is, since it is true, (9) when the angle 
 CPC is measured by 5 of the circumference, it is likewise true 
 ■when the angle between the equal components is measured by ^, 
 f T' IT' ^^' 0* *1'C circumference, where the series may be continued 
 sine limite. 
 
 3. If the proposition be demonstrated for the three angles a, b, 
 and a — b, it will be true for the angle a 4-'^ » that is, if we take two 
 equal components c and c', making with their resultant x, angles 
 =a -\- b, we shall have a'=2c cos. (a-f6). Thus, if in fig. 5 the 
 angles CPR, C'PR are each equal to a, and cPC, CPrf, c'PC, 
 C'Pd', each equal to b: conceiving two forces rfP, d'P each equal 
 to c, their resultant will, by hypothesis, be =2c cos. (a~-b), because 
 dPR=a — b ; and this quantity subtracted from the resultant of 
 c, c', d, d', will give x. But c and d have their resultant C act- 
 ing in CP, and =2c cos. b; the same thing holding with respect 
 to c' and d', we have two forces equal to C and equivalent to one, 
 which is 2C cos. a or 4c cos. a cos. b ; whence a:=4c cos. o cos. ft 
 
 — 2c cos. (a—b). But cos. a cos. b=i cos. (o+^y+d cos. 
 (a — b). (See Gregory's Trig., p. 44, art. 20). Which value of 
 cos.o cos. b, substituted for it in the preceding equation, gives 0'= 
 2ccos.(o+ b). So that the proposition when true for a, b, and a — b, 
 is true for a-\-b. 
 
 4. Let b be taken as small as we please in the series, 1, -j-V, 2^:j , -Jj- 
 &c. and let a be the preceding term in the series, then o, 6, a—b,a-\-b, 
 are 26, b, b, and 36, respectively, in each of which the proposition 
 holds: again, if a=Sb, a-^b=4b; if fl=46, 0+6 = 56, &c. So 
 that the theorem is demonstrated for all angles in the series 6, 26, 36, 
 46, 56, &-C., in which 6 may be taken of a magnitude less than,any 
 one which can be assigned. Consequently, the theorem is true with 
 respect to any rhombus whatever ; for let any rhombus be proposed, 
 which it is affirmed is an exception to this proof; we can, it is ob- 
 vious, by choosing 6 lower than any assigned angle, and taking a 
 suitable multiple of it, approach nearer the aecepted angle than by 
 any assignable difference, that is, we show that our theorem is ap- 
 plicable to the angle itself. j* ., 
 
 (12.) Prop. Any two forces having the ratio o^Hfe sides of a 
 rectangle, and whose directions coincide with those -^sMes, hav^ for 
 tlieir equivalent the diagonal of that rectangle. 
 
 Let the two forces C, C (fig. G,) act in the directions CP, CPS 
 which comprise the right angle P : complete the parallelogram^ 
 CPC'R, and draw its diagonals; parallel to CC, draw cc' termi- 
 nated by Cc, C'c', which are drawn parallel to the resulting diagonal. 
 Conceive c and c' to be two equal forces acting in the equal lines cP,
 
 CON'CURRING FORCES IN ONE PLANE. 21 
 
 c'P, opposite to each other, and consequently annihilating each 
 other's effects; then cPDC and c'PDC being rhombi, the force CP 
 is the equivalent of cP, DP, and C'P that of c'P, DP, by the preced- 
 ing proposition. Therefore, the components CP, CP are the same 
 in effect as the opposite ones cP, c'P together with DP, DP ; that 
 is, the equivalent sought is 2DP or RP, the diagonal of the paralle- 
 logram. 
 
 Cor. Since RP : rad. :: CP : cos. CPR :: C'P : cos. C'PR, we 
 have the resultant equal to either component divided by the cosine 
 of the angle which it makes with the resultant. 
 
 (13.) Prop. Any two forces whatever have their equivalent ex- 
 pressed in magnitude and direction by the diagonal RP of the paral- 
 lelogram constructed on the lines CP, C'P, which represent these 
 forces. 
 
 Having completed the parallelogram CPC'R (fig. 7,) on the given 
 sides, draw cc' perpendicular, and Cc, C'c' parallel, to the diagonal ; 
 demit also CD, C'D' perpendicular to the diagonal: then will Cc 
 PD, C'c' PD' be rectangles, and the triangles CRD, C'PD' equal in 
 all respects, consequently Cc=DP, RD=D'P, and cP=c'P. The 
 addition of the equal forces c, c' , acting in the opposite directions 
 cP, c'P, will make no difference in the state of the system ; and 
 since the components DP, cP, have CP for their resultant, and the 
 components D'P, c'P, the resultant C'P, (by the preceding proposi- 
 tion,) we may, instead of the original forces CP, C'P, substitute the 
 forces cP, c'P, DP, D'P, of which the two former destroy each 
 other's effects, and the latter DP, D'P, are manifestly equal to RP ; 
 that is, the resultant of the two forces CP, C'P, is equal to the dia- 
 gonal RP of the parallelogram. 
 
 " Thus have we," observes Dr. Gregory, " by a series of connect- 
 ed* propositions, demonstrated that which is justly reckoned the 
 most important in the theory of Statics, and which is now commonly 
 spoken of under the title of the Parallelogram offerees. The de- 
 monstration here given is commenced upon the same principle (9) 
 as that proposed by D^Membert, in the Memoirs of the French Aca- 
 demy for 1769 : it was somewhat simplified by Francceur in his 
 Mechanics ; but what is here offered, at the same time that it is more 
 concise than the demonstration of Francoeur, is freed, it is hoped, 
 from some objectionable positions into which that author has certainly 
 fallen." 
 
 The foregoing proposition of the parallelogram of forces, has also 
 been established by some eminent mathematicians by processes 
 purely analytical, and deduced from some obvious principle neces- 
 sarily involved in the question itself, as for example, that if two equal 
 forces P, P concur at an angle e, the direction of the resultant must 
 bisect this angle, and moreover its intensity must be some function
 
 22 ELEMENTS OF STATICS. 
 
 of P and 5, which is no more than saying that the intensity of 
 the resultant must in some way depend on the intensity of its equiva- 
 lent components and on their mode of action. This is the condition 
 from wiiich Poisson sets out his analytical investigation. We have 
 given it with some little modification at the end of the volume. 
 
 (14.) From what has now been proved, it follows tliat when the 
 intensities and directions of any two concurring forces are given, 
 the determination of the third ibrce equilibrating these will be re- 
 duced to the determination of the diagonal of a parallelogram, from 
 liaving the two sides and included angle given ; or still more simply, 
 it will be reduced to the determination of the third side of a triangle, 
 from having two sides and the included angle given ; for if PC, PC 
 are the two given forces (tig. 7,) then, by drawing CR equal and 
 parallel to PC', the diagonal PR will be just as well determined as 
 if we had constructed the parallelogram CC, so that instead of the 
 sides and diagonal of a parallelogram, we may always represent 
 three equilibrating forces as well in direction as in intensity by the 
 three sides of a triangle taken in order as PC, CR, RP; so that as 
 these sides are as the sines of their opposite angles, we may say 
 that the intensity of any one of three equilibrating forces is propor- 
 tional to the sine of the angle between the directions of the other 
 two. If we represent the sides PC, PC by P, Q, and the angle 
 CPC of their direction by a, then, since in the triangle CRP the 
 angle C is the supplement of a, we shall have this analytical ex- 
 pression for R, viz., R3=P3-)-Q»-f 2PQcos.o, so that the funda- 
 mental theorem of statics, when expressed algebraically, is precisely 
 that which is also the fundamental theorem of plane trigonometry. 
 
 (15.) Knowing how to compound two forces, we may easily 
 compound several or determine a single force which will balance 
 them, and that either by geometrical construction or by analytical 
 representation. Thus, suppose four forces PC,, PC^, PC 3, PC^, 
 concurred at P, then we miglit proceed geometrically as follows : 
 Draw in the plane of the two forces PCj, PC 2, the line CjRj, equal 
 and parallel to PC,, then PR, will, from what has already been 
 proved, be the equivalent of PC,, PCg, and may therefore be sub- 
 stituted for them in the system. Again, in the plane of the two 
 forces PR,, PC 3, draw R,Rjj equal and parallel to PC 3, then, as 
 before, PRj will be the equivalent of PR,, PC 3, that is of the three 
 forces PC,, PC 3, PC 3. Lastly, in the plane of the two forces PR3, 
 PC4, draw RgRg, parallel and equal to PC^, and we shall then 
 determine PR3, which must be the equivalent of the whole system. 
 From this construction it is obvious, that if commencing at the point 
 of concurrence P we draw successively PC,, CjR,, RjRj* Rgl^s' 
 equal and parallel to the several forces, then the line PR3, which 
 closes the polygon PC.RiR^RjP, will represent the resultant of
 
 CONCURRING FORCES IN ONE PLANE. 23 
 
 the system, and this in whatever planes the competent forces act, 
 since in our construction we have not confined these forces to any 
 particular planes. Should the concurring forces be in equilibrium, 
 the last of the points Rj, Rg, R3, &c. will fall on P, making the 
 resultant 0. 
 
 This graphical method of compounding forces, will not, however, 
 answer the purposes of computation, and we shall therefore now 
 seek a general analytical expression for the resultant of any number 
 of concurring forces, confining ourselves first to those only which 
 are situated in one plane. 
 
 Determination of the Resultant of any Number of concurring 
 Forces situated in one Plane. 
 
 (16.) We have already seen how any two concurring forces may 
 be compounded into one, but before we can conveniently compound 
 a greater number we must first reverse this process, and know how 
 to resolve any single force into two concurring forces acting in cer 
 tain proposed directions. 
 
 Let PR (fig. 8,) represent any given force acting on P, and let it 
 be required to resolve it into two others concurring in P, and acting 
 in the directions PC, PC, so that PC may make a given angle a 
 with PR, and PC' may make a given angle j3 with PR. Parallel to 
 PC draw RC, then (13) making PC equal to RC, PR will repre- 
 sent the force of which PC, PC represent the components, that is, 
 PC, PC Avill be the components sought. Their analytical values 
 will be obtained by determining trigonometrically the two sides PC, 
 CR of the triangle CPR, from having the side PR and interjacent 
 angles given; we see, therefore, that the resolution of a given force 
 in any two arbitrary directions may always be effected. That it 
 may be effected in the easiest way possible, it is requisite that the 
 proposed directions be perpendicular to each other, for then the 
 analytical operations are expressed simply by the equations, PC = 
 PRcos.a,PC'=PRcos. )3, where a and j3 are complements of each 
 other. When, therefore, we have to decompose a force into two 
 others, and are at liberty to choose their directions, we shall always 
 on the score of simplicity, assume these directions perpendicular to 
 each other. 
 
 (17.) Let it now be required to compound into a single resultant 
 the several concurring P, P^, Pg, P3, &c. (fig. 9). 
 
 Assume any rectangular axis AX, AY, the first AX making with 
 the direction of the several forces the angles a, Oj, a^, aj, &c., and 
 the second AY making with the same directions the angles 3, /Sj, 
 |3o,]83, &c. Then by resolving each of the proposed forces into 
 two others acting along the assumed axes, we shall have for the sum 
 X of all the components kx, Ax^, AXj, &c. acting along kx,
 
 24 ELEMENTS OF STATICS. 
 
 P COS. a + P, COS. Oj-f-Pj COS. Oj+Pj COS. 03+<fec. = X .... (1), 
 
 and for the sum Y of all the other components, that is those acting 
 along AY 
 
 Pcos. /3 + P, cos. ^.-fPj cos. /Sj-I-Pa COS. l3^-^iic. = Y .... (2), 
 
 so that these two sums, that is the single force X acting along AX 
 and the single force Y acting along AY, may be substituted for the 
 proposed system of forces ; hence the resultant of these two will be 
 the resultant of the original system. But the resultant R of two 
 forces X and Y acting at right angles, being the diagonal of the rec- 
 tangle XY, is R=v' X^-f-Ys . . . . (3), this, therefore, is the gene- 
 ral expression for the resultant of any system of concurring forces 
 acting in a plane. 
 
 (18.) It might at first sight appear, that since the angles /3, p,, 
 &c., are the complements of a, o,, (fee, it would be better to write 
 in the first member of (2) the expressions sin. o, sin. Oj, &;c. in- 
 stead of cos ji, cos. i3,, &;c. ; such, however, is not generally the 
 case, although this change would do very well for the particular ar- 
 rangement of the forces exhibited in the figure, for it is easy to see 
 that this arrangement is such as to cause all the components acting 
 along each axis to conspire. If one of the forces P^ were directed out 
 of the angle YAX, as in fig. 10, its component Ax^ would obviously 
 oppose the conspiring forces Ax, Ax^, Aa*,, and therefore, agreea- 
 bly to (6), its analytical representation should carry a contrary sign, 
 which it actually does do when we give it the form P3 cos a^, but 
 not M'hen written P3 sin. ^, although each form represents the 
 same linear magnitude ; similar remarks apply when P3 is situated 
 in either of the angles XAY', X'AY'. Regarding then the compo- 
 nents which act in the directions AX, AY as positive, and those 
 which act in the opposite directions, AX', AY' as negative, the se- 
 veral terms in (1) and (2) will have their proper signs involved in 
 those of their cosines. 
 
 (19.) In order to completely determine the resultant R, we must 
 know its direction as well as its intensity (3). Now putting a, b, 
 for its inclination to Ax, AY, we know that 
 
 X Y 
 
 X=R cos. a, Y=R cos. b .'. cos. a= -t^-> cos. i= -rj-, 
 
 either of which equations makes known the direction of the resultant, 
 so that the resultant will be completely represented by the equations 
 
 R=^X2 + Y2,cos. a =4-'? • • • • (4). 
 If the proposed forces are themselves in equilibrium, then R=0, so 
 that we must then have ^X^-^Y^—O or X^ + Y^ =0 ; but as every
 
 CONCURRING FORCES IN ONE PLANE. 25 
 
 square is essentially positive, the sum of two cannot be unless 
 each separately is 0, so that when the forces are in equilibrium, we 
 must have X=0, Y=0, showing that each system of components 
 must be in equilibrium, and this is obviously true whatever be the 
 inclination of the axes of the components. 
 
 (20.) As a particular example of the preceding general theory, 
 let us suppose four forces, P, P,, Pj, P3, concurring in a point 
 A, of which the intensities are respectively denoted by the num- 
 bers 2, 3, 4, 5, and let the angles included by their directions 
 be PAP, =30°, P,AP2 = 15°, P^kP^=75°. First assume, as 
 above directed, two rectangular axes AX, AY, and, as their position 
 is arbitrary, let us for greater simplicity suppose one of them AX 
 to coincide with AP ; then the inclinations of the several forces to 
 the assumed axes will obviously be as follows : 
 PAX= 0°=a .-. cos. a =1 
 P,AX= 30°=ai COS. ttl=i^/3 
 P2AX= 45°=a, COS. o,=i^/2 
 P3AX = 120°=a3 COS. a3=—hy/B 
 
 PAY=90°=/3 .-. COS. /3 =0 
 PiAY=60°=/3, COS. ^,=1 
 P2AY=45°=i3^ COS. /3, = i^2 
 P3AY=30°=|33 COS. 133=1 v/3 
 consequently, the two general equations 
 
 P COS. a+P^ COS. a, +P2 cos. a2 +&C. = X 
 P COS. j3-i-Pi COS. |3i -I-P2 COS. /32 -}-&c.= Y 
 become in this case 
 
 2+3^3+2^/2 — #y3=^X 
 0+f+2v'2 + fV3=Y; 
 hence, the numerical values of X and Y being thus determined, the 
 
 value of ^=y/X2-i-y2 is known, and thence also of 
 
 X Y 
 
 cos. a = -TT-j or cos. b = -r^-. 
 
 Determination of the Resultant of any Number of concurring 
 Forces situated in different Planes. 
 
 (21.) It was necessary before we could compound together seve- 
 ral forces acting in one plane, first to determine the resultant of two, 
 or to establish the parallelogram of forces, so likewise before we 
 can treat the more general case, or compound together forces acting 
 in different planes, we must first know how to determine the result- 
 ant of three. This however is a very easy matter, it has indeed 
 C 4
 
 26 ELEMENTS OF STATICS. 
 
 been aecomplisheil jToomelrically already (15), and not only for 
 three but for any number of forces. But let there be three P, P,, 
 P2, concurring in A, (Hit. 11,) and represented by the lines AB, 
 AC, AD; then we know from the article just referred to, that it 
 we draw BE in the same plane with, and equal and jiarallel to, AC, 
 and then EF in the same plane with, and equal and parallel to, AD, 
 the line AF, which closes the twisted quadrilateral ABEFA, will 
 represent the resultant. Now the three lines AB, BE, EF, are ob- 
 viously the three edges of a parallelopiped BG, of which AF is the 
 diagonal ; hence the lines representing three concurring forces not 
 in the same plane form the edges of a parallelopiped whose diago- 
 nal is their resultant. It is manifest that if any force AF be pro- 
 posed, and we draw from A three lines AB, AC, AD, in any direc- 
 tions whatever, not all in the same plane, nor yet any two in the 
 same plane as AF, we may construct a parallelopiped having these 
 three lines for edges, and AF for its diagonal, the opposite edges 
 meeting in F. Now the forces represented by the edges of this 
 parallelopiped meeting in A, have the given force AF for their re- 
 sultant, therefore this given force has these three for its compo- 
 nents, so that any force may be decomposed into three conctir- 
 ri)ig forces acting in any proposed directions, provided all three 
 are not in 07ie plane nor any two in the same plane as the proposed 
 force. 
 
 This decomposition will be most easily effected, analytically, 
 when the directions of the components are at right angles ; for if 
 a, i5, and y, represent the inclinations of the proposed force R to 
 these several rectangular directions, then the three components will 
 be obviously expressed by R cos. o, R cos. ji, R cos. y . (1), for in 
 fact, these components are no other than the projections of the ori- 
 ginal force on three rectangular axes, calling these several projec- 
 tions or component forces, X, Y, Z, we have, by adding their 
 squares, X2+Y2-fZ2=R3 (cos.2 a+cos.a |3 + cos.2 y), 
 but (.^nal. Geom. p. 228, cos.^ a +cos.a j3+ cos.'' y=l . . (2), 
 .-. R=v/X"2TV= + Z2 
 
 It thus appears that when we wish to decompose a given force R 
 in three rectangular directions AX, AY, AZ, making with R any 
 proposed angles a, /3, y, we shall have, for the analytical values of 
 the components, the expressions 
 
 X=Rcos.a, Y=Rcos.(3, Z=Rcos.y .... (3), 
 and when on the other hand we wish to compound three given rec- 
 tangular forces X, Y, Z, the intensity of the resultant will be given 
 by the expression R=v/X2+Y=+Z* . . . . (4), 
 and its direction by the expressions 
 
 X Y ' Z 
 
 COS. a = -^, C0S.3=-T7-, COS.yrs-rg i*^. (5). 
 
 K K K ' '
 
 CONCURRING FORCES IN GENERAL. 27 
 
 Two of these latter equations are however sufTicient to fix the 
 position of the resultant ; since, on account of the necessary condi- 
 tion (2), any one of the cosines become fixed when the other two 
 are ; thus, when a and /3 are determined, we get y by the equation 
 
 COS. y=v/l cos. 2 a COS. 2^ .... (6). 
 
 We need not embarrass ourselves here with any inquiry about the 
 ambiguity of the sines of the cosines in (5), arising from the ambi- 
 guity of the sine of the radical R in (4) ; for, as we know that the 
 resultant must necessarily lie within the angle formed by X, Y, Z, 
 the angles a, /3, y, which it forms with these lines, must always be 
 acute. 
 
 (22.) Let us now proceed to determine the resultant of any num- 
 ber of concurring forces P, Pj, Pg, P3, &c., situated in space, and 
 acting in given directions. 
 
 Through the point of concourse A draw three rectangular axes 
 AX, AY, AZ, and let us call 
 
 o , ,3 , y , the angles which P makes with these axes, 
 «i'/5,,yi, .... P, 
 
 »2'l^2'y2' • • . • P2 
 &C. &C. 
 
 then, by decomposing each force according to these axes, Ave have 
 P COS. a , P cos. i3 , P cos. y for the components of P 
 
 PjCOS. a^, PjCOS. /3i, PjCOS. y, P^ 
 
 PgCOS.aa, P2COS./32, PsCOS.ya P^ 
 
 &c. &c. 
 
 Adding together all the forces which act in each axis, the three sums 
 X, Y, Z, will represent three rectangular forces acting in given di- 
 rections, which may be substituted for the proposed system, the 
 values of these three forces being 
 
 P COS.a + Pj COS. ttj-f 1*2 ^^^- a2+*^^' = ^^ 
 
 P cos./3-i-PiCos. fSj+Pj cos./32 + &c. = Y I . . . . (1). 
 
 P cos. y-fPj C0S.yj+P2 COS. y2+&C. = Zj 
 
 Having thus reduced the system of forces to three, we have, for the 
 
 intensity of the resultant, the expression R=v'X'^ + Y^-j-Z2 . (2), 
 
 and for the angles a, b, c, which it makes with the axes, the expres- 
 
 X , Y Z ,^, 
 
 sions cos. a=— — , cos. 6=— 5-, cos.c=— =3- .... (.3). 
 K K K 
 
 In this way, therefore, we may completely determine the resultant 
 of any system of forces situated in space, when we know the inten- 
 sity of each force, and the angles its direction makes with three 
 rectangidar axes. The cosines in (1) necessarily carry with them 
 the proper sines as in art. (18). 
 
 When the system of concurring forces is in equilibrium, then, 
 since R=0. we must have X=*+Y^+Z^=0 ; but as every square is
 
 28 ELEMENTS OF STATICS. 
 
 essentially positive, this cannot be unless X=0, Y=0, Z=0, that 
 is to say, we must luive (aqua. 1,) 
 
 P cos.o+Pj cos. aj 4-^2 <^os. 02+&c.=01 
 P cos.>3-f-Pi cos.,3,-fP2 cos. f32+«fcc.=0 I . . . . (4). 
 P cos. y+Pj cos.yi+P2 cos. 7'2+<fec.=0 J 
 These, therefore, are the equations of equUibrium of a system of 
 concurrin? forces P, Pj, Pj, &c situated any how in space. 
 
 (23.) The projection of any line in space on two rectangular 
 axes, may obviously be found by first projecting the line on the 
 plane of those axes, and then projecting this projection on the axes 
 themselves. Hence, if a system of forces in equilibrium be project- 
 ed on the plane of xy, the forces represented by these projections 
 will be also in equilibrium, seeing that the components of these 
 forces will be X=0 and Y=0. Now the position of one of the 
 rectangular planes, as the plane of xy, is always arbitrary ; on 
 this account therefore, and on account of the equations (4) it follows, 
 that ivhen any number offerees arc in equilibrio, their projections 
 upon any plane or ttpon any line will also be in equilibrio. 
 
 (24.) As the resultant R or AF (fig. 11,) of any system of forces 
 is the diagonal of the parallelopiped whose edges are the components 
 X, Y, Z, or AB, AC, AD, it follows that X, Y, Z, are no other 
 than the co-ordinates of the point F, (Anal. Geom. p. 222.) Know- 
 ing, therefore, the co-ordinates of a point F in the line AF passing 
 through the origin A, we know enough to enable us to write the 
 equation of this line or of the resultant. These equations are (Anal. 
 
 7a Z 
 
 Geom.p.226-7,)z=-r^a', z=^^y. If we remove the origin of 
 
 xL jL 
 
 the rectangular axes without altering the directions of the axes, so 
 that the co-ordinates of the point of concourse A may be x', y', z\ 
 
 2 
 then the equation of the resultant will be z — z'=-— - (x — x' )■, 
 
 A. 
 
 2 
 
 z—z' =-r^(y — y'), or the line will be equally represented by 
 
 combining either of these equations with that of the third projection, 
 VIZ. x—x =^(y—y ). 
 
 Having now established the general theory of the equilibrium of a 
 free point acted upon by any number of forces any how situated, 
 we shall devote a short chapter to the application of this theory to 
 particular examples.
 
 PROBLEMS ON CONCURRING FORCES. 29 
 
 CHAPTER II. 
 
 PROBLEMS ILLUSTRATIVE OF THE PRECEDING THEORY. 
 
 (24.) We have already adverted (4) to the propriety of represent- 
 ing balanced forces by means of weights, from tlie circumstance that 
 any influence tending to move a free point may be always counter- 
 acted and rendered nugatory by the opposing influence of some 
 weight communicating to the point, by means of a cord attached to 
 both. All, therefore, that has been established in the preceding 
 chapter respecting tlie equilibration of forces acting on a point, ap- 
 plies when these forces are weights acting on a point through the 
 intervention of cords, provided only that we consider these cords to 
 be themselves without weight or thickness, to be inextensible, and 
 to be perfectly capable of moving over the fixed points or puUies, (as 
 at fig. 13,) employed to direct the influence of the weights, with 
 perfect freedom. The consideration of a system of weights thus 
 acting on a free point through the intervention of cords and fixed 
 points, introduces the consideration of two new modifications of 
 force, viz. Tension and Pressure. 
 
 By the tension of a cord is to be understood its tendency to 
 stretch under the influence of an appended weight ; as this tendency 
 varies with the weight, this is taken as its measure ; so that when 
 we speak of the tension of a cord as a force, we always mean the 
 weight which produces that tension. The pressure on a fixed point 
 is measured by, or is equal to, that force which must be applied to 
 it, when supposed free, to keep it in equilibrium. 
 
 Problem I— A cord PACBP, passes over two fixed points or small 
 pulleys A, B in the horizontal line AB, and two given equal weights, 
 suspended at the extremities P, P, support a third given weight W. 
 It is required to determine the position of C, (fig. 12). 
 
 The point C is kept in equilibrium by the equal tensions (P or P , ), 
 of the strings CA, CB, and by the weight W acting vertically. 
 Hence, by resolving the forces in the directions of two axes CX, 
 C Y, the one parallel and the other perpendicular to AB, the forces 
 in each axis must destroy each other. Hence, taking first the com- 
 ponents in CX, we have the condition 
 
 P COS. a + Pj COS. a, =0 or P cos. a — P, cos. a = 0, 
 from which we immediately infer that as P=Pi , a=a', and there- 
 fore /3=i3,, so that the triangles ACY, BCY are equal, and CY 
 bisects AB. Again, because ]3=/3j, the components in CY are 
 2P cos. /3, and Win the opposite direction, therefore the second condi- 
 tion is 
 
 c2
 
 30 ELEMENTS OF STATICS. 
 
 w 
 
 2P COS. |3 — "\V=0 .-. COS. /3 = sin. B= -rr^. 
 
 This equation is sufficient to determine the point C or the line EC ; 
 but, to avoid any trigonometrical computation, let us put for cos. ^ 
 
 YC BC v/BV^+YC2 2P 
 Its equal ^, then yu = YC = "W 
 
 YC3 W2 ^4p2 \V3 
 
 The solution of this problem may be conducted differently as fol- 
 lows : Parallel to CB draAv AE, and produce the perpendicular CY 
 till it meets in E ; then the triangle ACE thus constituted will have 
 its three sides in the directions of the three balancing forces, and 
 will, therefore, be proportional to them ; hence, as the tensions of 
 CA, CB are equal, the sides CA, AE are equal, so that the perpen- 
 dicular AY bisects CE ; also 
 
 op A P 
 
 P : W : : AC : EC=2YC .-. -^=^, 
 
 W YC 
 
 and the remainder of the solution may be as above. 
 
 If we suppose W=0, then tlie above expression for YC shows 
 
 that YC=0, as it obviously ought to be; that is, the cord will be 
 
 brought into a horizontal line AB. iJut upon no other hypothesis 
 
 will tliis be the case, except, indeed, we suppose the weights P to 
 
 be infinitely great, for however small W be assumed, yet so long as 
 
 P is of finite magnitude YC will have a finite value, and can never 
 
 be accurately 0, so that it is impossible for any two weights P, Pj, 
 
 however great, acting as in the figure, to draw a third weight W ever 
 
 so small up to the horizontal line AB. The same is true if instead 
 
 of a small weight W attached to a cord without weight, we consider 
 
 the cord itself to have weight : we may therefore say with Professor 
 
 Whewell 
 
 " Hence no force, however great, 
 Can stretch a cord, however fine, 
 Into a horizontal line 
 Which shall be accurately straight." 
 
 As W increases from 0, YC increases and becomes infinite when 
 W=2P, so that when the weight W is cither equal to or greater 
 than 2P, there can be no equilibrium, for W will continually descend 
 drawing up the weights P, P^. 
 
 Problem II. — Suppose the weights P, P,, are unequal, and that 
 the line AB, instead of being horizontal, makes a given angle BAB' 
 with the horizontal line AB' : to determine the position of C, 
 (fig. 13.)
 
 PROBLEMS ON CONCURRING FORCES. 31 
 
 The solution will perhaps be most easily obtained without re- 
 solving the forces; thus, draw AE parallel to CB, meeting the verti- 
 cal line CE in E, then the three sides CA, AE, EC, being in the 
 directions of the three equilibrating forces P j , P,W, are proportional 
 to them ; hence, knowing the proportion of tlie three sides of the 
 triangle AEC, we may determine its angles : we may, therefore, 
 consider the angles ACE and AEC=ECB as found; consequently 
 the angle CAD, the complement of ACE, becomes known, and DAB 
 being given the angle CAB becomes known ; hence, in the triangle 
 CAB, we have the side AB and the angles C and A to determine the 
 two sides AC, BC. 
 
 Thus, suppose P=4t^, Pi,=3tfe, and W=5j^ : also AB=6 feet, 
 and the angle DAB=30°, then the angles of a triangle whose sides 
 are AC=3, AE=4, EC=5, are CAE=90°=BCA, AEC=36° . 
 54'=ECB; hence DCA=90°— ECB = 53" . 6', and consequently 
 ABC=DCA— 30°=23° . 6'. We thus have AB=6 feet, ACB = 
 90% ABC =23° . 6' , whence AC =2 • 354, and BC=5 • 518 feet. 
 
 Problem III. — Two equal weights P, P^, balance themselves over 
 any number of fixed pulleys : to determine the pressure on each, 
 (fig. 14.) 
 
 Each of the points A, B, C, &c. are kept in equilibrium by three 
 forces, viz. by the equal tensions on each side of it and by the pres- 
 sure it sustains, which latter is therefore equal and opposite to the 
 resultant of the two equal tensions P. Hence, calling the angles at 
 A, B, C, &c. a, b, c, &c. we have 
 
 the pressure on A=2P cos.ia 
 B=2Pcos.|6 
 C=2Pcos.ic. 
 &c. &c. 
 
 Problem IV. — A cord ACB of given length is fastened to two 
 hooks A and B, and a weight W is at liberty to slide by means of the 
 ring C freely upon this cord ; at what point will it rest? (fig. 15.) 
 
 It is obvious that if the two hooks were in a horizontal line, as 
 A', B the weight W ought to settle itself at a point symmetrically 
 situated with respect to the two points A', B, that is CA'B will be 
 an isosceles triangle, and therefore the angles A'CH, BCF, made 
 with the horizontal line HF, will be equal. But if the hook be at A 
 instead of at A', then the force in CA being the tension of CA, or the 
 pressure upon the hook A, we may consider the hook to be removed 
 and a force equivalent to this pressure to be applied at A; but it 
 matters not at what point of its direction a force is applied, so that 
 C will continue undisturbed if the force be applied at A', that is, it 
 will make no difference as to the position of C, whether the hook be
 
 32 ELEMENTS OF STATICS. 
 
 at A or at A', the tension of CA being the same throug-hout; hence 
 the angles ACII, BCF are equal, and the tension of CA equal to 
 that of CB. Produce AC to meet the vertical ED in D, then the 
 sides of the triangle BCD, being in the direction of the forces, are 
 proportional to them ; this triangle is moreover isosceles having CB 
 = CD on account of the equal tensions of CB, CD, or of the equal 
 angles BCF, ACH ; hence AD is equal to the length of the string. 
 We thus have given 
 
 AB=a, AD=/, EAB=a .-. AE =a cos, a., E B=a sin. o 
 
 „^ „„ v^ /'■' — o^'cos.'o — a sin.o 
 
 ED=^//2 _ a^ cos.» a ••. BF=^^ ^ ; 
 
 also, since, DE : EA : : BF : FC 
 
 a cos. a «Rin.,i 
 
 .*.rC= — - — 1- 
 
 2 ( >//2 o^cos.^o' 
 
 these values of BF, FC determine the point C, 
 
 As to the pressure p on the hook B or A, we have 
 
 BF : BC : : nV : J3, but BF : BC : : ED : DA 
 
 DA ,,. / 
 . », VV= w 
 
 • -P- 2ED 2^//"^^=^^ cos.^tt 
 
 Or the pressure may be found by first determining the angle D by 
 means of the sides AE, AD; this angle being equal to BCY or AC Y, 
 we have, by calling it /3, and resolving the equal tensions p along 
 
 W 
 
 the axis CY, 2d cos. i3=W.-. n= 
 
 -^ ^2 cos. ^ 
 
 In the very same way the problem may be solved, when, instead 
 of a weight W acting vertically, any power acting obliquely be at- 
 tached to the ring, for the lines BA', AE being drawn perpendicu- 
 lar to the direction of this force and ED parallel to it, the above rea- 
 soning becomes then obviously applicable to this case. 
 
 The question may be viewed in rather a difl'erent manner from 
 that above, by considering that as the cord ACB is of constant 
 length, the point C, before arrivmg at a state of rest, must describe 
 the arc of an ellipse, and, moreover, that the place of rest must be 
 at the lowest point possible; hence the horizontal line HF must be 
 a tangent at C to the ellipse whose foci are A and B, seeing that 
 every other point in this ellipse is above that line ; hence, by the 
 property of the ellipse, the angles BCF, ACH are equal, and, con- 
 sequently, the tensions are equal, because their components in the 
 directions CF, CH must be equal. 
 
 Itis very clear, although we have not assumed it above, that because 
 the cord passes freely through the ring, the tension of the part CB 
 must be communicated to the part CA, for nothing hinders this com- 
 munication, so that the cord will be equally tense throughout.
 
 CONCURRING FORCES IN ONE PLANE. 33 
 
 Problem V. — A cord of given length passes over two pullies, 
 and one of its extremities P is put througli a small ring or noose at 
 the other extremity C ; a given weight AV is then attached to P : it is 
 required to determine the tension of the string when in equilibrium, 
 as also how much of the cord will hang below the ring, (fig. 16.) 
 
 The tension of CP is measured by the weight W, and this same 
 tension must be communicated to the parts CB, BA, AC, since the 
 cord passes freely through the ring and over the pullies ; but when 
 three equal concurring forces are in equilibrium, the angles formed 
 by their directions are each 120° ; hence, drawing the horizontal line 
 AD, we have in the isosceles triangle ADC, the angles A, D, each 
 equal to 30°, and the angle C equal to 120° ; consequently, as the 
 position of AB, with respect to the horizon, that is, the angle BAD, 
 is known, we know in the triangle BAC the side AB and the angles 
 A and C which is sufticient for the determination of AC, BC, and, 
 therefore, of the place of C ; also the perimeter of this triangle be- 
 ing taken from the whole length of the string, leaves CP the dis- 
 tance of the weight from the noose. 
 
 If, instead of the loop or ring at C, the extremity C were firmly 
 fastened by a knot to BCP at a given distance from the other ex- 
 tremity P, then the tension of CP would not be freely communi- 
 cated to CA or CB, but whatever tension CA had, the same would 
 be communicated to AB and BC, for the rope being freely moveable 
 over A and B, could not rest so long as either of these tensions pre- 
 vailed. The equal tensions of CA, CB, as also the position of the 
 knot C, will be determined in this case as in problem IV. 
 
 Problem VI. — The extremities of a given cord are fastened to two 
 hooks given in position, and to a given point in it is applied a power 
 P acting in a given direction : to determine the pressure upon the 
 hooks (fig. 17.) 
 
 As the point C in the given cord ACB is given, as also the line 
 AB, therefore the three sides of the triangle ABC are given, and, 
 consequently, the three angles; also, as the direction of YCP is 
 given, the angles at Y are given; hence the two exterior angles ACP, 
 BCP are given. If, therefore, we draw CX perpendicular to CY, 
 the angles a, a' will be known, so that calling the pressures on B 
 and k, p and/)', we shall have, by the conditions of equilibrium, 
 p cos. a — jo'cos. a'=0,p sin. a-fjo'sin. a'=P, 
 
 P cos. a' Pcos.a' 
 
 .'. »= = 
 
 COS. a sni. a'-f-sin. a COS. a' sin. ACB 
 
 , P COS. a P COS a 
 
 COS. a sin. a' + sin. a cos. o' sin. ACB 
 The pressures/),/)', therefore, are to each other as the sines of the
 
 34 ELEMENTS OF STATICS. 
 
 angles .3,, (3, or of the angles ACP, BCP; but this much we might 
 immediately have inferred from the property tliat when three forces 
 equiUbrate, each is proportioned to the sine of the angle between 
 the directions of the other two. 
 
 CHAPTER III. 
 
 ON THE FUNICULAR POLYGON AND CATENARY. 
 
 (25.) If a cord be kept in equilibrium by means of several forces 
 P, Pj, Pj, P3, &c. acting at the knots /j j , p^, p^,&,c. the figure /),, 
 p^, Pj, (fee , which it forms itself into, is called the fiiniadar po- 
 lygon (fig. 18). We propose here to investigate the conditions of 
 equilibrium of such a figure. 
 
 And first it is obvious, that the several points /J j , 7^2 ' ^^- ^^^ ^^^^^ 
 kept in equilibrium by the three forces which concur there; it is 
 equally obvious that the tension of the string;;,, yj^ being the same 
 throughout, there is the same pressure upon the knot /jj, as upon the 
 knot/)j, but exerted in the opposite direction, and the same of any 
 two consecutive knots /J^, p^; p^, p^. Sic. Hence, if the three 
 forces which equilibrate /),, were applied io p^, the equilibrium of 
 P2 would remain undisturbed; but of the forces thus acting on P2 
 two would destroy each other, since, as just observed, the tension 
 of /jj P2 presses the points /),, /jj ^^''th equal force but in opposite 
 directions ; we may, therefore, consider but four forces acting on p^ , 
 viz. the forces inp^ Pj? and in/jj Ps together with those inpj P, 
 and in /J, P,. Again, if the four forces equilibrating /Jj' ^^ trans- 
 ferred to /J3, the equilibrium of p^ will remain undisturbed, and, as 
 before, the two forces due to the tension oi p^ p^ will destroy each 
 other, and thus the point pg will be kept in equilibrium by five forces 
 of which only one, viz. that in p^ p^ will be a tension, the others 
 being the forces P, PjjPj, P3, originally applied to the cord at the 
 knots p^, P^^Pj. Proceeding in this way from knot to knot, it is 
 plain that when we shall have arrived at the last knot or at the ex- 
 tremity of the cord that the point will be kept in equilibrium by the 
 concurrence of all the forces originally distributed along the cord, at 
 the points/jj,;j2' «^^- the directions of these forces being preserved. 
 
 From all this it follows then, that for the funicular polygon to 
 exist, the intensity and directions of the several forces acting at the 
 knots, must be such, that if they were all applied to one point they 
 would keep it in equilibrium, and that to find the direction and ten- 
 sion of the 71 th side of the polygon it will only be necessary to as-
 
 If Off e 34 ■ 
 
 -'^
 
 FUNICULAR POLYGON, 35 
 
 certain what would be the direction and intensity of the resultant of 
 all the forces acting on the a — 1 preceding knots, if they were all 
 to concur. 
 
 It thus appears, that in the funicular polygon the conditions of 
 equilibrium are the very same as if the forces all concurred, or were 
 to be transferred parallel to themselves to a single point, so that if 
 we assume three rectangular axes, and call, as before, the angles at 
 which the several directions of the forces are inclined to these, a, 
 ttj ttj, &c. /3, j3i, ^21 ^^' 7' 72' 73' <^^'' t^^^ conditions necessary 
 to the existence of the funicular polygon will be 
 
 P cos. a + Pj cos. ttj-l-Pj COS. a2+&c. = 01 
 P cos. i3-i-Pi COS. /Si+Pg COS. 132+ &c. = I . (1). 
 P cos. y-j-Pi COS. Yi+Pa cos. yg-f <^C-=0j 
 When the forces all act in one plane, then two axes taken in this 
 plane will be sufficient, as one of these equations then becomes 
 identically 0, the conditions being 
 
 P COS. a + Pi cos. ttj+Pa COS. a2+&C.=.0 ) ,^. 
 P COS. /3 + Pi COS. /3i -fP2 COS. 13^ -i-&c.=0 5 ' ^'^■^' 
 
 To construct the polygon in any particular case, we must know 
 not only the intensities and directions of the several forces, but also 
 their points of application ; when these are known, we may easily 
 construct the successive sides of the polygon ; thus the resultant of 
 P, P^ being determined, we shall thence have the direction and in- 
 tensity of the force in p^ p^, and the point /;2 being known we thus 
 have the side p^ P2'i i'^ ^^^^ manner the resultant of Pj and the force 
 in /?2 Pi-, just determined, will make known the intensity and direc- 
 tion of the force inp^, p^, so that, as the point p^ is given, wemay 
 construct the second side pg Pa^ ^^^^ ^o on till the polygon is com- 
 pleted. 
 
 If any one of the forces were attached to the cord not by a fixed 
 knot, as we have hitherto supposed, but by a moveable ring, then in 
 the equilibrated state of the system the tensions on each side of the 
 ring would be equal, and would therefore form equal angles with the 
 force on the ring. 
 
 In this way may the absolute tension between any two proposed 
 knots be determined, but if we wish merely to find the ratio of the 
 tensions of any two sides of the polygon, then, recollecting that each 
 of three equilibrating forces acting on a point is proportional to,the 
 sine of the angle between the other two, and calling the several ten- 
 sions t, t^, t^, &;c. (see fig. 18,) we have 
 
 t sin. Oj t^ sin. a^ t^ sin. a^ . 
 
 t^ sin. a ' ^2 sii^* ^2^ ^3 sin. 04' 
 Multiplying these equations together, and omitting the factors com- 
 mon to both numerator and denominator, we have generally
 
 39 ELEMENTS OF STATICS. 
 
 t sin. a. sin. a, sin. a, . . . . sin. ff,„_i 
 
 — = -. —. ^—. : .... (3). 
 
 /" sm. a sin. Oj sm. a^ . . . . sin. 021,-2 
 Should, therefore, the zngXes a, a^,a^., &Lc.he equal throughout, 
 the tension will be uniform throughout, and, conversely, if the ten- 
 sion be uniform throughout, the angles must be all equal ; when, 
 therefore, the angles are equal, tlic uniform tension of the cord is 
 measured by either of the extreme forces P,P„ , wliich are necessarily 
 equal in intensity, as they measure the equal tensions of the extreme 
 sides of the polygon. 
 
 (26.) Tlie most important case of the funicular polygon is that in 
 which the several forces P,, Pj, &lc. (fig. 19,) are weights, acting in 
 the same vertical plane upon fixed points of the cord when suspend- 
 ed at its two extremities P, P„ , (fig. 19), we shall therefore consider 
 this case in particular, and first we may remark, that the polygon so 
 formed will lie wholly in the vertical plane of the forces, for of the 
 three equilibrating forces concurring in /),, two, viz. those in;>,, 
 P, and in p.^ Pj, are in the vertical plane; therefore the third, or 
 that in 7) J, 7^2' n^ust be in the same plane ; also, this last and that in 
 p^ P2 being in the vertical ])lane, the force in/^j P^ must be in that 
 plane, and so on. Let us then draw in this plane the horizontal and 
 vertical axes PX, PY ; the angles aj, Oj, &c. which the directions 
 of the forces make with the first of these axes, are each equal to 90°, 
 and the angles /i,, iSj, &;c. made with the other axis, are each 0; 
 hence, denoting the sum of all the weights Pj, P,, &c. by AV, the 
 equations of equilibrium (2) at page 35 become, in this case, 
 
 PCOS. a+PnCOS. a„=0 ~> ,.. 
 
 Pcos./3 + P„cos. ^„4-W=0 5 •••'K^h 
 where P and P„ are the pressures on the two points of suspension ; 
 these pressures are therefore readily determinable if the angles a. 
 On , that is the directions of the extreme cords are given, there beinw 
 no necessity to know the situation of the knots, nor yet the separate 
 forces P,, P,, &c., but only their sum W. All this, indeed, may 
 be determined from the equations themselves ; thus, let t represent 
 the tension of any side of the polygon, and let us put a for the angle 
 it makes with the horizontal axes, and b the angle it makes with the 
 vertical axis. 
 
 The tension t may be considered as exerting a pressure upon the 
 knot at that extremity of the proposed side which is farthest 
 from the point P, so that substituting this pressure for P„ in the 
 equations (1) and calling the sum of the weights between P and this 
 knot w, we have 
 
 P cos. a + ^ COS.« = ) , , 
 
 P COS. fi-\-t COS. b-\-w=Q ^ . . . . (.-ij, 
 two equations from which the two unknowns t and a may be deter-
 
 THE CATENARY. 37 
 
 minea, and thus the intensity and direction of the force in any side 
 of the polygon ascertained ; and, from knowing the intensities and 
 directions of the forces in two adjacent sides, Ave find the intensity 
 of the vertical force at the angle by taking the resultant. 
 
 As to the ratio of any two tensions, it is involved in the general 
 expression (3) of last article, Avhich, because in the case under con- 
 sideration a^ is the supplement of a2, ci^ the supplement of a^, and 
 
 , ^ t sin. 02 71-1 • 1-1 ^ sin. flfon'-i 
 
 so on, reduces to — = —- in like manner — = r^ , 
 
 tn sin. a In' sin. a 
 
 .•. — =^-^ — ^nL • so that the tensions of any two sides of the poly- 
 
 tji sin. (l2n' — 1 
 
 gon are reciprocally as the sines of the angles loliich they forvn, 
 ivith the vertical axis. 
 
 Since these angles are the complements of those which the same 
 sides form with the horizontal axis, we may substitute the cosines 
 
 of these latter for the sines of the former, or because cos.= we 
 
 sec. 
 
 may say that the tensions are directly as the secants of their incli- 
 nation to the horizon. 
 
 (27.) TTie Catenary. By referring to equations (1), last article, 
 we see that they express the conditions of equilibrium of these three 
 forces acting at their point of concurrence, viz. the force P inclined 
 at an angle a to the horizon, the force P^ inclined at an angle an , and 
 the vertical force W. Hence, in our polygon (fig. 19), if we pro- 
 duce the directions of the pressures P, P„, that is the extreme sides 
 of the polygon, the point O in which they meet must be the point 
 of concourse of which we speak, at which the vertical weight W 
 and the pressures P, P„ acting maintains the equilibrium of O ; these 
 pressures are therefore the same as if all the weights acting at the 
 angles of the polygon were collected and applied at O, it would 
 therefore not be improper to consider W so applied as the resultant 
 of all the original weights. 
 
 If we suppose in our polygon the weights to be attached at equal 
 distances, the less these distances are taken the greater will be the 
 number of sides of the polygon, and, consequently, the figure Mali 
 approach the more nearly to a curvilinear form, which form it must 
 actually assume when the distances between two consecutive weights 
 become 0, that is, when the weights act upon every point of the cord ; 
 now this is the same as considering every point itself to be weighty, 
 so that the curve, of which we speak, will be that which a perfectly 
 flexible physical line or chain actually assumes, when suspended 
 at its extremities. It is called the catenary curve, (fig. 20.) 
 
 The direction of the pressures on the points of suspension, will, 
 obviously, be tangents to the catenary at those points, and, from 
 D
 
 38 ELEMENTS OF STATICS. 
 
 what has been said above, it appears that the point O, in which 
 these directions meet, would be kept in equilibrium by the pressures 
 or the tensions of the lines OP, 0P„ , and by the whole weight W 
 of the chain suspended at O. 
 
 (28.) Let us seek the equation of the catenary, supposing that 
 the cord or chain is uniformly heavy throughout, that is that the 
 lengths of any two portions are to each other as their weights. Ta- 
 king the horizontal and vertical lines PX, PY, for axes of co-ordi- 
 nates, we shall have for any point M, Vm=x, mM=^y and PM=5 ; 
 and the portion s of the cord is held in equilibrium by the tensions 
 at P and M, acting in the directions OP, OM of the tangents at those 
 points and also by the weight of s; or the point O is held in equi- 
 librium by the same tensions and the vertical weight s, the relation 
 therefore between s and the tensions is determined by the relation 
 
 sin. POM s 
 between the sines of the angles about 0, that is (14)- . '.\ =—, 
 
 where p represents the pressure on P. Now 
 sin. POM=sin. (lOM-j-POI) 
 
 =sin. lOMcos. POI-1-cos.IOMsin. POl 
 =sin. 7nM0 sin. a — cos. mMO cos. a. 
 
 consequently, since mM0=M05 
 
 sin. POM. 
 
 —■ — ,,,^ =sm. a»— cot. mMO cos. a ; 
 
 sin. MOs 
 
 but (Bi^. Calc. p. 113,) cot. mMO=-^ hence 
 
 s . dy dy ... 
 
 — =sin.a -r- cos. a .•. S=^p sin. a — p -^ cos. a . . . ( 1 J, 
 
 p dx ^ ^ dx ^ ^ 
 
 which is the differential equation of the catenary. 
 
 We may obtain a differential equation, involving only x and y, 
 
 provided we differentiate this with respect to x and substitute for 
 
 IL its equal -^ =1 1 _,. ^^ for ^^e thus have 
 
 d^y 
 dx^ 
 
 ■^d^=-p'''-''d^''-'==-p'''-\\'';^ 
 
 4 
 
 Jdij<' dHj 
 
 1 -f -r^ = — pC0S.o-r4 
 
 dx^ 
 
 The numerator of the fraction in the second member of this equa- 
 tion will obviously be the differential of the denominator if we mul- 
 tiply it by dy ;* by doing this, therefore, and then integrating the 
 
 * This is the same as multiplying both sides by ~ and then multiplying by dx, 
 
 dx 
 
 to prepare each side for integration.
 
 THE CATENARV 39 
 
 equation, we have y=. — p cos. a ^ 1 + -~ + c, from which we 
 
 dy \/ (c — y)^ — o'^cos.^a 
 get -^ = — ! ^ i- .... (2). 
 
 ° dx pCOS. a ^ ■" 
 
 It remains to determine the constant c, for which we have this con- 
 dition, viz. that when x=0 and2/=0, that is at the point P,—r- = 
 tan. a, so that for this point the equation (2), just deduced, is 
 
 sin. o \/(c^ — w^ cos.*^ a) 
 
 tan. a= =^-^ i-, 
 
 cos. a p cos. a 
 
 orjjsm. a=v'c3 — p^cos.^'a 
 
 .'. c^=p'^ (a'm.^ a + cos. 2 a) =p^ .'. c=p : 
 hence the differential equation of the catenary (2) is 
 
 dx pcos.a ^ 
 
 dym 
 If we substitute this value of-;— in the equation (1), there results 
 
 s=p sin. a— '^( p — y)^ — p^cos.^a. (4), which expression, for the 
 length of the arc, proves that it is rectifiable. 
 
 In order to obtain an equation between x and y, independently of 
 differentials, let us put in (B) p — 2/=-, p cos. a=a, then dy= — 
 dz and the equation reduces to 
 
 dz 
 dx= — a — — . . . (5) ; to render this rational we must as- 
 
 k/ z^ — a^ 
 
 sume x/^TZr^ = z — z', from which we get the equation 2zz'= 
 
 a^-\-z'^, which differentiated gives 
 
 zdz'4-z'dz:= z'dz'r. = :-= — d log. z'; 
 
 _ z—z' z' ° 
 
 that is, from (5) dx=ad log. z' .•• x=a log. z'-\-c that is restoring 
 the value of z',a,'=a log. f z — .y/z^ — a* | -fc ; or, restoring the values 
 of z and a 
 
 a7= ;j COS. a log. [(p — y) — \/{p—yY — jo'^cos.^al+c (6). 
 The constant c may be determined from the condition that a;=0 
 when .V=0, the origin being at P, this condition gives 
 c= — p cos. a log. {p{\ — sin. a) I , and thus the equation (6) is 
 
 1 <(P—y) — v^V^ — 2nv-fP^sin.2a, , , 
 
 2;= » cos. a log. \— — ^ ^ ^-^ ' ^ \ . . Cj^ 
 
 ^ ^ ^ p(l— sin. a) ^ ^ ^ 
 
 In order to determine the lowest point in the catenary, or that point 
 at which the tangent of the inclination to the horizon is 0, we must put
 
 40 ELEMENTS OF STATICS. 
 
 fly 
 
 -p=0 in (3), which will give for y the value 3/=/) (1 — cos. a)... (8), 
 
 und this put for y in the equation (7) gives 
 
 x=p cos. a log. '-. . . . (9) ; also the length s of the cord 
 
 1 — sin. a ^ 
 
 hanging between the point of suspension P and the lowest point is, 
 by equation, (4) s=p sin. a . . . (10). 
 
 (29.) If botli points of suspension P, V „, are in the same hori- 
 zontal line, the portion of the cord between P and the lowest point 
 will, obviously, be equal in length, and symmetrical in figure to that 
 between P„ and the lowest point: hence the value of s in (10) will 
 be half the length of the cord. When, therefore, we know the 
 horizontal distance D of the points P, P„ and the length L of the 
 cord, we may by help of the last three equations determine the angle 
 a and the tension p : thus dividing (9) by (10) we have 
 D COS. o , COS. o ,,,^ 
 
 T=— loff- -; ■■ ; .... (11) 
 
 L sm. a ° 1 — sm. a ^ ^ 
 
 an equation from which, the unknown quantity a, may be determined 
 
 by approximation, after which/) will be given by (10), viz. 
 
 knowing, therefore, o and p, we may readily find the tension t at 
 
 any point of the cord; for if s be the length, hanging between P 
 
 and this point, then from equations (2) art. 26 
 p COS. a-\-t COS. a=0 1 
 p COS. ji-i-t COS. 6 + s=0 l. . . (13) 
 
 also cos.^ a-\-cos.^ b^l j 
 
 from which, cos. a and cos. b being eliminated, there results for t 
 
 the value t=^\/p'^ — "Ipn sin. o -f-*^ 
 
 = v7j^cos.'^a+(/> sin.a — sy .... (14). 
 This expression we may simplify and render independent of p ; 
 thus, substitute for s the value in equation'(lO),and we shall thus 
 have for the tension a at the lowest point A, 
 
 a=p COS. o=, (equa. 12), 5 L cot. a . . . (15), 
 consequently, by substitution, the general expression for ^ is 
 
 f=^|L^cot.''a+(iL — s)» .... (16) 
 and from this we may get the value of cos. o, by means of the first 
 of (13). It thus appears that when a flexible cord, or chain of 
 given length, is suspended from two points, at a given distance from 
 each other, in the same horizontal line, we may always determine 
 its tension and direction at any point. 
 
 (30.) The general equations of the curve (4) and (7), as also the 
 expression (14) for the tension at any point, will become simpler in
 
 THE CATENARY. 41 
 
 form, if the origin of the axes be taken at tne lowest point A of 
 of the curve ; for confining ourselves to the consideration of the 
 branch A P„, we may view A and P„ as the two points of suspen- 
 sion of the cord A P„, in Avliich case a=0, and as y, Avhich has 
 heretofore been measured downwards, will now be measured up- 
 ■wards, we must change the sign which it carries in the preceding 
 formulas when we wish to adapt them to this arrangement of the 
 axis. Calling the tension at A, a, vie thus have, by equation (4), 
 s=,y2ay=y^ .... (1), als o from e quation (7) 
 
 or since by the equation just deduced 
 
 ^ a^-\-s'=^ a"-+2ay+y"'= a+y, 
 we may put the last equation under the form 
 
 x=a\og.\ \. . . . (3). 
 
 The expression (14) for the tension t at any point will be 
 t=^ d'+s'' .... (4). 
 
 All these equations involve the unknown tension a, but this may 
 be determined by trial from (3), since we know the values of .r and 
 s in one case, viz. .r=iD and S:=5L. By means of this value 
 of s and a, thus determined, we may obtain y, that is the length of 
 A A', or the distance of the origin A from the middle of P P„, and 
 knowing thus the position of the axes, the curve may be con- 
 structed from its equation (2). 
 
 We shall now give an example of the preceding formulas. 
 
 Problem I. — (31.) The length of a heavy flexible chain is just 
 double the horizontal line, joining the points of suspension to de- 
 termine the pressure on these points, the inclination a to the hori- 
 zon, &c. 
 
 It is obvious, from equation (11, p. 40), that it is suflicient to 
 know the ratio of the length L to the distance D, in order to de- 
 termine the angle a. We shall not, however, employ this formula, 
 but that above marked (3), as this is of more easy application. If 
 then, in this formula, we suppose s equal to half the length of the 
 chain equal to 1, then x, which is half the horizontal distance be- 
 tween the points of suspension, must be i ; moreover a will then 
 be cot. a (equa. 15, p. 40) : hence we shall have 
 
 a = « log. { ^1 ; the logarithm here indicated being hy 
 
 perbolic, it will be convenient to convert it into a common loga- 
 D 2 6
 
 42 ELEMENTS OF STATICS. 
 
 rithm which requires that we multiply its value by '43429, so that 
 •21715==alog. f^tl_ifLl_i|. Now the first side being little 
 
 more than |, a near value of a at once presents itself, viz a=:^ 
 ^•2, which substituted in the second member gives '2 log. 10-099 
 =•20086; a result which is rather too small; let us, therefore, 
 take a a little larger, making it a=i="25; the second member will 
 then be k log. 8- 1231 =-22743, which is a little greater than the 
 true result. Hence by the known rule of trial and error, since the 
 differences of the results are nearly as the differences of the sup- 
 positions which have led to them, we have 
 •22743=-20086=-02657 : -22743— •21715=-01028 : : -05 : -0194 
 
 .•.a=-25— •0194 = -2306 nearly. 
 To obtain a still nearer approximation to the truth, let us now as- 
 sume a=-23, then the second member of the equation in a is 
 •23, log. 8-8092=-21735, and as this is a little too great, let us, 
 finally, put a=-22 and we have -22 log. 9-1996=-21203 ; conse- 
 quenUy, -21735— •21203=-00532 : 21735— -21715 
 
 = -0002 : : -01 : -00037, .-. a=-23— -00037=-22963 
 Seeking now in the tables for the natural cotangent correspond- 
 ing to this number, Ave find for the angle a, or the inclination of 
 the chain to the horizon at cither point of suspension, the value 
 o=77°, 4', therefore putting 1=^ L, the pressure on these points is 
 
 (equation 12), »=-: = --; and the tension at the lowest 
 
 ^ ' "^ sm. tt -97463 
 
 point A is (equation 15) a=l cot. o= -22963/ 
 
 Also for the distance of A, below the horizontal line P P„, we 
 
 have (equation 8), 2/=p (1 — cos. a) = -79638/. the depth of th$ 
 
 lowest or middle point. 
 
 Problem II. — A heavy flexible chaui, 100 feet in length and 
 weighing 1000 lbs. is suspended at its extremities to two fixed 
 points, in the same horizontal line, 95 feet H in. asunder. It is re- 
 quired to determine the greatest depth of the curve, the tension at 
 the lowest part and the tensions at the points of suspension. 
 
 In this example the ratio ^ is -95125; hence, as in the former 
 
 problem, we shall have to determine, a, or rather cot. a, from the 
 equation 
 
 •43429x-95125=-41312=alog. fii^^'-tii. 
 
 a 
 
 After a little consideration we find that c:=|=l-75 is a near 
 
 value, substituting this, therefore, in the last member, we have 
 
 |log. 1*7232 =-4 1359; this being a little greater than the true re-
 
 THE CATENARY. 43 
 
 suit, let US take a?= 1-7, and we then have 1-7 log. l-7484=-41249 ; 
 
 consequently, 
 
 •41359— •41249=-0011 : 41359— 41312 = -00047 : : -05 : -02136 
 
 .-. a=:l-75— •02136=1-72864 nearly. 
 Again, let a=l-73, then 1-73 log. 1-7331 =-41316. 
 Comparing this result with that obtained by the first supposition, 
 we have the proportion 
 •41359 — •41316='00043 : -41359- •41312=-00047.-:-02:-02219 
 
 .-. a=l-75— -02219 = 1-72781, 
 this number corresponds to the natural cotangent of 30°, 4', there- 
 fore, for the inclination a we have a=30°, 4' 
 
 / 50 
 
 .•• pressure, a= = =99-8 ft.= 998 lbs.* 
 
 ^ -^ sm.a -50101 
 
 also tension at A, a=/ cot. a=50x 1*72781 =86-4 ft.=8641bs. and 
 
 distance of A from PP„, y=p (1 — cos. tt) = 13-4 ft. 
 
 Problem III. — A chain of given length, 2 / hangs freely over two 
 given points, in ahorizontal line, in what position will itrest?(fig. 21). 
 
 When the chain is at rest it is plain that what in the former pro- 
 blems was the pressure upon the points of suspension P, P„ , will 
 here be equivalent to the weight of either PP' or of PkP„' the 
 parts hanging vertically, these parts are, therefore, equal to each 
 other, and to what we have hitherto called p ; hence, calling half 
 the length of the catenary s, the expression for I will be (equa. 10, 
 p. 40,) l=:s-^p=p (sin. a-fl) ... (1) ;' also the expression for half 
 
 COS (X 
 
 PPn or /' is (equa. 9), /'=pcos. alog. ': .... (2) ; divi- 
 
 l"^— sin. a 
 
 ding this by the last equation we have 
 
 I' cos. o . cos. o sin. j3 sin. |3 
 
 / 1-f-sin. a °" 1 — sin. tt ~ 1 -f cos. i3 °'l •—cos. /3 
 
 =tan. 1 i3 log.^^j^^-^t = — tan. i /3 log. tan. 1 /3 ; 
 
 the first member of this equation being given, it follows that to 
 determine /3, the angle at which the chain is inclined to the vertical, 
 we have only to find by trial a number such that when multiplied 
 by its logarithm, the product shall be equal to a given number. 
 
 * Because the weight of a foot of the chain is 10 lbs. 
 
 xppj sin, g \/l — C OS.23 I] — cos. 
 
 1 + cos. ^ 1 + cos. o \ 1 + cos. /S —^^' i ^ 
 
 Cyoutig^s Trigonometry, p. 37.) In like manner, 
 
 sin, a _ II + cos. s _ 1 
 
 1 — cos. g, ~\\ -fToT^ tan. ;^ '^
 
 44 ELEMENTS OF STATICS. 
 
 When this is found, a becomes known, and from the above equa- 
 tion (1), we getp= — — -, which gives the length of that part 
 
 sm. a ^ 1 
 
 of the chain which hangs vertically on each side of the curve. 
 
 Problem IV. — Given the distance 2/' between two fixed points in 
 the same horizontal line, to determine the length of the shortest 
 chain that can remain suspended, as in the preceding problem. 
 
 We have just seen that -j= — tan. \ ,3 log. tan. A /3 ; and as / is to 
 
 be a minimum, -j must be a maximum, /' being constant, that is to 
 
 say, calling tan. i ^, x, — x log. x= — log. x^ =log. — = max. .'.— 
 
 x^ X 
 
 max, or x^ =min. 
 
 This equation is solved at p. 74 of the Differential Calculus, where 
 
 the value of x is found to be a;= — = =tan. i 3 
 
 e 2-7182818... -• 
 
 .-. cotan. i ,3=2-7182818..., .-. | ,3=20°, 12' .-. /3=40°, 24'. 
 
 It appears, therefore, that in this case we must have 
 
 -r= log. -i=- .-. /=cZ'=2-7182818/', 
 
 / e ° e e 
 
 so that if the distance 21' between the fixed points be 10 feet, then 
 
 the length 2l of the shortest chain, which will suspend itself by 
 
 hanging over them, will be 27'182818 feet. 
 
 For the length of each part of the chain hanging vertically we have 
 
 p=-^ = = =1 I (1 +tan.'' i ^) 
 
 ' sin. tt+l cjs.,3-(-i 2cos.aA/3 ^ ^ "r 2 f-y 
 
 =} / (IH — -), and for the distance of the lowest point in the cate- 
 nary from the horizontal line 
 
 y=p (I — COS. a)=p — I r^ r-.p — / tan. i/3 
 
 ^ ^ ^ ^ ^ l+sm. a ^ ^ 
 
 For further particulars respecting the curves formed by flexible lines, 
 acted on by different forces, as also respecting those which elastic 
 lamina; assume under like influences, we must refer the student to 
 Professor JVhewelVs Mechanics, chap. x. and xi., (the first edition 
 of this work is here referred to,) where these matters are very elabo- 
 rately treated, and at great lenerth.
 
 ECIUILIBRIUM ON A SURFACE. 45 
 
 CHAPTER IV. 
 
 ON THE EQUILIBRIUM OF A POINT ON A CURVE OH SURFACE. 
 
 (32.) If a material point be placed upon a curve surface, and be 
 kept in that place by the mutual action of any number of forces ap- 
 plied to it, the resultant of these forces must be in the direction of the 
 normal to the surface, and must be equivalent to the pressure whicli 
 the surface sustains. For, if the resultant had any other direction, 
 we might decompose it into two, one in the direction of the normal, 
 and the other in the direction of a tangent to the surface ; the first 
 of these would be opposed by the resistance of the surface, but the 
 second, being unopposed, would cause the point to move. Consider- 
 ing, therefore, the resistance which the surface opposes to the nor- 
 mal force, as one of the system of forces acting upon the proposed 
 point, we may altogether disregard the surface and view the point 
 as a free point kept in equilibrium by the system of forces P, Pj, 
 Pj, &c. and N ; and, hence, we have the same equation of condition 
 as in (22), that is putting 6, 0', 9", for the angles which N forms 
 Avith three rectangular axes, we have 
 
 N cos. 9 -f P cos. a -f- Pi COS. ai + Pg cos. Og -f &c.=0 
 N COS. 9' -j- F cos. i3 -f Pj cos. ^^ -j- P^ ^^^- ^'2 + &c, = 
 
 N COS. (9"-f- P COS. y + Pi cos. yi + P2 ^^S. y^ -f &C.=0 ; 
 
 or, putting as at (22) X, Y, and Z, for the sums of the components 
 along the respective axes, the three equations may be written 
 N cos. Q -f X=0, N cos. 9' + Y=0, N cos. 9"+Z=0 . (1) 
 Now we must here remark that the angles 6, 9', 9", which de- 
 termine the direction of the force N, are entirely dependent on the 
 equation of the surface, and on the co-ordinates of the point to which 
 the forces P, P,, &c. are applied. Knowing, therefore, the equa- 
 tion of the surface, and the position of the point, we may always 
 determine the direction in which the resultant of the applied forces 
 P, Pi, &c. must necessarily act to ensure the equilibrium. Thus, 
 the equation of the surface being m=F (x, y, z,)=0, we have {Bif. 
 
 Laic. p. 166,) COS. 9=u ^-, COS. 9 =v-7-»cos. 9 =v-7-, 
 ax ay ciz 
 
 where v=- 
 
 dx^'^dy'^'^dz^' 
 
 The values of cos. 9, cos. 6', cos. 9", being thus found, if we sub- 
 stitute them in the equations (1) and then eliminate N between each 
 two, we shall obtain two equations expressing the conditions which 
 must exist among the applied forces and their inclinations to the
 
 46 ELEMENTS OF STATICS. 
 
 axes, in order that their resultant may be a normal force. Indeed, 
 to obtain these equations, we need not take the trouble to first cal- 
 
 1 , 1 . • du du , dii . , ,, 
 
 culate V, but may simply substitute -:-, -;-, and -7-» for 6,6 ,6 ,rc- 
 
 dx dy dz 
 
 spectively, because v disappears with N. It hence appears, that 
 by means of the equation of the surface, and the position of the 
 point, the equations of equilibrium, originally three, become re- 
 duced to two. If, indeed, one of the axes of reference, as the axis 
 of z, coincide with the normal, and, consequently, originate at the 
 point, then to find these two equations it will not be necessary to 
 know the equation of the surface; for then the equations (1) will be 
 P cos. a-j-Pj cos. aj-f P2 cos. 02-f<S:c.=0 
 Pcos. /3 + Pi cos. /3i-f-P2 COS. /32 + &c.=0 
 N + P COS. v+Pi COS. 71 +P2 COS. 72 H-&c.=0 : 
 the two first of which are all that are requisite to establish the equi- 
 librium, for if these hold, the third must necessarily hold, since the 
 intensity of the normal forces or pressure may be any whatever, 
 without disturbing the equilibrium. This third equation is neces- 
 sary, however, to determine the resultant of the applied forces, or 
 the intensity N of the whole normal pressure. 
 
 (33.) Having thus briefly noticed the conditions necessary for 
 the equilibrium of a point on a surface, viewing the matter in the 
 utmost generality, we shall now consider the equilibrium under 
 more particular circumstances, taking those cases only which are 
 likely to present themselves in nature, w'here the acting forces are 
 gravity or weight. 
 
 In this point of view, where we are to consider the means of re- 
 taining a heavy point on a given surface, the problem becomes very 
 much simplified from the following considerations. A heavy body 
 (considered as a point) placed upon a surface, will, unless it presses 
 entirely in the direction of the normal, tend to move on that surface 
 towards the horizon, and this tendency will be in a certain determi- 
 nate direction, viz. in that direction which is nearest the perpendicu- 
 lar to the horizon. 
 
 Now if through the point at which the body is placed a tangent 
 plane to the surface be drawn, and from the same point a tangent 
 line, perpendicular to the horizontal trace of the tangent plane, 
 this tangent line will, obviously, be shorter than any other drawn 
 from the same point to the same trace ; hence this tangent line must 
 be more nearly perpendicular to the horizon than any other through 
 the body's plane, and, consequently, in the direction of this line 
 the body will tend to move ; the very same conditions, therefore, 
 which would be necessary to counteract the tendency to move on 
 the surface, would be necessary to counteract the tendency to move
 
 EQCILIBRIUM ON A SURFACE. 47 
 
 if the point were placed on this line, instead of on the surface ; we 
 may, therefore, in seeking the conditions of equilibrium, substitute 
 the straight line of which we are speaking, for the curve surface. 
 The position of this line is always determinable from the equation 
 of the surface, for the trace is found by putting in the equation of 
 the tangent plane, 2=0, supposing the plane oi xy to coincide with 
 the horizon, and the line sought will be represented by the equations 
 which characterize the perpendicular to this trace through the 
 given point. In the vertical plane through this line must the forces 
 act, so that we may resolve each into two, one acting in this line, 
 and the other in a line perpendicular to it; the sum of the forces 
 acting in this latter line, be it what it may, will be counteracted by 
 the resistance of the line, so that to establish the equilibrium it will 
 merely be necessary that the sum of the forces, acting in the in- 
 clined line, be 0. 
 
 Hence we shall need but one equation of condition ; and, indeed, 
 in whatever two rectangular directions the forces be resolved, the 
 conditions of equilibrium will always be expressed in a single equa- 
 tion. For calling the resistance of the line R, and the sum of the 
 components of the other forces X and Y, we know that the condi- 
 tions of equilibrium are 
 
 P X _ p 
 CX+Rcos. a=0) J cos. a I /-* V 
 
 ^Y+Rsin. a=0 5 -'-I Y ^ ^ r '(A-) 
 sin. a 
 Now each ofthese equations exists sepam/c/y, whatever be X, orwhat- 
 ever be Y, because R is always equal, and opposite to the pressure 
 
 X Y 
 
 or—: > whatever this may be; and, as the equilibrium 
 
 cos. a sm. a 
 
 requires that they exist together, it is" merely necessary that we 
 
 X Y 
 
 have =- or X tan. a— Y=0 • (B.) 
 
 cos.o sm. a 
 
 If we take the axes of components the one parallel and the other 
 perpendicular to the horizon, then the angle a (fig. 22) will be ob- 
 tuse, and COS. a= — cos a'= — sin. i and the equation of condition 
 just deduced is, therefore, in this case, X + Y tan. i=0 • (C) ; i 
 being the inclination of the line of support to the horizon. It must 
 be remembered that when this equation is satisfied, and we wish to 
 determine the resistance R or the pressure on the line, we must recur 
 to one of the equations (A). 
 
 From what has now been said, it appears that the equation (C) 
 expresses the conditions of the equilibrium of a heavy body upon 
 any curve surface, i being the inclination of its tendency to move,
 
 48 ELEMENTS Of STATICS. 
 
 in virtxie of its weight, to the horizon ; and the horizontal and ver 
 tical axes of components being taken in the vertical plane of this 
 tendency. 
 
 We shall now proceed to the solution of a few problems. 
 
 Problem I. — (34.) Given the inclination i, of the straight line 
 AC, to tlie horizon AB, and the weight of a heavy body W to de- 
 termine what weight, P acting in a given direction, W M will be 
 sufficient to sustain W on the line (fig. 23). 
 
 Here are three forces acting at W, viz. the weight W in the ver- 
 tical direction WY, the resistance of the line AC acting in the per- 
 pendicular direction WR, and the weight P acting in the direction 
 WM, and all these directions are given ; hence, as one of the forces 
 W is given, we have enough to determine the other two. 
 
 Let us call tlie given angle CWM, (, then CWQ being equal to 
 90 + 1, wc have MWQ=90+i + f, which call 0, then we have sin. 
 MWQ=sin. e, sin. MWR = cos. t, sin. RWQ=sin. QWN= sin. i, 
 consequently, calling the resistance AVR, R, 
 
 _R sin.g ^ sin. e_ cos. (i-fQ 
 
 AV COS. ( ' ' COS. f cos. ( 
 
 P sin. i sin. i 
 
 AV COS. s cos, t 
 
 If the power P act along the plane, then f=0, and, consequently, 
 
 P Sin* i 
 in this case, R=W cos. {, P=W sin. i .-. :=r= — '—.' 
 
 K cos. I 
 
 If the power act in a direction parallel to the horizon then t= — i: 
 
 hence R=AV :=AV sec. i, P=AV — ^.•.— =sin. i. 
 
 COS. I cos. I a 
 
 If the power act in a direction perpendicular to the horizon, then 
 
 fi = 180°, also COS. f =sin. i, therefore, R=0, P=AV ; that is, there 
 
 must be no pressure upon the line, and, therefore, the power P must 
 
 be equal to the whole weight which it sustains. If we suppose the 
 
 power to act perpendicularly to the line, then {=90°, and cos. 
 
 (i-f-{)=sin. I, and the general formulas give 
 
 sin. i sin. i 
 
 R = AV — = QD,P = AV — = OD. 
 
 These equations show that the equilibrium cannot be maintained 
 under these circumstances, unless an infinite pressure is exerted 
 on the line requiring an infinite power P. On reviewing the fore- 
 going results, it appears that the power P, necessary to support a 
 weight AV on an inclined plane, will be the least possible when it 
 acts in the direction of this plane : indeed it is plain from the
 
 EQUILIBRIUM ON A SURFACE. 49 
 
 Sin. t 
 general expression P=W — '—, that P will be the least possible, i 
 
 remaining the same, when cos. t==l. 
 
 If we had solved this problem by the method of resolution, 
 taking for axes the lines WC, WR, as recommended at the former 
 part of last article, then the single equation of equilibrium of which 
 we have there spoken would have been P cos. f — W sin. i=0 . (1) ; 
 from which we immediately get the value of P sought, viz. 
 
 p=w'^"''. 
 
 cos. £* 
 To determine the pressure we must employ the equation furnished 
 by the other component forces, viz. those acting in WR, this equa- 
 tion is R+P sin. t — W cos. i=0 (2.) 
 
 „ „, sin. i sin.s — cos. i. cos. f 
 or R=W =0 
 
 COS. f 
 
 „ „, cos. i cos. ( — sin. i sin. t „^ cos. ( i 4-s) 
 .'. R=W = W i^ <-. 
 
 cos. £ COS. i 
 
 By employing the second mode of resolving the forces, that is ac- 
 cording to horizontal and vertical axes, we should have for the con- 
 ditions of equilibrium the equation (C), in this case, 
 
 P cos. (s+i) -f-lP sin. (,+z_W?!^.=0. 
 ^ ^ cos.e 
 
 whence P cos. ? cos.*^ i+P cos. i sin.^ z=: W sin. i 
 
 .-.Pcos. .=Wsin.i.-.P = W— -. 
 
 COS. s 
 
 For the resistance R we must employ one of the equations (A) ; 
 
 1 • 1 /. 1 T^cos. (s+i) _, ,,, COS. (f-fi) , ^ 
 
 taking the first we have P r-^^—: — i=K= W ^ ^ as before. 
 
 sm. I COS. f 
 
 Problem II. — Given the position of the line AC, and of the 
 pulley M, as also the weights of W and P, to determine where- 
 abouts AV must be placed that the equilibrium may be possible 
 (fig. 24). 
 
 The perpendicular MM' is given because the position of M and 
 of AC are given. Call this perpendicular a; then by equation (1) ; 
 last proposition, 
 
 W . . . ^P^—WHm.H 
 
 cos. a = — -sm. I .', sm. t= . 
 
 P p 
 
 Now WM sin. f=MM' = a 
 
 ...WM= « P'^ 
 
 sin.f sin.tVP^— W^sin.^i 
 an equation which determines the place of W. 
 E 7
 
 50 ELEMENTS OF STATICS. 
 
 Problem III. — Two weights W, W, attached to the extremi- 
 ties of a siring, which passes over a fixed puUy, mutually support 
 each other on two inclined planes (fig. 25), to determine the rela- 
 tions between W, W', the tension of tlic string, and the pressures 
 on the planes. 
 
 Here each weight is supported by the tension of the string, which 
 is the same throughout ; this tension will then be the same, as re- 
 gards each body, as tlie power we have hitherto called P. 
 
 If, therefore, we designate the angles concerned in one of the 
 
 planes as m prob. I., we have r^= ^ 1 
 
 VV COS. t 
 
 In like manner, for the other plane we have 
 
 R' cos. (i' + f') P sin. i' _ W _sin.i' cos. r _ 
 
 W cos. t' ' W cos. t' ' * W' sin.icos.f'' 
 
 which equations exhibit the relations required. 
 
 If the pulley be fixed at tlie intersection of the planes, so that 
 
 the string acts in each plane, then f — 0, t' — and 
 
 W sin. i' CA , . , . , • , • i i- 
 
 ——=— — :-=-—--; that IS, the weights are m this case as the lines 
 W sin. i CA' ^ 
 
 on which they rest. 
 
 It has been already observed that when the weight rests on a 
 point of a curve, the conditions are the same as if it rested on the 
 tangent line through that point; the inclination of this line to the 
 horizon, which it is necessary to know, may be determined when 
 we know the equation of the curve, referred to vertical and hori- 
 zontal axes, and the co-ordinates of the point where tlie body is 
 placed. If we resolve all the forces whicli are applied to the point, 
 in the directions of the axes, and call the inclination of the tangent 
 line to the axis of x, i, then we know that the conditions of equi- 
 librium will be expressed by the single equation (C), at page 47, 
 viz. X + Y tan. i=0; but, if x, y are the co-ordinates of the point, 
 
 we know (jDiff". Calc. p. 113,) that tan. i—-r'i hence the equation 
 
 ^ ux 
 
 of condition is X-fY -^=0 .... (1.) 
 dx 
 
 If a weight W be supported on a curve by means of anotlier 
 weight P (fig. 26), hanging vertically, the two weights being con- 
 nected by a flexible string passing over a pulley, then it will be 
 most convenient to take the vertical line MX as axis of x, and the 
 horizontal line MY as axis of y. In this case we shall have the 
 following values for the forces X and Y, viz. for X we shall have 
 the weight W diminished by P cos. MWn=P cos. WM7/1, and 
 for Y we shall have — P cos. MWw, that is, putting MW=r,
 
 EQUILIBRIUM ON A SURFACE. 51 
 
 X=W— P-, Y=— P^; 
 r r 
 
 substituting these values in the above equation, we have 
 
 T r ax r "^ ax 
 
 hni,s\ncer^=x^-{-y^.-.r -Y-^x-\-y-~-; hence the equation ol equi- 
 
 Ubrium is W-P-^=0 (2). 
 
 Problem IV. — A given weight W rests upon a circular arc, as 
 in fig. 26, being supported by another given weight P, by means 
 of a string passing over a pulley, fixed at a given point in the ver- 
 tical line MX, passing through the centre C : to determine the po- 
 sition of W. 
 
 Referring the curve to the vertical and horizontal axes MX, MY, 
 and calling MC, x', we have, for the equation of the curve, x^ — ■ 
 2x'x-\-x^^y^=:r" ; for substituting /, the length of the string MW, 
 for x^-\-y", I- — 2x'x=r^ — x'^. Hence differentiating with respect 
 
 dl dl x' 
 
 to x,l — x'=0.-. —j—=-j- , so that the general equation of 
 
 equilibrium (2) is, in this case, , 
 
 W-P— -0-/-— • 
 
 which gives the position of W. Or, because 
 
 pa ^' y3 
 
 l^=x--\-y''=r^A-1x'x — a:'^ .•.a^=Mm={-^^^+l }— — -—p. 
 
 Problem V. — Instead of a circle let the curve of support be an 
 hyperbola with its transverse diameter vertical, the pulley being in 
 the centre (fig. 27). 
 
 The equation of the curve is a^ y"^ — b^ x^= — a^ b^ ; and the 
 expression for /, the distance of any point in it from the centre, is 
 {Anal. Geom. p. 143-4,) 
 
 a^-i-b^ ■ a^4-b^ 
 
 l^= — - — x^ — 63_g2 ^2 — 53 . g2 being put for — ^— . 
 a^ ^ ^ a^ 
 
 Hence, by differentiating, /-y- =6^3: .*. — — =—j—, 
 
 and the equation of equilibrium is, therefore, 
 
 W_P^==0,...-f=cos.AM=^,.
 
 G2 ELEMENTS OF STATICS. 
 
 This fixes the position of W ; or if we substitute for / its value in 
 terms of e and x, as given above, we shall have 
 
 X^ = ^ X"— TT . .-. x= — — = M7». 
 
 It appears from this expression that the equilil)rium is impossible 
 if W is less than Pe ; and if W=Pe the point of rest must be at 
 an infinite distance. 
 
 Problem VI. — It is required to find a curve such that a given 
 weight P hanging over the pulley may balance another given weight 
 W at every point of it (fig. 28). 
 
 We have here to find a curve such that the equation 
 
 W-P^=0,„rW-P'^!+^=O 
 ax ax 
 
 may exist not only at one particular point, as in the preceding cases, 
 
 but at every point (x, y) of the curve. This equation, therefore, 
 
 can be no other than the diff'erential equation of the sought curve. 
 
 Hence, multiplying by dx and integrating, there results 
 
 Wa; — Pr+C=0, or Wa:— P ^/x^+ y-|-C=0, or 
 
 pa_W2 , 2WC C^ 
 
 y'+ — p7— ^' — pi-^'-- pr=o • • • • (1); 
 
 for the equation of the curve sought. In order to simplify this 
 equation, let us remove the term containing the first power of x, 
 which is done by substituting for x, in this equation, the value (See 
 
 WC 
 Anal. Geom. p. 173-4, )a:=.p^ wi""^-^' which leads to the equa- 
 
 tion, Y^ pj X2= (2); and this equation 
 
 characterizes an hyperbola related to its principal axes. For the 
 distance c between the centre and focus of this hyperbola, we have 
 
 WC 
 
 {Anal. Geom., p. 171,) c=^^rp; p^; but this is the distance of 
 
 the new origin from the primitive origin, and the primitive origin 
 IS on the pulley ; hence the pulley is at the focus of the hyperbola. 
 By putting first Y=0 and then X=0 in the equation (2), we 
 have for the semi-axes of the hyperbola, 
 
 >A P^ B ^ 
 
 '*^— W»— P«' (W— P»)t ' 
 
 in which equations C is arbitrary.
 
 PARAtLEL FORCES. * 53 
 
 SECTION II. 
 
 ON THE EQUILIBRIUM OF A SOLID BODY. 
 
 (35.) Having considered pretty much at large the equilibrium of 
 forces, acting upon a free point, it is time now to examine the more 
 general case in which forces act upon different points, all connected 
 together in an invariable manner, as we shall here suppose the parts 
 of a solid body to be. We shall divide the theory into two parts ; 
 first, considering the forces which act upon the body to be all 
 parallel, and then considering them to act in any manner whatever. 
 
 CHAPTER I. 
 
 ON PARALLEL FORCES 
 
 (36.) Let us first consider two parallel forces P, P,, acting at the 
 extremities of a straight line AB, (fig. 29,) and let it be required to 
 determine what must be the intensity, and where the point of ap- 
 plication, of a single force, which, acting on the line, shall have the 
 same effect as these two. Let us represent the parallel forces by 
 the lines AP, BPj, and let us apply to the extremities of the line 
 any two equal but opposite forces AM, BMj ; these will destroy 
 each other, and will, therefore, have no effect on the system. 
 Hence, instead of the two forces AP, BPj, acting on the line, we 
 may consider as acting the four forces AM, AP, BMj,BPj, or the 
 resultants of these, AR, BRj. We have thus exchanged our two 
 parallel forces for two oblique forces, meeting in some point C. 
 Considering the lines to be all rigid, we may transfer the points of 
 application of these forces to their point of concurrence C, making 
 CE=AR, and CEi=BRj, so that these concurring forces, acting 
 on C, have the same effect on the rigid line AB, with which they 
 are connected by the rigid lines CA, CB, as the original forces P, 
 Pj. Let us now resolve the forces CE, CEj, into their original 
 components Cm, C^, and C?n,,Cj9j ; then since the two Cm, Cm^, 
 are equal and opposite they destroy each other, so that the system 
 will be reduced to the two conspiring forces C;;, Cp^, or to the 
 single force C/j + Cp^, which is equal to AP + BPj ; thus we have, 
 for the intensity of the resultant of the two forces P, P^, P+Pj = 
 R ; and as this force may be applied at any point, in its direction 
 e3
 
 M ELEMENTS OF STATICS. 
 
 CO, will bp that point in the proposed line to which it must be 
 applied ; the situation of this point is thus determined. By similar 
 triangles, 
 
 C0_^ OBE/?i 0B_ Cp.Ep, _ Cp 
 
 A0~1e^ ^" CO "(1^1 '"■ AO~ Ep.i:p,~Cp^ ' 
 that is OB : AO : : Cp : Cp^, or Cp, Cp^ being equal to P, Pj ; 
 OB : AO:: P:P,. 
 
 Hence we conclude that the resultant of two parallel forces is 
 also parallel, is equal to their sum, and acts at that point which di- 
 vides the distance between them into parts reciprocally proportional 
 to their intensities. This point therefore is fixed however the di- 
 recticn of the components may vary. 
 
 The proportion, just deduced, gives also (see fig. 30,) 
 AB: OB:: R : P 
 AB:AO::R:Pi; 
 therefore P, Pj, and R are to each other, respectively, as OB, Oa, 
 AB, that is to say, that any two of the three forces are to each other 
 reciprocally as their distances from the third, so that when any 
 three of the six quantities concerned, viz. the three forces and the 
 three distances are given, the other three may be determined by the 
 successive application of this theorem. The same theorem then 
 serves to divide a given force R into tAvo others parallel to it, acting 
 at given distances OA, OB, on each side of 0. And lastly, it serves 
 also to determine the intensity and point of application of that force 
 Pj (fig. 31) which will keep in equilibrium t!ie line AO, acted upon 
 by two opposing parallel forces P, R', whenever such equilibrium 
 is possible. This qualification is necessary, because there is one 
 case in which two parallel forces, acting on opposite sides of a line, 
 cannot be equilibrated by any third force, viz. the case in which the 
 iwo forces are equal; for it is plain that if R' be equal to P, that R, 
 which is equal and opposite to R', cannot be the resultant of P, and 
 any other force Pj, because if it were we should have P + Pj =R, 
 whereas P alone is equal to R : hence Pj must be 0, and therefore, 
 by the theorem, tlie point of application B must be infinitely distant, 
 so that no single force can keep the line AO at rest when its extremi- 
 ties are solicited by equal parallel forces acting in opposite directions. 
 The tendency of these forces will plainly be to cause the line to turn 
 about its middle point, this being at rest. 
 
 From what has now been said it follows that the resultant of two 
 opposite forces, P, R', applied to different points. A, O, is equal to 
 their difference P/, acting parallel to them in the direction oj the 
 greater, and that its point of application B is given by the pro- 
 portion. 
 
 P', : R' :: AO : AB .•.AB=^;1a0= p, ^ p AO. 
 
 r 1 K — "
 
 PARALLEL FORCES. 55 
 
 From knowing how to compound two parallel forces acting upon 
 a straight line, we are enabled to compound any number so acting. 
 The resultant will, obviously, be equal to the algebraic sum of the 
 components, affixing opposite signs to those which draw in opposite 
 directions. As to the point of application of this resultant we shall 
 not stop to determine it for this particular case, but shall proceed to 
 consider the theory of parallel forces in all its generality. 
 
 (37.) Let P, Pj, Pj, &c. be parallel forces, applied to any sys- 
 tem of points. A, Aj, Aj? &c. any how situated in space, but inva- 
 riably connected by rigid lines, (fig. 32,) and let it be required to 
 determine the resultant of this system both in intensity and position. 
 
 The most obvious mode of proceeding is this, viz. first to com- 
 pound two of the forces P, P^, and to substitute for them their re- 
 sultant R; then to compound this with the third force Pg, and to 
 substitute for the two R, P^, that is, for the three P, P^, P^, their 
 resultant Rj, and so on till the system is reduced to the two paral- 
 lel forces R„ — 1, P„, of which the resultant will be that of the 
 whole system ; and will, therefore, be equal in intensity to the 
 sum of the components. In this process of composition the several 
 partial resultants R, Rj, Rg, &c. are not only determined in inten- 
 sity, but the point of application in the line joining the points, acted 
 on by the two components, is in each case determined. Every such 
 point would remain fixed, however the direction of the component 
 parallel forces might vary, provided their respective intensities did 
 not vary (36) ; and, therefore, the point of application of the final 
 resultant would remain fixed, however the direction of the system 
 of parallel forces might vary provided they retained their respective 
 intensities ; it is through this point, therefore, that the resultant of 
 the system must always pass under every change of direction ; it is 
 hence called the centre of these parallel forces. 
 
 (38.) Let now there be assumed any three rectangular axes, and 
 let us represent by 
 
 X, y, z, the co-ordinates of the point A 
 •^i» ?/if ^i» • • • • Aj, 
 
 •^2 '2/2' ^2' • • • • A.2» 
 
 &c. &;c. 
 
 and by X, Y, Z, those of the centre of tlie parallel forces ; we shall 
 prove that these latter are severally 
 
 - p + p^+p^... 
 
 y_ Py+P,yr+P,y, . 
 P+P.+P^... 
 
 ^- P4-Pi+P, .... J 
 
 ^...(1).
 
 56 ELEMENTS OF STATICS. 
 
 For let us first consider only two forces P, Pj, acting on the points 
 A, A, (fiff. 33), of which the abscissas are OA'=x AO, '=x\, and 
 let C be the centre of these two forces, its abscissa being OC ' '=X' ; 
 then if ff, c, Oj, be the projections of A, C, A^, on the plane of xy, 
 we shall have, on account of the parallels, 
 
 A,C : AC :: a^c : ac ::A.\C=x^ — X' : A'C=X' — x; 
 but (36), A,C : AC :: P : P, .-. x,— X' : X'—x :: P : Pj 
 
 .•.(P+PJX'=Px+P,x, ...X' = -^-^^. 
 
 Let us now proceed with the two forces (P+Pj) and P2, after 
 having joined their points of application C, A,, exactly as we have 
 proceeded with P and Pj, and, calling the abscissa of the centre of 
 our new forces X", the result must be 
 
 (P+P.+P,)X"=(P+PJX'+P,x, ; 
 or, substituting for X' the value just obtained 
 
 (P+P.-f-P^) X"=Vx-^V,x,+F^x^ 
 Px-\-V,x,-^P ^x^ 
 
 ' "^ - P + P1+P3 • 
 
 Proceeding in this manner till we arrive at the centre of all the 
 parallel forces, of which the abscissa is X, we shall have, finally, 
 Fx-^V,x,-\-P,X2 ... 
 P+P1+P2... 
 as announced ; and if for the axis of x we substitute successively 
 the axes of y and of z, we shall have the similar equations 
 
 V Py+Piyi+P^y^.-. 
 ^- p+p^+p^... 
 
 Pz + PlZ,+P2~2 • ■ • 
 P + P,+P, ... ' 
 
 and thus we may always determine the co-ordinates of the centre 
 
 when we know those of the points of application of the system of 
 
 parallel forces, as well as the several intensities of those forces. 
 
 Calling the resultant of the forces R, the preceding equations give 
 
 RX=Px+Pi x.+P^x^ ' ") 
 
 RY=P^ + P,3/,+P,3/, . 1.(2). 
 
 RZ=Pr+P, r,+P,^3 . J 
 
 We may here remark that it is possible so to place the axes of 
 
 co-ordinates, that two of the three equations (2) will suffice to fix 
 
 the position of the resultant of the system ; for let one of the axes, 
 
 as the axis of z, be taken parallel to the direction of the forces, 
 
 then, as the resultant itself will be parallel to the same axis, its 
 
 position will be known if we only know where it meets the plane 
 
 of xy, that is, if we know the X, Y, of any point in it ; hence the 
 
 two first of equations (2) are sufficient to determine the line in 
 
 which the resultant acts, and this is all we want to know, since on
 
 PARALLEL FORCES. 57 
 
 whatever point in this line it acts, the effect is the same. Under this 
 arrangement of the axes, therefore, the equations necessary for the 
 determination of the resultant in intensity and position are 
 
 R=P+P,+P2 + P3+ • ■) 
 
 RX=Pa^+P, a^.+P^ X2 + P3 x,+ • y • ■ (3). 
 RY=Pi/+P, 2/1 +P3 2/2+P3 2/3+ • J 
 (38.) The product of any force, by the perpendicular distance of 
 the point on which it acts from any plane, is called the moment of 
 that force with respect to the plane ; thus Fx is the moment of the 
 force P with respect to the plane of YZ, because x is the distance 
 of the point A on which it acts from that plane. Hence we learn 
 from either of the three equations (2) just given, that the moment 
 of the resultant of a system of parallel forces in reference to any 
 plane is equal to the sum of the moments of the components in 
 reference to the same plane ; the algebraical sum being always un- 
 derstood, regard being had to the signs of the forces as well as to the 
 co-ordinates of the points on Avhich they act. It is easy to see how 
 the foregoing results become abridged when the forces all act in one 
 plane, as also when the several points on which they act are in one 
 straight line ; in the former case only one co-ordinate plane is ne- 
 cessary, viz. the plane in which all the points are situated ; in the 
 latter case only one axis is necessary, viz. the line in which the 
 points are situated, so that either one or two of the foregoing gene- 
 ral equations may in particular cases become superfluous. 
 
 The preceding theory will enable us readily to determine the con- 
 ditions of equilibrium of a system of parallel forces ; for let us as- 
 sume the axis of z parallel to the direction of the forces, then, since 
 the sum of the forces, that is the resultant R, is 0, we have, by the 
 equations marked (3), 
 
 P + P,+P,-f P3-f . =0 
 Px+P^x,-{-F,x,+F,x,+ . =0 [- . (4). 
 Py+P,y,+P,2/,+P3 2/3+ 
 which are the equations necessary to establish the equilibrium, and 
 they express these conditions, viz. 
 
 1st. The sum of the forces must be equal to 0. 
 2d. TTie sum of their moments, in reference to each of two per- 
 pendicular jAanes parallel to their direction, must be equal to 0. 
 
 (39.) Before terminating this chapter, we should remark, that a 
 more concise notation is frequently employed to express the equa- 
 tions (1), (2), &c. thus the equations (1) are written 
 Y S (Pa;) ^ s (Pv) „ S (Vz) , , .... 
 
 sTpT sTpT SfTT' character S signifymg 
 
 the sum of the whole system of quantities of the form of that to 
 which it is prefixed. In like manner the equations (2) may be 
 
 8 
 
 '::}
 
 58 ELEMENTS OF STATICS. 
 
 written RX = S (Pa-), RY=s (Py), RZ=r (Tz), 
 
 and the equations of equilibrium according to this notation are 
 
 2 (P)-0, 2 (Px)=0, S (Pt/)=0, 
 provided the two perpendicular planes to which the moments are re- 
 ferred are parallel to the direction of the forces. We shall now pro- 
 ceed to some interesting and important applications of the theory de- 
 livered in this chapter. 
 
 CHAPTER II. 
 
 ON THE CENTRE OF GRAVITY. 
 
 (40.) Experience teaches us that all bodies within our reach tend 
 towards the earth, to which, if abandoned to themselves, or left un- 
 supported, they would fall in a vertical direction. The reason why 
 smoke and vapours in general do not fall to the earth, is that they 
 are not left unsupported, being indeed borne up by the air in the 
 same way that a piece of wood is borne up by the water in a vessel, 
 and prevented from reaching the bottom as it would do if this sup- 
 port were removed. This universal tendency of all bodies to the 
 earth, proves the existence of a soliciting power whose influence 
 extends equally to every body with which we are surrounded. By 
 the tendency here spoken of we mean the disposition to move, and 
 that this is the same in all bodies, great and small, when all support 
 is taken away, has been fully and frequently established by the most 
 convincing experiments ; thus if a very small particle be placed be- 
 side a large mass in a vessel exhausted of air, they will, when let 
 go, continue beside each other during the whole time of descent, and 
 will both strike the bottom of the vessel at the same instant, so that 
 if it were possible to destroy the cohesion among the particles of 
 matter, in virtue of which it becomes a solid mass, thus enabling 
 each particle to obey whatever force acted upon it individually, yet 
 the tendency to move being exactly the same in each, no one parti- 
 cle could in descending displace any other, so that the same ar- 
 rangement would be preserved during the descent as if all the par- 
 ticles cohered. We call this soliciting power of the earth, which 
 we see is altogether independent of the mass on which it acts, the 
 FORCE OF GRAVITY, or simply gravity, thus naming an influence, the 
 nature of which we know nothing only as regards its effects, and 
 this is in fact all that we here require to know of it. 
 
 It may be proper here to caution the student against a ver}' com- 
 mon misapplication of the term gravity. We use incorrect language
 
 CENTRE OF GRAVITY. 59 
 
 when we speak of the gravity of this body, or the ^avity of that, for 
 gravity is not of the body but of the earth, and is always the same 
 at the same place, exerting the same effect on all bodies however dii- 
 ferent, that is, producing in all the same tendency to move. The 
 weight of a body furnishes us with no information respecting the 
 force of gravity, but only with respect to the number of its constituent 
 particles, for if one body is double the weight of another this does 
 not arise from any variation in the force of gravity, but because there 
 are twice the number of particles in one body that there are in the 
 other, and each particle is influenced alike ; so that it will require 
 double the effort to support one that it requires to support the 
 other ; the lighter body may, however, have more external surface 
 or appear under gi-eater bulk than the heavier, but then the pores 
 which separate the component particles M-ill be proportionally larger. 
 
 Understanding by the weight of a body the effort necessary to 
 prevent its falling, we may correctly say that the weight of a body 
 is the resultant of all the efforts (or weights) which gravity im- 
 presses upon its component particles ; as these component efforts 
 are all directed in parallel lines, their resultant must be equal to their 
 sum, and act in their common direction and at a point which will be 
 the centre of these parallel forces, and which in the present case is 
 called the centre of gravity of the body. The determination of this 
 centre in different bodies may be effected by the application of the 
 theory delivered in the preceding chapter, provided we suppose, as 
 we shall here do, that the bodies proposed are perfectly homoge- 
 neous, so that the effort necessary to counterbalance the influence of 
 gravity on any part of the body will be proportional to the mass of 
 that part. 
 
 Before proceeding to particular applications of the theory, we may 
 as well here notice the distinguishing characteristics of the point 
 which we have called the centre of gravity, and which are direct in- 
 ferences from that theory ; these are, 1st, that if the centre of gra- 
 vity be supported, the whole body will be in equilibrium, because 
 the resultant of all the forces which act on it will be opposed in 
 whatever position the body be plaped : moreover every body kept 
 in equilibrium by a single force, must have its centre of gravity 
 in the line of direction of that force. 2d. The sum of the pro- 
 ducts of each particle of a body into its distance from any plane, 
 the distances on opposite sides taking opposite signs, is equal 
 to the product of the whole mass into the distance of its cen 
 tre of gravity from the same plane ; so that if a plane divide a 
 body into symmetrical halves, it must pass through the centre of 
 gravity. The first of these properties points out an experimental 
 method of finding the centre of gravity of a body : thus, let 
 the body be suspended by a string attached to any point, it will
 
 60 ELEMENTS OF STATICS. 
 
 arrange itself so that this string would, if we could continue it, pass 
 through the centre of gravity. In like manner, if it were suspended 
 from any other point, the line of the siring would also pass through 
 the centre of gravity, consequently the intersection of these two 
 lines would determine that centre. If the body have a Hat surface, 
 we may lav it on a horizontal table pushing it more and more over 
 the edge till it just balances itself, in which position the centre of 
 gravity will be vertically over the edge of the table ; if, then, we 
 mark the line of the edge on the body, and proceed in the same way 
 with the body in another position, we shall thus have a point in the 
 same vertical as the centre of gravity, and to which if, as a support, 
 an indefinitely slender vertical rod were applied, and the table re- 
 moved, tlie body would remain in equilibrium. Or if it were to be 
 suspended by this point, the flat surface would assume a horizontal 
 position- 
 
 (41.) We shall now investigate general analytical expressions 
 for the determination of the centre of gravity of any body whatever. 
 
 Let ABC, Sic. [fig. 34,) represent any solid body, the component 
 particles of which we shall call P, P,, V^^Sic. and their sum or 
 the mass of the whole body, B. Then, if G be the centre of gra- 
 vity of this body, and the body be referred to three rectangular 
 planes, the distance of G from the plane of zy will, by equation (1), 
 
 page 55, be X=GH= ^'''^^^'^^^^^'^^"^ " ' ' (1). 
 
 The numerator of this fraction consists of the sum of the in- 
 numerable particles P, P^, P2,&c., multiplied by their respective 
 distances from the plane of zy; but although the terms are innu- 
 merable, yet their sum may be accurately determined by the aid 
 of the integral calculus. In order to this determination, let CN be 
 any increment of the body, then the corresponding increment of 
 the expression under consideration, that is of the numerator of (1), 
 will be equal to the sum of all the particles in the slice CN, mul- 
 tiplied by their respective distances from the plane of zy. Now, 
 calling the increment MN of the abscissa, //, it is obvious that 
 however small we take //, that' is how^ever slender the slice CN 
 may be, the sum of which we have just spoken will always be 
 comprised between these two, viz. the sum of the same particles 
 when multiplied each by the distance AM=.2', and the sum when 
 multiplied each by AN=ar+A ; that is, putting S for the expres- 
 sion we are considering, and A S, A B, for the corresponding incre- 
 ments of this and of the body, A S will always be intermediate 
 between .r A B and (.r-f /<) aB, but the ratio of these is 
 
 (.t+ZOaB , . ^ ,. . 
 
 ii — = 1 m the limit, 
 
 xaB
 
 CENTRE OF GRAVITY. 61 
 
 or when h, and consequently A B, is 0; therefore the ratio of the 
 intermediate quantity A S to either must in the limit be 1 ; that is, 
 
 ^—=1 in the limit, that is, -^=].-,f/S=:rrfB .-. S=/j;rfB ; 
 hence the expression (1) is 
 
 X=GH = -/i^] 
 
 B I 
 
 In like manner S. . . . . (A), 
 
 fyclB 
 
 Y=-^ 
 
 B 
 
 equations from which the co-ordinates of the centre of gravity of 
 B may be determined when the equation of B is known. 
 
 But it must be observed, that though in all these equations rfB 
 signifies the differential of the body, yet it is not to be represented 
 in all by the same analytical expression : for regard must be had 
 to the position of the slice A B, as this will in general be different 
 in its three positions, parallel to the rectangular planes ; and there- 
 fore also, in general, the expressions for dB will all three be dif- 
 ferent; but this will be shown more clearly when we come to 
 apply the formulas to the determination of the centres of gravity 
 in surfaces and solids, (art. 43.) 
 
 It is seldom, however, requisite to employ all three of these 
 equations for that purpose ; much will depend upon a happy ar- 
 rangement of the co-ordinate axes : thus, if we know the position 
 of a plane that will divide the body into symmetrical halves, then 
 we know that the centre of gravity must lie in this plane, (p. 59 ;) 
 taking, therefore, this for one of the co-ordinate planes, it is clear 
 that two of the equations (A) will suffice to determine the centre. 
 If we know two perpendicular planes, of Avhich each divides the 
 body into symmetrical halves, and in all bodies of revolution any 
 two planes through the axis of revolution will do this, then we 
 also know the line in Avhich the centre lies, and hence one of the 
 above equations will be sufficient. Should the body be merely a 
 lamina of matter, so thin, indeed, as to be taken for a plane sur- 
 face, then, by choosing the axis in this plane, more than two of 
 the foregoing equations can never be requisite, and but one if one 
 of the co-ordinate axes divide the figure symmetrically into halves ; 
 and the same is obviously true if the body be considered merely as 
 a plane line. If the curve be of double curvature, all three of the 
 equations will generally be necessary. 
 
 Let us now proceed to the actual determination of the centres of 
 gravity in given figures, considering in order lines, surfaces, and 
 solid bodies. 
 F
 
 62 ELEMENTS OF STATICS. 
 
 Determination of the Centre of Gravity of a Plane Line. 
 
 (42.) When the body may be considered as a line lying in one 
 plane, we shall put 2 « for B ; and supposing first that the line is 
 symmetrically situated with respect to the axis of x, that is, that the 
 centre is in this axis, we shall have by the first of (A) this expres- 
 sion for the distance of the centre from the origin, viz. 
 
 x=-' — = • 
 
 s s 
 
 But when the line is not symmetrical with respect to the axis, 
 then we must determine the Y of the centre of gravity as well as 
 the X, and this, by the second of equation (A), is 
 
 s 8 dx 
 
 is given in terms of x by the equation of the line. 
 
 , Problem I. — To determine the centre of gravity of a given 
 straight line. 
 
 fxdx 7> oc^ 
 
 In this case we have X=^- = =5 x, therefore, represent- 
 
 x X 
 
 ingthe whole line by a, we have, when x=a,X=\ a, so that the 
 centre of gravity is at the middle point, as indeed is obvious without 
 calculation. 
 
 Problem II. — To determine the centre of gravity of the contour 
 of any polygon. 
 
 Let us represent the sides of the polygon by Pj, Pj? Pg* <fec. 
 and let the co-ordinates of the angular points be 
 
 then the co-ordinates of the middle points of the sides will be 
 ^\+x^ y^+y^ . a^2+a^3 3/2+^3 . ^^ 
 2 ' 2 ' 2 ' 2 ' 
 and these points are, by last problem, the several centres of gravity 
 of the sides, so that we now have to find the centre of gravity of the 
 weights P, Pj, Pj, &c. acting at these points, and for this we have 
 the equations 
 
 2 /^p _j_p f p _|_ p ■) 
 
 Y_ Pt {y,+y.)+^Xy.+y,)+'P. Jy-^-i-yS-^ - • • Pn (y.+yi) . 
 
 2(P, + P, + P3+ p.)
 
 CENTRE OF GRAVITY. 63 
 
 It is obvious that if the polygon be regular, the middle point, or 
 the centre of the inscribed circle, will be the centre of gravity, and 
 •f*!' Pa' P3' *^^' ^^^^^ ^^ ^^^ equal ; hence if the polygon have n sides 
 
 an equation which expresses this geometrical property, viz. that if» 
 from the corners, and from the centre of a regular polygon, perpen- 
 diculars to any line in its plane be drawn, the sum of the perpendicu- 
 lars from the corners will be equal to as many times that from the 
 centre as there are sides to the polygon. 
 
 Problem III. — To determine the centre of gravity of a circular arc 
 BAC=2s (fig. 35). The equation of the curve referred to the axes 
 AX, AY is 
 
 y^=2rx — a;2 
 
 ' ' <^^ ^2rx—x^ ' \ "^rfx^ \^2rx—x'^ 
 •'• ^=T/v7^r^=y ^-^'^^'^^^^'+'^ (Int.Calc.pm-S). 
 
 s ^ ^-^ J s s 
 
 so that the distance of the centre of gravity from the centre of the 
 
 circle is a fourth, proportional to the arc, the radius, and the chord 
 
 of the arc. 
 
 When the arc is a semicircle the chord is double the radius, and 
 
 then 
 
 2r r 
 
 0G= = -= -63662^ 
 
 3 . 141593 . . 1 . 57079 
 
 and when it is a whole circle, then, y being = 0, OG is 0, as we 
 otherwise know to be the case. 
 
 It may be here observed, that in this solution we have not sought 
 for the arbitrary constant necessary to complete the above integral, 
 nor need we seek for it in any case, because it is always the definite 
 integral that we want. Since the body whose centre of gravity we 
 require is necessarily limited, it is between its limits that our integral 
 is to be taken, and thus limited it can require no correction, (Int. 
 Calc. p. 90-1.) 
 
 Problem IV. — To determine the centre of gravity of the arc of <i 
 cycloid. 
 
 The diflferential equation of this curve is, (see Int. Calc. p. 115,)
 
 64 ELEMENTS OF STATICS. 
 
 dy WT—y 
 
 dx \ y 
 where r is the radius of the generating circle 
 
 For the whole curve y^2 r, so that the distance of the centre of 
 gravity G from the vertex is equal to one third of the diameter of 
 the generating circle. 
 
 Determination of the Centre of Gravity of a Plane Area. 
 
 (43.) When the body is considered as a plane area w, the first of 
 the equations (A) becomes, by substituting u for B, 
 fxdu fxydx 
 
 A= = — r— J . . . . ( 1 U 
 
 which is of itself sufficient to determine the centre when the axis 
 passes tlirouffh it, that is when this axis divides the figure into 
 symmetrical halves. But suppose we require the centre of gravity 
 of the part ii on one side the axis, that is of the area ABD (fig. 36), 
 then, in addition to X = AM, we must also know Y = MG. 
 
 Now P being any point (x, y), we are not to substitute in the ex- 
 pression for Y, instead of dB, the value ydx, which we put in the 
 expression for X ; because now we want, agreeably to the general 
 investigation, not the differential of APN, but that of APN'D ; the 
 former was correctly expressed by PN . dx, but the latter must be 
 PN' . dy=(AJ) — AN) dy=(a — x) dy, a being the distance of the 
 bounding ordinate from the origin ; hence 
 ^J-ydu^ f{a—x)ydy 
 
 « fi.a—x) dy ^ ^' 
 
 It appears then that the expression for du depends upon the manner 
 in which we conceive u to be generated, that is whether we consi- 
 der the increment \ii to be paralel to the axis of y or to the axis of 
 ' X, or, which is the same thing, whether we consider in the equation 
 of u the independent variable to be x or y. We may, however, ex- 
 press du in a form which will leave it optional with us which shall 
 be the independent variable ; this form is du=fdx dy, where the 
 integral sign applies to either of the difi'erentials under it, regard 
 being had in the integration to the limits between which the va- 
 riable is comprised. Thus, in the case we are considering, if we 
 take the integral sign to apply to dx, and integrate between the limits 
 r=a and x=x, then the above equation is the same as du=(a — x) 
 dy ; but if we consider the integral sign to apply to dy, then the inte-
 
 CENTRE OF GRAVITY. 65 
 
 gral y being between the limits y=y and 2/=0 (see fig. 36), the 
 above expression is the same as du=ydx. Hence, by writing du 
 in the above general form, the expressions for X and Y will be 
 
 ffdydx' ffdy'dx 
 
 in which the integrations may be performed in any order. If in the 
 
 first of these we integrate for 3/, first, we shall have the form (1) 
 
 above : and if we integrate the second for x first, we shall have the 
 
 1 fu^dx 
 form (2), but if we integrate this for y first, then Y= " -^ — . (3). 
 
 Jydx 
 
 When the area is bounded by straight lines only, the case will be 
 
 too simple to require the aid of the calculus, as in the first of the 
 
 following problems. 
 
 Problem V. — To determine the centre of gravity of a plane tri- 
 angle ABC (fig. 37). 
 
 Bisect any side as AB by a line CD from the opposite angle, then 
 the centre of gravity must be in this line ; for it will bisect any line 
 a h whatever drawn parallel to AB, so that, taking any point in da, 
 there shall always be a corresponding point in (/ b, equidistant from 
 CD, or from a plane through CD perpendicular to the plane of the 
 triangle ; hence the sum of the moments on one side this plane will 
 be equal to the sum of the moments on the other side ; hence the 
 centre of gravity being necessarily in the plane of the figure, must 
 be in the line CD. In like manner, if AC be bisected by the line 
 BE, the centre will lie in this line, hence it is at the point G where 
 it intersects the former. 
 
 We might immediately infer from this, that the three lines from 
 the angles of a triangle bisecting the opposite sides necessarily meet 
 in a point. To determine the distance of this point from C let us 
 draw ED, which, as it bisects the two sides AB, AC, must be paral- 
 lel to CB and equal to half of it ; hence the triangles EGD, BCD are 
 
 BC CG 
 
 similar .•.——-=-— --=2 .-. CG=|C|!I; to express this distance 
 ED GU) ^ ^ 
 
 analytically, put a, b, c, for the sides respectively opposite to the 
 
 angles A, B, C, and e for the line CE, then, (Geometry, p. 38-9,j 
 
 2e=^\2a''-\-2h^—c^\ .-. CG==i ^{2«2-f 26^—0^} 
 
 in like manner BG=^ ^Z \'2a'' +2c''—b''\ 
 
 kG=\ y/l^b'^+^c^—cv'] 
 
 Adding together the squares of these expressions there results the 
 
 equation 3 (AG^+BC-f CG«) =AB2 + BC2+CA'^ ; that is, in any 
 
 plane triangle the sum of the squares of the sides is equal to three 
 
 times the sum of the squares of the distances of the vertices from 
 
 tlie centre of gravity of the triangle. 
 
 f2 9
 
 66 ELEMENTS OF STATICS. 
 
 If three equal bodies be placed at the vertices of a triangle, their 
 centre of gravity will coincide with tlie centre of gravity of the tri- 
 angle ; for the centre of gravity of the two equal bodies A, B wiU 
 be at D, and the weight at this point will be 2 A ; hence, joining 
 D, C, the centre of gravity of 2 A and C = A will be a point G, 
 
 ^ ^ 2 A CG 
 such that — -— ^ T^-TT . 
 
 A (jrD 
 
 Again, the centre of gravity will still be the same point if the 
 equal weights be placed at the middle points D, E, F ; for the mid- 
 dle M of DE will l)e the centre of gravity of two of them, and the 
 centre of these and the third will be a point G, such that MG = 
 5GF and MG is equal to hGF, because, by similar triangles, 
 
 BF GF 
 EGM,BGF, j^=^, = 2. 
 
 The centre of gravity of any plane polygon may be found by first 
 finding the centres of gravity of its component triangles, and then 
 the common centre of gravity of all these points loaded with their 
 respective triangles. 
 
 Problem VI. — To determine the centre of gravity of a circular 
 segment (fig. 38). 
 
 From the equation of the curve, taking the centre for origin, and 
 putting OD = a, we have y=^\r^ — x^ \ 3 
 
 ' ' u u u ' 
 
 If the segment is a semicircle a = 0, and u = 1-57079 r", hence, in 
 
 this case, X= J^ = ^-^ = -42441 r. 
 
 Problem VII. — To determine the centre ofgravity of the common 
 parabola (fig. 39). 
 
 From the equation of the curve, y = \/[pa:| 
 
 _f xy dx _ /xf dx 3 3 
 
 fydx fxk dx b 5 
 
 If we require the centre of gravity of the semi-parabola ABa:, then 
 Y, which for the whole parabola is 0, will have to be calculated by 
 the second formula (3), that is, we shall have 
 
 ^^ fy^dx i fxdx 3 .c , 3 3 „ 
 
 ^=2ijdi=P' 27^rrf^=F^''^^^=T2^ = ¥^^- 
 
 Determination of the Centre of Gravity of a Surface of Revolution. 
 
 (44.) Let the axis of x be the axis of revolution, then it will pass 
 through the centre of gravity of the body, and therefore the first of
 
 CENTRE OF GRAVITY. 67 
 
 the general equations (A) will be sufficient to determine it. Putting 
 S for the surface, this equation becomes, (see Int. Cede. p. 138,) 
 
 f xd^ fxy (Is 
 
 J\. — ^ - * 
 
 /^W'^^'^-'^ 
 
 S fyds 
 
 fy 
 
 
 dx 
 
 Problem VIII. — To determine the centre of gravity of a spheric 
 surface (fig. 40). 
 
 By the circle, x^-\-y'''=r'^ 
 
 dy^ x'^ , , fxyds f xdx 
 
 •'• -T-7 = —^ •'• y(^^ = ^d^ •'• -"h^i =r,— • 
 
 dx^ y^ ^ fyds f dx 
 
 Hence, integrating between x=r and a:;=OC=a, we have 
 
 X=il!:!irj^=i(r+ a) =0G ; 
 r — a 
 
 hence G is at the middle of CB. 
 
 Problem IX. — To determine the centre of gravity of a conical 
 surface (fig. 41). 
 
 Taking the centre of the cone as the origin, we have for the re- 
 volving line OB the equation 
 
 y=ax.'.-^=a^.-.yds=a .^{l+a^] dx 
 
 fxyds ^fx^dx^2^^ 1.0C. 
 fy ds fx dx 3 3 * 
 
 Hence G is at the distance of one third the height from the base. 
 
 Determination of the Centre of Gravity of a Solid of Revolution. 
 
 (45.) Putting V for B in the general expression for X, and we 
 
 fx dY fxy^ dx 
 have lor any volume oi revolution X = =^-^r;: — = - , „ , • 
 •^ V fy'^ dx 
 
 Problem X. — To determine the centre of gravity of a segment of 
 a spheroid (fig. 42). 
 
 Taking the origin at A, the vertex of the generating ellipse, we 
 have for its equation ■ .: . f ■ 
 
 „_6^ fxy^ dx f(2ax — x^) xdx 
 
 y-=-,{^ax—x ) .-. X=-p-^-= j-^2ax'-x') dx ' 
 
 .^ ^ax^ — kx* 8ax—3x^ 
 
 that is AG=^— ^ — r— = — — • 
 
 ax^ — la^ 12 a — 4x 
 
 For a hemispheroid we have a:=aand X=fa.
 
 08 ELEMENTS OF STATICS. 
 
 As the foregoing expression for AG is independent of b, it re- 
 mains the same when b=a, or >vhen the body is a sphere, so that 
 if a sphere be described on either axis of a spheroid, any segments 
 cut off by a plane perpendicular to this axis will have the same cen- 
 tre of gravity. 
 
 (46.) The general formula employed in this article will, after a 
 slight modification, serve to determine the centre of gravity of any 
 volume generated by the motion of a varying surface along a fixed 
 axis, perpendicular to its plane, and passing through its centre of 
 gravity, as the axis of x, provided only that in every position this 
 surface is the same function of x. For, calling the generating plane 
 K, the differential ^/B of the body generated will be K dx, (Int. 
 Calc. p. 144,) and hence the formula in last article will be 
 
 /K xdx 
 
 JKdx 
 We shall give two examples of the application of this form of the 
 general equation. 
 
 Problem XI. — To determine the centre of gravity of a pyramid 
 or cone. 
 
 Let AB be the axis, which call b, and the area of the base HA, 
 a ; then, since the area of any two sections perpendicular to the axis 
 are as the squares of their distances from A, we have 
 b^_a_ Tz_<^^'' 
 
 substituting this value of K in the formula above, there results 
 fKx dx _ fx^ dx _ 3a,-* _ 3 
 ~ /K dx ~ /r» dx ~~4x^~ T^' 
 hence AG is equal to I the altitude of the cone or pyramid. 
 
 If the solid is the frustum or trunk of a cone or pyramid of which 
 the distances of the two ends from A are respectively b', b, then, in 
 the expression for x, we must integrate between these values of x, 
 that is 
 
 fl.Kxdx f\.x^dx 3 b*—b'* 
 
 /*, K rfa; /*, x"" dx 4 b^—b'^ 
 
 Problem XII. — To determine the centre of gravity of any seg- 
 ment of an ellipsoid. 
 
 Calling the principal semiaxes a, b, c, the equation of the surface 
 is a''6^2;'+aV^y''^-6^c'a:*=a''6"c^ and the equation of a section 
 at the distance x from, and parallel to, the plane of yz, is 
 
 b' z'+c' y'= ^-- ^
 
 CENTRE OF GRAVITY. 69 
 
 irom which we get these functions of x for the semiaxes of the ge- 
 nerating ellipse, viz. 
 
 ,, bs/laJ'—xH , c^\d' — x''\ 
 
 = ■ -1 C = ■ -1 
 
 a a 
 
 and therefore the area of the generating surface in any position is, 
 
 (Int. Calc. p. 124,) rtb' c' =^ {d'—x^)=K 
 
 X- 
 fKxdx_fx (a^— a?==) dx_h a^ x''--ix*_ Ga^— Sa?^ 
 
 X. 
 
 fKdx f(a^—x^)dx a^ x—^ x^ 12a''~4x^ 
 This expression being independent of 6 and c, shows us that ellip- 
 soids having one common axis, have a common centre of gravity for 
 all the segments cut off by a plane perpendicular to that axis ; which 
 is an extension of the property noticed at (45). 
 
 (47.) Having now given, in succession, all the most usual for- 
 mulas for the determination of the centre of gravity, together with 
 their practical illustration, we shall terminate by merely writing 
 down the forms which the general equations (A) take when we wish 
 to apply them to a surface, or a volume which is not of revolution, 
 nor yet symmetrical with respect to an axis. 
 
 The general expression for the differential of a surface S, referred 
 to three rectangidar planes, is, (Int. Calc. p. 151,) 
 
 where it is optional for us to consider the differential dS to be taken 
 either with respect to x or with respect to y ; hence putting this for 
 dB in the general equations referred to, or rather writing the coeffi- 
 cients under the radical in the more abridged form p', q', we have 
 these expressions for the co-ordinates of the centre of gravity of the 
 surface, 
 
 ff X y/ l+p'^+q"". dx dy 
 ffs/ T+p^+q'\ dx dy 
 
 X^j./j^y_ 
 
 Y^ //y ^ l+p'^+q'^- dx dy 
 ff^ l+p'^'-^-q'^. dx dy 
 
 ffx/l+p"'+q'^' dx dy' 
 Again, the general expression for the differential of a volume V is, 
 (Int. Calc. p. 148,) dY=f z dy dx; in which it is optional whe- 
 ther we consider the differentiation to be performed with respect to 
 the independent variable x or y; but we may render this expres- 
 sion still more general, as well as more symmetrical by putting fdz 
 for z, for then the differential dY=ff dz dy dx may be considered
 
 70 ELEMENTS OF STATICS. 
 
 as taken relatively to either of the three variables .t,j/, r, we please. 
 Hence the expressions for the co-ordinates of the centre of gravity 
 of the volume are 
 
 fffdz dy dx ff ^ dx dy 
 
 y^Mj/JliIiu!^ or //-y- '^y ^^ 
 
 fff dz dy dx ff z dy dx 
 
 2.=IILll^-^y— or iff^^^y^^^ 
 
 fffdz dy dx ffzdydx ' 
 On Guldin's Theorem, or the Centrobaryc Method. 
 
 (48.) The expressions for Y, at articles (42) and (43), furnish a 
 very remarkable theorem for the determination of the surfaces and 
 volumes of bodies of revolution. The expressions referred to im- 
 mediately give the equations 
 
 2 rt Ys=2 rt fyds and 2 rt Y f ydx=7t f y^ dx ; 
 the former relating to the curve line, the latter to the surface. Now 
 2rtY is the circumference of which Y is the radius ; it, therefore, 
 expresses the circumference which would be described by the cen- 
 tre of gravity of the line 5 if it were to revolve round the axis of x : 
 but 2 ft fyds expresses the area of the surface which would be 
 engendered by this revolution: hence, 1. The surface getierated by 
 the revolution of a curve round an axis is equal to the length of 
 that curve multiplied by the circumference described by the centre 
 of gravity. 
 
 Again, 2 rt Y, in the second of the above equations, being equal 
 to the circumference which would be described by the centre of gra- 
 vity of the surface s, if it were to revolve round the axis of x, and 
 nfy^dx being the expression for the very volume which would 
 thus be generated, it follows that, 2. Tlie volume generated by the 
 revolution of a plane surface round an axis is equal to the area of 
 that surface multiplied by the circumference described by its centre 
 of gravity. 
 
 These two propositions comprise the theorem of Guldin, and 
 their application to the determination of the surfaces and volumes 
 of bodies constitutes the Centrobaryc method. By this method we 
 see that when we know the length of the generating line, or the area 
 of the generating surface, as also the distance of its centre of gravity 
 from the axis of revolution, the value of the surface, or solid gene- 
 rated either by a whole or a partial revolution, may be at once found. 
 Also any two of these three things being given, viz. the generatrix, 
 the distance of its centre of grarity from the axis, and the magnitude 
 generated being given, the third may be found.
 
 CENTRE OF GRAVITY. 71 
 
 As an example of this method, suppose we wanted to know the 
 volume of a paraboloid of revolution : Let a be the axis or height of 
 the generating semi-parabola, and b its base ; then the distance of 
 its centre of gravity from the axis §6, so that the circumference ge- 
 nerated by this centre is Ibn. Again, the area of the generating sur- 
 face is, (Int. Calc. art. 67,) §c6, consequently, multiplying these two 
 quantities together, have, for the volume sought, 
 
 3 2 1 
 
 V=— - 6 rt x-^ ai = -roi^ rt ; 
 
 4 3 2 
 
 this volume is, therefore, I that of its circumscribing cylinder. 
 
 Suppose now that we wished to determine by this method the cen- 
 tre of gravity of a semicircle of radius r. We know that the area 
 of this semicircle is k r^ri, and that the volume of the sphere, gene- 
 rated by it is ^ r^rt ; hence, as this expression must be equal to the 
 former multiplied by the circumference described by the centre of 
 gravity sought, we have for this circumference, the value 
 
 ill^=lr; .•.|-r--2rt = -42441r=X; 
 ir^ 7t 3 3 
 
 and this is a more simple way of determining the centre than that 
 
 employed in problem VI. 
 
 We shall conclude the present chapter Avith a few miscellaneous 
 
 examples for the exercise of the student. 
 
 1. For the centre of gravity of a parabola of the nih. order, whose 
 
 . . , , . • ^ 2n+l 
 equation is a"— i2/=x" ; the expression is X=- -x. 
 
 2. For the distance of the centre of gravity of a semi-ellipse whose 
 axes are 2a, 2b, from the base or minor axis, the expression is 
 
 3 rt ' .. -•'" 
 
 ■ ' 2 
 
 3. For a paraboloid of revolution, whose altitude is a, X=—a. 
 
 o 
 
 4. For a segment of hyperboloid, whose altitude is a, 
 
 _, Sa4-Sx 
 
 ^ = T^ 7-^' 
 
 12a+4a; 
 
 5. The convex surface of a conic frustum, or trunk, is found by 
 Guldin's theorem to be equal to half the sum of the circumferences 
 of the ends multiplied by the slant height. "JX^ * - 
 
 SCHOLIUM. 
 
 It has been shown in the outset of this chapter, that for a body to 
 be supported, it is absolutely necessary that its centre of gravity lie 
 in a vertical line, passing through the base on which the body stands ;
 
 72 ELEMENTS OF STATICS. 
 
 or if the boily stand on props or legs, this vertical line must pass 
 through the area, whirh a string stretched round these legs would 
 enclose. The space thus enclosed by the feet of the human body 
 is. obviously, but small, and M'hen we consider the very various 
 positions in standing, stooping, walking, &;c. which we can easily 
 and safely assume, and the great rapidity with which we can pass 
 from any one of these positions into another, we cannot fail to be 
 impressed with the wisdom and bounty of the great Creator, who, 
 by such admirable disposition of the limbs and joints of the body 
 lias rendered this small space sufficient for its support in all these 
 various attitudes. A person in danger of falling experiences an 
 irresistible impulse to overtake, as it were, tlie point where the line 
 of direction seeks to meet the horizontal plane, and hence the long 
 strides which he is impelled to make when violently pushed in 
 the back, the feet endeavouring to overstep the point alluded to ; so 
 Avhcn standing and inclining the body forward, till the line of di- 
 rection is about to fall beyond our toes, we cannot help putting for- 
 M'ard our foot to overstep it. Children who have not the same 
 command over their limbs as grown persons, and who have, more- 
 over, a less sense of danger, do not always use the best means to 
 recover their stability when about to fall, but then in the more 
 hazardous circumstances they usually take care to secure for them- 
 selves a more spacious base than grown persons ; thus in walking 
 up or down stairs, we commonly see children employ both their 
 hands and feet, thus securing for themselves a very considerable 
 base, out of which the line of direction is not easily forced. 
 
 If a body rest on a single point, it is necessary that the vertical 
 line through it should pass also through the centre of gravity, but 
 " in certain cases a body resting upon a single point may yet have a 
 disposition to recover from any derangement, and to resume its ver- 
 tical position. Thus if the base be a plane, and the bottom of the 
 body rounded, but such that the centre of gravity lies below the centre 
 of curvature, the mass may rock backwards and forwards, but will 
 soon regain its erect site. Let O (tig. 43,) be the centre of the incur- 
 vation at the end of the body, and G or o- its centre of gravity lying 
 in the axis AO. Conceive the body to be rolled on its horizontal 
 plane from A to A', the point which touched A will merge into o, 
 and the axis will come into the position aO'. Now if the centre of 
 gravity G stood above 0, it would evidently in the position G' lean 
 beyond the vertical A'O', and the body would fall over : but if the 
 centre of gravity were at g below 0, it would still in changing to 
 g' lie within the vertical A'O', and, consequently, the body would 
 roll back to its first position." Leslie's Natural Philosophy, p. 55. 
 
 It may be remarked, that when, as in the case just adduced, the 
 body resists the tendency to overturn it, and* returns to its first po-
 
 EQUILIBRIUM OF A SOLID BODY. 73 
 
 sition of equilibrium, this equilibrium is called stable ^ but when it 
 yields to every tendency to overturn it, and falls, the equilibrium 
 from which it has been disturbed is called unstable. The equili- 
 brium of an ellipse resting on the extremity of its minor diameter 
 is stable, but the equilibrium when it rests on the extremity of the 
 major diameter is unstable. 
 
 CHAPTER III. 
 
 ON THE EQUILIBRIUM OF A SOLID BODY ACTED UPON BY FORCES 
 APPLIED TO DIFFERENT POINTS AND IN DIFFERENT DIRECTIONS. 
 
 (49.) We come now to consider a body or system of material 
 points connected together in an invariable manner, when in a state 
 of equilibrium from the action of any system of forces whatever, 
 and to determine what the conditions are which must necessarily 
 characterize such a system. The reasonings, therefore, of this 
 chapter will be so general that the results to which they lead will 
 comprehend in them all the particulars hitherto deduced respecting 
 the equilibrium of forces acting under certain restrictions, wdiether 
 through the intervention of a solid body, or upon a single point. 
 The more important of these particular deductions are, however, 
 essential to the establishment of the general theory, so that we are 
 not to expect that the discussion of the general proposition, on 
 which we are about to enter, will be independent of its particular 
 cases already considered, but that it will on the contrary be in a 
 great measure founded upon them. We shall find it convenient to 
 divide this proposition into two parts, taking first the case where 
 the component forces all act in one plane, and afterwards considering 
 them to act without restriction. 
 
 I. When the forces are all situated in one plane. 
 
 (50.) Let the plane of the forces P, P^, P^, &c. be taken for the 
 plane of xy, and let their points of application be (x, y), (x^, y^), 
 (Xg, 2/2),. &c., also let the inclinations of the forces to the axis of x 
 be as usual, a, a^, 02, &c,, and to the axis of y, j3, j3-^, 8^, &;c. ; 
 these latter angles being the complements of the former. Let now 
 each force be decomposed into two parallel to the axis, and we shall 
 thus have instead of 
 
 P the components P cos. a, P cos. /3 
 Pj . . P COS. ttj, P cos. /Sj 
 
 Pa . • . P COS. ttj, P COS. ^2 
 &-C. • &c. 
 
 G 10
 
 (0; 
 
 74 ELEMENTS OF STATICS. 
 
 SO that the points are now acted upon by two systems of parallel 
 forces, and, supposing that each system has a single resultant, it 
 will be parallel to the components, and it follows that the original 
 system may be replaced by two forces X and Y, of which the in- 
 tensities are 
 
 X=P COS. a-fPi COS. ttj-fPj, cos. 03-f&.C. > 
 Y = P cos. /3-j-Pi cos. /3i-|-P2 COS. iSj+'Stc. 5 ' 
 thus when we know the right-hand members of these equations, as 
 we are here supposed to do, we know also the intensities and direc- 
 tions of the two forces X, Y, but not as yet their points of applica- 
 tion. 
 
 To determine these let y' be the distance of the force X, from the 
 axis of X, to which it is parallel, and x' be the distance of Y from 
 the axis of y, then we know (38) that 
 
 Xy'=yVcos.a+y^V^ cos.a^+y^ P^ cos.a^-f &c. ? , . 
 
 Yx'=ar Pcos. /3-f-a;i P, cos./3i-far2 Pacos.^g-f&c. 5 ' • ' •\'^J'' 
 from which, as the values of x, a?,...., y, y,.... are supposed to be 
 known, x' and y' become known, and two lines drawn parallel to 
 and at these distances from the axes will coincide with the direc- 
 tions of the forces X, Y, and as a force may be applied at any 
 point of its direction, we may consider the intersection of these two 
 lines to be the common point of application of the two forces X, Y, 
 so that the original system is thus reduced to two determinate forces, 
 acting on a determinate point in known directions ; and, lastly, these 
 are reducible to a single determinate force R, by the equations 
 
 R=^/fX^-f YM cos. a=|, cos. b=l .... (3); 
 
 a and b being the inclinations of R to the axes of x and y. 
 
 It is thus proved, that when a system of forces, acting with given 
 intensities, and in given directions, upon a given system of points 
 invariably connected, has a single resultant, its intensity, direction, 
 and point of application may all be determined by means of the 
 given quantities. 
 
 (51.) It should be remarked here, that the equations (2) which 
 enable us to determine the point (x', y') of application of the result 
 ant, after we have found X and Y, furnish us with more information 
 than we absolutely require, for it would be quite sufficient for us to 
 know the situation of any point through which the resultant passes, 
 for knowing this the position and intensity of it would be given by 
 equations (3), and the effect of it will, we know, be the same to 
 whatever point it be applied. Again the position thus determined 
 ought, obviously, to be independent of the co-ordinates x, y ; a?,, y^ ; 
 (fee. because it would remain the same to whatever points the forces
 
 EQUILIBRIUM OF A SOLID BODY. 75 
 
 P, P , <fcc. are applied, provided we take them in their directions. 
 What is here said amounts to this, viz. that we ought to be able to 
 determine thy perpendicular distance of the resultant from a given 
 fixed point, when we know the perpendicular distances of the com- 
 ponents from that point, and this determination we shall be able to 
 effect by means of the equations (2). 
 
 The given fixed point from which the distances of the directions 
 of the several forces are to be estimated, we shall, for simplicity, 
 consider to be the origin and the product of each force by the per- 
 pendicular on its direction, we shall call the moment of that force 
 with respect to the proposed point. This being premised, let us 
 substitute for X and Y, in equations (2), their values (3), viz. 
 Rcos.aand R cos. b, and then subtract the one equation from the 
 other, and we shall thus have 
 
 R (?/' cos. O' — x' cos. 6)=P {y cos. a — x cos. j3) + 
 
 Pi (2/1 ^0^- «■! — ^1 ^^^- ^1)+ ^^^ 
 Let us inquire into the meaning of the first side of this equation. 
 Suppose m (fig. 44) to be the point of application of the resultant 
 'R=:m n, then mB=a:', mA = y', and let perpendiculars be 
 drawn from 0, A, B, on the resultant, then it is plain that ?/'cos.a 
 =A.p, x' COS. ^=B q ; also, if Os be drawn parallel to 7np the 
 triangles, m^/B, OsA, will be in all respects equal, and, therefore, 
 As=By, therefore Aja — 65'= Or, which call r, consequently 
 R (?/' cos. a — x' cos. b)='Rr^moment of R . . . . (4). 
 In like manner, for any other force P„, we must have 
 
 Pn (3/« <^os. a„ — x„ cos. /3„)=P„;?„=momen? o/P„ (5) ; 
 
 jo„ representing the perpendicular from O on the direction of P„ ; 
 hence the foregoing equation is the same as 
 
 nr=Fp+V,p^+F,p,+F,p,+ &c (6); 
 
 so that the moment of the resultant is equal to the sum of the mo- 
 ments of the components. 
 
 In the figure to which our reasoning has been applied we have so 
 taken the direction of the force R that the angles a and b may have 
 positive cosines : but the inference (4) will always be the same 
 whatever be the directions of R, thus in fig. 45, where cos. b is 
 negative, the same reasoning as that employed above shows us that 
 Or=Bq-\-Ap, and thus the same conclusion (4) is obtained. 
 
 (52.) The foregoing general theorem is analagous, in terms, to 
 that given at (38), for the moments of a system of parallel forces 
 with respect to a plane ; but the student must carefully observe that 
 there is no analogy between the theorems themselves ; for the mo- 
 ment of a force, with respect to a point, is altogether distinct from 
 its moment with respect to a plane ; this latter moment depends on 
 the point of application of the force, and is independent of its direc-
 
 76 ELEMENTS OF STATICS. 
 
 lion; whereas the moment, with respect to a point, on the contrary, 
 
 depends on the direction of the force, and is independent of its 
 
 point of application. 
 
 The second side of the equation (6) is supposed to be entirely 
 
 known, and this is only supposing that the intensities P, Pj, &c. 
 
 are known, and the several directions in which they act, or which 
 
 is the same thing, the equations of these directions. Thus, suppose 
 
 the equation of P„'s direction is known to be 
 
 t"os. /3„ , , , 
 
 y= -x-\-b .'. b cos. a,=v cos. a„ — X cos. 3.., 
 
 ^ cos. o„ " -^ 
 
 X and y being co-ordinates of any point whatever in the given di- 
 rection ;• if we take any particular point (x„, y^) we have 
 
 b cos. a„ = i/„ COS. o„ — x„ cos. ^3,, ; 
 therefore, by equation (5), last article Pn=b cos. o„ ; and b and 
 COS. a„ are, by hypothesis, known both as to sign and quantity ; 
 hence p^ is known both as to sign antl quantity. Hence the seconil 
 member of the equation (6) is entirely known when the intensities 
 and directions of the forces are known ; also R is known, from tlie 
 same data, by the equations (1) and the first of (3) in article (50), 
 consequently, r, and, therefore, the position of the resultant R is 
 known. By using the notation employed at (39) the expression 
 for r will be 
 
 and the equations necessary for the determination of the resultant, 
 both in intensity and direction, will be those marked (1) and (6) in 
 last article ; that is the three following 
 
 X = 2 (P COS. o), Y = 2 (P cos. j3), Rr=X (P/)); 
 the first two giving R=v/|X--f-Y*}. 
 
 (53.) When the system is in equilibrium then we have X=0, 
 Y=0, and, consequently, R=0, so that the equations of equili- 
 brium are 2 (P cos. a)=0, S (P cos. /3)=0, 2 (Pp)=0; or, in 
 their more expanded form, 
 
 P cos, a + Pi cos. a^ + Pg cos. aj + &C. = 0") 
 
 P COS. ^+Pj cos. /3j+P3 cos. ,3a+&;c.=0 I . . (I). 
 
 ^P +Pi/'i +P2/'. -f&c.=0j 
 We have already shown how the signs and values of the several 
 products in the last of these equations are to be determined from 
 the data of the problem ; but, in the case of equilibrium, it some- 
 times is convenient to determine the signs from other considerations 
 less analytical. Let us suppose the origin of the axes A (fig. 46), 
 which we have taken for the centre of moments, to be connected 
 with the system by means of the rigid perpendiculars/), p^, p^, &c. 
 on the directions of the several forces ; then the whole system be-
 
 A •!£-
 
 E<1UILIBRIUM OF A SOLID BODY. 77 
 
 ino- at rest the point A will be at rest, and may, therefore, be con- 
 sidered as fixed. Now the tendency of the forces P, P^, P^, as 
 marked in the figure, is to make the extremities of p, p^, p^, on 
 which they act, to turn round A in the same direction, and, conse- 
 quently, the tendency of the whole system with which these points 
 are invariably connected is to turn round A ; to prevent motion, 
 therefore, the remaining forces Pg, P^, &c. must impress on tlie 
 system an equal tendency to turn round A in the opposite direction ; 
 we ought, therefore, to attach to these opposing tendencies contrary 
 signs, that is, if we consider the forces P, P^, P^, &c. conspiring 
 in one direction to be positive, we ought to consider the forces Py, 
 P^, &c. conspiring in the opposite direction to be negative. In the 
 former method of determining the sign of any moment P„ p^^, we 
 considered P„ to have always the same sign, or to be positive, so 
 that the moment always took the sign of p^. If, on the contrary, 
 we attach the sign to P„, it ought, obviously, to be in conformity 
 to the principle just stated, and then we must consider p^ to be al- 
 ways positive 
 
 (54.) Of the preceding equations of equilibrium we may now 
 remark that the two first establish the condition that there must 
 exist some fixed point. A, in connexion with the system, round 
 which, however, the system may revolve ; the thii-d equation, in 
 conjunction with these, must then establish the remaining condition 
 of equilibrium, viz. that there can be no motion of rotation. If the 
 body or system, on which the forces act, be not entirely free, but 
 be only at liberty to move in any direction about a fixed point, then 
 the last of the above equations will be sufficient to establish the 
 equilibrium, the fixed point being taken for the centre of moments. 
 
 If of the equations (1) only the last has place, then all we -can 
 infer is that the system has no tendency to move round the point 
 taken for the centre of moments, and since we here suppose that 
 the value of R, viz. R=v^{X^-f Y^,5 is something, but that Rr or 
 2 (P/j) is 0, we must conclude that r==0, so that the point taken 
 for the centre of moments must be on the resultant, in order that 
 the equation S (P/;)=0 may have place, and for every point on 
 the resultant it obviously will have place ; so that if any point on 
 the resultant were to be fixed, this point would sustain the whole 
 pressure of the system. 
 
 //. TVhen the forces are situated in different planes. 
 
 (55.) Let us now consider the forces P, P^, P^, &c. as acting, in 
 any manner whatever, upon a solid body, or upon a system of points, 
 invariably connected. Let the inclinations of P to the axes of 
 X, y, and z, be as usual a, (3, y ; the inclinations of P^, a^, |3j, y^ ; 
 and so on. Moreover let the co-ordinates of any point in P's di- 
 G 2
 
 78 ELEMENTS OF STATICS. 
 
 rcction be .r, y, and z ; those of any point in P,'s direction, a-j, y^, 
 2^, &c ; and let all these directions be produced till thev pierce one 
 of the co-ordinate planes, say the plane of xy, excluding for the 
 present those forces which may be parallel to this plane. Let us 
 call the co-ordinates of the point where P pierces the plane of xy, 
 x\ and y', then, for the equations of this line, we shall have, 
 {Anal. Geom., p. 225-6,) 
 
 y—y-=bzS ''•■'<}' 
 where a and b denote the tangents of the angles which the projec- 
 tions of the proposed line on the vertical planes make with the 
 axis of z. Now between these constants a, b, and the given incli- 
 nations of the line itself to the axes, there exists the relations, 
 {Anal. Geom. p. 228,) 
 
 cos. a=a cos. y , COS. /3=i6 cos. y 
 cos. a , COS. (3 
 
 .-. a= , b— ; 
 
 COS. y COS. y 
 
 hence substituting these values in the equations (1), and putting z 
 =0, we have, for the co-ordinates, x', y', of the point where P 
 pierces the plane of xy, the values 
 
 , X COS. y ZCOS. a a; P COS. y 2 P cos. a 
 
 cos. y P COS. y 
 
 (2) 
 
 y COS. y 2 COS. 3 ■»/ P COS. y 2 P COS. j3 
 
 w'=" ^- . . . . (3j ; 
 
 "' COS. y P COS. y ^ ^ 
 
 at this point (x', y' ), since it is in P's direction, we may consider 
 P to be applied ; let us then do so and decompose P thus applied, 
 according to the three axes, and we shall then have, instead of the 
 original force P, these three, viz. P cos. a, P cos. /3, P cos. y ; the 
 first two acting in the plane of xy, and the third acting parallel to 
 the axis of 2, and all upon the point (x', y' ) where P's original 
 direction pierces the horizontal plane. Hence the moments of the 
 vertical force P cos. y with respect to the vertical planes, are, by 
 taking account of equations (2) and (3) just given, 
 
 X' P COS. y = P COS. y 2 P COS. o 
 
 y' P COS. y=.y P COS. y — 2 P cos. (3. 
 Now whatever we have said respecting the force P and its com- 
 ponents, equally applies to any other force P „, that is, it is equiva- 
 lent to the three component forces P„ cos. a„, P^cos. 3 ^, P^ oos.y^ ; 
 acting on the point where P „'s direction pierces the horizontal 
 plane : the first two being in that plane and parallel to the axis of 
 X and y, and the third force being vertical, or parallel to the axis of 
 2 ; moreover the moments of this last force, in reference to the 
 vertical planes, must be 
 
 a^« P„ t'o^- r„ — ^™ P„ <^os. a, and j/„ P„ cos. y„ — 2„ P„ cos. f3„ ;
 
 EQUILIBRIUM OF A SOLID BODY. 
 
 79 
 
 C « ti m « 
 2 -C ■ - - 
 
 s* g S 
 
 IC (M <D o 
 
 o 5^ ;::3 n c 
 
 ■O *i 
 
 3 s-C « n « 
 cr m o rt -1?^.^ 
 
 52 « o J3 -S "^ c3 
 
 S 2 
 
 o o "^ 
 
 
 « si ^^ § 
 
 c — -^ CL, I . 2 
 
 <u - a" s c -^ ~ 72 
 
 o >~. OJ 3 cj u 
 
 rx.^— 0) (U -H O t) ^ 
 
 S oj ~ .. I-" o '■-" 
 
 <u i- ^ ■;:;; ,o =« t„ '~^ 
 
 <» „, " ^ •-' t_ 
 
 ^ 2 X^ cu « K o 
 
 rt O rt ^ 
 
 (L> S 
 
 03 a; 
 
 m 3 
 
 
 6K ^. 
 
 o 
 
 -ii „, .tj a< -S 
 
 <D 3 c C . '=5 . _ 
 
 'T3X 
 
 3 O O O 
 
 'o.-f: II II 11 
 
 o o cr 
 
 5j 3 M OJ ^ 
 
 t) ca 3 -5 
 
 ^ s ^ 
 
 O ,3 
 
 = W^^ §" + + + 
 
 ^a,QH 
 
 (N w (>< 
 H »< t« 
 
 8 ?« ^- 
 
 oc aj* M 
 
 O o o 
 
 u o " 
 M <s ** 
 
 « IN J^ 
 
 +++ 
 
 7i 
 
 t*-i ^ 3 
 3 J2 QJ 
 
 
 O 03 « li. 
 
 OS - 3 OJ 
 
 .23 S- x'o Sf^ 
 
 g?^SpH§q;SS® 
 ^T'„^~^-i203 "» 
 
 ■"" " ■ a> 3 rt 
 
 P^t: o— j-g^ o 
 
 s . ^ S 
 
 O 03 
 
 C3. d ra o ;! 
 
 3 -3 ~ rt 
 
 
 rt 03 y 
 
 3 2 CL ?^ " ^ 
 
 ^ o o-> o — 
 
 >.- 
 
 « C 23 ?^ o !-i 
 
 ti P i^ "Z ^-^ ^ 
 S Pi 
 
 2 ^ 
 
 >.■? >,^ « 
 
 02 O X"^ CJ 
 
 ° o o o o 
 
 02 'tf M N M 
 
 g gpHpHOH 
 
 o o w oj o 
 
 ^ to 
 
 Oh « 
 
 X 8 "2. ?- 
 
 
 a*o 
 
 2 c .Si II S 
 
 -►J -J. 3 O Qj 
 
 ^ .^ 3 >^'-^ > 
 
 05 t, 
 
 rt 
 
 O) 
 
 r— < 4^ 
 
 _ O 
 
 .2 o a> 
 
 >5 03 
 
 V 
 
 03 -^^ 
 
 3 
 
 3 WrC 
 
 O 03 
 
 « rt « 
 
 P-l V 02 
 - 3 -C 
 '>irt *" Ph Oh Ph 
 
 II " ^+ + + 
 
 O) CO 03 02 
 O O O 
 
 o o « 
 
 PiPhPi 
 
 f^ tT ^ 
 
 11 1_ 
 
 S" i^ i^ 
 
 QJ « ^ ^ 
 _3 02 .3 ^3 
 
 '43 
 ^ .13 42 
 
 O 03 j2 S" Jo 
 
 bc-3 r3 2 *• 
 
 3 ~ « <U 
 Pi ^ -3 
 
 ft 
 2 S O 03 
 
 rt ^ 
 
 Ph 
 
 .2 ^ 
 
 + + + 
 
 -3 S S « « 2 2 g-^ 
 
 fioH*-^— 0-T352 
 
 3 -^ 03 03 '3 W 
 
 cr'- 
 
 O 
 
 02 "m ^ 
 
 2 P 
 
 ^PhPhPl, 
 
 3 O 
 
 W « 
 
 CO « -^ 03 f. 
 
 S S rt -^ 
 
 sS>3t,2?'-So 
 cr > t« S3 — K 2 
 
 C .3 rg o ^ '3 .2 « 
 r*^ '§ o 3 2 <^"S £ 
 
 -a 
 
 o .22 -o 
 -Q a, 
 
 m O (u 
 
 « 02 ^ 
 
 <*4 rt c3 3 3 .*-* •-- 
 
 l;; '!« 
 
 02 g 
 
 S =S ~ — ' r- 
 
 2^= ft^« «^.2i 
 J ^ ^^ 2 ^ .S g- ^ 
 
 yS03;»-tiMi:t.*'3 
 •^^^,'^£'002302 
 
 o2^ 03 g^:2-=« 
 
 "S g 3 O O « O
 
 80 ELEMENTS OF STATICS. 
 
 not concur, the body would not rquilibrato, but woxild have a ten- 
 dency to revolve round a fixed point in the middle of the line, 
 joining the points of application of the two resultants. As, how- 
 ever, the equation under consideration, the first of (6), thus far fixes 
 a point in the system, it prevents the body from tending to move in 
 the direction of the axis of x. Applying the same reasoning to each 
 of the other two equations of the group (6), we infer that they are 
 sufficient to prevent in the body all tendency to motion in the direc- 
 tion of the axes of y and z, so that the entire system cannot tend to 
 change its place, that is, there can be no motion of translation. 
 
 There may, however, exist, in conjunction with these conditions, 
 a tendency to rotation in three distinct directions, and these three 
 tendencies must be counteracted, before the equilibrium of the body 
 can be established. It must be these tendencies then that are coun- 
 teracted by the remaining three conditions (7) ; but let us examine 
 into the meaning of these equations, arid, for this purpose, it will 
 be sufficient to consider the first term y P cos. a — x P cos. j3. The 
 forces P COS. a, P cos. f3, act perpendicularly to each other on the 
 point to which P is original^- applied, and in a plane perpendicular 
 10 the axis of z ; y is the distance of that point, in the axis of 2, 
 which is in this plane, from the direction of the force P cos. a, and 
 X is the distance of the same point from the direction of the force 
 P cos. ,3 ; so that if these distances were rigid lines, these two forces 
 would tend to turn the system about this point, that is about the axis 
 of z in contrary directions ; hence the expressions y P cos. a and 
 X P COS. j3 are the moments of the forces P cos. a and P cos. ;3, with 
 respect to the axis of z, or, which is the same thing, with respect 
 to the point where their plane cuts the axis of z. Hence the 
 gi'oup of equations (7) intimate that in the ease of equilibrium, 
 the sicm of the moments of the component forces, ivith respect to 
 the three axes, are severally ; understanding by the moment of a 
 force, with respect to an axis, the product of that force by the dis- 
 tance of the axis from its direction, the force itself always acting in 
 a plane perpendicular to the axis. AVe have thus a right to infer, 
 seeing that equations (6) and (7) fully establish the equilibrium, 
 that when there is no motion of translation, and when moreover the 
 moments of rotation about three rectangular axes are each 0, the 
 moments of rotation about any other axes whatever must be 0. 
 
 It remains to consider the elfect of those among the original forces 
 which may be parallel to the plane of xy, since the decomposition 
 we have hitherto employed supposes the forces all to meet this plane. 
 Suppose P to be parallel to the plane of xy ; at its point of applica- 
 tion apply also two equal but opposite vertical forces Q, — Q, then 
 one of these being directed towards the plane of xy, the component 
 of this and P will meet that plane.
 
 EQUILIBRIUM OF A SOLID BODY. 81 
 
 Call this resultant P', then, instead of the force P, we shall have 
 the two forces P' and Q acting at the same point, and to which the 
 preceding theory is applicable. Now the horizontal components of 
 P' are the same as the horizontal components of P, seeing that P' 
 is composed of P and a vertical force ; and therefore the two first 
 of equations (6) and the first of (7) remain unaltered, whether we 
 substitute the two forces P' and Q for P or not. For the third equa- 
 tion of (6), these two forces furnish the terms P' cos. y+Q= — Q 
 -}-Q=0, the same as the original force P for which cos. y is 0. 
 For the second of equations (7), these same forces furnish the terms 
 (x P' cos. y — zP' COS. a)=( — xQ — z P COS. a) and (x Q, — 0), 
 the aggregate of which is the terra — z P cos. a, the same that would 
 be given by the original force P, for which cos. y is ; and pre- 
 cisely in the same way is it to be shown, that there will be no dif- 
 ference in the last equation whether we substitute for P the forces 
 P', Q, or take the force P itself. Hence the foregoing equations 
 establish the equilibrium, however the original forces may act on 
 the system. 
 
 (57.) When the body to which the forces are applied is not free, 
 but is only at liberty to move in any direction round a fixed point, 
 then, taking this point for the origin, the equilibrium will be esta- 
 blished if the equations (7) have place, for these equations forbid all 
 tendency to move in the only way in which the body is at liberty 
 to move. 
 
 In such a case the pressure on the point must be equal in inten- 
 sity to the resultant of all the applied forces, or to the resultant of 
 the same forces if they were all to be transferred parallel to their 
 directions to the fixed point itself, since, as is plain from equations 
 (6), it will require the same pressure or opposing force to preserve 
 the equilibrium in one case as in the other. Calling the pressures 
 on the point, estimated in the several directions of the axes of x,y, 
 and z, X, Y, and Z, we have, from equations (6), 
 
 X= 2 (P COS. a), Y= S (P COS. |3), Z=: S (P cos. y). 
 
 When the body can move only round a fixed axis, then, taking 
 this for the axis of z, all tendency to this motion will be prevented 
 if the single equation 
 
 (y P cos. a — X Pcos. p) + (2/i Pi cos. a^ — .r, Pj cos.sj-f 
 (2/2 P2 COS. ttg ' — X2 P2 COS. |32)-}-&c.^0, exist. 
 
 We may consider the fixed axis to be secured at its extremities 
 by two fixed points or pivots, and then the general equations of 
 equilibrium will furnish us with expressions for the pressures sus- 
 tained by these pivots. Call the pressures upon one of the points, 
 (0, 0, Z',) in the directions of the axes, X', Y', Z' ; and the 
 pressures upon the other point, (0, 0, Z",) X", Y", Z" ; then, 
 these being the forces which, with those applied to the body, secure 
 
 11
 
 82 ELEMENTS OF STATICS. 
 
 the equilibrium of the system, we must liavc, between them and 
 the applied forces, the conditions 
 
 X' + X"=— 2 (P COS. a),Y' + Y"=— 2 (P COS. ,3), and 
 Z' + Z"= — 2 (P COS. y) . (1), in order that the equations (6) may be 
 satisfied ; and likewise the conditions 
 
 Z'X' + Z" X"=—l: (xF COS. y — zT COS. a) I ,„x 
 
 z' Y"+z" Y"= — 2 Cy P cos. y — r P cos. ,i)S""^ ^' 
 in order that the equations (7) may be satisfied. 
 
 The third of the equations (1) shows that the axis is pressed in 
 the direction of its length by the force 2 (P cos. y) ; which is, there- 
 fore, divided between the two pivots, although there is nothing to 
 teach us in what proportion. The remaining four equations of con- 
 dition enable us to determine, when the intensities and directions, 
 as well as the points of application of the applied forces, are given, 
 what efforts they exercise on each pivot to carry away the axis. 
 
 (58.) Before concluding the present chapter, we shall briefly 
 advert to the remarkable analogy which exists between the theory 
 of moments and the projections of plane figures in geometry; and 
 first we may remark, that the several terms which constitute the 
 equations (7), and each of which we have called the moments of 
 two component forces with respect to the axis perpendicular to their 
 plane, may each be considered as the moment of the projection of 
 the force itself on the plane perpendicular to the same axis, Avith 
 respect to the origin A. Thus let us take any force P„, and repre- 
 sent it by a part of its direction estimated from the point of appli- 
 cation ; then the projection of the line P„ on the plane of xy may 
 be considered as the projection on that plane of the proposed force. 
 It is obvious that the components of this projection parallel to the 
 axes of X and y must be the very same as the components of P^ 
 parallel to these axes, that is, this projection is the resultant of the 
 two forces P, cos. a„ and P„ cos. j5„ ; but when two forces act in a 
 
 n - n n ' n ' 
 
 plane, the moment of the resultant in reference to a fixed point is 
 equal to the sum of the moments of the components, or to their dif- 
 ference if they tend to turn the system in contrary directions about 
 this point, and the moment of our two forces is y^ P^ cos. o„, and 
 x^ P„ COS. /3„. Hence, calling the resultant of these forces, or the 
 projection of P„, P'^ ; and;;',, the perpendicular upon it from the 
 centre of moments, A, we have 
 
 /''n P'n=3/n P;" COS. a„ — X„ P„ cos. |3„ , 
 
 and similar expressions obviously have place for the moments of 
 the projections on the other two planes. 
 
 As the projection of a line on a plane is the product of the line 
 by the cosine of its inclination to the plane, or by the sine of its 
 inclination to a perpendicular to the plane, the projection P'„, of 
 J'„ on the plane of xy, will be expressed by the product P^ sin. y„;
 
 EQUILIBRIUM OF A SOLID BODY. 83 
 
 and, in like manner, the projections P",, and P"'„, of the same 
 line on the planes xz and yz will be P"„ =P„ sin. /3„, P"'„ = 
 P sin. a„ ; hence, calling the perpendiculars on these projections 
 from the origin, or centre of moments, 'p'\ and ^9'",^, we may write 
 the equations (7) in this more concise form, viz. 
 /J' Psin.7+/)', P,sin.yj+7/3 P^sin. y^+&c. = 01 
 
 p" P sin. i3+p", PiSin./3j-f/>"3 P^sin. |3^-f &c.=0 K (7). 
 
 /j"'Psin. a4-7)"\ P^sin.aj+jt;"'2PgSin. a3 + &c.=0 j 
 
 If /),j be the perpendicular from the origin, or centre of moments, 
 upon the line P„ in space, then /)„ P,^ will be double the area of the 
 triangle whose base is P,^ and vertex the origin ; also /)'„ P'„ will 
 be double the area of its projection ; we may say, therefore, that the 
 moment of the projection of any force, is equal to the projection of 
 its moment. 
 
 As the moment /)' P' of the projection of any force P on the 
 plane of xxj, or the moment jo" P", of the projection on the plane 
 of rz, or the moment jo'" P'", of the projection on the plane oi yz, 
 is in each case equal to double the projection of the triangle whose 
 base is P and vertex the origin, or centre of moments, it fol- 
 lows, from the theory of projections, {Anal. Geom. p. 256,) that if 
 p"" P"" represent the moment of the projection of the same force 
 on any fourth plane, passing through the centre of moments, we 
 must "have S {p"" P"")=cos. 5 S {p'V)-j-cos. b' S {p" P") + 
 cos. b" ^ (p'" P'"), where 6, 6', 8", are the inclinations of the new 
 plane to the planes of j5' P', of/j" P", and of p'" P'" respectively. 
 
 It also follows from the same theory, (Anal. Geom. p. 256,) that 
 if any number of forces be projected on three rectangular planes, 
 and the moments of the projections on each plane, regarding the 
 origin as the centre, be collected into one sum, the squares of the 
 three sums thus furnished will be constant for every system of rect- 
 angular planes having the same origin. 
 
 For the determination of the principal plane, or that in which 
 the moments of projection of any given forces amount to the great- 
 est sum, see the Analytical Geometry, p. 256 et seq.
 
 84 ELEMENTS OF STATICS. 
 
 CHAPTER IV. 
 
 PROBLEMS ON THE EQUILIBRIUM OF A SOLID BODY. 
 
 Problem I. — (59.) A bent rod or lever ACB is suspended at C, 
 about which point it is free to move in a vertical plane, and weights 
 P, P3 arc attached to its extremities : to find the position in which 
 it will rest (fig. 47). 
 
 Let us take the fixed point C for the centre of moments, then it 
 will be sufficient for the equilibrium that the moments of the applied 
 forces balance each other. These applied forces are first, the weight 
 Pj of the arm CA acting at its centre of gravity, the middle point 
 a ; secondly, the weight P acting at A ; thirdly, the weight P^ of 
 the arm CB acting at the middle point b ; and lastly, the M'eightPg 
 acting at B : these forces all have vertical directions, and the mo- 
 ments of the two latter oppose the moments of the two former. 
 Hence, drawing the perpendiculars from C on the directions of the 
 forces as in the figure, we have this equation for the conditions of 
 equilibrium, viz. 
 
 P . C;)+P, . C;;,=Pj . Qp,+y, . Cp,, 
 and it is from this equation that the required position, that is, the 
 angle ACp or the angle BCpg, must be determined. Put o for the 
 angle AC;;, a for the angle BC/)^ and e for the given angle ACB ; 
 also, call the given length AC, 2 o, and the given length BC, 2 a', then 
 
 Cj9=2a COS. a, C/;j=a cos. o, Cp^=a' cos. a'=a' cos. (a-f 6)> 
 C/>3=2rt' COS. a'= — 2a' cos. (a-fe); 
 hence the equation of equilibrium is the same as 
 
 (2P + P,) a COS. a=— (P„+2P3) a' cos. (a + e), 
 where a is the only unknown quantity. Or, since 
 
 cos. (a+e) sin. o sin. 6 — cos. o cos. 9 
 
 ^^ = = tan. asm. 9 — cos. 9, 
 
 cos. a cos. a 
 
 the equation reduces to 
 
 (2P + PJ a=P3+2P3) «' (tan. o sin. e — cos. e) 
 (2P-f-PJ«+(P.+2P3)a'cos.e 
 
 •• ^"•'* (P,+2P3) a' sin. 
 
 If tlie extremities of the lever carry no weights 
 
 P. a + P„ a' COS. 9 
 
 tan. a= p; — ^. . 
 
 PjjO sin. 9 
 
 These results would remain unaltered, though the two arms CA, 
 
 CB were of difierent thickness ; but when they are equally thick, 
 
 P a' 
 since their weights must then be as their lengths, or, since -^^ = — ,
 
 EftDILIBRlUM OF A SOLID BODY. 8^ 
 
 the last expression may be put under the form 
 
 ff^+a'^ COS. e 
 
 tan. a= — ^7— ^ . 
 
 a^ sm. d 
 
 Problkm II. — (60.) An oblique cylinder stands on a horizontal 
 plane, its inclination to which is 60°, perpendicular height 4 feet, 
 and diameter of the base 3 feet. Required the diameter of the 
 greatest sphere of the same material as the cylinder that will hang 
 suspended from the upper edge (fig. 48,) without overturning the 
 cylinder. 
 
 The centre of gravity G of the cylinder is at the middle of its 
 axis CD ; hence the acting forces are the weight of the cylinder in 
 the vertical direction GE, and the weight of the sphere in the ver- 
 tical direction B'P and these two forces are just sufficient to prevent 
 any tendency to motion about B ; hence the equation of equilibrium is 
 
 BE X cylinder=BF x sphere. 
 
 Now by trigonometry sin. GDE : DGE : : GE : DE=2v/5 
 
 .-. BF=2DE=4v^|, BE=BD — DE=| — 4^i ; 
 
 hence, substituting the volumes of the cylinder and sphere for their 
 
 weights to which they are proportional, the equation of equilibrium is 
 
 (I— 4v/5)x32x -7854 x4=4^^x§X -7854 xdiameter 1 ^ ^ 
 .-. diameter=2 • 006 feet. 
 
 Problem III. — (61.) One extremity C of a heavy rod is move- 
 able about a fixed point in a vertical plane (fig. 49), and to the other 
 extremity B is fastened a cord which goes over a pulley A in a ho- 
 rizontal line with C, and supports a weight P equal to half the 
 weight of the rod : required the position in which the rod will rest. 
 
 The forces which prevent motion about C are the tension of the 
 cord BA, measured by the weight P, and the weight 2P of the rod 
 acting vertically at the middle of CB. 
 
 Draw BF perpendicular to AC, and CE perpendicular to AB, 
 then the equation of equilibrium is 
 
 P . CE=2P . i CF=P . CF .-. CE=CF. 
 Put BC=fl, AC =6, and AF=a', then the right-angled triangles 
 AEC, AFB being similar, we have 
 
 EC'' _BF^ ^ {b—xY_ a^—jb —xY _ a^~^b^-\-2bx — x'' 
 AC^~ BA^*"' ¥~ ~x^+a^—{b — xy~ aF^—b^+2bx 
 
 an equation which reduces to 
 
 2bx^ — {'ib^—a^) x^'—'Zb (a^'—b) x=0. 
 
 One root of this equation is x=0, and the other two, as given by the 
 
 quadratic, x^ — — x=a^ — b^ . (1) 
 
 H
 
 86 ELEMENTS OF STATICS. 
 
 are x=b —^{a±^{Sb^+a^)\ (3). 
 
 Let us examine into tlie nature of these as connected with the posi- 
 tions of the rod. Tlie first root .r=0 gives the position in fig. 50, 
 which position, however, the rod cannot take if AC is greater than 
 CI3, that is if b exceeds a ; but when b does not exceed a, then this 
 will be one of the position-s of equilibrium, and it may be remarked 
 that whatever be the weight of the rod moveable about C, tlie other 
 end B will always be supported in every position by a vertical force 
 equal to half that weight. 
 
 There cannot be any other position of equilibrium for the same 
 weight P between the lines CB, CA, because in no such position can 
 the condition of equilibrium, viz. CE^CF, have place ; hence no ne- 
 gative root of (1) can be consistent with the conditions of the question. 
 
 To determine in what circumstances the two roots are admissible, 
 put a=n6, then the values of x will be 
 
 which for ?i = l gives .r=0, and a'=:|6 ; when n is less than 1, 
 then it is obvious that both values of x are positive ; hence, when 
 a is not greater than b, there are always two positions of equilibrium 
 determinable from the two roots or values of x in (2). As one 
 of these values of x is always greater than b, the corresponding po- 
 sition of the rod will be as in fig. 51. When n is greater than 1, 
 one of the above values of x is always positive and the other always 
 negative ; hence, when a is greater than b, there is but one position 
 of equilibrium (fig. 52) determinable from the positive root or value 
 of X in (2), but then there is another position determinable from 
 x=0, that is, the extremity B will rest in the vertical line from the 
 pulley, as in fig. 50. 
 
 Problem IV. — (62.) AD and BC (fig. 53) are two heavy bars 
 moveable in a vertical plane about their extremities A, B in the 
 horizontal line AB ; required the position in which they will rest 
 by leaning against each other. 
 
 Call the weight of the bar AD, acting vertically at its middle 
 E, P ; and the weight of the bar BC, acting vertically at G, P^ ; then 
 the forces acting on AD, to turn it round A, are P in the direction 
 EK, and the pressure of CB in the direction DL perpendicular to 
 BC ; call this pressure P^, then the bar AD being at rest, we must 
 have P . AK=P3 . AL, also the forces acting on BC, to turn it round 
 B, are Pj in the direction GH, and the pressure P^ in the perpen- 
 dicular direction LD ; hence, Pj.BH=P3.BD, these two are 
 therefore the equations of equilibrium. Eliminating P^, we have the 
 single equation
 
 EaUILlBRlUM OF A SOLID BODY. 87 
 
 P.AK.BD=P,.BH.AL. 
 Put AB=a, AD=6, BC=c, and BD=a?; then 
 
 ATT KT? A hb{b'-^a^—X') 
 
 AK=AE • COS. A= ^^ — —- 
 
 2ab 
 
 BH=BG.cos.B=: 
 
 AL=AD . cos. D = 
 
 2 ax 
 
 2x 
 Hence, by substitution, we have 
 
 2b{b''+a''—x^)V=bc{a^+x^~b^) (b^-\.x^ — a'>)F^; 
 an equation of the fifth degree, ffom which, when numbers are put 
 for a, b, and c, the vahies of x may be determined, (see Algebra, 
 p. 210.) 
 
 Problem V.— ;-(63.) A given rod or beam, not of uniform thick- 
 ness, has one end suspended by a cord of a given length, fixed at 
 a given point above an inclined plane of a given inclination, and 
 the other end of the beam is sustained by the inclined plane ; it is 
 required to determine the position of the beam, weight sustained 
 by the cord, and pressure against the inclined plane when the beam 
 is at rest (fig. 54). 
 
 Let P be the given point, AP the string, AB the position of the 
 beam when at rest, with its end B on the given inclined plane BK. 
 
 The forces acting on the beam are the tension of the strino- in 
 the direction AP, the weight of the beam acting vertically at the 
 given point G, its centre of gravity, and the pressure at B acting 
 in the given direction BD', perpendicular to the plane. Let PA, 
 D'B, be produced till they meet in D, then, as the resultant of the 
 forces acting in these lines must pass through D, and as the direc- 
 tion of this resultant is vertical, it follows that the vertical line DG 
 must pass through the centre of gravity. Hence to determine the 
 position of the rod, draw AH, PK, parallel to D'D, and AM paral- 
 lel to BK, also produce BA to N ; then the position will be ascer- 
 tained, if we can find the angle PCK=PAM=0. The known 
 quantities are BG=fl!, GA=b, PK=c, PA=/, CBQ=i=inclination 
 of the plane ; if, therefore, we call the unknown angle ABK, or 
 NAM, ^, we shall have 
 
 PM=;sin. e, AH=KM=(a+6) sin. p 
 .-. PM — KM=/ sin. 6-\-{a-\-b) sin. p=c .... (A). 
 If, therefore, we can obtain another equation involving no other 
 unknown quantities besides e and ^, this latter may be eliminated, 
 and thence e determined. Now two different expressions for BD, 
 involving only these unknown quantities, may readily be obtained
 
 88 , ELEMENTS OF STATICS. 
 
 from the two triangles ABD, GBD, which have this side in com- 
 mon ; for observing that 
 
 PAN=BAD=9 — t 
 
 ADB=APM=90°— » 
 
 BDG=KBQ=/ 
 
 BGD=comp. QBG=90°— (i+t); 
 
 we have, from the triangle ABD, 
 
 sin. ADB : sin. BAD : : AB : BD ; 
 
 that is, cos. 9 : sin. (9 — ^) : : a+6 : BD . . . . (1) ; 
 
 and in the triangle BGD, we have 
 
 sin. GDB : sin. DGB : : BG : BD ; 
 
 that is, sin. i: cos. (i+t) '• • a- BD .... (2). 
 
 Equating now the two expressions for BD, furnished by (1) and 
 
 ,„. , (a+b) sin. (e — *) o cos. (i+*) 
 
 (2), we have l-J—i ^^ lL= —^-^^-^ 
 
 COS. e sm. t 
 
 .'. (a+6) sin. i sin. (s — ^)=a cos. B cos. (i+$) .... (B) ; 
 hence, by means of equations (A) and (B), 6 may be determined, 
 and thus the position of the beam found. 
 
 It remains to determine the tension T of the string, and the pres- 
 sure P on the inclined plane. In order to this draw GV parallel 
 to PA, then, since the sides of the triangle GVD are respectively 
 parallel to the three forces, they, or the sines of their opposite 
 angles, are proportional to these forces ; that is, calling the force in 
 GD, or weight of the beam AV, we have 
 
 sin. GVD : sin. VDG=sin. z : : W : T 
 sin. GVD : sin. VGD=cos. 9 : : W : P ; 
 
 , _, sin. i „^ _. COS. (94-i) ^.. 
 
 consequently, T= W, P= ^ ^ M V. 
 
 COS. e COS. e 
 
 Problem VI. — (64.) A given beam AB is supported by strings 
 which go over pullies C, D, and have given weights P, P^, attach- 
 ed to them, to find the position of equilibrium (fig. 55). 
 
 Produce the strings till they meet in g, then the vertical ^E will 
 pass through the centre of gravity G of the beam, and if GF be 
 drawn parallel to D^ the three sides of the triangle GFg will be 
 proportional to the three equilibrating forces, and these are all 
 given. Put AG=fl, GB=&, the inclination of CD to the horizon- 
 tal line CK, or DK', i ; these quantities are also given. Let a 
 represent the angle DCA, and (3 the angle CDg, then sin. BDPj= 
 cos. (/3+z)=sin. FG^, and cos. ^CK=cos. (a — i)=sin. F^G ; 
 
 P cos. (3-\-l^ 
 
 hence, from the triangle GFe-, we have -r-= '-p :{ ... (1) 
 
 ^ * Pj COS. (tt — t) ^ 
 
 P _ COS. (fi+i) _cos. (i3-fi) 
 W^sin. (180°— p — a) ~ sin. (^+a)'
 
 EQUILIBRIUM OF A SOLID BODY. 89 
 
 From these two equations the unknown angles a and /3 may be 
 determined. But to find the position of AB, we must also know 
 the angle A'=S. From G draw the perpendiculars Gp, Gp^, on 
 the strings, then Gp=GA sin. A=a sin. (a — 6), and 
 
 Gjo,=GB sin. GBp^=b sin. (f3+6), 
 hence a sin. (a — 5) P=6 sin. (i3+6) P^ 
 
 ^ F_ bsm.(i3 + 6) 
 
 " P, a sin. (a — 6) ^ ^' 
 
 from which equation S may be determined, a and /3 having been 
 previously found. The quantities thus found enable us to find the 
 two unknown sides of the triangle Co-D, and those of the triangle 
 AgB, and thence the distances CA, DB. 
 
 Problem VII.— (65.) A given beam AB hangs by two strings, 
 CA, DB, of given lengths, from two given fixed points C, D ; to 
 find the position in which it will rest (fig. 56). 
 
 Here instead of the tensions we have the lengths a', b' of the 
 strings C A, DB ;. hence, using the same notation and the same 
 reasoning as in the last problem, we get two difljerent expressions 
 (1), (2), for the ratio of these tensions ; that is, we have one equa- 
 tion between the unknowns a, j3, and 8 ; it will, therefore, require 
 two more equations to determine them ; these two may be readily 
 obtained, since we may deduce two different expressions for the 
 perpendiculars from A and B on the strings. Thus, if we draw 
 AM parallel to BD, it is plain that the perpendicular from A on DB 
 will be equal to that from C on DB, minus that from C on AM ; 
 that is, calling CD, c, the expression for this perpendicular will be 
 
 c sin. |3 — a' sin. (a+/3) ; 
 but the expression for the same perpendicular is also AB sin. B, 
 that is {a+b) sin. (i3+S) 
 
 .*. c sin. /3 — a' sin. (a+/3)=(ff+|3) sin. (|3 + 8) .... (1). 
 
 Again the perpendicular from B on kg is equal to the perpendi- 
 cular on it from D, minus that from D on BN parallel to A"-, the 
 expression for this perpendicular is, therefore, 
 c sin. a — b' sin. (a+/3) ; 
 but the expression for the same perpendicular is also AB sin. A, 
 that is, («+6) sin. (a — 6) 
 
 .*. c sin. a — b' sin. (a+/3)=(o + 6) sin. (a — 6) . . . (2). 
 These equations, in conjunction with that before mentioned, viz. 
 with .cos^+lJ^^sin.O + g) _ ^ ^ 
 
 sm. (/S+a) asm. (a — b) ^' 
 
 are sufficient to determine the unknown quantities sought. 
 
 Problem VIII.— (66.) A given solid hemisphere, with its convex 
 part upon a smooth inclined plane of given inclination, is kept 
 h2 12
 
 90 ELEMENTS OF STATICS. 
 
 from sliding by a string of given length having one end fastened to 
 a given point, and the other end attached to the edge of the hemis- 
 phere : it is required to determine the point where the hemisphere 
 touches the plane when at rest, the pressure on the plane, and the 
 tension of the string (fig. 57). 
 
 Let P be the given point, PI the string, IFM the hemisphere, C 
 its centre, and CT perpendicular to MI, then the distance CG of 
 its centre of gravity is i CT. The forces acting on the body are 
 the weight W in the vertical direction GR, the pressure P in the 
 direction FO perpendicular to the inclined plane AB, and the ten- 
 sion T in the direction IP : this last direction must when produced 
 meet in O the point of concurrence of the other two forces. To 
 determine the point F, draw GN, IL, and PE, perpendicular to 
 CF, ID perpendicular to PE, and PH to AB, then the given quan- 
 titles 3,re 
 
 CG=a, CF=b, PH=EF=c, PI--=/, cot. angle BAR= 
 
 cot. angle GON=? ; 
 
 and we wish to find HF or PD and IL. Put x andy for the sine 
 
 and cosine of the angle GCN, or CIL, then we have 
 
 CN=ff.y, GN=a.r, CL=bx, lA=by 
 
 .'. 0N=/ • GN=/rt.r, CO=CN — ON=fl (y — tx) 
 
 LO=CL'— CO =bx — a (y — tx), LE=CE — CL= 
 
 lY)=b — c — bx. By the similar triangles OLI, IDP, 
 
 OL : LI : : ID : DP .-. DP=M^I^iZlM ; 
 
 ox — a (y — tx) 
 and since DP+DP2=IP''= I" 
 
 ...(6_c_6x)«+s'-^i^^=^l«=P. 
 ^ ^ ' o.r — a (y — tx) ' 
 
 This equation joined lo x'^-^-y" =1 is sufficient to determine a; 
 and y, and thence DP and LI. The sides LI, LO, being now 
 known, the angle LOI is known ; hence, taking the centre of mo- 
 ments at L we have 
 
 OLsin. LOIxT=OLsin. LOG • W ; andLOG=Z.A, 
 
 ... T= -i' "' ^^ W ; also, P=T cos. LOI+W cos. LOG 
 sm. LOI 
 
 = (sin. LOG • cot. LOU- cos. LOG) W. 
 
 Problem IX. — (67.) To determine the position in which a para- 
 boloid ABC will rest upon a horizontal plane (fig. 58). 
 
 Suppose P to be the point on which it rests, then the pressure be- 
 ing in the vertical direction PG, is in the normal to the surface at 
 P, and, moreover, passes through the centre of gravity G. Taking 
 the axis AX, AY the equation of the vertical section, through AX 
 
 and PG is y^=2px ; therefore, the subnormal NG is NG=y ^ =
 
 EQUILIBRIUM OF A SOLID EODV. 91 
 
 p ; but, calling the altitude AX of the paraboloid a, the distance AG 
 
 2 2 
 
 is (ex. 3, p. 71) AG=fa, .•. x+p =— a .-. x= — a — p ; 
 
 o o 
 
 this therefore is the abscissa of the point on which the body rests, 
 
 hence, the tangent of the inclination XRP of the axis to the hori- 
 
 dv 
 zon, that is, the value of -^, for the point P is 
 
 tan.e=g=^/A=^|^^3(|«_;,)|(l). 
 
 Besides the point thus determined, there is no otlier on which we 
 can place the body where the normal shall be equal to the distance 
 of the centre of gravity from the horizontal plane, which equality 
 must exist in order that the body may rest on that point, making, 
 however, one exception, viz. when the point is the vertex A ; for 
 at tliis point the normal being in the line of direction of the centre 
 of gravity, the body would necessaril}^ rest on it, be the length of 
 the normal what it may. In order that the body may be unable to 
 rest on any other point besides this, the altitude must be such that 
 between the vertex and base there shall be no point whose noi'mal 
 shall equal the distance of the centre of gravity from the plane, 
 therefore the condition is that the result (1) may be either impossi- 
 ble or infinite, which requires that 
 
 3 3 
 
 « <-2-;'' or a=— ;) ; 
 
 when, therefore, a=^p we see that the vertex is the point at which 
 the normal measures its distance from the centre of gravity. Hence 
 if any segment of a paraboloid, whose altitude does not exceed ^p, 
 be placed any how on a horizontal plane, except indeed on its base, 
 it must always restore itself to an upright position and rest, when it 
 does rest, on its vertex. 
 
 Problem X. — (68.) To determine the pressures exerted by a 
 door on its hinges or on the two pivots upon which it hangs. 
 
 Call the weight of the door acting at its centre of gravity P the 
 distance of the centre of gravity, that is, of the direction of the force 
 from the vertical axis x, and the distance between the pivots z' — z"; 
 then there being no pressures in the direction of the axis of y, the 
 equations 1 at page 81, become, in this case, 
 
 X' + X"= — PCOS. a=0, Z' + Z"= — P cos. y=— P (1) 
 
 .-. X'=— X"; 
 hence the equations (2) furnish only (z' — z") X'= — Fx 
 Pr Vr ' 
 
 '•' X'= , , .-. X"= -——T' . . . (2). 
 
 z ' — z' z — z ' 
 
 The equation (1) shows that the door presses with its whole weight
 
 92 ELEMENTS OF STATICS. 
 
 P, the vertical line of the hinges ; and the equations (2) express the 
 horizontal forces on the hinges, the lower hinge being pushed in- 
 wards, and the upper hinge drawn outwards, each with a force equal to 
 Px 
 
 Problem XI. — (69.) One end of a uniform beam of weight W 
 is moveable round a fixed point in a vertical plane, and to the other 
 end is attached a string which passes over a pulley, and is loaded 
 with a given weight P ; the fixed point and pulley are in a horizon- 
 tal line, and tlieir distance asunder is equal to the length of the 
 beam, which, however, is not given ; required the angle between 
 the beam and horizontal line when the beam is at rest, 
 
 P P^ 1 
 
 cos..=--±(-— + -)i. 
 
 Problem XII. — (70.) A cone of marble, the axis of which is 
 twenty feet, and base diameter, six feet, stands on the edge of its 
 base, the axis making an angle of 00° with the plane of the horizon ; 
 what must be the direction and intensity of the least force applied 
 to its vertex that will just sustain the cone in that position ; the 
 weight of the cone being 284 cwt. 1 
 'J he least force is !• 377 cwt. acting perpendicular to the lower side of the cone. 
 
 Problem XIII. — (71.) What will be the height of the greatest 
 segment that can be cut off a prolate spheroid whose longer axis is 
 double the shorter, by a plane perpendicular to the longer axis, so 
 that it may be unable to rest upon any point of its convex surface 
 except the vertex. ? 
 
 Height := semi, trans. X (3 — \/5). 
 
 Problem XIV. — (72.) A beam of given weight W, rests with one 
 end against a vertical wall and the other upon an inclined plane ; 
 calling the inclination of the beam to the horizon i, the pressure 
 against the wall P, and the thrust against the inclined plane T, to 
 determine the intensities of P and T. 
 
 p^ W .^^v/ hin.«z - fjcos.''z^ y^ 
 2 tan. V sin. i. 
 
 Problem XV. — (73) A ladder of uniform thickness and weight 
 w pounds, is placed against a vertical wall, and at a given inclina- 
 tion i to the horizon, a person weighing W pounds ascends the lad- 
 der, and it is required to determine the pressure at the top and the 
 thrust at the bottom of the ladder when the person arrives at any 
 given height.
 
 ON THE MECHANICAL POWERS. 93 
 
 CHAPTER IV. 
 
 ON THE MECHANICAL POWERS. 
 
 (74.) Having now considered pretty fully the theory of the equi- 
 Jibrium of forces, applied to different points of a solid body, it is 
 proper that we should speak of those cases in which the forces are 
 not immediately applied to the body, on which their influence is ul- 
 timately exerted, but to some intermediate body contrived for the 
 purpose of transmitting this influence in the most advantageous 
 manner. These contrivances are called Machines, and the simple 
 elements or constituent parts of all machinery are called the mecha- 
 nical povjers. These are six in number, and are as follow : the Le- 
 ver, the TVlieel and axle, the Pulley, the Inclined Plane, the Screw, 
 and the Wedge. 
 
 The Lever 
 
 (75.) A lever is a rigid bar or rod, moveable about a fixed point 
 or fidcrum, and it is divided into three different kinds, depending 
 on the position of this fulcrum with respect to the applied force, and 
 the body to be influenced by it ; thus, if the fulcrum be between 
 the force or power and the body or weight, as in fig. 59, the lever 
 is said to be of the first kind ; if it be situated so that the weight is 
 on the middle, as in fig. 60, the lever is of the second kind, and 
 when, as in fig. 61, the power is in the middle, the lever is of the 
 third kind. In the figures the rods are straight, but they would still 
 be levers if bent or curved. 
 
 That a lever acted upon by any forces may be in a state of equili- 
 brium, it is, obviously, merely necessary that the sum of the mo- 
 ments, taking the fulcrum as the centre, be equal to ; this condi- 
 tion, therefore, comprises the whole theory of the lever, so that 
 when, as is commonly the case, the equilibrating forces are a weight 
 
 W p 
 
 W and a power P, then the condition is W p^=Vp •'•-p- =— (0 ; 
 
 r p^ 
 
 hence the weight and power are to each other, reciprocally, as their 
 distances from the fulcrum. If, as is usually the case, the object be 
 to balance W with the least power possible, this must act so that jo 
 may be the greatest possible, and, therefore, the power must act per- 
 pendicularly to the lever. 
 
 When the weight w of the lever itself is to be taken into con- 
 sideration we must view it as a third force acting at its centre of gra-
 
 94 ELEMENTS OF STATICS, 
 
 vity ; if we call the distance of this new force from the fulcrum g, 
 then the ahove equation will be W p^:izH'g=Fp . (2); the upper 
 or lower sign being used according as this force tends to favour or 
 to oppose W. 
 
 (76.) The mechanical advantage of the single lever may be con- 
 siderably increased by combining several together, so that the power 
 of one may be communicated to another. 
 
 Thus in the system of levers, represented in fig. 62, if we call 
 the arms which are on the same side of the fulcra as the power p, 
 Pi,p.^, «fec. and the other arms p',p\, p\, &c. and, moreover, call the 
 powers acting at the extremities of the latter arms Pj, P^, &,c. then 
 in the case of equilibrium we shall have the equations 
 P;;=P,/;', Pj>,=Pj^', l\p^=PsP2'^ <^c. 
 and multiplying these together, 
 
 Fpp,p^""=l\p'p\p'^-"; 
 or, the last power P„ being the weight W, 
 
 PpPiP2"-=^P'P\P\'—-^ 
 so that the power is to the weight, as the product of tlie arms on 
 the side of the weight, to the product of the arms on the side of the 
 power. 
 
 Problem I. — (77.) A body is weighed successively in the two 
 scales of a false balance, in the one scale it balances a weight /j, in 
 the other a weight q, required the true weight of the body. 
 
 Suppose the lengths of the arms of the balance to be a and b, and 
 let X represent the true weight of the body, then its moment in one 
 scale is ax, and in the other bx, and, by the problem, 
 
 ax^bp, bx=aq ; 
 multiplying these together we have, abx^ = abpq .'. x^ y/pq; 
 hence the true weight is a mean proportional between the two false 
 weights. 
 
 Problem II. — (78.) The common steelyard (fig. 63) is a bar AB, 
 moveable about a fulcrum O ; P, the body to be weighed, is hung 
 at the shorter arm A, and a given weight W is moved along the other 
 arm till it balances P ; then the weight of P is known from the place 
 of W ; to find how the distances OC increase with regard to the 
 weight P. 
 
 Let G be the centre of gravity of AB, and Gg vertical meeting 
 the horizontal line AOC in g. Call the weight of AB, w, OA.=p, 
 OC=Pj, Og=g, then, by equation (2) above, 
 
 IDcr Pp
 
 ON THE MECHANICAL POWERS. 95 
 
 hence, if we take Ox =-^y we have 3^0=-^^ ; hence xC varies 
 ' W W 
 
 as P ; and, proceeding from x, equal additions of distance to xG 
 correspond to equal additions of weight to P : that is, if xQ> be 
 graduated with points 1,2,3, &c. at equal successive intervals from 
 X, W placed at these points will successively balance the weights 
 1, 2, 3, &c. 
 
 Problem III. — (79.) A homogeneous lever AB (fig. 64) of the 
 second kind is equally thick throughout; it is required to deter- 
 mine what must be the length of the arm AB, that a given weight 
 W, acting at the extremity of the arm AC, may be supported by 
 the least power P possible, taking into account the weight of the 
 lever itself. 
 
 The lever being homogeneous, and equally thick throughout any 
 portion of its length, may be taken to represent the weight of that 
 portion ; hence, calling the length AB of the lever x, its weight will 
 be X, acting at G, its middle point; therefore, putting AC=/;, we 
 have, by equation (2), 
 
 W »,+ ^a; • x=Vx .-. P= ^-i-f d x. 
 ^ X 
 
 To determine for what value of x this expression for P is a mini- 
 
 rf P W» 
 
 mum, we have —, — = f^ 4-^=0 ; from which we immediate- 
 
 dx x^ 
 
 ly geta;^=2W/?j .•. x=y/\2 Vf p^], the value required. 
 
 It must be remembered that we have taken lengths to represent 
 weights, so that, whatever our unit of length has been, the M^eight 
 of that unit must be considered as the unit of weight, thus if we have 
 measured/)^, and therefore x, in inches, then the numerical expres- 
 sion for W will be the weight W, divided by the weight of an inch 
 of the lever. 
 
 By substituting this value of x in the foregoing expression for P, 
 we have, for the intensity of this power when acting at the greatest 
 advantage, 
 
 P= :^p=+W{'i'^PA=^/\^'^P^]' 
 V^2 W j9j 
 
 The Wheel and Jlxle. 
 
 (80.) This machine is in reality only a modification of the lever; 
 it consists of two parts, a cylinder called the axle, and the surround- 
 ing circle or wheel connected with it, having its centre in the axis 
 of the cylinder about which the whole turns (see fig. 65).
 
 96 ELEMENTS OF STATICS. 
 
 This machine is not complete in itself like the lever, requiring the 
 addition of cords, or chains, or some other intermediate body, to 
 communicate with the forces engaged. 
 
 The more immediate object of this machine is to support or raise 
 a weight W, suspended to a rope wound about the axle, by means of a 
 power P applied to the circumference of the wheel. But in number- 
 less applications of this machine the axle does not communicate di- 
 rectly with the weight by a cord : but, by being surrounded with teeth, 
 as in fig. 66, acts upon a toothed wheel, and the axle of this last 
 upon another toothed wheel, and so on, the power being thus trans- 
 mitted to the weight W. The effect of a power thus transmitted 
 we shall consider presently, examining first the more simple case 
 just noticed. 
 
 Let the radius of the wheel be p, that of the axle p^, then (72) 
 
 W p 
 
 1 Pi 
 
 the weight and power being to each other, reciprocally, as their dis- 
 tances from the fixed axis, as in the lever. 
 
 If the equilibrating forces do not act tangentially, as we here sup- 
 pose, then instead o{ p, p^ representing the radii, they will represent 
 the distances of the directions of the power and weight from the 
 axis. 
 
 It is obvious that the greater be the wheel the less will be the 
 power requisite to support or move a given weight : and that a con- 
 tinually decreasing power may have a uniform effect upon a constant 
 weight, it must act upon a series of continually increasing wheels, 
 constantly keeping up the above proportion, the radii of the wheels 
 varying inversely as the powers: thus P and P' being any two 
 %'alues of the powers we have 
 
 — -t. ^- Pi E. _Z' 
 
 ^ ~ Pi'¥i~ Px" P'~^' 
 
 In this way the varying power exerted by the main-spring of a 
 
 watch while uncoiling is made to produce a uniform efl^ect ; this 
 
 power acting on a series of varying wheels (fig. 67) called ihefuzee. 
 
 It should be remarked that the macliine we are now considering 
 is virtually unchanged, though the wheel be stripped of its rim and 
 the power be applied at the extremities of the spokes. 
 
 It should be further remarked that the thickness of the rope, when 
 considerable, must not be neglected in estimating the conditions of 
 equilibrium ; for we ought to consider the forces to be transmitted 
 along the middle or axis of the rope, and, therefore, the radius of 
 the rope should be added to that of the wheel, and of the axle, so 
 that in the above expressions p and p^ are the distances of the axes 
 of the ropes from the axis of the machine.
 
 ON THE MECHANICAL POWERS. 97 
 
 (81.) When the axle is toothed it is called a ;jin«on, and the teeth 
 its leaves. If a power be in equilibrium with a weight by means 
 of a system of toothed wheels and pinions, as in fig. 66, then we 
 shall iind that the power will be to the weight as the product of the 
 radii of the pinions to the product of the radii of the wheels. 
 
 For let the radii of the axle or pinions be r, r^, r^, &c. and those 
 of the wheels R, R^, R^, &c. then the power P acting on the first 
 wheel equilibrates a weight or power P^, on the pinion, expressed 
 
 PR 
 
 by Pj= ; this then is the power applied to the second wheel 
 
 and which, therefore, comnaunicates to its pinion a power expressed 
 
 by P.= -^^ = ZM.; 
 
 r^ r r^ 
 
 this IS ttie power applied to the third wheel, and continuing this 
 computation it is plain that the power which the nth pinion or axle 
 
 Tk P R R, R„ .... Rn_i , 1 T>, • 1 
 
 acts IS P„ = - — ' ; and consequently, Vn. is the 
 
 r r^T^ . . . . Tn—\ 
 
 weight which the power P will balance on the wth axle. 
 
 Instead of expressing this result in words, as above, we may say, 
 since the radii are as the circumference, and these, again, as the 
 number of teeth they carry, that the power is to the wciight as the 
 product of the numbers expressing the leaves to each pinion to the 
 product of the numbers expressing the teeth to each wheel ; the 
 number for the first wheel, which is plane, expressing the number 
 of teeth it could carry, and, in like manner, the number for the last 
 axle being that expressing the number of leaves it would carry. 
 
 For further particulars respecting tooth and pinion work, and, in- 
 deed, respecting machinery in general, the student is referred to 
 Professor Gregory's Treatise of Mechanics, a work abounding with 
 valuable information. Much interesting matter will also be found 
 in Dr. Lardner's elegant volume on Mechanics, in the Cabinet Cy- 
 clopaedia. 
 
 SCHOLIUM. 
 
 (82.) The student will have observed that the foregoing theory 
 of toothed wheels is founded on the supposition that the power 
 communicated from the tooth of one wheel to that of another is in 
 the direction of a tangent to the circle on which this latter is raised, 
 as in the plane wheel and axle, and that such may really be the di- 
 rection of the power, a particular figure must be given to the teeth, 
 at least to the working sides of them. This figure is that of the invo- 
 lute of the circle on which they are raised. Thus, let IHF, KE6 
 I 13
 
 98 ELEMENTS OF STATICS. 
 
 (fig. 68) be the wheels to whicli the teeth are to be accommodated, 
 the acting face GCH of the tooth a must have the form of tlie curve 
 traced by the extremity II of the flexible line F«H, as it is unwrap- 
 ped from the circumference ; and, in like manner, the acting face of 
 the tooth b must be formed by the unwrapping of a thread i'rom the 
 circumference of the circle KE6. T!ie line FCE drawn to touch 
 both circles will cut the surfaces of the two teeth in C, the point 
 where they touch each other, at a point in the common tangent to 
 both circles, and the force arising from their mutual pressure will 
 always act in the direction of the circumference of the wheels at E 
 and F. But, continues Dr. Gregory, whose words we have bor- 
 rowed in the preceding description, although Roemer, Varignon, De 
 la Hire, Camus, Euler, Emerson, Kaestner, and Robison, have 
 turned their thoughts to this object, and some of them have stmck 
 out rules of ready application in practice, it is to be regretted that 
 these rules have been little followed by practical mechanics, most of 
 whom have, in this case, been more inclined to follow a set of hack- 
 iiied rules handed down from one workman to another, although com- 
 pletely destitute of scientific principle. Even watchmakers, in whose 
 constructions a little more than common skill and nicety in the exe- 
 cution might be expected, are but few of them acquainted with any 
 rules founded upon the deductions of accurate theory ; but commonly, 
 we are informed, give to their teeth the shape assumed by a horse 
 hair when held bent between the fingers, a method so vague that it 
 is difficult to conceive how it came to be adopted. 
 
 The Pulley. 
 
 (83.) A pulley is a grooved wheel moving freely on an axis, and 
 fixed in a case or block. It communicates applied force in conjunc- 
 tion with a cord which the groove receives. 
 
 The Jixed pulley we have already employed in various parts of 
 this work for the sole purpose to which it is applicable, viz. for the 
 purpose of changing the direction of a force acting by a cord, and 
 although the fixed pulley is, for this purpose even, an important 
 instrument, yet as it docs not afford any mechanical advantage in 
 the way of accumulating force it offers no theory for discussion. It 
 is different with the moveable pulley (fig. 60,) to the block of which 
 the weight is fastened, which is sustained between the power P, 
 acting at one end of the cord, and the pressure on a fixed hook Q at 
 the other end ; the tension of the cord being uniform, it is obvious 
 that the intensity of P must be equal to the strain or pressure on Q. 
 Let us examine the relation which these equal powers bear to the 
 weight. Continue the directions of the ropes PC, QD, till they 
 meet, unless they should be parallel ; then, since the system of
 
 ON THE MECHANICAL POWERS. 99 
 
 forces P, Q, W is in equilibrium, the point E, wliere the directions 
 of two meet, must be in the direction of the third, and the resultant 
 W of the equal forces P, Q will be 
 
 W=2P cos. a .... (1); 
 a being equal to half the inclination of the cords from P and Q. 
 Instead of the tabular cosine we may introduce into this expression 
 the cosine corresponding to the radius r of the pulley, provided we 
 
 write '- — instead of cos. a ; this cosine is, obviously, the line 
 
 r 
 
 C??, being the sine of the angle COn the complement of a; hence, 
 
 dividing each side of the expression thus changed by P, we have 
 
 W 2 cos. a 
 
 P r • • • v-/' 
 
 that is, the power is to the weight as the radius of the pulley to the 
 chord of that arc of it which is in contact with the rope. 
 
 If the cords from P and Q are parallel, then the equilibrium being 
 the effect of three parallel forces, the middle force W must be equal 
 to the two P and Q; in this case, therefore, W=2P . (3); and it 
 i.3 obvious that half the circumference of the pulley must be in con- 
 tact with the rope. 
 
 It may be observed from the expression (1) that tiie moveable 
 pulley affords a mechanical advantage only so long as the inclina- 
 tion of the cords from P and Q is less than 120°; for at this angle 
 cos. a=5, and at greater angles cos. a is less than 5. The great- 
 est advantage is gained when the cords are parallel. 
 
 (84.) The advantage gained by a single moveable pulley may be 
 multiplied to any extent by employing a system of pullies, as in 
 fig. 70, thus, representing the tensions of the several ropes by the let- 
 ters annexed to them in the figure, we have from equation (1) above 
 \Y=2t, cos."^a 
 t^ =2 t^ cos. ftg 
 ta =2 t^ cos. a. 
 
 tn-i=^t„ cos. a„=2 P cos. a„ ; 
 hence, multiplying these equations together and expunging the 
 factors common to each side of the resulting equation, we have 
 W=2" P (cos. a. cos. a^. COS. a^ . . . . COS. a„) .... (4). 
 P being the power and n the number of pullies. 
 
 If the several cords are parallel, as in fig. 71, this equation be- 
 comes W=2" P (5) ; and, if the angles are all equal to each other, 
 W=2«P cos.^tt .... (6). 
 
 In what is here said the weights of the several pullies have been
 
 100 ELEMENTS OF STATICS. 
 
 neglected, but in strictness these should be taken into account. If 
 they increase the weights on the ropes which pass round them by 
 the several quantities A, A,, A^, «fcc., then the above series of equa- 
 tions will be 
 
 W + A =2/, cos. o, 
 
 /, -f Aj=2/^cos. a^ 
 
 &c. &c. 
 
 (85.) Instead of attaching the several ropes to immoveable points, 
 as in fig. 71, they are all in another arrangement of the system 
 fastened to the weight, as in fig. 72, the ropes being parallel. The 
 relation between the power and weiglit in this system is at once 
 seen from looking at the figure ; thus the first pulley or that which 
 first receives the power supports twice P, the second, therefore, 
 supports four times P, the third eight times P, and the nth supports 
 2" times P, and all of this except P is tlie weight ; hence deducting 
 P the weight supported is W = (2« — 1) P.. . . (7). 
 
 In this system, as well as in tliat exhibited in fig. 71, each pul- 
 ley is connected with two parallel branches of rope, of which each 
 branch bears half the weight attached to the pulley ; but if three 
 parallel branches of rope be connected with each pulley, as in the 
 systems exhibited in figures 73 and 74, each branch will bear a 
 third of the attached weight ; hence, if ?i be the number of these 
 systems of threes, we shall have instead of the equations (5) and (7), 
 
 W=3'' P and W = (3'' — 1)....P; 
 the first equation belonging to the arrangement in fig. 73, and the 
 second to that in fig. 74. 
 
 (83.) We have hitherto supposed each pulley to be attached to a 
 separate string ; but if only one string is employed, as in the sys- 
 tems represented in fig. 75, then, as this string is uniformly tense 
 throughout, the common tension being P, the weight, which is 
 equal to the sum of the tensions when the branches of the rope are 
 parallel, must be equal to 2P times the number of moveable puUies ; 
 that is, W=2n P, W including the weight of the lower block. 
 
 77ie Inclined Plane. 
 
 (87.) This machine is simply a plane surface inclined to the ho- 
 rizon. 
 
 The theory of this machine is unfolded in problem I. page 48, 
 
 Avhere it is shown that if i be the inclination of the plane, and e the 
 
 angle which the direction of the power makes with the plane, the 
 
 sin. t 
 relation between the power and weight is P=W '■ — . When

 
 ON THE MECHANICAL POWERS, 101 
 
 the direction of the power is parallel to the plane, the machine af- 
 fords the greatest mechanical advantage possible, for then 
 
 COS. £ = 1 and P=W sin. i, 
 so that in this case the power is to the weight as the altitude of the 
 plane to its length. 
 
 But when the direction of the power is horizontal or parallel to 
 the base of the plane then £=i, and the relation is P=W tan. z, 
 which shows that the power is to the weight as the height of the 
 plane to the base. 
 
 The Screiv. 
 
 (88.) Before we can investigate the mechanical power of the 
 screw we must ascertain exactly the form of the spiral surface 
 which it presents. The generation and equation of this surface has 
 been explained in the Diff". Calc. p. 200 ; but it Avill be proper to 
 repeat, in part, that explanation in this place. Let us conceive then 
 a rectangle to be rolled round a vertical and cylindrical column 
 which it just embraces, the line which was the diagonal of this 
 rectangle will form itself into a winding curve, called a helix, and 
 it will make just one turn round the column, its horizontal projec- 
 tion being a circle ; if immediately above this another equal rectan- 
 gle be applied to the cylinder, the vertical edges, when brought 
 together, being in a line with those of the first, the diagonal of this 
 will form a continuation of the helix, and, in this way, will be ex- 
 hibited on the surface of the cylinder, the trace of the winding sur- 
 face which forms the screw. 
 
 If now, beginning with the bottom, we were to strip off these 
 
 rectangles one after the other, turning the cylinder round at the 
 
 same time, so that all of tliem might be ranged in the same vertical 
 
 plane, we should, obviously, have the figure presented at fig. 76, 
 
 tlie uniform straight line AB being the developement of the helix ; 
 
 we may, therefore, say that this curve is formed by winding round 
 
 an upright cylinder an inclined straight line AB, always preserving 
 
 its inclination constant ; if we consider this inclined line to be the 
 
 edge of an inclined plane, then the surface, thus wound round the 
 
 cylinder, will be the surface of the screw. This surface, therefore, 
 
 differs from the inclined plane in no respect but in its winding 
 
 course, and it will, obviously, require just as much power to sustain 
 
 a weight on the winding surface as on the straight surface, so that 
 
 if W be any weight on the surface, and a power acting horizontally 
 
 P BC be ^ * . , 
 support it, we must have —- = ——=— — ; but Ac is the circum- 
 W AC Ac 
 
 ference of the cylinder which carries the screw, therefore calling 
 
 the radius of it r, and the height be, which measures the interval 
 
 between each turn of the screw and the next, h, we have 
 
 i2
 
 102 ELEMENTS OF STATICS. 
 
 but if the power P, instead of being applied directly to W, is ap- 
 plied to the arm of a lever, at R distance from the fulcmm, and if 
 the direction of W be at r distance on the other side, then the value 
 
 of P Avill be P — , and in this case the condition of equilibrium will 
 r 
 
 be P-=W JL- ... P=W J^ .... (2). 
 
 r 2 7t r 2 rt K 
 
 Now this equation expresses the power of tlie screw ; for the 
 weight to be balanced or raised, the resistance to be overcome, the 
 pressure to be sustained, &c. is always a force in the direction of 
 the axis of the cylinder, and acting upon the inclined plane wind- 
 ing round it; it is balanced by a power P, acting perpendicular to 
 the same axis, at the extremity of a lever (see tig. 77) whose ful- 
 crum is in the axis, and, therefore, at the distance of the radius of 
 the cylinder from the winding surface or thread of the screw. The 
 weight is throAvn on the thread by its being connected with a nut 
 or internal screw N, which is a spiral-grooved case fitted to receive 
 the external screw. 
 
 In fig. 77, the whole pressure on the screw is thrown upon the 
 thread within the nut N, and the power applied at the extremity 
 of the lever R must bear to this pressure or weight the relation (2) 
 above, in order to balance it, or, which is the same thing, a power 
 something greater than this must be applied to move the lever. 
 The relation (2) when expressed in words is this, viz. The power 
 is to the \veight or resistance as the interval between two adjacent 
 turns of the thread to the circumference of the circle described by 
 the power ; and this relation we see is altogether independent of 
 the radius of the cylinder, and, therefore, also of the degree of pro- 
 tuberance of the thread ; it varies only with the distance h between 
 the turns or contiguous spires of the thread, and with the inclina- 
 tion of the thread to the axis of the cylinder ; hence so long as this 
 distance and this inclination is preserved, it matters not what form 
 be given to the surface of the thread, nor how protuberant it be 
 made. 
 
 We see from the expression (2) that there are two ways in which 
 the power of the screw may be increased ; first by diminishing the 
 distance h between the turns, or, secondly, by increasing the length 
 of the lever R ; it is, however, not strictly correct to say that the 
 power of the screw is increased by this latter change, for this in 
 fact remains unaltered : it is the applied power that is here in- 
 creased.
 
 ON THE MECHANICAL POWERS. 103 
 
 The Wedge. 
 
 (89.) The wedge is a triangular prism, AC (fig. 78), chiefly em- 
 ployed for splitting or separating bodies ; for this purpose the edge 
 AB of the wedge is introduced between the bodies to be separated, 
 and the power which drives it is applied to the head DC ; this 
 power, when the machine is in equilibrium, must balance the re- 
 sistances opposed to its entry, and which can only act on the edge 
 AB, and on the two faces DB, CA. Indeed in a state of equili- 
 brium there can be no resistance opposed to the edge of the instru- 
 ment, as is obvious, so that this state is preserved by three forces, 
 viz. the power P applied to the head, and the resistances P^, P^ 
 acting against the faces. To determine the conditions of equili- 
 brium of forces, thus acting, we must introduce hypotheses, not 
 only unsupported by observation and experiment, but in direct con- 
 tradiction of them ; thus, for three forces to keep a body in equili- 
 brium, it is absolutely necessary that when one of the three is 
 withdrawn, the body, through the influence of the other two, should 
 move unmolested in a direction opposite to that of the force with- 
 drawn ; in the wedge, therefore, when the pressure or impelling 
 power is withdrawn from the head the pressures on the sides should 
 expel it from between the resisting surfaces ; this, howevei', is al- 
 most universally contrary to experience : in the cleaving of wood, 
 for instance, the wedge may be driven to any extent between the 
 resisting sides (fig. 79), and will usually remain there without sen- 
 sibly receding, although the power be removed from the head, the 
 friction being fully equal to balance the expelling forces acting on 
 the faces. 
 
 The mathematical theory of this machine is founded on the 
 hypothesis, that the resisting surfaces as well as the faces of the 
 wedge are perfectly smooth, or, which is the same thing, that the 
 friction is nothing, whereas in practice the friction is every thing, 
 the wedge would be comparatively useless without it. Seeing, 
 therefore, the great eflfect of friction in this machine, which is suf- 
 ficient to maintain the equilibrium even Avhen the applied power is 
 withdrawn, it is obvious that no deduction from the mathematical 
 theory of it can be of much practical utility. It is true, indeed, 
 that in all other machines, as well as in the wedge, friction always 
 opposes a hindrance to the full effect of the applied powers to pro- 
 duce motion, and that, therefore, the deductions of pure theory, 
 where these hindrances are not taken into account, require some 
 modification before they can agree with the results of actual expe- 
 riment, but then, by polishing or lubricating the acting surfaces, 
 these hindrances may be more and more diminished, and the results 
 of practice be made to approa^'ti nearer and nearer to the deductions
 
 104 ELEMENTS OF STATICS. 
 
 of theory. In the application of the wedge, however, it is in most 
 cases impossible, even if it were desirable to diminish in the small- 
 est degree the friction on its faces, it is this that hinders the pres- 
 sures on the faces from driving the wedge back, and is, therefore, 
 a power which greatly favours the efficacy of the machine ; as the 
 impelling force is usually applied at intervals, by means of repeated 
 blows, and not in the form of a continued pressure, the friction 
 serves to hold the wedge where the last blow had driven it. 
 
 (90.) Abstracting from the influence of friction, the equilibrium 
 of the wedge may be thus investigated. Let any arbitrary length, 
 DE, (fig. 80,) represent the power applied perpendicularly to the 
 head of the wedge, and draw DM, DN perpendicular to the faces 
 AC, BC, and complete the parallelogram IK, having DE for its 
 diagonal ; DI and DK, or IE, will then represent the pressures P^, 
 Pj against the faces of the wedge, so that the three equilibrating 
 powers are as the three sides of the triangle DEI, or as the three 
 sides of the similar triangle ABC, that is, 
 
 P : P, : P, : : AB : AC : BC ; 
 or, calling the length of the edge C, /, 
 
 P : P, : P, : : ABx/ : ACx/: BCx/; 
 which proportion, obviously, implies that the power on the head 
 of the wedge and the equilibrating pressures on the faces are pro- 
 portional to the areas of the head and faces, on which they re- 
 spectively act. 
 
 SCHOLIUM. 
 
 To the foregoing theory of the simple machines we shall append 
 the following judicious remarks from Venturoli. 
 
 The false opinion which persons unskilled in the nature and the 
 power of machines are apt to conceive, often encourages empty 
 errors and mischievous deceptions. One of the most common of 
 these conceits is that of considering machines as available to in- 
 crease and multiply the force of agents, which is not always true. 
 To form a just notion of the aid which may be expected from ma- 
 chines, looking to the uses to which they are most commonly put, 
 we shall divide them into two classes ; those intended simply to 
 sustain a weight, and those intended to draw it, or raise it equably. 
 
 In machines of the first class, both the effect of the machine and 
 the immediate effect of the power can only be estimated by the 
 weight sustained. This being understood it is evident that the 
 machine increases the eflfect of the power ; so that, for example, 
 a force of 10 lbs. will sustain by means of a lever, 100 lbs., pro- 
 vided that the arm of the force be ten times as long as that of the 
 weight.
 
 ox THE MECHANICAL POWERS. 105 
 
 If it be asked how the force can ever produce an effect so much 
 greater than itself, we shall perceive, if we consider well, that the 
 force 10 does not really sustain the whole weight 100, but only the 
 tenth part of it. Let the lever be supposed to be of the second kind ; 
 the force 100 may be resolved into two, the one equal to 90 which 
 acts upon the fulcrum, and the other equal to 10 which acts at the 
 point of application of the power. The first is entirely sustained 
 by the prop, and the power sustains the second alone. Archimedes 
 required only a fixed point to hold the terraqueous globe in equili- 
 brium. If he had found it, says Carnot, it would not in reality have 
 been Archimedes, but the fixed point, Avhich would have sustained 
 the earth. 
 
 In machines of the second class neither the effects of the machine 
 nor that of the power can be estimated simply by the weight raised ; 
 otherwise the measure of the effect would be altogether vague and 
 indeterminate. In fact any force, however small, may carry a weight 
 of any assignable magnitude however great ; if it only be granted 
 that the weight admits of being divided and of being carried, one 
 piece at a time. Wherefore it is necessary to take into account 
 the time also in which the power can carry the weight through a 
 given space, or the velocity with which the weight is carried ; and on 
 this account it is that the effect is measured by the product of the 
 weight multiplied by the velocity. 
 
 Now upon this principle we have already shown that the machine 
 does not increase the effect of the force. If a man with a force equi- 
 valent to 10, raise, by means of a machine, a weight of 100, he 
 moves with a velocity ten times as great as that of the weight, and 
 does as much as if operating without any machine he carried those 
 100 at ten journeys, loading himself with 10 at a time. In a word, 
 what is gained in the quantity of the weight moved is lost in the ve- 
 locity ; and the effect remains the same. 
 
 Between the two classes of machines, above described, there is then 
 this characteristic difference, that the first add to the effect of the 
 power, the second do not add to it. 
 
 There is another difference, not less remarkable, respecting the 
 resistances of friction, and of ropes, and other resistances. In ma- 
 chines of the first class these resistances are all of them advanta- 
 geous to the power* and themselves also sustain their portion of the 
 weight ; whence there remains so much the less of it for the power 
 to support. On the contrary, in machines of the second class, the 
 resistances are all of them detrimental to the power, and form part of 
 the weight to be overcome : whence, on this account, a force is re- 
 
 * Because the weight, before it can move, must overcome these resistances as 
 well as the power. 
 
 14
 
 106 ELEMENTS OF STATICS. 
 
 quired greater than that which would be required in the immediate 
 application of the power. — VenturoWs Mechanics, part ii. p. 164. 
 
 CHAPTER V. 
 
 ON THE STRENGTH AND STRESS OF BEAMS. 
 
 (91.) It is obvious that in all the practical operations of me- 
 chanics, and more especially in the raising of structures, it is of 
 great importance to know the weight or stress each component part 
 is fitted to bear without endangering the stability of the whole ; and, 
 consequently, numerous experiments to ascertain the strength of 
 materials, particularly of beams and bars, have at various times been 
 undertaken by scientific men. Into a detail of these experiments we 
 do not, however, propose to enter, but merely to present to the student, 
 in a short compass, some of the more interesting and valuable par- 
 ticulars furnished by theory, and confirmed by the experiments ad- 
 verted to. 
 
 If a uniform rod or bar of any substance be suspended by one ex- 
 tremity, and loaded at the other till it is on the point of being torn 
 asunder, we ought to expect, independently of actual experiment, 
 that the weight would be proportional to the tranverse section ; for, 
 if this bar were conceived to be divided longitudinally into any num- 
 ber of equal strips, no reason could be assigned why one of these 
 should support a greater portion of the weight than either of the 
 others, so that each would support an equal part of the weight sup- 
 ported by the whole, just as an assemblage of parallel ropes divide 
 the weight of an appended body equally among them. AH experi- 
 ments on lateral strains prove this deduction to be correct, and to 
 be quite independent of the figure of the section, requiring only 
 uniformity and equality in the texture of the bodies compared, so 
 that we may lay it down, as a general law, that in bars of the same 
 material the lateral resistances are as the areas of their traverse sec- 
 tions. 
 
 (92.) When the bar or beam is supported in a horizontal position, 
 then the law of resistance, which it opposes to fracture by an in- 
 cumbent weight, is more difficult to establish, because here we do 
 not see so clearly how the resisting forces exert themselves, nor in 
 what degree. It was laid down by Galileo that if a beam were sup- 
 ported at its extremities, as in fig. 81, and loaded by a weight at the 
 middle, that all the fibres of the beam would exert equal resistances 
 to prevent fracture, and that when these were overcome the section
 
 THE STRENGTH AND STRESS OF BEAMS. 107 
 
 would tend to turn about that boundary of it in contact with the 
 weight, viz. about AB. As all the fibres exert equal resistances, 
 and in the direction of their lengths these resistances will be so many 
 equal and parallel forces which may, therefore, be considered as 
 concentrated in the centre of gravity of the section, so that denoting 
 the resistance of a single fibre by k, and considering the section to 
 be a rectangle of breadth b, and depth h, kbh will express the sum 
 of the resisting forces, and as this acts at the centre of gravity its 
 moment to turn the section about AB will be 
 
 kbhx\h= — 
 
 (93.) The hypothesis of Leibnitz agreed with that of Galileo, 
 as regards the axis about which the section would turn, but it dif- 
 fered from it as regards the equal resistances of the fibres through- 
 out the whole fracture ; for, according to Leibnitz, the forces ex- 
 erted by the fibres were directly proportional to their distances from 
 the axis of the section, so that the middle fibre exerted but half the 
 force of the extreme fibre, therefore, calling the force of this k, the 
 
 sum of the forces would be ~-^, and the centre of such a system 
 
 of parallel forces being at |/i, the moment to turn the section would 
 kbh^ 
 
 ~^: 
 
 Now it may be remarked that as far as regards the comparative 
 strength of rectangular beams of the same material, or of beams 
 generally, which have only rectangular sections when cut traversely 
 it matters not which of these hypotheses be adopted, for both equally 
 warrant the inference that the laiv of resistance is as the breadth, 
 multiplied by the square of the height or depth; and this law, 
 which has been confirmed by numerous experiments, immediately 
 leads to an inference of considerable practical importance, viz. that 
 a beam is much more efficient when placed with its narrower side 
 uppermost, that is, so that its breadth may be less than its depth ; 
 for if we call the breadth b, and the depth or height h, then the re- 
 lative strengths, when b and h are alternately uppermost, are ex- 
 pressed by bh^ and hb^, and, consequently, in the former position 
 
 the beam is — times as strong as in the latter, that is, as many times 
 
 as strong as the depth contains the breadth. This important fact is 
 always attended to in buildings, the joist, rafters, &c. being always 
 placed with the narrower side uppermost. 
 
 It has been supposed above that the segments of a fractured beam 
 tend to turn about the line where the fracture terminates ; but, from
 
 108 ELEMENTS OF STATICS. 
 
 f'xperimpnts recently undertaken by Mr. Barlow, it appears that AB 
 (fig. 82) is not the line about which the section tends to turn, as Ga- 
 lileo and Leibnitz had supposed, but that the tendency is to turn 
 about a line entirely within the section, so that the fibres on that 
 side of the line where the fracture begins are extended, and those on 
 the other side compressed ; this axis Mr. Barlow calls the ncnlrcd 
 axis, dividing the section into the area of tension and the area of 
 compression, and he calls the centre of tension or of compression 
 that point in the area of tqnsion or of compression where all the 
 forces in that area should be collected to have the same effect, or the 
 same moment, with respect to the neutral axis. 
 
 The existence of a neutral axis somewhere within the area of frac- 
 ture, was maintained by Mariotte, James Bernoulli, and Professor 
 Robison ; but Mr. Barlow appears to have been the first who set 
 about the determination of this axis by actual experiment. (See the 
 historical sketch of former theories, prefixed to Mr. Barlow's Essay 
 the Strength and Stress of Timber.) 
 
 The general conclusion from these experiments was this, viz. 
 " The centre of tension and the centre of compression, each co- 
 incided with the centre of gravity of its respective area : and the 
 neutral line, which divides the two, is so situated that the area of 
 tension into the distance of its centre of gravity from the neu- 
 tral axis is to the area of compression into the distance of 
 its centre of gravity from the same line, in a constant ratio for each 
 distinct species of wood, but approximating in all towards the ratio 
 of three to one. 
 
 (94.) This theorem being established, says Mr. Barlow, it is evi- 
 dent that we may thence, without any specific numbers for exhibit- 
 ing the actual resistance of the fibres, compute the proportional 
 strengths of differently formed beams ; and of the same formed 
 beams in different positions ; of which we will give one example 
 by way of illustration. 
 
 Problem I. — Let a square beam be fixed with one end in a wall, 
 first in a direct position, viz. with its sides perpendicular and hori- 
 zontal ; and, secondly, with its diagonal vertical to find the ratio of 
 its strength in these two positions. 
 
 Conceive ABCD (fig. 83) to denote the beam in its first position, 
 EF the neutral axis, EABF the area of tension, and t the centre of 
 tension or centre of gravity of that area, EDCF the area of com- 
 pression, c its centre of gravity, and G tlie centre of gravity of the 
 whole area of fracture, and the same letters will denote the similar 
 quantities, in fig. 84, which represents the section of the beam in 
 its second position.
 
 THE STRENGTH AND STRESS OF BEAMS. 109 
 
 Then, by the preceding theorem, we have (fig. 83), 
 
 area AEBFxn?x3=«rea EDCFxnc, 
 « and in fig. 84, area EBFxn^x3=«re« EFCDAxwc ; 
 both which, from the property of the centre of gravity at (p. 59), 
 are reducible to 
 
 (fig. 83,) area AEBFx2 nt^area ABCDxn G 
 (fig. 84,) area EBF x2 nt=area ABCDxn G. 
 For the sake of simplifying the computation, let the side of the 
 square =1, n)^=x, or nt^kx, then nG = d — x ; the area AEBF 
 =a^, and the area ABCD^l, whence our first equation gives 
 x^=k — oc, or x^-\-x^h. ; whence x= — 5±\/3 = -366, which de- 
 notes both the depth of tension Hn, and area of the same AEFB ; 
 
 consequently, •366x—^r— ='066978, the numerical expression for 
 
 the resistance to tension, on which depends the strength of the 
 beam. It remains now to compute the same for the second position 
 of the beam, as in fig. 84. 
 
 Here if we denote wB by x, nt=l x, and the area EBF=a?^ also 
 area ABCD=1, as before; whence our second equation becomes 
 a'^^|x=2^/2 — a?, or a;»+fx=|v'2 = 1-0606. 
 From which we readily obtain a'=-578 nearly, 
 jj2:_. 33408, and x^^Ks ^=5 a:='=-06436, which is the numerical va- 
 lue of the tension in this position of the beam. The strength of the 
 beam, therefore, in the latter position is to that in the former in the 
 ratio of -06436 : -06697 ; or as the numbers 643 : 669 nearly, which 
 accords with experimental results, and, in a similar way, may the 
 strengths of difl'erently formed beams be compared. We shaU now 
 consider the straining effects on beams differently supported, and 
 loaded by weights at different parts. j;^ 
 
 Problem II. — A beam of timber AB (fig. 85) is fixed with one 
 end in a wall, and loaded with a weight W at the other end ; to 
 determine the efficacy of this Aveight to break the beam. 
 
 At the place where the beam has the greatest tendency to break, 
 the broken piece will tend to turn round the neutral axis ; if the 
 distance of this axis from W be /, then AV will express the energy 
 of W to produce this eff'ect ; but AV will be greatest when / is 
 greatest, that is when this denotes the whole length of the project- 
 ing beam ; hence the beam will tend to break close to the Avail, and 
 ZW will express the strain there. The strain varies therefore as the 
 length of the beam, W being the same. 
 
 Problem III. — A beam rests loosely on two props. A, B (fig. 86), 
 and is loaded at a given point C by a given weight W : to determine 
 the stress at C. 
 K
 
 110 ELEMENTS OF STATICS. 
 
 Of course the tendency to I)reak will be at C, and we may, there- 
 tore, assimilate this to the preceding case by conceiving the beam 
 to be fixed in a wall up to C, and to be strained by a force equal to 
 the pressure upon the prop at the other end. 
 
 Now by (36) the pressure on the prop A is expressed by 
 
 P=£1LW: 
 AB 
 
 this then being the force that strains the projecting beam AC, its 
 
 AC . CB„, 
 energy is — —^ — W. 
 
 If the load be in the middle of the beam, the product AC . CB 
 becomes the greatest possible, being the square of half the length ; 
 hence a beam will be less able to support a weight at its middle 
 than if it be placed at any other part. The strain obviously varies 
 as the product of the distances of the weight from the props. 
 
 It may be further remarked, that when the weight acts at the 
 
 AC^ 
 
 middle, the stress, being =— — — W=| AB . W, is one-fourth the 
 
 stress which the same weight W would produce acting at the ex- 
 tremity of a projecting beam equal to AC, or it is equal to the whole 
 stress on a projecting beam equal to AB, the weight at its extremity 
 being =i W, or a projecting beam equal to half AB and loaded with 
 i W at its end, will sufler half the stress at the wall. 
 
 Problem IV. — To determine the stress on a projecting beam, 
 and on a beam resting on props when the weights are distributed 
 uniformly over them. 
 
 Let AC be the projecting beam and w its weight, including the 
 uniformioad ; this weight will act at half the distance AC from the 
 wall, and therefore the stress is k AC . w, which is just half what 
 it would be if w were placed at the extremity. 
 
 When the beam rests on props (fig. 87), the pressure on each 
 
 prop is half its weight W, this, therefore, is the force acting at P 
 
 which tends to fracture the beam at C with an energy expressed by 
 
 i W . AC ; 
 
 but there is another force exerted, viz. that due to the weight w of 
 
 the portion CA, and which, by last case, opposes the former with 
 
 an energ}' expressed hy ^ w . AC ; hence the expression for the 
 
 stress on C must be i (W — jv) AC : to eliminate w we have, on 
 
 account of the uniformity of the beam, 
 
 AC 
 AB : AC : : W : w=^W, 
 Ao 
 
 SO that the expression for the stress will be
 
 THE STRENGTH AND STRESS OF BEAMS. Ill 
 
 AB^C-AC^ AC^BC 
 
 ^ ' AB ^~2 AB • 
 
 Hence in this case also, as in problem III., the stress varies as the 
 product AC . BC. 
 
 The strain at the middle point of the beam is § W . AB, just 
 half what it would be if the whole load were placed there, (pro- 
 blem III.) 
 
 SCHOLIUM. 
 
 (95.) By means of these problems it will be easy to find the most 
 economical forms for beams, either projecting or supported at the 
 ends, so that they may in no part possess superfluous strength, 
 that is, that the strength in every part may be exactly in proportion 
 to the stress there. Thus if a projecting beam of uniform breadth 
 is to support a weight at its extremity, it will be equally strong 
 throughout if the vertical sides are in the form of a parabola (fig. 
 88); for, by (prob. II.), the stress varies as AC=a:', and the 
 strength of the beam, being (93) as the breadth into the square of 
 the depth, and the breadth being constant, varies simply as CD^= 
 y^ ; hence, in order that the strength and stress may be throughout 
 
 in a constant ratio, AC must vary as CD^, that is, — — ^=constant 
 
 AC 
 
 =a or y'^=ax, the equation of a parabola. But the shape need not 
 necessarily be parabolic in order to insure uniformity of strength, 
 as it will depend in a measure upon the nature of the vertical sec- 
 tions : thus, if these sections are required to be all squares, then 
 the breadth and depth being every where the same, the strength 
 will vary as the cube of the depth, and hence AC should vary as 
 CD', that is, y^=ax, the equation of the cubical parabola, which 
 must therefore be the form of the tapering beam. 
 
 Again, if the depth of the beam is to be constant, then AC should 
 vary as the breadth, and therefore the upper and under faces of the 
 beam will be triangles. If a beam supported on two props is to be 
 uniformly strong, and at the same time uniformly broad, it will be 
 necessary to form the vertical sides elliptical, for the breadth being 
 constant the strength will vary as the square of the depth, and, by 
 (prob. IV.) the stress at any point C varies as AC . CB ; hence the 
 square of the depth at C must be in a constant ratio to AC . CB, 
 which requires that D (fig. 89,) be always in an ellipse whose axis 
 is AB, (Jnal. Geom. p. 122.) 
 
 (96.) It may be moreover remarked, that, by means of the fore- 
 going expressions for the strain or tendency to produce fracture, 
 combined with the results of experiment, we may determine the
 
 112 ELEMENTS OF STATICS. 
 
 actual weight which any sfiven beam will support in given circum- 
 stancns. Thus, suppose it is f<>und by experiment that a beam of 
 breadth h, depth h, and lenffth /, just breaks with a weight w at its 
 middle, and that it is required to determine what weight W will just 
 break another beam of like materials whose breadth is B, depth H, 
 and length L. In each case the tendency to resist fracture is just 
 balanced by the tendency to produce it ; the expressions, therefore, 
 for these two tendencies must be equal, and therefore the ratio of 
 the tendencies to resist fracture in the two beams must equal the 
 ratio of the tendencies to produce fracture. Now the tendency to 
 resist fracture is what we understand by the strength of the beam, 
 and the tendency to produce fracture is the stress ; hence, equali- 
 zing the two ratios spoken of, we have (prob. III.) 
 
 B . H" :! L . W B^. H^ /. w 
 
 b.h^ ~ U.io ■** ~t.h' ' L ' 
 
 the weight required. It is obvious that from the same equation we 
 may deduce the length L when W is given, and also that the equa- 
 tion remains the same whether W, w act in the middle or at the 
 end of each beam. 
 
 The expression here given may serve to compare the strength of 
 any beam in a model with that of the corresponding beam in the 
 structure. Thus, suppose the beam which we have considered to 
 have been submitted to experiment to belong to the model, and the 
 other to be the corresponding beam in the structure whose like 
 dimensions are n times those of the former, then the foregoing ex- 
 pression for W will be W=n^ iv, which will be the greatest pos- 
 sible load tiie beam in the structure can bear, including of course its 
 own weight. Now if the weights alone of the two beams are 
 respectively p and P, then, since these must be as the cubes of their 
 like dimensions, we must have P=n^j3, consequently the beam in 
 the structure so far from bearing a load, will but just support its 
 own weight if we make it so large that n^ p^n^w, that is, if n= 
 
 — : we see, therefore, how erroneous it would be to estimate the 
 P 
 
 strength of a large beam in a structure by that of a similar small 
 beam in a model, regarding only the comparative dimensions of 
 each ; for, by increasing the magnitude of the large beam, without 
 in the least changing the relative proportions of the two, we should 
 nevertheless render it at length too large to support even its own 
 weight, although the model agreeably to which it has been formed, 
 might be able to support a load many times its own weight. There 
 is, therefore, necessarily a limit to the magnitude of all structures, 
 even indeed to the magnitude of the animal structure, and to trees, 
 beyond which limit they would be unable to support their own
 
 81 A- 82 
 
 83 
 
 H 
 
 84- 
 
 ^^m 
 
 S3 
 
 89 
 
 "^~W1lJ|_ 
 
 .It 
 
 ■H^''ilijlijr
 
 THE STRENGTH AND STRESS OF BEAMS. 113 
 
 weight ; we accordingly find men of enormous magnitude, as O'Brien 
 " the celebrated Irish Giant," to be so weak that they are scarcely 
 able to walk about. 
 
 In connexion with these remarks may be mentioned the curious 
 question, proposed by Mr. Emerson, among the mechanical pro- 
 blems annexed to his algebra ; the question is this : Supposing, 
 with Borelli, that a strong man can bear but261bs. at arm's end, and 
 that the weight of his whole arm is equivalent to 41bs. at arm's end ; 
 from the length of his arm being given, to find the dimensions of that 
 man's arm that can bear no more than its own weight. 
 
 This problem is immediately solved by means of the relation n= 
 
 w 
 
 — , deduced above, w representing here the weight 26+4 or 30lbs., 
 
 the weight of the common man's arm and load, and p representing 
 the weight 41bs. of his arm alone, so that n=7 5 : this, therefore, 
 is the number of times any dimension of the large man's arm must 
 contain the corresponding dimension of the common man's arm ; 
 let us then suppose the common man's arm to be a yard long, the 
 length of the other man's arm, to just support itself, must be 7^ 
 yards, and, as the body is, in well proportioned persons, about tAviee 
 as long as the arm, we therefore conclude that a man upwards of 15 
 yards high would not be able to stretch out his arm. 
 
 Problem IV. — To determine the relative strengths of beams 
 loaded in the middle Avhen their ends are loosely supported, and 
 when they are firmly fixed in two vertical walls. 
 
 When a beam is loosely supported and acted upon by a weight 
 at the middle, this iveight is equally divided between the two props, 
 but at these points there is no strain ; when, on the contrary, the 
 ends of the beam are firmly fixed in immovable walls, then it is the 
 strain on the middle which is equally divided between the two ex- 
 tremities ; that is to say, the fibres in the section at each wall are 
 strained half as much as those at the middle section. The whole 
 of the weight, therefore, is not expended here, as in the former 
 case, in straining the middle of the beam, but a portion is employed 
 in straining each end half as much. Now, whatever weight strains 
 the middle, | of this will (by prob. III.) strain each section at the 
 wall half as much ; hence, if we represent that part of the weight 
 which strains the middle only, by 4, the part which strains the 
 ends will be 2, and therefore the Avhole straining weight will be 6, 
 so that the weight 6 will produce no more stress on the middle of 
 the beam thus fixed, than the weight 4 when the ends rest loosely 
 on props ; hence the relative strengths of fixed and loose beams are 
 as 6 to 4 or as 3 to 2, Avhich relation Mr. Barlotv has verified by 
 experiment. 
 
 k2 15
 
 114 ELEMENTS OF STATICS. 
 
 It may be observed, that the weiglit 8 uniformly distributed over 
 the beam would produce the same strain in the middle as the weight 
 4 applied there (prob. IV), and the weight 4 uniformly distributed 
 will strain the ends as much as the weight 2 applied to the middle : 
 hence, if the load be uniformly distributed over the fixed beam, it 
 will be no more strained with the weight 12 thus disposed, than 
 with the weight 6 acting at the middle, so that here, as in the other 
 cases, the efliciency of the beam is doubled by spreading the weight 
 uniformly over it. 
 
 SCHOLIUM. 
 
 (97.) The result of the preceding investigation, although con- 
 firmed by Mr. Barlow's experiments, differ materially from tlie con- 
 clusions deduced by other philosophers, as Girard, Emerson, and 
 Jiobison, who find the comparative strengths of supported and fixed 
 beams to be as 1 to 2, and not as 2 to 3. Emerson's reasoning on 
 this point is as follows : 
 
 Suppose DA=AC (fig. 90,) and BE=BC, and let P be the 
 weight which would break the beam when resting on A and B. 
 Suppose the beam cut through at C, and let 5 P be laid upon D, 
 whilst I P remains at C ; then the pressure at A will be =P, there- 
 fore the beam will also break at A having the same stress there as 
 it had at C. For the same reason, if ^ P be applied to E, CE will 
 break at B. Consequently, if 2 P be applied to C, the beam being 
 whole, and the ends D, E fixed, the beam will break at A, C, and 
 B ; and, therefore, bears twice the weight, or 2 P at C, before it 
 breaks. 
 
 Now the foregoing reasoning appears to assume, that before the 
 beam can break at C, the strain on A and on B must be sufficient 
 to break the beam at those points also ; yet it is shown that the 
 beam will break simultaneously at these three points, if, besides the 
 weight P acting atC, the points D, C, E, be each loaded with 
 iP; hence, to enable the middle point C to yield to the pressure of 
 P, it is only necessary that the fibres at A and at B be half as much 
 strained as they are by the influence of i P acting at D, C, and E ; 
 because but half the strain of the fibres at A are, in virtue of this 
 influence, in the direction of AC, the other half being in the direc- 
 tion of AD ; and, in like manner, but half the strain at B is in the 
 direction of BC, so that if, in addition to P acting at C, as much 
 more weight is added as will produce these half strains, the parts 
 AC, BC will be deflected sufficiently for the beam to break at C ; 
 we have, therefore, to add to P only half as much as would produce 
 the whole strains at A and B, that is, instead of P we should add 
 ^P, making the whole breaking load ^ P, which is the same result
 
 THE STRENGTH AND STRESS OF BEAMS. 115 
 
 as before obtained, and the coiTectness of which Mr. Barlow's ex- 
 periments confirm. 
 
 Mr. Barlow remarks on this subject, " in every experiment that I 
 made after the complete fracture in the middle, the two fragments 
 had been so little strained at the points of fixing, that they soon after 
 recovered their correct rectilinear form ;" and, in order to show the 
 foundation of the error which all theorists have made, in assuming 
 that the fixed beam would break simultaneously in the middle and at 
 the walls, he further adds, " If the beam instead of being fixed at 
 each end were merely rested on two props, and extended beyond 
 them on each side equal to half their distance, and if weights w, w' 
 (fig. 91,) were suspended from these latter points each equal to one 
 fourth the weight W, then this would be double of that which would 
 be necessary to produce the fracture in the common case ; for, divi- 
 ding the weight W into four equal parts, we may conceive two of these 
 parts employed in producing the strain or fracture at E, and one of each 
 of the other parts as acting in opposition to iv and iv' , and by these 
 means tending to produce fractures at F and F'.' 
 
 " This is the case which has been erroneously confounded with 
 the former, but the distinction between them is sufficiently obvious ; 
 because, here the tension of the fibres, in the places where the strains 
 are excited, are all equal ; whereas in the former the middle one was 
 double of each of the other two."* 
 
 Venturoli, in his valuable book on Mechanics, says, in the words 
 of Dr. Creswell's translation, " The beam would sustain a load con- 
 siderably greater, if, instead of being simply placed upon two props, 
 it were immoveably fixed in stone-work at both its extremities. 
 For, it that case, it cannot break unless le gives way in three places 
 at the same tirae."t 
 
 Problem V. — To determine the dimensions of the strongest rect- 
 angular beam that can be cut out of a given cylindrical tree. 
 
 Let r be the radius of the base of the cylinder, and x and y the 
 breadth and depth of the required beam, then, as the strength varies 
 as xy^, this quantity must be a maximum ; hence 
 
 Also, as the diagonal of the rectangle is equal to the diameter of the 
 circle, we have 
 
 ^•+!,.=4r».-.x+3,|=0.-.| = -^; 
 
 • Essay on the Strength and Stress of Timber, third ed. p. 149. 
 f VenturoWs Mechanics, Part 11., p. 60 .
 
 116 ELEMENTS OF STATICS. 
 
 2 x" 
 
 hence, by substitution, y =0 .-. 3/^=2 x'=4 r' — x* 
 
 2 r 
 .-. 3 a:''=4 r^ .-. x=— — , y=2r v/|, the dimensions required. 
 
 For further information on the subject of this chapter, anil more 
 especially for an account of the various experiments that have hither- 
 to been made to determine the strength of materials, the student is 
 referred to Professor Gregory's valuable Treatise of Mechanics ; to 
 the second volume of Sir David Brewster's edition of Ferguson's 
 Lectures ; to Mr. Barlow's work on the Strength and Stress of 
 Timber, as also to his treatise on Mechanics in the Encyclopaedia 
 Metropolitana ; to Part II. of Creswell's translation of Venturoli ; 
 and lasdy, to Professor Leslie's instructive volume on the Elements 
 of Natural Philosophy. 
 
 Perhaps we ought to remark before closing this chapter, that in 
 all the foregoing investigations on tlie stress of beams, we have not 
 taken into account the deflection from the horizontal line which the 
 force produces before it actually breaks the beam. By reason of 
 this deflection the energy of the breaking force is not, strictly speak- 
 ing, expressed by the intensity of the force multiplied by its distance 
 measured along the beam from the section of fracture, but by the 
 intensity into the perpendicular distance of the fracture from its di- 
 rection ; this perpendicular distance is equal to the former distance 
 multiplied by the cosine of the angle of deflection, and therefore, by 
 introducir\g this cosine as a factor into all the foregoing expressions 
 into which the moments of the straining forces enter, they will be- 
 come rigorously correct ; but except in very long beams, or in very 
 elastic ones, the deflection is too small to render this modification of ' 
 much consequence : Mr. Barlow, however, has not neglected its 
 influence in his important inquiries on this subject. 
 
 END OF THE ELEMENTS OF STATICS.
 
 PART 11. 
 
 ELEMENTS OF DYNAMICS. 
 
 SECTION I. 
 
 ON THE RECTILINEAR MOTION OF A FREE POINT. 
 
 (98.) Having considered the general theory of equilibrating' 
 forces, we come now to Dynamics, the second principal division of the 
 science of Mechanics, and which comprehends the theory of unba- 
 lanced forces. Dynamics, therefore, considers bodies in a state of 
 motion, while Statics has to do only with bodies at rest ; in this 
 first section we shall confine ourselves to the consideration of recti- 
 linear motion only, but in the opening chapter we shall lay down a 
 kw general and fundamental principles which always hold, whatever 
 be the path of the moving point, and which, in fact, will be found to 
 comprise the whole theory of its motion. 
 
 CHAPTER I. 
 
 ON THE FUNDAMENTAL EQUATIONS OF MOTION. 
 
 (99.) By the inertia of matter is meant its incapability of altering 
 the state into which it is put by any external cause, whether that 
 state be rest or motion. 
 
 It is manifest that if a body at rest receive an impulse in any di- 
 rection,* it will, if entirely at liberty to obey that impulse, move in 
 that direction, and with a uniform rate of motion ; for as we suppose 
 the body to be entirely uninfluenced by any other cause, and since it 
 is incapable of exertion itself, it is plain that for whatever reason we 
 could suppose the motion to slacken at any point of its path, for 
 the same reason we might suppose the motion to quicken. The 
 body will, therefore, continually move at a uniform rate in the di- 
 rection impressed upon it, that is, if notiiing extraneous interferes 
 with its motion. 
 
 * The body is here considered as a single point, or else as recei^'ing its im- 
 pulse towards the centre of gravity, so that no rotation is impressed on it. 
 
 117
 
 118 ELEMENTS OF DYNAMICS. 
 
 (100.) Wc have just spoken of the rate of a body's motion : we 
 estimate this, when the motion is uniform, by the space the body 
 passes over in some determinate portion of time, as in one second, 
 which indeed is the portion generally assumed for the unit of time ; 
 so that when we observe a moving body to pass uniformly over ten 
 feet every second of time, we express the rate of its motion I)y say- 
 ing that it moves with a velocity of ten feet, or, for greater brevity, 
 that its velocity is ten feet, and this is what we are to understand by 
 the equation v = \0 feet, space being taken as the measure, or repre- 
 sentative, of velocity. 
 
 Suppose now that t represents, not the time, but an abstract num- 
 ber expressing the number of seconds elapsed sinc« the commence- 
 ment of the uniform motion, and let s denote the corresponding space 
 passed over by the body, then we obviously have the three equations 
 
 * s 
 
 v=—, s=tv, t=—, 
 t V 
 
 so that any two of the three quantities s, t, v, being known, we 
 may immediately find the third. 
 
 But if t" is not reckoned from the commencement of motion, but 
 only after a certain space s' has been described, then, s being the 
 whole space gone over from the commencement, the three equa- 
 tions will be 
 
 s — s' s — s' 
 
 v= — - — , s=s -^vt, t=- 
 
 t V 
 
 These equations, or indeed any one of them, comprehend the 
 whole theory of the motion of a body acted on by a single impulse, 
 or influenced by any cause which produces uniform motion. We 
 shall give an instance of their application. 
 
 Two bodies a, b (fig. 92), animated by the velocities v, v' set out 
 simultaneously from the points A, B, and move in the same direc- 
 tion AC ; to determine the time of their coming together. 
 
 Suppose they come together at the point C, then 
 
 AC=r/, BC = y' /, that is, calling AC, s, and AB, s' 
 
 s' 
 
 s=vt, s — s'=v t .wt — s =v t .'.t= -, 
 
 V — V 
 
 that is, the abstract number expressing the units of time will be that 
 which arises from dividing the space between the points of starting 
 by the difference of the spaces denoting the velocities. 
 
 It may be remarked here, that whatever be the nature of the in- 
 fluence which produces uniform motion, and which we have above 
 called an impulse, we have a right to conclude that its effect will 
 be proportional to its intensity ; in other words, that such in- 
 fluences, acting on the same body, or on equal bodies, are propor- 
 tional to the velocities they produce
 
 FUNDAMENTAL EQUATIONS OF MOTION. 119 
 
 For if a body receive a certain velocity in consequence of a cer- 
 tain impulse, it ought obviously to acquire double that velocity if at 
 any point of its path that impulse be repeated in the same direc- 
 tion, but if this second impulse take place at that point from vi^hich 
 the body set out, it must unite with the first impulse, so that the 
 consequence of a double intensity of impulse will be a double ve- 
 locity in the body, and, in like manner, a triple intensity will pro- 
 duce a triple velocity, and so on. 
 
 (101.) Let us now consider the circumstances of variable mo- 
 tion, and let us first ascertain the expression for the velocity of a 
 body so moving at any epoch t". If we first assume that the ve- 
 locity which the body has at t" continues uniform from t" to t'\., 
 then, calling t^ — t, A /, and the increment of the space or s^ — s, 
 
 A S 
 
 A 5, we have for the velocity at t"',v^= , however small A t, 
 
 A t 
 
 and consequently A s, which depends on it, may be ; but if no in- 
 terval of time A t" exists so small, during which the velocity does 
 not vary, then the above equation is true only when A t, and con- 
 sequently A s, becomes ; hence, by the principles of the diffe- 
 rs 
 rential calculus, we have in this case v=-r • Cl)> which is there- 
 at 
 
 fore a general expression for the velocity of a moving body at any 
 time t" however its motion may vary, and of course it applies also 
 when the motion is uniform, for then 
 
 AS ds 
 
 r= =-— -^ constant .... (2). 
 
 At dt ^ ^ 
 
 (102.) It is obvious that if the velocity of a moving body con- 
 tinually vary, it must be influenced by some continuous cause, 
 however this cause may itself vary in efficiency ; for from the in- 
 stant the cause ceases to act, that instant the body ceases to vary 
 in velocity in consequence of its inertia. We call the cause of 
 variable motion, whatever it really be, force: an accelerative force 
 if the velocity continually increase, and a retardive force if the ve- 
 locity diminish. We shall, in our general reasonings, consider the 
 force as accelerative, because in order to adapt our conclusions to 
 retardive forces, it will be necessary merely to prefix to the ex 
 pression for F the negative sign. Let us now investigate this 
 expression ; and first we must remark, that as the effect of a con- 
 stant accelerative force is obviously to generate constant increments 
 of velocity in equal times, if we agree as heretofore to represent 
 causes by their eflfects, we shall obtain the expression for F by di- 
 viding the increment of the velocity by the units in the increment 
 of the time, measured from any epoch t", that is,
 
 120 ELEMENTS OF DYNAMICS. 
 
 F=^....(l): 
 
 such then is the expression for a constant accelerative force, A t" 
 being any interval of time from /", and A v the corresponding aug- 
 mentation of velocity. The velocity of the body in this case is with 
 propriety called a uniformly accelerated velocity. 
 
 (103.) But suppose that F is not a constant force ; then if from 
 any epoch t" there is an interval A /" so small that F remains un- 
 changed throughout it, the expression just given will in that case 
 represent the intensity of the force acting at the epoch /", and 
 continuing unabated and unaugmented during the interval A t". 
 If, however, choose A /" as small as we will, F still changes during 
 the interval, then Ave shall express this fact by saying that the in- 
 terval A/", during which F remains constant, is ; hence, for a 
 
 dv 
 continually varying force, the expression is F=-^ (2), and this may 
 
 be regarded as a general expression for the accelerative force whether 
 
 it be constant or variable, for when it is constant 
 
 x^ ^v dv ,„. 
 
 F= =-r-=constant .... (3). 
 
 At dt ^ -^ 
 
 We may give a different form to the general expression for F, for 
 
 ds „ dU 
 smce i»=^- .'. F=-r— . . . . (4). 
 
 dt df* ^ ^ 
 
 We have seen (equa. 1,) that the expression for F at any epoch 
 t" is equal to the increment of the velocity that tvould he generated 
 in any number of seconds after that epoch (if F were thence to 
 cease to vary,) divided by that number ; that is, F, estimated at any 
 epoch t" , is equal to the increment of velocity that would be gene- 
 rated by that force constantly acting during one second. But the 
 velocity of a moving body at any epoch is measured by the space 
 it would pass over in the succeeding second, if its motion were 
 thence to become uniform ; hence the force acting upon a moving 
 body at any epoch t", is measured by the space the body would 
 pass over in the 2d second of time after t", provided it were to pro- 
 ceed during that second with the increment of the velocity generated 
 during the 1st second. 
 
 It thus appears that both velocity and force may be measured by 
 space, and therefore that in every dynamical inquiry, where the 
 mass is not considered, the only concrete quantity concerned is 
 space, for, as before observed, t denotes an abstract number, viz. the 
 number of units or seconds in the time t". 
 
 (104.) It should be remarked here, that the forces of which we 
 have just spoken are in no respect influences of a different kind 
 from those considered in statics ; they merely manifest themsehes
 
 FUNDAMENTAL EQUATIONS OF MOTION. 121 
 
 differently by producing different effects, and it is to the effects only 
 that we look in estimating these influences. The statical effect of 
 a force applied to a body is pressure or weight, and we accordingly 
 represent the force, in statics, by pressure or weight. The dyna- 
 mical effect of the same force is accelerated velocity, and ac- 
 cordingly we represent the force by velocity ; or, since space mea- 
 sures velocity, we represent it by space. These different modes 
 of estimating the same force, therefore, naturally present them- 
 selves upon observing their effects ; but, for all the purposes of com- 
 parison, it matters not, as was observed in Statics (4), by what we 
 represent the efficiency of any force, taking care only always to 
 keep up the proportion between the forces and their representative 
 quantities. Thus there would be no impropriety, if there were no 
 inconvenience, in representing an accelerative force by a weight, 
 provided we always proportioned the weight to the efficiency of the 
 force ; and this leads us to a remark of some importance, viz. that the 
 pressure or weight produced by the action^ of a force on anybody, 
 is to the pressure or weight produced by the action of any other 
 force on the same body, as the acceleration producedby the former 
 force is to the acceleration produced by the latter : for it is plain 
 that the ratio of the two forces must be the same abstract number 
 however they are represented ; so that if we know the two pressures 
 or the two weights wliich the forces are fitted to produce, and also 
 the acceleration which one is fitted to produce, we know also the 
 acceleration which the other is fitted to produce. 
 
 (105.) In ail the foregoing investigations it should be remarked, 
 that we have put entirely out of consideration the nature of the path 
 which the moving body describes. All that we have said as to the 
 velocity of a body regards its rate of motion along the path, whether 
 straight or curved, in which it happens to move, and has nothing to 
 do with the manner in which that motion has been produced ; for 
 however it moves, and by whatever agency, the same velocity is 
 always expressed by the same linear space. So too with regard to 
 the moving influence itself, or the force ; this also has been esti- 
 mated without any reference to the path along which it impels the 
 body ; but it should be observed, that a force, when commencing 
 its influence on a body, may find that body already in motion, and, 
 as is easy to conceive, may act on it so as to divert it from its ori- 
 ginal path, and cause it to describe some other ; in such a case the 
 body may be moving under the influence of two forces, or under the 
 influence of an impulse and a force ; but still there must exist, or 
 at least we can conceive, some single force which if immediately 
 applied to the body, at any instant of time, would give it the same 
 motion that it actually has at that instant in virtue of the combined 
 influences alluded to. Now it must be remembered that it is this 
 L 16
 
 122 ELEMENTS OF DYNAMICS. 
 
 single and equivalent force which F represents in the foregoing 
 
 equations, and which, when its intensity is the same, is always 
 
 measured by the same linear space or length of path, be this path 
 
 whatever it may. By the path of a body, urged by an accelerative 
 
 force, is meant the track of its centre of gravity. 
 
 Another circumstance of importance deserves to be mentioned 
 
 ds 
 here, viz. that the general expression -j-forthe velocity at any point 
 
 of the path is no other than the differential coefficient of the variable 
 path s taken relatively to the independent variable /. It is from this 
 circumstance, as we shall hereafter see, that we are enabled to de- 
 termine the path of a moving body from knowing its velocity at any 
 point of it in quantity and direction. In like manner the general 
 expression for the force is the differential coefficient of the velocity 
 taken relatively to the same independent variable, and this expres- 
 sion combined with that for the velocity, leads, as we shall presently 
 
 dv 
 see, (equa. D,) to the expression F:=v-j-, which is sufficient to 
 
 determine the force which influences the body, when we know 
 what function the velocity v is of the space s. 
 
 (106.) It will be expedient, for the convenience of reference, to 
 collect together here the fundamental equations of motion now esta- 
 blished, introducing such slight modifications of form as may tend 
 to facilitate their practical applications in our future inquiries, and 
 deducing from them such brief inferences as may be of more espe- 
 cial interest or importance. It will be best to keep distinct those 
 equations which refer to constant forces, or to motion uniformly ac- 
 celerated, from those which refer to variable forces, or to motion 
 not uniformly accelerated. 
 
 I. TFhen the accelerating Force is constant. 
 
 Referring to equation (2) we have 
 
 dv=F dt .-. v=Yt+c .... (A). 
 
 If t become when v does, that is, if t is measured from the com- 
 mencement of motion, the constant c vanishes, and we infer from 
 this expression for v, that the velocities acquired in any times, reck- 
 oning from the commencement of motion, are proportional to the 
 times themselves. 
 
 Introducing the value v=Ft in the equation (4), we have 
 ds=Ftdt .-. s=d Ft'>=h't .... (B), 
 no constant being added, because s vanishes with t. From this 
 equation we infer, that the spaces measured from the commence- 
 ment of motion are proportional to the squares of the times. We
 
 FUNDAMENTAL EQUATIONS OF MOTION, 123 
 
 may further remark here, that if the acquired velocity r=F/ were 
 to continue uniform during the time it', the space passed over in 
 that time would be, (100), s=:iFt^; hence the space described 
 from the commencement of motion is equal to that which would be 
 described in half the tirpe by the body moving uniformly with the 
 acquired velocity. 
 
 If we eliminate t by means of the equations (A), (B), disregard- 
 ing the constant c, we shall have 
 
 v^=2Fs .-. i; = v/"2Fs (C) ; 
 
 showing that the spaces described from the commencement are prO' 
 portionals to the squares of the acquired velocities. 
 
 The foregoing equations are, obviously, sufficient to determine 
 any two of the quantities F, t, s, v, when the other two are given, 
 the time being supposed to be reckoned from the beginning of the 
 motion. But when this is not the case, and the time is supposed 
 to commence not till the body has acquired a given velocity v^, then 
 regard must be had to the constant in (A) ; the value of this constant 
 is plainly 0=?;^, because by hypothesis v^ is what v becomes when 
 t=0. Equation (A) will, therefore, here be v=Ft-\-v^ .... (A') ; 
 equation (B) will be 8 = 5: Ft^-\-v^t . . . . (B') ; and by eliminating 
 t from these two, we have for (C) the equation 
 
 ij2=2 Fs+?;,2 .'.V \/"2Fs+t>7. . . . (C). 
 
 II. When the accelerating Force is variable. 
 
 From equations (4) and (2) we have 
 
 u=-Tr, F=-Tr .'. —=-r-''.Fds=vdv 
 at dt V ds 
 
 .:fFdsr-riv^....(D); 
 
 which equation is sufficient to determine the velocity v when we 
 
 know what function the force F is of the space s, or it is sufficient 
 
 to determine this function when we know the functions. From 
 
 the same equation, also, 
 
 ^=/f 
 
 dv (E) ; 
 
 which makes known the space described when we know what func- 
 tions V and F are of this space. 
 
 And, lastly, from the equation [4) t= I — . . . . (F) ; which 
 
 determines the numerical value of t. 
 
 Having established these equations, we shall now proceed to ex- 
 hibit their practical application, more especially to those motions 
 which are presented to us in nature.
 
 124 ELEMENTS OF DYNAMICS. 
 
 CHAPTER II. 
 
 ON THE RECTILINEAR MOTION I'RODUCED BY A CONSTANT FORCE. 
 
 (107.) The most remarkable and important instance of the action 
 of a constant force is that wliich nature presents to us in what we 
 have called gravity, being that force, in virtue of which all bodies 
 near the earth fall to its surface, with a uniformly accelerated ve- 
 locity, in a vertical direction. 
 
 Numerous and very accurate experiments have fully established 
 the fact, tliat the velocity of a falling body, when all resistance is 
 removed, is uniformly accelerated, and that its direction is that of a 
 vertical line, or a normal, to the earth's surface at the point M'here 
 it falls. Such experiments, liowever, made at any particular place 
 on the motions of bodies falling from a small elevation, are not suffi- 
 cient to warrant the conclusion that gravity is really a constant force 
 in the acceptation in which we use the expression. All that we can 
 fairly infer from them is that, at the same place, and within the 
 range of small elevations, no sensible variation of force is discovera- 
 ble, and that, therefore, within the limits of our experiments, at 
 least, gravity may be considered as a constant force. But to ascer- 
 tain the real nature of gravity, by means of such experiments as 
 these, it is obvious that they ought to be repeated in various parts of 
 the earth, and at great elevations as well as small. This indeed has 
 accordingly been done, and it has been always found that a heavy 
 body carried to the summit of a high mountain loses part of its 
 weight, shewing, therefore, that gravity acts with less intensity at 
 the summit than at the base of the mountain ;* and, on the contrary, 
 it has always been found that the body increases its weight when car- 
 ried into those latitudes which are nearer to the centre of the earth. 
 These results of observation are doubtless sufficient to show that 
 gravity is not a constant force, and, moreover, that its variation de- 
 pends, in some way, upon the distance from the centre at which it 
 acts. But it is doubtful, chiefly on account of the comparatively 
 small elevations attainable by man, and partly on account of the im- 
 
 • At an elevation of a mile above the surface of the earth, the intensity of gra- 
 vity is diminished part ; and a pendulum clock, beating seconds at 
 
 •' 1977-291 f ' »^ . b 
 
 the level of the sea, would lose 21-898 seconds a day at this altitude, a quantity 
 not to be overlooked. Any traveller, having leisure and the proper apparatus, 
 might try the experiment in the barrack on Mont Cenis, or at the Hospice of St. 
 Bernard. — HerscheVs Physical Astronony. Ency. Met-
 
 ON RECTILINEAR MOTION. 125 
 
 perfection of instruments, whether from such experiments the real 
 law of the variation of gravity could have ever been safely inferred. 
 The discovery of this law, as well, indeed, as of that which retains 
 the planets in their orbits, was in fact the result, not of experiment, 
 but of conjecture ; but then it was the conjecture of Newton. 
 
 He was the first who conceived the splendid idea, and who after- 
 wards fully verified and established the important fact, that the at- 
 tractive force, not only of the earth but of every body in the solar 
 system, decreases in intensity in the same proportion as the square 
 of the distance from the centre of the attracting body increases. 
 This, therefore, is the law of universal gravitation, and which, as 
 Sir John Herschel beautifully observes, governs equally " the fall of 
 a leaf and the precession of the equinoxes." 
 
 The investigation of this law is not fitted for this place ; it belongs 
 indeed, to Physical Astronomy, but we propose to touch upon it 
 hereafter, at present we confine our attention to those motions which 
 take place near enough to the surface of the earth to render the vari- 
 ation of gravity inappreciable. We shall shortly see that the ex- 
 pression for tlie force of gravity at the earth's surface is about 32 
 feet, and, from the observation in the note, it appears that at a mile 
 above the surface this value is diminished only by about the 2000th 
 part, which is too small to affect sensibly the circumstances of the 
 motion of a falling body computed on the hypothesis that the force 
 suffers no variation at all. 
 
 On the vertical Motion of heavy Bodies. 
 
 (108.) Let g represent the force of gravity, then, for the space 
 descended by a heavy body in t seconds, we have by (B) the ex- 
 pression, 8=1 gt^ ; and consequently, the space descended in one 
 second is s=lg. Now this space has been ascertained, by very 
 accurate experiments, to be in the latitude of London I6J3 feet, 
 very nearly ;* hence 
 
 16^Vft.=i^.-. 5-=i=32ift.; 
 this, therefore, is tlie expression for the force of gravity at the earth's 
 surface, and in vacuo. 
 
 1 . To determine the space through which a heavy body will de- 
 scend in four seconds at the latitude of London, and also the velocity 
 it will acquire. 
 
 * From the most recent experiments in the latitude of London, the value oi g 
 is found to be 193-14 inches, which is rather greater than 32i feet, this latter 
 being indeed the value of gravity at about the latitude of 45°. The number 32^ 
 is, however, still retained in most of our elementary books, and will serve equally 
 well for the purposes of practical illustration. 
 l2
 
 126 ELEMENTS OF DYNAMICS. 
 
 Using g for F the expression (B) gives for the space 
 s=h £'/»=! 6-r'jX4«=257^ ft. ; 
 also the equation (A^ gives for the velocity i;=g-/=32^x4 = 128| ft. 
 
 2. To determine in what time a heavy body will descend 400 
 feet. 
 
 From(B)/= l-?i=^rp? = 4' 
 ^ ^ ^ g >V/ 321 
 
 76 
 
 77 
 hence the time is 4^f seconds. 
 
 3. If a body be projected downwards, with a velocity of 30 feet, 
 in a vertical direction, how fir will it fall in four seconds? 
 
 By equation (B') s = k gt^'+v^ /= IBy^X 16+30x4 = 3771 
 feet. 
 
 4. A body is projected vertically upward with a velocity of 120 
 feet, how high will it ascend in 3 seconds ? 
 
 Here since gravity retards the motion of the body, it must be 
 considered as negative, and we have, from equation (B') 
 
 s= — i ^^2+t), f= — 16^^X^^+120x3=2151 feet. 
 
 5. To what height above the surface of the earth will a body 
 ascend which is projected vertically upward with a velocity of 100 
 feet? 
 
 It will, obviously, ascend to the same height that it must fall 
 from, to acquire a velocity of 100 feet ; hence, from equation (A), 
 
 100 
 
 ^=5-^ •••^=373-1- =3 -11; 
 
 and from equation (B), s=^ vt=\bbk feet. 
 
 7. With what velocity must a body be projected to reach a height 
 of 579 feet ? 
 
 From equation (C) i> = ^5 2 ^s| = ^/ 5 641x579 1 = 193 feet. 
 
 8. With what velocity must a body be projected downwards from 
 the top of a tower, whose height is 150 feet, so that it may arrive at 
 the bottom in two seconds? 
 
 Calling the velocity v^, equation (B') gives s=5 gt^-\-t\t 
 
 s 150 
 .•.Vj= lgt=—^ 16yVx2=42| feet. 
 
 9 Suppose a body is let fall from a height of 300 feet, and that 
 two seconds afterwards another body is let fall from a height of 200 
 feet, in what time will the former overtake the latter ? 
 
 Let us suppose that the second body will have been in motion x 
 seconds when the first -overtakes it, then the first will have been in 
 motion .r + 2 seconds ; consequently, the space described by these- 
 ''ond will be s=^\ gt^=^\Q^:^^^; and, therefore, the space described
 
 ON RECTILINEAR MOTION. 127 
 
 by the first must be 16 ^Va^^+lOO, but this space is also 
 
 5=1 ^/==16yWa?+2)^ consequently, 16^2^?^+ 100 =16x^(^^+2)^ 
 
 107 
 .-. 100=641 a:+64^.-.a:=-j^; 
 
 hence they will meet if |^ of a second. 
 
 10. How far must a body fall to acquire a velocity of 90 feet ? 
 
 Ans. 125-9 feet. 
 
 11. What space was described in the last second by a body which 
 had fallen 7 seconds ? Arts. 209^^ feet. 
 
 12. With what velocity must a body be projected into a well 350 
 feet deep, that it may arrive at the bottom in 4 seconds ? 
 
 Ans. 231 feet. 
 
 0)1 the Motion of Bodies along inclined Planes. 
 
 (109.) When a body is placed on an inclined plane the force of 
 gravity produces a certain pressure, represented by the weight of 
 the body : if we resolve this vertical pressure P in two directions, 
 the one along the plane, and the other perpendicular to it, the former 
 component will be P sin. ^, taking i for the inclination of the plane 
 to the horizon, and, to prevent the body from moving down, this is 
 the force or pressure which must be counterbalanced. As, therefore, 
 P represents the force of gravity in the vertical direction, and P sin. 
 i the force in the direction of the plane, and moreover, as g repre- 
 sents the vertical acceleration, we shall have for the acceleration 
 down the plane, (see p. 121,) P : P sin. i :: g : g sin. i ; hence 
 the body is urged down the plane by the constant force, ^'=^sin.i; 
 
 and, therefore, substituting in the formulas (106) this value of fif' 
 for F, they will then comprise the whole theory of motion down an 
 inclined plane, whether the body have an initial velocity up or down 
 the plane or not. 
 
 If I represent the length of the plane, and h its height, then 
 
 Tt h 
 
 sin.i=-y; hence the accelerating force is g-j- '•> and, therefore, the velo- 
 city acquired in descending down the whole length /, that is in de- 
 scending through the space s=/, by the influence of this force must 
 be (C), v= v^ {'^S^\ ; which expression, being independent of /, 
 shows that the velocity acquired in descending down all planes of 
 the same height is equal to the velocity acquired in falling through 
 that height. 
 
 The velocities of two bodies, the one falling through the perpen- 
 dicular height, and the other falling through the length of the plane, 
 are respectively i;=g-^, u'=g-/' sin. i ; but, as these velocities are
 
 128 ELEMENTS OF DYNAMICS. 
 
 equal, we must have gt^gt' s'ln.i .'. /=/' sin. i; so that the time 
 of falling through the height is to the time of Hilling through the 
 length, as sin. i to 1. But if we wish to know what extent of length 
 is gone through by the one body, while the other goes through the 
 whole height, then referring to the expressions for the spaces, we 
 have 
 
 5=1. gP=i gn, s'=^gt"' sin. i ; 
 and these also are to each other as 1 to sin. i. If, therefore, from B 
 (fig. 93) we draw the perpendicular BD, AD will be the length gone 
 through by one body, while the other falls through the height AB, 
 because AB : AD : : 1 : sin. ABD=sin. C=sin. i. If we draw the 
 vertical DB' and BB' perpendicular to DB, then the time of falling 
 through DB' would equal the time of falling through DB, but 
 DB' = AB, therefore the time of falling through AB is equal to the 
 time of falling through either of the inclined planes AD, BB'. Hence 
 this remarkable property of the circle, viz. : If from the extremities 
 A, B, (fig. 94,) of the vertical diameter AB, cords be drawn, a body 
 would fail through either of them in the same time that it would fall 
 through the vertical diameter. 
 
 (110.) AVe shall now add an example or two of motion on an in- 
 '•lined plane. 
 
 1. The length of an inclined plane is 60 feet, and its inclina- 
 tion 30°, what velocity would a body acquire in falling down it for 
 2" ? 
 
 Substituting, g sin. i, for F, in the equation (A), we have 
 v=gt sin. z=32ix2x|=32i feet. 
 
 2. How long would a body be in falling down an inclined plani 
 whose length is 100 feet, and inclination 60° ? 
 
 Substituting g sin. i for F, in the equation (B), we have 
 
 / = f : := I r = r-TTT-, ! ^TT = 2*6 SBCOnds. 
 
 \^gsm.i \32ixlN/3 x/116^VX3v'3^ 
 
 3. If a body be projected up an inclined plane whose length is 
 ten times its height, with a velocity of 30 feet, in what time will the 
 velocity be destroyed ? 
 
 The time is necessarily the same as would be required to produce 
 a velocity of 30 feet in a body falling from rest down the same 
 
 plane ; hence making the substitution of ^ -^ for g, in the equa- 
 
 *■ /AN u , •^^ 30x10 
 
 tion (A), we have t = — —= ^ ~, — i=9-3 seconds. 
 ^ ' gh 321x1 
 
 4. A body is projected up an inclined plane whose height is ^th 
 of its length, with a velocity of 50 feet. Find its place, and the 
 velocity, after 6" have elapsed.
 
 ON RECTILINEAR MOTION. 129^ 
 
 Here the force g -j- retards the motion of the body, and must, 
 
 therefore, be considered as negative : hence, from equation (B'), we 
 have 
 
 h 1 
 
 s=v^ t — h g-— T-i^=50x6 — 16 rVx — X36=203^ feet, 
 
 and from equation (A), v= — g—j-t= — 32i x-6-X6= — 32J- feet ; 
 
 .*. 50 — 32i = 17|- feet, the velocity required. 
 
 5. How long would a body be in falling down an inclined plane 
 whose height is to its length as 7 to 15, to acquire a velocity of 20 
 feet? ^ns. 1*3 seconds. 
 
 6. Required the length of a plane whose inclination is 30° that 
 will cause a body let go at the top, to acquire a velocity of 500 
 feet when it reaches the bottom. ^ns. 7772 feet. 
 
 It should be remarked, that in what is here said about motion 
 along an inclined plane, friction is entirely disregarded ; the body 
 being supposed to slide freely down the plane without suffering the 
 least impediment. 
 
 (111.) The two problems following are added as a further illus- 
 tration of the motions of bodies under different modifications of 
 gravity, and, also, as an additional application of the principle stated 
 at (p. 121). 
 
 Problem I. — Two weights W, W^, connected by a thread 
 passing over a small pulley C, as in fig. 95, are placed upon the two 
 inclined planes CA, CB ; to determine the circumstances of their 
 motion. 
 
 The vertical pressure of the whole mass, produced by the force 
 of gravity, is W + W^; the acceleration which would be produced 
 by the same force is g. Again, the pressure of W in the direction 
 WA, or which is the same thing the tension of tlie thread, AV^ C, 
 is W sin. i ; also the pressure of W^, in the direction W^ B, is 
 Wj sin. ij ; hence the system must move in virtue of the difference 
 of these two pressures and to find with what acceleration F we have 
 (p. 121,) 
 
 W + W, : W, sin. i,-W sin. i : g : : ^^ '' V, + W '"'" ' ^=^ ' 
 this, therefore, is the expression for the accelerative force which 
 urges Wj down the plane CB, and which, consequently, draws W 
 up the plane AC ; and, therefore, substituting this expression in- 
 stead of F in the equations (A) and (B), at art. (106), we have, for 
 the velocity acquired and space passed over at the end of i seconds, 
 after the commencement of motion, the expressions 
 
 17
 
 130 ELEMENTS OF DYNAMICS. 
 
 W, sin i, — W sin. I W, sin. /, — W sin. { ,„ 
 
 v= — "■/: s= — s:t' 
 
 W,+ W * ' 2(W,+ VV) ^ 
 
 If the two planes were vertical, then the problem would be to de- 
 termine the motion when the two weights hang vertically at the 
 ends of a tliread passing over a pulley ; since, therefore, in this 
 •ase, sin. i and sin. i^ are each unity, we have 
 
 F W, — W W, — W AV, — W 
 
 If only one of tlie planes were vertical, the problem would be to 
 determine the motion when one weight W^, hanging freely, draws 
 another W up an inclined plane. In this case sin. ii=l 
 W, — Wsin. z 
 
 w,+w * 
 
 If one of the planes were vertical and the other horizontal, the pro- 
 blem would be to determine the motion when Wj, hanging vertically, 
 draws W along a horizontal plane. In this case sin. ?=0 
 
 Problem II. — A given weight Wj is to draw another given weight 
 W up an inclined plane of given height h ; required the length / of 
 the plane in order that the time of ascent may be the least possible. 
 
 The inclination of the plane being represented by i, as usual, we 
 
 have sin. i=-t-> and the above expression for F may, therefore, be 
 written 
 
 hence, by equation (B), the expression for s or the length / is 
 w I W/i 
 
 2 ' \(W,/— W/t)^' 
 
 w^ /+w/ 
 
 this expresses the time when / as well as h is given. To deter 
 mine, therefore, the value of this expression when a minimum, we 
 must put the first differential coefiicient derived from it, equal to 0, 
 / being the independent variable ; or, we may omit the radical, as 
 also the constant factors before differentiating, (Diff. Calc. p. 8,) 
 and we shall theri only have to make 
 
 /« W /_W/t W, W/i 
 
 wr/^^WA=™'"- ••• —7^ =-7 jr=^^- 
 
 W, 2W/i ^ , 2WA
 
 ON RECTILINEAR MOTION. 131 
 
 On the Motions of Projectiles. 
 
 (112.) Altliougli we do not intend to consider the general theory 
 of curvilinear motion in the present section, yet it will be advisable 
 to discuss here that particular case of it which we observe in bodies 
 when projected obliquely into space, near the earth's surface. We 
 know that every body so projected is influenced by two distinct 
 causes, viz. the primitive impulsion of which the effect is to give 
 the body some determinate and uniform velocity in a straight line, 
 and the force of gravity, of which the effect is continually to draw 
 down tlie body in a vertical direction ; these verticals tend to the 
 earth's centre, but throughout the path of a projectile they may 
 without sensible error be considered as parallel. On this hypo- 
 thesis, and abstracting for the present, as in the case of falling bodies, 
 from the resistance of the air, we may easily determine the curve 
 which the body describes. There are, indeed, two methods of 
 solving very readily this problem ; one method is first to express 
 by means of horizontal and vertical co-ordinates the equation of the 
 straight line which the impulsion would compel the body to describe, 
 if gravity did not act, and then to diminish the ordinate y by the 
 deflection which gravity would cause for the time t" . Thus, as- 
 suming the point of departure as the origin of the horizontal and 
 vertical axes, we have, for the initial direction of the body, the 
 equation 
 
 y=ax ; 
 but, in the time t" gravity diminishes this value of y by | gf^, 
 ,•. y=:ax — i gt^ .... (1). 
 
 If Vj be the velocity of projection, v^ wall express the linear 
 space which in the absence of gravity the body would pass over in 
 1" ; hence in t" it would pass over i\ t. Now the action of gra- 
 vity being always vertical, it is obvious that this force cannot at all 
 affect the horizontal advance of the moving body, so that, coitcs- 
 ponding to any time t", the abscissa will be the same whether 
 gravity act or not; but, from what has just been said, this abscissa, 
 in the absence of gravity, is v^t cos. d, o being the angle of eleva- 
 tion of the piece ; hence 
 
 a?=u t cos. 6 .'. t= .... (2). 
 
 t^l COS. 9 
 
 Substituting this value for t, in the equation (1), we have, for the 
 equation of the path, 2/=tan. 0. x . (3); which 
 
 Z V -^ COS. B 
 
 shows that the path of the projectile is a parabola, and that the 
 rectangidar axes are parallel to those of the curve (Anal. Geom. p. 
 183), so that the vertex of the parabola is the highest point of it.
 
 132 ELE>1EXTS OF DYNAMICS. 
 
 Tf h denote the height due to the velocity i',, that is to say, the 
 height from which a body must fall vertically to acquire this velo- 
 city, then since (C) v^=^/\2. gh\ llie equation may be written 
 
 V=tan. e X -—r T—x^ .... (4) 
 
 ^ Ah cos.« e ^ 
 
 The other method of obtaining the equation of the path to which 
 we have alluded is this. Taking the same origin as before, let the 
 direction of projection be taken for the axis of y, and a vertical line 
 drawn downwards for the axis of x ; then v^ being the initial velo- 
 city, as before, we have 
 
 x = l gt\ y=v, t=fy\2 gh\ .... (1); 
 h being the height due to the initial velocity. 
 
 Eliminating f we get 1/^=4 hx .... (2) ; the equation of the 
 parabolic path, and from which it appears that h is the distance of 
 ihe origin, or point of projection, from the focus of the parabola,* 
 and as this is equal to the distance of the same point from the di- 
 rectrix, it follows from equation (1) that the velocity at any point 
 of the curve is equal to the velocity acquired in falling vertically 
 from the directrix to that point. Having thus determined the nature 
 of the path of a projectile, we shall now subjoin a few general 
 problems arising out of this determination. 
 
 Problem I. — (113.) To determine the angle of elevation 9, for 
 which the range AB may be the greatest possible. 
 
 The general expression for the range or horizontal distance is 
 
 the value of x, given by equation (4) for y=0, that is, it is 
 
 x=4 h cos,'' B tan. 0=2 h sin. 2 (1) ; 
 
 and as this expression is to be a maximum, we must have 
 
 dx 
 
 — =4 h COS. 2 6=0. -.6=45° (2); 
 
 (1 9 
 
 which gives from (1) x=2 h . (2), for the greatest range. 
 
 As sin. 2 9=sin. 2 (90° — O), it follows from the general expres- 
 sion (1) for the range, that the range is the same for 90° — 9 as for 
 6, that is the ranges are the same whether the initial direction forms 
 an angle below the line of 45° or an equal angle above it, (fig. 96.) 
 
 Problem II. — Knowing the range of a shot with a given charge 
 of powder and a given elevation of the piece, to determine the range 
 at any otiier elevation. 
 
 Suppose we know the maximum range R, or that due to the ele- 
 vation of 45°, then from equation (2), above, the height due to the 
 velocity of projection, is h=t R ; hence this is the value of h, for all 
 
 • Any point in the path may, obviously, be considered as the point of projection.
 
 ON RECTILINEAR MOTION. 133 
 
 elevations with the same charge. Calling, therefore, the range due 
 
 to any other elevation e, r the expression for its value, will be r = 
 
 R sin. 2 6. 
 
 Thus any range is known by means of the maximum range. Or 
 
 if we know any range r corresponding to the elevation e, then to 
 
 determine the range r' corresponding to another elevation 9' , we 
 
 have the two equations r=R sin. 2 e, ?''=R sin. 2d', 
 
 ,. . _ , r' sin. 2 e' , sin. 2 e' 
 
 to eliminate R ; hence — =—. — r — .'. r =■—. — - — r. 
 r sin. 2 e sin. 2 
 
 Problem III. — Given the angle of elevation and the initial velo- 
 city, to determine the time of flight, and the greatest height of the 
 projectile. 
 
 Returning to equation (1), art. (112), we have, when 2/=0, 
 I2 ax _ I 2 Uni. 9. X 
 
 but, by problem I., x=4h cos.^ 9 tan. e ; hence by substitution 
 t =2 sin. 9 I . . . .(1); which expresses the time of flight. 
 
 To determine the greatest height above the horizontal plane we 
 must find the maximum value of y, from equation (4) art. (112), 
 for which purpose we have the equation 
 
 -^=tan. — =0 .... (2) 
 
 dx 2 h COS. 2 9 ^ ' 
 
 .•. x=2h C0S.2 9 tan. e=h sin. 2 .... (3) ; 
 
 which, by equation (1) prob. I., is half the whole range: putting 
 
 this value for x in the equation of the curve, we have for y 
 
 y=2 h sin.® 6 — h sin.^ 9—h sin.^ 9 . . . . (4), 
 
 which expresses the greatest height. 
 
 If 61=45°, sin.'' 9 = h .'• y=i h, so that (prob. I.), the greatest 
 
 height is one-fourth of the range. 
 
 The expression (2) denotes the tangent of the angle which the 
 
 curve makes with a horizontal line at any point (a:, y). 
 
 Problem IV. — Given the initial direction to determine the velo- 
 city, so that the projectile may pass through a given point. 
 
 Let [x', y') be the given point, then by the equation of the curve 
 (p. 131), 
 
 s^x'^ X 
 
 V =tan. 9. x' ^= .•, v.= 
 
 g- 
 
 2v^^cos.^9 ' ' ^ COS. 0% 2 (tan. e- x' — y') 
 
 Problem V. — When the velocity of projection is given, to deter- 
 mine the direction so that the projectile may pass through a given 
 point. 
 
 M
 
 134 ELEMENTS OF DYNAMICS. 
 
 By substituting in equation 4, art. (112), sec. '9, or rather 1 + 
 
 tan.'fl, for 
 
 1 , l+tan.= e ,„ 
 
 ;,— , we have 7y'=tan. 9. x' — ; x " 
 
 cos.*» *^ 4 A 
 
 tan.'' 9 tan. 9=. 1 ; this quadratic solved 
 
 x' x"^ 
 
 4 h Ah y' 
 
 x' ' x"^ 
 
 for Ian. gives tan. 9= ^-^ ; • . . . (1) ; 
 
 so that there are two difTerent directions whenever the problem is 
 possible, except when 'ih^=4lnj' — x"^, or (2h — y'y=x'"-\-y'^, in 
 which case there is but one direction, but when 
 
 {2h—y'y>x'-'+y", 
 the problem becomes impossible under the proposed conditions. 
 
 The time elapsed from the instant of projection till the projectile 
 reaches the proposed point is, by equation (1), art. (112), 
 \2y' + 2Um.9.x'\ 
 
 '=4- 
 
 Problem VI. — To determine the range on an oblique line passing 
 through the point of projection, and also the time of flight. 
 
 Let i be the inclination of the oblique line to the horizon, then 
 its equation is ^'=tan. i. x' . . . . (1); combining this with the 
 equation of the projectile we shall obtain the abscissa of the point, 
 where it meets this line, by the equation 
 
 • , , ^''^ 
 
 tan. t . X =tan. 9. x ; -— 
 
 4 A COS.* 9 
 
 •v 4 A COS. e sin. (e — i) 
 
 .'. x'=4h COS.'' 9 (tan. e — tan. i)= ^-^^ -. . .(2); 
 
 ^ ^ cos. t 
 
 and, consequently, the oblique range will be 
 
 x' 4h COS. 9 sin. (9 — -i) ,„, 
 
 r= := —7-^ . . . . (3) ; 
 
 COS. I COS.'' t 
 
 and the time may be found from equation (2), last problem, by sub- 
 stituting for x' and y' the values (1) and (2) in this. This substi- 
 
 /tan. — tan. i) 8 /j COS. e sin. (e — i). 
 tution gives t=s/\- ■ ^ -\ 
 
 ^ * g COS. I 
 
 COS. I g 
 
 Problem VII. — To determine the greatest range on an oblique 
 plane, and the greatest height above it. 
 
 The angle of elevation, which belongs to the greatest range, will 
 be that which renders the expression (3), last problem, a maximum, 
 or, since i is constant, we must have
 
 ' ON RECTILINEAR MOTION. 135 
 
 2 COS. e sin. (9 — i)=max. 
 
 =sin. \d + {9 — i)\ — sin. {e — {o — i)] 
 
 =sin. (2 6 — i) — sin. i 
 
 .-. sin. (2 9— z)=max. .-.2 9 — i=90° .-. e = k (90°+i). 
 
 Putting, therefore, this value of 6 in the expression (3) for the 
 
 range, we have for the maximum range R 
 
 P_ 4 h COS. h (9D°+0 sin. | (90° — i) 
 
 cos.^ i 
 
 2h(\ — sin. i),^ ,, , m • ^„^ 2h 
 
 =—r ^-z^(Bt. Young's Trig, art 26) = ~^- 
 
 To determine the greatest height M'P above AB' (fig. 97), we 
 must make PM — MM' a maximum, or, since MM'=a;tan. i, 
 
 M'P=tan. e. X — —5 • tan. ia?=max. 
 
 4 h cos.*^ Q 
 
 X 
 
 .'. tan. — — T tan. 2=0 
 
 2 h cos.39 
 
 _, „ . " -N 2 A cos. 5 sin. (5 — i) 
 
 .'. x=2 h cos.2 9 (tan. e — tan. i) = 7-^^ . 
 
 cos, I 
 
 This value of x being substituted in the above expression for M'P, 
 
 , . , . . sin. (e — i) . , 
 
 which, since tan. e — tan.z= ^^ -, is the same as 
 
 cos. 9 cos. I 
 
 X sin. {9 — I) X 
 
 cos. 9 COS. % 4 h COS. 9 
 
 gives for M'P, when a maximum, the value 
 
 -^ _2 A sin. (9 — i) sin. {9 — i) sin. {9 — *')>_'* ^i"-^ (^ — **) 
 
 COS. i * COS. i 2cos.i cos.^ £ 
 
 (114.) Collecting together the principal results of the preceding 
 
 propositions, we have the following formulas : 
 
 I. When the Plane is horizontal. 
 
 f9 h 
 
 Range=2 A sin. 2 9, time=2 sin.e-^ — 
 
 Greatest range =2 h 
 Greatest height =h sin.^ 9. 
 
 II. When the Plane is oblique. 
 
 — ^ -COS. esin. (9 — i) 
 Range =4 h ^ 
 
 ^ COS.^ t 
 
 COS. t ^ £•
 
 136 ELEMENTS OF DYNAMICS. 
 
 2/j 
 
 Greatest ranffe=- 
 
 l-|-sin. i 
 
 ^ ... hsu\*(d— i) 
 
 Greatest heiffht= ^--; — - . 
 
 ^ cos.'i 
 
 These equations contain the whole tlieory of projectiles in vacuo ; 
 tliey may all be deduced, independently of analysis, by the aid of 
 common geometry, and a few well known properties of the para- 
 bola. See the second volume of Br. Huttoii's Course of Mathe- 
 matics. 
 
 CHAPTER III. 
 
 ON THE RECTILINEAR MOTION, PRODUCED BV A VARIABLE FORCE. 
 
 (115.) We shall now proceed to show the application of the 
 general formulas, at art. (106), to cases of rectilinear motion, pro- 
 duced by forces varying in intensity according to some known law. 
 This variation is generally according to some function of the dis- 
 tance of the moving body from the fixed point, which is regarded as 
 the centre of force, although, in some cases which nature presents, 
 the variation is also dependent upon other circumstances ; as, for 
 instance, when the motion takes place, not in free space, but in a 
 resisting medium, where, it is obvious, the body will be hindered 
 from obeying the full influence of the attracting force hj-^ a resisting 
 force, varying in some manner with the velocity. These particu- 
 lars will be considered in prob. III. 
 
 Problem I. — (116.) To determine the vertical motion of a heavy 
 body towards the earth ; the force of gravity varying inversely as 
 the square of the distance from the centre. 
 
 Call the radius of the earth r, the distance of the body from the 
 centre at the commencement of motion a, and the distance at any 
 time r', after the commencement a; ; then by the hypothesis the 
 intensity of the force F, at the time t", will be given by the propor- 
 tion 
 
 — • — ::<>•: F .". F= — i?"= .* 
 
 Having got an expression for the force, the next object is to de- 
 duce that for the velocity. Referring to equation (D), we have 
 
 |.2 p. Ay. 2 r^ £* 
 
 J = — |-C ; the constant C will depend upon 
 
 • See note in next page.
 
 ox RECTILINEAR MOTION. 137 
 
 the initial velocity of the body, that is, upon the velocity which it 
 has at the distance a, where gravity begins to act ; if this velocity 
 
 is 0, then 2__|_C==0 .-. C= 2_ ; and thus the velocity 
 
 a a ^ 
 
 of the body has at any time t", that is, after having fallen from the 
 
 distance a to the distance x is completely determined, it is v^=: 
 
 2r^ g 2r^ g 2r^ g(a — x) .1 , j 
 
 —= ^-^ ; and when the body arrives at the 
 
 X a ax ■' 
 
 surface of the earth, that is, when x-=r, it will have acquired a 
 velocity expressed by 
 
 ,1rg{a — r) , • , .^ ■ ■ a ■. ■ « — '^ 
 v=^ — 2-):^ ; which, II a is infinite, since 
 
 is then 1, becomes ?;=v' 2 rg; so that the velocity can never be so 
 great as this, however far the body may fall, and, hence if it were 
 possible to project a body vertically upwards with this velocity, it 
 would go on to infinity and never stop. Of course this is on the 
 supposition that there is no resisting medium nor other disturbing 
 force. Taking the radius of the earth at 3965 miles, the last ex- 
 pression for V will be t) = 6*9506 miles ; so that if a body were to 
 he projected upwards, with a velocity of about seven miles a second, 
 and were to experience no resistance, it would never return to the 
 earth. 
 
 It remains now to determine the time t" ; and for this purpose we 
 
 have the equation -jj-z=v=.r\ _ ; from which we get 
 
 dt ax 
 
 s/a 
 
 rv/2o-\ a — X 
 
 ■dx 
 
 t=:r^^^f^=~-dx; 
 
 r-s/l g y/\ax — x'^\ 
 s/a p — X 
 
 'Tx/'ig J Vax — x^ 
 this integral may be immediately found by means of the general ex- 
 — 
 
 * In the expression — for the velocity at (p. 123), s is the space passed 
 
 d (a — x) dx 
 
 dt ~ dt '■ 
 dx dv </2 X 
 
 case, V = — — , and — = F ^ — "Ti"' ^^^' generally, whenever F tends to di- 
 
 ..-.,. ds 
 
 minish the space s, the increment A s being always negative, the expression — , 
 
 d^s 
 for the velocity, as also the expression -j-r- for the force must, obviously, be like- 
 
 dt^ 
 
 wise negative. 
 
 h2 18
 
 138 ELEMENTS OF DYNAMICS. 
 
 pression at the top of page 46 in the Integral Calculus, or we may 
 
 proceed thus : to the numerator of the expression to be integrated 
 
 add 2 adx, and then subtract the same quantity from it, and we 
 
 shall thus convert the differential into two others, of which one 
 
 will be immediately integrable by the rule for powers ; thus we 
 
 iadx — xdx ^ adx 
 
 shall have the two expressions — — — ; 
 
 ^ x/\ax—x-'\ y/{ax—x''\ 
 
 the integral of the first of these is obviously ^\ax — x"] ; that of 
 
 the second is the elementary integral, 5 a coversin."^ — x (Int. 
 
 %*' I i/iiA^ ii ' 2x a 
 
 J Calc. p. 10), or, which is the same thing, 5 a cos.-^ ( ). 
 
 Consequently 
 
 t= ,^ x/ox — a^^ + l acos. -1 ( ) ; 
 
 which expresses the number of seconds elapsed in moving from the 
 distance a to the distance x, from the centre of attraction. This 
 expression needs no correction, because a — x, and t become at 
 the same time. When x — r, that is, when the body arrives at the 
 surface of the earth, the number of seconds elapsed will be 
 
 ^=-^-- ^/ar — r'' +^fl cos. "»( ) ; 
 
 which is evidently infinite, when a is, although the velocity, as we 
 have before seen, is finite. 
 
 If the attracting body be considered as merely a point, then x may 
 at length become 0, and the whole time of falling to the centre from 
 the distance a will be expressed by 
 
 t= — — — xiacos. 1 — 1=- 717— a-. 
 
 r^2g 2rV2^ 
 
 If the body fall from any other distance a' the expression for t would 
 be similar, so that t^ : t'^ : : a^ : a'^, that is, the squares of the times 
 of falling from rest to the centre offeree are as the cubes of the 
 distances from which they fall. 
 
 Problem II. — (117.) To determine the motion of a body at- 
 tracted towards a fixed centre, the force varying directly as the dis- 
 tance. 
 
 This is the law of force which would attract a material point at 
 liberty to move along a perforation from the surface to the centre of 
 the earth. For the universal law of nature being this, that every par- 
 ticle of matter attracts with a force varying in intensity inversely as 
 the square of the distance at which it acts, it follows that the at- 
 tractive forces of homogeneous spheres must vary directly as their
 
 ON RECTILINEAR MOTION. 139 
 
 masses, or as the cubes of their radii, and inversely as the squares 
 of the distances of their centres from the attracted point. Now it is 
 shown, by writers on Physical Astronomy, that the attraction of a 
 sphere is the same as if its entire mass were concentrated in its cen- 
 tre ; it is shown, moreover, that a particle placed any where within 
 a spherical shell will remain at rest, being equally attracted in all 
 directions. If, therefore, a particle be placed below the surface of 
 the earth, and at the distance x' from the centre, it will be moved 
 only by the force which resides in the inner sphere of radius x, as 
 it is kept at rest by the influence of the shell, whose thickness is 
 r — x; hence, from what has just been said, 
 
 which shows that the force varies directly as the distance x. Having 
 thus got the value of F we have, as in last problem, 
 
 ^t;2= f 5-xdx^—-^X^ + C. 
 
 «/ r 2 r 
 
 If the body be merely dropped into the hole, v will be 0, when x=7-, 
 
 .-. 0= — ^r^+2 C .-. C = h gr .: V =J^{r^—x^) - (1); 
 
 which is the velocity of the body at any distance x from the centre 
 when the body reaches the centre, that is, when a:=0, the velocity 
 isv=^\gr\'---{2). This velocity must be spent before the body 
 will stop, and as the motion after passing the centre will be retarded 
 according to the same law, as it was before accelerated, the body 
 will continue to move till it reaches the opposite point of the earth's 
 surface, where it would stop ; but being again attracted by the 
 same force as at first, it will return and pass through the centre to 
 the point of departure, and will thus move backwards and forwards 
 continually. 
 
 To determine the time from the departure of the body till its arri- 
 val at the distance x from the centre, we have, as usual, 
 
 -^=''=J7('"---) ■••*=■ 
 
 r dx 
 
 •V — - 
 
 g V\t'—x^] 
 ,r p dx T X 
 
 S J ^\r^—x^\ g r 
 
 which requires no correction, since ^=0, when x=r. For the time 
 of reaching the centre we have, by making a?=0, 
 
 By making x=- — r we have, for the time of passing through the
 
 140 ELEMENTS OF DYNAMICS. 
 
 whole diameter t-=Tty/ — •••• (4) ; which is twice the time of fall- 
 
 ing to the centre as it ousjht to be. 
 
 If the body do not bepin to move from the surface of the earth, 
 but from some point within it at the distance r', instead of r, from 
 
 the centre then — r', will be the force at the commencement of mo- 
 r 
 
 tion instead of g-, and, therefore, substituting this for g, and r' for r, 
 the expression for t will be ;=v'— xcos. -*— ; and, consequently, 
 whena:=r', we have for the time of reaching the centre 
 
 the same as the expression (3). Hence, at whatever point within 
 the surface of the earth the body be placed, it will reach the centre 
 in the same time. 
 
 In order to find this time, take the radius r, of the earth, equal 
 to 3965 miles, and we shall have r '=21' 7"i, which will be the 
 time occupied in passing to the centre, however near to it the body 
 be placed. 
 
 It is obvious that if g represented the energy of any other force, 
 instead of gravity, at the distance r from the centre, the reasoning 
 and conclusions would be the very same, so that when a body is 
 attracted from a state of quiescence by any centre of force, varying 
 in intensity directly as the distance, the whole time of passing to' 
 the centre will be the same from whatever point the motion com- 
 mences, whether from a point infinitely distant, or from a point in- 
 finitely near. Hence the body would pass over an infinite space in 
 a finite portion of time, but then, by hypothesis, the force at the 
 commencement of motion must be infinitely great. 
 
 We shall now consider the motion of a body near llie surface of 
 the earth, taking into account the resistance of the air, which we 
 have hitlierto neglected. 
 
 Problem HI. — (118-) To determine the vertical motion of a 
 heavy body near the earth's surface, considering the resistance of 
 the air to vary as the square of the velocity. 
 
 If we represent the resisting force at any time t'\ after the com- 
 mencement of motion by f, and the velocity generated by u, then, 
 by hypothesis f=niv^, m being constant for all velocities, and 
 which can be determined only from experiment. Hence the force 
 F, accelerating the body, is F:=^ — inv^ ; so that here we have F 
 as a function of the velocity, and not of the space as heretofore ; 
 therefore, since
 
 ON RECTILINEAR MOTION. 141 
 
 .^. ^ dv dv ^ dv 
 
 (D)F=-y-, we have - =g — mv^ .-, dt=- 
 
 dt dt g — mv^ 
 
 The second meniber of this equation may be integrated by the me- 
 t!iod of rational fi&tions ; or we immediately see that the expression 
 
 • , / ^^^ , dv \ I 
 
 IS the same as ( — — - — -^ 1 ; ;— ) x — — i-; consequently, 
 
 \ gi-\-mi V gi — imv/ 2 g-3 
 
 / P dv n dv \ 1 
 
 \J gk-\-mh V *^ gh — mk v) 1 g\ 
 
 1 -^ "'^'- '■ 
 
 {log. (^+m3 u) — log. (^ — w^v)|+C 
 
 1m\ gh 
 
 = '°g-^^r ••••(!); 
 
 the constant being 0, because the time and the velocity begin to- 
 gether. 
 
 This determines the time of the motion necessary to generate a 
 given velocity. To find the velocity when the time is given it will 
 be necessary to disengage the equation from logarithms, putting, 
 therefore, e for the hyperbolic base, we have, since log. e=l, 
 
 o. J, , 1 gi+mhv it\/ mg gh+mhv .. 
 
 'It y/ nig log. e=log. ^ i — .-. e —-^ , — . (2) : 
 
 from which v may be found when t is known. 
 
 Since e exceeds unity, the first member of this equation increases 
 with t, and is infinite when t is, consequently, as t approaches to 
 infinity the denominator gi — mkv, must approach to 0, but when 
 
 a- 
 
 it is actually 0,u= v' -- • (3); so that the longer the body is in mo- 
 tion, that is, the greater the space through which the body moves 
 in a medium, varying in resistance as the square of the velocity, 
 and, towards an attractive force, varying in\-€rsely as the square of 
 the distance, the nearer will the velocity approach to constancy. 
 
 Having found the velocity we may readily determine the space s, 
 through which the body has passed to acquire it. For, by (D), 
 vdv=Fds=-{g — mv^) ds. 
 
 vdv 1 , . V ^ 
 
 -log. {g — 'mv^)-\-C. 
 
 '=/- 
 
 g< — mv^ 2m 
 
 To determine C put s=0, then v=0, 
 
 .'. 0= log. 2-+C .-. C=- — log. g 
 
 1 mv^ 
 
 .'. s= log. (1 ). 
 
 2m ^ ^ g ^
 
 142 ELEMENTS OF DYNAMICS. 
 
 If the space be already known the acquired velocity may be found 
 by this equation. 
 
 If the body is projected with a velocity v , in the resisting medium 
 in opposition to the force, then the motion becomes retarded, both 
 gravity and the force ^nr" conspiring to stop the body ; hence the 
 tttardive force is F= — g — mv^ 
 
 dv . ,, dv 
 
 dt ° g+mv" 
 
 V gm ^ s 
 
 We may determine C from tlie circumstance that at the com- 
 iiiencement of motion, that is, when <=0, v^=v^, so that 
 
 0=--L^ftan.-J-^i,,|+C; 
 V gm ^ S 
 
 therefore, subtracting the former equation, from this, we have 
 
 t=- 
 
 0^m ^ g ^ g 
 
 ^/ gm yg \g 
 
 from which equation v may be determined for any proposed time 
 t". It remains now to deduce the expression for the corresponding 
 space ; for this purpose we must employ the formula (E), which 
 gives 
 
 /v , /» vdv 
 -=- dv .'. — s= I ; 
 
 that is, — s=- — log. (fi'+mv*) + C. As s=0, whenu=y,, we 
 2m 
 
 have 0= — log. {g^mv,^)+C 
 
 1 Iff -l-wio ' 
 
 .-. s= llog. (g-\-mv, ") — log. (g+mv')=- — log. \^—^ — ^L 
 
 2m ' ^ ^° ^ ' ^ ^^ '1m ^ ^g frnv"" * 
 
 When the retardive force has destroyed the motion, that is, when 
 
 1 7HV ^ 
 
 v=0 we shall have s=- — log. \l-\ —} ; which expresses the 
 
 height to which the body will reach when projected with the velo- 
 city V. It will not acquire so great a velocity in returning to the 
 earth, because the accelerating force is only g — mv^.
 
 THE MOTION OF A FREE POINT. 143 
 
 SECTION II. 
 ON THE THEORY OF CURVILINEAR MOTION. 
 
 (119.) We now come to discuss the general theory of the mo- 
 tion of 2. free point, or material particle, independently of any re- 
 striction as to the nature of the path it is compelled to take. 
 
 We say point instead of body, because we do not propose to 
 take into consideration, in the present section, any circumstances 
 of the motion which may be dependent upon the mass of the 
 moving body. Whenever, therefore, in the course of this section, 
 we speak of the motion of a body, it must be noticed that we con- 
 sider the acting forces to apply themselves equally to all the par- 
 ticles of the body, and that these particles exert no power them- 
 selves sufficient to modify the motion. This, indeed, is the 
 hypothesis upon which the investigations in the preceding section 
 are founded ; where we have considered the motions of bodies 
 chiefly in reference to the force of gravity. 
 
 But, in fact, as already hinted at (118), all bodies in nature exert 
 a mutual influence on each other, and the intensity of this in- 
 fluence varies with the mass from which it emanates. Two bodies 
 then, M and m, at liberty to obey their mutual attractions, ap- 
 proach each other in virtue of the force resident in M, combined 
 with the force resident in m ; and, therefore, the distance of their 
 centres at any time will be the same as if the sum of the attractions 
 due to M and m were combined in M, and, instead of the other 
 body m, a single particle were placed at its centre ; so that we 
 should thus have to consider only the motion of a single particle or 
 free point acted on by a single centre of force, viz. the centre of 
 M's attraction. We may, therefore, after having attributed a proper 
 value to the attractive force tending to M's centre, disregard the 
 mass of the moving body m ; and when, in the course of the present 
 section, we speak of the motion of a body about a fixed centre of 
 force, we consider the influence of the moving body to have been 
 transferred to that centre as above, and, therefore, the body to move 
 as a free point. 
 
 Let us now proceed to investigate the general equations of mo- 
 tion.
 
 144 ELEMENTS OF DVNAMlCS. 
 
 CHAPTER I. 
 
 ON THE GENERAL EQUATIONS OF THE MOTION OF A FREE POINT. 
 
 (120.) When a material particle moves in a curve line, it is ob- 
 vious that its direction at any point of its path is in the tangent at 
 that point, alonff which, if it were there left to itself, it would pro- 
 ceed with a uniform velocity : and its curvilinear course is kept up 
 only by the continual influence of some force or forces which at 
 every instant deflect it from the rectilinear course it tends to pur- 
 sue, in virtue of its inertia. 
 
 As far as efiects are concerned we may consider a body thus 
 moving to be impelled along the curve by an accompanying force, 
 varying in intensity conformably to the circumstances of the mo- 
 tion, and we know that the value of this force at any time t", 
 during which the arc s has been described, will be expressed by 
 
 ~dF' 
 If this same force had a statical effect, or which is the same 
 thing, if it were directly opposed by an equal force F, we might 
 then, instead of the first force, substitute three others acting on the 
 point and in the directions of three rectangular axes ; if o, (3, y, be 
 the angles which these form, with the line along which the point 
 tends to move, then the values of these three forces will be 
 
 F cos. a, F cos. ^, F cos. y ; 
 so that tlie point will have the same tendency to move under the 
 influences of these three forces as under the influence of the ori- 
 ginal force F ; these, therefore, are fitted to produce the same ac- 
 
 celeration -r— as F ; let us see what acceleration each alone is fitted 
 or 
 
 to produce. In order to this let us first remark that since 
 
 dx (is dx d^ s d^ x , . 
 
 -— =C0S. a, -pCOS. a =-p.'.— 7— COS.a =— :— . (1); 
 
 ds dt dt df' dt"" ^ ' 
 
 then, by the principle stated at p. 121, 
 
 -, ^ d^s d^s d^x ,„, 
 
 r : t COS. o : : —r— : -;— cos. o= — — .... (2) ; 
 
 d^ X 
 
 hence the acceleration which F cos. a is fitted to produce, is -^-^, 
 
 and, in like manner, the accelerations due to F cos. /3 and F cos. y 
 
 rf^V .d'z 
 are —r— , and -r-^ ; x, y, and z being the co-ordinates of the point
 
 THE MOTION OF A FREE POINT. 145 
 
 at the instant t". Thus tlien the consideration of the curvilinear 
 motion of a material point in space is reduced to the consideration 
 of the rectilinear motions of its three projections along three rec- 
 tangular axes, and which describe the rectilinear spaces x, y, s, 
 Avhile the body itself describes the curve s. Calling the forces 
 along these axes X, Y, and Z, we have 
 
 and these are the general equations of the motion of a free point. 
 
 mi T • • ,.1 • • dx dy ,dz , , 
 
 Ihe velocities of the proiections are-r-, -r^, and^-, and the ve- 
 ^ "^ dt dt dt 
 
 ds 
 locity of the point itself is-^, but (Diff. Calc. p, 215,) 
 
 rfs" _ dx"" +dy''-i- dz^ , , ^ 
 
 IF ~ df^ ""^ ^ ' 
 
 from which expression it follows that if the lines which represent 
 
 the velocities -7- , -^, -r-, be taken for the edges of a rectangular 
 dt dt dt ^ ^ 
 
 ds 
 parallelopiped, the diagonal of it will represent the velocity— of 
 
 the point in space ; and it equally follows from the equation (2), 
 
 combined with the two similar equations furnished by the other 
 
 projections, that if the lines which represent the accelerative forces 
 
 d^ X d^ y d^ z 
 
 —fji -yf 5 -T-j, be taken for the edges, the diagonal will represent 
 
 d^ s . 
 the accelerative force y^, which acts upon the point in space. 
 
 It thus appears that the velocity which a body actually has may 
 always be decomposed into three, directed according to three rec- 
 tangular axes ; and the accelerative force, by which a body is ac- 
 tually influenced, may always be decomposed into three, directed 
 according to three rectangular axes ; and conversely such a system 
 of velocities, or of forces, may be always compounded into one. 
 When the motion is in a plane curve, one of the components is of 
 course 0. 
 
 If we differentiate the equation (4), relatively to the independent 
 variable t, we shall have 
 
 ds^ d^'x. dx-\-d^y . dy-^d'^z. dz 
 
 "^IF-^ dP ' 
 
 but, in virtue of equations (3), the second member of this equation is 
 
 2{^dx+Ydy+Zdz); 
 consequently, returning to the integral, we have 
 N 19
 
 146 ELEMENTS OF DYNAMICS. 
 
 (1) . . . . ^=,«=2/(X (/^ + Y ch, + Z dz) .... (2). 
 
 We infer, therefore, that when the component forces X, Y, Z, are 
 known functions of the co-ordinates .r, y, z, for every point of the 
 body's path, or trajectory, as it is called, and when, moreover, 
 these functions are such as to render \dx-\-\dy-\-'Aiiz an exact 
 differential,* then the velocity of the moving body will be deter- 
 minable from this equation. The complete integral of (2) will in- 
 volve an arbitrary constant, which can only be determined from 
 previously knowing the velocity at some known point of the tra- 
 jectory, which is the same as if we knew the point at which the 
 motion commenced. 
 
 Suppose the general integral of (2) were 
 v'^f{x,y,z,) + C; 
 if we knew that the velocity, at the point (a, b, c,), were t\ wc 
 should have v^"=f(a, b, c,)-j-C 
 
 .:v'-v,'^=f(x,y,z)-f{a,b,c). 
 From this equation it appears that when we know the functions that 
 X, Y, Z, are of x, y, z, and the velocity at any point (a, b, c), we 
 may find tlie velocity at any other point [x, y, z,) merely from 
 knowing its co-ordinates, without requiring to know either the form 
 of the curve between the two points (a, b, c), (.r, y, r), or the time 
 of describing it ; in bodies, therefore, which move in curves, re- 
 turning into themselves, the velocity is always the same at the same 
 point. 
 
 (121.) It is an important fact that tlie differential in equation (2) 
 is always exact whenever the body moves under the influence of a 
 force emanating from a fixed centre, or, indeed, when any number 
 of fixed centres act on the body, provided always that the intensity 
 of each force is a function of the distance of the point, or body, on 
 which it acts. 
 
 Let there be but one such centre, then placing the origin of the 
 co-ordinates there, and calling r the distance of the moving body 
 from it, at any point of its path, we may express the force acting 
 on it by/r, the form of the function / being known ; and, by re- 
 solving this force according to the three axes, we have the compo- 
 nents 
 
 X=./r.f Y=/r.f Z=/r.f 
 
 Substituting these expressions in the general equation (2) we have 
 
 * In order to this these functions must satisfy the conditions 
 dX dY (IX iJZ dY dZ 
 
 -^=^'-^= di' 17=^' ^^'' ^«'- ^'^^ P- '"^-^
 
 THE MOTION OF A FREE POINT. 147 
 
 r2==2//r . Z^L_^_Z ; but since r^=::X^+y^+z^ .-. rdr 
 
 = x dx-{-y dy-\-z dz, therefore, by substitution, ij^ = 2/"/)' . dr 
 .... (1) ; so that the velocity is always determinable. 
 
 The same may be shown generally as follows : 
 
 Having chosen the axes of reference, let the co-ordinates of one 
 
 of the fixed centres be a, b, c, and let R be the force exerted by this 
 
 centre on a point at the distance r from it. Now the cosines of the 
 
 angles which r makes with the axes are, severally, 
 
 X — ci xi "— h z '•"—* c 
 
 , , : so that the three components of R are 
 
 r r r 
 
 x— a ^y — b z—c 
 r r r 
 
 In like manner, for the components of R^, R^, &;c., the forces simul- 
 taneously acting on the point from a second, a third centre, (fee, 
 
 we have R, -, U,- -, R, 
 
 T r r 
 
 nf-4^, Ry--=^, E.17^ 
 
 r„ r„ Ta 
 
 R,£=±., R„^_=^, R.i=:fa; 
 
 n n n 
 
 so that if we add together each of these three vertical columns of 
 components, we shall have the expressions for X, Y, and Z, to be 
 introduced into the general formula (2). But, before performing 
 this addition, we should remark, that since 
 
 ... r„ rfr„=(.r— a„) dx-\-(y—b^) dy + {z — cj dz; 
 consequently, if we divide this equation by i\, and multiply by R„, 
 and then put n successively equal to 0, 1, 2, &c., we shall have 
 
 Rrfr=R f/.r+R ^ dy+- dz 
 
 r r ^ r 
 
 R, (7r,=R, ^ dx+Yi^ y- — ^ dy+- — ^ dz 
 
 R„ dr=n,,'^~^dx+^y—^dy+^—-^ dz 
 
 .-. R(/r+R,rfr,... + R„rfr„= Xdx + Y dy +Zdz; 
 hence, if R„ be a function of r„ whatever be n, the first member of 
 this equation will be always integrable, and Ave shall have 
 v'^=2fIldr+2fR^dr +2/R„rfr„
 
 148 ELEMENTS OF DYNAMICS. 
 
 (122.) We shall notice in this place a remarkable property con- 
 nected with the motion of a body about a single centre of force, viz. 
 that tlie trilineal spaces described by the radius vector or line joining 
 the moving point and fixed centre are to each other as the times of 
 describing them, and this whatever be the law according to which 
 the intensity of the attractive force varies. 
 
 When a body acted upon by a single centre of force moves in a 
 curve line, we may consider such motion to arise from a primitive 
 impulsion given to the body which would alone have caused it to de- 
 scribe a straiglit line, but being continually acted upon by a force out 
 of this line it is deflected from this path at tlie very commencement 
 of motion, leaving it a tangent to the path it actually takes at the 
 point of projection ; as moreover nothing draws the body out of 
 the plane in which the centre of force and line of projection are 
 situated, the path of the body must be a plain curve. Hence, placing 
 the origin of the rectangular axes at the centre of force S (fig. 98), 
 and taking P for the place of the body at any time t", we have 
 
 X= — F cos. a, Y= — F cos. p 
 
 r r 
 
 the negative signs being used because F tends to diminish the spaces 
 X and y ; consequently the equations of motion are 
 d^ X X d" y y 
 
 To eliminate F multiply the first of these equations by y and the 
 
 second by x, and subtract the products ; there results 
 
 yd' X - xd' y _^ 
 
 TlF ~" • • • • ^^' 
 
 an expression altogether independent of the value of F, so that 
 
 whatever results from it will hold even when F is repulsive. 
 
 Now it is easy to perceive that the numerator of this expression 
 
 is the differential of ydx — xdy, therefore, multiplying by dt and 
 
 iidx xdu 
 
 integrating, we have - — -—^ —= C .... (2) 
 
 .-. fydx — f xdy=:Ct-\-Q^, 
 or2fdyx — xy=Ct + CK . . . (3). 
 Let us inquire into the geometrical signification of the first member 
 of this equation. The term 2 fydx obviously expresses twice the 
 area SP'PM, and the term xy is twice the triangle PSM, conse- 
 quently the equation (3) is the same as 
 
 2 sector SP'P=C/+C,. 
 Suppose t" to commence when the body is at P', then since when 
 /— -0, the sector=0 .'. C, =0, consequently
 
 THE MOTION OF A FREE POINT. 149 
 
 c,^ r^. ^. sector SPP' ,_ 
 2 sector SP F'=Ct .-. ==| C, 
 
 that is, as announced above, the sectors described are proportional 
 to the times of describing them, and therefore equal areas are de- 
 scribed about S in equal times: this remarkable and important 
 property is called the general principle of equal areas. 
 
 (123.) In order to complete the theory of curvilinear motion, it 
 may be as well to repeat here the general expression given at (103) 
 for the force fitted to produce the motion which a body actually has 
 at any instant if immediately urged by that force in the direction of 
 its motion, that is, in the direction of a tangent to the curve at the 
 point where it is at that instant. Calling such a force S, its value is 
 
 S = ^alsoi w''=/Srf5. 
 
 The force S may be called the tangential force ; we see from the 
 second equation that upon this force the velocity of the body in its 
 path wholly depends, and it is wholly expended in producing this 
 velocity ; if, therefore, all the forces which influence the body at any 
 particular point were decomposed, each into two, one in the direc- 
 tion of the tangent, and the other in the direction of the normal, the 
 sum of the former components, that is, the tangential force, would 
 determine the velocity and direction of the body's motion at that 
 point. It follows, therefore, that if among the forces which act 
 upon a body there be any which always act in the direction of a 
 normal to its path, the components of these must necessarily destroy 
 each other in the expression (2), (p. 146,) for the velocity along the 
 curve ; because, as just observed, this velocity is wholly due to the 
 tangential force. 
 
 (124.) Before closing this preliminary chapter, we shall briefly 
 show how the equations of motion investigated in the preceding 
 section, are to be deduced from the more general theory laid down 
 in the present chapter. 
 
 As a first application, let it be required to determine the motion 
 of a point moving from the effect of an impulsion only, then, as 
 there is here no acceleration, the equations of the motion are 
 
 d'x d'y d-z 
 dt^ 'HF 'dt^ ' 
 
 multiplying each by dt and integrating, we have, for the velocities 
 in the directions of the axes, the expressions 
 
 dx dy . dz 
 
 dt dt dt 
 and therefore (equation 4, p. 145,) the velocity along the path is 
 n2
 
 150 ELEMENTS OF DYNAMICS. 
 
 ds 
 
 v=x/\a'*+b'+c'^'\ which is constant ; that is, — =C .-. s=C/ + C,, 
 
 at 
 
 Cj being the space already passed over when /" commences. 
 
 From equations (1) we can prove that the path of the body must 
 necessarily be a straight line ; for, multiplying each by dt and inte- 
 grating, they give x^=(tt-\-u', y=bt-\-b', z=ct-\-c', by means of 
 which eliminating /, and there results the following general rela- 
 tions among the co-ordinates, viz. 
 
 a a'c — ac' b , b' c — be' 
 c c ^ c c 
 
 As a second application, let it be required to determine the motion 
 
 of a projectile in space. Here the equations of the motion are 
 
 d'' X d'^y 
 
 -J— =0, —i~= — gi therefore, multiplying by dt and integrating, 
 
 we have for the components of the velocity, — =C, -^=C' — gt, 
 
 multiplying by dt and integrating again, we have, 
 
 x=Ct, y=C't — kgt^; or, putting C'=aC, 
 y=ax — ^gt", as before found. 
 
 CHAPTER II. 
 
 ON THE MOTION OF A BODY CONSTRAINED TO MOVE ON A Cn'EN CURVE. 
 
 (125.) In this chapter we shall show the application of the fore- 
 going general theory to the circumstances of constrained motion, 
 the moving body being prevented from obeying the influence of the 
 applied forces through the intervention of a rigid line. 
 
 When a material point is thus compelled to move on a curve, the 
 curve offers at every point passed over a certain resistance to the 
 motion in the direction of the normal, and it is in consequence of 
 this resistance that the wonted path of the body is continually di- 
 verted and its motion confined to the curve. This resistance, there- 
 fore, may be considered as a normal force continually acting on the 
 moving body, and which, combined with the other forces on the 
 body, produces the motion which actually has place. Omitting the 
 normal force and taking the components X, Y of the others on 
 which alone tlie velocity along the curve depends (p. 149), we have, 
 by equation (2, p. 146), 
 
 v'=2f{Xdx+Ydy)....(l), 
 by means of which the velocity at any proposed point (x, y) of the
 
 CONSTRAINED MOTION. 151 
 
 curve may be found, X and Y being functions of the co-ordinates. 
 As shown at (p. 146), the expression after integration will take the 
 form v^ — '^i^=f{.^i y) — ^/(«» ^)^ («> b) being the point where the 
 velocity is known to be v^. 
 
 As this result is independent of the normal pressure, that is, as it 
 remains the same whatever tliis pressure may be, we infer that it 
 must remain the same whatever the curve between (a, b) and {x, y) 
 may be, so that as long as the applied forces remains the same, and 
 the velocity of the "body at the point (a, b) remains the same, the 
 velocity at any other point {x, y) will be the same by whatever path 
 it arrives at it. 
 
 Precisely the same conclusions would follow if we had supposed 
 the motion to be on a curve of double curvature instead of on a 
 plane curve, as is very obvious from the equation at p. 146. 
 
 If the body move on the curve in virtue of an original impulse 
 merely, then, since there are no acting forces, X=0 and Y=0, and 
 consequently v^^v^, which shows that the primitive velocity will 
 be preserved and continued unchanged whatever be the curve along 
 which it moves. 
 
 Let us suppose the body to move down a curve in consequence 
 of the action of gravity, then we have, by taking the axis of X vertical, 
 
 X= — g, Y=—0.\v^=2fXdx=v^^ — 2gx. 
 To determine v^ let h be the height above the origin from which 
 the body begins to descend, that is, the ordinate of the point at 
 which the velocity is 0, then 
 
 0=v^''~2gh .'.v^''=2 gh .'.v^=2 g {h — x) .... (2). 
 
 In the general case of this problem we have seen that the velocity 
 depends on the co-ordinates x and y of the j)oint arrived at, but is 
 independent of the path to it ; in this particular case we see that 
 the velocity depends only on the ordinate x of the point arrived at, 
 being independent both of the abscissa of the point and of the path, 
 and thus any point in a straight line parallel to the horizon will 
 be arrived at ivith the same velocity if the body descend from a 
 fixed point above it along any line or ciirve whatever, the velocity 
 being that which would be acquired by falling freely through the 
 vertical height. 
 
 It immediately follows from this, that when the body has arrived 
 at the lowest point of the curve, its acquired velocity will be suffi- 
 cient to carry it up the ascending branch (if the curve have one,) to 
 the same height as it descended from, whether the two branches be 
 similar or not, although the times of descent and ascent will be dif- 
 ferent if the branches be different in form. 
 
 To determine the time requires that we know the curve of descent, 
 in which case we have, by the general expression (D), art. (106), v=-
 
 152 ELEMENTS OF DYNAMICS. 
 
 (Is J, (h ... , . 
 
 — .'. (It =—, that IS, m the case ol gravity. 
 
 If ^'* p ds , . 
 
 ' =^\2g{h — x)\'''^=J ^\2g{h — x)\ ••••/ >' 
 
 (126.) "We shall shortly give an example or two of this kind of 
 constrained motion ; but we shall first investigate a general expres- 
 sion for the resistance, or normal force, at any point of the con- 
 straining curve, when this curve is given. 
 
 Let APM (fig. 99,) be any given curve on which a material point, 
 P is compelled to move when acted upon by forces whose compo- 
 nents are X and Y. Let PN represent the normal force or the re- 
 sistance which the body receives when at P and call it R ; the com- 
 ponents of this force are R cos. NPC and — R cos. NPC ; conse- 
 quently, taking into account all the forces which act upon the body, 
 the equations of its motion are 
 
 ^=Y-R-] ""^^' 
 dt'' da J 
 
 Multiplying the first of these by-p, and, the second by -p, and sub- 
 tracting the second from the first, we have 
 
 dyd^x — dxd'^y ^^dy ^dx 
 
 ds dt^ ds ds ^ ' 
 
 consequently, since (DiJ^. Calc. p. 136,) the general expression for 
 the radius of curvature y at any point (x, y) is 
 
 r=-r^Ji j—JT-^ i* follows that 
 
 dyd^x — dxd^y 
 
 dx dy 1 ds^ dx dy v^ 
 
 ^-^ 'ds~^ is'^'^ w-^ rf^ ^ rfi+7 • • • • ^^^- 
 
 Now the expression X -^ — Y -r- is the result obtained by resolving 
 
 CIS CIS 
 
 the forces X and Y in the direction of the normal, and which, there- 
 fore, if the body were at rest, would, when taken negatively, denote 
 the resistance of the curve ; but being in motion, the curve suffers 
 
 zn additional resistance expressed by — . We must here remark, 
 
 7 
 however, that we have considered, in the above reasoning, the re- 
 sistance R to be offered by the concave side of the curve ; but if, 
 on the contrary, the body press against the convex side, then R will 
 not be the sura but the difference of the resistances, that is, the re-
 
 CONSTRAINED MOTION. 153 
 
 sistance expressed by — must be subtracted from the resistance 
 
 '^ 
 which the curve would oppose if the body were at rest.* 
 
 When the body is retained on the curve by the force of gravity 
 
 only, then Y=0 and X= — g\ therefore, in this case, 
 
 ^2 
 
 As the pressure =p — , at any proposed point, depends solely upon 
 
 the velocity at that point, it would remain the same if this velocity 
 were produced by a primitive impulse only, and the motion to be 
 uninfluenced by any acting forces, as indeed is plain from the ex- 
 
 pressionR= ± — , for the resistance, when X =0 and Y=0. It 
 
 is readily seen, therefore, that the pressure of which we speak 
 arises entirely from the inertia of the moving body, or its tendency 
 to move when at any point of the curve in the direction of a tan- 
 gent and with its acquired velocity ; this tendency necessarily 
 causes it to exert a pressure against the deflecting curve, and which, 
 as we have just seen, requires the curve to oppose the resistance 
 
 ± — m addition to the resistance necessary to oppose the normal 
 
 r 
 effect of tne actmg forces. 
 
 ^2* 
 
 A distinct name is given to the normal force or pressure =p , 
 
 whose action on the body thus tends to repel it from the centre of 
 curvature at that point of its path where its velocity is i; ; it is called 
 the centrifugal force. 
 
 If the curve on which the body moves is a circle, and if we con- 
 ceive that at the instant the velocity is v an attractive force expressed 
 
 1^2 
 
 by — be placed at the centre, then, if at the same instant all the 
 
 other forces were destroyed, the body would continue to move in 
 the same circle and with the same velocity v ; for the repelling force 
 tending to increase the distance of the body from the centre, is 
 just balanced by the attractive force tending to confine the body to 
 the curve, and moreover as i;, if the body were left to itself, would 
 continue unchanged throughout the curve (p. 151), the force which 
 
 * The student will not fail to remark that the upper sign applies when w« 
 consider the body to be moving on the convexity of the curve, in which case the 
 pressure due to the acting forces is obviously diminished by this ; and the lower 
 sign applies when the body moves on the concavity, in which the pressure is 
 necessarily increased by the same quantity. 
 
 20
 
 154 ELEMENTS OF DYNAMICS. 
 
 counteracts the pressure arising from v at one point of the circular 
 path will be competent to do so at every point ; as, therefore, all 
 pressure on the curve is destroyed, the motion of the body cannot 
 be affected if the rigid curve were removed and the body left un- 
 constrained. 
 
 This fact is at once deducible from the expression (2) for R, and 
 indeed in a more general form ; for in order that R may be 0, wliich 
 is the same as saying in order that the motion may continue un- 
 changed though the resisting curve be removed, we see that the 
 normal effect of the applied forces must be equal and opposite to 
 
 the force — : so that if a force always expressed by this were al- 
 
 y 
 ways to act at the centre of curvature, corresponding to the radius 
 y, the rigid curve, be it what it may, might be removed without 
 changing the trajectory. By this arrangement it is plain that the va- 
 
 riable force — must itself move so as to describe the evolute of the 
 
 curve described by the body ; the evolute of a circle is a point, viz. 
 the centre. 
 
 As the central force, or as it is usually called the centripetal 
 
 force, necessary to retain a body in a circle is F = , and since, 
 
 7 
 if /" be the time of one revolution, we must have, in consequence 
 
 2 rt Y 4 rt^Y 
 
 of the uniform velocity, w = .•.F = — ^, which expresses 
 
 alike the intensity of either the centripetal or the centrifugal 
 force. 
 
 In like manner, for any other circle of radius y^, and time <,", 
 the centripetal or centrifugal force is 
 
 F -ll'll . FF ■■^■^^^ 
 r,_ ^^^ ..r .r,.. ^, .^^, , 
 
 hence, 1st. When the circles are equal, the centripetal or centri- 
 fugal forces are inversely as the squares of the times ; and 2d. When 
 the times of revolution are equal, the forces are as the radii of the 
 respective circles. 
 
 Let us apply these results to an interesting problem, viz. to 
 the determination of the centrifugal force at different places on the 
 earth's surface, from knowing the time of one rotation on its 
 axis. 
 
 The earth, by means of its diurnal motion, carries round with it, 
 with a uniform velocity, every point on its surface in 86164 se- 
 conds. At the equator the radius y is 20921185 feet, therefore the 
 centrifugal force at the equator is
 
 CONSTRAINED MOTION. 155 
 
 „ 4rtV 4rt=20921185 ,,,„,, ^ 
 
 As this force opposes the force of gi'avity, it follows, that if it did 
 not exist, that is, if the earth did not revolve on its axis, the force 
 of gravity instead of being what it really is, viz. g':=32-08818 at 
 the equator, would be G = g" + •1112447, and thus the Aveight of 
 
 •1112447 , . 
 
 any body would be a noof ^" P^""^ more than it really is. 
 
 32^08818 
 
 The ratio of G to F being 
 
 32 • 1994247 : •I 112447 or 289 : 1 nearly, 
 
 .■.F^^....m. 
 
 Now every parallel to the equator being carried round in the same 
 time t" as the equator, we have, by representing the centrifugal 
 force in the parallel whose latitude is I and radius y, by F^, 
 r:T. ::F:F, 
 
 because it is evident that yj=y cos. /. 
 
 The force G of gravity is not diminished by the whole of the 
 centrifugal force F^, except at the equator, because this force acts in 
 any parallel PAP' (fig. 100,) not in the direction Fp opposite to 
 gravity, but in the direction Pr, if, therefore, we decompose the 
 force Pr in the perpendicular directions Pjo, Fq, both, as well as 
 Pr, being in the plane of P's meridian, we shall have, for the force 
 Pp opposing gravity, 
 
 P;?=Pr cos.p'Pr=F^ cos. POQ=F, cos. /; 
 hence the expression for the diminution of gi'avity is (equa. 2,) 
 
 -^cos.^/; 
 
 which therefore varies as the square of the cosine of the latitude. 
 
 The other force Fq, being tangential, tends to draw the particles 
 
 of the revolving body from the poles towards the equator, and to 
 
 cause it to assume the figure of an oblate spheroid ; the expression 
 
 for this force is 
 
 G C 
 
 Fq=F^ sin. ^=-^g^sin. / cos. Z=— — sin. 2 I, 
 
 which therefore varies as the sine of twice the latitude. 
 
 From the foregoing principles, let it now be required to deter- 
 mine the time in which the earth must perform its diurnal revo- 
 lution in order that the centrifugal force at the equator may be ex- 
 actly equal to the force of gravity, or in order that a body may have 
 no weight there.
 
 156 ELEMENTS OF DYNAMICS. 
 
 Let F, represent the time of rotation, corresponding to which the 
 centrifugal force is G, then, as tlie centrifugal forces in the same 
 circle are inversely as the squares of the times, we have, (equa. 1,) 
 
 Hence if the diurnal rotation of the earth were performed in a 17th 
 part of the time really occupied, or if it were to turn round 17 times 
 as rapidly, bodies at the equator would lose all their w^eight, and 
 would, therefore, if placed at a small distance above the surface, 
 remain suspended without any visible support ; if the earth were to 
 revolve still more rapidly than this, no body could remain on its 
 surface : every thing would be repelled from it by the centrifugal 
 force. This inference it must be remembered is, as well as that 
 implied in equation (2), on the supposition that the earth retains its 
 spherical figure during its rotation, which however is not strictly 
 correct on account of the oblique influence of the centrifugal force 
 tending to elevate the equator and to depress the poles, thus giving 
 to the earth a spheroidal form. It is demonstrated that a fluid 
 spheroid of the same density as that of the earth, cannot remain 
 in equilibrium if it revolve in a shorter time than ^' 25' 26".* 
 
 (126.) The general equations (2) and (3), at page 151, contain 
 all that is necessary for the determination of the motion of a body 
 down any given curve by the action of gravity; or indeed of any 
 force g acting in parallel lines. The theory of the pendulum, a 
 highly important subject, is established by their aid, and to this 
 theory w^e shall apply them in the following chapter. We may re- 
 mark, however, before entering upon this, that the expression (1) 
 for the velocity at page 150, is general for all hypothesis of the 
 acting forces, and when this velocity is determined and the curve 
 
 ds 
 given, the time will be given by the equation t =f — , in the par- 
 ticular case, however, where the body is acted upon by a single 
 centre of force, varying according to some function of its distance, 
 then, for the determination of the velocity, it will be proper to use 
 the expression (1) at page 147, taking the integral with a negative 
 sign if the force be attractive, tending to diminish the co-ordinates, 
 and taking it with a positive sign if repulsive ; the axes of reference 
 too must here originate at the centre of force. 
 
 * See Professor Airy''s Tracts, iiage 150, second edition; or the Thiorie 
 Analytique du Systeme du JMonde, of Poiit^coxilant ; torn, ii., page 400.
 
 ON THE SIMPLE PENDULUM. 157 
 
 CHAPTER III. 
 
 ON THE SIMPLE PENDULUM. 
 
 (127.) A SIMPLE pendulum is considered to be a material point, 
 attached to a thread or rod without weight, and oscillating about a 
 fixed axis connected with the other extremity of the rod. Such a pen- 
 dulum, it is evident, can have no physical existence, yet it is con- 
 venient to discuss the theory of such an imaginary pendulum, be- 
 cause, as will be shown in a subsequent chapter, whatever be the 
 oscillating body, there may always be found a point at which, if a 
 single particle were placed and connected by a rod, without weight, 
 to the point of suspension, the oscillations of this simple pendulum 
 would be performed in the same time as those of the compound 
 body ; and all the circumstances of its angular motion would be the 
 same, and thus any pendulum may be reduced to an equivalent 
 simple pendulum. 
 
 The moving point which we here consider, is confined to the 
 curve in which it moves by the thread, the accelerating force being 
 gravity ; hence, the tension suffered by the string at any point of 
 the path, must be equivalent to the pressure which would be sus- 
 tained by the curve at that point if it were rigid, and the moving 
 point were unconnected with the thread. The constraining forces 
 being equivalent, the theory developed in the preceding chapter be- 
 comes immediately applicable to the motions of simple pendulums ; 
 these motions, although usually in circular arcs, may nevertheless 
 oe in any curves whatever, for the thread as it oscillates to and fro 
 may be forced to wrap itself about curves springing from the point 
 of suspension, and thus the material point will be forced to describe 
 curves which are the involutes of these ; and in this way we may 
 have circular pendulums, cycloidal pendulums, &LC. Of these, how- 
 ever, the circular pendulum is the most simple and important. 
 
 (128.) It wiU contribute to the convenience of the student to 
 bring together in this place those formulas distributed in the pre- 
 ceding chapter, which are required in the problems we are about 
 to give ; these formulas are as follow : 
 
 velocity = v = \/ ^g(h — x)' ♦ • • (A), 
 
 /» ds* ,^^ 
 
 time = ; ==/ — (B), 
 
 J s/%g{h—x) 
 
 * If we consider the arc s in this expression to be the arc of descent, this, 
 being measured from the origin or lowest point, must diminish as the time in- 
 
 o
 
 158 ELEMENTS OF DYNAMICS. 
 
 centrifugal force =/= — =— ^^ . . . .(C), 
 
 y y 
 
 _ dy 2sr(h — x) ,^, du . 
 
 tension = T = ^-^H — ^— ^ • (D), or, since -p expresses the 
 
 cosine of the angle which the normal makes with the axis of x, if 
 we call this angle a we may write the last expression thus : 
 
 2 o- (// — x) 
 tension = g-cos. a -\ — ^— ^^ (E), 
 
 y 
 which form will be sometimes most convenient to use, (see prob. IV. 
 following.) 
 
 Problem I, (129.) To determine the time of oscillation in a cir- 
 cular pendulum (fig. 101). 
 
 Taking the origin of the vertical and horizontal axes at the 
 lowest point of the curve, and calling the radius of the circle or the 
 length of the rod r, we have, for the equation of the path, 
 - « . dy"" (r — x)' 
 
 .*. dS: 
 
 dx^ 2rx — x^ 
 dy^ , rdx 
 
 rfx2* ^\ 2rx—x^\ 
 
 consequently, by equation (B) above, the expression for the time is 
 
 /ds _ ** /* d""^ 
 
 VVig(ih—^)'~V\2f\ J :yj{h:^{2rx—x^)l' 
 The differential expression under the integral sign may be put 
 
 under the more convenient form — ; — ^ -; r^(l — :r-)~^ 5 
 
 ^\2rl^ \nx — x^p 2r 
 
 which shows, that if the second factor be developed by the binomial 
 
 theorem, the difTerential in question will be reduced to a series of 
 
 3?" dx 
 
 others all of the form — rr r. which we know to be an in- 
 
 ^\hx — x^\ 
 
 tegrable form. 
 
 These details we have entered into at length in the Integral Cal- 
 culus, page 93, and the result is, that the proposed integral, taken 
 between the necessary limits, that is from x =0, the lowest point 
 of the curve, to x=h, the point of departure, is 
 
 creases, and therefore ds in this formula is negative ; but if we refer to the arc 
 of ascent, the time and arc increase together, and ds is positive, we shall there- 
 fore always consider the formula ais applied to the ascending arc, since the time 
 will be the same whether the body descend from the height x to the lowest point, 
 or ascend from the lowest point to the height .r ; or whether it descend from h to 
 X, or ascend from x to h. On this hypothesis, therefore, the integral (B) com- 
 mences at x^=x and ends at x=h ; for the descending arc, on the contrary, the 
 integral commences at x=A and ends at h=x.
 
 ON THE SIMPLE PENDULUM. 159 
 
 dx 
 
 f 
 
 y/\(Ji — x) {2rx — x^)\ 
 
 consequently for 2 T', or the time of a complete oscillation, we have 
 
 ^ ^* ^^2^ 2r^^2.4^ hr^ ^ "^■<^:!, 
 
 by means of which the time may be approximated to, to any degree of 
 accuracy. The expression h is the versed sine of the arc of descent, 
 
 or of half the whole path, and— is the versed sine of a similar arc 
 
 r 
 
 to radius 1, and therefore of the inclination of the rod to the verti- 
 cal in its initial position ; the smaller this inclination is the more 
 convergent will the foregoing series be. Suppose, for instance, the 
 initial inclination were 5°, then the versed sine of this being "0038053, 
 the second term of the series would be only 
 
 , 1 , -0038053 ^^„, , 
 (-)« 2 =-0004757, 
 
 and if the pendulum vibrated but one degree on each side of the 
 
 vertical, then, since the versed sine of 1° is '0001523, the second 
 
 ,, , , , 1 , -0001523 
 term of the series would bebut(— - y = -000019, and, 
 
 Ai Ai 
 
 by supposing the arc of vibration less and less, the expression for 
 the time of an oscillation would continually approximate to 
 
 2f==rtv/— ••••(2), 
 E 
 which will differ insensibly from the true time when the arcs of vi- 
 bration are very small, that is, not exceeding about 4° ; and there- 
 fore, for all arcs between this and 0, the times of vibration of the 
 same pendulum will not perceptibly differ, that is, in very small 
 arcs the oscillations may be regarded as isochronal, or as all per- 
 formed in the same time. 
 
 Since the time occupied by a body falling freely through the height 
 
 ^ r is expressed by^— , it follows, from the foregoing expression, 
 
 that in pendulums of such limited ranges or amplitudes as we have 
 supposed, the time of vibration is to the time of falling freely 
 through half the length of the rod a* 3-14159 fo I.
 
 160 ELEMENTS OF DYNAMICS. 
 
 It is an important matter to know exactly the length of a pendu- 
 lum wliich will vibrate seconds, and combined with experiment the 
 the f^oneral expression (1) will cnal)le us to determine this length 
 with perfect accuracy for any given arc of vibration. 
 
 Thus, let r, r' be the lengths of two pendulums vibrating in arcs 
 
 h h' 
 of the same number of degrees, then, since — = — the series within 
 
 r r 
 
 the brackets will be the same for each pendulum ; hence, the times 
 
 of oscillation of these pendulums will be to each other as 
 
 r r' 
 
 ftx/— to }t^ — or as ^r to >/r': 
 g g 
 
 the times of oscillation are therefore as the square roots of the 
 lengths. Let the pendulum r make n oscillations in the same time 
 t" that the pendulum r' performs n' oscillations, then the respective 
 
 t" t" 
 times of a single oscillation will be — and—, which are to each 
 ® n n 
 
 other as — to — ; hence, by the proportion just deduced, 
 
 r : r :: — - : — — : : n " : 7i', 
 
 that is, the lengths of pendulums vibrating in similar arcs are to 
 each other inversely as the squares of the num,ber of oscillations 
 made by them in the same time. 
 
 Now a seconds' pendulum must vibrate t times in t", if, therefore, 
 we take a pendulum of any length r, and count the number n of vibra- 
 tions it makes in any time t", we shall find the exact length r' of the 
 seconds' pendulum vibrating in a similar arc by this proportion, viz. 
 
 n^ r 
 t':n'::r:r' =— .... (3). 
 
 If the length of the seconds' pendulum be thus determined for very 
 small arcs, we may thence, by help of the expression (2), determine 
 the force of gravity at the place where the experiment is made for, as 
 
 l=Hy-.:g=^^r'-i4). 
 
 Now in the latitude of London r' has been found to be =:39 . 14 
 inches, consequently ^=7t^x39 . 14 in. =32*19 feet, the force of 
 gravity in the latitude of London. 
 
 Knowing the length of the seconds' pendulum, it will be an easy 
 matter, from the foregoing theorems, to find the time of vibration of a 
 pendulum of any other length, or the lengUi of a pendulum vibrating 
 in any other time. Thus, the length of the seconds' pendulum being 
 r', and that of any other r, we have by the first of those theorems, 
 this expression for the number t of seconds, this last will vibrate in,
 
 ON THE SIMPLE PENDULUM. 161 
 
 viz. t = f— , ••• r = r' t^. Suppose, for example, we wanted 
 to know the time of an oscillation of a pendulum 20 feet long, 
 
 we should then have t = f— - — — - = 2.5 nearly, so that the time 
 ^yoU . 14 
 
 would be about 2 seconds and a half. 
 
 Again, if we wanted to know the length of a pendulum that 
 should oscillate once in ten seconds, then we have 
 r=39 . 14x10^=3914 . inches. 
 We may also readily determine the number of seconds lost or gain- 
 ed in a day by lengthening or sliortening a seconds' pendulum by 
 any proposed quantity ; for, from the equation (3), we have 
 
 _r' P 
 
 where r' is the length of the seconds' pendulum, and f =86400, the 
 number of seconds in 24 hours, n being the number of times the 
 altered pendulum r oscillates in 24 hours. Suppose r=r'-{-p, and 
 n=t — q, p will then be the error in length of the altered pendu- 
 lum, and q the consequent deficiency in the number of vibrations, 
 or the loss in seconds, and we shall have 
 
 r +p= -^ =r +2r — + &c. 
 
 ^^ t2—2tq-\-q"- ^ t 
 
 If the loss amount to but a few seconds, the powers of — may ob- 
 viously be neglected without sensible error, and we shall thus have 
 ;j=2r^ -^ and 0= -r — 
 
 the first equation showing the increase of length corresponding to 
 a given loss, and the second showing the loss consequent upon a 
 given increase of length ; and the expressions hold when the pen- 
 dulum is diminished hyp; q then expressing the gain. 
 
 Hitherto we have considered the pendulums compared to oscil- 
 late at the same place ; but it is a very important inquiry to deter- 
 mine the lengths of pendulums oscillating seconds at different places 
 on the earth's surface, as such a determination readily leads to the 
 discovery of the true figure of the earth. If we represent by G and 
 g the intensities of gravity at any two places, and by r and r' the 
 lengths of the corresponding seconds' pendulums, then, by equation 
 
 G £• 
 
 (4), we shall have >'=— , »''=-^» so that the intensity of gravity 
 
 at any places varies as the length of the seconds'' pendulum at 
 those places. But it is manifest that the intensity of gravity must 
 2 21
 
 162 ELEMENTS OF DYNAMICS. 
 
 depend upon the figure and constitution of the earth, and accord- 
 ingly it is proved, by the writers on Physical Astronomy, that, 
 considering the earth to be a homogeneous spheroid of equilibrium, 
 the intensity of gravity must vary as the normal, so that, from the 
 foregoing equations, the normal to the earth's surface at any place 
 varies as the length of the seconds' pendulum at that place ; aud 
 thus when the lengths of the pendulum for any two known latitudes 
 are accurately ascertained by experiment, sufficient data will be 
 furnished for determining the ratio of the earth's polar and equato- 
 rial diameters, or for finding the ellipticity or spherical compression, 
 as it is called, and by which is meant the ratio of the difference of 
 these two diameters to the greater. But we shall make the deter- 
 mination of this ratio from the proposed data a distinct problem. 
 
 Problem II. — (130.) To determine the compression or ellipti- 
 city of the earth by means of seconds' pendulums. 
 
 Let a, b represent the equatorial and polar semi-diameters, e the 
 
 eccentricity, and c= , the compression; then 
 
 a^ — h^ a — b a-\-b 
 
 e»= — - = X— !— =c 2 — c . 
 
 cr a a 
 
 Now c is itself but a small fraction, and the square of it is too small 
 to be worth regarding in this inquiry, so that we may consider the 
 compression to be expressed by c=k t^. 
 
 Let X, /t' be the latitudes at which the lengths of the seconds' 
 pendulums are /, /', then, since (Diff. Calc. p. 134,) the expres- 
 sions for the normals at these latitudes are 
 
 6'' 1 , ^^ i 
 
 ^= T ' (1 — c'^sin.U)!' ^ "" "^ ■ (1— e'^sin.U')! ' 
 we have /:/'::(l — e* sin. ^ A,)~l : (1 — e'^sin. ^ A-')~i- , that is, 
 by expanding the two last terms by the binomial theorem, and 
 omitting the square and higher powers of e^ sin. " A, on account of 
 their excessive sraallness, 
 
 /:/':: 1 + | e*sin. =A: \-\-h e^sm.^X' 
 ::l+csin. =»A :l + csin. »A' 
 
 /sin. ''A' — /'sin. -A ' ■ , •> , • „, 
 — sm. ^ A — sm.°A 
 
 which expression would be the value of the compression, if, as we 
 have supposed, the earth were of uniform density. Such, how- 
 ever, is not the case, yet the conclusion just obtained will enable 
 us to deduce the true compression, whatever be the law of the
 
 ON THE SIMPLE PENDULUM. 163 
 
 earth's density, by the aid of the following very remarkable propo- 
 sition discovered by Clairaiit, viz. " Whatever be the law of the 
 earth's density, if the elliptieity of the surface be added to the ratio 
 which the excess of the polar above the equatorial gravity bears 
 
 to the equatorial gravity, their sum will be—-, m being the ratio 
 
 of the centrifugal force at the equator to the equatorial gravity."* 
 Now the ratio which the excess of the polar above the equatorial 
 gravity bears to the equatorial gravity, is no other than the ellipti- 
 eity c, as determined upon the hypothesis of homogeneity ; for, 
 calling the polar gravity G and the equatorial gravity g, and re- 
 collecting that these are as the normals, we have 
 
 G : s^::b:~.-. ^— =c; 
 
 a g a 
 
 consequently, whatever be the law of the earth's density if we 
 call the elliptieity or compression C, we have 
 
 C + c=— -=— • — — (page 109,) and therefore 
 
 I 
 
 c= 
 
 1-F 
 
 578 i . .., . ,^ 
 — sm. "/, — sm. ^A 
 
 From experiments at Madras A=13° . 4' . 9" , 1=39 • 0234 
 From experiments at Melville Island 
 
 A'=74° . 47' . 12" , l'=39 • 2070 , 
 5 
 from which c=-0053478 and as =-0086505, therefore 
 
 C = -0033027=— . 
 302 
 
 From this value it appears that the equatorial diameter of the earth 
 
 exceeds the polar by about the 300th part of its whole length ; that 
 
 is, these diameters are to each other as 300 to 299, and this ratio 
 
 agrees almost exactly with the ratio as determined by means of the 
 
 actual measurement of degrees. See Dr. Gregory'' s Trigonometry, 
 
 page 231 ; and Mry's Tracts, page 186. 
 
 Problem III. — (131.) To determine the time of oscillation of a 
 cycloidal pendulum (fig. 102). 
 
 When the axes originate at the extremity of the base ot the 
 
 * See Professor Airy's Tracts, p. 174.
 
 164 ELEMENTS Ot DYNAMICS. 
 
 cycloid we have found for the length of any arc s (Int. Calc. p. 
 115,)» 
 
 s=4;- — 2^j2r(2r — a-); (1); 
 
 but if we measure s from the vertex, tlion, since the length of the 
 semirycloid is 4 r, we sliall have, by subtracting the foregoing ex- 
 pression from this, and then removing the origin to the vertex, that is, 
 substituting — x-f 2r for x, we have, in the inverted cycloid (fig. 103,) 
 
 s=2y/^^.: ds= J^dx .'. t=r — 
 
 \x J ^2g (h—x) 
 
 r n (Ix r . _ 2 , ^ 
 
 = v/ — / = a/ — versm. ^-pX+C; 
 
 gJ ^ hx—x'' S " 
 
 I r 2 . 
 
 hence, from x=x to x=h, /= — In — versin.~*^-x I , which 
 
 \^ A 
 
 expresses the number of seconds in descending from the altitude h 
 to the altitude x ; therefore the time of descent to the lowest point 
 
 r 
 A at which a:=0 is t^n^/— .... (2). 
 g ' 
 
 This expression being independent of h, is very remarkable, in- 
 asmucli as it proves that the time of descent to the lowest point is 
 always the same from whatever point in the curve the body be- 
 gins to descend. The oscillations in a cycloid are, therefore, al- 
 ways isochronal. 
 
 The cycloidal pendulum must oscillate between the two equal cy- 
 
 cloidal cheeks SB, SO, about which the thread SP wraps itself; the 
 
 length of the pendulum being equal to that of the curve SB, or which 
 
 is the same, of the semi-cycloid BA, and this from equation (1), is 
 
 by putting y=2r, s = 4r, so that calling the length of the pendulum 
 
 / the expression for a complete vibration is, from equation (2), 
 
 / 
 2t=7t^ ; 
 
 or 
 
 O 
 
 which is the same as the expression given in last problem for the 
 time of vibration of a pendulum of the same length, in a very small 
 circular arc. 
 
 Problem IV. — (132.) When a body vibrates in a circular arc, to 
 determine the tension of the string at any point (fig. 101). 
 
 • An obvious error has crept into the formula here referred to ; instead, of 
 
 s=^'irf~^w^-\- c 
 
 it should have been 
 
 v^/; 
 
 =— 2v'2r(2r — y)4-C, 
 
 ^ir—y 
 and C is determined from the condition that »=0 when y=0.
 
 ON THE SIMPLE PENDULUM. 165 
 
 Here the cosine of the angle », or PXA, which the normal makes 
 with the axis of X, is obviously ; — . Hence, by the formula for 
 
 the tension, we have 
 
 r — X , 2^(A — x) r + 2/t — 3x 
 
 tension = 2* ^^-^ '- = 2r 
 
 ^ r r * r 
 
 at the lowest point, or where a;=0 tension =«• ; which, if 
 
 the body fall from /j=r, becomes 3g ; that is, the body, when it 
 
 comes to A, is acted upon by three times as much force as it would 
 
 be if at rest there, and, therefore, the body stretches the string 
 
 with three times its weight. The tension is, obviously, greatest at 
 
 this point. 
 
 In order to determine the point at which the tension is 0, we 
 
 r-\.2h — 3x ,^ r-{-2h , , . 
 
 have g = .•. x = — - — , the abscissa of the point 
 
 at which the pendulum, let fall from the height h, produces no 
 strain on the point of suspension. 
 
 To determine the point at which the strain is the same as when 
 the body hangs at rest, we must equate the expression for the ten- 
 sion with g, we thus find x=^h. 
 
 Problem V. — (133.) To determine centrifugal force and the ten- 
 sion of the string in the cycloidal pendulum. 
 
 We have already seen (prob. H.) that in the cycloid the expres- 
 
 „ . dx . dx X 
 
 sion tor sin. a, or -^, is -7- = »/ -- 
 
 ds ds 2r 
 
 J 2 r—x 
 X 
 
 dy ,,, . „, hr — X dy 2 r — 
 
 .-. COS. a or-# = v/U — sin. ''I a = .-. ,-= . p 
 
 ds ^ ' ^ \1 2 ?• dx S 2r 
 
 dy 
 
 ' ' dx 
 By means of these expressions we may find the radius of curvature 
 at any point (^x, y,) from the formula, Diff. Calc. p. 124-5. It is 
 _ ds^ _ d^y _ 2 r I — r 
 
 '^~~~dx^~ d3^~'~^'x'^ ~Xs/\xJi~r^x)\ 
 ^__ =2^ [2r(2r — x); 
 hence the expression for the centrifugal force is 
 f=—= g {h—x) . 
 •^ r ^\2r{2r—x)\' 
 
 and, for the tension, we have 
 
 \2t—x\_^ g(h--x) 
 
 2r '^ ^ \2 r {2 r—x) i'
 
 166 ELEMENTS OF DTN'AMICS. 
 
 At the lowest point, or where x=0, both the centrifiigal force 
 and tension are obviously greatest ; the expressions for them being, 
 
 •/ - 2r ' -" + 2r 
 
 Problem VI. — (131.) To determine the time of gyration of a 
 conical pe.dulum. 
 
 When tlie pendulum, instead of vibrating in a vertical plane, is 
 made to pass over, or generate, a conical surface, as in fig. 104, it 
 is called a conical pendulum. 
 
 The motion of such a pendulum is due to three forces, viz. the 
 tension /, of the string AC ; the force of gravity g, in the direction 
 AD, and the centrifugal force/, in the direction AB ; and these three 
 forces keep the body, A, at the same constant distance r from S ; 
 hence, resolving the tension t in the directions AS, AE, we have 
 
 t COS. a=/=— (art. 12G), t sin. a=^ ; therefore, 
 
 _,^ sin. a gr a gr ^ gr" 
 
 puttmg CS = a ; = ^ot~ = —,.\ tJ«=-5— , 
 
 ^ ^ cos. a v* r V* a 
 
 but t" being the whole time of gyration, we have 
 
 V' = —7, — = -^ ••• f =2 rtj- ; 
 /^ « \j g 
 
 which expression, being independent of r, shows that the periodic 
 
 time varies as the square root of the altitude of the conical surface 
 
 described, whatever be the length of the pendulum, or the radius of 
 
 the base of the cone. 
 
 We have seen (prob. I.) that the time in which a pendulum of 
 
 length a vibrates in a small circular arc is expressed by t 
 
 hence the time of gyration of a conical pendulum is exactly double 
 the time in which a simple pendulum, whose length is the height of 
 the cone, would vibrate in a small circular arc. 
 
 ""^7'' 
 
 CHAPTER IV. 
 
 ON CENTRAL FORCES. 
 
 (135.) The motions of bodies, acted on by central forces, is a 
 branch of the general theory, of so much importance, in the system 
 of the world, that it will be proper to give it a distinct considera-
 
 ON CENTRAL FORCES. 167 
 
 tion, and to present the equations of motion with no more generality 
 than may be requisite, in order to comprise the theory of a body's 
 motion, when urged by a single centre of force. 
 
 It will be convenient here to put aside the use of rectangular co- 
 ordinates, and to employ polar co-ordinates originating at the centre 
 of force, so that P (fig. 105,) being the place of the body at any 
 time t"y and S the centre of force, the point P will be determined 
 from knowing SP=r, and the angle PSX=u. 
 
 The law of force, supposed to be some function of r, will be 
 known when the general relation between r and w, independently of 
 t", is known, and, conversely, this general relation, or the equation 
 of the orbit, will be known, when the law of force is known ; this 
 we shall now proceed to show. 
 
 Let us suppose the force R to be attractive, then we know (121) 
 
 ds ds^ 
 
 that the velocity -7-, in the orbit, will be -^ = — 2fUdr . . .(1). 
 
 Now. whatever be the independent variable, it is shown (Int. Calc, 
 p. 118) that {dsY=r'' {d^y-\-{drY ; let then t be the independent 
 variable, and we have, from (1), 
 
 dJ^ dr^ ^ .^ J .^x 
 
 We moreover know (Int. Cede. p. 131) that the polar expression 
 for the area generated in t", viz. the area XSP is ifr^du, and we 
 have seen (122) that this area is proportional to t, that is, 
 
 fr^do^=ct.'.r^~=c (3). 
 
 As « is the angular space passed over, -7- expresses the angular 
 
 velocity, and the equation (3) shows that this angular velocity varies 
 inversely as the square of the distance of the body from the centre 
 of force. If we substitute in (2) the value of 
 
 -, in (3), we have —+'-=-2fRdr=:v- . (4). 
 
 This equation, as it contains only the two variables r and t, which 
 
 we see are at once separable, will enable us, when R is given, to find 
 
 the general relation between r and t, and thus the distance of the 
 
 body from the centre at any given time. 
 
 Again, by the same two equations, viz. (2) and (3), we may eh- 
 
 df 
 minate r and — , when ("Rdr is found, and we shall thus have a dif- 
 
 dt •' 
 
 ferential equation, involving only o and t, from which the general 
 relation between « and t may be determined ; and thus the position 
 of the body at any time completely found. The two general ex-
 
 168 ELEMENTS OF DYNAMICS. 
 
 pressions for /, thus obtained, will, when put equal to each other, 
 obviously represent the path of ihe body. But this will be better 
 done by eliminating at once (It* from the equations (2) and (3), by 
 which we get for tlie diffeiential equation of the orbit 
 
 c' 1 rfr* 
 
 —. l-r. -rrr+M = — 2/"R(/r=r2 .... (5) ; which will become 
 
 somewhat simplified by putting u for — , as it then takes the form 
 
 or which is the same thing, v^-=c^\-^---\-u'^ \ . . . . (7.) 
 
 — f/M has been per- 
 formed, is the difTerential equation of the orbit, which, by integra- 
 tion, will become the algebraical equation of the curve. If, how- 
 ever, the orbit is already known, and we require to determine the 
 law of force, the operation will be easier, as no integration will be 
 necessary ; thus, by differentiating (6), we have 
 
 These equations, or in fact the single equation (5), contains the 
 whole theory of central forces, at least as far as regards the nature 
 of the orbit, the law of the force, and the velocity of the body at 
 any point. When the time enters into consideration the equations 
 (2), (3), and (4) become useful. 
 
 (136.) Having established these equations, it would be easy now 
 to deduce from tliem a variety of other forms, but we shall not de- 
 tain the student by so doing. One or two transformations, how- 
 ever, deserve to be noticed, on account of their utility. 
 
 Looking to the terms within the brackets in equation (5), as coit- 
 nected with the angle P, (fig. 106,) we observe that they denote 
 
 • We must caution the student against supposing that we here depart from the 
 theory laid down in the Differential Calculus, by seeming to view dt as a finite 
 quantity instead of absolutely nothing. It must be remembered that in every for- 
 mula, containing only first differential coefficients, the independent variable may 
 
 be always considered as entirely arbitrary ; that is, every coefficient such as — , 
 
 may always, without at all altering its value, be changed into the more general 
 
 form -rj\< (see Diff. Calc. p. 99.) In the above, therefore, we tacitly suppose 
 
 this change to be effected, and eliminate, in fact, the finite quantity {dt).
 
 ON CENTRAL FORCES. 169 
 
 {Diff. Calc. p. 119)* -^—-{.1= . I 
 
 Now, if from the pole a perpendicular p be demitted upon the tan- 
 gent at P, its length will be /J=r sin./^P ; hence we may transform 
 the equation (5) into 
 
 an expression of remarkable simplicity. 
 
 (137.) Another useful and simple form is obtained by introducing 
 the expression for the chord of the osculating circle drawn from P 
 through the pole. This expression is thus deduced : from the 
 centre of curvature C (fig. 107,) draw CM perpendicular to the 
 radius vector, then PM will, obviously, be half the chord of curva- 
 ture ; upon the tangent PT demit the perpendicular ST:=jo : then 
 the angle TPM being equal to the angle PCM the triangles STP, 
 PMC, are similar, therefore, 
 
 chord=2 PM=2^^^-=2y^ (1). 
 
 bP r ^ ^ 
 
 Now if we join SC=a, and draw the perpendicular ST'=PT, then, 
 of whatever curve S is the focus and CP^y the radius of curva- 
 ture, we always have, (6'eom. p. 35,) • 
 
 cLt (It 
 
 a2=r2 +y2_2y» .-. 0=2r-i 2y .-. 2y=2r-j-; 
 
 dp dp 
 
 dr 
 and, substituting this in (1), we have chord=2p-^ ; and, conse- 
 
 quently, the equation (10), art. (136), becomes by substitution 
 
 2^2 fly, 
 
 R=— : — r- Moreover, since from (10), v^='Rp—=Ry,h. chord; 
 chord ■' ^ dp 
 
 and since we likewise know, if the force R were to remain constant 
 
 and to draw a body let fall from P along the chord of curvature, 
 
 that when it should have fallen through | the chord we should have 
 
 t'^=RX2 chord, we infer that the velocity in the curve, at any point, 
 
 is the very same as the body tvould acquire in falling through one 
 
 fourth the chord of curvature, supposing the force, at that point, 
 
 to remain constant. 
 
 (138.) The force F, which placed at S, would compel the body 
 
 at P to revolve round the centre, at the same distance SP, with the 
 
 angular velocity, it actually has when at P, is called the centrifugal 
 
 force at P, or rather the centrifugal force is equal and opposite to 
 
 dr 
 * At the top of the page here referred to, the expressions r — and — r2 should 
 
 dh> 
 be interchanged 
 
 P 22
 
 170 ELEMENTS OF DYNAMICS. 
 
 this. The expression for the angular velocity is, as we have al- 
 ow 
 ready seen, — which is no other than the actual velocity in the 
 
 small circle which a point in the radius vector at the unit of dis- 
 tance from S describes as SP revolves ;• the velocity, therefore, due 
 
 flw 
 
 to the force F, of which we speak, is r~j ; but when the body moves 
 
 in a circle the centrifugal force is expressed by the square of the 
 velocity divided by the radius, so that we have here 
 
 F=r-^.-. F=^ (equa. 3, art. 135). 
 
 If the force R, really placed at S, were but just sufficient to retain 
 the body at P in a circle, either of these values of F would express 
 its intensity; whatever additional influence, therefore, R actually 
 exerts, or whatever influence short of this R exerts, it must be 
 wholly employed in diminishing, or increasing the radius vector SP ; 
 
 this portion of force, therefore, is truly represented by .±-^— ; 
 
 the upper sign applying when the radius vector increases, and the 
 lower when it decreases ; or omitting, as usual, the signs before 
 the difl^erential coefficients, we have, for the intensity of the central 
 or centripetal force R, 
 
 ''='-W-lF=^-liF ■ ■ ■ ■ ^'^- The force ^ 
 
 dr 
 is called the paracentric force, and the velocity — , due to it, the 
 
 paracentric velocity. This velocity we may readily express for any 
 point in terms of the coordinates, independently of /. Thus sub- 
 stituting the value of — 2/Rrfr, in equation (5), art. (135) in the 
 equation (4), which precedes it, and we have 
 
 dr" c" dr" 
 
 ■^=^--^= (paracentnc velocity)". 
 
 The paracentric force is (equa. 1) the difference between the 
 centripetal and centrifugal forces. ' 
 
 We shall now proceed to illustrate this theory. 
 
 On the Motions of the Planets. 
 (139.) Before Newton's discovery of the law of universal at- 
 
 • For as the angle w is described in /", the point to which we allude describes 
 the arc 1 X u ; hence the velocity, being the diBerentiai coefiicient of the space, 
 
 with respect to the time, is — . 
 dt
 
 ON THE MOTIONS OF THE PLANETS. 171 
 
 traction the paths in which the planets revolve about the sun had 
 been ascertained by observation ; and the following laws, discovered 
 by Kepler, and afterwards called Kepler'' s laivs, were known to be 
 true. They are the three following : 
 
 I. TTie radius vector of every planet describes about the sun as 
 a pole, equal areas hi equal times. 
 
 II. The path of every planet is an ellipse, having the sun in 
 one of its foci. 
 
 III. TTie squares of the times of revolution are as the cubes 
 of the mean distances from the sun, or as the cubes of the major 
 axes of the orbits. 
 
 (140.) From these facts, revealed by observation, let us now de- 
 duce the law of attractive force, on which they must necessarily 
 depend. In order that nothing may be assumed in this inquiry, let 
 us, before we bring into application the preceding theory, show 
 that this force, whatever it be, must be directed towards the sun. 
 In order to this take the centre of the sun for the origin of the rect- 
 angular axes, these being in the plane of the orbit, then the com- 
 ponents of the force acting on the body at any time <", in directions 
 parallel to these axes, will be 
 
 d^x d ^v 
 
 -p^=X, — -^=Y, from which we get, as at art. (122), 
 
 We have, moreover, seen that fydx — f-^dy is the double area 
 described by the radius vector about the sun, during the time t" 
 (122), and since by Kepler's first law, this area is always propor- 
 tional to the time, we may generally represent it by ct, c being 
 
 ydx — xdv 
 constant ; hence, we have, by differentiation, — = c ; 
 
 d . (vdx ^— xdv^ 
 and, differentiating again, we have — j - '^ = . (2) ; 
 
 Y V 
 
 consequently, (1) Xy — Ya?=0 .-. — =— ; that is, the lines repre- 
 
 A. X 
 
 sented by X, Y, are proportional to those represented by x, y, 
 (fig. 108), and, therefore, PR is in the same line as PS, that is, 
 the resultant of the forces on P is in the direction PS, so that the 
 planet P moves under the influence of a central force at S, the place 
 of the sun. We infer that this force must be attractive and not re- 
 pulsive, because, from the second law, the curve PP' is always con- 
 cave to S, and, therefore, P is drawn from its wonted path Fp to- 
 wards S. 
 
 (141.) Having established this, we have now only to compare 
 the polar equation of the orbit with the equation (8), at page 168,
 
 172 ELEMENTS OF DYNAMICS. 
 
 in order to dcterniine the value of R the force of attraction on the 
 planet. In the ellipse referred to the focus, the expression for r, is 
 
 (Anal. Geoin. p. 165,) r=—^ —; in which a is the serai-ma- 
 
 ^ l-fecos.e 
 
 jor axis, OB, (fig. 110,) e the ratio of the eccentricity OS to the 
 semi-major axis and d the variable angle P8B. But, instead 
 of SB, let us take any other fixed axis, SX, making with SB the 
 angle BSX=a, then calling PSX, w, g=qj — a, and, therefore, 
 j.^ a(l—e') .1^ ^ ^ l+fcos.(co— g) ^ _ ^ ^ 
 1 -f e cos. (co — a) ' ' r a ( 1 — €'■') • • • • v ;• 
 
 Differentiating this, with respect to the variable angle «, we have 
 du e sin. (« — a) _ 
 d^^ ~~a(l—e'') ' 
 
 , I-/.. • • . d^U eCOS. (co a) .^^ 
 
 and differentiating again, -j-t-= 77-^^ — 77-^' (2) ; adding this to 
 
 doi^ u ( 1— C") 
 
 equation (1) above, we have 
 
 —J—, + u = — — — . Hence, by equation (8), page 168 
 
 "~a (1— e-) ~ a(l— e*") "r* ^ ^ 
 
 The coefficient of — , in this expression for the central force is 
 
 constant for the same orbit ; hence every planet is retained in its 
 orbit by an attractive force residing in the sun, and varying 
 in intensity inversely as the square of the distance at which it 
 acts. 
 
 (142.) For the velocity at any point, we have 
 
 3 c* fdr_ 2 c'' 1 
 a{\—c-)J r^^a(l— O* 
 to determine the constant C we must know, a. priori, the velocity 
 at some given distance ; or we must know, at what distance and 
 with what velocity the planet is originally projected into space. 
 Calling this primitive velocity v^, and the corresponding distance 
 fj, we have 
 
 2c2 , 1 1 
 
 hence the velocity is greatest when r is least, that is, at that extre- 
 mity of the major diameter which is nearest to the sun: this point 
 is called the nearer apse, the curve being there perpendicular to the 
 radius vector. At the opposite point, or farther apse, the velocity 
 is least, since at this point r is greatest. 
 
 ^rr^i 3 c* pdr 2 c'' 1 ^ 
 
 ■' ail — e-)J r^ a (I — e") r
 
 ON THE MOTIONS OF THE PLANETS. 173 
 
 It may be remarked that the coefficient —7- -— , which occurs 
 
 •' a(l — e^) 
 
 in the foregoing expressions for R and v, is the value of the at- 
 tracting force at the unit of distance from the centre, being what R 
 becomes when r=l; 5C always represents the space described by 
 the radius vector in one second of time. 
 
 (143.) The equation (9) at page 169, will furnish a simpler ex- 
 pression for the velociiy than that just deduced ; for the perpendi- 
 cular p from the focus on the tangent being {Anal. Geom. p. 139.) 
 
 6V a^ri — e^)}- 
 r,2_ __v J_ 
 
 ' 2 a — r 2 a — r 
 
 2_ c^_ c^, 2 a— -r 
 
 ''' ^ ~p~'a2(l— e^' r ^' ^ 
 
 It will be easy to compare this velocity with that Avhich a body 
 would have revolving in a circle at the same distance r, and about 
 the same centre of force ; for, calling this velocity v', we know 
 
 that R = .*. w'^=Rr ; hence, referring to the value of R, in equa- 
 
 r 
 
 tion (3) above, we have 
 
 1 • IT 1 ■ 1 3 o — )' 1 
 
 vel." m ellipse : vel.^ in circle : : : — : : 2 a — r : a ; 
 
 ar r 
 
 that is, as the distance of P from the empty focus of the ellipse to 
 
 the semi-major axis. 
 
 From the same expression for v^ we learn that the velocity at the 
 mean distance, r=:a, that is, at the extremity of the minor axis, is 
 a mean proportional between the velocities at the apsides or at the 
 distances r=r', and r=2a — r'. 
 
 (144.) In the expression for the force (3) the quantities a, e, c, 
 which enter, are different for different planets ; we cannot conclude, 
 therefore, from this expression, whether, like terrestrial gravity, this 
 force is independent of the magnitude and constitution of the body 
 attracted, that is, whether at the same distance r, or at the unit of 
 distance, R has, in all cases, the same value ; but Kepler's third 
 law, which we have not hitherto used, enables us to establish this 
 point. Let T" be the time of a complete revolution of any planet 
 P ; then, c being the double area described in 1", cT will, by the 
 first law, be the double area described during a whole revolution, 
 that is, it will express twice the area of the ellipse ; but the area of 
 this ellipse is (Jnt. Calc. p. 123 art. 68) rf a^^/ \ 1 — e^; ] 
 
 ....T=.,«v!.-e'!.-.^^ = -45^. 
 
 In like manner with respect to any other planet P' 
 p2
 
 174 ELEMENTS OF DYNAMICS. 
 
 a'(l— e'^) T' 
 
 But, by Kepler's third law, 
 
 T : T' : : a" : a'~ 
 
 and each of these expressions denotes the influence of the central 
 force on each of the planets P, P', at the unit of distance, these in- 
 fluences, therefore, being the same, the force is of a similar nature 
 to that of terrestrial gravity, influencing all bodies alike at the same 
 distance from the centre of force. 
 
 (145.) It may here be remarked that the same law of force (equa. 
 3) would be established if we did not know, from observation, that 
 the paths of the planets were ellipses, but only that tliey were conic 
 sections ; for the equation (3) would remain the same except as re- 
 lates to the value of e which is either equal to, less than, or greater 
 than 1, according as the curve is a parabola, an ellipse, or a hyper- 
 bola. Hence if a body move in a conic section, iti virtue of a 
 central force at the focus, it must vary inversely as the square of 
 the distance at which it acts. 
 
 (146.) Let us now proceed to the inverse problem, viz. to the 
 determination of the orbit which a body must necessarily describe 
 about a central force, which varies inversely as the square of the 
 distance at which it acts ; this being the law of force which, as we 
 have before remarked, was discovered by Newton to be that which 
 governs the planetary motions. 
 
 Let R =— , /i being the intensity of the force at the unit of dis- 
 
 tance, or when r=l, then substituting this expression for R in the 
 general equation (5), page 168, we have for this particular law of 
 force the following expression for the velocity, viz. 
 
 ..=£!jl.i!l + ,.=?^+C..(l); 
 
 r^^ r'' rfu2^ ^ r ^ ^ ^ 
 
 the integral of which is the equation of the orbit. 
 
 But, by article 143, equation (2), when the orbit is a conic section 
 ^_ c" 1 dr"" _ c" 2a — r 
 
 ^-75l7i--^+M-„,(i_,a)- —;. (2;; 
 
 the integral of this equation is, therefore, the equation (1) p. 172, 
 involving the arbitrary constant o, and this, be it observed, is true 
 whatever be the values of the constants a and e. But if these be de- 
 termined so that 
 
 (^'•••"=M.-W''-'=- -«••'">' 
 
 the two equations (1), (2) will, obviously, become identical; hence
 
 MOTIONS OF THE PLANETS. 175 
 
 with these conditions equation (1) will also be the integral of (1) 
 above, and, in which c and a are fixed by the equations (3) and (4) : 
 (1), therefore, is the equation of the orbit sought. 
 
 We are warranted, therefore, in inferring the converse of the pro- 
 position at the close of last article, viz. that if the central force vary 
 inversely, as the square of the distance, the body must describe a 
 conic section, having that force in its focus. 
 
 (147.) Having thus determined the nature of the orbit, let us en- 
 deavour to ascertain its form, which will require the determination 
 of the constants c and C, both of which depend upon the circum- 
 stances of the initial motion of the body. 
 
 C may be determined from knowing the initial velocity and dis- 
 tance, that is, the impulsion with which the body is launched into 
 space, and the distance of the point of projection from the centre of 
 force, call these respectively v^ and r^, then from equation (1) 
 
 C=V-?^..(l). 
 
 To determine c requires that we know not only the point and velo- 
 city of projection, but also its direction ; or the angle it makes with 
 the radius vector at that point. Call this angle 6, then the initial 
 velocity, in the direction perpendicular to the radius vector, is Uisin.0, 
 
 which is, therefore, equal" to ''it-; but equation (3), p. 167, 
 
 „ dat , 
 
 fi^ — =c ; consequently, 
 
 c=.r^ Vi sin. 9 . . . . (2). 
 Hence the constants which enter into the equations (3) and (4) 
 are determined in terms of the initial quantities ; substituting 
 
 them in those equations, we get, from (4), a = i~^lTT ' ^'^^' 
 
 from (3), e= ^1 - - =^1 + j^- {v,-- -). 
 
 By means of these two equations the orbit may be constructed ; 
 its form may be completely determined by the second equation alone, 
 since the form depends entirely upon the value of e. It will be an 
 ellipse if e<;l, an hyperbola if ei>\, and a parabola if e=l ; that 
 
 is, the orbit will be an ellipse if Uj ^ <C — ; an hyperbola if u^ ^>> — ; 
 
 a parabola if v ^ ^ = — ; 
 
 so that the same central force which governs the planetary motions, 
 and causes them to describe ellipses, would have been equally com- 
 petent to cause them to describe either hyperbolas or parabolas, but
 
 176 ELEMENTS OF DYNAMICS. 
 
 not any other curves. For aught we know to the contrary, there- 
 fore, there in:iy be planets governed by the same attractive influence, 
 and moving in hyperbolic or parabolic orbits, and which, therefore, 
 continually proceed onward in space without ever returning. 
 
 It is a singular fact, that, as the foregoing expression for a, the 
 semi-major axis of the orbit, is independent of d, the angle of pro- 
 jection, the major axis of the orbit, will be of the same length what- 
 ever be the angle of projection ; the minor axis, however, will vary 
 with this angle, seeing that its sine enters into the expression for the 
 eccentricity. 
 
 As to the absolute lengths of these axes, they are at once deter- 
 mined when the initial conditions of the motion are given by means 
 of the equations (3) and (4), since the constants which enter them 
 are known in terms of these given conditions by equations (1) and 
 
 (2)- 
 
 The velocity of the body at any point of its orbit is given by 
 
 the equation (2) art. (146), and, by substituting this expression, 
 for V in the general equation (5), art. (135), we readily get a dif- 
 ferential expression for the angle to corresponding to that point, and, 
 finally, the value of du,, given by this expression, substituted in the 
 equation (3) of the same article, will furnish a differential equa- 
 tion between t and r, and thence the time of arriving at the pro- 
 posed point becomes known. 
 
 In all these results the quantity h enters, which quantity denotes 
 the value of the accelerative force at the unit of distance from the cen- 
 tre of attraction ; or rather it expresses the number of these units 
 in the linear space which measures the attractive force at the unit 
 of distance. Now the attractive forces of the sun and planets, corres- 
 ponding to any proposed distance, vary directly as their masses, there- 
 fore, whatever energy the unit of mass exerts at the unit of distance 
 the mass M of the sun exerts M times as much, and the mass 
 m of the planet, m times as much ; the whole force, therefore, 
 which the sun regarded as fixed, exerts on the planet at the unit 
 of distance, is (art. 119) M-|-?)i times as much as that exerted by 
 the unit of mass at the unit of distance. Considering this latter to 
 be the unit of force, and representing it by 1, accordingly M+m 
 will truly represent the attractive power exerted on the planet at 
 the unit of distance, and, therefore, at r such units distant tlie ex- 
 
 pression for the attractive force will be — , which is the value 
 
 h . 
 of —in the preceding formulas when we consider the action of the 
 
 sun on a single planet only. We may also express the intensity 
 of the solar and planetary attractions in terms of terrestrial gra-
 
 ON THE MOTION OF A SOLID BODY. 177 
 
 vity ; thus, calling the radius of the earth r^, and m^ its mass, we 
 have, since the attractions are directly as the masses and inversely 
 as the squares of the distances, 
 
 m, M Mr,'' 
 — : —: : sr : sr, 
 
 the attractive force of the sun at any distance r ; and, in like man- 
 ner, the attractive force of the planet is — —„^ ; hence their united 
 
 influence is ^ . g, the factor which multiplies g being an 
 
 abstract number. This number will, of course, remain the same if 
 we deliver the mass and distance each from its peculiar unit, re- 
 garding M, wi, r^, &c. to be numbers merely, in which case writing 
 
 M+m r,^ \ r/ . , 
 
 the expression thus . -^—k\ we see that —^2* is that which 
 
 we have taken above for the unit of attractive force ; it is plainly 
 the expression for the attraction of the mass 1 at the distance 1 . 
 
 We here close the second section ; for, to pursue these interest- 
 ing inquiries further, we should be compelled to pass by matters 
 more especially entitled to a place in a treatise on Mechanics. We 
 have, however, given thus much of the first principles of Physical 
 Astronomy, because we Avere unwilling altogether to omit touching 
 upon a subject so highly calculated to excite the inquiries of the 
 student, and because, moreover, we had indulged a hope of being 
 able to unfold these first principles with more simplicity than is 
 usually done. 
 
 SECTION III. 
 ON THE MOTION OF A SOLID BODY. 
 
 (148.) Hitherto we have limited our investigations almost 
 solely to the motions of a single point : or, if we have at any time 
 introduced the consideration of a moving body, it has usually been 
 on the hypothesis, that the influence by which it moved was dif- 
 fused uniformly through its mass, acting alike upon every particle ; 
 so that if any portion of the body were to be removed, all the in- 
 fluence, in virtue of which that particular portion moved, would be 
 taken away too, and the remaining part of the mass would go on 
 precisely in the same way as it would have done if accompanied by 
 the part subtracted, and precisely as it would do if reduced to a sin- 
 
 23
 
 178 ELEMENTS OF DYNAMICS. 
 
 gle particle. It is in this way that the forces of attraction are uni- 
 formly difTused through the masses of the attracted bodies, and to 
 this description of forces our attention has been almost entirely 
 directed hitherto. We have not, however, wholly overlooked 
 those motions which result from pressure constantly pushing for- 
 ward a mass of matter, not tliat we mean to mark any difference, 
 as far as effects are concerned, between pressure dynamically con- 
 sidered, and an attractive force ; for, as already observed at (104), 
 the motion produced by an attractive force may be considered as 
 due to the body's own pressure, as is manifest ; but if it were to 
 be urged by a pressure less than this, we ought obviously to ex- 
 pect a diminished acceleration ; and if it were urged by a greater 
 pressure, we should look for an increased acceleration ; but, as just 
 remarked, particular cases of this kind have already been consider- 
 ed, viz. in problems I. and II. at page 129 ; thus referring to the 
 first of these problems, we find that the whole weight to which 
 motion is given, is VV-fWj, which represents the whole pressure 
 of the mass moved ; but the motion is due to a less pressure, viz. 
 to Wj sin. I, — W sin. '^ i, and accordingly we find a diminished 
 acceleration. 
 
 (149.) The pressure which thus moves a body is called a moving 
 pressure, or rather a moving force ; the acceleration due to such a 
 ibrce, or the force competent to produce any proposed acceleration, 
 may be determined by the principle at page 121 : thus, calling the 
 whole pressure or weight of any mass ^I, W, the acceleration due 
 to this pressure g, and that produced by any other pressure or 
 
 F 
 moving force, F, we have o' : F : : W : W — =the moving force, 
 
 o 
 
 which expression obviously represents the weight which the mass 
 M would have if the accelerative force which impressed that weight 
 were F. 
 
 The weights of bodies obviously vary with their mass, or with 
 the quantity of matter they contain, and also, as the accelerative 
 force which impresses weight; that is to say, weight varies con- 
 jointly as the mass and the accelerative force ; we may, therefore, 
 in these inquiries substitute for any weight W the quantity to 
 which it is always proportional, viz. the quantity Mg"; M being the 
 mass or rather the number of times the body contains some fixed 
 unit of mass, and "■ being the value of gravity, or the force which 
 impresses weight on the mass. Representing the weight W by 
 M^", and putting ^ for the moving force, we have, by the foregoing 
 expression, 
 
 *=MF=M -j-=M-7— , 
 ^ dt dt*
 
 ON THE COLLISION OF BODIES. 179 
 
 sijpposing, as we here do, that each particle of the moving mass 
 Jias a common velocity at every instant. 
 
 As it immediately follows from this that F=— , we see that the 
 
 acceleration is as the moving force or pressure directly, and as the 
 mass inversely. 
 
 For the velocity at any time t", we have, supposing ^ constant, 
 
 dv^— (It .•. v= :^ t, so that the velocity acquired in any time i", 
 
 is the same, so long as the ratio, — , of the moving force to the 
 
 mass moved is the same. 
 
 It is usual to call the product My of the moving mass by its ve- 
 locity the momentinn, so that as accelerative force is represented by 
 the differential coefficient of the velocity relatively to the indepen- 
 dent variable /, the moving force is represented by the differential 
 coefficient of the momentum relatively to t. Instead of mass, how- 
 ever, we may, if we please, always substitute weight, since, as re- 
 marked above, these are always proportional. 
 
 We shall terminate these introductory remarks by observing, that 
 all the deductions at art. 105, respecting the acceleration, velocity, 
 space, and time, apply equally here ; and all the formulas em- 
 ployed there become suited to the present inquiry M'hen -rj- is sub- 
 stituted in them for F ; provided, as before mentioned, that the mov- 
 ing force impresses a common velocity on all the particles of the 
 mass moved, so that the acceleration of the whole may be that of 
 each particle. 
 
 CHAPTER I. 
 
 ON THE COLLISION OF BODIES. 
 
 (150.) In the present chapter we propose briefly to consider 
 the circumstances of the motions of bodies moving in certain di- 
 rections, from the effects of impulsion and impinging against each 
 other. 
 
 Let us suppose that the quantity of matter which we assume for 
 the unit is projected by any given impulsion ; it will move with a 
 constant velocity always proportional to the intensity of the force 
 of impulsion (p. 118); this velocity, therefore, will always cor-
 
 180 
 
 ELEMENTS OF Dl-XAMICS. 
 
 rectly rcprpsent tho intensity of the force. If we take two such 
 units, and two such imj)ulsions act on them simultaneously, and in 
 the same direction, the relative positions of tlic moving masses will 
 be always preserved, so that if the two units were actually blended 
 into one mass, and the two impulsions be thus made to coalesce and 
 form a double impulsion, the same velocity as before would be im- 
 pressed on the mass ; and it is plain that in like manner if M such 
 units be blended into one mass, which receives an impulsion of M 
 times the intensity we have supposed applied to the unit, still the 
 same velocity would be impressed ; consequently, when any body 
 whose mass is M moves from the effect of an impulsive force, the 
 correct expression for the intensity of that force must be My, tjie 
 mass into the velocity, that is the momentum, measures the impulsive 
 force. 
 
 Knowing then how to estimate the intensity of impulsive force, 
 we may proceed to consider the circumstances of 
 
 Direct Impact. 
 
 (151.) We shall first consider the bodies which impinge to be 
 entirely inelastic, or of such a nature that they are blended by the 
 impact into one mass, and our object will be from knowing the 
 forces of the impinging bodies to determine the motion of the united 
 mass. 
 
 Problem I. — Two inelastic bodies M, M, move in the same 
 straight line with velocities t", v^ : to determine the velocity after 
 impact. 
 
 The impulsive force on M is Mr ; that on Mj is M,Vj, and these 
 two combined form the impulsive force \»hich moves the blended 
 mass M + Mj ; but if V be the velocity of this mass, the impulsive 
 force on it must be (M + MJ V, 
 
 .'. (M+MJ V=Mt;+M,f, .-. V -±^^ . . (1) 
 
 the velocity required. If the body M^ were moving in a direction op- 
 posite to M so as to meet it, then v^ should be taken negatively, and 
 
 in that case, V= — ,, \ i— ^ . (2), and the sign of V thus deter- 
 M + M, ^ ^ ^ 
 
 mined will point out the direction in which the mass moves after 
 
 impact. 
 
 Mu 
 If the body M, be at rest, then, since v, =0, V^^^ — r-r . (3). 
 •^ * ' 1 ' M + M, ^ ^ 
 
 In all cases the momentum (M + MJ V after impact is the sum 
 
 of the momenta before impact, those being taken with opposite 
 
 signs which act in opposite directions ; so that the momentum lost
 
 ON THE COLLISION OF BODIES. 181 
 
 by the one body by the collision, is precisely that which is gained 
 by the other. In the last of the above cases (3), the momentum of 
 the whole mass being (M-fMj) \ = Mv, and the momentum gained 
 by M, which was at rest being M^Vj this must be the momentum 
 lo'st by M. 
 
 This loss of force in M shows that the mass M^ opposes a re- 
 sistance to the communication of motion, and M^V expresses the 
 value of that resistance, or the impulsive force necessary to balance 
 it ; this result is quite independent of the weight of the resisting 
 body, seeing that we here consider only its mass ; it is, therefore, 
 the consequence of its inertia, which is therefore proportional to 
 the mass. 
 
 (152.) Let us now consider the impact of two elastic bodies 
 which, as in the former case, move so as to impinge at some point 
 in the common line described by their centres of gi'avity. 
 
 Bodies of this class yield to the force of impact, and sutler a com- 
 pression, and therefore a change of figure ; and the elasticity is that 
 inherent force Avhich the body exerts to recover its original form. 
 If the force thus exerted at every point throughout the whole depth 
 of the impression, while the body is recovering its form, is equal to 
 the impressing force at that point, the oi'iginal form of the body must 
 be perfectly restored, and the elasticity is then said to be perfect.* 
 In this case, whatever velocity one body lost during the action of 
 the force of compression, it afterwards lost just as much more during 
 the action of the force of restitution ; and whatever velocity the other 
 body gained during the compression, it gained as much more during 
 the restitution ; for the compressing and restoring forces (which 
 are no other than continued pressures, although operating for an 
 exceedingly short time) are equal, and therefore equally oppose the 
 motion of one of the bodies, and equally favour the motion of the other. 
 
 It is easily seen that the same would be true if only one of the 
 bodies were perfectly elastic, and the other perfectly hard, or inca- 
 pable of receiving an impression. 
 
 When the force of restitution is not equal to that of compression, 
 but only the eth part of it, the elasticity is said to be imperfect, and 
 e measures its relative intensity ; it is competent to communicate 
 only the eth part of the velocity due to perfect elasticity. 
 
 Problem II. — Two elastic bodies M, M^, moving with velocities 
 V, v^, strike with direct impact: to determine their velocities after- 
 wards. 
 
 So long as the compression continues, the bodies move as one 
 
 * The restitution is moreover considered to occupy the same length of time as 
 the compression. 
 
 o
 
 182 ELEMENTS OF DYNAMICS. 
 
 mass, and therefore with the velocity V = — rr— ^t^-^s so that M will 
 
 lose the velocity v — V and M, will lose the velocity v^ — V, and 
 these would express the velocities communicated, but in opposite 
 directions, by the force of restitution, if the elasticity were perfect; 
 but let it be only the fth part of perfect, e being a fraction, then the 
 additional velocity lost by M will be e (v — V), and byM,, 
 e(v^ — V); hence the velocities after the impact will be 
 
 of the body M, V =v — (1-f e) (v — V) (1); 
 
 of the body M,, V"=u,— (1+e) (v^—Y) (2); 
 
 or, substituting for V its value above, 
 
 V=..-(l+e)4^-^) .... (4). 
 
 If c=0, that is, if the bodies are inelastic, then, as is plain from the 
 equations (1) (2), the velocities after impact are V' = V, V"=V, as 
 we know they ought to be. If e=l, that is, if the elasticity is per- 
 fect, then, from equations (3) and (4), 
 
 If we multiply each mass by the velocity of it after impact, we have, 
 when the bodies are perfectly elastic, 
 
 MV' + MiV"=Mu+M, v^ (7) ; 
 
 hence the sum of the momenta before impact is the same as the 
 sum after impact. 
 
 If we subtract the equation (2) from (1), taking e=l, we have 
 \'—Y"=v — 2v — v^-\-2v^=v^—v .... (8); 
 hence the difference of the velocities is the same both before and 
 after impact. 
 
 When the bodies are perfectly equal, as well as perfecdy elastic, 
 they exchange their velocities, each moving after the impact with 
 the velocity the other had before the impact ; this follows from put- 
 ting M=Mi, in the equations (5), (6) ; if, therefore, one body be 
 at rest, the other, which strikes it, will impart to it its entire velo- 
 city, and rest in its place. 
 
 From the equations (7) and (8) we have M (V'^-i')=M, 
 (Vj — V"); V'-|-t' = i",+V" ; multiplying these together there re- 
 sults MCV'" — v2) = M, (I'l" — V"'')or 
 
 MV'^ + M, Y"-=Mv'+M,v,^ (9) ; 
 
 that is, the sum of the products of each body into the square of its 
 velocity is tlie same, both before and after impact. The mass into
 
 ON THE COLLISION OF BODIES. 183 
 
 the square of the velocity is called the vis viva, or living force; 
 SO that in the collision of perfectly elastic bodies there is no loss 
 of vis viva. 
 
 If one of the bodies is an immoveable mass we may consider it 
 in the light of a mass M j at rest, and infinitely large ; on which 
 supposition equation (3) gives for the velocity of M after impact 
 V'= — ev .... (10); that is, as we might expect, M would re- 
 bound with the same velocity with which it struck the immoveable 
 mass. It is true that, conformably to our hypothesis, the immove- 
 able mass is supposed to be perfectly elastic as well as the imping- 
 ing body ; but the motion would be the same if it were perfectly 
 hard, because the force of restitution would still be the same as that 
 of compression, and the whole of this would be exerted to repel the 
 body. 
 
 On Oblique Impact. 
 
 (153.) When the centres of gravity of two impinging bodies do 
 not move in the same straight line, yet if at the instant of collision, 
 the shock which each receives be directed towards its centre of 
 gravity, the effects will be calculable by the preceding formulas. 
 Thus suppose two spherical bodies M, Mj, (fig. Ill,) move so that 
 their centres describe the lines PA, QB, and that they strike each 
 other at the point C ; we may decompose each of the velocities 
 with which the bodies impinge into two, one in the direction of a 
 common tangent to the bodies at C, and the other perpendicular to 
 this tangent ; the components perpendicular to the tangent will be 
 those to which the impact is entirely due, the other components will 
 merely express the lateral velocity, or that parallel to the tangent 
 plane, which the respective bodies had before impact, and which, 
 as nothing opposes it, they must still retain. Hence, having de- 
 termined the velocities consequent upon the direct impact due to the 
 effective components of which we have spoken, by means of the 
 formulas in the preceding problems, we shall then have to compound 
 these velocities, each with what the body originally had in the di- 
 rection parallel to the tangent plane at their point of concourse. 
 
 As an example, let us take the case where one of the spheres is 
 at rest and infinite in magnitude, or, which is the same thing, let 
 the body struck be an immoveable plane : let the velocity given to 
 the impinging body be v, the angle its direction makes with the 
 plane, that is, the angle of incidence a, then the components of the 
 velocity are v cos. a, v sin. a; the latter of these being perpendicu- 
 lar to the plane produces the impact ; therefore,' if e measure the 
 elasticity of the body, the velocity, after the impact, will be (equa. 
 10) Y'=:ev sin. a, and the direction will be perpendicular to the 
 plane ; hence the velocity, after impact, will be the resultant of the
 
 184 ELEMENTS OF DYNAMICS. 
 
 velocities v cos. a, and e v sin. a, of which the directions are per- 
 pendicular to each otiicr ; therefore the velocity of reflection will 
 be Vx/{cos.^a-\-€^ sin. -a J ; and the angle a' of reflection will be 
 
 ev sin. a 
 
 tan. a = =e tan. a ; 
 
 V COS. a 
 
 so that if the elasticity be perfect, the velocity and the angle of re- 
 flection will be respectively equal to the velocity and the angle of 
 incidence. 
 
 For a more enlarged view of the theory of percussion and impact 
 the student may consult Dr. Gregory's Mechanics, vol. i. and the 
 Mecanique of Poisson, tom. ii. ; and for a variety of examples re- 
 ference may be made to Bridge'' s Mechanics, p. 150 et seq. We 
 omit practical examples here, which, however, are very easily 
 framed, in order to make room for more important matter. 
 
 CHAPTER II. 
 
 THE PRINCIPLE OF d'aLEMBERT. 
 
 (154.) Inquiries concerning moving forces may be sometimes 
 considerably facilitated by the aid of a very simple and very gene- 
 ral proposition, first introduced into Dynamics by D\^lcmbert: it 
 may be regarded as a dynamical axiom, and may be announced as 
 follows : 
 
 If there be any system of bodies A, B, C, &;c. which in virtue 
 of the forces applied to them would, if entirely free, receive the 
 several velocities a, h, c, Sic. but which, on account of their mutual 
 connexion, receive instead the velocities o, ji, y, Sic. then it is evi- 
 dent that if the velocity a, impressed on A were resolved into two, 
 of which one is the velocity a, actually received, and the other some 
 velocity a' ; and if, in like manner, the velocity b impressed on B 
 were resolved into two, viz. the actual velocity j3, and some other 
 /3' ; and if a similar decomposition be effected for each impressed 
 velocity, the forces due to the component velocities a', ji', y'. Sic. 
 if severally applied to the bodies A, B, C, &c. of the connected 
 system would keep the system in equilibrium. 
 
 For by the decomposition, which has been effected, all the mo- 
 tion which actually has place must be due to the other components, 
 and, therefore, thoise of which we speak must destroy themselves in 
 consequence of mutual actions of A, B, C, Sic. on each other. 
 
 It follows, therefore, that there must always be an equilibrium 
 between the impressed forces and the actual, or effective, forces,
 
 THE PRINCIPLES OF D ALEMBERT. 185 
 
 these latter being taken opposite to their real directions, that is to 
 say, if to the body A we apply the force originally impressed, and 
 also the force due to the actual motion of A, this latter being oppo- 
 site to its real direction ; and if we do the same with all the other 
 bodies B, C, &c. the whole system will be kept in equilibrium. 
 For although the effective force on either of the bodies is not equi- 
 valent to the impressed force, yet, as we have just seen, the im- 
 pressed forces may be considered as resulting from the effective 
 forces, combined with another set which destroy each other; hence 
 as these equilibrate, the system must equilibrate by the combined 
 action of the impressed forces with the effective forces taken in op- 
 posite directions. 
 
 The forces here spoken of are, of course, moving forces or sim- 
 ply momenta; that is, continued pressures, or pressures of but 
 momentary duration. 
 
 As an illustration of the foregoing general principle we may take 
 the problem already solved, at page 129, viz. to determine the mo- 
 tions of two weights W, W^, along inclined planes, placed back to 
 back, the weights being connected by a thread. 
 
 Let us first ascertain the impressed forces, or those in virtue of 
 which the bodies would move if unconnected, these are evidently 
 for W, W sin. i, and for W^, W^ sin. i. Let us now determine 
 the effective or actual forces ; for this purpose call the velocity of 
 W, V ; and that of W^, v^ ; then as we know (149) that the mo- 
 tive force is always equal to the mass multiplied by the accelera- 
 
 W dv* W* dv 
 
 tion, the effective forces are on W, -^ and on W,, ~. 
 
 g dt g dt 
 
 Now, by the foregoing principle, if these forces taken with con- 
 trary signs, that is, taken negatively, be simultaneously applied to 
 the respective bodies with the former forces, the system will be in 
 equilibrium ; that is, there will be an equilibrium if to the two 
 bodies W, Wj at rest, there be applied the respective forces 
 
 w • . , W rfi; , „r . . W dv^ 
 
 W sin. t. A • —r and W , sm. % , . — -; 
 
 g dt ^ ^ g dt' 
 
 therefore, as these must pull the thread in opposite directions, we 
 
 must have the equation 
 
 .' W rfu . . Wj dv^ 
 
 W sm. I H -- = W 1 sin. i, • —-•. 
 
 g dt ^ 1 g dt' 
 
 * Regard must, of course, be paid to the signs of the acting forces ; so that 
 if we consider a force applied to a body in one direction to be positive, we must 
 consider any other force applied in an opposite direction to be negative ; there- 
 fore, as we here suppose the effective force on W to pull it up the plane, we 
 must tak^t negatively, because we have taken the impressed force on it, which 
 would pull it do-wn positively. 
 
 q2 24
 
 186 ELEMENTS OF DYN.tMICS. 
 
 therefore, since v is necessarily equal to t^, we have 
 5 _^_|. Zj! ^ = W, sin. L — W sin. i 
 dv W. sin. i. — W sin. i 
 
 " dt W + W, ^ ' 
 
 •which is the value of the accelerative force. 
 
 This solution it may be observed is not so simple as that ^ven 
 upon different principles at page 129 ; but it may serve to illus- 
 trate D'Alembert's principle. It may not, however, be amiss to re- 
 mark here that the solution of this and of other similar problems 
 may be readily obtained by equating two different expressions for 
 the effective moving force. Thus in the present problem the effective 
 moving force is obviously Wj sin. i^ — W sin. i ; it is also 
 W dv W, dv^ 
 
 and, therefore, we have the same equation as above. 
 
 We shall give another illustration of D'Alembert's principle in 
 this place. 
 
 Two weights Wj, W, are attached, the one to a wheel, and the 
 other to its axle, to determine their motions. 
 
 The impressed forces, or those in virtue of which the bodies 
 
 would move, if free, are the weights themselves ; the effective forces, 
 
 or those in virtue of which they actually move, are as in the last 
 
 problem 
 
 W dv W dv 
 
 • -7- on W, supposing this to ascend, and — - • -r 
 
 g dt ' il^ » g dt 
 
 on Wj, hence the system will be in equilibrium when to the bo- 
 dies W and Wj there are applied the respective forces 
 
 „, W dv ,„, W, dv. 
 
 W+ — . — and W, --ir'^ 
 
 g dt ^ g dt 
 
 and as these acting at the extremities of the radii r and R, of the 
 axle and of the wheel, tend to turn the system in opposite direc- 
 tions about the axis, we must have the equation 
 
 W dv W dv 
 
 ,(VV+^.|) = R(W,-^.i^)....(l); 
 
 the relation between v and v^ is easily determined, for, as the wheel 
 must turn in the same time as the axle, the velocities of the 
 weights attached to them must be as their radii, 
 
 R dv. R dv .„. 
 
 consequently the equation (1) is the same as
 
 ON THE MOMENTS OF INERTIA. 187 
 
 W R2 Wr dv 
 
 ' ^ g r ^ g Ut 
 
 dv_ R r W^—r^ W ^ 
 *'• lt~ W, R'^ + W7^ ^' 
 which expresses the accelerative force on the ascending weight, and 
 therefore for the velocity and space we have 
 
 _ RrW, — r^ W _ R r W, — r^V 
 
 ^~ W, RM- Wr ^^' * ~ 2(W,R^+ Wr)^ * 
 For the acceleration of the descending weight we have, in virtue 
 
 . , , dv, R^ W, — R r W 
 
 of the equation (2), ^ = ^y r. ^ ^y,.. S 
 
 R2 W, — R r W R2 W, — Rr W ^, 
 
 • ^ ^ ~ W, R2 -f VVr^ * ' ^ 2 (W, R'^ + Wr' 
 These two examples may suffice for the present to illustrate the 
 application of D'Alembert's principle. We shall have frequent oc- 
 casion to refer to it again in the course of the following chapters. 
 
 CHAPTER III. 
 
 ON THE MOMENTS OF INERTIA. 
 
 (155.) In treating of the rotation of a solid body, we shall always 
 have to take account of that portion of the impressed forces Avhich 
 must necessarily be employed in overcoming the inertia of the 
 system, and which, therefore, are not effective in producing motion. 
 In a body free to move in any direction, the inertia to be over- 
 come by the impressed force before motion can ensue, is obvi- 
 ously as the mass to be moved ; but when the body is compelled 
 to turn about an axis, the amount of inertia will obviously vary 
 with its distance from that axis. This resistance to motion about 
 any axis is called the moment of inertia of the system with respect 
 to that axis : and we shall see in the next chapter, that this mo- 
 ment is expressed by the sum of the products of all the particles 
 of the body into the squares of their respective distances from the 
 axis of rotation; that is, m, m', m", &c. denoting the component 
 particles of the mass M, and r, r', r", &c. their respective dis- 
 tances from the axis of rotation ; then, using the rotation already 
 employed at page 57, we shall prove that, with respect to that axis, 
 mom,ent of inertia = 2 (mr^). 
 
 At present we shall apply ourselves merely to the determination 
 of this expression in particular cases, as preparatory to the theory
 
 188 ELEMENTS OF DYNAMICS. 
 
 of rotation, to be delivered in the succeeding cliapters. But to 
 render the expression suitable for calculation, we must change it into 
 the form fr^dM, to which it is shown to be equivalent by precisely 
 ihe same reasoning as that employed at art. 41, to which the stu- 
 dent may turn. 
 
 If the whole mass M of the revolving body could be collected 
 into a single point at some distance k from the axis, then the 
 moment of inertia of the system, that is of this point, would be 
 simply k^ M, and this will be the same as the moment which ac- 
 tually has place, provided we so determine k that 
 
 k^M=fr^dM.:k=^-'—^, 
 
 so that if we can determine A-, which is called the radius of gyra- 
 tion, we shall at the same time know the moment of inertia. 
 
 It is frequently of consequence to know what mass M' ought to 
 be placed at any proposed distance k' from the axis, so that the 
 moment of inertia of the point thus loaded may equal that of the 
 mass M. In order to this, we must evidently have A:'-M'^A-'M 
 
 .-. M'=f-. M. 
 
 1. To determine the radius of gyration of a slender rod of length 
 / revolving about its extremity. 
 
 Putting r for any distance measured on the line from the axis, 
 
 T^ dr r' 
 
 we have by the formula k=y/— — = v/^j that is for the whole 
 
 line, or when r=l, k=l .^ I =•57735 I. 
 
 If the rod revolve about any point in it of which the distances from 
 the extremities are a and b, then, by taking the above integral be- 
 tween the limits r = — b, r=^a, we have, for tlie line a-\-b, 
 _ I «^ + 63 
 
 2. To determine the radius of gyration of a circle revolving ab- 
 out an axis through the centre, and perpendicular to its plane. 
 
 Putting R for the radius of the circle, its area will be ft R% also 
 at the distance r from the axis the area is n r^, therefore, in this 
 
 J,, ^ , ,7 Artfr^dr F" ^ 
 
 case, a M = 2 rt rdr, so that «=v — i— - — := ^ . — -, 
 
 rt K- \ K^ 
 
 that is, when r=R, A:=R y/k^=l R ^/'Z, and the result would ob- 
 viously be the same for a right cylinder revolving round its axis. 
 The moment of inertia is A:'^M = 5 jt R*. 
 
 3. To determine the radius of gyration of a circular annulus or 
 flat ring, the axis of rotation passing through the common centre 
 of the circles perpendicular to their plane.
 
 ON THE MOMENTS OF INERTIA. 189 
 
 Calling the radii of the inner and outer circles R and R', we 
 have, for the area of the annulus, the expression n R'^ — rt R^ 
 
 If, instead of the radius R', we take any variable radius r, inter- 
 mediate between R and R', then the expression for the annulus will 
 be M=rtr''— rtR'' .-. dM=2Krdr 
 
 _ 2fr^dr __^ _r| 
 
 ■" " ~ R'^_R^ ~2 (R'^-R^y 
 and this, taken between the proposed limits of r, that is from r=R 
 
 ..=«.,.. .=^-^_«;-....=.jw.(,, 
 
 when R=0 the annulus becomes a circle, and the expression for 
 k agrees with that in last example ; when R'= R the expression is 
 k=R .... (2), which applies when the annulus becomes merely 
 a circumference. It is easy to see that the expressions (1) would 
 remain tlie same for a cylindric shell or wheel whose thickness is 
 R' — R revolving about its axis, and the expression (2) applies when 
 the shell is of insensible thickness. 
 
 4. To determine the radius of gyration of the circumference of a 
 circle revolving about its diameter. 
 
 The distance of any point (x, y) from the axis of rotation is y, 
 the origin being at the centre, and the proposed diameter being 
 taken for the axis of x. Moreover, since 
 
 J-——, we have dM = ds= -dx .'. y'^ dM.= Rydx 
 ax y y 
 
 ,, 'Rfydx R X area of circle ?< R^ , -P2 
 
 s circumference 2 « R 
 
 The student cannot fail to have remarked the similarity between 
 these problems, and those for finding the centre of gravity, and he 
 must further observe that in determining the distance k of the centre 
 of gyration from an axis, the same regard must be paid to the man- 
 ner in which d M is taken that was necessary in determining the 
 centre of gravity, as the form of this diflerential will vary with the 
 position of the axis from which k is measured. If the body is a 
 plane area, and if, moreover, the axis from which k is measured is 
 in that plane, then, taking this axis for that of x, and putting (/M 
 under the general form dM=fdxdy, as at page 64, the expression 
 for A;^M will be, since what we have before called r is now y, 
 
 km=ffy^dxdy=f^. 
 
 Or this expression for the moment of inertia of a plane area, with 
 respect to an axis in it, may be deduced from considering the area 
 to be generated by the ordinate y moving along the axis of x ; for 
 as y generates the whole area, so must the momentum of y generate
 
 190 ELEMENTS OF DYNAMICS. 
 
 the whole momentum ; in the one case the generatrix is y, and 
 the quantity generated fydx; in the other case the generatrix 
 (see ex. 1,) 
 
 is ^ and therefore the quantity generated must be / ~ — . 
 3 *J o 
 
 5. Applying this formula, to the circle revolving about its dia- 
 meter or axis of x, we have, from the equation of the circle, 
 y^^2rx — x^ .'. 2ydy={2r — 2x) dx 
 
 .-. k=M = -r—r-'^^j!M==^rtr*(Int. Calc, p. 42), 
 
 this is for the semicircle ; but if instead of integrating from y= — r 
 to y^r, we had integrated from y= — to y^O, we should have 
 had for the whole circle A:'^M = 4 nr*; as in the whole circle M 
 is twice as great as in the semicircle, k^ is the same for both, viz. 
 
 To determine the moment of inertia of a solid of revolution, 
 with respect to the fixed axis, it will be most convenient to view 
 the solid as generated by the motion of a circle which continues al- 
 ways perpendicular to the fixed axis while the centre describes this 
 axis, as explained at page 144 of the Integral Calculus. In this 
 point of view, we may obviously consider the whole moment of in- 
 ertia to be generated by the moment of inertia of the generating 
 circle ; calling the variable radius of tliis y, the fixed axis being 
 that of X, the generating moment will, by ex. 2, he i ny*; hence 
 the whole moment generated must be k 2M = 5 ttfy*dx. 
 
 6. Suppose tlie solid were a sphere of radius a, then 
 2/2 -= 2 (IX — X- .•. y* dx = (2 ax — x^)^ dx 
 
 .•.A:2 M = hnf{2ax—x^) ''dx=H x^ {^ a^ — h ax -^^\x^), 
 and this integral between the limits x = 0, a? = 2a, gives for the 
 
 8 2 
 
 whole sphere k^ 'M = -— rt a' , .• . k'^ =— a^ . In like manner we 
 ^ 15 5 
 
 should find for a cone revolving about its axis A;2= ~^r^, r being 
 
 the radius of the base. 
 
 For a paraboloid the expression is k^ = l r^. 
 
 (156.) It is useful to know how to find the moment of inertia, 
 with respect to an axis, by means of the known moment with 
 respect to some other axis parallel to it. We shall, therefore, now 
 show how this is to be done. 
 
 Let AZ be the axis (fig. 112,) for which the moment of inertia 
 isy>*(ZM, and let A'Z' be the axis parallel to it for which the mo- 
 ment of inertia fr'^dM. of the same mass M is to be determined. As- 
 
 X il-^. .'. « : r: ' . 'v ''f / s: ■'■ i ' '- '^^^ 
 

 
 ON THE MOMENTS OF INERTIA. 191 
 
 sume AZ to be the axis ofz, and AX, AY to be those of a: andy; then 
 for every particle m of the body, the corresponding value of Z?n or 
 of its projection Am' is r^=x''-\-y^. In like manner, the distance 
 of the two axes being a, if we call the co-ordinates AA', A'B of this 
 axis a, i3, we shall have a^ = a^ + Z^'^- Now the distance of the 
 particle m from A' Z', that is of the point (x, y) from the point 
 (a, /3), is 
 
 r'^={x—ay+{y-~^y 
 
 — x^-\-y"-—1 a X—2 ^y-\-a^-\-^'^ 
 = r^ — 2 a X — 2 ^y-\-a^, therefore 
 /r'2 rfM=/r2 dM—2 ajx rfM— 2 ^ fy dM-^-a^M; 
 or, putting X and Y for the co-ordinates of the centre a gravity of 
 the system, Ave have (art. 41,) 
 
 fr'^ dM=fr^ (M+a^M — 2M (aX-f^Y). 
 When the original axis passes through the centre of gravity, then, 
 since Y=0 and X^O, the formula becomes 
 
 fr'" dM=fr"-dM-\-a''M, or M A:'=^=M [k^+d") ; 
 hence to the moment of inertia, estimated from an axis through the 
 centre of gravity, we must add the product of the mass by the 
 square of its distance from the new axis, and the sum will be the 
 new moment of inertia. 
 
 The foregoing expression for the moment of inertia about any 
 axis, shows that of all axes taken parallel to each other, that which 
 passes through the centre of gravity of the body will be the one in 
 reference to which the moment of inertia is the least ; k is in this 
 case called the principal radius of gyration. 
 
 1. Take a straight line or slender rod I revolving about an axis 
 at the distance a from the middle ; then, by the first example of last 
 article, the moment about the middle is •y'j P, therefore the moment 
 about the proposed axis is k'^ M=y^2 l^-{-a^ I. 
 
 2. Let a circle revolve about any line in its plane. 
 
 The moment round a diameter parallel to the line is, by ex. 5, 
 irtr*,and consequently, calling the distance of this diameter from 
 the proposed line a, we have 
 
 k^ M=k rt r*+a^ M .-. A;2==| j-^^a^ 
 The result is necessarily the same when the proposed axis is out 
 of the plane of the circle. 
 
 3. Required the moment of inertia of a cone revolving about an 
 axis through the vertex, and perpendicular to the axis of the cone 
 (fig. 113). 
 
 Let VA be to AB as 1 to n, then calling the distance VA' of any 
 variable section x, we have A' P=/ia; for the radius of that section, 
 and for its moment of inertia the expression, as found in last ex- 
 ample, is
 
 192 ELEMENTS OF DYNAMICS. 
 
 i rt7i* x*-\-x'^M=i rtn*x*+rtn^x*=7tn^{4 n«+l)a;*; 
 hence, for a solid generated by this circle, the moment is 
 
 which applies to the cone of altitude x. 
 
 CHAPTER IV. 
 
 ON THE ROTATION OF A SOLID BODY ABOUT A FIXED AXIS. 
 
 (157.) Suppose a body of a known form and mass to be freely 
 moveable about a fixed axis, passing through A (fig. 114), and per- 
 pendicular to the plane on which the figure is represented. 
 
 Suppose an impulse or shock to be given to the body thus cir- 
 cumstanced, and that it is in the direction BC, perpendicular to the 
 plane AB, drawn through the fixed axis ; if the impulsion were 
 oblique to this plane we should only have to take account of that 
 component of it which is perpendicular to the plane, because the 
 other being directed towards the fixed axis, would be counteracted 
 by its resistance, and its effect destroyed. 
 
 If, when the impulse was given at B, we conceive the mass to 
 have been perfectly free, and to have .been concentrated in the point 
 B, or, which is the same thing, if we conceive the impulse to have 
 been directed towards the centre of gravity of an equal mass M, 
 perfectly free, then, v being the velocity communicated, Mv would 
 be the momentum communicated; this expression, therefore, pro- 
 perly represents the intensity of the impulse or the force impressed 
 on the systems (150). The effect of this impulse upon the parti- 
 cles of the body, under consideration, is to cause each to describe a 
 similar arc of a circle in the same time, the radius of any one. being 
 Am=r ; each particle will, therefore, move with the same angular 
 velocity about the fixed axis, and this, therefore, may be called the 
 angular velocity of the whole mass ; let it be represented by w. 
 Now the only force impressed on the system is Mi», acting at B, 
 and the forces witli which the component particles vi, in', in". Sic. 
 move, arise from their mutual connexion with each other ; we may, 
 therefore, apply to this inquiry the principle of D'Alembert, and 
 thus obtain an equation between the impressed and the actual 
 forces. As the particles all have the common angular velocity u, 
 their actual velocities are rw, r'w, r" u. Sic. and, consequently, their 
 moving forces, or momenta, mru,, m' r' u, m" r" w, &;c., and these 
 are the forces which acting at the distances r, r', r", &c. from the 
 axis of rotation, must equilibrate with the force Mv, acting at the 
 distance R=AB, when this latter acts in the opposite direction ;
 
 CENTRE OF OSCILLATION. 193 
 
 consequently, multiplying each force by its distance from the axis, 
 we must have the equation 
 
 lVTT?i) 
 
 mr^i^+m' r' ^ o>+m" r" ^ ui-\-ikc.=MRv .-. co=— — ; — ■. (1) ; 
 
 2 {r- m) ^ ' 
 
 which implies that the angular velocity is equal to the moment of 
 the applied force, divided by the moment of inertia, just as the 
 linear velocity v, which the same force is competent to produce on 
 an equal mass, is expressed by that force, M.v, divided by the mass 
 moved, by which mass the inertia of the body is always repre- 
 sented : so that as M denotes the resistance to progressive motion, 
 X (r^in) denotes the resistance to angular motion. This expression, 
 therefore, is fitly called the moment of inertia of the revolving body. 
 (158.) Let us now suppose that instead of an impulse giving ro- 
 tation to the body, every particle of it is actuated by an accelerative 
 force ; these forces may be all different, but, as before, we shall 
 consider them as acting in planes perpendicular to the fixed axis. 
 Calling the several forces acting at m, m', m", &c. F, F', F", &c. 
 and /J, p',p", the perpendiculars from the axis on their directions; 
 the applied motive forces will be mF, m' F', m" F", &c. and u 
 
 being the angular velocity, or rw the absolute velocity of m, r — 
 
 dcj 
 will be its acceleration, so that the actual motive forces are mr — , 
 
 m' r' —, m" r" — , &c. ; hence, by D'Alembert's principle, these 
 
 forces, taken in the reverse direction, balance the former, so that 
 their moments give the equation 
 
 («ir^+m' r'^+m" r'^)—=mFp-\-m' F'p'-{-m" F" p"-\-&,c. 
 
 . (/^^ S (Fpm) _ 
 
 "dt 2 {r"- m) ^^ ' 
 
 that is, as before, the angular acceleration is equal to the moment 
 of the applied moving forces, divided by the moment of inertia, 
 just as the linear acceleration which the same forces sF • 2w is 
 competent to produce in an equal mass, 2?72 is expressed by that 
 force divided by the mass moved, that is, by 2»i. 
 
 Let us now apply the result just obtained to the theory of the 
 compound pendulum. 
 
 Centre of Oscillation of a vibrating Body. 
 
 (159.) When a heavy body vibrates about a horizontal axis, by 
 the force of gravity, the mass is considered as a compound pendu- 
 R 25
 
 194 ELEMENTS OF DYNAMICS. 
 
 lum, and it is an important problem to determine what must be the 
 length of a simple pendulum which shall perform its oscillations 
 in the same time about the same axis of suspension ; or rather 
 to determine what simple pendulum must be substi luted for the 
 compound mass, in order that the vibrations of both may be the 
 same as regards velocity, acceleration, and time. 'J'he foregoing 
 general formula enables us to solve this problem ; for as the acce- 
 lerative force acting upon every particle of the mass is the same, 
 viz. F^g, the formula, as applied to this case, may be written 
 
 Now let G (fig. 115,) be the centre of gravity of the oscillating 
 body, M, and AB a vertical plane through the horizontal axis of 
 suspension, which axis we here suppose to l>e perpendicular to the 
 plane of the paper ; let GP be perpendicular to this plane when 
 the body is in any proposed position, that is, when the plane AM, 
 through the centre of gravity and perpendicular to the plane of the 
 paper, makes any angle GAB=0 with the plane AB ; then we 
 know, by the theory of the centre of gravity (41), that 
 
 MxGV=mp-\-m'p' +m"p"-\-, &c. 
 that is, putting a for the distance AG of the axis from the centre of 
 gravity, since GP = a sin. e ; M . a sin. = S {mp), substituting, 
 therefore, this value in the numerator of the expression (1) above, 
 we have 
 
 rfu M . a sin. 6 M . « sin. B a sin. 6 . 
 
 l/?^"T(r2wy~^M|F+a2^==pq:^ ••••()' 
 
 where k represents the radius of gyration, in reference to an axis 
 parallel to that of suspension, and passing through the centre of 
 gravity of the body, and where a is the distance of these two axes. 
 Suppose now that the mass were removed, and that in its stead 
 there were suspended a single particle at the distance / from the 
 axis, then 2 (r^m) would become simply I'^m, and in order that the 
 angular acceleration of this simple pendulum may be the same as 
 that of the mass, for which it has been substituted, we must deter- 
 mine / from the equation, 
 
 M . a sin. e ml sin. 6 - a 1 
 
 "^ {r^m) ~ Wm ' ' 2 (r^n) ~~ 7 
 
 this, therefore, expresses the length of the simple pendulum, whose 
 vibrations will agree with that of the compound pendulum : it is 
 equal to the square of the radius of gyration measured from the axis, 
 divided by the distance between the axis and centre of gravity.
 
 CENTRE OF OSCILLATION. 195 
 
 That point in the body, wliich is at tlie distance / from the axis, is 
 
 called the centre of osdllation of the compound pendulum. There 
 
 are, it is evident, an infinite number of such centres, or of points at 
 
 the distance / from the axis, but we here more especially mean that 
 
 point which is in the line passing through the point of suspension 
 
 and the centre of gravity of the mass. 
 
 As OG is the distance of the centre of gravity G, from the centre 
 
 of oscillation O, it follows, from the equation (3), that the distance 
 
 A- 2 
 between these two centres is 0G= — .... (4); therefore, since 
 
 a 
 
 so long as the plane of the body's vibration remains the same h must 
 
 remain the same, it follows that with this condition the distances 
 
 between the centres of gravity and oscillation are inversely as the 
 
 distanrps between the centre of gravity and point of suspension. 
 
 Moreover, if the centre of oscillation were made the point of 
 suspension, then to find the corresponding value of / we must put 
 ttfe value (4) for a in (3) which substitution, as it alters not the 
 value of /, shows us that, provided we do not alter the plane of the 
 body's vibration, the centre of oscillation and point of suspension 
 are convertible ; that is, if we convert the centre of oscillation into 
 the point of suspension, the point of suspension will become the 
 centre of oscillation. 
 
 Also the distances of the axis of suspension from the centres of 
 gravity, of gyration, and of oscillation, are in continued proportion, 
 
 for a : \/\a^-\-k^\ : : ^\a^-\-h^\ : a-\ — . We may, likewise, infer 
 
 from the value of Z, what the distance a of the centre of gravity from 
 
 the axis must be, in order that for the same plane of vibration the 
 
 time in which the body performs its oscillations may be the least 
 
 possible ; for as the equivalent simple pendulum will vibrate the 
 
 quicker the shorter it is, we shall merely have to determine a so 
 
 that we may have 
 
 A:^ . . dl ^ ¥ 
 
 l=a-i — =a mimmurn. .-. -7- = !' =0 .-. a=k: 
 
 a da a'^ 
 
 so that the axis of suspension must pass through the principal cen- 
 tre of gyration. Several other particulars might be deduced from 
 the foregoing investigation, but we shall mention here but one more, 
 which is that if the compound mass consist of several distinct bodies, 
 the centre of oscillation of the whole will be found by taking the 
 continued product of each mass into the respective distances of its 
 centres of oscillation and of gravity, from the axis of suspension, 
 adding the products together, and dividing the sum by the product 
 of the whole system into the distance of the common centre of 
 gravity from the axis.
 
 196 ELEMENTS OF DYNAMICS. 
 
 For calling the several masses M, M', &c. and the length of the 
 equivalent pendulum L, we have, by the preceding theory, 
 
 1 = 
 
 M . a 
 
 M' {a'^-j-k'") 
 
 ^~ M'.a' 
 
 ^.•.2(/.M.a)=slM(o'+A-^)|. 
 
 &c. &c. 
 
 As the second member of this equation expresses the moment of 
 inertia of the whole system, it follows that 
 
 ^- 2(M.a) ••••^^^ 
 which establishes the proposition, since the denominator of this 
 fraction is equal to the whole compound mass into the distance of 
 its centre of gravity from the axis. This same formula will, obvi- 
 ously, serve for any vibrating mass when we find it convenient to 
 consider it in separate parts. 
 
 (160.) We shall now give a few examples of the determination 
 of the centre of oscillation in different bodies. 
 
 1. To determine the centre of oscillation of a slender rod or 
 straight line suspended at any point. 
 
 Let a, b, be the lengths, on contrary sides, of the point .of suspen- 
 
 sion, then (155) for k^ we have, k^=— — — — -. 
 ^ ' a-\-h 
 
 Again, the centre of gravity being in the middle of the line, its 
 distance from the point of suspension is ^ (o — h)\ hence (3), 
 ,_ 3 (o' + ^") _2jo=— _o^+^) 
 — A (a3_6i)— 3(« — 6) ■ 
 If the rod is suspended at its extremity, then 6=0, and l=\a., or 
 two thirds the length. If it is suspended at its middle, then a-=b, 
 and /= Gc, that is, the centre of oscillation is at an infinite distance, 
 and, therefore, to perform one vibration would require an infinite 
 length of time, and this is tlie same as saying that no vibration at 
 all could take place ; indeed, in whatever position about the centre 
 of motion the rod be placed, it would obviously rest there, seeing 
 tliat its centre of gravity would be supported. 
 
 If 6 = do, then l:=a, or two thirds the whole length, the same 
 distance as when the rod is suspended at its extremity ; so that in 
 both these cases the oscillations will be performed in the same time, 
 as, indeed, ought to be the case, because the centres of suspension 
 and of oscillation are merely interchanged, (p. 194.) 
 
 2. To determine the centre of oscillation of an angular pendulum 
 (fig. 116,) composed of two equal slender rods AB, AC. 
 
 Bisect the arms AB, AC, in ^ and y ; these points will be their
 
 CENTRE OF OSCILLATION. 197 
 
 centres of gravity ; bisect the line joining them in G, and this will 
 be the centre of gravity of the system, and AG^Ag* sin. g, or AG 
 = 5acos. e. 
 
 Again, the distance of A from the centre of oscillation of the part 
 AB or of AC, is equal to | AB = |fl, therefore, by the formula (5), 
 
 L= ^'^ , ^—=% a sec. 6 ; 
 
 2 a . i a cos. 9 
 
 hence because L, the length of the equivalent simple pendulum, in- 
 creases with 6, it follows that the time of vibration of an angular 
 pendulum may be increased without limit, by merely increasing the 
 angle between the arms ; when 6=0, that is, when the arms close, 
 and form but one rod := a, the corresponding value of L is |a, and 
 when 0=90° or when the arms open into a straight line, L is in- 
 finite. 
 
 When the time of vibration is given, we may easily determine 
 the corresponding angle of the arms when their lengths are known, 
 for, from the given time, L will be determined, and thence 
 
 3L 
 
 sec. e=- — . 
 2 a 
 
 Thus if the time is one second, then L =395, and if «=15 inches 
 
 sec. 0=75° • U'i .-. /_ BAC = 150° • 23'. 
 
 3. To 'determine the centre of oscillation of a sphere. 
 
 Let r be the radius of the sphere, then the mass is | rtr^, and the 
 square of the principal radius of gpation is (p. 189-190,) A:^=|r^ ; hence 
 
 ( = «+-=»+-. -....(I); 
 
 a being the distance of the centre of the sphere from the point of 
 suspension. 
 
 If the axis of suspension were a tangent to the sphere, we sliould 
 have r = a, therefore, in that case, I = r -\- fr. 
 
 From the expression (1) we get for a, w^hen Z is given, 
 
 a = n ± x/ [W—ir^; 
 so that a sphere may be suspended at two different distances from 
 the centre, and yet vibrate in the same proposed time ; if, however, 
 I /* = I 7-^, that is, if the time of vibration is to be that which be- 
 longs to the simple pendulum, Z =2?- ^|, then there is but one 
 suitable distance for the centre from the axis of suspension, viz. 
 the distance a = r ^ |. 
 
 4. Suppose the bob of a clock pendulum to consist of two 
 spheric segments joined at their bases : to determine the distance 
 of the centre of oscillation from the centre of the bob. 
 
 Let r be the radius of the sphere, and x the height of one of the 
 segments, then the moment of inertia of the two segments is (ex. 6, 
 r2
 
 198 ELEMENTS OF DYNAMICS. 
 
 p. 190), MA-« = 2 ;t x^ (I r« — 5 rx +j\x''), and the volume of the 
 two segments is 
 
 M = 2 , .. (r- J .) .-. i'= ^^S^^ 
 ^ ' a a (r — j x) 
 
 and this is the distance of the centre of gravity of the bob from 
 
 the centre of oscillation. 
 
 If instead of the radius r of the sphere, the radius r' of the base 
 
 of each segment is given ; then since r'" = (2 r — x) x 
 
 .-. r= — ■ and, by substitution, - = 7 .. ,„ , — rr , 
 
 a added to either of these expressions will give the distance of the 
 centre of oscillation from the point of suspension. 
 
 5. To determine the centre of oscillation of a cone suspended 
 at its vertex. 
 
 Putting (as in ex. 3, page 191), x for the altitude of the cone, 
 and nx for the radius of its base, we have, for the moment of inertia, 
 the expression -} rt 7i^ x^ (? n^ -^ 1), also the distance of the centre 
 of gravity from the vertex is (4G,) I x, consequently, 
 
 ^^W^x^ {kn- -\-\) ^ \ ^n- x^ {k n- + \) ^ , ^^,^ ,^ 
 M.ix h x.nn^x^'.ix * * ' 
 
 6. In a similar manner is found for the centre of oscillation of a 
 circle vibrating edgewise, and suspended at the distance <t from its 
 
 centre, / = a + :;-• 
 2« 
 
 7. And when the circle vibrates flatwise I = a -\- ■—. 
 
 4a 
 
 On the Centre of Percussion. 
 
 (161.) When a solid body revolves about a fixed axis, the {ore4 
 with which it would strike a fixed obstacle would vary with the 
 .situation of the point of impact, as also would the shock received 
 by the immoveable axis ; but there is a point at which, if the im- 
 pact take place, the obstacle will receive the whole force of the 
 moving body, and the axis will receive none, so that, if at the in- 
 stant of impact the axis were to be annihilated, the body would 
 still remain at rest. This point is called the centre of percussion. 
 In order to ascertain the situation of this centre, let us refer to the 
 expression (1) for <o, u being here considered to represent the an- 
 gular velocity at the instant of impact ; let also the moving force 
 due to the impact, and which we before represented by Mv, be 
 called/, and the distance of the direction of impact from the axis 
 
 <d 2 (r^ in) 
 D, the formula referred to then gives / = ^^ -; this ex-
 
 CENTRE OF SPONTANEOUS ROTATION. 199 
 
 presses, therefore, the force of percussion at the distance D. Now 
 if this force is equal to that of the whole moving mass, no portion 
 of it will be expended in straining the axis, and D will in that case 
 measure the distance of the centre of percussion from the axis. 
 
 To determine what the whole force of the revolving body is, Ave 
 must add together the forces due to the component particles ; that 
 due to any one, m, is mru> ; hence, calling the whole force F, we 
 have 
 
 ^ ^ ^ ^ D 2 rm) 
 
 consequently the distance of the centre of percussion from the axis 
 
 is equal to the distance of the centre of oscillation from the axis ; if, 
 therefore, the impact be perpendicularly directed to the plane pass- 
 ing through the axis of suspension and centre of gravity, then the 
 centres of percussion and oscillation will be in the straight line par- 
 allel to the axis of suspension. 
 
 If the plane through the centre of gravity, and perpendicular to 
 the fixed axis, divide the body symmetrically, it is plain that what- 
 ever force directed in this plane strike the body, the axis may sutler 
 a direct shock, but it will not be twisted ; in such bodies, therefore, 
 the centre of percussion coincides with the centre of oscillation, be- 
 cause at this point the impact will neither strain nor twist the axis. 
 But in other cases, impact at the centre of oscillation, although it 
 would occasion no direct strain on the axis, may yet tend to twist 
 it, as it is easy to conceive, (see Br. Gregory^ s Mechanics, vol. l.p. 
 300 ; and Francoeur^s Mecanique, p. 352.) 
 
 Between the centre of gyration of a body revolving about a fixed 
 axis and the centre of percussion, we may remark this difierence, 
 viz. the centre of gyration is the point in which, if the whole revolv- 
 ing mass were concentrated, the same angular motion would be 
 generated by any force as before, and therefore, an obstacle meeting 
 the line which connects this point with the axis of motion, will be 
 struck with the same force as it would be by the revolving mass, at 
 whatever point of the line the impact take place. But the centre of 
 percussion is that particular point which would strike an obstacle 
 with the whole force of the revolving mass. 
 
 Of the Centre of Spontaneous Rotation. 
 
 (162.) Intimately connected with the centre of percussion is that 
 of spontaneous rotation. 
 
 We have just seen that when a body revolves on an axis, there 
 exists a point at which a fixed obstacle would receive the whole of 
 its force, so that if this same force were to strike the body when at
 
 200 ELEMENTS OF DYNAMICS. 
 
 rest in the same point, it would produce in it a rotatory motion round 
 the same axis, even though that axis had l)ocn removed ; the axis 
 about which a quiescent body, when struck in a direction not pass- 
 ing tlirough tlie centre of gravity, thus spontaneously revolves, is 
 called the axis of spontaneous rotation. 
 
 Instead then of considering the body which receives the impulsion 
 to be retained by a fixed axis, let us suppose it to be perfectly free ; 
 or, to render the inquiry the more general, let us first consider a 
 system of material particles 7n, m' , m'\ Sic. entirely free and uncon- 
 nected, and moving with the parallel velocities i-, v', v", &c. and let 
 us inquire what will be the motion of the centre of gravity of the 
 system. 
 
 Through this centre conceive a plane parallel to the directions of 
 the impvdsions ; as, at the commencement of the motion, the sum of 
 the moments of /n, ?n', m", &c. with respect to this plane, is 0, (p. 
 59,) and as the bodies preserve their respective distances from the 
 plane througliout the motion, it follows that the sum of the moments, 
 with respect to this plane, must be the same at every instant ; hence, 
 the sum being always 0, the centre of gravity of the system must ne- 
 cessarily move always in this plane. Conceive another plane also pa- 
 rallel to the directions of the impulsions, and passing, in like manner, 
 through the centre of gravity of the system, but making an angle with 
 the former plane, then, as before shown, the centre of gravity will al- 
 ways move in this plane, it must, therefore, describe the line of 
 intersection of these planes, that is, the centre of gravity of the sys- 
 tem describes a straight line parallel to the directions of the impress- 
 ed velocities. 
 
 Conceive now a plane perpendicular to the directions of the ve- 
 locities, and design by e, e', e" Sic. the distances of the points m, m' 
 m", Sic. from this plane at the commencement of motion: at the 
 end of the time t", their distances will be e-\-vt, e'-j-uV, e"-\-v"t. 
 Sic. ; let us take the moments with respect to this plane, then a, r 
 being the respective distances of the centre of gravity from the plane 
 at the commencement of motion, and at the end of the time t", we 
 shall have the following equations (41) 
 
 (m-\-m'-\-7n"-\-Sic.) a=me-\-m'c'-{-7n"e"-\-Sic. 
 (yn-f-m' -f m"-|-&;c.) .r=m(e-|- i'/)-f m'(e' -f u'f) -f&c. 
 Subtracting the first from the second we have 
 
 (77i-fm'+?n"-f-&;c.) {x — a)={mv-\-m'v'-\-m"v"-{-Sic.) t, 
 which shows that the space rr— o, described by the centre of gravity, 
 is proportional to the time ; hence the centre of gravity of the sys- 
 tem moves uniformly. It must be remembered, that in the foregoing 
 equations those values of e, e', Sic, of w, v', Sic, are taken nega- 
 tively which are measured in opposite directions to those considered 
 as positive.
 
 CENTRE OF SPONTANEOUS ROTATION. 201 
 
 X — fl 
 
 As is the velocity of the centre of gravity, it follows, that 
 
 if the whole mass of the system were concentrated there, its mo- 
 mentum or quantity of motion would be 
 
 X "^^ a ' 
 
 {m-\-m' -\-m" -\-&Lc.) =mv^m'v' -\-m"v" -]-&lc. 
 
 by the equation just deduced ; hence the second member of this 
 equation represents the intensity of the impulsion that must be ap- 
 plied to the whole mass of the system, when concentrated in the 
 centre of gravity, to make that centre move, as it actually does ; we 
 conclude, therefore, that the centre of gravity moves with the same 
 velocity as if all the impulsions ivere immediately impressed on it, 
 or, which is the same thing, the centre of gravity moves as if the 
 whole mass of the system loere concentrated in it, and all the 
 forces were applied to it in directions parallel to those they really 
 take. 
 
 If the impressed velocities were not parallel, the same thing would 
 also have place. For if we decompose each of them into three 
 others parallel to rectangular axes, we may apply to each of the 
 three groups of parallel velocities the foregoing reasoning, and thence 
 infer that the centre of gravity would move in the direction of each 
 axis, as if the forces parallel to that axis were immediately applied 
 to that centre ; and therefore its motion in space being compounded 
 of these, it moves as if all the impulsions distributed through the 
 system were directly applied to the centre of gravity or to the whole 
 mass concentrated there. 
 
 Let us now suppose the bodies in the system to be invariably 
 connected together, as in the case of one solid mass. 
 
 Let P, P', &c. represent the impulsive forces which act on the se- 
 veral bodies, decompose each force into two others ; the one due to 
 the motion which it actually produces, and the other due to the mo- 
 tion destroyed on account of the mutual connexion of the bodies in 
 the system ; so that F, F', &c. may be the forces which produce 
 their full effect, and ff, &c. those which are destroyed by the mu- 
 tual action of the parts of the system ; thus F and/ will be the com- 
 ponents of P ; F', andy"' the components of P' ; &c. In virtue of 
 the forces F, F', &c., which are fully effective, the motion must be 
 the same as if these forces only acted on the system, all connexion 
 between its parts being destroyed, so that from what has been 
 proved above, the centre of gravity ought to move as if all the forces 
 F, F', Sec. were immediately applied to it, in directions parallel to 
 those which they actually take. As to the forces f,f, &c. they are 
 mutually destroyed when they act upon the several parts of the sys- 
 tem, and consequently satisfy the six equations (6), (.7), page 79; 
 
 26
 
 202 ELEMENTS OF DYNAMICS. 
 
 but when transported, parallel to themselves, to the centre of gravity, 
 they ought, for much greater reason, to be mutually destroyed, since 
 then the equations (4) page 28, suflice to establish their equilibrium. 
 Hence the centre of gravity moves as if the several impulsions 
 were immediately applied to it. 
 
 (163.) Let us now examine the motion of a body which receives 
 an impulsion that does not pass through the centre of gravity ; the 
 motion of translation would, we know, from what is proved above, 
 be the same as if the impulsion were applied in a parallel direction 
 to the centre ; but, beside this, there would be impressed a motion 
 of rotation, precisely the same as would have been impressed by 
 the same force if a fixed axis had passed through the centre of gra- 
 vity. This double motion, arising from a single impulsion, may be at 
 once shown to take place as follows. Let P (fig. 117) represent the 
 impulsive force, and, perpendicular to its direction, draw GA from the 
 centre of gravity ; at an equal distance GB, on the other side of G, let 
 two forces iP and IP, equal and opposite to each other be applied, 
 these will have no effect on the system, so that we may consider the 
 motion which the body actually takes, in consequence of the single 
 force P, to be the result of the three forces acting as in the figure at A 
 and B. The motion of translation is due to the force P, considered as 
 acting at G, or, which is the same thing, this motion is due to the 
 force 5 P acting at A, and to d P acting at B, in direction BS ; the 
 remaining forces, therefore, that is the force |P at A, and the oppo- 
 site force 5 P at B, in direction BR, are those to which the rotatory 
 motion is due ; the tendency of these forces is to turn the body about 
 G, being symmetrically situated with respect to it, and the value of 
 the forces to produce this effect is at A, GAXjP, and at B, GBxiP, 
 and as these forces turn the system in the same direction their whole 
 effect is 
 
 GAx|P+GBxiP=GAxP; 
 
 which is the effect due to the impressed force P to turn the body 
 about a fixed axis through s. 
 
 It follows, therefore, that ?W/m a body is acted upon by anyim- 
 pidsive forces, of ivhich the resultant does not pass through the cen- 
 tre of gravity, the body tcill have, in consequence, a double mo- 
 tion; 1, the centre will move as if the forces were immediately ap- 
 plied to it ; 2, the body ivill turn as if this centre were absolutely 
 fixed. 
 
 Let P (fig. 118) be the momentum, or quantity of motion, im- 
 pressed on the body, r its distance OG from the centre of gravity 
 of the body M ; then, for the velocity of translation due to this force, 
 
 P 
 
 we have w=jrj:. Again, for the angular velocity ca due to the same
 
 ROTATION OF A SOLID BODY. 203 
 
 force P, acting at the distance GO from the centre of motion, we 
 
 Pr 
 
 have (157) io=^rj7^; consequently, the absolute velocity of any 
 
 point in the body is compounded of these two, viz. 
 
 progressive velocity, ^=^ 1 
 
 P r vr r (^)' 
 
 angular velocity, «=^ . p=p J 
 
 r being the perpendicular distance of the centre of gravity of the 
 mass M, from the direction of the applied force P, and k being the 
 principal radius of gyration. These results may be expressed in 
 words, as follows, viz, the progressive velocity is equal to the mov- 
 ing force, divided by the mass of the body, and the angular velo- 
 city is equal to the m,oment of the force divided by the moment of 
 inertia. 
 
 If, however, the body is not free to revolve about its centre of gra- 
 vity, but is constrained to turn about some other point moving uni- 
 formly, the angular velocity will be different. It will be easy, how- 
 ever, to estimate it, for as the tendency to turn about the centre of 
 gravity is the same on whichever side of it the impulsion be given, 
 provided only it act at the same distance from it and in contrary di- 
 rections, we may obviously consider the angular motion which 
 accompanies the progressive motion of the point, to be the result of 
 a force acting in a direction opposed to the progressive motion, and 
 at the opposite side of the centre of gravity, but at the same distance 
 from it as the centre of motion. The value of the impulse to which 
 the motion of the body is due,rfwill be known from knowing the 
 progressive velocity of the centre of motion and the mass moved. 
 We shall give an illustrative example of this hereafter (at prob. IV., 
 chap. VII.); at present we consider the body as entirely free, in 
 which case we observe the following particulars. 
 
 At the instant the impact is given, the point departs in the di- 
 rection Oh, its initial velocity being equal to the sum of its pro- 
 gressive and angular velocities as their directions coincide, and the 
 same is, obviously, the case with any other point in GO. With the 
 points in GO', on the other side of G, it is different, for although 
 they have the same angular velocities as the corresponding points 
 in GO, yet, turning in the contrary direction, their absolute velo- 
 city of any one of them is equal to the difference between its pro- 
 gressive and angular velocities; that is to say, every point O' in 
 the line GO' has, in virtue of the angular motion of the system, a 
 velocity backward, and, in virtue of the progressive motion of the 
 system, a velocity forward ; this latter is the same for all the points 
 in the body, and equal to that of the centre of gravity, but the
 
 204 ELEMENTS OF DYNAMICS. 
 
 backward velocity of every point in GO' varies witli its distance 
 from G, being at G, and increasing regularly as the distance in- 
 creases; there must in consequence be some point in GO', either 
 within or without the body of which the velocity backward is pre- 
 cisely equal to the velocity forwards ; this point then is for an in- 
 stant at rest while all the other points of the body are in motion ; 
 so that the whole system, when the initial motion is given, turns 
 spontaneously round it ; this point is hence called the centre of 
 spontaneous rotation. Its situation is readily determined from the 
 condition which characterizes it, which is, that calling its dis- 
 tance from G, r ', and progressive velocity of the system v, 
 
 V fc^ 
 
 V — r'td=0 .*. r'=— , or, from equation (1) above, r=— ; hence, if 
 
 w ^ \ y J. 
 
 (fig. 119) be the point thus determined, its property, with respect 
 
 to the point of impact 0, is that OC = r-| — .... (2) ; which 
 
 proves (159) that the centre of spontaneous rotation coi)icides with 
 the centre of suspension corresponding to the point of percussion, 
 considered as the centre of oscillation, and is entirely independent 
 of the intensity of the applied force. 
 
 (164.) In order to determine at what distance GO, from the 
 centre of gravity, the impulsion must have been given to produce 
 the actual progressive and rotatory motions observed in any body, 
 
 we have, from the equation (1) above, r= ; or, if V be the ro- 
 
 V 
 
 tatory velocity of any point at the distance R from the centre of 
 
 V . k' V 
 
 gravity, then since "=p-» we have r=-^. — . (1). 
 
 Applying this to the double motions of the planets, we may deter- 
 mine at what distance, from the centre of each, the original impul- 
 sion must have been impressed by the hand of the Creator to 
 cause their actual motions of progression in space and rotation on 
 their axes. 
 
 Taking the earth for example, we know that it performs its revo- 
 lution on its axis in a sidereal day, by which rotatory motion every 
 point on the equator passes over about 25020 miles. 
 
 Also its orbit, or a path of about 596904000 miles, is passed 
 over by its progressive motion in 366 sidereal days, hence the ratio 
 
 V. , V 596904000 ^ „ 
 
 — isherel -. — = = 6o. 3 ; 
 
 V V 25020x366 
 
 and, considering the earth a sphere, we have (p. 189) A:*=|R"; 
 hence, by substituting these values in tlie formula (1), we have for 
 the distance r from the centre of tlie sphere at which the impulse
 
 ILLUSTRATIVE PROBLEMS. 205 
 
 T? 1 
 
 was given r := — - — — ; that is, about the -— — part of the radius 
 Loo ' Zi lt)o 
 
 distant from the centre. 
 
 It is very probable that not only the planets but that also the 
 sun may thus derive its motion from a single primitive impulse, and 
 if so, he, in common with the planets, must also have a progress- 
 ive motion in space ; this cannot, indeed, be rigorously proved. 
 " But," to use the words of Dr. Robison, as quoted by Professor 
 Gregory, " the very circumstance of his having a rotation in 27d. 
 7h. 47m. makes it very probable that he, with all his attending 
 planets, is also moving forward in the celestial spaces, perhaps 
 round some centre of still more general and extensive gravitation : 
 for the perfect opposition and equality of two forces necessary for 
 giving a rotation without a progressive motion, has odds against it 
 of infinity to unity.* This corroborates the conjectures of philo- 
 sophers, and the observations of Herschel and other astronomers 
 who think that the solar system is approaching to that quarter of 
 the heavens in which the constellation Aquila is situated." 
 
 CHAPTER IV. 
 
 PROBLEMS ILLUSTRATIVE OF THE PRECEDING THEORY. 
 
 (165.) We shall now proceed to illustrate the theory delivered 
 in the two preceding chapters, by showing its practical application 
 in a few miscellaneous problems. Several of these are those se- 
 lected by Mr. Barlow in his Treatise on Mechanics, in the Ency- 
 clopedia Metropolitana. 
 
 Problem I. — Let AB (fig. 120) denote an axle turning on two 
 fulcrums at A and B ; RS a wheel of given diameter to which is 
 attached, by a cord wound round its circumference, a given weight 
 AV; conceive ;>,p', ^>", /j'", to be given weights, fixed to the axes 
 by inflexible lines or wires, Cjo, CjO', &c. and let it be required to 
 determine the circumstances of the motion of the descending weight. 
 
 We shall, in the first place, consider this problem under its most 
 simple form, viz. we shall suppose the wheel RS, the axle AB, and 
 
 * It does not appear to us, however, tLat any weight should be attached to 
 this assertion, founded on the doctrine of chances, and which can strictly apply 
 only to the case of two impulsive forces, directed at random towards opposite 
 parts of a spherical body. Whatever primitive motions the Almighty may have 
 designed to impress on the sun, the impulses it must have received could not fail 
 to be those precisely competent to produce the intended effect. 
 
 s
 
 206 ELEiMENTS OF DYNAMICS. 
 
 the lines or wires Cp, Cp', &c. as divested of inertia, qv as offer- 
 ing no resistances to angular motion ; so that W and the four 
 masses p, p', p", p'", will be the only weights in the system, the 
 latter being all equal, placed at equal distances from the axle C, 
 and at right angles to each other. 
 
 Let Cp, Cp' &c.=:r; the radius of the wheel =r' ; the sum p 
 +/)' + p"H-/)"'=P, and the given weight =W; then the moment 
 
 P 
 of inertia of P will be r^ — , and the moment of W, the moving 
 
 force, will be r'W, consequently the acceleration of W, being equal 
 to r' times the angular acceleration, will be 
 
 W P r' ^W 
 
 r'^W-j- r'* 1- r^— that is, F= ,,-. rr sr, ; and the velocity of 
 
 g g r'^W+r^P® -^ 
 
 W, after any time t", will be 
 
 r'nv 
 
 D^ F/ = erf • 
 
 and the corresponding space descended will be 
 
 and thus the circumstances of Ws motion are all known. 
 
 Let us now take into consideration the inertia of the wheel and 
 axle; call the weight of the former W' and that of the latter W", 
 also let r" be the radius of the axle; then (page 187,) 
 
 VV W" 
 
 the moment of inertia of the wheel = 5 r'* — ,axle= 5 r"* , 
 
 • . . S , S 
 
 the square of the radius of gyration being in both cases A:*=2r*. 
 
 The motive force being still the same as before, we have, for the 
 acceleration of W, 
 
 p r^^ 
 
 r'MV4-dr'2 W' + |r"2 VV"+r-P^' 
 
 The accelerating force being thus known, the space, velocity, time , 
 «fcc. are determined by the usual formulas for constant forces. 
 
 If we suppose the system of small weights/?, p', p", &,c. to be 
 replaced by a solid body of revolution, as in fig. (121) the principles 
 of the calculation will be still the same ; for the moment of inertia 
 
 P . 
 
 of the solid P will as before be A:'—, k being the principal radius of 
 
 gyration as measured from the geometrical axis. Thus in fig, 121, 
 let P denote a sphere whose radius is 3 feet, and weight 500 lbs. ; 
 the weight W:=50 lbs., and the radius of the wheel 6 inches, or | a 
 foot, and of which the weight, as well as that of the axle, are sup- 
 posed inconsiderable with respect to the other parts of the system ;
 
 ](U / 
 
 ,s 
 
 \ 
 
 Ix'' 
 
 " -•. 
 
 
 " --■ 
 
 ,
 
 ILLUSTRATIVE PROBLEMS. 207 
 
 and let it be required to determine the time in which the weight W 
 will pass through any given space, as, for example, 50 feet. 
 
 In the sphere k^=fr''; hence the expression for the acceleration 
 of Wis 
 
 p_ r_-W 4^X5 _ 402^V 
 
 ,.'aW+|r^P ^ Ix50 + fx9x500 ' 18121 
 
 hence the time of descent is 15". 
 
 Problem II. — Let ABC (fig. 121) represent a wheel and axle, its 
 weight w, having a given weight W applied to the circumference of 
 the axle, and P applied to the circumference of the wheel in order 
 to raise W ; it is required to assign the space described by the ele- 
 vated weight W from rest, in any given time. 
 
 Let the radius of the axle be r, that of the wheel R, and the prin- 
 cipal radius of gyration of the wheel k ; then, for the moment of 
 inertia of the whole system, we shall have the expression 
 , w ^P W 
 
 ^3 _ ^ R2_ + ,.3 _ . 
 or or or 
 
 O o _ S 
 
 Now the actual weight, which, applied at the point D, gives mo- 
 tion to this, is not the whole weight P, since part of this is employ- 
 ed in balancing the weight W ; to know what this part is, we have, 
 
 Wr 
 by calling it P', the equation Wr=P'R .-. P'= — p— so that only 
 
 the weight P ~ = 
 
 is actually employed in moving the system, and as this weight acts 
 at the point D we must multiply it by R to obtain its moment, and 
 dividing R times this by the mass moved, or by the whole inertia 
 we have, for the acceleration of P, the expression 
 PR2_WRr 
 k^w+n^P + r^W ^ • • ' '^ >• 
 Now as the acceleration of P is to the acceleration of W as the ra- 
 dius R to the radius r, we have 
 R . • • PR^ — WRr PRr — Wr'' ^ 
 
 which expresses the acceleration- of the ascending weight. 
 If R=r the acceleration of either weight will be 
 
 p W 
 
 F= — - R2 2- 
 
 A;^w;-fR2(P-|-W) ^* 
 
 It should be remarked, that if the mass moved, W, have no weight
 
 208 ELEMENTS OF DYNAMICS. 
 
 but inertia only, or rather if its weight is otherwise supported, and 
 its inertia only has to be overcome by tlie ni:'.chine, as for instance 
 when it is to be moved along a perfectly smooth horizontal plane, 
 then, in the niinierators of the foregoing expressions, we must put 
 
 ■w=o. 
 
 Problem III. — Let ABC (fig. 121,) represent a wheel and axle 
 of given weight moveable about a horizontal axis which passes 
 through S ; and suppose a known weight W is applied to the cir- 
 cumference of the axle, to be raised by a given force P applied to 
 the circumference of the wheel ; to assign tlie proportion of the 
 radii of the wheel and axle, so that the time in which tlie weight W 
 ascends through any given space shall be a minimum. 
 
 Since the ratio only of the radii of the wlieel and axle is required, 
 let the radius of the axle be r, and that of the wheel xr ; the weight 
 of the wheel iv and k as before, the principal radius of gyration of 
 the wheel, and of which the value we know is (p. 188) k"=i x^r". 
 Then, substituting xr for R and ^x^r'' for A:*, in equation (2) 
 of the preceding problem, we have, for the acceleration of W, the 
 
 I» X W 
 
 expression -— ; — — ——{^ g ^ and consequently 
 
 {iw+i\)x^-]-w^ " ^' ig(yx—\y) ^' 
 
 This expression is to be a minimum, and consequently the quantity 
 under the radical will be a minimum ; therefore, dividing this by 
 
 o 
 
 the constant 7—, and putting for brevity J9 for ^ U)-\-V, we have 
 
 ■is 
 
 V; ^r-. = a minimum ; 
 
 hence, by difTerentiating, 
 
 2px{Px—W)—V{px^-^W)=0 .'. Fpx" — 2Wpx=V\\ 
 
 w ^\^ w , 
 
 If the weight of the wheel be too inconsiderable to deserve notice, 
 
 Wrb x/^W'^ + PW^ 
 
 then jO=P, and in this case x= '— (2), and if, 
 
 moreover, P=W, we have .r=l zb ■v/2. 
 
 Suppose, for example, ABC to represent a cylindrical wheel, the 
 radius of which is required, but of which the weight is 20 lb ; and 
 let the radius of the axle be 1 inch ; the weight, W, 100 lb., and the 
 weight P, 33 lb. ; to find the radius of the wheel. 
 
 Here 5 w-\-F=p=4S\h., therefore, by equation (1), 
 
 100 , 100^ 100'
 
 ILLUSTRATIVE PROBLEMS. 209 
 
 Consequently, since the radius of the axle is 1 inch, the radius of 
 the wheel must be 6*43 inches. 
 
 For other such problems as this, the student may consult Dr. 
 Gregory's chapter on the " Maximum Effects of Machines." 
 
 Problem IV. — In the wheel and axle, when a given weight P 
 acting at the distance R raises a weight W acting at the distance r 
 from the geometrical axis, it is required to assign the pressure sus- 
 tained by the axis, the weight of the wheel and axle, and the fric- 
 tion of the cord not being considered. 
 
 Suppose P and W to be at their respective extremities of the ho- 
 rizontal line passing through the centre of motion, and in that situa- 
 tion let O and G be the respective distances of the centres of oscil- 
 lation and gravity from the centre of motion ; then, since the angu- 
 lar velocity of any revolving system is the same as if its whole mass 
 were concentrated in its centre of oscillation, we may consider 
 P + W to be placed at the distance O from the centre of motion ; 
 and, since the force of any point in the revolving system is propor- 
 tional to its distance from the axis of motion, we have 
 
 0:G ::P-f W:^(P + W), 
 
 which is the force or pressure with which the centre of gravity de- 
 scends ; or in fact the force with which the whole mass descends. 
 Part of the whole pressure P + W of the system is thus supported 
 
 C 
 
 by the axle, and the other part, which we have just^ seen is — 
 
 X(P-|-W), is employed in producing the motion which actually has 
 place ; consequently, that part of the pressure sustained by the axle 
 must be 
 
 P+W-^(P+W). 
 
 It remains, therefore, to find the values of G and O, which are 
 _ PR — Wr _ PR^-fW r" 
 
 - P + W~' ~" PR_Wr ' 
 
 hence, by substitution, we have for the pressure j), 
 
 ' ^ PR^ + Wr^ PR^-fWr^^ 
 
 If R = r, as in the case of the single fixed pulley, then the 
 
 4PW 
 pressure is p = p ^^ . 
 
 Problem V. — Let A, B (fig. 122,) represent a single moveable 
 pulley by means of which the power P elevates the weight W; 
 s2 27
 
 210 ELEMENTS OF DYNAMICS. 
 
 then, having given P and W, together witli the weights of the equal 
 cylindric pulleys A and B, it is required to assign the space which 
 the descending weight P describes in a given time, the weight of 
 the moveable pulley being included in the weight W. 
 
 Let us refer the whole inertia of the system to the point p, so 
 that we may consider tlie force which moves P to be burdened with 
 the mass of P, and with the additional mass representing the inertia 
 of the other parts of the system, this mass being all accumulated 
 at /;, or, which is the same thing, incorporated in P. 
 
 In the first place the inertia of the pulley A, whose weight call 
 Q, is the same as that of half its mass placed at/j, (see page 188.) 
 
 In like manner the inertia of the pulley B is the same as that of 
 half its mass placed at q ; or, since the rotatory velocity of B's 
 circumference is only half the velocity of A's circumference, the 
 mass of half B at y has the same inertia as the same mass placed 
 
 at half the distance Op, or finally as J times the same mass 
 
 placed atj3, (p. 188.) Hence, as far as the inertia of the pulleys is 
 
 Q Q 
 
 concerned, the equivalent mass to be placed at/) is 5 . f- ^.— . 
 
 Again, as the velocity of W is half that of P, it move's as if it 
 hung at half the distance Op' from the centre of the pulley A, 
 where, as in the case of the pulley B, it would offer the same in- 
 ertia as ^ its mass placed at p' or p ; hence the inertia of the weight 
 
 P W 
 
 is represented by the mass 1- — placed at p, so that the whole 
 
 mass moved by the force which moves P is 
 
 P + ^W-f IQ+jQ _ SP-f 2W-f5 Q 
 
 g ^g ' . 
 
 To determine now the force or weight which moves this mass, 
 we must find how much of the applied weight P is employed in 
 merely balancing W ; this is easily done, because, as W is equally 
 supported by the two branches of rope/)'^, Oq' , one half of W is the 
 portion of P employed in balancing W ; hence the moving force is 
 
 and consequently for the acceleration of P we have 
 
 2P_W 8P+2W-f5Q 8P — 4W 
 
 F = 
 
 2 ■ Sg 8P + 2VV + 5Q' 
 
 8P_4W 8P — 4W ff/» 
 
 ^gt^s- 
 
 8P+2W+5Q*' 8P + 2W-f5Q 2 
 Problem VI. — In a system of pulleys contained in two equal and
 
 ILLUSTRATIVE PROBLEMS. 211 
 
 separate blocks a single string goes round them all : it is required 
 to determine the acceleration of P fastened at the extremity of the 
 string, and drawing up a weight attached to the lower block. 
 
 Let there be altogether n pullies, Q being the weight of each, 
 and let the weight of the lower block, together with its attached 
 load, be W, and let us, as in the preceding problem, ascertain what 
 mass in addition to its own must be incorporated in P to supply the 
 place of the inertia of the system. 
 
 The pulley which first received the cord from the ascending 
 
 weight turns with — th of the velocity of the pulley which delivers 
 
 the cord to the power, and therefore, as in the last problem, the 
 
 1 Q 
 
 mass at P, which will be a substitute for its inertia, is . — . 
 
 2 . /i^ g 
 
 The circumference of the next pulley in the lower block revolving 
 
 twice 'as fast as the foririer, the mass due to its inertia will be 
 
 4 Q 
 
 - — • . — , and so on ; hence, for the pulleys, the mass to be substi- 
 2.«^ g ' ■' 
 
 tuted will be 
 
 2.nr g ^ ' 2n^ g 6 
 
 Again, the velocity of W being — th of P's velocity, the mass at 
 
 P, which represents its inertia, will, as in the preceding problem, be 
 
 1 W 
 
 — . — ; hence the inertia of the weights is represented by the mass 
 
 P 1 W 
 
 — I . — , and therefore the whole mass moved is 
 
 g n^ g 
 
 — 4.— ^^ \ ^ ^ 2 n'' + 3 n + 1 
 g n^' g 2ng ' 6 ^ ' 
 
 W 
 
 The weight at P which balances W is — and consequently that 
 
 n, 
 
 W n P W 
 
 which moves the system is P = , and therefore, di- 
 
 n n 
 
 viding this by the mass moved, as above expressed, we have for 
 
 ^T, T. 12wrnP— W)^ 
 
 the acceleration of P ; F == — 7 — rr ,.,,, ^ — ttt^- — -;: 7t> 
 
 12 (nM^ + W) + Qn (2 n« + 3 n + 1) 
 
 from which, as in the preceding problem, the expressions for the 
 
 velocity and space generated in any time t" are immediately dedu- 
 
 cible. 
 
 Problem VII. — A wheel, whose interior and exteror radii are
 
 212 ELEMENTS OF DYNAMICS. 
 
 fj, r^, rolls down an inclined plane (fig. 123), of which the friction 
 is just sufficient to prevent sliding : to determine the circumstances 
 of the motion. 
 
 Let i be the inclination of the plane, and F the effective accele- 
 rative force down it, then, putting tv for the mass of the wheel, Fw 
 will represent the efFective moving force. But the impressed mov- 
 ing force is ivs^ sin. i, minus the resistance of the friction, which, 
 as it diminishes the amount of the moving force which the body 
 would otherwise have, we may represent by an opposing moving 
 force ; let us then call it w'g sin. i. Now if P be the point in con- 
 tact with the plane at D, where the motion is supposed to have com- 
 menced, then, since in any time t", DP' = P'P, it follows that the 
 rotatory acceleration of P is also F ; and, consequently, the accele- 
 ration of a particle at the unit of distance from C, that is the angu- 
 
 F 
 
 lar acceleration, is — ; hence wk' being the moment of inertia of 
 
 F 
 
 the wheel round C, — ivk'' will be the actual moment of the sys- 
 
 tem round C. But the only impressed moment is that arising from 
 
 the friction, it is, therefore, r^ w'g sin. i ; hence, by the principle 
 
 of D'Alembert, we have, by equating the impressed and effective 
 
 forces, 
 
 r. , ,^ . . Fwk^ 
 
 Fw={w — wjgsin.t; =r^w gsm.i; 
 
 eliminating w', we have Fw r^'^-^-Fw k^=wrj^^ g sin. t 
 . p^ g sin, i r^" 
 
 which expresses the acceleration of the centre C down the plane, 
 
 r *-4-r ^ 
 and in which k'>=-!--~-, (ex. 3, page 188.) 
 
 If the cylindrical wheel be indefinitely thin, or when r^=r^, 
 F=5 g sin. i. 
 For the time of describing a given space s, we have, 
 
 g sm. t r^ ' 
 
 2 5 '^ sin. X T ^ 
 and for the velocity acquired, v^=Ft=^ \ — "^ ' ^ \, which 
 
 expresses also the rotatory velocity of every point on the outer cir- 
 cumference. 
 
 To determine the absolute velocity of any proposed point P of 
 the outer circumference in space, or in its cycloidal path, we must 
 compound together these two velocities ; so that, calling the abso- 
 lute velocity V, we have
 
 ILLUSTRATIVE PROBLEMS. 213 
 
 V=2 V cos, 5 w P m=2 v x tab. cos. | arc. P^; m ; 
 and the degrees in Vpnl are known, since the length 2 s of the arc 
 PP'm is known ; therefore the absolute motion of P is determined 
 both as to velocity and direction. 
 
 \i p be diametrically opposite to the point of contact P', it ap- 
 pears from this expression for V, that the absolute velocity of znf 
 point P at any time varies as the tabular cosine of half the arc Vp ; 
 at/3 this velocity is gi-eatest, being =: 2v ; and at P' it is least, 
 being for an instant 0, that is, P' is the centre of spontaneous rota- 
 tion of the body. The last conclusion, viz. that the point of. con- 
 tact can have no motion along the plane, is an immediate conse- 
 quence of the conditions of the problem, for if it had any motion 
 along the plane, the body would slide on that point, whereas the 
 friction is supposed to be sufficiently gi-eat to prevent sliding down 
 the plane. 
 
 If, in consequence of any initial impulse, the rotatory velocity 
 exceed that of translation, P' will no longer be the centre of spon- 
 taneous rotation, but will have a velocity backward greater than that 
 forward, so that the body will move up the plane and will continue 
 to do so till this excess of rotatory velocity is destroyed by the fric-- 
 tion ; when the body, after being for an instant stationary, will re- 
 verse its progressive motion, and roll down the plane with a velo- 
 city equal to that of rotation. If the velocity of translation were 
 made to exceed that of rotation the body would partly roll down 
 the plane and partly slide. These deductions are on the supposi- 
 tion that the friction of the plane is just sufficient to prevent sliding 
 when the initial velocity of the body in progression is equal to that 
 in rotation. 
 
 Rotatory motion of this kind may be produced by the uncoiling 
 of a thread or riband previously w'ound about the body ; the ten- 
 sion of the thread supplying the place of friction. 
 
 Problem VIII. — A sphere, whose mass is P, is placed on the 
 slant side of a smooth prism, whose mass is Q, and a fine thread or 
 riband is fixed to the prism at B (fig. 124,) and coiled round the 
 sphere in the plane of its vertical gi-eat circle, the object of it being 
 to cause the sphere to roll and not slide down the plane. The base 
 AC of the prism, as well as the horizontal plane on which it is 
 placed, is perfectly smooth, so that it moves along the plane OC, 
 in consequence of the pressure of the rolling sphere. It is required 
 to determine the tension of the string, at any time the pressure on 
 the prism, and the path described by the point of contact P. 
 
 In this case the rolling body has two motions, viz. one down the 
 inclined plane, and the other in a horizontal direction, as well as 
 the rotatory motion about iis centre. As in last problem, the rota-
 
 214 ELEMENTS OF DYNAMICS. 
 
 tory acceleration of any point in tlie circumference, must be equal 
 to the acceleration of the point of contact P down the plane ; but 
 this rotatory acceleration is wholly produced by tlic tension T of 
 the thread, it will, therefore, be expressed by dividino^ this force by 
 the moment of inertia of the sphere, that is, the acceleration of the 
 point of contact is T-^fP,(p. 190-1.) Let us now deduce another 
 expression for this acceleration of the point P from tlie actual space 
 BP passed over by it, and for this purpose put ON=x, OA=.rj, O 
 being the place of A at the commencement, or when P is at B ; let, 
 moreover, BC=o, AC=6, AB = c; then the space BP is 
 
 BP=(Xj+6 — a:) sec. A=(Xj-f 6 — ^)7:' 
 
 and therefore the second differential coefficient, with respect to the 
 time, must express the acceleration down the plane ; that is, 
 c ,d^x.^ d^x 5'V 
 J^~dF~~dF^^2¥ ^ ^* 
 
 The first of these differential coefficients denotes the acceleration 
 of the point A or of the entire prism in a horizontal direction, and 
 the second denotes the acceleration of the point P, or of the sphere 
 in a horizontal direction. Now the motive forces to which these 
 accelerations are due, are equal and opposite, therefore, calling p the 
 pressure on the prism, or — p, the resistance against the sphere, we 
 have for the horizontal force on the sphere 
 
 dt^ ' c c ^ ^ 
 
 dt^ ^ c c ^ ' 
 
 Also, for the vertical force on P, we have 
 
 dt^ 
 Consequently, since y «= ^{x — x{) .... (5), 
 the equations (1) and (4) give 
 
 Also the equations (1), (2), (3), give 
 
 and from these two equations we obtain for p and T the values 
 
 6cP(2P+ 7Q) g 
 
 ^^7 a^ (P+ QJ+y^"(2P'+7Q) 
 
 ^ey = _P,.+^l+T4....(4). 
 
 a
 
 MOTION OF A SYSTEM OF BODIES. 215 
 
 2acF(P+Q)g 
 7a^(P + Q)+6H2P+7Q) 
 both of which are constant quantities. 
 
 As to the path described by the point of contact P, it is imme- 
 diately deducible from the conditions of the problem ; for, as both 
 bodies commence their motion together, the horizontal momentum 
 of the sphere is equal and opposite to that of the prism, that is, 
 
 that is, (equa. 5,) 
 
 hence the path is a determinate straight line. 
 
 For a very complete and elegant solution to this problem, con- 
 sidered under different modifications, the student may consult a 
 paper by Mr. Mason, in the twentieth number of Leybourn's Ma- 
 thematical Repository. 
 
 CHAPTER V. 
 
 ON THE MOTION OF A SYSTEM OF BODIES ACTED ON BY ANY 
 ACCELERATIVE FORCES WHATEVER. 
 
 (166.) We propose in this chapter to investigate some very 
 general and remarkable theorems which apply to the motion of a 
 system of bodies acting in any arbitrary manner on each other, 
 and each influenced by any accelerative forces. 
 
 Let m, m^, m^, &lc. represent the masses of the different bodies 
 in the system, x, y, z ; x^, y^, z^, &c. the rectangular co-ordinates 
 which mark their position, X, Y, Z, the components of the accele- 
 rative forces on m, X^, Y^, Zj, the components of those on m^, &c. 
 then the motive forces applied to any one, as m, will be inX, mY, 
 
 d^ X d^v 
 mZ, and those which actually have place will be m—j-^, wi-r^, 
 
 (la z 
 
 m ; hence the differences of the impressed and effective forces 
 
 dt^ 
 resolved in the directions of the co-ordinates are 
 d^x ^ d^y ^^ d^z 
 
 and, in like manner, for each of the other bodies m^, m^, &c. we 
 get similar expressions for the differences between the impressed
 
 216 
 
 ELEMENTS OF DYNAMICS. 
 
 and effective forces, and we know, from the principle of D'Alembert 
 (154), that if these dilfercnces alone acted on the system it would 
 he kept in equilibrium. But when any forces keep a system in 
 equilibrium these forces must fulfil the conditions (6) and (7), at 
 page 79 ; hence, in the present case, we must have these two 
 groups of equations, viz. 
 
 2(m-^)=2:(mX) 
 
 (A) 
 
 ^ 'dV' ^^^ (77?^X — »nxY) 
 
 , xd^z — zd^x, , rt vx 
 
 l{m ^^ )=2 (ma:Z — mzx) L 
 
 S (w 
 
 dV 
 zd^y — yd^z 
 
 (B). 
 
 rf/= 
 
 )== '2,{mzY — myT.) { 
 
 These two groups of equations, which contain the conditions of 
 the motions of any system of bodies under any circumstances, fur- 
 nish several general principles of motion ; one or two of these we 
 shall proceed to develope. 
 
 Let X, Y, z, represent the co-ordinates of the centre of gravity of 
 any system of bodies m, m^, m^, Sic. ; then (39) we shall have 
 S (mx) 2 (my) S (mz) _ 
 
 I,{m) 
 
 2 (w) 
 
 2 (in) 
 
 and taking the second differential co-efficienls with respect to t 
 
 d' 
 
 S (m 
 
 
 
 dt'' 'd" z 
 
 , d'^z 
 
 rf/a 2 (m) dV' 2 (m) dt» 2 
 
 Comparing these with the equations (A) we have 
 d" x_2 (??iX) rf" Y 2 (mY) rf" z_ 2 (mZ) 
 
 (m) 
 
 (C); 
 
 dP 2 (m) ' rf/^ 2 (m) ' rf/2 2 (m) 
 or putting M for the whole mass, m-\-m^-\-m^-\-, &c. of the system, 
 we have 
 
 M^=2(mX),M^=2(7nY), M^=2(7nZ); 
 
 These equations show that if the whole mass of the system were 
 to be concentrated into the centre of gravity, and this centre to 
 have the same acceleration as in the actual state of the system, the 
 moiino^ force of the whole mass at the centre would be the same
 
 MOTION OF A SYSTEM OF BODIES. 217 
 
 as the entire moving force of the actual system ; but S (?nX) being 
 independent of x, x^, x„, &c. is independent of the distances be- 
 tween the several bodies m, m^, m^, &c. and, therefore, would 
 remain the same if these masses were united in a single point, 
 provided only the same accelerative forces were applied parallel to 
 their actual directions ; hence the motion which the centre of gra- 
 vity actually has is the same as it would have if all the system were 
 united there, and the forces applied to it parallel to the directions 
 they really have ; whence this general principle of motion, viz. 
 
 T7ie centre of gravity of a system of bodies, acted upon by any 
 accelerative forces and mutually influencing each other, moves in 
 space as if the system were united into that centre, and the forces 
 which solicit the bodies were directly applied to it. 
 
 (167.) When the system is acted on by no other forces than the 
 mutual attractions of its parts, the second members of the equa- 
 tions (A) must vanish ; for, as the action of any two of the bodies 
 is mutual, each will impress on the other the same motive force ; 
 and as these forces are opposite to each other, it follows that if the 
 bodies were connected with each other by rigid rods, these, on 
 account of the sequel at opposite pressures at their extremities, 
 would be held in equilibrium, whence all motion in the system 
 would be prevented by the forces applied to its several parts thus 
 connected together ; hence these forces must fulfil the six equations 
 of equilibrium, so that we must here have 
 
 2 (mX)=0, 2 (mY)=0, S (rnZ)=0 .... (D) 
 2 (myX'—mxY)=0, 2 (m^rZ— ?ncX)=0,2 (mzY — ?m/Z)=0 . (E); 
 and, consequently, from the equations (C), we get 
 
 d^x d^x fPz 
 
 ———^0, , ^ =0, -——=0, of which the integrals are 
 dt^ dt^ dt^ '^ 
 
 x=a-\-bt, Y=a'-|-67, z=a"-j-b"t 
 where a, b, a', b' a", b", are the arbitrary constants, introduced by 
 the integration. 
 
 If we eliminate t from any two of these equations there will re- 
 sult a linear equation either between x and v, or between x and z, 
 or else between y and z ; it follows, therefore, that the centre of gra- 
 vity of the system must describe a straight line, and its velocity at 
 any time will be 
 
 -J 
 
 *"+*'+''^- = ./16»+6-.+6-^ 
 
 df" 
 
 which, being constant, shows that the motion of the centre of gra- 
 vity of a system of bodies, whose motions are entirely due to their 
 mutual influences, is both rectilinear and uniform. 
 
 This is called the general principle of the conservation of the 
 centre of gravity. If no primitive impulse be given to the centre 
 T 28
 
 2J8 ELEMENTS OF DYNAMICS. 
 
 of gravity of a system, then b, b', b", will be each = 0, and there- 
 fore, v=0, or the centre will be at rest. 
 
 (108.) We have seen at (122,) that the differential expression 
 ydx — xdy,* is the difTerential of double the area described by the 
 projection on the plane of xy of the radius vector of m ; hence the 
 sum of the products of each body into the differential expression for 
 the area it traces out on this plane in any time will be expressed by 
 5 2 (mydx — mxdy). In like manner the corresponding sums for 
 the other planes will be 5 S {mxdz — mzdx), and 52 {mzdy — 
 mydz) ; or calling these several sums 5 dS, k <iS', 5 dS", and diffe- 
 rentiating, we have 
 
 2 {myd 'x — mxd "y ) = rf "S 
 
 2 {mxd ^z — mz </ «x) = rf 2 S ' 
 
 •zlmzd'^y — 7)iyd^z)=d^S" 
 
 rf'S 
 and, therefore, from the equations (B) 2 (myX — inxY)=-T—- 
 
 2 {mxZ — mzX) = —^— ; 2 (mzY — myZ)= -j- . 
 
 Now when the system is influenced only by the mutual actions of 
 its parts, we have seen (equa. E,) that the first members of these 
 equations are each=0, consequently, 
 
 *^ =» • -rf^ =" • l«r ■•• S=<^' . S=e'< , S"=c"( . 
 
 This result establishes the general principle of the conservation of 
 areas, viz. that in any system of bodies, moving in virtue of their 
 mutual actions on each other, the sum of the products of each body 
 into the projection, on any plane, of the areas, described by its ra- 
 dius vector, is proportional to the time in which those areas are 
 described. 
 
 (169.) We shall investigate one more general principle, which, not 
 being deducible from the general equations (A) and (B), requires 
 that we consider the motion of the system in another point of view. 
 
 In every system of bodies, the motion of each is due to the force 
 impressed on it, combined with those which arise from the action 
 of the other parts of the system ; the simultaneous action of all these 
 forces determine the motion of any one body m; and, therefore, taking 
 the components of these forces, viz. 7nX, mY, mZ, where X, Y, Z 
 are the accelerations of the body in the directions of three rectangu- 
 lar axes, we must have, at any time t", the equations. 
 
 * The sectorial area to which this expression appHes is measured from the axis 
 of y, as at art. 122 ; but for the complement of this area, or that measured from 
 the axis of x, the corresponding expression is, of course, xdt/ — tfdx. It is easy 
 to see that the law of &reas, established in this article, holds in either case.
 
 MOTION OF A SYSTEM OF BODIES. 219 
 
 m — ,— = m X, m —,i^ =m Y , m -5— =m Z 
 at^ at-' cir 
 
 &c. &c. &c. 
 
 Multiplying now, the equation in x by 
 
 -T— , the equation in y by 2 -y, the equation in z by 2 — ; 
 
 dx 
 and, in like manner, the equation in x^, by 2 -^, and so on, and then 
 
 adding them together and integrating, we shall evidently have the 
 equation 
 
 dx'+dy'+dz' , dx\ + dy\-\-dz\ , . 
 
 2fm (X dx+Y dy+Z rfz)+2/m,(X, fZ:r,+Y, c/y,+Z, (/zj+&c. 
 
 or, in the usual notation, 
 
 dx^-\-dv'^-\-d'-^ 
 2 (m ^^^/ )=2 2/m (XfZa;+Y f/«/ + Z rfz) 
 
 that is, s (mi;'^)=2 ^fm(X dx + Yf/y+Z rfz) . . . . (F), which 
 is similar to the equation (2) at page 146, when the system consists 
 of but a single point ; the second member is also integrable in like 
 circumstances, that is, when X, Y, Z, &c., are functions of x, y, z, 
 &c. and fulfil also the conditions named at page 146. 
 
 If these conditions have place, then, as at the page referred to, 
 S(mD^) — •B(mv\)=f(x,y, z, x^,y^,z^. Sic.) — 
 f {a, b, c, a^, b^, c^, &c.) 
 the product mv'^ of the mass into the square of its velocity is the vis 
 viva, or living force, so that when the second member of (F) is an 
 exact differential, we infer that the sum of the living forces of the sys- 
 tem generated in moving from one position to another, depends 
 solely upon the position left and that arrived at, and is independent 
 of the paths which the several bodies have taken, and of the time of 
 describing them. This is the general principle of the conservation 
 of living forces. 
 
 This principle always holds, that is to say, the second member 
 of (F) is always an exact differential when the bodies move in vir- 
 tue of their mutual attractions. To prove this, let r be the distance 
 between two of the bodies of the system m, m^ ; then 
 
 y-={^x-xj^{y-yj+{z-z,y . . . . (1) ; 
 let Rbe the function of r which expresses the intensity of the attrac- 
 tive force exercised by the unit of m on m^, and by the unit of m^ 
 on m ; then the whole attractive force of m on m^ will be Rm ; and 
 the whole attractive force of m^ on m will be Rm^ . The compo- 
 nents of the forces Rm will be (see page 147,)
 
 220 ELEMENTS OF DYNAMICS. 
 
 R,„^ ^ , Rm^ ^, Rm- 
 
 r ' r ' r 
 
 and these, in virtue of (1) above, are the same as 
 
 (ir dr dr 
 
 Km -f- , Km -7- , Km -r- . 
 dx ay dz 
 
 In like manner, for the components of the force Rm^ , we have 
 
 „ rf?* „ </r dr 
 
 Rm,-^-.Rm.^,Rm,-^. 
 
 This last set of components liave the same signs as the former set, be- 
 cause the force m, R is opposite to 7n R, and the coefficients -; — , — — , 
 
 dr , . . . . . . , „ . dr dr dr 
 
 - — , obviously mvolve opposite signs to the coemcients -y-, — , — . 
 
 Substituting the foregoing component forces for X, Y, Z, in the 
 
 expression S {in X dx-\-m Y dy-\-m Z dz), we have 
 
 ^dr , ^- dr , _ rfr , dr 
 
 mmi R-T- (te+mmj R-7- dij-\-mm^ R -^ dz+m^mKj-^ tu", 
 
 dr dr 
 
 -|-mimR-^ — (/(/i-f m^ 7?j R -^(/ci=mmi Rrfr, 
 
 the first side of this equation being mm^ limes the total ditTerential fj 
 of J", considered as a function of the six variables in equation (1) 
 above. The same reasoning applied to every two bodies in the 
 system, must lead to a similar result, so that, 
 
 2 S / (m X dx+ mYdy fm Z dz)=-i 1 . ?n f" , / R dr , 
 which is always integrable, since, by hypothesis, R is a function 
 of r. 
 
 CHAPTER VI. 
 
 ON THE COMPOSITION OF ROTATORY MOTIONS ; AND ON THE PRINCIPAL 
 AXES OF ROTATION OF A SOLID BODY. 
 
 On the Composition of Rotatory Motions. 
 
 (170.) If a body receive simultaneously impulses which are sepa- 
 rately competent to produce rotation about different known axes, 
 the result will be a rotation about a new and determinable axis. It 
 will be sufficient to consider three given axes at right angles to each 
 other, and the problem we propose now to solve is this, viz. when 
 the body tends to turn at the same instant about three rectangular 
 axes, AX, AY, AZ, with the respective angular velocities «, «', w" ; 
 to determine the position of the axis about which it actually turns, 
 and the angular velocity of the rotation.
 
 PRINCIPAL AXES OF ROTATION. 221 
 
 Let m represent any particle of the body (fiff. 125,) and let us 
 first consider the motion of it about the axis AY ; this will be in a 
 circle pq7n, and we shall suppose it in the direction of these letters. 
 The plane of the circle pqm being parallel to the plane of xz, the 
 particle has no motion in the direction of AY, and its motion in the 
 directions of AX, AZ will obviously be the same as if the circle 
 coincided with the plane of xz, we may then, in estimating these 
 motions, suppose such to be the case. Now as m turns towards the 
 plane of xy, its co-ordinate x increases, and its co-ordinate z di- 
 minishes ; hence the differential of x, with respect to the time, will 
 be positive, that of z negative. The absolute velocity of in about 
 AY will at any time t", corresponding to the co-ordinates x, y, b(^ 
 in the direction of the tangent mr, taking, therefore, this line to 
 represent it, its components in the directions of AX, xlY, will be 
 ms, and ma. As the expression for the absolute velocity is Am . a>' 
 we have Am . u'=^mr, and since ms=^mr sin. inrs=mr sin. A= 
 to'. A?»sin. A=u,'.mh=ia' z, this, therefore, expresses the velocity in 
 the direction of AX which is due to the rotation about AY. In like 
 manner, ma=mr cos. rma=m7- cos. A^u' . Am cos. A=u>'x is the 
 velocity in the direction AZ to be taken negatively ; hence the ve- 
 locities in the directions of AX, AZ, due to the rotation of any par- 
 ticle at the point (x, y, z), at any time t" is w' z and — ut'x. 
 
 Consider now the rotation of the particle m about the axis AX 
 (fig. 126,) from Y towards Z, and applying precisely the same rea- 
 soning, we get for the velocities in the directions of Z and Y the 
 values toy and — i^z. And finally, applying the same reasoning 
 when the rotation is about the axis of z from X towards Y (fig. 
 127), we have, for the velocities in the directions of AY, AX, w" a: 
 and — "i"2/- 
 
 It follows, therefore, that when all these rotations have place 
 simultaneously, we have, by adding together the above partial velo- 
 cities along the axes, the following expressions for the whole velo- 
 cities in those directions, viz. 
 
 dx , 
 
 -^=.z-. y 
 
 -^=w X — wZ 
 
 at 
 dz 
 
 (1)- 
 
 Now we may determine from these expressions about what axis 
 the body actually turns at the instant t", when the foregoing mo- 
 tions have place simultaneously ; for as every particle in the axis 
 of instantaneous rotation is motionless, we must have for all of them 
 t2
 
 222 ELEMENTS OF DYNAMICS. 
 
 dx ^ (hi ^ dz ^ 
 
 ,¥=«• i=»' .¥=»■ 
 
 that is, u,' z — u"?/=0, ^^" X — wr=0, wy — u'a:=0, 
 and these three equations obviously characterize a straight line in 
 space passing through tiie origin of the original axes ; two of these 
 equations are sufficient to determine the position of this line, as for 
 instance the equations 
 
 a:=— z, y=—r, z . . . . (1), where -;-, and — ^ 
 
 are the trigonometrical tangents of the angles which the projections 
 of the instantaneous axis make with the axis of z ; hence if a, /3, y, 
 denote the angles which the instantaneous axis of rotation makes 
 with the axis of x, y, and z, we have {Anal. Geom. p. 228,) 
 
 COS. a= .( ^ , ,« , , 771^ cos- /3= 
 
 cos. y = ; -, 
 
 and thus the position of the required axis is determined in terms of 
 the known angular velocities. 
 
 To determine the angular velocity of the body about this axis, we 
 need consider only the angular velocity of any single particle chosen 
 at pleasure ; let us take a particle on the axis of x ; if from it Ave 
 draw a perpendicular p to the instantaneous axis, then the distance 
 of the particle from the origin being x, we have 
 
 p=x s\n.a=x<y\l — cos.*^ o= — ^--r -,. 
 
 Now as for this particle 2/=0, 2=0, in the second members of (1), 
 we have, for its absolute velocity, 
 
 v=V^ 
 
 ^x^-\-d\l^-\-dz^^ , ,„ 
 
 and, consequently, for the angular velocity v, we have 
 
 v=-=y/\^^^J^^J'^\ (2), 
 
 p 
 so that the three angidar velocities w, to', u", about three rectangu- 
 lar axes, are equivalent to the singular angular velocity, 
 
 about an axis inclined to these three, at angles whose cosines are 
 the expressions for cos. a, cos. 3, cos. y, above. 
 
 It is obvious, from what has now been said, that when a body 
 revolves about any axis given in position, and with a given angular 
 velocity, we may always resolve the motion into three partial 
 rotatory motions about the three rectangular axes of co-ordinates, 
 these component motions being in the directions assumed at the
 
 PRINCIPAL AXES OF ROTATION. 223 
 
 outset of this article. For the equations of the axis of rotation com- 
 pared with the equations (1), and the expression for the angular 
 velocity compared with the expression (2), will furnish three equa- 
 tions among the unknowns w, w', u", which are sufficient to fix their 
 values. Hence, to whatever combination of rotatory motion the 
 rotation which actually has place be due, we may always con- 
 sider it as the resultant of the three angular motions above con- 
 sidered. 
 
 On the principal Axes of Rotation. 
 
 (171.) There are some remarkable properties belonging to cer- 
 tain axes of rotation, passing through any point in space, which 
 well deserve notice; for instance, whatever point be chosen, there 
 always exists one axis in reference to which the moment of inertia 
 of the revolving body is a maximum, and another for which the 
 moment of inertia is a minimum ; these two axes are always per- 
 pendicular to each other, and they, together with a third axis througli 
 the same point, perpendicular to them both, are called the three prin- 
 cipal axes of rotation, passing through that point. When the point 
 chosen is the centre of gravity of the body, these axes have each 
 of them a property peculiar to themselves, which is that neither of 
 them will suffer any pressure from the rotation of the body round 
 it, so that, when this rotation has once commenced, if the axis were 
 to be withdrawn, the rotation would, nevertheless, continue as if it 
 were there. 
 
 To establish these interesting properties it will be requisite, first, 
 to obtain a general expression for the momentum of inertia of a re- 
 volving body, in reference to any axis whatever. 
 
 In order to this, let AB (fig. 128,) be the axis in reference to which 
 the moment is to be determined, and assume a point on it A for the 
 origin of the co-ordinates. Let /n be a particle of the body, and of 
 which the co-ordinates are x, y, z ; let the perpendicular niB = r, 
 and the distance A.in of the particle from the origin 6 ; also call the 
 angle mkB, e and let a, /3, y ; a', i3', y', be the angles which AB, 
 and Am, make with the axes of x,y, z; we shall thus have 
 
 82 = a;3 4-y-|-23. ... (1) 
 
 cos. e = cos. a cos. a' + cos. j3 COS. j3'-|-cos. y cos. y' ; 
 
 therefore, since cos. a' = — , cos. 3' =— , cos. y' =— , we have, 
 
 by substitution, 5 cos. « = x cos. a-\-y cos. |3-f z cos. y . (2). 
 
 Now in the right-angled triangle mAB we have r^ = 5^ sin. ^ s= S'' 
 — 5^ COS. ^ s ; therefore, substituting in this value of r^ the expres- 
 sions (1) and (2), we have r^ = (1 — cos.^ a) x^-{-(l-{-cos. ^^) y"* 
 -f- (1 — cos. '^y) z^ — 2xy COS. a COS. (3 — 2 xz cos. » cos. y —
 
 224 ELEMENTS OF DYNAMICS. 
 
 2 yz COS. /3 COS. y, that is, r'=:x' sin. ' a+y" sin. ' /3-f-2' sin. " ■y — 
 2 ary cos. a cos. /} — 2 xz cos. o cos. •y — 2 yc cos. ,3 cos. y. 
 
 From this equation we immediately get the expression for the 
 moment of inertia 2 (r^ m) of the body, in reference to the axis 
 AB, for putting for brevity 
 
 2 {x^m)=k, 2 (v2 w)=B, 2 (z2m) = C 
 2 (art/m)=D, 2 (a'2-m)=E, 2 (yzw)=F 
 the foregoing equation gives 
 
 2 {r^m)=k sin.2a + B sin.* ;3 + C sin.^y J _ 
 
 — 2D cos.acos./3 — 2 E cos. a cos. y — 2Fcos.|3cos.y5 ^ ' 
 which is the general expression for the moment of inertia of the 
 mass M, in reference to any axis AB through the origin of the co- 
 ordinates, and inclined to them at the angles a, /3, y. The position 
 of the rectangular axes of co-ordinates originating at A, being ar- 
 bitrary, if we could so fix them that they should give D=0, E=0, 
 F=0 .... (4), the general expression (3) would be considerably 
 simplified, being for such axes reduced to the first line ; that is, we 
 ijhould then have, 
 
 2 (r27/i)=Asin.2a-fBsin.2^-fCsin.2y .... (5). 
 We shall presently show that there really does exist for every 
 point A three rectangular axes, for which the conditions (4) have 
 place, these being what are called the principal axes of rotation. 
 But, before we enter upon the proof of this, it will be expedient to 
 show how to transform the equation (3) into another, into which 
 there shall enter, instead of A, B, C, the expressions for the mo- 
 ments of inertia around the three axes chosen for those of the co- 
 ordinates. Now as the distance r' of the particle m from the axis 
 of .r is v' \y^-\-~^ |» its distance r" from the axis of y, v/la^+^*|» 
 and its distance r'" from the axis of r, ^\x'^-\-y''\, it follows that 
 if we put for the moments of inertia about these axes the symbols 
 A', B', C', we shall have 
 
 2 (r'^ »i)=2 (y^-\-z^) m=B + C=A' 
 
 2 (r"2m)=2 (x'^+z^) 7n=A+C=B' 
 
 2(r""'m)=2 (ar^-fy'') w=A+B=C'. 
 When, therefore, these three moments of inertia are known, the 
 moment, with respect to any other axis AB, will be obtained by 
 substituting in the equation (3) the values of A, B, C, in terms of 
 A', B', C', deduced from these expressions; the result of this sub- 
 stitution is 
 
 2 {r^m)=h A' (sin. ''/S-fsin.^y — sin.'^ a) 
 -f i B' (sin.= o-fsin.= y — sin. ^ ^)4-4 C (sin.* o+sin.«^ — sin.^ y). 
 But since cos. * a -|-cos. * ,3-|-cos. * y=l .... (6), or 1 — sin. * o 
 -f-1 — sin.* i3-\-l — sin.* y =1, or sin.- a+ sin.* /3 -f sin.* y=2; 
 this equation is the same as 2 (r*m)=A' cos. * a + B' cos. ' /3-f- 
 C'cos.^^ (7) ; which shows that the moment of inertia, with
 
 7?7 
 
 Jl'l _ /'/„/, II 
 
 -TFl 
 
 in
 
 PRINCIPAL AXES OF ROTATION. 225 
 
 respect to any axis, is equal to the sum of the products, arising from 
 multiplying each of the moments, with respect to the principal axes 
 by the squares of the cosines of the inclinations of the proposed axis 
 to these. On account of the relation (6) between these inclinations, 
 we need introduce but two of them into the expression (7), which 
 may be written 
 S(r2m)=A'+(B'— A')cos.2 ^+(C' —A') cos.^ y . . . . (8). 
 The quantities A', B', C', are necessarily positive, being formed 
 from squares multiplied by masses : hence, if A' is the smallest of 
 the three, every term in this equation will be positive, and, what- 
 ever the arbitrary angles j3, y may be, we must always have, in this 
 case, 2 (r^ni) > A', that is to say, no other line AB through the 
 origin can be found, about which, if the body revolve, the moment 
 of inertia can be so small as when the body revolves about that 
 principal axis which we have taken for the axis of x. But, if A' is 
 the greatest of the three quantities, then S (r^m) <;A', for every 
 value of /3 and y, so that then the principal axis in question will be 
 that for which the moment of inertia is greater than for any other 
 axis through the same origin. A' may, however, be neither the 
 greatest nor the least of the three moments A', B', C, but may be 
 intermediate between B', C'; but we may make either of these 
 three quantities stand first in the equation (8), by eliminating 
 from (7) that angle which multiplies the quantity we wish to stand 
 alone, by means of the relation (6), thus if y be eliminated instead of 
 a, as above, then C' will stand first, and the same conclusions will 
 then apply to the axis of z instead of to the axis of x ; hence, of 
 the three moments of inertia relative to the principal axes, one of 
 them is a maximum, and another a m,inimi(m. 
 
 This conclusion, however, is on the supposition that the princi- 
 pal moments are all unequal ; but it may happen that this is not the 
 case ; let us suppose then that two of them are equal, as A'=B', 
 then the equation (8) becomes 
 
 S (r2m)=A' + (C' — A') cos.^'y ; 
 where it is evident that if A'> C, A' will always be > S (r^m), 
 provided y is not 90°, and-, with the same condition, A' will always 
 be < 2 (r'^m), if A' <CC' ; but, when y=90, whatever a and /3 be, 
 then always 2 (r^m)=A'; we infer, therefore, that when the prin- 
 cipal moments of inertia relative to the axis of x and y are equal, 
 the moments are all equal, for every axis in the plane of xy, and 
 the moment relative to the axis of z, will be a maximum or a 
 minimum, according as A'<C', or A'>C'. 
 
 If A'=B' = C', then 2 (r2m)=A', so that, in this case, all the 
 axes through the origin are principal axes. It follows, therefore, 
 that when more than three principal axes can oass through any 
 point, an infinite can pass through that point, 
 
 29
 
 226 ELEMENTS OF DYNAMICS. 
 
 (172.) It is time now to show tliat tliroiigli every point of space 
 three principal axes may always he tlrawii ; that is, three axes in 
 reference to which the equations (4) have place. From the pro- 
 perties just developed we shall, obviously, he led to one or other 
 of these principal axes, if they exist by determining for what values 
 of the arbitrary and independent angles o, p, the general expression 
 (3), for the moment relative to any axis, becomes a maximum or a 
 minimum ; so that, for the determination of the position of the axes 
 of greatest or least moment or of the suitable values of a and li, we 
 should have, by the theory of maxima and minima, the two equa- 
 
 dx(r'm) „ dx(r'm) „ ... ^ ■ . ^ , 
 
 lions -. ^=0, ^i ^=0; which are sufficient to fix the 
 
 da 0/3 
 
 values of a and ,3. The performance of the actual differentiation, 
 here indicated, is an easy matter ; but the subsequent elimination 
 of a or of j3, in order to obtain a final equation involving only one 
 of these quantities, is a very tedious and troublesome operation, 
 which, at length, conducts to a complicated cubic equation. In- 
 stead, therefore, of employing this method of investigation, we 
 shall, after the example of Professor Whcwell, adopt the following 
 shorter and more elegant process from Lagrange. The problem 
 is to find the position of three axes of rectangular co-ordinates, 
 x', y', z', such that 
 
 2 (x' y' m)=0, 2 {x' z' m)=0, 2 {y' z' m)=0 .... (A). 
 Let the three fixed axes, in reference to which the required ones 
 are to be determined, be those of x, y, z, both systems having the 
 same origin. 
 Let x' make with x, y, z, angles whose cosines are a, b, c 1 
 
 y' . . • . .a', b', c' [-.(1); 
 
 z' . . . . a",6",c"J 
 
 then the angles contained between x and y, between x and ?, and 
 between y and z, being right angles, and, consequently, their co- 
 8ine8=0, we have (Anal. Geom. p. 228, art. 182.) 
 a a' +bb' +cc' =0~] 
 aa"-\-b'b"+cc"=0 
 a'a" +b'b"-\-c'c"=0 
 and (Anal. Geom. p. 228, art. 181,) ^ .... (2); 
 a« -f6'» +0" =1 
 a'»+6'« -j-c'^ =1 
 a"3_j-6"3-|_c"«=l. 
 and these are the six equations of condition, which must be ful- 
 filled by the constants (1). Now we have already seen that the 
 general expression for the moment of inertia, with respect to any 
 axis AB, making the angles a, )3, y, with x, y, z is
 
 PRINCIPAL AXES OF ROTATION. 
 
 227 
 
 2(r''m) = Asin. ''a +B sin. '2)3 + sin. « y. 
 
 — 2D COS. a COS. /3 — 2E COS. a COS. y — 2 F cos. j3 cos. y. 
 But if this same axis AB make the angles a', fi', y', with x', ii' z' 
 then since, by hypothesis, 2 {x'y'ni) =0, &c. the moment of in- 
 ertia, in reference to it will be 
 
 2 {r^m)=k sin. 2a' + B'sin. 2;3'-|-C'sin. ^y'; 
 where A', B', C, stand for S {x'^m), 2 {y'^m,) 2 (z"^m.) 
 
 These two expressions for S (rVi) are, therefore equal. The 
 latter is the same as 
 
 A' + B' + C — A'cos. 2 a'— B'cos.2|3' — C'cos.2 y', 
 but A' + B' + C' = S ( x'^+y'-^^z"') m=s {x"~+y''+z^) m ; and this 
 last expression is what we have before represented by A + B + C ; 
 hence, substituting in the first expression, A — A cos.^a for 
 Asm.i^tt, B — B cos. 2 a for B sin. ^ /3, and C — cos. ^ y for C sin. ^ y, 
 and then equating the two, we have 
 
 A + B + C — (A COS. 2 a+B cos.2|3 +C cos." y) 
 — (2D COS. a cos. ,3+2 E cos. a cos. y+2 F cos, p cos. y) 
 =A + B + C — (A'cos. 2a' + B'cos.2/3' + C' cos.^ y') .... (3.) 
 Now since (Anal. Geom. p. 228, art. 182,) 
 
 COS. a'^a cos. a+6 cos. )3 + c cos. y 
 cos. ^'■:=a' cos. a+6' COS. ^-\-c' cos. y 
 COS. y'=a" cos. a+6" cos. j3 + c'' cos. y ; 
 the last term in (3) becomes 
 
 A' (a^ COS. ^ a-\-b^ cos. 2 p + c'^ cos. ^ y 
 
 + 2 «& cos. tt COS. 3+2 ac cos. a cos. y+26c cos. /3 cos. y) 
 
 + B' (a'2 COS. 2 a+6'2 cos. « |3 + c' 2 cos. " y 
 
 +2 ft' 6' cos. a cos. j3+2 a' c' cos. a cos. y+2 b' c' cos, j3 cos. y) 
 
 + C' {a"^cos. a+6"=*cos.2/3+c"2cos.2y 
 
 +2 a" 6" cos. a cos. ,3+2 a" c" cos. acos. y+2 b" c" cos. /3cos.y); 
 
 and this expression must be identical with 
 
 A COS. "a+B COS. ^/3 + B cos, "y 
 
 + 2 D COS. a COS. i3 + 2 E cos. a cos. y + 2 F COS. /3 COS. y ; 
 
 hence, equating the co-efficients of the like terms, we have 
 
 A'a2+B'a'^+C'a"2 =A (1') 
 
 A'b^ +B' b'^ +C'b"^ =B (2') 
 
 A'c^+B'c'^+C'c"^ =C (3') 
 
 A'ab+B'a'b'-i-C'a"b"=T) (4') 
 
 A'oc+B'a'c' + C'a"c"==E .... (5') 
 A'6c-fB'6'c'+C'6"c"=F. , . . (6') 
 These six equations, combined with the six marked (2), are 
 sufficient to determine the twelve unknown quantities which enter 
 them, but we shall only require to determine four of them, viz. 
 n, b, c, and A', and shall therefore eliminate the rest. In order to 
 this, add together (1') a, (4') b, (5') c, and we have
 
 228 ELEMENTS OF DYNAMICS. 
 
 A'a(a«+i«-f c«)+B'a' (aa'+bb'-\-cr') + C a" (aa"+bb" + cc") = 
 
 Aa+Dft+Ec; 
 or, by the conditions (2), page 226, 
 
 A'fl=Ao+D6-fEc') 
 Similarly (2') 6 + (6') c+(4') a gives A'6=BZ; + Fc-t-Dfl [ . . (4). 
 
 . . (3') c + (5') a + (6') 6 A'c=Cc + Ea + F6 J 
 
 These three equations, together with the condition a'-f 6'+c*= 
 1, are sufficient to determine a, b, c, and A'. 
 By the first two (A' — A) a — Db — Ec =0 
 (A' — B)6— Fc — Da=0 
 from which, by eliminating c, we have 
 
 KA' — A)F+ED^ «— |(A' — B)E + FD|6=0 
 ■ ^,_ (A'-A)F +ED 
 ••^-(A'_B)E + FD"';"^^^' 
 also eliminating b from the same two equations, 
 
 KA' — A)(A — B) — D-^ a— KA' — B)E+FD| c=0 
 . . (A'-A)(A'-B)-D^ 
 
 ••'= (A'-B)E+F1) ""•••' ^^)- 
 
 Substituting these values of b and c in the third of equations (4), 
 that is, in (A' — C)c — Ea — F 6=0, there results 
 
 (A'-A)(A'-B)-D^ (A'-A)F+ED 
 
 C^ -^^ (A'-B)E + FD ^-* (A^ITBTEirFD-"' 
 or, (A' — A)(A'— B)(A' — C) — 
 
 {(A' — A) F^'+CA— B) E*-|-(A' — C) D«| — 2 FED=0 (7), 
 
 a cubic equation in A'. This equation being of the third degree 
 has necessarily, at least, one real root, and consequently, the values 
 of a, b, and c, as determined from the equations (5) and (6) com- 
 bined with the equation a^-^b^-^c'^=l, are real, so that there exists 
 at least one principal axis, viz. the axis of x', and its position is 
 determined by these three equations. 
 
 Returning now to the equations (1'), (2'), &c. and making the 
 same combinations of them as before, only using now a', b', c', 
 instead of a, b, c, we shall, obviously, be led to the same cubic 
 equation (7), except that B' will occupy the place of A' ; and if we 
 use a", 6", c", instead of a, b, c, the resulting cubic will differ 
 from that above only in having C in place of A'. Thus although 
 the first of these cubics determine three positions for the axis of x' 
 (one at least being real) ; the second, three positions for the axis 
 of y' ; and the third, three positions for the axis of z' ; yet these 
 systems of threes must coincide, and can, therefore, furnish only 
 three distinct and different directions for the axes of x', y', and z', 
 given by the three roots of (7). 
 
 It remains to show that these roots are all real. Suppose two of
 
 PRINCIPAL AXES OF ROTATION. 229 
 
 them to be i mpos sible, and, therefore, of the form m + n'^'^l, 
 and m — n\/ — 1 ; the quantities a, b, c, are possible, when the root 
 A' is so, and for one of the impossible roots the corresponding quan- 
 tities a', b', c', will be of the form p+q "^ — 1, p'+q' v^^H^, 
 p"-]-q"->/ — 1, an d for the other root, a", b", c", will be of the 
 formju — q >/ — 1,;j' — q' ^^ — \,p" — q" ^ — 1. Now as 
 a' a"-j-b' b" -\-c' c"=0 
 .•.p''-i-p">+p"^+q^+q'^-\-q"^=0, 
 which is absurd, because the sum of a series of squares can never 
 be 0. Hence the three rectangular axes, each having the property 
 (A), really exist through whatever point we require them to be 
 drawn, because the origin may be arbitrary. 
 
 (173.) But the existence of the three principal axes may be es- 
 tablished in another way, after having shown, as above, that one 
 exists. For suppose the axis of x, whose position is arbitrary, to 
 coincide with this principal axis ; then we must have o=l, b=0, 
 c^O, and these values substituted in the three equations (4) in a, b, c, 
 above, give A'' — A^O, D=0, E=0, so that the cubic equation (7), 
 from which the values of A' are to be deduced, becomes, by putting 
 these values for D and E, 
 
 (A' — A) (A' — B) (A' _ C) — (A' — A) F==0 
 
 .-. (A' — B) (A — C) — F^=0, 
 or A'^ — (B + C) A' + BC - F=^=0 (1). 
 
 This being a quadratic will furnish two values for A', and to de- 
 termine the corresponding inclinations a, b, c, we have (4) the two 
 equations A' b^Bb-\-Fc-{-Da,a"-\-b^-{-c^=l ; but D=0, and, since 
 the axis to be determined must make a right angle with that of x, 
 we must have a=0, therefore 
 
 (A' — B)6=Fc; b^-i-c^ = l, 
 so that if d represent the inclination of the required axis to that of 
 
 , , c A' — B 
 
 y, then o=cos. 6, c=sm. 9, .*. tan. 0=— = — — — 
 
 _ 2 tan. _ 2 F (A' — B) 
 •'• t^"- 2 ^=TZ-[^Te~F^ _(A' — B^ ' 
 the denominator of this fraction will be given by adding the identi- 
 cal equation (C _ B) A' — (C — B) B=(A' — B) (C — B) to the 
 equation (1), for there results 
 
 (A'_B)2 — F^=(A' — B) (C — B) (2). 
 
 2 F 
 .-. tan.2 9=g--^ (3), 
 
 and, as the same tangent belongs to two arcs of which the differ 
 ence is 180°, therefore there are two values for 9, of which the dif- 
 ference is 90°, so that besides the principal axis, which has been 
 U
 
 230 ELEMENTS OF DYNAMICS. 
 
 made coincident with that of x, there are two otlicrs in tlio plane of 
 zy, inclined to the axis of y in the angles e and 90-f-e, or perpen- 
 dicular to each other. 
 
 AVhen we know the position of one of the principal axes, taking 
 it for the axis of x, the position of the other two becomes deter- 
 minable from tlie equation (3), just deduced. 
 
 (174.) Let us now prove, that if a body revolve about one of its 
 principal axes passing through the centre of gravity, this axis will 
 suffer no pressure from the centrifugal forces of the several particles. 
 
 Let the body revolve about the axis of z, then every particle m 
 will describe about this axis the circumference of a circle of radius 
 s/\x'^-\-y- 1 and, therefore, if u be the angular velocity of the sys- 
 tem, « \/^x^+i/2 \ will express the rotatory velocity of any particle 
 7/1 whose co-ordinates are x,y \ but the centrifugal force being equal 
 to the square of tlie velocity, divided by the radius, its general ex- 
 pression here is co^ \/\x'^-^y'^\ and consequently the strain which 
 any particle m produces on the axis is m (^^ y/ \x'^ -\-y- \ ; if this 
 force be resolved in directions parallel to x and y, the two compo- 
 nents will be mu>^ X and mu'^y, and the moment of these forces, to 
 turn the body about the axis of ^ and o( x, will be inu^'xz and rriw^yz, 
 and therefore, of the forces exercised by all the particles, the mo- 
 ments will be 
 
 u'^ 2 (w xz) and o^^ 2 (myz) .... (1), 
 if these be each 0, there will be no effort used by the centrifugal 
 forces to incline the axis of z towards the plane of xy ; such is the 
 case, therefore, when the axis of rotation is a principal axis ; hence, 
 in this case, the only effect of the forces mu^x and mts'y on the axis, 
 is to move it parallel to itself, or to translate the body in the direc- 
 tions of X and y ; the aggregate of these forces is 
 
 w^ 2 (m x) and u« 2 (m y) . . . . (2), 
 and if these be each 0, the forces will use no effort to move the 
 body or to press the axis : and they are when the axis passes 
 through the centre of gravity ; we conclude, therefore, that when a 
 body revolves about one of its principal axes passing through the 
 centre of gravity, the rotation causes no pressure whatever upon 
 the axis, which may, therefore, be removed without at all affecting 
 the motion of the body, the rotation once impressed continuing per- 
 manent. On this account the principal axes through the centre of 
 gravity are called the axes of permanent rotation, or by some, the 
 natural axes of rotation. 
 
 (175.) If the initial rotatory motion of the body be not about a 
 permanent axis of rotation, the effects of the centrifugal forces on 
 the axes cannot be destroyed, inasmuch as the foregoing conditions 
 cannot obtain ; these forces, therefore, will alter the axis of rotation, 
 and the body will at every instant of its motion, if free, turn about
 
 PRINCIPAL AXES OF ROTATION. 231 
 
 a different axis, called the instantaneous axis of rotation ; and it 
 may be proved that if this axis does not at the commencement of 
 motion coincide with a permanent axis, it can never coincide with 
 one afterwards, so that whenever we observe a body to revolve 
 about one axis during any time, however short, we may conclude 
 that it has continued to revolve about that axis from the commence- 
 ment of the motion, and that it will continue to revolve about it for 
 ever, imless checked by some extraneous obstacle. 
 
 These particulars will be more completely established in the fol- 
 lowing articles. 
 
 We may further remark here, that when a body revolves about 
 any one of the principal axes passing through the fixed point, which 
 is taken for the origin, although this point be not the centre of gra- 
 vity of the body, yet the expressions (1) will still be 0, so that the 
 revolving body will use no effort to cause the axis to turn about the 
 origin in any direction ; but as the expressions (2) will not be 0, the 
 axis must sustain a pressure in the directions of x of y, which would 
 cause a tendency in the axis of z to turn about those of y and x, 
 unless these pressures were wholly exerted upon the fixed point or 
 origin ; that is, unless the resultant of all the pressures passed through 
 this point ; such, therefore, in virtue of the former conditions, must 
 be the case ; so that through any given fixed point in a body, there 
 may always be drawn three axes around which the body may turn 
 uniformly without changing its original axis of rotation, although it 
 would be at liberty to do so, as it is free to move in any direction 
 about the fixed point. In order, therefore, that a body retained at 
 rest by a single fixed point may, by means of an impulse, receive a 
 permanent motion of rotation, it is necessary and sufficient that the 
 impulse be such as to cause a percussion on one of the principal 
 axes of the body, through the point, equivalent to a single force ap- 
 plied to this point perpendicularly to the same axis. 
 
 It remains now for us to prove the assertion above, viz. that the 
 instantaneous axis of rotation can at no instant coincide with a per- 
 manent axis unless the body has continued to revolve about this 
 axis from the commencement of the motion, and in order to this it 
 will be convenient, first, to ascertain the equations which express 
 the general theory of a body's rotation about its centre of gravity, 
 and then to discuss those particular forms of them which arise from 
 supposing the rotation to take place about the principal axes. 
 
 (176.) The group of equations (A, B,) at art. (166), which ex- 
 presses the conditions of the motions of any system of bodies mutu- 
 ally connected, and each acted upon by any accelerative forces, 
 obviously holds when the system constitutes a solid body ; we may 
 regard them, therefore, as embodying the analytical theory of the
 
 232 ELEMENTS OF DTNAHICS. 
 
 motion of a solid body, of which each particle is acted upon by any 
 accelerative forces. 
 
 The first three of the equations referred to, completely determine 
 the progressive motion of the body in space, or the path described 
 by its centre of gravity, furnishing for this purpose the requisite 
 equations 
 
 rf'x _ 2(?nX) rf'Y _ s(mY) d^z _ s (rnZ) 
 
 dt^ ~ AT"' (//2 ~ M ' "^ M~' 
 
 where the sign S includes under it all (he particles m, m,, m,, &c. 
 
 of the mass M, which are acted upon by the accelerative forces X, 
 
 X„X,; Y, Y,, Y„«fec. 
 
 The remaining three equations (B) must be those which deter- 
 mine the rotatory motion of the body round this moving centre ; or 
 if the centre of gravity remain fixed, and the body be free to move 
 round it in every direction, then the three equations (B) must be 
 sufficient to determine the circumstances of the rotatory motion 
 arising from the action of the same accelerative forces. 
 
 The rotatory motion thus produced, on the supposition that the 
 centre of gravity is fixed, must, since the progressive and rotatory 
 motions are independent of each other, be really that which accom- 
 panies the progressive motion the body actually has when this cen- 
 tre is free, and which progressive motion is that which the centre 
 would have if all the accelerative forces acted upon it as a single 
 free point, so that the absolute motion in space of ai particle of 
 the body is compounded of these two. 
 
 Supposing then the centre of gravity of the body t be fixed, if 
 we place there the origin of the co-ordinates, all that .oncerns the 
 rotatory motions of the body will be comprised in the equations 
 
 . y d" X X d^ y . ,^ _. . 
 
 S ( -^^^^ m ^^w)=S (Xym — Yx m) 
 
 ^^~di^'^ 'W^^^^ (ZxTW — Xzm) 
 
 z d^ y 11 d^ z 
 S(-^m-^^^^^m)=S(Yz/n — Zy7n). 
 
 Let us represent, as at (170) the angular velocities of the body 
 about the three axes of co-ordinates by w, w', w", then, instead of 
 
 the coefficients -y-j , -r-j , we may introduce their values from equa- 
 tion (1), page 221, so that the first of the preceding equations will 
 be, by transposing, 
 
 _ /"ir V N d(o"x — iiz) d(u'z — «"y) 
 X(Ya?m — Xi/m)=2x— i ^ -m — sy — ^^ — -3- ^^m.
 
 PRINCIPAL AXES OF ROTATION. 233 
 
 If we actually perform the differentiation here indicated, and always 
 
 , . . dx dy dz . . 
 substitute for — , ^, — , their values as given by the equations re- 
 ferred to, we shall have 
 
 (2co"2+co'2— co2) 2 (xym) 
 — (»""+-^) S (y^ m) — («"«'+ £)^{xz m)+ 
 
 o «' S (x^—y^) m. 
 Now let us suppose the axes of co-ordinates to be the principal 
 axes of the body, then we know that 
 
 2 (xym)=0, (xzm)=0, {yzm)=0; 
 hence, putting 
 
 2(y2+z2)wi=:A, 2G'r2+z2)m=B, ^(x^ -\-y'') m=C 
 .-. 2(a?2-^2) 7n=B — A, 
 the foregoing equation becomes simply 
 
 2:(Ya?m — X2/m) = C-^+(B — A) ««', 
 
 and, in a precisely similar way, we obtain 
 
 2 (X z m— Z a; wi)=B-^ -f (A — C)«w" 
 
 2(Zym — Yzw)=A^+(C — B)«'«". 
 
 (177.) Suppose now that no accelerative forces, X, Y, &c. solicit 
 the body, these equations become in that case 
 
 A J+(C — B)m'«"=0^ 
 B-^+(A— C)««"=0 
 
 }>....(.). 
 
 dt 
 
 C^+(B— A)««' = Oj 
 
 In these equations A, B, C, are constant for the same body, and 
 putting for abridgment 
 
 B— C ^ C— A ,^ A— B ^ 
 
 they become rf« = L o' «" dt, dot' = M « «" dt, da" = Nu «' dt. 
 Multiplying these severally by u, «', «", and putting ua'co"dt=:d^, 
 we have «rf« = L df, «'a«' = M d^, w"t?»" = N d^, and the in- 
 tegrals of these are 
 
 u2 30
 
 234 ELEMENTS OF DYNAMICS. 
 
 u" = 2L ^ + a\ u'" = 2M ip + 6«, «'"» = 2N <}. + c« . . . . (2), 
 where a, b, c, are what u, u', u" become when /, which is the in- 
 dependent variable in the function f, becomes ; that is, those con- 
 stants are the initial rotatory velocities about the axes. 
 
 Consequently, substituting these values foruw'co" in the equation 
 
 dt= 7-77-' w^ shall have, for the determination of *, correspond- 
 
 ing to any time t", in functions of /, the differential equation 
 
 v/K2L<}'-fa2X2M.}.+62)(2N4. + c2)f 
 Suppose now that an initial velocity a is given to the body about 
 one only of the principal axes, then 6 •= 0, c = 0, and this expres- 
 
 sion becomes dt =^^^^~^ ■ f^^^r^f^' ^hat is, replacing 
 2L^+a® by its value w', and d^ by its value— jt^ , 
 
 . 1__ ^" 
 
 ^ ^/JiMNf «» — a"' 
 and the integral of this is 
 
 ^ ^ ^ * 2a ^ «-f a 
 
 .-. e'"'C. c2<"'>/mn= "~" .... (3). 
 lo-f-a ^ ^ 
 
 Now the constant C must be determined so that when /=0, w may 
 be =a, that is, the first member must vanish for t=0 ; hence e*"^ 
 must =0, or C = — co, consequently, there must always be uz=a, 
 and therefore (2) always $=0, w'=0, a)"=0 ; consequently, as be- 
 fore shown, the impressed velocity about one of the principal axes 
 of rotation continues perpetual and uniform. 
 
 We see, moreover, from the equation (3), that if the instanta- 
 neous axis of rotation does not coincide with a principal axis at 
 the commencement of motion, it cannever afterwards coincide with 
 it : for if we suppose the coincidence to take place at any epoch, and 
 that the angular velocity is then a, then, measuring t" from that 
 epoch, the foregoing equation must give u>=a when /=0, which re- 
 quires, as shown above, that C= — od, and therefore w must be al- 
 ways =a, for every value, positive or negative, of t. 
 
 (178.) Let us now see what must be the conditions, in order that 
 the instantaneous axis of rotation, if it do not accurately coincide 
 with one of the principal axes, may yet always be very nearly co- 
 incident. 
 
 Let us suppose the axis of instantaneous rotation to be nearly co-
 
 PRINCIPAL AXES OF ROTATION. 235 
 
 incident with the axis of z ; then considering the angular motion to 
 be the resultant of three, about the three principal axes, the veloci- 
 ties o, w', about the axes of x and y, are, by hypothesis, to be very 
 small in comparison with the angular velocity to" about the axis of 
 z, because the body turns almost entirely about this axis. The ex- 
 pression for the sine of y, the angle which the instantaneous axis 
 makes with that of z, is (p. 222,) 
 
 sm. y= — ^— ^ ! i— 
 
 Now, on account of the smallness of both w and co', the tliird of 
 the equations (1) p. 233, becomes C -^=0 very nearly ; so that 
 
 «", the velocity round the axis of z, is very nearly constant. Call 
 it w"=w, then the remaining equations of (1) become 
 . dui ,-, „, 
 
 B-^+(A-C)nw=0j 
 By differentiating the first of these, we have 
 
 but, from the second, — = — - — n^ ; hence, by substitution, 
 
 rf. (A-C)(B- C) 
 dt^ + AB "" "-^ 
 
 or, putting for brevity the coefficient of io=/2, 
 
 ^+;»„=o....(2). 
 
 The integral of this equation is, (see Int. Calc. p. 233,) 
 
 1 o 
 
 t-{-c'=-rsm. — 1 — . 
 / c 
 
 If at the commencement of motion, or when t=0, « were accu 
 
 lately 0, the constant c' would necessarily be ; as it is, however, 
 
 c' must be very small : calling Ic', k, we have 
 
 «=c sin. (It-\-k) .'. — =lc . cos. {lt-\-k) ; 
 
 and, substituting this last expression in the first of (1), we get for 
 , , , , Ale . cos. (It4-k) 
 
 « the value W == rrr ^^-^rrr ^. 
 
 n{B — C) 
 Here also we may observe that if, at the commencement of the 
 motion, the instantaneous axis accurately coincided with the axis 
 of z, c would necessarily be 0, for otherwise w and «' would never
 
 236 ELEMENTS OF DYNAMICS. 
 
 be both ; so that we again infer wliat has been otherwise proved, 
 viz. that w and u' if at the beginning are always 0. But, if, as 
 we here suppose, u and w' are not accurately at tlie commence- 
 ment, then c, instead of being 0, must be very small ; consequently, 
 if / involves no impossible quantity, w and w' must always be small, 
 however great / may be, for be this as great as it may the factors 
 sin. (lt-\-k), cos. {lt-\-k), can never exceed unity. 
 
 The expression for / is l=^n^ \ — -\ ; where A, 
 
 B, C are essentially positive ; hence that the expression may be 
 possible A — C, B — C, must be cither both positive or both nega- 
 tive ; that is, C, the moment of inertia with respect to that axis 
 about which the axis of instantaneous rotation perpetually oscillates, 
 must be either the least or the greatest of the three moments A, B, C. 
 
 If we were to integrate (2) on the hypothesis that / is imaginary, 
 or l^ negative, the resulting expressions for to, w', would be expo- 
 nentials, t entering as an exponent ; their values would, therefore, 
 increase continually with /, showing that the supposition of these 
 quantities continuing small, after the commencement, is inadmis- 
 sible. 
 
 We conclude, therefore, that when a body commences to revolve 
 about a principal axis, it will perpetually do so : when it happens 
 that any extraneous cause deranges this uniform rotation a little, the 
 body will, nevertheless, always have the new axes of rotation, whicli 
 it perpetually turns about very near to the original axis, provided 
 this happened to be either the axis of greatest or of least moment ; 
 but if it happened to be the axis of mean moment, then, however 
 trifling the derangement may have been, its effects will increase with 
 the time, and the body will depart altogether from its original mo- 
 tion. We hence say that the rotation about the principal axis of 
 greatest or of least moment is stable, while the rotation about the 
 axis of mean moment is unstable. 
 
 CHAPTER VII. 
 
 MISCELLANEOUS DYNAMICAL PROBLEMS. 
 
 Problem I. — (179.) If a body revolve about any centre of force 
 S, (fig. 129,) and if the velocity at an apse A is v, the expression 
 for the sector ASP, described by the radius vector in any time / 
 from A, will be ASP = | AS . v . t, required the proof. 
 
 Taking S for the origin of the rectangular axes, and SA for tlie
 
 MISCELLANEOUS DVNAMICAL PROBLEMS. 237 
 
 axis of X, the components of the velocity in the curve are always 
 
 dx . dy . , , . A 
 
 dt dt' ^^ ^P^^' v<^locity in the curve being in a parallel 
 
 direction to the axis of y, and being in the direction of x, we have, 
 at this point, ^=0, -^=v; therefore, since (122) ^'^^yzZjI^^c, 
 we must have at A, a;t;=AS . v=c .-. |cf =sector ASP=i AS .v . /. 
 
 Problem II. — To determine the curve, such that if it revolve 
 about its axis placed vertically, with a given angular velocity, a 
 heavy ring at liberty to slide along it shall remain wherever it is 
 placed, (fig. 130.) 
 
 Let CBA be the curve, and u the given angular velocity, and let 
 w be the point where the ring is placed : draw RN a normal to the 
 curve and RM perpendicular to the axis •? also let ST be a tangent 
 at R, and call the angle NRM, &c. The absolute velocity of R is 
 
 MR^ <o^ 
 
 MR. w, and, therefore, the centrifugal force is '- — =MR . w^ 
 
 ^ MR 
 
 in the direction RR' ; the component of this, in the direction RT 
 of the curve, is MR . u^ sin. a, and the component of the force g of 
 gravity in opposition to this, that is, in the direction RS,is gcos.a; 
 hence, in order that the ring may rest, these two forces must 
 balance each other, .•. MR . co^ sin. a.=.g cos. a 
 
 MN 2- 
 
 •■• ^^ • "'mr="' • ^^=^ •■• ^^=5' 
 
 that is, the subnormal is constant, and, hence, the curve is a parabola 
 with its vertex downward. Upon similar principles we may deter- 
 mine the concave surface which a fluid presents when a rotatory 
 motion is given to the vessel which contains it. This surface is a 
 paraboloid. 
 
 Problem III. — An upright cylinder standing on a smooth hori- 
 zontal plane has a string coiled several times round it in the plane 
 of its base ; one end is fixed to the cylinder, and to the other is 
 attached a body P, to which a velocity is given in the direction of 
 the string; to determine the motion, (fig. 131.) 
 
 Let V be the progressive velocity of the cylinder at any instant, 
 
 then M being its mass, Mu will be the impulsive force due to this 
 
 velocity ; as it acts at the extremity of the radius r of the base, and 
 
 in the direction of a tangent, it will communicate to the circum- 
 
 Mw • r^ 
 ference of the cylinder the rotatory velocity =2v; therefore, 
 
 the absolute velocity of every point of the unwound string must, at
 
 238 ELEMENTS OF DYNAMICS. 
 
 that instant, be 3v, which, therefore, expresses the correspondinc 
 velocity of P ; hence, at that instant, the whole momentum of the 
 system is Mu + 3Pi;; but if the given velocity originally commu- 
 nicated to P be I',, the momentum communicated will be Fi\, 
 
 Vv 
 .-. Mv+3 Pu=Prj .-. v= ri— Tu ' *'^'^' therefore, is the velocity 
 
 of the cylinder in progression, at any instant, and twice this is the 
 rotatory velocity of the circumference ; the velocity is, therefore, 
 uniform, so that the motion of the cylinder is wholly due to the 
 initial impulse it receives from P ; P, therefore, never afterwards 
 acts on the cylinder. 
 
 Problem IV. — A uniform straight rod AB (fig. 132,) is placed in 
 an assigned position upon a smooth horizontal plane, and one end 
 of it, B, is drawn uniformly along the straight line CD with a given 
 velocity u ; it is required to find the position of the rod at any time, 
 and its angular velocity. See note at page 257. 
 
 Let G be the centre of gravity of the rod, then the uniform 
 motion of B along the straight line CD may be considered as the 
 consequence of an impulse at G, in the direction GD', parallel to 
 CD. As the progressive velocity thus generated is v, the value of 
 the impulsive force is 2a- v, a representing the mass of half the 
 rod or of AG. But, as this force really acts at B instead of G, 
 there would be generated in addition to the progressive velocity a 
 uniform angular velocity about G, if B were not constrained to 
 continue on the line CD ; as it is the angular motion must be about 
 B. Now the same force which applied at B produces any angular 
 motion of the rod about G, would, obviously, if applied at A, and 
 in an opposite direction, produce the same angular motion ; what- 
 ever angular motion, therefore, has place in the present case, is due 
 to the force 2 av applied, at the commencement of motion, to the 
 point A in the direction AC'. Calling, therefore, the angle ABY, 
 which the rod makes with the perpendicular BY at the beginning 
 «j, we have, for the constant angular velocity about B, 
 du 2ayxBE 2ayx2 «cos. w, ' 3 v 
 dt^i{2ay~^ p ^1' '^ *^°^' "^ ' 
 
 hence, at any time /", the rod will make with BY an angle w equal 
 
 V 
 
 to I < • — cos. Wj, and the length of path gone over by B will have 
 
 been equal to tv, so that the position and angular velocity of the 
 rod at any time is completely determined. 
 
 The curve traced out by any point in the moving rod is, ob- 
 viously, a species of cycloid, for each point describes uniformly 
 the circumference of a circle, whose centre B is uniformly moving
 
 MISCELLANEOUS DYNAMICAL PROBLEMS. 239 
 
 along a straight line ; that point in the rod whose rotatory ve- 
 locity is equal to the progressive velocity of B will describe the 
 common cycloid ; for if with centre B a circle be described throutrh 
 this point, and then a tangent to it be drawn parallel to CD, this- 
 circle, by rolling on the tangent so that its centre, or the point of 
 contact, may move with the proposed velocity of B, will obviously 
 cause every point in the circumference to revolve with that same 
 velocity, and thus the point in question will trace the path which 
 it actually has, and Avhich must, therefore, be the common cycloid. 
 
 Problem V. — Suppose a heavy particle is placed at a given 
 point in a perfectly smooth narrow tube of a given length, and sup- 
 pose the tube to be whirled about one end as a centre with a given 
 angular velocity in a horizontal plane ; it is required to determine 
 the velocity and direction of the particle when it leaves the tube ; 
 the motion being solely generated by the revolving tube, (fig. 13.3.) 
 
 Let SA=a be the tube in its first position, and B the place of the 
 particle. Call SB, b, and the distance SP of the particle at any 
 time t", r ; let also v represent the uniform angular velocity of the 
 tube, and SC its position, when the particle quits it. 
 
 As no centripetal force acts on the particle, its motion along the 
 tube is entirely due to the centrifugal force rv^ (art. 138), that is 
 
 — — = rv^. 
 dp 
 
 Multiplying this by 2 dr, and integrating, we have 
 
 dr^ dr^ 
 
 — =r^v^+C, also 0=bv^+C .-. —=v^ (r^ — b') . (1) ; 
 
 dt'^ dr 
 
 hence, when r=a, or when the body arrives at the mouth C of the 
 
 tube, its velocity in the direction CE of the tube is v v/{«" — b^ \ 
 
 also its velocity in the direction CD, perpendicular to this is va ; 
 
 hence its velocity in its path at that point is the component of these, 
 
 viz. V 'y{2a^ — b^\ and for the angle ECF, which the direction 
 
 makes with the tube, we have 
 
 „„„ CD va a 
 
 tan.Z-ECF=^=^^^^,_^,, = -^^-^-. 
 
 This angle, added to the angle S, will give the position of CF, 
 with respect to AS. If T represent the time in which the tube 
 passes from the position SA to SC, then since v is the angle 
 described in one second, Tv will express the angle S, and to find 
 T, we have, from the equation (1), 
 
 1 r-{-^\r' — b»\ ^ 1 , a+^{a-'^b'\^ 
 
 t=-\og. .^^^^ ^ .-. T=-xlog 
 
 a+^\a^—b 
 
 V " b 
 
 so that the angle S is expressed by log.
 
 240 KLEMENT8 OF DYNAMICS. 
 
 Pnoni-EM VI. — Two bodies P and Q (fijr- l^^O ^re plarod on a 
 smootli horizontal plane, and connected by a perfectly flexible and 
 inextensible thread, which passes freely through a small rinj^ R in 
 the plane ; a given velocity is communicated to 1* in a given di- 
 rection : it is required to find the equation of the curve described by 
 P on the plane, the length of the thread being indefinite. 
 
 Call the radius vector RP, r, and let P, Q represent the masses 
 
 d'r 
 of the bodies, then -r-j being the acceleration of Q, the motive 
 
 d'r 
 force of Q must be Q —j-^ ; hence P must be drawn towards the 
 
 centre R by this motive force ; the accelerative force on P is there- 
 
 Q(i* r 
 fore — — r- which may be considered as the centripetal force at R 
 P (If* ^ * 
 
 which retains P in its orbit. 
 
 rf'r . 
 Now we know that the paracentric force — r-j is equal to the 
 
 difTerence between the centripetal and the centrifugal force — , (see 
 
 art. 138,) therefore 5^-f 1^^=| .... (1). 
 
 Multiplying by 2 dr and integrating, 
 
 O dr^ 1 1 c^ r'^ r * 
 
 or, substituting for the square of the paracentric velocity its value 
 
 / 5 f^ r s > I P_1_Q" 1 
 
 at page 170, we have — ^ — = I „ ^ • J . « being the 
 
 angle between r and any fixed line RX. Consequently, 
 
 . _ | P + Q c,dr 
 
 -^ P ' rx/lr* — c,''!' 
 
 JP-fQ r 
 
 — — — . sec— J — . . . . (3), the 
 
 polar equation of the orbit. 
 
 The constants c^ and c^ are readily determined from the initial 
 conditions of the motion ; thus let v, be the initial velocity, and o 
 the angle which its direction makes with the original position of the 
 string r, and let the value of r be then a ; the initial velocity in the 
 direction of the radius vector will consequently be u cos. a, hence, 
 
 (If 
 putting this for -y- in the equation (2), we have, for the determina- 
 tion of Cj , this equation, observing that c, which denotes double the
 
 MISCELLANEOLS DYNAMICAL PROBLEMS. 241 
 
 sectorial area described about R in one second, and which is con- 
 stant, is expressed by a v^ sin. a, 
 
 . . , . r 1 .V, 1 1 1 ,P + Qcos.«a 
 
 which ffives for — the value — = — v" — :^^ . 
 
 ^ Cj c^ a Psin.^a 
 
 To determine c^ let the radius vector coincide with the assumed 
 line RX at the commencement; then u=0 when r=a, and con- 
 sequently, 
 
 -J 
 
 P + Q , P + QcOS.^a 
 
 — — — . sec.— 1^/- 
 
 P sin.^ a 
 
 Problem VII, — A given weight W (fig. 135,) hangs at the middle 
 point 71 of a string PCnC'P' passing over the pulleys C, C, in the 
 same horizontal line, drawing up the two equal weights P, P', hang- 
 ing at the extremities of the cord ; to determine the motion of W. 
 
 Draw nD perpendicular to the horizontal line CC' then CD=C'D 
 
 Let CD=a, Dn=x, Cn=y, then — =cos. CnD=cos. C'nD 
 
 y 
 
 Now to the point n there are applied the force P, in the direction 
 nC, the force P' in the direction nC', and the force W in the di- 
 rection nW. 
 
 The actual, or effective forces, in these directions, are respectively 
 
 — . -7^ , — . —r^ , — . -; — , and of these it must be observed 
 g dt^ g dt^ g dP 
 
 that the first two have contrary directions to the corresponding ap- 
 plied forces ; for the applied force P acts from n towards C, where- 
 as the actual force is from C towards n, since W descends ; conse- 
 quently, taking the effective forces in opposite directions to those 
 which they really have, there must, by the principle of D'Alembcrt, 
 be an equilibrium among the forces 
 
 ^ g ' dV' ' ^ g' dV^' g ' dt^' 
 
 acting at n, in the directions ??C, n(^' nW. Consequently the verti- 
 cal components of these must equilibrate, that is, 
 
 ^^^^ g' dt^-> 2/ -^^ g' dt^ "-^^^' 
 The relation between x and y is y"^ =a^-\-x^ .'. — = -r^ , , , (2) 
 
 X 
 
 therefore, substituting this value of — , dividing by 2P, and putting 
 
 y 
 
 W 
 
 for abridgment — — =7n, we have 
 
 At 1 
 
 X 31
 
 242 ELEMENTS OF DYNAMICS. 
 
 "'l^^'^'^'d'^y^^ (r«</:r — f/j/) (3); 
 
 and, by integration, im — +i i7^=S {nix — y) + c . . . (3) ; 
 
 dy^ x^ dx~ 
 or, since in virtue of the condition (2), —r- = — . —r-^ this equa- 
 
 ^ ^ rfr y^ dt^ ^ 
 
 dx^ „ 2 s; (mx — v)+2c 
 tion gives — r— =v — ^^r-; — -t ; 
 
 which expresses the square of the velocity of W. The determina- 
 tion of c depends upon the initial position, when the velocity is ; 
 thus c^g (?/i — mx^), where x■^^ and y^ are the values of a; and y, 
 at the commencement of motion. 
 
 To find when the velocity is again we have/rom (3), by intro- 
 ducing this value of c, the equation g [mx — y)-\-g (^/i — wxJ^O ; 
 or, substituting for y its va\\iey/\a'^-\-x^], and reducing, we have 
 the quadratic equation 
 
 (1 — m^) x~ — (2 my I — 2m^x-i) x — (y^ — mx.^)--\-a^=0. 
 One of the roots of this equation is by hypothesis, x=x^, and, if 
 we represent the other by x^, we have, by the theory of equations, 
 _ 2m (y,—mx,). _ 2my,~(m'^+l) a^ i 
 
 If m=l, that is, if W=2P, W will continually descend, and never 
 become stationary, as remarked at page 30, and the same will oi 
 course happen if W > 2P ; hence, for the system to become sta- 
 tionary at any time after the commencement of motion, W must be 
 less than 2P, and the distance below the horizontal line, at which 
 this will take place, will be given by the above value of x^ , so that 
 the weight W will move continually backward and forward between 
 the two points x^, x^. Whenever the relation between 2P and AV 
 is such as to render x^ negative, it will be impossible for the system 
 to become stationary at any time after the commencement. 
 This problem may be solved otherwise as follows : 
 Since the components of the forces P, P' in 7iC, wC are together 
 
 X 
 
 equal to 2 P — , a portion of W, equal to this, is expended in balanc- 
 ing the equal weights P, P', hence the moving force is only W — 
 
 X 
 
 2P — , and this, divided by the inertia opposed to motion at the 
 
 d'^x 
 point n, must give the acceleration -j-^ of W. The inertia due to 
 
 W 
 
 W is its own mass — ; the inertia of the other bodies must be ex- 
 
 g 
 pressed by such a mass placed at n, or incorporated with W, as
 
 MISCELLANEOUS DYNAMICAL PROBLEMS. 243 
 
 when multiplied by the acceleration of W will give the same mo- 
 tive force in the vertical direction that the real bodies P, P', acting- 
 along nC, nC have in the vertical direction. The motive force of 
 
 P + P', or 2P, in the direction nW is 2 — . — . — •" ; to determine 
 
 g y dt^ 
 
 therefore what mass M, having the acceleration -r-^ , must have the 
 
 same motive force, we have 
 
 M-^ = 2^ ^ i^.M='>^ ^i'}!y^£-^^ 
 
 dt' g y dl' ^ S ' y dt^ ' di" 
 
 Dividing, therefore, the motive force W — 2 P — by the whole in- 
 
 y 
 
 d^x 
 ertia at n, we have for the acceleration — — the equation 
 
 dp 
 
 g (W— 3 P — ) 
 
 ^ ^ y ^ d^x 
 
 ^^'^^^J^'dF"^dF-> 
 
 therefore, putting as before 
 
 . W dy . X d^x d^y ^ 
 
 m for , and -^ lor — , g(mdx — dy) = m — i dun 
 
 2p dx y^^ ^^ dt^ ^ dt^ '^' 
 
 which is the same as equation (3) before determined. 
 
 Problem VIII. — Two given bodies P, Q (fig. 136,) are connect- 
 ed together by a string, which passes over a fixed pulley at a given 
 distance from a smooth horizontal plane. It is required to deter- 
 mine the circumstances of the motion when P is drawn alono- the 
 plane by the descending weight Q. 
 
 Let the velocity of Q be v, and the velocity of P, M ; then, call- 
 ing PB, X, and PC, s, CQ, y, and the angle P, a, we have 
 V ds X 
 
 u dx ' s ' 
 
 hence the actual motive force of P, in the direction PC, and of 
 which the force in PB is the component, is 
 du F 1 _ du P u 
 
 dt g cos. o dt g V ' ' ' ' ^ '' 
 the impressed force on P is gravity and the resistance of the plane 
 so that no moving force is impressed. 
 
 Again, the actual force of Q is —. — — , and the impressed 
 
 dt g 
 
 force is the weight Q. Hence, by D'Alembert's principle, the 
 
 force (1,) acting in the direction CP, equilibrates the force
 
 244 ELEMENTS OF DYNAMICS. 
 
 acting in the direction CB, that is, 
 
 at V ° at 
 
 .-. P u du-\-Q, V dv=Q g V dt=Q, g dy 
 and integrating this equation, P Ji'^ + Q y^=2 Q g (y — c), c being 
 the value of y at the commencement of motion, therefore, since 
 
 we have v'^=: ~ j^ for the square of the velocity of Q, 
 
 P s^+Q x^ 
 
 » and y being known in terms of x from the known length of CB, 
 and of the string ; and the velocity of P is then found from the pre- 
 ceding equation. 
 
 The problem may be solved otherwise as follows : 
 Having found, as before, that the motive force of P is 
 du F u 
 dt ' g ' V * 
 we have, for the determination of the mass M, Avhich, when accele- 
 rated with Q has the same motive force, or offers the same resistance 
 to motion, the equation 
 
 -^ dv du F u ,, P M du 
 dt dt g V g V dv 
 
 Q 
 hence the whole inertia of the system being M -| — , we have for 
 
 the acceleration of Q 
 
 Q Qgv dv , ,^„ s 
 
 — ^ =^T (P^Se 123.) 
 
 M + — Qv +F u — ^ 
 
 g dv 
 
 Consequently Q^ dy=Q v dv-^F u du, as before. 
 
 Problem IX. — A perfectly flexible chain is wound round a cylin- 
 der, supported with its axis parallel to the horizon. Then the 
 weight and dimensions of the cylinder being given, as also the 
 weight and length of the chain : it is required to determine the time 
 in which the chain, impelled by the force of gravity, will unwind 
 itself; a given length being unwound at the commencement of the 
 motion. 
 
 The moving force here always acting is due to that part of the 
 chain which hangs down, and the resistance to be overcome is the 
 rotatory inertia of the cylinder, and of the mass of chain which en-
 
 MISCELLANEOUS DYNAMICAL PROBLEMS. 245 
 
 velopes it. As, (page 189,) the square of the radius of gyration of 
 the cylinder is gR'-', and that of a mere circumference of the same 
 radius R, R^, it follows that the system would ofler the same resist- 
 ance to rotation, if for the cylinder we substitute an indefinitely 
 slender ring of the same radius, containing half the mass of the cy- 
 linder, and the whole mass of the unwound chain. 
 
 Let then 2W denote the whole weight of the cylinder, and w 
 that of the chain ; let the length of the chain be /, a the length hang- 
 ing down at the commencement of the motion, and x the length 
 hanging down at any time t" . Then the weight of any part being 
 
 as its length, we have I : x : : iv : —r-, which expresses the moving 
 
 force acting at the extremity of the radius at any time t" ; the value 
 
 wx 
 of this force so acting to turn the system is R— r-j and this, divided 
 
 by the moment of inertia, will give the angular acceleration (158), 
 and consequently the acceleration of the extremity of the radius, or 
 of the descending chain, is 
 
 „„ wx „„ ,W+?t' , . d^x wx 
 R2 — r- -T- R2 ( — -?— ), that IS, -~= ^-ttttt r- 
 
 I g dt^ * i{yi+w) 
 
 dx 
 Multiplying this by 2-^, we have 
 
 2 d^^ X dx wx dx 
 
 dt^ ' ~dt^ ° l{W+zv) dt' 
 This, multiplied by dt and integrated, gives 
 
 dx^ „ tvx^ ^ 
 
 dP ^ /(W+w) ^ 
 
 To determine C we have the condition v=0 when x=a, 
 
 w a^ 
 
 ■■■ »=^7(W+1^+ ^ • 
 
 hence, subtracting this equation from the former, 
 
 w Ix^—a^) dx , w (x^— a^) 
 
 l{yj + w) dt ^ '^ /(W + w;) ^' 
 
 consequently 
 
 dt- ^/ \ \ TJ^Zr^a-y 
 
 the integral of which is (Int. Calc. art. 18), 
 
 which is the number of seconds occupied in unwinding the length 
 X — a, the part a being unwound at the beginning ; and when x = 1 
 x2
 
 246 ELEMENTS OF DYNAMICS. 
 
 we have, for the number of seconds occupied in unwinding the 
 whole chain, 
 
 Problem X. — To a point in the circumference of the base of an 
 upright cylinder, standing on a smooth horizontal plane, is fastened 
 one extremity of a string, and to the other a weight P. Now a 
 given velocity is communicated to P in a direction perpendicular to 
 the string: to determine the circumstances of the motion (fig. 137). 
 
 Let A be the point to which the string is fastened, and B the point 
 of contact at any time t" ', let also AOB=o, BP=:z, AO==o, ten- 
 sion of the string =T, then, taking ON, NP for the rectangular 
 co-ordinates of P at the time t" , we have, since the acceleration of 
 
 T 
 
 P in the direction PB is — , these equations of its motion in the 
 
 directions of the co-ordinates, viz. 
 
 cPx T . d^y T 
 
 ^ = -p«"-"'^ = -p- COS. «....(!), 
 
 T 
 
 the negative signs being used because the accelerative force — , being 
 
 in the direction PB, tends to diminish the co-ordinates. 
 
 Multiplying the first of these equations by cos. co, and the second 
 
 by sin. w, and subtracting, we have 
 
 d^'x . d^y ^ 
 
 cos. u> —r- sm. to -—-^=0 . . . . (2 ). 
 
 dt'' df' ^ ^ 
 
 Again, the angular velocity of the string, since OBN'=o, is — ; 
 
 and, therefore, the absolute velocity of P in direction PQ, of its 
 
 path, is z — , and therefore its components, in the directions of the 
 
 co-ordinates, are z cos. w . — , and — z sin. w -j-, consequently, by 
 
 differentiating, 
 
 d^x dz doi . dJ^ d^a 
 
 —r-r=-rCOS. CO —^ jjsm.co -7- — ^zcos. «^—. 
 
 rf/2 dt dt dp dP 
 
 d^y dz . du> doi" . d^a 
 
 —rr= r- sm. co -; z cos. co -7- — z sm. 10 -r-r-' 
 
 dp dt dt dt'' dp 
 
 Substituting these in the equation (2), there results 
 
 dz fZco rf^co dz (?^co da 
 
 di'di'^^'dp'^ '' l'^~'dP~dt'
 
 MISCELLANEOUS DYNAMICAL PROBLEMS. 247 
 
 and inteerratiiiff each side, W.— = W. -t- ,•. — =-— -, the anofular 
 
 z ° dt 2 dt ° 
 
 velocity of the string at any time t". To determine c let v^ be the 
 initial velocity of P when z:=6, then the initial angular velocity is 
 
 V C V 
 
 -T^=y- .'. c=Uj ; consequently, — = angular velocity of the string, 
 
 ^1= absolute velocity of P; hence, P continually moves with its 
 
 initial velocity. As the centrifugal force of P is that to which the 
 
 V 2 Pv 2 
 tension of the string is due, we have, T=P — =j — ^—, I being the 
 
 whole length ABP of the string. 
 
 Problem XI. — To determine the path of a projectile in a resist- 
 ing medium. 
 
 Let R represent the resistance of the medium in opposition to 
 the motion of the body ; then the forces acting upon it, in the di- 
 rections of its horizontal and vertical co-ordinates, are 
 d^_ dx d^y _ dy 
 
 dP~ ds' dt"~ ^ ds' 
 
 which are the equations of the motion ; and by means of which, 
 when the law of resistance is known, the nature of the trajectory 
 may be determined. The law generally received is, that the re- 
 sistance varies as the square of the velocity ; assuming, therefore, 
 this to be the case, the foregoing equations are 
 
 d"^ X dx d^y „ c?V • ds' 
 
 — — = — mv'^ ^-, -Tfr = — ff — mv^ -r, or, smce 1;^= __ 
 dt^ ds dt^ ^ ds dt^ 
 
 the equations are the same as 
 
 d^x_ ds^ dx d^y_ ds" dy 
 
 'dF~~^di^' d^^'di^ ^~^dP \F* 
 
 As the second members of each of these equations contain only 
 
 first differential co-efficients, and as the values of these co-efficients 
 
 remain the same, however we change the independent variable, 
 
 (Biff. Calc. p. 99,) the equations may be written 
 
 d^x 
 
 d^x ds dx d^y ds dy ^ ^ dt 
 
 -dF^-'^Tt • W IF^-^-'^dt' di ' ^'^ •••^=-^^* 
 
 dt 
 
 Integrating this equation we have 
 
 . dx , _, rfa? 
 
 log.^=-m.-fC.-.-^=c6— ....(2), 
 
 e being the base of the hyperbolic system of logarithms, and c the 
 number whose logarithm is C. 
 
 The determination of c will depend upon the initial conditions
 
 248 ELEMENTS OF DYNAMICS. 
 
 of the motion ; let the initial velocity be v^, and a the angle it 
 makes with the axis of x. The component of v^, according to this 
 
 dx 
 axis, is VjCos.a, and this same component is what -j- becomes 
 
 when s=0, that is to say v cos. a=ce°=c; hence, substituting 
 
 dx 
 this value of c in the preceding equation, we have^=Vj cos. o 
 
 c— "^ .... (3). 
 
 dy 
 Again: multiplying the first of equations (1) by -^, the second 
 
 by -j-, and taking their difference, we have 
 
 d^x dy d^y dx_ dx 
 IF ' dt~~dF ' Tf^H' 
 
 d-^ 
 (It* doc ddc 
 
 dividing each side by ~r the result will be — — • =^; 
 
 or, putting y' for -^, — — . ^ = g, an equation containing only 
 
 first diflferential co-efficients, and which we may, therefore, write 
 thus (Biff. Calc. p. 99) 
 
 ^(dx)(dy') = g(dt-)', 
 the parenthesis intimating that the independent variable is arbitraiy. 
 
 Also, from equation (3), (dt^) = —^ ^2^5 hence, by sub- 
 
 stitution, — v^- cos. ^ » e-2"" —^ = g . . . . (4). 
 
 Now y^ = v/1 -f (2/'2) . • . (dx) = .J ^^ - , substituting 
 this value of (dx) in (4) and omitting the parentheses, we shall 
 
 p- g2ms ^g 
 
 have the equation Vl -\- y'^ • dy' = r— , 
 
 •^ "^ v^ cos.^o 
 
 of which the integral is 
 
 y' VT+Y^ + log. (2/' + vT+y^) = c - ^T^iX " * * ^^^' 
 
 dy 
 In order to determine C we must revert to the value of?/' or -^, 
 
 at the commencement of the motion ; this value is y' = tan. a, cor- 
 responding to which we also have a; = 0, 2/ = 0, 5 = 0;
 
 MISCELLANEOUS DYNAMICAL PROBLEMS. 249 
 
 tan. a V 1 + tan.'^o + 
 
 log. (tan. a + v^l + tan. ^a)+- 
 
 J 
 
 hence C 
 
 t Inor. ftan. n. -4- V I -1- Ian. "n.)-*- 
 
 m V/ COS.^( 
 1 
 
 therefore the value of C in the foregoing equation may be regarded 
 as known. 
 
 To determine a differential equation between x and y we may 
 eliminate e^™* by means of equation (4) and (5) ; we shall thus 
 have 
 
 dy: . 
 
 dx-- 
 
 m ly'^i + 2/" + log. (2/' + v/ 1 + 2/'^) — C! 
 
 also, since dy = y'dx, 
 
 . dy== y^y 
 
 m \y'^\ + y'^ + log. (y' + >/ 1 + y'^) — C \ 
 These equations are too complicated to admit of integration in 
 finite terms, otherwise we might now obtain the values of x and y 
 in functions of ?/', and then, eliminating y', we should have a single 
 equation in x and y which would be the equation of the path sought. 
 As it is, we can only obtain an approximation to the form of the 
 curve described, which approximation may, however, be carried to 
 any degree of exactness. For the method of effecting the actual 
 construction of the trajectary the student may refer to Mr. Barlow's 
 Mechanics, in the Encyclopoedia Metropolitana, or to Venturoli's 
 Mechanics, and for the general theory of motion in a resisting me- 
 dium, he may consult the second book of Mr. Whewell's Dynamics. 
 We shall here terminate this miscellaneous collection of prob- 
 lems, and must refer the student, for a more extensive variety, to 
 the Ladies' and Gentlemen's Diaries, and to Leybourn's Mathe- 
 matical Repository, works which cannot be too strongly recom- 
 mended to the attention of the mathematical student, and to which 
 our obligations are due for several of the foregoing examples. 
 
 32
 
 250 ELEMENTS OF DYNAMICS, 
 
 NOTE. 
 
 Page 21. 
 
 PoissorCs Proof of the Parallelogram of Forces. 
 
 (180.) When two equal forces act on a point according to differ- 
 ent directions, their resultant, whatever it be in intensity, must ne- 
 cessarily bisect the angle between these directions, as shown at art. 
 (7) ; and to determine the intensity of this resultant, M. Poisson pro- 
 ceeds as follows : 
 
 Let mA, mE (fig. 138,) be the directions of the components, 
 
 whose common value call P; also let 2a,' represent the angle AtwB, 
 
 then mC being the direction of the resultant, we shall have AmC 
 
 =BmC^a:. The intensity of this resultant can depend only on 
 
 the quantities P and x, of which, therefore, it is some unknown 
 
 function. Representing, then, by R the value of the resultant, we 
 
 shall have R=y"(P, .r). In this equation R and P are the only 
 
 quantities of which the numerical value varies with the unit of force 
 
 ■p 
 
 that may have been chosen ; their ratio r— is independent of this 
 
 P \ 
 
 unit ; whence we may conclude that it must be simply a function 
 of X, and consequently that the function / (P, x) is of the form 
 P . ^x. We therefore have 
 
 R=P . ^x, 
 aud the question is reduced to the determination of the form of the 
 function ^x. 
 
 In order to this, let us draAV arbitrarily through the point m, the 
 four lines mA', /nA", mB', wB" ; suppose the four angles A'mA, 
 A"mA, B'mB, B"mB, equal among themselves, and represent each 
 of them by z ; this done, decompose the force P, directed accord- 
 ing to mA, into two equal forces, directed according to mA' and 
 mA", that is to say, regard this force P as the resultant of two equal 
 forces whose value is unknown, and which acts in the given direc- 
 tions mA', mA". Representing the common value of these compo- 
 nents by Q, we shall have 
 
 P=Q . ^2, 
 
 for there ought to exist among the quantities P, Q, and z, the same 
 relation as among the quantities R, P, and x. Decompose now the 
 same force P, acting in the direction mB, into two forces Q, acting
 
 NOTE, 251 
 
 in the directions wB' and m'B" ; the two forces P are thus replaced 
 by the four forces Q ; the resultant of these must, therefore, coincide 
 in magnitude and direction with the force R, Avhich is the resultant 
 of the two forces P. Now, calling Q' the resultant of the two 
 forces Q, acting in the directions mA' and mB', and observing that 
 A.'mC=B')nC=x — z, this force Q' will take the direction mC, 
 and we shall have 
 
 Q'=Q.^{x—z). 
 In like manner, the resultant of the two other forces Q, which act 
 in the directions mA." and mB", will take the direction mC, since 
 this line divides the angle A"?nB" into two equal parts, and because 
 A"mC=B"mC=x+z, we shall have 
 
 Q" = Q . t (x+z), 
 Q" representing this resultant. The two forces Q' and Q" being 
 directed according to the same line mC, their resultant, which is 
 also that of the four forces Q, will be equal to their sum ; we must 
 therefore have 
 
 R=Q' + Q" 
 but we already have R=P • fX^Q, ' ^z • ^x ; substituting then this 
 value of R and those of Q' and Q" above, and then suppressing the 
 common factor Q, there results 
 
 ^x ' ^z=^ {x — z)-\-^{x-{-z). 
 
 This is the equation which we must now solve in order to obtain 
 the value of ^x, or, which amounts to the same thing, that of qiz. 
 This is effected in a very simple manner by the following considera- 
 tions. 
 
 Let us develope f (x — z) and q> [x-\-z), according to the powers 
 ofz, by means of Taylor's theorem; let us substitute these two se- 
 ries in our equation, and then divide all its terms by ^, we shall 
 thus have 
 
 ^ ^ ^ ^x dx^ 2 ^ ^x dx* 2-S'4^ ^ 
 
 Now as ^z ought not to contain x, x cannot enter into the coet 
 ficients 
 
 d^ ^x d* ^x 
 ^x dx^ ' fx dx* ' 
 all these quantities, therefore, must be constants, that is, indepen- 
 dent of the variables x and z. Let b be the value of the first, we 
 have 
 
 d^ AX , 
 
 whence, by successive differentiation, we have 
 rf* ^x , d^ AX - d* AX
 
 252 ELEMENTS OF DYNAMICS. 
 
 d^ ^x ,d^^o: ,, d^ ^x ,, 
 
 dx^ dx^ ^x dx^ 
 
 &c. &.C. &c. 
 and, consequently, 
 
 *z=2U4- ■ 1 f-&c.L 
 
 or else in replacing b by another constant — a' which is allowable, 
 a^z\ a* z* a^ z^ , « , 
 
 We recognise the series within the parenthesis to be the de- 
 vdopement of cos. az ; then fz=2 cos. az, and putting x in place 
 of z 
 
 ^x=2 cos. ax 
 .'. R=2 P cos. ax. 
 To determine the quantity a, which we know to be independent 
 of X, we may observe that when .r=90° the two forces P are di- 
 rectly opposite ; their resultant R is then ; so that we must have 
 
 cos. (a-90°)=0; 
 which requires that a be an uneven whole number. This whole 
 number must be 1 ; for if we had a > 1, for example a=S, the re- 
 
 90° 
 sultant R would become for x =:— — the two forces P would then 
 
 o 
 
 equilibrate without being opposite, which is impossible. We shall 
 have, therefore, 
 
 R=2 P* cos. X. 
 This result establishes the property otherwise deduced in art. 11, 
 viz. that any two equal forces have for their resultant the diagonal 
 of the rhombus constructed on the straight lines which represent 
 them in magnitude and direction, and the property may now be 
 generalized as in the text ; the equation above deduced being the 
 only thing connected with the general theorem of the parallelogram 
 of forces of difficulty to establish.
 
 Fa0o ?jS.
 
 ( 253 ) 
 SOLUTIONS AND NOTES BY THE EDITOR. 
 
 Problem XI. — Page 92. — It will here be convenient to use fig. 
 49, (plate p. 90,) which is used in the solution of prob. Ill, p. 85, 
 observing that in the present question AC=CB and .-. that the per- 
 pendicular CE falls upon the middle of AB and bisects the angle 
 ACB, which is here denoted by e. Then, as in the problem cited, 
 
 CF 
 
 we shall have P . CE=W • -— -, (1), by using W instead of his 
 
 2P ; but CE=CB • cos. -, and CF=CB . cos. 6, .-. by substitu- 
 
 tion and reduction (1) becomes 2 P cos. - = W cos. 9, (2) ; but (see 
 
 9 
 
 Young's Trigonometry, p. 37,)* we have cos. e=2 cos.= - — 1 .-. 
 
 P el 
 
 by subst. and reduc. (2) becomes cos.^ — — ^r^ cos. — = — , hence by 
 
 9 V P'' 1 - 
 
 quadratics we have cos.-= — ^ ±(^^^+ ^T as required.! 
 
 Note. — It may be well to observe that the sign -f is to be used 
 
 Q 
 
 before the radical in the value of cos. — when the rod lies as in the 
 
 figure ; but the sign — is to be used when it lies in an opposite di- 
 rection. 
 
 Problem XII. — Let us imagine the cone to be placed as in the 
 problem. Let P denote the force sought, and/) the perpendicular 
 from the edge on which the cone stands to the line of its direction ; 
 regarding the edge as a fixed point round which the cone can turn 
 freely in a vertical plane passing through its axes. Now by putting 
 284=:W, and p'=: the perpendicular from the edge to a vertical line 
 passing through the centre of gravity of the cone, we shall have as 
 in the last problem, P . p=W . p', (1) ; hence since P . j9=const. 
 and that P is to be a minimum, p must be a maximum : but it is 
 easy to see that p is a maximum when it coincides with the lower 
 side of the cone 
 
 * American edition. 
 
 f The answers given by Mr. Young to problems XI. XII. and XIII. are incorrect. 
 
 Y
 
 254 ELEMENTS OF DYNAMICS. 
 
 .'.p=^\20^+3^\=^{i00 + 9\ = s/{^09l, 
 
 it is also easy to tind />' =0-1 96-^-2 ; hence, by restoring the value 
 
 of W and substituting the values of/), p', (1), gives 
 
 -3 0-196x142 , _^ ^ 
 
 P= — ==1-377 cwt. 
 
 ^409 
 
 acting perpendicular to the lower side of the cone. 
 
 Note. — Since averticalline through the edge passes between the 
 centre of gravity and vertex of the cone, P acts towards the hori- 
 zon ; in the contrary case it must act in an opposite direction. 
 
 Problem XIII. — See fig. 58, p. 98. Let ABC denote the seg- 
 ment sought, having G for its centre of gravity and GP for a ver- 
 tical line passing through it. Then by what was shown at p. 72 when 
 the segment is at rest, GP must pass through the point of contact 
 P of the segment with the horizontal plane RP. Now put a=half 
 the transverse axis, and AX=x, then by prob. X, p. 68, 
 _ Sax — Sx'^ 
 
 but since P is to coincide with A (per prob.) we have AG=GP, .-. 
 
 Any Sx^ 
 
 the normal GP = —-^ — ; but see Young^s Analytical Geom. 
 
 (edition by Williams, just published by Carey, Lea & Blanchard, 
 Philadelphia,) p. 135, we have GP(=AG)=| a. Hence, by com- 
 paring the values of GP we have a = — , or by reduction 
 
 o CI — ^ OC 
 
 x^ — Sax^ — a^, .-. by quadratics x= — (3 — ^5) as required. 
 
 Problem XIV. — Let normals to the wall and inclined plane be 
 drawn through the extremities of the beam ; then when the beam is 
 at rest they will evidently intersect each other in a vertical line 
 which passes through the centre of gravity of the beam, for the re- 
 sultant of — P, — T, which denote the reactions of the wall and 
 plane must equal W and act directly opposite to it. Let the line of 
 common section of the plane of the normals and horizon be taken 
 for the axis of x. and a vertical line through the extremity of the 
 beam which is on the inclined plane be taken for the axis of y ; now 
 it is evident that the reactions of the plane and wall act in the di- 
 rections of their respective normals, .-. put z=the angle made by 
 the normal to the inclined plane with the horizon, and we shall have 
 — T cos. z-|-P=0, (1) ; — T sin. z + W=0, (2) for the resultant 
 of the forces which act on the beam when resolved in the direc- 
 tionsof the axes of x and y respectively, supposing the beam to
 
 SOLUTIONS AND NOTES. 255 
 
 be kept at rest by the forces which affect it, (see art. 19, p. 24.) 
 Again, since the beam is uniform, its centre of gravity is at its 
 middle point, and hence, we shall evidently have, tan. z=2 tan. i, 
 
 W 
 
 (3) ; by (1) and (2) we have, tan. z = — -, T = ^ f W^ + P^ | ; 
 
 hence, and by (3), P = -^.; T = ^(^^"-^ ^'+^' ^"^^LJ) . W 
 2 tan. t sm. i 
 
 as required. 
 
 Problem XV. — Let / denote the length of the ladder, rf = the dis- 
 tance which the man has ascended on it, and d' = the distance of 
 the common centre of gravity of the ladder and man from the lower 
 end of the ladder measured on it, then by the nature of the centre 
 of gravity (art 40, p. 60,) we have 
 
 „ i I w + dW .,, 
 
 now (art. 36, p. 50,) the resultant of w and W = lo -f- W , and it 
 passes through their common centre of gravity, (or at the distance 
 d' from the lower end of the ladder,) and is vertical to the horizon. 
 Let a straight line be draAvn from the lower end of the ladder to 
 the point of intersection of a normal to the wall at the upper end 
 of the ladder, and th6 vertical through the common centre of gra- 
 vity of w and W ; then imagine a plane to be drawn at right an- 
 gles to the line (thus drawn) through the lower end of the ladder ; 
 hence we may consider the ladder as resting on the wall and 
 plane, like the beam in the last problem ; hence, by using the 
 same notation and proceeding in the same manner as in the last 
 problem, we have, 
 
 tan. z = -^, (2); T = ^ ( W'^ + P^) . (3). 
 
 By putting ^^ + W = W ; we also evidently have 
 
 / 
 
 tan. z = —r- X tan. i, (4), 
 a 
 
 hence by (2) and (4) we have 
 
 d'^ 
 ^, W' ^/(sm.''^-f-^cos.«^) 
 
 as required; where P^the pressure against the wall, and T= the 
 thrust at the bottom of tlie ladder.
 
 256 SOLUTIONS AND NOTES. 
 
 Solution of Question VII. proposed at page 211. 
 
 Let i, r^, r^, g, w, wk^, denote the same things as in the author's 
 solution; then the accelerating force of gravity down the plane 
 = o- sin i. Let us suppose that the plane is perfectly smooth, and 
 that the wheel is acted upon by the motive force wY at the point of 
 contact P' towards D ; put ?=the time from the origin of the mo- 
 tion, a:=the space described by the centre down the plane, and e = 
 the angle described by the wheel around its axis. Then by (p. 202) 
 the centre moves in the same manner that it would do if the forces 
 were immediately applied to it without changing their direction 
 
 .•. g s'm.i — F=--7Y' (0' ^^^^ ^^^^ wheel turns in the same manner 
 
 that it would do if the centre were absolutely fixed, but the force 
 
 impressed to turn the wheel is ivF whose effect evidently=tt'Fr^, 
 
 d^o d^d 
 
 and the effect produced =^-Sr^rfm=-^—i^Z;^(r=the distance of any 
 
 rfr dt^ 
 
 element of the wheel from its centre); hence by (art. 154, p. 184,) 
 ^.F=A:^g, (2) .-. by (1) r,g sin.z=r,^+i^^^ .... (3). 
 
 Multiply (3) by r^, then if we suppose r^e=nx, (4), (where 
 
 d^O d^x 
 n=a constant number), we shall have r^--r-= n—^, and (3) will 
 
 . d^x_ r/g- sini dx_ r^gsinA _ r/g- sini t^ 
 
 ^'''"^ ^~ 77"-H^' •■• dt~ u'+nk^ ' ^ ^' ^ ~ ?•/ -fTiP "2' ^^^' 
 where x, e, t, are supposed to commence together, whence all the 
 circumstances of the m.otion become known. If n = 1, the wheel 
 rolls without sliding, the force wY performing the office of the 
 friction supposed by the author in the case which he has considered, 
 //fl (11* ii y* 
 
 If n is >1, then ^2^. = ^"5;= *^^ velocity of rotation is > -^ = 
 
 the velocity of the centre = the velocity of translation, yet the 
 wheel does not move up the plane as the author says it will, near 
 the end of his solution ; hence his remarks are incorrect.
 
 SOLUTIONS AND NOTES. 267 
 
 Solution of Question IV. proposed at page 238. 
 
 Suppose (with the author), that B moves with the given velocity 
 
 V towards D, then the space whicii it will describe in the time t 
 
 (from the origin of the motion), = vt ; put 6 = the angle ABC, 
 
 (e'= its value at the origin), r = the distance of any element dm, 
 
 of the rod from the end B, M = the mass of the rod, R = BG, 
 
 g = 32. 2ft. (= the accelerating force of gravity), dt = the constant 
 
 element of the time, x = the distance of dm (estimated on CD), 
 
 from the initial position of B, then evidently x = vt — r cos.^, (1). 
 
 rd^d 
 Now g cos.edt dm = the force impressed on d7n, and r— = the 
 
 force received by it (in the instant dt) in a direction perpendicular 
 to the rod, (supposing the forces tend to diminish e) ; .♦. the motion 
 
 lost by dm, = dm (-j- — \- g cos.edt j and the momentary effect of 
 
 this force to turn the rod about B (by the principle of the lever), 
 
 d^d 
 = dmr (^-77--\- g cos.edt), hence by art. (154) we shall have by 
 
 using S as the sign of integration relative to the mass of the rod, 
 
 d^O d^d 
 
 Sdmr(r-j- + g cos.edt) =0, or -r^ Sr^dm + g cos.eSrdm = 0, 
 
 but by art. (42), (155) Sn/m = RM, Sr''dm — —-M, hence by 
 
 o 
 
 d^e 32" 
 substitution and reduction, we have -77^+ -^cos.e = 0, (2), multiply 
 
 de^ 3a" 
 ^2) by de and take the integral and we have-j--| — igsin.e := c = 
 
 dt 2R 
 
 contrast. (3). 
 
 Suppose that at the origin tJj8 velocity V was impressed on the 
 centre of gravity G of the rod in the direction GD', then by chang- 
 ing r in (1) into R we have x = vt — R cos. 9 (4) for the value of 
 X which corresponds to the centre of gravity, hence evidently 
 
 dx ,^ „ . ,d9' , ... , . , . de'^ , V — V ,„ 
 
 -=V=t; + Rsm.. ^(at the ongm), which gives-^=(j^-^)» 
 
 but (3) at the origin gives 
 
 _+ _ sm..'= c, hence (^^^) + 2|sm..'=c, 
 
 hence, and by (3) dt =——-——-— (5) ; 
 
 -Jl^-T — ;)+-^(sin.e'— sin.0) 
 >IVRsin.6i7 2R^ ' 
 
 2 Y
 
 268 SOLUTIONS AND NOTES. 
 
 the integral of (5) will give t in a function of 6 and known quanti- 
 ties, .•. reciprocally 6 will be found in a function of t and known 
 quantities, hence x as given by (4) will be exhibited in terms of t 
 and known quantities, ••• we have the value of a: which corresponds 
 to the position of the centre of gravity at any time, {t) and by put- 
 ting r = in (1) we have x=: vt which gives the position of B at 
 the same time, and the position of the rod is determined in all re- 
 spects ; and it may be observed that the sign + is to be used in (5) 
 before de when V is greater than v, but the sign — in the contrary 
 case. 
 
 Again, if the second term of the quantity under the radical in (5) 
 is much smaller than the first, the integral may be readily found in 
 a rapidly converging series ; but if the second term is infinitely 
 small relative to the first term, it may be neglected, and we shall have 
 
 V V 
 
 (when V is > v), .5 — : — ,dt = de and by integration, and correction 
 
 V — V 
 
 9=i9'-{-:^—. ■/, which will also give the value of e when v is >V 
 
 these are the results which the author should have found in the case 
 
 (of the general problem), which he appears to have considered ; and 
 
 it is easy ta see that every thing else required in this case is found. 
 
 Note. — It is easy to see by (1) that the point B moves always 
 
 with the velocity v ; for it gives ^= v -\- r sin e-^- , but at the point 
 
 dx 
 B, r=0, hence -^= the velocity of B = r, as it ought to do. 
 
 THE END.
 
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