OS- -r\! ■\' REESE LIBRARY OF TlIK UNIVERSITY OF CALIFORNIA Rccclveii . iiicssions > Y LONDON: X'^ CHARLES GRIFFIN AND COMPANY, stationers' hall court. 1875. /i-3^ J' PEEFACE. This book is designed as an Introduction to more abstruse works on Engineering and Mechanics, and in particular to those of the late Professor Rankine. Its study demands only a previous acquaintance with the ordinary Rules of Arithmetic, and with the Elementary Alge- braical Notation. A few pages have been devoted to the Differential and Integral Calculus, as these have been used in different parts of the book, their application having been in every instance explained. Professor Rankine's Manual of Applied Mechanics has been taken as the model for this work, the only alteration being the treating of the Theory of Motion before that of Force, as more in harmony with modern practice, and as proposed by himself for the present purpose. The general design of the work having been indicated, it only remains for me to explain briefly how my name has been con- nected with that of Professor Rankine on the Title-page, and also in what condition it was left at the time of his recent lamented death. 1 was Professor Rankine's Assistant, and lectured for him during his illness, and it was whilst on a visit which his death suddenly terminated, that the arrangement was made which connected me with him in the task. My duty was simply to assist him in its preparation. On my mentioning to him that the amount of labour I should have to do hardly justified my IV PREFACE. name appearing with his as joint-author, he replied, that, owing to his state of health, more of the work might devolve upon me than I expected. The issue has proved the correctness of his surmise. As to the state of the MS. at the time of his death, two hundred pages had been already completed, and the general scope and plan of the work decided upon. I need hardly say that his wishes have been implicitly carried out in every respect, so far as lay in my power. The work has been completed at the request of Professor Rankine's Executrix, and at that of the Publishers, at whose desire also I have undertaken the superintendence of New Editions of his other Scientific Manuals, some of which have already been submitted to the Public. E. F. B. Glasgow, October, 1873. PREFACE TO SECOND EDITION. This Second Edition has been carefully revised, and some additions have been made to the text. E. F. B. London, October, 1874. CONTENTS. MATHEMATICAL INTRODUCTION. Article Arithmetical Rules. Page 1. To find the Prime Factors of a Given Number, ... 1 2. To find the Greatest Common Measure (otherwise called the Greatest Common Divisor) of Two Numbers, ... 1 3. To Reduce the Ratio of Two Num- bers to its Least Terms, . 2 4. ToExpress the Ratio of Two Num- bers in the Form of a Continued Fraction, .... 2 6. To form a series of Approxima- tions to a Given Ratio, . < 2 G. Logarithms — Definitions, . . 4 6-16. Logarithms, . . • 4, 5 17. Antilogarithms, ... 6 Trigonometrical Rules. 18. Trigonometrical Functions De- fined, . . ... 6 19. Relations amongst the Trigono- metrical Functions of One Angle, A, and of its Supple- ment, 7 20. The Circular Measure of an Angle, 8 21. Trigonometrical Functions of Two Angles, ... 8 22. Formulaj for the Solution of Plane Triangles, ... 8 23. To Solve a Right-angled Triangle, 9 24. To Express the Area of a Plane Triangle in terms of its Sides and Angles, . , . .10 Rules of the Differential and Integral Calculus. 25. Definitions, . . . .10 26. Rules for finding Differential Co- efficients, . . . .11 27. Illustration of the Differential Calculus to Geometry, . .12 28. The Integral Calculus the Inverse of the Differential, . . 13 29. Approximate Computation of Integrals, . . . .13 Rules for the Mensuration op Figures and Finding of Centres of Magnitude. Section 1.— Areas oj Plane Surfaces. Article Page 30. Parallelogram, . . , .16 31. Trapezoid, .... 16 32. Triangle, . . - . . 16 33. Parabolic Figures of the Third Degree, . . . .16 34. Any Plane Area, . . .17 35. Circle, 21 36. Area of a Circular Sector, . . 22 Section 2.^Volumes of Solid Figures. 37. To Measure the Volume of any Solid, 22 Section 3. — Lengths of Curved Lines. 38. To Calculate the Lengths of Cir- cular Arcs, . . . . 23 39. To Measure the Length of any Curve, 25 Section 4. Geometrical Centres and Moments. 40. Centre of Magnitude— General Principles, .... 41 . Centre of a Plane Area, . 42. Centre of a Volume, 43. Centre of Magnitude of a Curved Line, 44. Special Figures, Elementary Mechanical Notions. Definition of General Terms and, Division of the Subject. 45. Mechanics, . . . . 3C AQ. Matter, 30 47. Bodies 30 48. A Material or Physical Volume, 30 49. A Material or Physical Surface, 30 50. Line, Point, Physical Point, Measure of Length 51. Rest 52. Motion, 53. Fixed Point, 54. Cinematics, 55. Force, 56. EquiHbrium or Balance, 67. Dynamics— Statics and Kinetics. 58. Structures and Machines, VI CONTENTS. PART I.— PRINCIPLES OF CINEMATICS, OR THE COMPARISON OF MOTIONS. Article Pagre 69. Division of the Subject, 33 Chapter I. — Motions of Points. Section 1. — Motions of a Pair of Points. 60. Fixed and nearly Fixed Direc- tions, 31 61. Motion of a Pair of Points, _ . 34 62. Fixed Point and Moving Point, 35 63. Component and Resultant Mo- tions, 35 64. The Measurement of Time, . 35 65. Velocity, . . . . .36 66. Uniform Motion, . . .37 Section 2. — Uniform Motion of Several Points. 67. Motion of Three Points, . . 37 68. Motions of a Series of Points, . 37 69. The Parallelopiped of Motions, . 38 70. Comparative Motion, . . 38 Section 3 — Varied Motion of Points. 71. Velocity and Direction of Varied Motion 39 72. Components of Varied Motion, . 40 73. Uniformly Varied Velocity. • 41 74. Graphical Representation of Mo- tions, . . , . 75. Varied Rate of Variation of Velocity, .... 76. Combination of Uniform and Uni- formly Accelerated Motion, . 77. Uniform Deviation, . 78. Varying Deviation, . 79. The Resultant Rate of Variation, 80. The Rates of Variation of the Component Velocities, . 81. The Comparison of the Varied Motions, .... Chapter II.— Motions of Rigid Bodies. Section 1. — Rigid Bodies, and their Translation. 82. The Term Rigid Body, . . 47 83. Translation or Shifting, . .47 Section 2. Simple Rotation. 84. Rotation or Turning, . . 47 85. Axis of Rotation, . . .47 86. Plane of Rotation, ... 48 87. Angular Yelocity, . . . 48 88. Uniform Rotation, ... 48 89. Rotation Common to all Parts of Body, 49 90 Right and Left-Handed Rotation, 49 91. Relative Motion of a Pair of Points in a Rotating Body, , 49 92. Cylindrical Surface of Equal Velocities, . . . .50 93. Comparative Motions of Two Points relatively to an Axis, . 50 94. Components of Velocity of a Point in a Rotating Body, . 60 Section 3. — Combined Rotations and Translations. 95. Property of all Motions of Rigid Bodies, 51 96. Helical iMotion, 97. To find the Motion of a Rigid Body from the Motions of Three of its Points, Special Cases, .... 99. Rotation Combined with Trans- lation in the same Plane, Rolling Cylinder ; Trochoid, Plane RoUing on Cylinder; Spiral Paths, Combined Parallel Rotations, . 103. Cylinder Rolling on Cyhnder; Epitrochoids, 104. Equal and Opposite Parallel Rotations Combined, 105. Rotations about Intersecting Axes Combined, . 106. Rolling Cones, 107. Comparative Motions in Com- pound Rotations, . 98. 100. 101. 102. 42 51 . 58 62 62 63 63 Section 4. — Varied Rotation. 108. Variation of Angular Velocity, 109. Components of Varied Rotation, Chapter III. —Motions of Pliable Bodies, and of Fluids. 110. Division of the Subject, ^^ B^CTio^l.-MotionsofFlexi^leCords. I JJa; ^^^^HSfby Surfaces of '' 111. General Principles. . . .05 Revolution 66 CONTENTS. Section 2. — Motions of Fluids of Constant Density. Article Pa 114. Velocity and Flow, . 115. Principle of Continuity, . 116. Flow in -A Stream, . 117. Pipes, Channels, Currents, and Jets, . - . . . 67 Article pag© 118. Steady Motion, ... 68 119. Motion of Pistons, . . . 68 Section d.— Motions of Fluids of Varying Density. 120. Flow of Volume and Flow of Mass, 69 121. The Principle of Continuity, . G9 PART II.— THEORY OF MECHANISM. Chapter I. — Definitions and General Principles. 122. Theory of Pure Mechanism De- fined, 123. The General Problem, . 124. Frame; Moving Pieces; Con- nectors; Bearings, 125. The Motions of Primary Moving Pieces, . 71 72 126. The Motions of Secondary Mov- ing Pieces, .... 127. An Elementary Combination, . 128. Line of Connection, . 129. Principle of Connection, . 130. Adjustments of Speed, 131. A Train of Mechanism, . 132. Aggregate Combinations, Chapter II.— On Elementary Combinations and Trains of Mechanism. 133. 134, 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. Section 1. — Rolling Contact. Pitch Surfaces, ... 74 Smooth Wheels, Rollers, Smooth Racks, . ... 74 General Conditions of Rolling Contact, . . . .74 Circular Cylindrical Wheels, . 75 A Straight Rack and Circular Wheel 75 Bevel Wheels, . . . .76 Non-Circular Wheels, . . 77 Section 2.— Sliding Contact. Skew-Bevel Wheels, . . 77 Principle of Sliding Contact, . 79 Teeth of Wheels, . . .81 Pitch and Number of Teeth, . 81 Hunting Cog, .... 83 A Train of Wheelwork, . . 83 Teeth of Spur -Wheels and Racks. General Principle, . 86 Teeth Described by Rolling Curves, .... 86 The Sliding of a Pair of Teeth on each other, . . .87 The Arc of Contact on the Pitch Lines, .... 88 The Length of a Tooth, . . 88 Involute Teeth for Circular Wheels, .... 88 The Smallest Pinion with Invo- lute Teeth, .... 89 Epicycloidal Teeth, ... 89 Teeth of Wheel and Trundle, . 90 Dimensions of Teeth, . .91 The Teeth of a Bevel Wheel, . 91 The Teeth of Non - Circular Wheels, .... 92 158. A Cam or Wiper, ... 92 159. Screws. Pitch, ... 92 160. Normal and Circular Pitch, . 93 161. Screw Gearing, . . .94 162. The Wheel and Screw, . . 95 163. The Relative Sliding of a Pair of Screws, . . . .95 164. Oldham's Coupling, . . 96 Section 3. — Connection hy fiands. 165. Bands Classed, ... 97 166. Principle of Connection by Bands, 97 167. The Pitch Surface of a Pulley or Drum, . . . .98 168. Circular Pulleys and Drums, . 98 169. The Length of an Endless Belt, 99 170. Speed Cones, . . . .100 Section 4. — Linlcworh. 171. Definitions, .... 172. Principles of Connection, . 173. Dead Points, .... 174. Coupling of Parallel Axes, 175. The Comparative Motion of the Connected Points, 176. An Eccentric, .... 177. The Length of Stroke, . 178. Hooke's UniversalJoint, '. 179. The Double Hooke's Joint, 180. A Click, .... Section 6.— Reduplication of Cords. 181. Definitions, .... 106 182. Velocity Ratio, . . .106 182a. The Velocity of any Ply, . 106 183. White's Tackle, . . . 106 101 101 101 101 102 103 104 104 105 105 viii CONTENTS. Section 6. — Comparative Motion in the ' ' Mechanical Powers." Article Page 184. Classification of the Mechanical Powers," . . . .107 Section 7. — Hydraulic Connection. 185. The General Principle, . .110 Article Pajje 186. Valves, 110 187. The Hydraulic Press, . .110 188. The Hydraulic Hoist, . . Ill Section 8. — Trains of Mechanism. 189. Trains of Elementary Combina- tions, Ill 190. The General Principles, . lt\^ritten under the third quotient, and so on. For example : Quotients, a, 5, c, d, &c. ^ ^. I n n' n" Fractions, =^, r, — , — , — ,; 10 m m' m n _0 + a a n _1 + bn^ n" _ n + cn' „ m~l + 0~P m~ + b m^ m" ~ jti + cm" 452 To take a particular case; let the given ratio be as before, -^vv, then we have the following series : — Quotients, 1 3 1 1 1 15 2 ^ . ] 1 4 5 9 14 219 452 ^''^^^^^^^' 1 i 3 4 7 n 172 355 Less or greater than K G L G L G L G given ratio, J The fractions in a series formed in the manner just described are called converging fractions, and they have the following properties : — First, each of them is in its least terms; secondly, the difference between any pair of consecutive converging fractions is equal to unity divided by the product of their denominators; for example, 9^_ 5 _ 36 -35 1 . 9 _ 14 ^ 99^^^ 1 ^^^^^^ ^^ 7 4 ~ 7 X 4 " 28' 7 11 7 x 11 77' are alternately less and greater than the given ratio towards which they approximate, as indicated by the letters L and G in the example; 2iT\d, fourthly, the difference between any one of them and the given ratio is less than the difference between that one and the next fraction of the series. Fractions intermediate between the converging fractions may be 4 MATHEMATICAL INTRODUCTION. found by means of the formula -= -, — ri where — and - , are •^ hm + kjn. m m any two of the converging fractions, and h and k are any two whole numbers, positive or negative, that are prime to each other. 6. Logarithms. Definitions. — The power of a number is the product of itself multiplied a certain number of times. The index or exponent of the power is the small figure placed above the right- hand corner, which denotes the number of times the multiplication takes place. The Logarithm of a number to a given base is the index of the power to which the base must be raised to be equal to the given number. That number of which the indices of the powers are the logarithms, is called the base of the system. A suffix denotes the base of the logarithm ; i^ a'' = n, x is the logarithm of the number n to the base a, or log„ n — x. Logarithms to the base 10 are called common logarithms. 7. The logarithm of 1 is 0. 8. The common logarithm of 10 is 1, and that of any power of 10 is the index of that power; in other words, it is equal to the number of noughts in the power; thus the common logarithm of 100 is 2; that of 1000, 3; and so on. 9. The common logarithm of -1 is - 1, and that of any power of •1 is the index of that power with the negative sign; that is, it is equal to one more than the number of noughts between the decimal point and the figure 1, with the negative sign; for example, the common logarithm of -01 is - 2 ; that of -001, - 3; and so on. 10. The logarithms given in tables, are merely the fractional parts of the logarithms, correct to a certain number of places of decimals, without the integral parts or indices; which are supplied in each case according to the following rules : — The index of the common logarithm of a number not less than 1 is one less than the number of integer places of figures in that number; that is to say, for numbers less than 10 and not less than 1, the index is 0; for numbers less than 100 and not less than 10, the index is 1 ; for numbers less than 1000, and not less than 100, the index is 2 ; and so on. The index of the common logarithm of a decimal fraction less than 1 is negative, and is one more than the number of noughts between the decimal point and the significant figures; and the negative sign is usually written above instead of before the index ; that is to say, for numbers less than 1 and not less than •!, the index is 1_; for numbers less than -1 and not less than -01, the index is 2 ; and so on. The fractional part of a common logarithm is always positive, and depends solely upon the series of figures of which the number consists, and not upon the place of the decimal point amongst them. logarithms — definitions. 5 Examples. Number. Logarithms. 377000 6'57634 37700 4-57634 3770 3-57634 377 2-57634 37-7 1-57634 3-77 0-57634 •377 1-57634 •0377 2-57634 •00377 3-57634 and so on. 11. The logaritLm of a product is the sum of the logarithms of its factors. 12. The logarithm of a power is equal to the logarithm of the root multiplied by the index of the power. 13. The logarithm of a quotient is found by subtracting the logarithm of the divisor from the logarithm of the dividend. 14. The logarithm of a root is found by dividing the logarithm of one of its powers by the index of that power. Note. — In applying these principles to logarithms of numbers less than 1, it is to be observed that negative indices are to be subtracted instead of being added, and added instead of being subtracted. 15. To avoid the inconvenience which attends the use of nega- tive indices to logarithms, it is a very common practice to put, instead of a negative index to the logarithm of a fraction, the complement (as it is called) of that index to 10; that is to say, 9 instead of 1, 8 instead of % 7 instead of 3, and so on. In such cases, it is always to be understood that each such complementary index has - 10 combined with it; and to prevent mistakes, it is useful to prefix - 10 + to it; for example, xT„_-u„„ Logarithm witli Logarithm with Is umber. Negative Index. Complementary Index. •377 1-57634 -10 + 9-57634 •0377 2-57634 -10 + 8-57634 •00377 3-57634 -10 + 7-57634 16. To find the fractional part of the common logarithm of a number of five places of figures; take from the table the logarithm corresponding to the first three figures, and the difference between that logarithm and the next greater logarithm in the table; mul- tiply that difference by the two remaining figures of the given number, and divide by''lOO; the quotient will be a correction, to be added to the logarithm already found. 6 MATHEMATICAL INTRODUCTION". Example. — Find the common logarithm of 37725. Log. 377, 57634 Log. 378, 57749 Difference, 1L5 X 25 ^100 Correction 29 Add log. 377, .' 57634 Log. 37725, 57663 Answer. 17. To find the natural number, or antilogarithm^ corresponding to a common logarithm of five places of decimals, which is not in the table; find the next less, and the next greater logarithm in the table, and take their difierence. Opposite the next less logarithm will be the first three figures of the antilogarithm. Subtract the next less logarithm from the given logarithm ; annex two noughts to the remainder, and divide by the before-mentioned difierence ; the quotient will give two additional figures of the required anti- logarithm. (The first of those figures may be a nought.) Example. — Find the antilogarithm of the common logarithm •57663. Next less log. in table, 57634 Next greater...... 57749 Difference, 115 Given logarithm, 57663 Subtract log. 377, 57634 Divide by diff'erence, 115)2900 Two additional figures, 25 so that the answer is 37725. Note. — The last two rules refer particularly to the tables in Eankine's Useful Rules and Tables, but are equally applicable to other tables. For instance, where the logarithm of a number of 5 figures is given in the tables; in these last two rules, for 3 read 5, and for 5 read 7. TRIGONOMETRICAL RULES. The following is a summary of the Principles and Chief Rules of Trigonometry : — Definition. — Every expression which in any way contains a number, or depends for its value upon the value of the number, is said to be 2i function of that number, as 2x, x^, log. x, tan x are all functions of x. 18. Trigonometrical Functions Defined.— Suppose that A, B, C TRIGONOMETRICAL FUNCTIONS. 7 stand for the three angles of a right-angled triangle, C being the right angle, and that a, b, c stand for the sides respectively opposite to those angles, c being the hypothenuse; then the various names of trigonometrical functions of the angle A have the following meanings : — • A ^ A ^ sm A = - : cos A = - : c c . c-h . . c - a Tersin A = : coversm A = : c ' c tan A = -y : cotan A = - : a A ^ A ^ sec A = 7 : cosec A = — . a The complement of A means the angle B, such that A + B = a right angle; and the sine of each of those angles is the cosine of the other, and so of the other functions by pairs. 19. Relations amongst the Trigonometrical Functions of One Angle, A, and of its Supplement, 180°-A:— • A n 2^ tan A 1 . sm A = Jl — cos^ A = T- = j-y '^ sec A cosec A A /I ^"TT ^o*^^ ^ ^ . COS A= Jl- sin^ A = T- = -——A > ^ cosec A sec A versin A = 1 - cos A ; coversin A = 1 - sin A; tan A = ^HL4: = _-L_=sin A-sec A= V^ec^A-l; cos A cotan A cotan A = -?^—r = r- = cos A • cosec A = Jcosec^ A - 1 ; sm A tan A sec A = T-= x/1 + tan2 A; cos A ^ cosec A = -; — T- = ^1 + cotan2 ^. sm A sin (180°- A) = sin A; cos (180°- A) = -cos A; versin (180° - A) - 1 + cos A =2 - versin A ; coversin (1 80° - A) = coversin A ; tan (180° - A) = -tan A; cotan (180° - A) = - cotan A ; sec (180° - A) = -sec A; cosec (180° - A) = cosec A. 8 MATHEMATICAL INTRODUCTION. 20. The Circular Measure of an Angle. — If a right line as radius by revolution about a fixed point at its extremity as centre, traces out an angle from a fixed position, the angle may be measured by the ratio of the arc to the radius; this mode of measurement is called circular measure. The U7iit of circular measure is the angle whose arc is equal to the radius, that is, 360° -- 2^ = (57° 17' 45" = 206265"). To compute sines, tfec, approximately by series; reduce the angle to circular measure — that is, to radius-lengths and fractions of a radius-length let it be denoted by A. Then A* A^ A'^ A^ A* A^ cos A=l- ^ + 2^:1-23^5:6 ^'^•=- 21. Trigonometrical Functions of Two Angles : — sin ( A ± B) = sin A cos B ± cos A sin B ; cos (A ± B) = cos A cos B :p sin A sin B ; ' J. / A J- T>\ t^^ ^ — tan B tan (A ± B) =-. r =^. ^ ' 1 qp tan A tan B 22. FormulaB for the Solution of Plane Triangles.~Let A, B, C be the angles, and a, h, c the sides respectively opposite them. I. Relations amongst the Angles — A + B + C = 180<'; or if A and B are given, C = 180° - A - B. II. When tlie Angles and One Side are given, let a be the given side; then the other two sides are , sin B sin C sin A ' sin A III. When Two Sides and the Included Angle are given, let a, b be the given sides, C the given included angle ; then To find the third side. First Method : c= J{a^ + b^-2abco3C)', Second Method : Make sin D = ^^^^ ' cos ^ . then a + b ^ c = {a + b) cos B. Third Method: Make tan E = hJj^. gin - ; then a-b 2 c = {a-b) sec E. TO SOLVE A RIGHT-ANGLED TRIANGLE. To find the remaining Angles, A and B. If the third side has been computed, sin A = -'-sin C: sin B = --sin C. c c If the third side has not been computed, ._ A + B ^ C ^ A-B a-h , C tan • — ~ = cotan — ; tan— ^r — = cotan ^ : ^ A Z a+b 2 . A+B A-B ^ A+B A-B A = -^— + -^; B==-^ —. TV. When the Three Sides are given, to find any one of the Angles, such as C — a2 + 52_c2 cos C 2ah or otherwise, let a + h + c j then cos 0_ /s{s-c)^ C_ / (.-a)-(^-5)\ 2-V ah ^^"'2-V ^6 ' ^ C ^ A(s-c) ^ C /{s-a cotan 17 = A / 7^ ^ tt; tan -5^ = A / ^ - 2 V (s - a) (s - 6) 2 V s{s ){s-b). s-c) ^ 2 Js{s-a){s-b){s-c) v^ = . ao Note. — In all trigonometrical problems, it is to be borne in mind, that small acute angles, and large obtuse angles, are most accurately determined by means of their sirms, tangents, and cosecants; and angles approaching a right angle by their cosines, cotangents, and secants. 23. To Solve a Right-angled Triangle. — Let C denote the right angle; c the hypothenuse; A and B the two oblique angles; a and 6 the sides respectively opposite them. Given, the right angle, another angle B, the hypothenuse c. Then A = 90°-B; a-c-cosB; 6 = c-sinB. Given, the right angle, another angle B, a side a, A = 90° - B; h = a' tan B; c = a • sec B. Given, the right angle, and the sides a, b, tan A = 7- J tan B = - ; c = J a^ + b\ b a ^ 10 MATHEMATICAL INTRODUCTION. Given, the right angle, the hypothenuse c; a side a, sin A = cos B = -; b= Jc^-a^. c Given the three sides, a, h, c, which fulfilling the equation c2 = a2 + b-y the triangle is known to be right-angled at C, • A <* • -D ^ sin A = -; sm B = ~. c c 24. To Express the Area of a Plane Triangle in terms of its Sides and Angles. Given, one side, c, and the angles. c^ sin A sin B Area = -^ • : — y^ . 2 sm C Given, two sides, b, c, and the included angle A. 6 c • sin A Area = ^ . Given, the three sides a, b, c. Let ^ = s; then Area = J \ s(s-a){s-b) s - g)> . RULES OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. 25. Definitions. — A function has already been defined. "When a function of one quantity is assumed equal to another quantity, both quantities are called variables, the one upon whose assumed value the other depends being called the independent variable, while the other, whose value depends upon it, is called the de- pendent variable. The expression y = ^x for instance denotes that the dependent variable y, depends for its value upon the independent variable x, or y is & function of x. A quantity, x, may be assumed to be made up of an infinite number of infinitesimal parts, dx, this expression meaning simply one of the small infinitesimal dififerences of which x is made up, i.e., x = n'dx, where n is assumed to increase without limit, and dx to diminish without limit, this process of considering a quantity to be diminished without limit is called differentiation. The quotient, if it has a limit formed by taking the difference of the function of a quantity, and the function of that quantity with a small increment, and dividing by the increment, is termed the differential coefficient of the function, with regard to the quantity dtix + dx) — (hx . , ,.«. . , — -i~ 13 the differential coefficient of <^ x with respect to X, this is generally written ' x; or otherwise the small increment DIFFERENTIAL AND INTEGRAL CALCULUS. 11 or decrement of the dependent variable divided by tliat of tlie independent variable, the former being a function of the latter, is called the differential coefficient, thus -^ is the diflferential co- CISC efficient of y with respect to x, it being always borne in mind that dii -J- is one quantity, which cannot be divided into a numerator dy^ and a denominator dx. 26. Rules for finding differential coefficients, — If 2/ = C (a constant); ~ = 0. di X The Differential Coefficient of the sum of functions is equal to the sum of the differential coefficients of the functions, or if 'v = w + y + z where all of these quantities are functions of Xj then dv _ dw dy dz dx ~ dx dx dx' In the same way to find the differential coefficient of the differ- ence, product, and quotient of functions of quantities. liv=y- z, then — - = -^ - —, where v, y, z, are functions of x. ^ ' dx dx dx' ' ^' ' ,^ ^, dv dw dy dz , liv = wyz,t\.eu-^^^-^^-yz + ^-wz + ^-wy, where Vj w, y, Zj are functions of x. dif dz Z ' — ^ — y 7/ Cm 1^ tL CG d CG If V = — , then — - = r , where v, y, z. are functions of x. z dx z^ If (t>x = nx, 'x = n; or otherwise let 'x = 7- If cjiX = x^^j (ji'x ■— nso" ~ ^, thus if ^x = xJ, 4>'x = 7x^. If 4>x = log^x, 'x = ^^, thus if x = logio X, 'x =^-10 = 1 _ -43429 °' 2-30258^" X ^ 1 f L I B R A U Y li^x = vo^x, d>'X=-. I li'x = a-\og,a. !| UN I YE li S I T Y OF If 'x = g^. }j YitZZlltZ^n.. i CALIFORxNIA. If ^x = tan X, ct>'x \ y cos- X Definition. — By sine "^ x is meant the angle whose sine is X, thus if 2/ = sine ~^x, x = sine y. MATHEMATICAL INTRODUCTION. If f a; = sia ^ x^ (p' x- ^ a; = cos "■ X, <^' X- - 1 ^x = tan x^" x = n (n - \)x^~^, f'" x = n(n-l) {n-2) x""'^ ; where " x stands for the second differential coefficient, (p'" x for the third differential coefficient. So let ^ £c = sin x, then "knot," j In treating of the general principles of mechanics, the foot 'per second is the unit of velocity commonly employed in Britain. The units of time being the same in all civilized countries, the propor- tions amongst their units of velocity are the same with those amongst their linear measures. Component and resultant velocities are the velocities of component and resultant motions, and are related to each other in the same MOTIONS OF A SERIES OF POINTS. 3< way with those motions, which have ah'eady been treated of in Article 63. 66. Uniform Motion consists in the combination of uniform velocity with uniform direction; that is, with motion along a straight line whose direction is fixed. Section 2. — Uniform Motion of Several Points. 67. Motion of Three Points. — Theorem. The relative motions of three points in a given interval of time ^v^ j are represented in direction and magni- tude hy the three sides of a triangle. Let O, A, B, denote the three points. Any one of them may be taken as a fixed point; let O be so chosen; and let O X, O Y, O Z, fig. 22, be axes traversing it in fixed directions. Let Aj and B-^ be the positions of A and B relatively to O at the beginning of the given interval of time, and Ag and Bg their positions at the end of that interval. Then A^ Ag and Bj Bg are the respective motions of A and B relatively to O. Complete the parallelogram A^^ Bj 5 Ag ; then because Ag b is parallel and equal to A^ B^, 6 is the position which B would have at the end of the interval, if it had no motion relatively to A ; but Bg is t he actual position of B at the end of the interval ; therefore, h Bg is the motion of B relatively to A. Then in the triangle B^ h Bg, Bj 6 = Aj Ag is the motion of A relatively to O, h Bg is the motion of B relatively to A, Bj Bg is the motion of B relatively to O; so that those three motions are represented by the three sides of a triangle.— Q. E. B. This Theorem might be otherwise expressed by saying, that if three moving points he considered in any order, the motion of the third relatively to the first is the resultant of the motion of the third relatively to the second, and of the motion of the second relatively to the first; the word ^^ resultant'' being understood as already ex- plained in Article 63. 68. Motions of a Series of Points. — Corollary. If a series of points he considen^ed in any order, and the motion of each point determined relatively to that which precedes it in the series, and if the relative motion of the last point and the first point he also deter- mined, then will those motions he represented hy the sides of a closed 38 PRINCIPLES OF CINEMATICS. polygon. Let O be the first point, A, B, C, &c., successive points following it, M the last point but one, and N the last point ; and, for brevity's sake, let the relative motion- of two points, such as B and C, be denoted thus (B, C). Then by the Theorem of Article 67, (O, A), (A, B), and (O, B) are the three sides of a triangle; also (O, B), (B, C), and (O, C), are the three sides of a triangle ; therefore (O, A), (A, B), (B, C), and (O, C), are the four sides of a quadrilateral ; and by continuing the same process, it is shewn, that how great soever the number of points, (O, N), is the closing side of a polygon, of which (O, A), (A, B), (B, C), (C, D), &c., (M, N) are the other sides. — Q. E. I). In other words, the motion of the last point relatively to the first is the resultant of the motions of each point of the series relatively to that preceding it. 69. The Parallelopiped of Motions. — In lig. 23, let there be four points, O, A, B, C, of which one, O, is assumed as fixed, and is traversed by three axes in fixed directions, O X, O Y, O Z. In a given interval of time, let A have the motion Aj Ag along or parallel to O X; let B have, in the same interval, the motion h B2 parallel to O Y, and rela- tively to A ; then B^^ ^2' *^® diagonal of the parallelogram whose sides are B^ 6 = Aj Ag and h Bg, is the motion of B rela- Let C have, relatively to B, the motion c Cg parallel to O Z ; then Ci O2, the diagonal of the parallelopiped whose edges are Aj Ag, b Bg, and c Cg, is the motion of C relatively to O, being the resultant of the motions represented by those three edges. This is a mechanical explanation of the composition of motions, leading to results corresponding with the geometrical explanation of Article 63. 70. Comparative Motion is the relation which exists between the simultaneous motions of two points relatively to a third, which is assumed as fixed. The comparative motion of two points is expressed, in the most general case, by means of four quantities, viz. : — (1.) The velocity ratio* or the proportion which their velocities bear to each other, tliat is, the proportion borne to each other by the distances moved through by the two points in the same interval of time. (2.). (3.) (4.) The directional relation* which is the relation be- tween the directions in which the two points are moving at the same instant, and which requires, for its complete expression, three * These terms are adopted from Prof. Willis's work on Mechanism. VELOCITY AND DIRECTION OF VARIED MOTION. 39 angles. Those three angles may be measured in dijfferent ways, and one of those ways is the following : — (2.) The angle made by the directions of the compared motions with each other. (3.) The angle made by a plane parallel to those two directions with a fixed plane. (4.) The angle made by the intersection of those two planes with a fixed direction in the fixed plane. Thus, the comparative motion of two points relatively to a third, is expressed by means of one of those groups of four elements which Sir William Rowan Hamilton has called " quaternions'^ In most of the practical applications of cinematics, the motions to be com- pared are limited by conditions which render the comparision more simple than it is in the general case just described. In machines, for example, the motion of each point is limited to two directions, forward or backward in a fixed path; so that the comparative motion of two points is sufficiently expressed by means of the velo- city ratio, together with a directional relation expressed by + or - , according as the motions at the instant in question are similar or contrary. Section 3. — Yaried Motion op Points. 71. Velocity and Direction of Varied Motion. — The motion of one point relatively to another may be varied, either by change of velocity, or by change of direction, or by both combined, which last case will now be considered, as being the most general. In fig. 24, let O represent a point assumed as fixed, X, O Y, O Z, fixed directions, and A B part of the path or orbit traced by a second point in its ^ p^^ 24. varied motion relatively to O. At the "* instant when the second point reaches a given position, suc h as P , in its path, the direction of its motion is obviously that of P T, a tangent to the path at P. To find the velocity at the instant of passing P, let A t denote an interval of time which includes that instant, and A s the dis- tance traced in that interval. Then A£ A« is an approximation to the velocity at the instant in question, which will approach continually nearer and nearer to the exact velocity as the interval A t and the distance A s are made shorter 40 PRINCIPLES OF CINEMATICS. acd shorter; and the limit towards which -—converges, asA « and A t are indefinitely diminished, and which is denoted by ds '' = Tt (^-^ is the exact velocity at the instant of passing P. In the language of the differential calculus, the space is a function of the time and the velocity is the differential coefficient of the space with respect ds to the time, thus s = (pt and -— = <^' ^ r= v. It will be seen here- after that, the velocity (v) itself is a function of the time {t). This is the process called " differentiation." Should the velocity at each instant of time be known, then the distance Sj - 5q, described during an interval oi' time t^ - t^, is found by integration (see Article 29), as follows : — (2.) 72. Components of Varied Motion. — All the propositions of the two preceding sections, respecting the composition and resolution of motions, are applicable to the velocities of varied motions at a given instant, each such velocity being represented by a line, such as P T, in the direction of the tangent to the path of the point which moves with that velocity, at the instant in question. For example, if the axes O X, O Y, O Z, are at right angles to each other, and if the tangent P T makes with their directions respec- tively the angles a, i3^ y then the three rectangular components of the velocity of the point parallel to those three axes are V cos et'j V cos /3j V COS y. Let X, y, z, be the co-ordinates of any point, such as P, in the path A P B, as referred to the three given axes. If a point p be assumed indefinitely near to the point P, its co-ordinates will be x + dx, y + dy, z + dz, and if ds have the already assumed value, dx, dy, dz, will be its projections on the three axes; that is, the lengths bounded by perpendiculars let fall from the extremities of ds on the three respective axes. Then it is well known that dx „ dy dz cos a = J- ; cos ^ = -T^; cos 7 = -J-; ds ds ds and consequently the three components of the velocity v l=~j are dx ^ dy dz ._ , vcos:, = -^^;vcm? = j-^;vcoBy=j-^; (3.) UNIFORMLY VARIED VELOCITY. 41 HOW by the Geometry of three dimensions COS^ ex. + cos^ /3 + COS^ y = 1. and hence these are related to their resultant by the equation ©'*©** e-3'=-- <'w 73. Uniformly-Varied Velocity. — Let the velocity of a point either increase or diminish at an uniform rate; so that if t repre- sents the time elapsed from a fixed instant when the velocity was Vq, the velocity at the end of that time shall be v = V(^^at\ (1.) a being a constant quantity, which is the rate of variation of the velocity, and is called acceleration when positive, and retardation when negative. Then the mean velocity during the time t is ^^^^^j^^^^_ ^2.) 'Oc^v V(.^Vr.-^at at 2- = ^ = ''<>+ 2 and the distance described is at'^ 8 = VQt + -j- (3.) If there be no initial velocity, that is, if the body start from a a (^ state of rest, then v = at and s = -^, and these equations are illus- trations of the use of the differential calculus; for first differentiate at?' s with respect to t in the equation s = — ^, and there is obtained -J-. {=v) = — ^— = atj which is the first equation, then differentiate v = at, and there is obtained ~r. = ci. To find the velocity of a cL t point, whose velocity is uniformly varied, at a given instant, and the rate of variation of that velocity, let the distances, A%, A^g, described in two equal intervals of time, each equal to At, before and after the instant in question, be observed. Then the velocity at the instant between those intervals is -^-Izt-^ (^o . and its rate of variation is Av As^-Asj^ ""-A-r-^Atf- ' (^O 42 PRINCIPLES OF CINEMATICS. where the variation of velocity = and the rate of varia- . 25. tion being either acceleration or retardation, as the velocity of the point is being increased or diminished, is that quantity divided by A^. 74. Graphical Representation of Motions,— Since in uniform motion the space is equal to the product of the velocity and time, and since in geometry a rectangular area is the product of a base line and perpendicular, an uniform motion may be represented by a rectangular area, as in fig. 25, where A B represents a certain number of units of time, and A C a certain number of units of velocity per unit of time. It will be noticed that in uniform motion, the velocity or number of units of velocity at each unit of time is the same, as at A, B, E. Varied motion and uni- formly varied motion may also be graphi- cally represented: in the first, the line C D will be a curve ; and in the second, the line CD will form a constant angle with AB; hence in varied motion any ordinate, E F, depends upon the abscissa A E, and the mean velocity is the mean ordinate of a figure so formed, or is the quotient of the area (space) divided by the base (time), whereas in uniformly-varied mo- tion, the space described depends upon the initial and final velo- cities alone, and not upon the intermediate velocities. Pig. 26 represents varied motion where the velocity at each point is re- presented by the ordinate at that point, and the mean velocity is equal to the area of the figure divided by the base A B. Eig. 27 represents uniformly- varied motion, and it is evi- dent that, in order to estimate the area of the figure ABCD, that is, the space, it is only necessary to consider the initial and final velocities. In these figures, if the velocity be null at any point, there will be no ordinate at that point: if the direction of motion change, this will be represented by a change of sign of the ordinate or velocity. There is another method of graphically representing the motion of a point: in this the abscissae represent the time, and the ordinatea D Fior. 27. ACCELERATED MOTION. 43 at each point the space passed over in the corresponding number of units of time, or the distance of the point from a certain datum point. In this case the space described in any number of units of time is equal to the difference of the lengths of the ordinates at the corresponding intervals, and the velocity is proportional to the quotient of the difference of the ordinates divided by the difference of the abscissse. 75. Varied Rate of Variation of Velocity.— When the velocity of a point is neither constant nor uniformly-varied, its rate of variation may still be found by applying to the velocity the same operation of differentiation, which, in Article 73, was applied to the distance described in order to find the velocity. The result of this operation is expressed by the symbols, ^ .^* dv _ dt _d^s ^ ^^dTt^ dt "J^'' and is the limit to which the quantity obtained by means of the formula 5 of Article 73 continually approximates, as the interval denoted by A ^ is indefinitely diminished. In the fraction d t ~~dr' ds'w, the limit of the difference of either of the spaces As in equa- tion (5), Article 73, and d • d s, \s, the limit of the difference of that difference, viz., Asg- A^^; that is, d in this fraction is represented by the minus sign ( -) in the other, and dshy the limit of either of the quantities Asp /\s^. Here in the language of the difierential calculus, the velocity [v) is a function of the time {t), and the acceleration {a) is the differential coefficient of the velocity with respect to the time, thus v = 't, or = -r-. Also the velocity, v, being the'differential coefficient of the space with respect to the time, see Article 71; the acceleration a is the 2nd differ- ential coefficient of the space with respect to the time, or v being ^'^, a = -4y" t. 76. Combination of Uniform and Uniformly Accelerated Motion. — Assume a pair of rectangular axes of co-ordinates. Let the uniform motion be represented by abscissse along O X, and the uniformly accelerated motion by ordinates parallel to O Y; let OB [ = x) = vt, represent the space described in the time ^ with a t^ the velocity v, and let C { = y) = — o-> represent the space de- PRINCIPLES OF CINEMATICS. scribed witli a uniform rate of acceleration, a, in the same time t, see Article 73, then a;'^ = v2^2 ^nd y — —-', .'. x^ = y ^, where the square of any abscissa bears a con- stant ratio to the corresponding ordi- nate, and the path of the point is known by Conic Sections to be a ^ig- 28. Parabola. The same follows for any axes of co-ordinates; but if the direction of the uniformly accelerated motion be that of the uniform motion or directly opposed to it, the resultant direction will be the same as that of either motion, or will be that of the greater component. 77. Uniform Deviation is the change of motion of a point which moves with uniform velocity in a circular path. The rate at which uniform deviation takes place is determined in the following manner : — Let C, fig. 29, be the centre of the cir- cular path described by a point A with an uniform velocity v, and let the radius U A be denoted by r. At the beginning and end of an interval of time A^, let A^^ and Ag be the positions of the moving point. Then the arc Aj Ag^t^A^; the chord A^ Ag = ■y A^ * ' Fig. 29. The velocities and directions at A^ and Ag are represented by the equal lines Aj^ Y^ = Ag Vg = '^j touching the circle at A^and Ag respec- tively. From Ag draw A, v equal and parallel to Aj Vj, and join Yg-y. Then the velocity Ag Yg ^"^^7 ^® considered as compounded of Ag-y and v Yg; so that v Yg is the deviation of the motion dur- ing the interval A^; and because the isosceles triangles AaV Yg, C Aj A2, are similar : — v^ ' At chord r arc ' deduced by substituting the value of Aj Ag already found; and the approximate rate of that deviation being the deviation divided by the interval of time in which it occurs, is v^ ^ chord r arc ' THE COMPARISON OF THE VARIED MOTIONS. 45 but the deviation does not take place by instantaneous changes of velocity, but by insensible degrees ; so that the true rate of devia- tion is to be found by finding the limit to which the approximate rate continually approaches as the interval A^ is diminished indefinitely. Now the factor — I'emains unaltered by that diminu- tion ; and the ratio of the chord to the arc approximates continually to equality ; so that the limit in question, or true rate of deviation, is expressed by ^.. (1.) r 78. Varying Deviation. — When a point moves with a varying velocity, or in a curve not circular, or has both these variations of motion combined, the rate of deviation at a given instant is still represented by Equation 1 of Article 77, provided v be taken to denote the velocity, and r the radius of curvature of the path, of the point at the instant in question. 79. The Resultant Rate of Variation of the motion of a point is found by considering the rate of variation of velocity and the rate of deviation as represented by two lines, the former in the direction of a tangent to the path of the point, and the latter in the direction of the radius of curvature at the instant in question, and taking the diagonal of the rectangle of which those two lines are the sides, which has the following value : — the first term of the quantity under the first radical is the square of d V 15 -1- in Article 73, and the second the square of - , Equation (1), Article 77. 80. The Rates of Variation of the Component Velocities of a point parallel to three rectangular axes, are represented as follows: — ^ d^jf^ d?z dt^' dt^ ' dt^' ^ '^ and if a rectangular parallelopiped be constructed, of which the edges represent these quantities, its diagonal, whose length is v{(S)'*(sy-(S)'} •« will represent the resultant rate of variation, already given in another form in Equation 1 of Article 79. 81. The Comparison of the Varied Motions of a pair of points 46 PKINCIPLES OP CINEMATICS. relatively to a third point assumed as fixed, is made by finding the ratio of their velocities, and the directional relation of the tangents of their paths at the same instant, in the manner already described in Article 70, as applied to uniform motions. It is evident that the comparative motions of a pair of points may be so regulated as to be constant, although the motion of each point is varied, pro- vided the variations take place for both points at the same instant, and at rates proportional to their velocities. 47 CHAPTER 11. MOTIONS OF EIGID BODIES. Section 1. — Rigid Bodies, and their Translation. 82. The term Rigid Body is to be understood to denote a body, or an assemblage of bodies, or a system of points, whose ffgure undergoes no alteration during the motion which is under con- sideration. 83. Translation or Shifting is the motion of a rigid body rela- tively to a fixed point, when the points of the rigid body have no motion relatively to each other; that is to say, when they all move with the same velocity and in the same direction at the same instant, so that no line in the rigid body changes its direction. It is obvious that if three points in the rigid body, not in the same straight line, move in parallel directions with equal velocities at each instant, the body must have a motion of translation. The paths of the different points of the body, provided they are all equal and similar, and at each instant parallel, may have any figure whatsoever. Section 2. — Simple Rotation. 84. Rotation or Turning is the motion of a rigid body when lines in it change their direction. Any point in or rigidly attached to the body may be assumed as a fixed point to which to refer the motions of the other points. Such a point is called a centre of rotation. 85. Axis of Rotation. — Tbeorem. In every possible change of position of a rigid body, relatively to a fixed centre, there is a line traversing that centre whose direc- tion is not changed. In fig. 30, let O be the centre of rotation, and let A and B denote any two other points in the body, whose situa- tions relatively to O are, before the turning, Aj, B^, and after the turnin g, Ag, Ba. Join Aj As, Yi^ ZO. ^ Bi B.2, forming the isosceles tri- angles O Aj Ag, Bi Bg. Bisect the bases of those triangles in C 48 PRINCIPLES OF CINEMATICSw and D respectively, and through the points of bisection draw two planes perpendicular to the respective bases, intersecting each other in the straight line O E, which must traverse O. Let E be any point in the line O E ; then E Aj Ao, and E Bi B^, are isosceles triangles ; and E is at the same distance from O, A, and B, before and after the turning; therefore E is one and the same point in the body, whose place is unchanged by the turning; and this demonstration applies to every point in the straight line O E ; therefore that line is unchanged in direction. — Q. E. D. In fig. 31, the same construction and reasoning being applied, the point E being supposed vertically above or below the point O, it is evident that the planes through O D, and intersect, and the axis will be represented by a straight line perpendicular to the plane of the paper through O and E. OoROLLARY. It is evident that every Fig. 31. line in the body, parallel to the axis, has its direction unchanged. 86. The Plane of Rotation is any plane perpendicular to the axis, such as any plane parallel to the plane of the paper, in fig. 31. The Angle of Rotation, or angular motion, is the angle made by the two directions, before and after the turning, of a line perpendicular to the axis, as A^ O Ag, or Bj O Bg, in fig. 31. 87. The Angular Velocity of a turning body is the ratio of the angle of rotation, expressed in terms of radius, to the number of units of time in the interval of time occupied by the angular motion. Speed of turning is sometimes expressed also by the number of turns or fractions of a turn in a given time. The rela- tion between these two modes of expression is the following : — Let a be the angular velocity, as above defined, and T the turns in the same unit of time ; then T- ^ • a=29rT; (710\* 2t= 6-2831852. -y^J^.j 88. Uniform Rotation consists in uniformity of the angular * The value of * may he easily remembered by taking the first three odd numbers twice each, and placing the six in a row, using the lirst three as the denominator, and the last three as the numerator of a fraction: we thus 355 obtain 113 | 355 = .. _; this is a nearer approximation than 3-14159, and ia generally much more easOy employed in calculation. ROTATING BODY. ^ 49 velocity of the turning body, and constancy of tlie direction of its axis of rotation. 89. Rotation common to all Parts of Body. — Since the angu- lar motion of rotation consists in the change of direction of a line in a plane of rotation, and since that change of direction is the same how short soever the line may be, it is evident that the condition of rotation, like that of translation, is common to every particle, how small soever, of the turning rigid body, and that the angular velocity of turning of each particle, how small soever, is the same with that of the entire body. This is otherwise evident by considering, that each part into which a rigid body can be divided turns completely about in the same time with every other part, and with the entire body. 90. Right and Left-Handed Rotation. — The direction of rota- tion round a given axis is distinguished in an arbitrary manner into right-handed and left-handed. One end of the axis is chosen, as that from which an observer is supposed to look along the direction of the axis towards the rotating body. Then if the body seems to the observer to turn in the same direction in which the sun seems to revolve to an observer north of the tropics, or in that in which the hands of a watch or clock revolve, the rotation is said to be right-handed; if in the contrary direction, left-handed: and it is usual to consider the angular velocity of right-handed rotation to be positive, and that of left-handed rotation to be negative; but this is a matter of convenience. It is obvious that the same rotation which seems right-handed when looked at from one end of the axis, seems left-handed when looked at from the other end. 91. Relative Motion of a Pair of Points in a Rotating Body. — Let O and A denote any two points in a rotating body; and con- sidering O as fixed, let it be required to determine the motion of A relatively to an axis of rotation drawn through O. On that axis let fall a perpendicular from A; let r be the length of that perpendicular. Then the motion of A relatively to the axis traversing O is one of revolution, or translation in a circular 'path of the radius r; the centre of that circular path being at the point where the perpendicular from A meets the axis. If a be the angular velocity of the body, that is, the velocity of a point situate at the distance unity from the axis of rotation, then the velocity of A relatively to the axis -traversing O is v = ar; (1-) and the direction of that velocity is at each instant perpendicular to the plane drawn through A and the axis. The rate of deviation of A in its motion relatively to the given axis is - = a^r; (2.) 50 PRINCIPLES OF CINEMATICS. in which the first expression is that already found in Article 77, and the second is deduced from the first by the aid of Equation 1 of this Article. It is evident that for a given rotation the motion of O relatively to an axis of rotation traversing A is exactly the same with that of A relatively to a parallel axis traversing O ; for it depends solely on the angular velocity a, the perpendicular distance r of the moving point from the axis, and the direction of the axis; all which are the same in either case. r is called the radius-vector of the moving point. 92. Cylindrical Surface of Equal Velocities. — If a cylindrical surface of circular cross section be described about an axis of rota- tion, all the points in that surface have equal velocities relatively to the axis, and the direction of motion of each point in the cylin- drical surface relatively to the axis is a tangent to the surface in a plane perpendicular to the axis. 93. Comparative Motions of Two Points relatively to an Axis. — Let O, A, B, denote three points in a rotating rigid body ; let O be considered as fixed, and let an axis of rotation be drawn through it. Then the comparative motions of A and B relatively to that axis are expressed as follows: — The velocity-ratio is that of the radii- vectores of the points, and the directional relation consists in the angle between their directions of motion being the same with that between their radii-vectores. Or symbolically : Let r-^, r^, be the per- pendicular distances of A and B from the axis traversing O, and v^ and v<2, their velocities; then V, r. A A - — — ; and v. Vo = r-.r^. 94. Components of Velocity of a Point in a Rotating Body. — The component parallel to an axis of rotation, of the velocity of a point in a rotating body relatively to that axis, is null. That velocity may be re- solved into components in the plane of rotation. Thus let O, in fig. 32, represent an axis of rotation of a body whose plane of rotation is that of the figure ; and let A be any point in the body whose radius- vector is O A = r. The velocity of that point being v ^ ar {a representing the ^ig- 32. velocity of a point situated at the distance unity from the axis of rotation), let that velocity be represented by the line AY perpendicular to A. Let B A be any direction in the plane of rotation, along which it is desired to find the com- ponent of the velocity of A; and let Z V A U = e be the angle made by that line with A V. From V let fall Y XJ perpendicular HELICAL MOTION. 51 to B A; then A U represents the component in question; and de- noting it by u, u = v cos & = ar ' cos 6 (1.) From O let fall B perpendicular to B A. Then Z A O B = Z Y A XJ = ^; and the right-angled triangles O B A and A U V are similar; so that AY : AU : : OX: O B'^r cos (2.) Now the entio-e velocity of B relatively to the axis O is a r cos e = u, (3. ) so that the component, along a given straight line in the plane oj rotation, of the velocity of any point in that line, is equal to the velo- city of the point where a perpendicular from the axis meets that line. Section 3. — Combined Rotations and Translations. 95. Property of all Motions of Rigid Bodies. — The foregoing proposition may be regarded as a particular case of the following, which is true of all motions of a rigid body. The components, along a given straight line ^?^ a rigid body, of the velocities of the points in that line relatively to any point, whether in or attached to the body or otherwise, are all equal to each other; for otherwise, the distances between points in the given straight line must alter, which is inconsistent with the idea of rigidity. 96. Helical Motion. — Rotation is the only movement which a rigid body as a whole can have relatively to a point belonging to it or attached to it. But if the motion of the body be determined relatively to a point not attached to it, a translation may be com- bined with the rotation. When that translation takes place in the direction of the axis of rotation, the motion of the rigid body is said to be helical, or screw-like, because each point in the rigid body describes a helix or screw, or a part of a helix or screw. Let Vx denote the velocity of translation, parallel to the axis of rotation, which is common to all points of the body; this is called the velocity of advance. The advance during one complete turn of the rotating body is the pitch of each of the helical or screw-like paths described by its particles; that is, the distance, in a direc- tion parallel to the axis, between one turn of each such helix and the next; and a being the angular velocity, so that — is the time of one turn (2 -tt being the space traversed in one turn by a point at the distance unity from the axis), the value of the pitch (or the space passed over, which is equal to the product of the velocity and time) is 52 PRINCIPLES OF CINEMATICS. P= -^'; whence Vi= ^"^ (1.) Let r, as before, be the radius- vector of any point in the body, and let V2 = ar (2.) denote its velocity/ of revolution, or velocity relatively to the axis, due to the rotation alone. Then the resultant velocity of that point is v=^/W^=cc-.^ {^, + r^} (3.) The inclination of the helix described by that point to the plane of rotation is given by the equation V 7) i = arc * tan • — = arc • tan • — — : (4.) that is, an angle whose tangent is equal to v-^ divided by v^, or to p divided by 2 s- r, the tangent of that angle beii?g the ratio of the pitch to the circumference of the circle described by the point rela- tively to the axis of rotation. 97. Problem. — To find the Motion of a Rigid Body from the Motions of Three of its Points. — ^ ,^ Let A, B, C, fig. 33, be three ^ points in a rigid body, and at a '^^ — ^ given instant let them have mo- " ^ tions relatively to a point indepen- dent of the body, which motions are represented in velocity and o -^———-U -p direction by the three lines A V^, ^ T — B Yj, C V^ It is required to find \ the motion of the entire rigid \ body relatively to the same fixed Ac point. r^.iC_ixr Through any point o, fig. 34, ^. draw three lines oa, oh, oc, equal ^«' ' and parallel to the three lines ^ A V^, B Vj, C Vg. Through a, h, and c, draw a plane a b c, on which let fall a perpendicular o n from o. Then o n represents a component, which is common to the velocities of all the three points A, B, C, and must therefore be common to all the Fig. 34. points in the body ; that is, it is a velocity of translation. From the points V„, V„ Y„ draw lines Y^W,, YT^I V/fJ,, equal and parallel to o n, but opposite in direction to it; and join SPECIAL CASES. < )> ^ 53 n, 'v^v A U„, B Uj, C Uc, which will all be parallel to the^ aa;6e planVJ)"^ that is, to the plane a b c. The last three lines will repr^^t the component velocities which, along with the common velocity^of^ translation parallel to o n, make up the resultant velocities of tn*^ three points. Through the point A draw a plane perpendicular to the component of its motion, which is parallel to a h c; that is, to A U„, and through B draw a plane perpendicular to B Uj. These two planes will intersect each other in a line ODE, which will be parallel to o n. The perpendicular distances of that line from the points A B being unchanged by the motion, it represents one and the same line in or attached to the rigid body, and it is there- fore the axis of rotation. A plane drawn through the third point, C, perpendicular to C Up, will cut the other two planes in the same axis : the three revolving component velocities Au;, BUT, cxj;, will be respectively proportional to the perpendicular distances, or radii-vectores, A D, B E, C F, of the three points from that axis ; and the angular velocity will be equal to each of the three quotients made by dividing the revolving component velocities of the points by their respective radii- vectores. This rotation, combined with a translation parallel to the axis, with a velocity represented by o n, constitutes a hdical or screw-like motion, being the required motion of the rigid body. — Q. E. I. 98. Special Cases of the preceding problem occur, in which either a more simple method of solution is sufficient, or the general method fails, and a special method has to be employed. I. When the motions of the points of the body are known to be all parallel to one plane, it is suflBcient to know the motions of two points, such as A, B, fig. 35. Let A O, BO, be two planes tra- versing A and B, and perpendicular to the respective directions of the simul- taneous velocities of those points ; if those planes cut each other, the entire motion is a rotation ; the line of intersection of the planes O, being the axis of rotation, and the angular velocity, are found as in the last Article. If the two planes are parallel, the motion is a translation. II. If three points in the same plane have parallel motions oblique to the plane, the motion is a translation. III. If three points in the same plane move perpendicidarly to the plane, as A B C, fig. 35 a, then if their velocities are equal, the motion is a translation; and if their velocities are unequal, the 54 PRINCIPLES OP CINEMATICS. motion is a rotation about the axis which is the intersection of the plane of the three points with the plane drawn through the extre- Fig. 36. Fig. 35 a. -^ Fig. 35 c. mities of the three lines which represent their velocities viz., through the points, V«, Yj, Y^\ the angular velocity being found as in Article 97. If the plane of rotation is known, then the simultaneous veloci- ties of two points, as A and B in figs. 35 b and 35 c, are sufficient to determine the axis O. ©9. Rotation Combined with Translation in the Same Plane.— Let a body rotate about an axis C (fig. 36), fixed relatively to the body, with an angular velocity a, and at the same time let that axis have a motion of translation in a straight path perpen- dicular to the direction of the axis, with the velocity u, represented by the line C U, It is required to find the velocity and direction of motion of any point in the body. From the moving axis draw a straight line C T perpendi- cular to that axis and to C U, and in that direction into which the rotation (as represented by the feathered arrow) tends to turn C U, and make u CT = - ...., (1.) Then the point T has, in virtue of translation along with the axis C, a forward motion with the velocity u ; and in virtue of rotation about that axis, it has a backward motion with the velocity a-OT=^u, equal and opposite to the former; and its resultant velocity is 0. Hence every point in the body, which comes in succession into the position T, situated at the distance - from the axis C in the direc- a tion above described, is at rest at the instant of its arriving at that position; that is, it has just ceased to move in one direction, and is about to move in another direction; and this is true of every SPIRAL PATHS. 55 point which arrives at a liTie traversing T parallel to C. Conse- quently the resultant motion of the body, at any given instant, is the same as if it were rotating about the line which at the instant in question occupies the position T, parallel to C, at the distance - ; and that line is called the instantaneous axis. To find the Cb motion of any poin t A in the body at a given instant, let fall the perpendicular A T from that point on the instantaneous axis; then the motion of A is in the direction AY perpendicular to the plane of the instantaneous axis and of the instantaneous radius-vector A T, and the velocity of that motion is v = a-AT (2.) 100. Rolling Cylinder; Trochoid.— Every straight line parallel to the moving axis C, in a cylindrical surface described about C with the radius -, becomes in turn the instantaneous axis. Hence a the motion of the body is the same with that produced by the roll- ing of such a cylindrical surface on a plane P T P parallel to C and to C U, at the distance -. a The path described by any point in the body, such as A, which is not in the moving axis C, is a curve well known by the name of trocJioid. The particular form of trochoid called the cycloid, is described by each of the points in the rolling cylindrical surface ; being such a curve as is described by a nail in the tyre of a revolv- ing wheel. 101. Plane Rolling on Cylinder; Spiral Paths.— Another mode of representing the combination of rotation with translation in the same plane as follows : — Let O, fig. 37, be an axis assumed as fixed, about which let the plane O C (containing the axis 0) rotate (right- handedly, in the figure), with the angular velocity a. Let a rigid body have, rela- tively to tJie rotating plane, and in a direc- tion perpendicular to it, a translation with the velocity u. In the plane O C, and at right angles to the axis O, take O T = -, in such a direction that the a velocity w = a-OT, which the point T in the rotating plane has at a given instant, shall be in the contrary direction to the equal velocity of translation Uj which the rigid body has relatively to the ro^fl-tini? 56 PRINCIPLES OV CINEMATICS. plane. Then each point in the rigid body which arrives at the position T, or at any position in a line traversing T parallel to the lixed axis O, is at rest at the instant of its occupying that position ; therefore the line traversing T j^arallel to the fixed axis O is the instantaneous axis; the motion at a given instant of any point in the rigid body, such as A, is at right angles to the radius-vector A T drawn perpendicular to the instantaneous axis; and the velocity of that motion is given by the equation v^a-ATT. All the lines in the rigid body which successively occupy the position of instantaneous axis are situated in a plane of that body, P T P, perpendicular to O C; and all the positions of the instan- taneous axis are situated in a cylinder described about O with the radius O T ; so that the motion of the rigid body is such as is pro- duced by the rolling of the i^lane P P o^^ the cylinder whose radius is O T = — . Each point in the rigid body, such as A, describes a a plane spiral about the fixed axis O. For each point in the rolling plane, P P, that spiral is the involute of the circle whose radius is U T. The simplest method of understanding the nature of this curve, is to wrap a cord round the perimeter of a cylinder, placed on a sheet of paper, to attach a tracing point to any point in the cord in juxtaposition with the cylinder, and then to unwrap the cord from the cylinder, keeping the cord always in the same plane parallel to the plane of the paper; the tracing point will trace the involute of a circle on the sheet of paper. For each point whose path of motion traverses the fixed axis O ; that is, for each point in a plane of the rigid body traversing O parallel to P P, the spiral is Archimedean, having a radius-vector increasing by the length u for each angle a through which it rotates; this spiral is traced by a point moving uniformly from the centre along the radius, while the radius itself revolves. 102. Combined Parallel Rotations.— In figs. 38, 39, and 40, let O be an axis assumed as fixed, and O C a plane traversing that axis, and rotating about it with the angular velocity a. Let C be an axis in that plane, parallel to the fixed axis O; and about the moving axis C let a rigid body rotate with the angular velocity b 7-elatively to the plane O C; and let the directions of the rotations a and b be distinguished by positive and negative signs. The body is said to have the rotations about the parallel axes O and C com- bined or compounded, and it is required to find the result of that combination of parallel rotations. Fig. 38 respresents the case in which a and 6 are similar in direction; fig. 39, that in which a and b are in opposite directions, COMBINED PARALLEL ROTATIONS. and h is the greater; and fig. 40, that in which a and h are in opposite directions, and a is the greater. Fig. 40. Let a common perpendicular O C to the fixed and moving axes be intersected in T by a straight line parallel to both those axes, in such a manner that the distances of T from the fixed and moving axes respectively shall be inversely proportional to the angular velocities of the component rotations about them, as is expressed by the following proportion : — :6: :CT:OT. ,(1.) When a and b are similar in direction, let T fall between O and C, as in fig. 38 ; when they are contrary, beyond, as in figs. 39 and 40. Then the velocity of the line T of the plane O C is ct • O T ; and the velocity of the line T of the rigid body, relatively to the plane O C, is 6 • C T, equal in amount and contrary in direction to the former; therefore each line of the rigid body which arrives at the position T is at rest at the instant of its occupying that position, and is then the instantaneous axis. The resultant angular velocity is given by the equation c^a + h; (2.) regard being had to the directions or signs of a and 6 ; that is to say, if we now take a and h to represent arithmetical magnitudes, and affix explicit signs to denote their directions, the direction of c will be the same with that of the greater ; the case of fig. 38 will be represented by Equation 2, already given ; and those of figs. 39 and 40 respectively by c = h-a'y c-=a — h (2 A.) The relative proportions of a, b, and c, and of the distances between the fixed, moving, and instantaneous axes, are given by the equation a:b:c: :CT:OT:OC (3.) The motion of any point, such as A, in the rigid body, is at ea^^- 58 PRINCIPLES OF CINEMATICS. instant at right angles to the radius-vector AT drawn from the point perpendicular to the instantaneous axis; and the velocity of that motion is v^^c-AT (4.) 103. Cylinder Rolling on Cylinder; Epitrochoids.— All the lines in the rigid body which successively occupy the position of instan- taneous axis are situated in a cylindrical surface described about C with the radius CT; and all the positions of the instantaneous axis are contained in a cylindrical surface described about with the radius O T ; therefore the resultant motion of the rigid body is that which is produced by rolling the former cylinder, attached to the body, on the latter cylinder, considered as fixed. In fig. 38, a convex cylinder rolls on a convex cylinder; in fig. 39, a smaller convex cylinder rolls in a larger concave cylinder; in fig. 40, a larger concave cylinder rolls on a smaller convex cylinder. Each point in the rolling rigid body traces, relatively to the fixed axis, a curve of the kind called epitrochoids. The epitrochoid traced by a point in the surface of the rolling cylinder is an epicycloid. In certain cases, the epitrochoids become curves of a more simple class. For example, each point in the moving axis C traces a circle. When a cylinder, as in fig. 39, rolls witliin a concave cylinder of double its radius, each point in the surface of the rolling cylinder moves backwards and forwards in a straight line, being a diameter of the fixed cylinder; each point in the axis of the rolling cylinder traces a circle of the same radius with that cylinder, and each other point in or attached to the rolling cylinder traces an ellipse of greater or less eccentricity, having its centre in the fixed axis O. In the examples shewn in figs. 41, 42, and 43 the ratio of the rolling-circle to the base-circle* is - , so that the epitrochoids are o three-lobed. Each figure shews an external and an internal epitro- choid, traced by rolling the rolling-circle outside and inside the base-circle respectively. The centres of the base-circles are marked A; those of the external rolling-circles, B; those of the internal rolling-circles, 6; and the tracing points of the external and in- ternal rolling-circles are marked C and c respectively. In fig. 41 the tracing-points are in the circumferences of the rolling-circles; and the curves traced are epicycloids, distinguished by having cusps at the points where the tracing-point coincides with the base-circle. In fig. 42 the tracing points are inside the rolling-circles; and the curves traced are prolate epitrochoids, dis- tinguished by their wave-like form. In fig. 43 the tracing-points • The fixed circle is called a base circle. 69 Fig, 41. Fii?. 42. CO PRINCIPLES OF CINEMATICS. are outside the rolling-circles; and the curves traced are airtate epitrochoids, distinguished by their looped form. An important property of curves traced by rolling is that at Fig. 43. every instant the straight line joining the tracing-point and the pitch-point, or point of contact of the rolling-curve and base-curve, is normal to the traced curve at the tracing point. The distance B C or be may in each case be called the tracing- arm. In mechanism for the tracing of epitrochoids (used chiefly in ornamental turning), the rolling and base-circles are the pitch- circles of a pair of spur-wheels, made with great accuracy. Elliptic paths traced by rolling form a particular case of internal epitrochoids. In fig. 44 is represented a rolling-circle, which rolls inside a base-circle of exactly twice its radius. Then (considering a quarter of a revolution at a time), while the centre of the rolling- circle traces a quadrant, B b, of an equal circle about A, a point D in the circumference of the rolling-circle traces a straight line traversing A, and a point C, inside the rolling-circle, traces a quadrant, C c, of an ellipse whose semiaxes are A C = A B + B C, and Ac = CI) = AB-BC; also a point C outside the rolling- EPITROCHOIDS. 61 but rigidly attached to it, traces a quadrant, C c', of an whose semiaxes are A C = B C + A B, and A c' = C D = circle, ellipse B C' - A B. The former may be called an internal, and the latter an external, ellipse. The proportions of the axes of either of them Fig. 44. may be indefinitely varied by adjusting the position of the tracing- point; but in every internal ellipse the sum, and in every external ellipse the difference, of the semiaxes is equal to the diameter of the rolling-circle; that is, to the radius of the base-circle. This is the principle of the mechanism commonly used for turning ellipses. It is evident that by having a number of tracing-points carried by one rolling-circle, several ellipses differently proportioned and in different positions may be traced at the same time. C2 PRINCIPLES OP CINEMATICS. 104. Equal and Opposite Parallel Rotations Combined. — Let a plane O C rotate with an angular velocity a about an axis O con- tained in the plane, and let a rigid body rotate about the axis C in that plane parallel to O, with an angular velocity - a, equal and opposite to that of the plane. Then the angular velocity of the rigid body is nothing; that is, its motion is one of translation only, all its points moving in equal circles of the radius OC, with the velocity a • O C. This case is not capable of being represented by a rolling action. 105. Rotations about Intersecting Axes Combined. — In fig. 45 let O A be an axis assumed as fixed; and about it let the plane A C rotate with the angular velocity a. Let O C be an axis in the rotating plane ; and about that axis let a rigid body rotate with the angular velocity h re- latively to the rotating plane. Yig. 45. Because the point O in the rigid body is fixed, the instantaneous axis must traverse that point. The direction of that axis is determined, as before, by considering that each point which arrives at that line must have, in virtue of the rotation about O C, a velocity relatively to the rotating plane, equal and directly opposed to that which the coincident point of the rotating plane has. Hence it follows, that the ratio of the perpendicular distances of each point in the instantaneous axis from the fixed and moving axes respectively — that is, the ratio of the sines of the angles which the instantaneous axis makes with the fixed and moving axes — must be the reciprocal of the ratio of the component angular velocities about those axes; or symbolically, if O T be the instantaneous axis, sin AOT:sinCOT: :h'.a (1.) This determines the direction of the instantaneous axis, which may also be found by graphic construction as follows : — On O A take O a proportional to a; and on O C take O b proportional to h. Let those lines be taken in such directions, that to an observer looking from their extremities towards O, the component rotations seem both right-handed. Complete the parallelogram Ohca; the dia- gonal O c will represent the direction of the instantaneous axis. The resultant angular velocity about this instantaneous axis is found by considering, that if C be any point in the moving axis, the linear velocity of that point must be the same, whether com- puted from the angular velocity a of the rotating plane about the fixed axis A, or from the resultant angular velocity c of the rigid VARIATION OF ANGULAR VELOCITY. 63 body about the instantaneous axis. That is so say, let CD, C E, be perpendiculars from C upon O A, O T, respectively; then a''CT) = c-CE; but CI) : CE : : sinZ A O C : sinZC T; and therefore sinZCOT :sinZAOC: laic, and, combining this proportion with that given in Equation 1, we obtain the following proportional equation: — sinZCOT rsinZAOT :sinZAOC ^ : : ja^ : h_ i c_ > (2.) : : Oa : 06 : Oc j that is to say, the angular velocities of the component and resultant rotations are each ■proportional to the sine of the angle between the axes of the other two; and the diagonal of the parallelogram O b c a represents both the direction of the instantaneous axis and the angu- lar velocity about that axis. 106. Rolling Cones. — All the lines which successively come into the position of instantaneous axis are situated in the surface of a cone described by the revolution of O T about O C ; and all the positions of the instantaneous axis lie in the surface of a cone described by the revolution of O T about O A. Therefore the motion of the rigid body is such as would be produced by the rolling of the former of those cones upon the latter. It is to be understood, that either of the cones may become a flat disc, or may be hollow, and touched internally by the other. For example, should Z A O T become a right angle, the fixed cone would become a flat disc ; and should Z A T become obtuse, that cone would be hollow, and would be touched internally by the rolling cone; and similar changes may be made in the rolling cone. The path described by a point in or attached to the rolling cone is a spherical epitrochoid; but for the purposes of the present trea- tise, ifc is unnecessary to enter into details respecting the properties of that class of curves. 107. Comparative Motions in Compound Rotations. — The velo- city ratio of two points in a rotating rigid body at any instant, is that of their perpendicular distances from its instantaneous axis ; and .the angle between the directions of motion of the two points is equal to that between the two planes which traverse the points and the instantaneous axis. Section 4. — Varied Eotation. 108. Variation of Angular Velocity is measured like variation of linear velocity, by comparing the change which takes place ia C4: PRINCIPLES OF CINEMATICS. the angular velocity of a rotating body, A a, during a given interval of time, with the length of that interval, A t, and the rate of variatio7i is the value towards which the ratio of the change of angular velocity to the interval of time, -^— , converges, as the A t length of the interval is indefinitely diminished ; being represented by -z-, and found by the operation of difierentiation. 109. Components of Varied Rotation. — The most convenient way, in most cases, of expressing the mode of variation of a rotatory motion, is to resolve the angular velocity at each instant into three component angular velocities about three rectangular axes fixed in direction. The values of these components, at any instant shew at once the resultant angular velocity and the direction of the instantaneous axis. For example, let a^, ay, a^, be the rectangular components of the angular velocity of a rigid body at a given instant, — rotation about x from y towards z, about y from z towards x, and about z from x towards y, being considered as positive; then a= JW + a^^ + a^^) (1.) is the resultant angular velocity, and cosoe = — ; cos/3=— ; cosy = ~; (2.) a a a are the cosines of the angles which the instantaneous axis makes with the axis of x, y, and z respectively. 65 CHAPTER III. MOTIONS OF PLIABLE BODIES, AND OF FLUIDS. 110. Division of the Subject. — The subject of the present chapter will be considered under the following branches: — I. The Motions of Flexible Cords. 11. The Motions of Fluids not altering in Yolurce. 111. The Motions of Fluids altering in Volume. Section 1. — Motions of Flexible Cords. 111. General Principles. — As those relative motions of the points of a cord which may arise from its extensibility, belong to the subject of resistance to tension, which is a branch of that of strength and stiffness, the present section is confined to those motions of which a flexible cord is capable when the length, not merely of the whole cord, but of each part lying between two points fixed in the cord, is invariable, or sensibly invariable. In order that the figure and motions of a flexible cord may be determined from cinematical considerations alone, independently of the magnitude and distribution of forces acting on the cord, its weight must be insensible compared with the tension on it, and it must everywhere be tight; and when that is the case, each part of the cord which is not straight is maintained in a curved figure by passing over a convex surface. The line in which a tight cord lies on a convex surface is the shortest line which it is possible to draw on that surface between each pair of points in the course of the cord. (It is a well-known principle of the geometry of curved surfaces, that the osculating plane or tangential plane at each point of such a line is perpendicular to the curved surface.) Hence it appears, that the motions of a tight flexible cord of invariable length and insensible weight are regulated by the follow- ing principles : — I. The length between each pair of points in the cord is constant. II. That length is the shortest line which can be drawn between its extremities over the surfaces by which the cord is guided. 112. Motions Classed. — The motions of a cord are of two kinds — I. Travelling of a cord along a track of invariable form; in which case the velocities of all points of the cord are equal. F GQ PRINCIPLES OF CINEMATICS. II. Alteration of the figure of the track by the motion of the guiding surfaces. Those two kinds of motion may be combined. The most usual problems in practice respecting the motions of cords are those in. which cords are the means of transmitting motion Vietween two pieces in a train of mechanism. Such pro- blems will be considered in Part II. of this treatise. Next in point of frequency in practice is the problem to be considered in the ensuing Article. 113. Cord Guided by Surfaces of Revolution. — Let a cord in some portions of its course be straight, and in others guided by the surfaces of circular drums or pulleys, over each of which its track is a circular arc in a plane perpendicular to the axis of the guiding surface. Let r be the radius of any one of the guiding surfaces, i the angle of inclination which the two straight portions of the cord contiguous to that surface make with each other, expressed in length of arc to radius unity. Then the length of the portion of the cord which lies on that surface is r i; and if 5 be the length of any straight portion of the cord, the total length between two given points fixed in the cord may be expressed thus : — L = 2 -5 + 2 -ri (1.) Let c be the distance between the centres of a given adjacent pair of guiding surfaces, s the length of the straight portion of cord which lies between them, and r, r', their respective radii; then evidently «= Jc^-(r±rf (2.) the i yrn > of the radii being employed, according as the cord r crosses I . i t c i. < -, , > the line of centres c. [ does not cross j The case most common in practice is that in which the plies^ or straight parts of the cord, are all parallel to each other; so that i = lSO° in each case, while a certain number, n, of the guiding bodies or pulleys all move simultaneously in a direction parallel to the plies of the cord with the same velocity, tt; where u represents the velocity of translation of the guiding surfaces, and v the longitudinal velocity of any point in the cord v-2n u (3.) Section 2. — Motions of Fluids of Constant Density. 114. Velocity and Flow. — The density of a moving fluid mass may be either exactly invariable, from the constancy or the adjust- ment of its temperature and pressure, or sensibly invariable, from the smallness of the alterations of volume which the actual altera- PIPES, CHANNELS, CURRENTS, JETS. 67 tions of pressure and temperature are capable of producing. The latter is the case in most problems of j)ractical mechanics affecting liquids. Conceive an ideal surface of any figure, and of the area A, to be situated within a fluid mass, the parts of which have motion rela- tively to that surface; and let u denote, as the case may be, the uniform velocity, or the inean value of the varying velocity, resolved in a direction perpendicular to A, with which the particles of the fluid pass A. Then Q = wA (1.) is the volume of fluid which passes from one side to the other of the surface A in an unit of time, and is called the Jlow^ or rate of flow, through A. When the particks of fluid move obliquely to A, let 6 denote the angle which the direction of motion of any particle passing A makes with a normal to A, and v the velocity of that particle; then u = v cos d (2.) 115. Principle of Continuity. — Axiom. When the motion of a fluid of constant density is considered relatively to an enclosed space of invariable volume which is always filled with the fluid, the flow into the space and thefloio out of it, in any one given interval of time,, 7}iust he equal — a principle expressed symbolically by 2-Q = (30 The preceding self-evident principle regulates all the motions of fluids of constant density, when considered in a purely cinematical manner. The ensuing articles of this section contain its most usual applications. 116. Flow in a Stream. — A stream is a moving fluid mass, indefinitely extended in length, and limited transversely, and having a continuous longitudinal motion. At any given instant, let A, A', be the areas of any two of its transverse sections, con- sidered as fixed; w, it', the mean normal velocities through them; Q, Q', the rates of flow through them ; then in order that the principle of continuity may be fulfilled, those rates of flow must be equal; that is, 1^ A = 2a' A' = Q = Q' = constant for all cross sections of the channel at the given instant; (1.) consequently, u A""- * ^"'^ or, the normal velocities at a given instant at two fixed cross sections are inversely as the areas of these sections. 117. Pipes, Channels, Currents, and Jets. — When a stream of 68 PRINCIPLES OF CINEMATICS. fluid completely fills a ^n;;e or tube, the area of each cross section is given by the figure and dimensions of the pipe, and for similar forms of section varies as the square of the diameter. Hence the mean normal velocities of a stream flowing in a full pipe, at different cross sections of the pipe, are inversely as the squares of the diameters of those sections. A channel partially encloses the stream flowing in it, leaving the npper surface free; and this description applies not only to channels commonly so called, but to pipes partially filled. In this case the area of a cross section of the stream depends not only on the figure and dimensions of the channel, but on the figure and elevation of the free upper surface of the stream. A current is a stream bounded by other portions of fluid whose motions are diflferent. A jet is a stream whose surface is either free all round, or is touched by a solid body in a small j3ortion of its extent only. 118. Steady Motion of a fluid relatively to a given space con- sidered as fixed is that in which the velocity and direction of the motion of the fluid at each jixed point is uniform at every instant of the time under consideration; so that although the velocity and direction of the motion of a given particle of the fluid may vary while it is transferred from one point to another, that particle assumes, at each fixed point at which it arrives, a certain definite velocity and direction depending on the position of that point alone; which velocity and direction are successively assumed by each particle which successively arrives at the same fixed point. The steady motion of a stream is expressed by the two conditions, that the area of each fixed cross section is constant, and that the flow through each cross section is constant, then the differential coefficient of a constant being equal to (see Article 26, page 11), '.t=«'t?=« (!•) If u represents the normal velocity of a fluid moving steadily, at a given fixed point, 'ii-^-' (^•) expresses the condition of steady motion. 119. Motion of Bistons. — Let a mass of fluid of invariable volume be enclosed in a vessel, two portions of the boundary of which (called pistons) are movable inwards and outwards, the rest of the boundary being fixed. Then, if motion be transmitted between the pistons by moving one inwards and the other out- wards, it follows, from the invariability of the volume of the enclosed fluid, that the velocities of the two pistons at each instant THE PRINCIPLE OF CONTINUITY. 69 will be to each other in the inverse ratio of the areas of the respec- tive projections of the pistons on planes normal to their directions of motion. This is the principle of the transmission of motion in the hydrauliG press and hydraulic crane. The Jlow produced by a piston whose velocity is u, and the area of whose projection on a plane perpendicular to the direction of its motion is A, is given, as in other cases, by the equation q^uA (1.) Section 3. — Motions of Fluids of Yauying Density. 120. Flow of Volume and Flow of Mass. — In the case of a fluid of varying density, the volume, which in an unit of time flows through a given area A, with a normal velocity u, is still repre- sented, as for a fluid of constant density, by Q = An; (1.) but the absolute quantity, or mass of fluid which so flows, bears no longer a constant proportion to that volume, but is proportional to the volume multiplied by the density. The density may be expressed, either in units of weight per unit of volume, or in arbitrary units suited to the particular case. Let g be the density; then the^ow; of mass may be thus expressed : — ^q = ^Au (2.) 121. The Principle of Continuity, as applied to fluids of varying density, takes the following form : — the flow into or out of any fixed space of constant volume is that due to the variation of density alone. To express this symbolically, let there be a fixed space of the constant volume V, and in a given interval of time let the density of the fluid in it, which in the first place may be supposed uniform at each instant, change from ^^ to ^2- Then the mass of fluid which at the beginning of the interval occupied the volume V, occupies Y o at the end of the interval the volume — ^ : and the difference of those volumes is the volume which flows through the surface bounding the space, outward if 53 is less than ei> inward if es is greater than ^j. Let t^ - t-^ be the length of the interval of time ', then the rate of flow of volume is expressed as follows : — Q= r\ ' (!■) 'uKlVKUSlTY OF PAET IL THEORY OF MECHANISM. CHAPTER I. DEFINITIONS AND GENERAL PRINCIPLES. 122. Theory of Pure Mechanism J) eMed.— Machines are bodies, or assemblages of bodies, which transmit and modify motion and force. The word " machine," in its widest sense, may be applied to every material substance and system, and to the material uni- verse itself; but it is usually restricted to works of human art, and in that restricted sense it is employed in this treatise. A machine transmits and modifies motion when it is the means of making one motion cause another ; as when the mechanism of a clock is the means of making the descent of the weight cause the rotation of the hands. A machine transmits and modifies force when it is the means of making a given kind of physical energy perform a given kind of work; as when the furnace, boiler, water, and mechanism of a marine steam engine are the means of making the energy of the chemical combination of fuel with oxygen perform the work of overcoming the resistance of water to the motion of a ship. The acts of transmitting and modifying motion, and of transmitting and modifying force, take place together, and are connected by a cer- tain law j and until lately, they were always considered together in treatises on mechanics ; but recently great advantage in point of clearness has been gained by first considering separately the act of transmitting and modifying motion. The principles which re- gulate this function of machines constitute a branch of Cinematics, called the theory of pure mechanism. The principles of the theory of pure mechanism having been first established and understood, those of the theory of the work of machines, which will form the subject of Part YI. of this work, which regulate the act of trans- mitting and modifying force, are much more readily demonstrated and apprehended than when the two departments of the theory of machines are mingled. The establishment of the theory of pui-e mechanism as an independent subject has been mainly ac- MOVING pieces; connectors; bearings. 71 complished by the labours of Professor Willis, whose nomenclature and metliods are, to a great extent, followed in this treatise. 123. The General Problem of the tlieory of pure mechanism may be stated as follows : — Given the jnode of connection of two or more movable points or bodies with each other, and with certain fixed bodies; required the comparative motions of the movable points or bodies : and conversely, v^hen the comparative motions of two or more movable points are given, to find their proper mode of connection. The term " comparative motion " is to be understood as in Articles 70, 81, 93, and 107. In those Articles, the comparative motions of points belonging to one body have already been consid- ered. In order to constitute mechanism, two or more bodies must be so connected that their motions depend on each other through cinematical principles alone. 124. Frame; Moving Pieces; Connectors; Bearings. — The/ram^ of a machine is a structure which supports the moving pieces, and regulates the path or kind of motion of most of them directly. In considering the movements of machines mathematically, the frame is considered as fixed, and the motions of the moving pieces are referred to it. The frame itself may have (as in the case of a ship or of a locomotive engine) a motion relatively to the earth, and in that case the motions of the moving pieces relatively to the earth are the resultants of their motions relatively to the frame, and of the motion of the frame relatively to the earth ; but in all problems of pure mechanism, and in many problems of the work of machines, the motion of the frame relatively to the earth does not require to be considered. The moving pieces maybe distinguished into primary and second- ary; the former being those which are directly carried by the frame, and the latter those which are carried by other moving jjieces. The motion of a secondary moving piece relatively to the frame is the resultant of its motion relatively to the primary piece which carries it, and of the motion of that primary piece relatively to the frame. Connectors are those secondary moving pieces, such as links, belts, cords, and chains, which transmit motion from one moving piece to another, when that transmission is not effected by imme- diate contact. Bearings are the surfaces of contact of primary moving pieces with the frame, and of secondary moving pieces with the pieces which carry them. Bearings guide the motions of the pieces which they support, and their figures depend on the nature of those motions. The bearings of a piece which has a motion of transla- /tion in a straight line, must have plane or cylindrical surfaces, exactly straight in the direction of motion. The bearings of rotat- 72 THEORY OF MECHANISM. ing pieces must have surfaces accurately turued to figures of revolu- tion, such as cylinders, spheres, couoids, and flat discs. The bearing of a piece whose motion is helictil, must be an exact screw, of a pitch equal to that of the helical motion (Article 96). Those parts of moving pieces which touch the bearings, should have surfaces accurately fitting those of the bearings. They may be distinguished into slides, for pieces which move in straight lines, gudgeons, journals, bushes, and. pivots, for those which rotate, and screws for those which move helically. 125. The Motions of Primary Moving Pieces are limited by the fact, that in order that different portions of a pair of bearing sur- faces may accurately fit each other during their relative motion, those surfaces must be either straight, circular, or helical ; from which it follows, that the motions in question can be of three kinds only, viz. : — I. Straight translation, or shifting, which is necessarily of limited extent, and which, if the motion of the machine is of indefinite duration, must be reciprocating ; that is to say, must take place alternately in opposite directions. (See Part I., Chapter II., Section 1.) II. Simple Quotation, or turning about a fixed axis, which motion may be either continuous or reciprocating, being called in the latter case oscillation. (See Part I., Chapter II., Section 2.) III. Helical or screw-like motion, to which the same remarks apply as to straight translation. (See Part I., Chapter IL, Section 3, Article 96.) 126. The Motions of Secondary Moving Pieces relatively to the pieces which carry them, are limited by the same principles which apply to the motions of primary pieces relatively to the frame. But the motions of secondary moving pieces relatively to the frame may be any motions which can be compounded of straight translations and simple rotations according to the principles already explained in Part I., Chapter TI., Section 3. 127. An Elementary Combination in mechanism consists of a pair of primary moving pieces, so connected that one transmits motion to the other. The piece whose motion is the cause is called the driver ; that whose motion is the effect, the follower. The connection between the driver and the follower may be — I. By 7'olling contact of their surfaces, as in toothless wheels. II. By sliding contact of their surfaces, as in toothed wheels, screws, wedges, cams, and escapements. III. By bands or wrapping connectors, such as belts, cords, and gearing-chains. lY. By link-work, such as connecting rods, universal joints, and clicks. AGGREGATE COMBINATIONS. 73 "V. By reduplication of cords, as in the case of ropes and pulleys. VI. By an intervening fluid, transmitting motion between two pistons. The various cases of the transmission of motion from a driver to a follower are further classified, according as the relation between their directions of motion is constant or changeable, and according as the ratio of their velocities is constant or variable. This latter principle of classification was employed by Professor Willis, in the first edition of his Principles of Mechanism, as the foundation of a ])rimary division of the subject of elementary combinations in mechanism into classes, which are subdivided according to the mode of connection of the pieces. In the present treatise, elemen- tary combinations will be classed ymmarily according to the mode of connection ; which is the classification employed by Professor Willis in the Edition of 1870. 128. Line of Connection. — In every class of elementary combina- tions, except those in which the connection is made by reduplica- tion of cords, or by an intervening fluid, there is at each instant a certain straight line, called the line of connection, or line of mutual action of the driver and follower. In the case of rolling contact, this is any straight line whatsoever traversing the point of contact of the surfaces of the pieces; in the case of sliding contact, it is a line perpendicular to those surfaces at their point of contact; in the case of wrapping connectors, it is the centre line of that part of the connector by whose tension the motion is transmitted; in the case of link-work, it is the straight line passing through the points of attachment of the link to the driver and follower. 129. Principle of Connection. — The line of connection of the driver and follower at any instant being known, their comparative velocities are determined by the following principle : — The respec- tive linear velocities of a point in the driver, and a point in the fol- lower, each situated anywhere in the line of connection, are to each other inversely as the cosines of the respective angles made hy the paths of the points with the line of connection. This principle might be other- wise stated as follows : — The components, along the line of connec- tion, of the velocities of any two points situated in that line, are equal. 130. Adjustments of Speed. — The velocity-ratio of a driver and its follower is sometimes made capable of being changed at will, by means of apparatus for varying the position of their line of con- nection, as when a pair of rotating cones are embraced by a belt which can be shifted so as to connect portions of their surfaces of different diameters. 131. A Train of Mechanism consists of a series of moving pieces, each of which is follower to that which drives it, and driver to that which follows it. 132. Agregate Combinations in mechanism are those by which compound motions are given to secondary pieces. 7i CHAPTER 11. OX ELEMENTAKY COMBINATIONS AND TRAINS OF MECHANISM. Section 1. — Rolling Contact. 133. Pitch Surfaces are those surfaces of a pair of moving pieces, which touch each other when motion is communicated by rolling contact. The line of contact is that line which at each instant traverses all the pairs of points of the pair of pitch surfaces which are in contact. 134. Smooth Wheels, Rollers, Smooth Racks. — Of a pair of pri- mary moving pieces in rolling contact, both may rotate, or one may rotate and the other have a motion of sliding, or straight transla- tion. A rotating piece, in rolling contact, is called a smooth wheel, and sometimes a roller; a sliding piece may be called a smooth rack. 13o. General Conditions of Rolling Contact. — The whole of the principles which regulate the motions of a pair of pieces in rolling contact follow from the single principle, that each pair of points in the pitch surfaces, which are in contact at a given instant, must at that instant he moving in the same direction tvith the same velocity; that this must be the case is evident from the rigidity of the bodies, for did the pair of points vary in velocity, it would follow that there was motion among the particles, or in a particle at least, of the body, which is contrary to the hypothesis of rigidity. The direction of motion of a point in a rotating body being per- pendicular to a plane passing through its axis, the condition, that each pair of points in contact with each other must move in the same direction leads to the following consequences: — I. That when both pieces rotate, their axes, and all their points of contact, lie in the same plane. II. That when one piece rotates and the other slides, the axis of the rotating piece, and all the points of contact, lie in a plane per- pendicular to the direction of motion of the sliding piece. The condition, that the velocities of each pair of points of con- tact must be equal, leads to the following consequences : — III. That the angular velocities of a pair of wheels, in rolling contact, must be inversely as the perpendicular distances of any pair of points of contact from the respective axes. TV. That the linear velocity of a smooth rack in rolling contact with a wheel, is equal to the product of the angular velocity of the A STRAIGHT RACK AND CIRCULAR WHEEL. 75 wheel by the perpendicular distance from its axis to a pair of points of contact. Respecting the line of contact, the above principles III. and IV. lead to the following conclusions : — Y. That for a pair of wheels with parallel axes, and for a wheel and rack, the line of contact is straight, and parallel to the axes or axis; and hence that the pitch surfaces are either plane or cylin- drical (the term "cylindrical" including all surfaces generated by the motion of a straight line parallel to itself). YI. That for a pair of wheels, with intersecting axes, the line of contact is also straight, and traverses the point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term " conical" including all surfaces generated by the motion of a straight line which traverses a fixed point). 136. Circular Cylindrical Wheels are employed when an uniform velocity-ratio is to be communicated between parallel axes. Figs. 38, 39, and 40, of Article 102, may be taken to represent pairs of such wheels ; C and O, in each figure, being the parallel axes of the wheels, and T a point in their line of contact. In fig. 38, both pitch surfaces are convex, the wheels are said to be in outside gearing^ and their directions of rotation are contrary. In figs. 39 and 40, the pitch surface of the larger wheel is concave, and that of the smaller convex ; they are said to be in inside gearing, and their directions of rotation are the same. To represent the comparative motions of such pairs of wheels symbolically, let OT-ri, CT-r^. be their radii : let O C = c be the line of centres, or perpendicular ) that for I gearing, c = ri±r2 (1.) distance between the axes, so that for outside inside Let a^, «2» be the angular velocities of the wheels, and v the common linear velocity of their pitch surfaces ; then •y = «! : r. : ^2 ± «i : «2 • ^i'} } " the sign ± applying to | ^^^^^^^ \ gearing. 137. A Straight Rack and Circular Wheel, which are used when an uniform velocity-ratio is to be communicated between a sliding piece and a turning piece, may be represented by fig. 36 of Article 99, C being the axis of the wheel, FTP the plane surface of the rack, and T a point in their line of contact. Let r be the radius ro THEORY OF MECHANISM. of the wlieel, a its angular velocity, and v tlie linear velocity of tlie rack; then v-r a. 138. Bevel Wheels, whose pitch surfaces are frustra of regular cones are used to transmit an uniform angular velocity-ratio between a pair of axes which intersect each other. Fig. 45 of Article 105 will serve to illustrate this case; O A and O C being the pair of axes, intersecting each other in O, O T the line of con- tact, and the cones described by the revolution of O T about O A and O C respectively being the pitch surfaces, of which narrow zones or frustra are used in i)ractice. Let Oj, (Xg, be the angular velocities about the two axes respec- tively; and let ii = ZAOT, i2 = Z.C0T, be the angles made by those axes respectively with the line of contact ; then from. ' the |)rinciple III. of Article 135 it follows, that the angular velocity- ratio is a.^ _ sin ^l ,^ . ' — '• " • } • \ ') % sm ?2 Which equation serves to find the angular velocity-ratio when the axes and the line of contact are given. Conversely, let the angle between the axes, Z A O C = % + *2 = J, be given, and also the ratio ~ ; then the position of the line of contact is given by either of the two following equations : — CTo sin / ~^ sm ^1 == — TT-^ :r — r^ -, '} J{ai + ai^'Ia^a^ cosj) i ^^ • • _ ^1 sin^ ^ j Sill Co —- T~: — 5 o 2^^ ~~ ^ 5 I J(ai + a2 + 2 aj^a^ cos J) J which are formed from equation (1) by substituting for % its value = (j - i^, and for i^ its value — {j - Zj). As this is the first instance of the use of Trigonometrical analysis, the method of formation of these equations will be ex- jjlaiued : — From Equation (1) it follows that- — sin /i • % = sin ?^ * % = sin(/-*i)-«2 = sin^* • cos ij • ^2 - cosj • sin i^ • a^ = sin j • J{1 - sin^ ^\) • a.2 - cosj • sin i-y ' a^. (See Trigonometrical Rules, Sections 19 and 21.) Squaring both sides, and transposing SKEW-BEVEL WHEELS. 77 siij- J • sin- ii • «| + (sin i^' 0^ + cor J * sin ij • a^Y = sin- ^;-,^2 J '(^2 s2 n sin^ ij 6^2 - co&^j ' sin^ z^ • a| + sm'' z^ • aj; + cos-^ • sm=^ ^^ • al + 2 sin 'ij • CTj • cos J • % — sin^J • al sin2 i\ • al + sin^ Zj ' al+2 sin ?*i • a^ • cos J • CTo = sin^^' • a?^ sin^ J • /TT? tto a£ + a| + 2 % • a^' cosj .*. sin CT2 ' sm ^ ^(^af + a| + 2 % • ^2 ' co«i)' o.Q Graphically, the same problem is solved as follows : — On the two axes respectively, take lengths to represent the angular velocities of their respective wheels. Complete the parallelogram of which those lengths are the sides, and its diagonal will be the line of contact. As in the case of the rolling cones of Article 106, one of a pair of bevel wheels may be a flat disc, or a concave cone. 139. Non-Circular Wheels are used to transmit a variable velocity-ratio between a pair of parallel axes. In fig. 46, let Cj, Co, represent the axes of such a pair of wheels ; Tj, T2, a pair of points which at a given instant touch each other in the line of contact (which line is parallel to the axes and in the same plane with them) ; and Ui, Ug, another pair of points, which touch each other at another instant of the motion ; and let the four points, T^, To, Ui, TJg, be in one plane perpendicular to the two axes, and to the line of contact. Then for every such set of four points, the two following equations must be fulfilled : — Fig. 4C. Cj C2 CiUi + C2U2 = CiTi + C2T2 arcTiUi^arcTsUa and those equations shew the geometrical relations which must exist between a pair of rotating surfaces in order that they may move in rollino: contact round fixed axes. Section 2. — Sliding Contact. 140. Skew-Bevel Wheels are employed to transmit an uniform Fin;. 47. Ficr. 48. THEORY OF MECHANISM Fior. 49. velocity-ratio between two axes which are neither parallel nor intersecting. The pitch surface of a skew-bevel wheel is a frustrum or zone of a hyperholoid of revolution. In fig. 47, a pair of large portions of such hyperboloids are shewn, rotat- ing about axes A B, C D. In fig. 48 are shewn a pair of narrow zones of the same figures, such as are employed in practice. A hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to whicli it is inclined at a constant angle. If two such hyperboloids, equal or unequal, be placed in the closest possible contact, as in fig. 47, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes A B, C D, in opposite directions. The axes will neither be parallel, nor will they intei-sect each other. The motion of two such hyperboloids, rotating in contact with each other, has some^times been classed amongst cases of rolling contact; but that classification is not strictly correct ; for although the component velocities of a pair of points of contact in a direction at right angles to the line of contact are equal, still, as the axes are neither parallel to each other nor to the line of contact, the velo- cities of a pair of points of contact have components along the line of contact, which are unequal, and their difference constitutes a lateral sliding. The directions and positions of the axes being given, and the required angular velocity-ratio, —, it is required to find the ohli- quities of the generating line to the two axes, and its radii vectores, or least perpendicular distances from these axes. In fig. 49, let A B, C D, be the two axes, and G K their common perpendicular. On any plane normal to the common perpendicular G K h, draw n 6 II A B, c c? 11 C D, in which take lengths in the following i)ro- portions : — complete the parallelogram hpeq, and draw its diagonal ehf; the line of contact EHF will be parallel to that diagonal. From p let fall p m perpendicular to h e. Then divide the common perpendicular G K in the ratio given by the proportional equation PRINCIPLE OF SLIDING CONTACT. 73 he:em:mh::GK:Gil:KB.; then the two segaients thus found will be the least distances of the line of contact from the axes. The first pitch surface is generated by the rotation of the line E H F about the axis A B with the radius vector G H = rj ; the second, by the rotation of the same line about the axis C T> with the radius vector H K = rg. To draw the hyperbola* which is the longitudinal section of a skew-bevel wheel whose generating line has a given radius vector and obliquity, let A G B, fig. 50, re- present the axis, G H JL A G B, the radius vector of the generating line, and let the straight line E G F make with the axis an angle equal to the obliquity of the generating line. H will be the vertex, and E G F one of the asymptotes,t of the required hyper- bola. To find any number of yjoints in that hyperbola, proceed as follows :- ^(G H2 + X W2). -Draw X W Y parallel to G H, cutting G E in W, and make X~^ Then will Y be a point in the hyperbola. 141. Principle of Sliding Contact. — The line of action, or of con- nection, in the case of sliding contact of two moving pieces, is the common perpendicular to their surfaces at the point where they touch; and the principle of their comparative motion is, that the components, along that perpendicular, of tJie velocities of any two points traversed hy it, are equal. Case 1. Two shifting pieces, in sliding contact, have linear velo- cities proportional to the secants of the angles which their directions of motion make with their line of action. Case 2. Two rotating pieces, in sliding contact, have angular velocities inversely proportional to the perpendicular distances from their axes of rotation to their line of action, each multiplied by the sine of the angle which the line of action makes with the particular axis on which the perpendicular is let fall. In fig. 51, let Ci, C2, represent the axes of rotation of the two pieces; Ai, A2, two portions of their respective surfaces; and Tj, Ta, a pair of points in those surfaces, which, at the instant under consideration, are in contact with each other. Let P^ Pa be the common perpendicular of the surfaces at the pair of points Ti, Ta; * The Hyperbola is the curve traced out by a point which moves in such a manner that its distance from a given fixed point (I), continually bears the same ratio greater than unity to its distance from a given fixed line (A B). t An ^ symptote is a straight line whose distance from a curve diminishes as the curve extends away from the origin. 80 THEORY OF MECHANIS3I. that is, the line of action; and let Ci Pi, C^ P2, be the common per- pendiculars of the line of action and of the two axes respectively. Then at the given instant, the components along the line Pi Pg of the velocities of the points Pi, P2, are equal. Let ?i, ^2, be the angles which that line makes with the direc- tions of the axes respectively. Let a^, a^, be the respective angular velocities of the moving pieces; then «i • Ci Pi • sin 2i = ^2 C2 Pa • sin 4; consequently, FicT. 51. a2 _ Ci Pi sin ^l «! Ca P2 sin 1*2 ^ ,(1.) which is the principle stated above. When the line of action is perpendicular in direction to both axes, then sin ix = sin ^2 = 1 ; and Equation 1 becomes (U.) ^2 _ ^1 Pi CIi Ca Pa When the axes are parallel^ ?i = ^2- Let I be the point where the line of action cuts the plane of the two axes; then the triangles Pi Ci I, P2 Ca I, are similar; so that Equation 1 a is equivalent to the following : — I Ca .(IB.) Case 3. A rotating piece and a shifting piece, in sliding contact, have their comparative motion regulated by the following prin- ciple: — Let C P denote the perpendicular distance from the axis of the rotating piece to the line of action; i the angle which the direc- tion of the line of action makes with that axis; a the angular velocity of the rotating piece; v the linear velocity of the sliding piece; ^ the angle which its direction of motion makes w^ith the line of action ; then v^a' C P • ^\ni- ^QG j (2.) AYhen the line of action is perpendicular in direction to the axis of the rotating piece, sin ■i = 1 ; and 'y = a-CP-seci = a'rC; (2a.) where I C denotes the distance from the axis of the rotating piece to the point where the line of action cuts a perpendicular from that axis on the direction of motion of the shifting piece. PITCH AND NUMBER OF TEETH. 81 142. Teeth of Wheels. — The most usual method of communi- cating motion between a pair of wheels, or a wheel and a rack, and the only method which, by preventing the possibility of the rotation of one wheel unless accompanied by the other, insures the preservation of a given velocity-ratio exactlj^, is by means of the projections called teeth. The pitch surface of a wheel is an ideal smooth surface, inter- mediate between the crests of the teeth and the bottoms of the spaces between them, which, by rolling contact with the pitch sur- face of another wheel, would communicate the same velocity-ratio that the teeth communicate by their sliding contact. In designing wheels, the forms of the ideal pitch surfaces are first determined, and from them are deduced the forms of the teeth. Wheels with cylindrical pitch surfaces are called spur wheels; those with conical pitch surfaces, hevel wheels; and those with hyperboloidal pitch surfaces, skew-bevel wheels. The pitch line of a wheel, or, in circular wheels, the pitch circle, is a transverse section of the pitch surface made by a surface per- pendicular to it and to the axis; that is, in spur wheels, by a plane i)erpendicular to the axis; in bevel wheels, by a sphere described about the apex of the conical pitch surface; and in skew-bevel wheels, by any oblate spheroid generated by the rotation of an ellipse whose foci are the same with those of the hyperbola that generates the pitch surface. The pitch point of a pair of wheels is the point of contact of their pitch lines; that is, the transverse section of the line of contact of the pitch surfaces. Similar terms are applied to racks. That part of the acting surface of a tooth which projects beyond the pitch surface is called the face; that which lies within the pitch surface, the Jlank. The radius of the pitch circle of a circular wheel is called the geometrical radius; that of a circle touching the crests of the teeth is called the real radius; and the difference between those radii, the addendum. 143. Pitch and Number of Teeth.— The distance, measured along the pitch line, from the face of one tooth to the face of the next, is called the pitch. The pitch, and the number of teeth in circular wheels, are regu- lated by the following principles : — I. In wheels which rotate continuously for one revolution or more, it is obviously necessary t/iat the pitch should be an aliquot part of the circumference. In wheels which reciprocate without performing a complete revolution, this condition is not necessary. Such wheels are called sectors. 82 THEORY OF MECHANISM. II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essential that the pitch should be the same in each. III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii, and inversely as the angular velocities. IV. Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth, and its reciprocal, the angular velocity -ratio, must be expressible in whole numbers. V. Let n, N, be the respective numbers of teeth in a pair of wheels, N being the greater. Let t, T, be a pair of teeth in the smaller and larger wheel respectively, which at a particular instant work together. It is required to find, first, how many pairs of teeth must pass the line of contact of the pitch surfaces before t and T work together again (let this number be called a); secondly, with how many different teeth of the larger wheel the tooth t will work at different times (let this number be called h) ; and thirdly, with how many different teeth of the smaller wheel the tooth T will work at different times (let this be called c). Case 1. If n is a divisor of N, a = Nj 6 = ^; c-1 (1.) Case 2. If the greatest common divisor of N and nh^d ix number less than w, so that n — md, N = M c?, then a = mN = M7i = Mmc?j 5 = M; c = m (2.) Case 3. If N and n be prime to each other, a = Nw; 6^N; c = n (3.) It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They, there- fore, study to make the numbers of teeth in each pair of wheels which work together such as to be either prime to each other, or to have their greatest common divisor as small as is possible con- sistently with the purposes of the machine. YI. The smallest number of teeth which it is practicable to give to a pinion (that is, a small wheel), is regulated by the principle, that in order that the communication of motion from one wheel to another may be continuous, at least one pair of teeth should always be in action; and that in order to provide for the contingency of a tooth breaking, a second pair, at least, should be in action also. For reasons which will appear when the forms of teeth are con- sidered, this principle gives the following as the least numbers of A TRAIN OF WHEELWORK. 83 teetli which can be usually employed in pinions having teeth of the three classes of figures named below, whose properties will be explained in the sequel : — I. Involute teeth, 25. II. Epicycloidal teeth, 12. III. Cylindrical teeth, or staves, 6. 144. Hunting Cog. — When the ratio of the angular veloctios of two wheels, being reduced to its least terras, is expressed by small numbers, less than those which can be given to wheels in practice, and it becomes necessary to employ multiples of those numbers by a common multiplier, which becomes a common divisor of the numbers of teeth in the wheels, millwrights and engine-makers avoid the evil of frequent contact between the same pairs of teeth, by giving one additional tooth, called a hunting cog, to the larger of the two wheels. This expedient causes the velocity- ratio to be not exactly but only approximately equal to that which was at first contemplated; and therefore it cannot be used where the exactness of certain velocity-ratios amongst the wheels is of impor- tance as in clockwork. 145. A Train of Wheelwork consists of a series of axes, each having upon it two wheels, one of which is driven by a wheel on the preceding axis, while the other drives a wheel on the following axis. If the wheels are all in outside gearing, the direction of rotation of each axis is contrary to that of the adjoining axes. In some cases, a single wheel upon one axis answers the purpose both of receiving motion from a wheel on the preceding axis and giving motion to a wheel on the following axis. Such a wheel is called an idle wheel: it affects the direction of rotation only, and not the velocity-ratio. Let the series of axes be distinguished by numbers 1, 2, 3, If Y\ is not greater than 6, and C lies between the prescribed minimum number of teeth (which, may be called ^), and its double 2 1, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required. Should B and C be inconveniently large, they are if possible to be resolved into factors, and those factors, or if they are too small multiples of them, used for the numbers of teeth. Should B or C, or both, be at once inconveniently large, and prime, then instead of the exact ratio ^, some ratio approximating to that ratio, and capable of resolu- \j tion into convenient factors, is to be found by the method of continued fractions. See Mathematical Introduction, page 2, Article 4. Should p, be greater than 6, the best number of elementary combinations is found by dividing by 6 again and again till a quotient is obtained less than unity, when the number of divisions will be the required number of combinations, m— 1. Then, if possible, B and C themselves are to be resolved each into m - 1 factors, which factors, or multiples of them, shall be not less than t, nor greater than 6^; or if B and C contain incon- veniently large prime factors, an approximate velocity-ratio, found T> by the method of continued fractions, is to be substituted for ^, as before. When the prime factors of either B or C are fewer in A TRAIN OF WHEELWORK. 85 number than m- 1, the required number of factors is to be made up by inserting 1 as often as may be necessary. In multiplying factors that are too small to serve for numbers of teeth, prime numbers differing from those already amongst the factors are to be preferred as multipliers; and in general, where two or more factors require to be multiplied, different prime numbers should be used for the different factors. So far as the resultant velocity-ratio is concerned, the order of the drivers N, and of the followers n^ is immaterial ; but to secure equable wear of the teeth, as explained in Article 143, Principle V., the wheels ought to be so arranged that for each elementary com- bination the greatest common divisor of N and n shall be either 1, or as small as possible; and if the preceding rules have beeu observed in the choice of multipliers, this will be insured by so placing each driving wheel that it shall work with a following wheel whose number of teeth does not coutain any of the same multipliers; for the original numbers B and C contain no common factor except 1. The following is an example of a case requiring the use of additional multipliers : — Let the required velocity-ratio, in its least terms, be B_360 C~ 7 • To get a quotient less than 1, this ratio must be divided by 6 three times, therefore m - 1 = 3. The prime factors of 360 are 2*2-2-3-3-5; these may be combined so as to make three factors in various different ways; and the preference is to be given to that which makes these factors least unequal, viz., 5 • 8 • 9. Hence, resolving numerator and denominator into three factors each, we have B_5-89 C ~ 1 • 1 • 7* It is next necessary to multiply the factors of the numerator and denominator by a set of three multipliers. Suppose that the wheek to be used are of such a class that the smallest pinion has 12 teeth, then those multipliers must be such that none of their products by the existing factors shall be less than 12; and for reasons already given, it is advisable that they should be different prime numbers. Take the prime numbers, 2, 13, 17 (2 being taken to multiply 7); then the numbers of teeth in the followers will be 13x1^:13; 17x1 = 17; 2x7 = 14. In distributing the multipliers amongst the factors of the num- erator, let the smallest multiplier be combined with the largest factor, and so on; then we have 83 THEORY OF MECHANISM. 17x5 = 85; 13x8 = 104; 2x9 = 18. Finally, in combining the drivers with the followers, those numbers are to be combined which have no common factor; the result being the following train of wheels : — 85 ^ 18^ 104 _ 360 14*13* 17 "T* 146. Teeth of Spur-Wheels and Racks. General Principle. — The figures of the teeth of wheels are regulated by the principle, that the teeth of a pair of wheels shall give the same velocity-ratio by their sliding contact, lohich the ideal smooth pitch surfaces would give hy their rolling contact. Let B^, Bq, in fig. 51, be parts of the pitch lines (that is, of cross sections of the pitch surfaces) of a pair of wheels with parallel axes, and I the pitch point (that is, a section of the line of contact). Then the angular velocities which would be given to the wheels by the rolling contact of those pitch lines are inversely as the segments I Ci, I Co, of the line of centres; and this also is the proportion of the angular velocities given by a pair of surfaces in sliding contact whose line of action traverses the point I (Article 141, Case 2, Equation 1 b). Hence the condition of correct working for the teeth of wheels with parallel axes is, that the line of action of the teeth shall at every instant traverse the line of contact of the pitch surfaces; and the same condition obviously applies to a rack sliding in a direction perpendicular to that of the axis of the wheel with which it works. 147. Teeth Described by Rolling Curves. — From the principle of the preceding Article it follows, that at every instant, the position of the point of contact T^ in the cross section of the acting surface of a tooth (such as the line Aj T^ in fig. 51), and the corresponding position of the pitch point I in the pitch line I B^ of the wheel to which that tooth belongs, are so related, that the line I T^ which joins them is normal to the outline of the tooth Aj Tj at the point Tj. Now, this is the relation which exists between tlie tracing- point Tj, and the instantaneous axis or line of contact I, in a rolling curve of such a figure, that being rolled upon the pitch surface Bj, its tracing-point Tj traces the outline of the tooth. (As to rolling curves, see Articles 100, 101, 103, and 106). In order that a pair of teeth may work correctly together, it is necessary and sufficient that the instantaneous radii vectores from the pitch point to the points of contact of the two teeth should coincide at each instant, as expressed by the equation rT, = rT,; (1.) and this condition is fulfilled if the outlines of the two teeth be traced by the motion of the same tracing-point, in rolling the same rolling curve on the saine side of the pitch surfaces of the respective icheels. TKE SLIDING OF A PAIR OF TEETH ON EACH OTHER. 87 T\\Q flank of a tooth is traced while the rolling curve rolls inside of the pitch line ; the face, while it rolls outside. Hence it is evident that the flanks of the teeth of the driving wheel drive the faces of the teeth of the driven wheel; and that the faces of the teeth of the driving wheel drive the flanks of the teeth of the driven wheel. The former takes place while the point of contact of the teetli is approachiyig the pitch point, as in fig. 51, supjDipsing the motion to be from Pj towards P2 ; the latter, after the point of contact has passed, and while it is receding from, the pitch point. The pitch point divides the path of the point of contact of the teeth into two parts, called the path of approach and the path of recess; and the lengths of those paths must be so adjusted, that two pairs of teeth at least shall be in action at each instant. It is evidently necessary that the surfaces of contact of a pair of teeth should either be both convex, or that if one is convex and the other concave, the concave surface should have the flatter curvature. 148. The Sliding of a Pair of Teeth on each Other, that is, their relative motion in a direction perpendicular to their line of action, is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C^ C2 to rotate backwards round Cj with the angular velocity a^, and the wheel 2 to rotate round Cg as before with the angular velocity a^ relatively to the line of centres Gi C2, so as to have the same motion as if its pitch surface rolled on the pitch surface of the first wheel. Thus the relative motion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the resultant motion given by the principles of Article 102; that is, a rotation about the instantaneous axis I with the angular velocity a^ + a^. Hence the velocity of sliding is that due to this rotation, about I, with the radius I T = r; that is to say, its value is r{ai + 02); (1.) so that it is greater, the farther the point of contact is from the line of centres; and at the instant when that point, passing the line of centres, coincides with the pitch point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of roll- ing contact. The roots of the teeth slide towards each other during the ap- proach, and from each other during the recess. To find the amount or total distance through which the sliding takes place, let ^1 be the time occupied by the approach, and t^ that occupied by the recess; then the distance of sliding is s= / ^r{ai+ a^ dt + / ^r^a^^a^ dt; (2.) •^0 -'0 or in another form, ii di denote an element of the change of angu- 88 THEORY OF MECHANISM. lar position of one wheel relatively to the other, ?\ the amount of that change during the approach, and i^ during the recess, then (ai + az) dt — di ; and ^rdi+ / 1 Wdi+ / rdi , (3.) -^ 149. The Arc of Contact on the Pitch Lines is the length of that portion of the pitch lines which passes the pitch point during the action of one pair of teeth ; and in order that two pairs of teeth at least may be in action at each instant, its length should be at least double of the pitch. It is divided into two parts, the arc of ap- proach and the arc of recess. In order that the teeth may be of length sufficient to give the required duration of contact, the dis- tance moved over by the point I upon the pitch line during the rolling of a rolling curve to describe the face and flank of a tooth, must be in all equal to the length of the required arc of contact. It is usual to make the arcs of approach and recess equal. 150. The Length of a Tooth may be divided into two parts, that of the face and that of the flank. For teeth in the driving wheel, the length of the flank depends on the arc of approach, — that of the face, on the arc of recess; for those in the following wheel, the length of the flank depends on the arc of recess, — that of the face, on the arc of approach. 151. Involute Teeth for Circular Wheels.— In fig. 52, let Cj, C^, be the centres of two circular wheels, whose pitch circles are B^, B2. Through the pitch point I draw the intended line of action Pj, Pg, making the angle C I P = ^ with the line of centres. Prom Cj, C2, draw ■(!•) CiPi^ICi'sin 0, C^^ = Ta,'sm0, perpendicular to P^ P2, with which two perpendiculars as radii, describe circles (called base circles) Dj, Dg. Suppose the base circles to be a pair of circular pulleys, connected by means of a cord whose course from jjulley to pulley is Pi I Pg. As the line of connection of those pulleys is the same with that of the proposed teeth, they will rotate with the required velocity-ratio. Now suppose a tracing-point T be fixed to the cord, so as to be carried along the path of contact P^ I Pg. That point will trace, on a plane rotating along with the wheel 1, part of the involute of the base circle Dj, and on a plane rotating along with the wheel 2, part of the involute Fig. 52. of the base circle D2, and the two curves so EPICYCLOIDAL TEETH. 89 traced will always toucli each other in the required point of contact T, and will therefore fulfil the condition required by Article 14G. All involute teeth of the same pitch work smoothly together. To find the length of the path of contact on either side of the pitch point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along P^ I P2, is less than the pitch in the ratio sin 6 : 1, for the former is proportional to r ' sin &, and the latter to r • &, and consequently that if dis- tances not less than the pitch x sin be marked off either way from I towards Pj and Pg respectively, as the extremities of the path of contact, and if the addendum circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice, it is usual to make the path of contact somewhat longer, viz., about 2^ times the pitch ; and with this length of path and the value of ^ which is usual in practice, viz., 75 J°, the addendum is about -^jj of the pitch. The teeth of a rack, to work correctly with wheels having invo- lute teeth, should have plane surfaces, perpendicular to the line of connection, and consequently making, with the direction of motion of the rack, angles equal to the before-mentioned angle 0. 152. The Smallest Pinion with Involute Teeth of a given pitch p, has its size fixed by the consideration that the path of contact of the flanks of the teeth, which must not be less than p - sin 6, cannot be greater than the distance along the line of action from I p the pitch point to the base circle, I P = r • cos 6. Then r = and substituting for I P its least possible value p * sin 6, hence the least radius is r=p tan 6; (1.) which, for ^ = 75 J°, gives for the radius r = 3-867/), and for the circumference of the pitch circle, p x 3-867 x 2 ^ = 24-3 p; to which the next greater integer multiple of jt?is 25^; and therefore twenty-jive^ as formerly stated, in Article 143, is the least number of involute teeth to be employed in a pinion. 153. Epicycloidal Teeth. — For tracing the figures of teeth, the most convenient rolling curve is the circle. The path of contact which a point in its circumference traces is identical with the circle itself; the flanks of the teeth are internal, and their faces external epicycloids, for wheels; and both flanks and faces are cycloids for a rack. Wheels of the same pitch, with epicycloidal teeth traced by the same rolling circle, all work correctly with each other, whatsoever may be the numbers of their teeth ; and they are said to belong to the same set. Por a pitch circle of twice the radius of the rolling or describing 90 THEORY OF MECHANISM. circle (as it is called), the internal epicycloid is a straight line, being in fact a diameter of the pitch circle ; so that the flanks of the teeth for such a pitch circle are planes radiating from the axis. For a smaller pitch circle, the flanks would be convex, and in- curved or under-cut, which would be inconvenient ; therefore the smallest wheel of a set should have its pitch circle of twice the radius of the describing circle, so that the flanks may be either straight or concave. In fig. 53, let B be part of the pitch circle of a wheel, C C the line of centres, I the pitch-point, E. the internal, and R' the equal external describing circles, so placed as to touch the pitch circle and each other at I ; let D I D' be the path of contact, consisting of the path of J^ approach D I, and the path of re- cess I D'. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch. The angle ^, on passing the line of centres, is 90°; the least value of that angle is 6 = /_01'D = A C I D'. It appears from experience that the least value of 6 should be about 60°; therefore the arcs D I = I D' should each be one-sixth of a circumference ; therefore the circumference of the describing circle should be six times the pitch. It follows that the smallest pinion of a set, in which pinion the flanks are straight, should have twelve teeth, as has been already stated in Article 143. 154. Teeth of Wheel and Trundle.— A trundle, as in fig. 54, has cylindrical pins called staves for teeth. The face of the teeth I'ig. 54. Fig. 55. of a wheel suitable for driving it, in outside gearing, are described by first tracing external epicycloids by rolling the pitch circle B^ of DIMENSIONS OF TEETH. 91 the trundle on the pitch circle B^ of the driving wheel, with the centre of a stave for a tracing point, as shewn by the dotted lines, and then drawing curves parallel to and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels. To drive a trundle in iiiside gearing, the outlines of the teeth of the wheel should be curves parallel to internal epicycloids. A peculiar case of this is represented in fig. 55, where the radius of the pitch circle of the trundle is exactly one-half of that of the pitch circle of the wheel; the trundle has three equi-distant staves; and the internal epicycloids described by their centres while the pitch circle of the trundle is rolling within that of the wheel, are three straight lines, diameters of the wheel, making angles of 60" with each other. Hence the surfaces of the teeth of the wheel form three straight grooves intersecting each other at the centre, each being of a breadth equal to the diameter of a stave of the trundle. 155. Dimensions of Teeth. — ^Toothed wheels being in general intended to rotate either way, the hacks of the teeth are made similar to the fronts. The space between two teeth, measured on the pitch circle, is made about one-fifth part wider than the thick- ness of the tooth on the pitch circle : that is to say, 5 thickness of tooth = r-T pitch, /» width of space = — pitch. ' The diflference of - of the pitch is called the hack-lash. The clearance allowed between the points of teeth and the bot- toms of the spaces between the teeth of the other wheel, is about one-tenth of tlie pitch. The thickness of a tooth is fixed according to the principles of strength ; and the hreadth is so adjusted, that when multiplied by pitch, the product shall contain one square inch for each 160 lbs. of force transmitted by the teeth. 156. The Teeth of a Bevel-Wheel have acting surfaces of the conical kind, generated by the motion of a line traversing the apex of the conical pitch surface, while a point in it is carried round the traces of the teeth upon a spherical surface described about that apex. The operations of describing the exact figures of the teeth of bevel-wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels, all those operations are to be performed on the surface of a sphere described 92 THEORY OF MECHANISM. about the apex, instead of oti a plane, substituting 2^oles for centres^ and great circles for straight lines. In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used : — Let 0, fig. 5Q, be the apex, and O C the axis of the pitch cone of a bevel-wheel; and let the largest pitch circle be that whose radius is C B. Perpendicular to O B draw B A cutting the axis produced in A, let the outer rim of the pattern and of the wheel be made a portion of the surface of the cone whose apex is A and side A B. The narrow zone of that cone thus employed will approach Fig. 56. sufficiently near to a zone of the sphere described about O with the radius O B, to be used in its stead. On a plane surface, with the radius A B, draw a circular arc B D ; a sector of that circle will represent a portion of the surface of the cone ABC developed, or spread out flat. Describe the figures of teeth of the required pitch, suited to the pitch circle B D, as if it were that of a spur-wheel of the radius A B; those figures will be the required cross sections of the teeth of the bevel-wheel, made by the conical zone whose apex is A. 157. The Teeth of Non-Circular Wheels are described by rolling circles or other curves on the pitch surfaces, like the teeth of cir- cular wheels; and when they are small compared with the wheels to which they belong, each tooth is nearly similar to the tooth of a circular wheel, having the same radius of curvature with the pitch surface of the actual wheel at the point where the tooth is situated. 158. A Cam or Wiper is a single tooth, either rotating continu- ously or oscillating, and driving a sliding or turning-piece, either constantly or at intervals. All the principles which have been stated in Article 141, as being applicable to sliding contact, are applicable to cams; but in designing cams, it is not usual to deter- mine or take into consideration the form of the ideal pitch surface which would give the same comparative motion by rolling contact that the cam gives by sliding contact. 159. Screws. Pitch. — The figure of a screw is that of a convex or concave cylinder with one or more helical projections called threads winding round it. Convex and concave screws are dis- tinguished technically by the respective names oi male and female, or external and internal; a short internal screw is called a nut ; and when a screw is not otherwise specified, external is understood. The relation between the advance and the rotation, which com- pose the motion of a screw working in contact with a fixed nut or NORMAL AND CIRCULAR PITCH. 93 helical guide, has already been demonstrated in Article 96, Equa- tion 1; and the same relation exists between the rotation of a screw about an axis fixed longitudinally relatively to the frame- work, and the advance of a nut in which that screw rotates, the nut being free to shift longitudinally, but not to turn. The advance of the nut in the latter case is in the direction opposite to that of the advance of the screw in the former case. A screw is called right-handed or left-handed, according as its advance in a fixed nut is accompanied by right-handed or left-handed rotation, I J^ A I when viewed by an observer yrom whom „ .. J-"''*^! l^^^^-l the advance takes place. Eig. 57 re- ; \ ^„^^^ r'-^.^S\'^ presents a right-handed screw, and fig. p\ \J[ K \ \^ i 58 a left-handed screw. ! i^,,.^^^^ r^^^^^^^l •' The 'pitch of a screw of one thread, -^--r"^ \ I ^ \.v. and the total pitch of a screw of any \<^\ \ ^J number of threads, is the pitch of the N I ' P helical motion of that screw, as ex- ' plained in Article 96, and is the dis- ^^S- ^7. Fig 58. tance (marked p in figs. 57 and 58) measured parallel to the axis of the screw, between the corresponding points in two consecutive turns of the same thread. In a screw of two or more threads, the distance measured parallel to the axis, between the corresponding points in two adjacent threads, may be called the divided pitch. 160. Noraial and Circular Pitch. — When the pitch of a screw is not otherwise specified, it is always understood to be measured parallel to the axis. But it is sometimes convenient for particular purposes to measure it in other directions; and for that purpose a cylindrical pitch surface is to be conceived as described about the axis of the screw, intermediate between the crests of the threads and the bottoms of the grooves between them. If a helix be now described upon the pitch cylinder, so as to cross each turn of each thread at right angles, the distance between two corresponding points on two successive turns of the same thread, measured along this normal helix, may be called the normal fitch; and when the screw has more than one thread, the normal pitch from thread to thread may be called the normal divided pitch. The distance from thread to thread measured on a circle described on the pitch cylinder, and called the pitch circle, may be called the circular pitch ; for a screw of one thread it is one circumference; for a screw of n threads one circumference The following set of formulae shew the relations amongst the differ- 94 THEORY OF MECHANISM. ent modes of measuring the pitch of a screw. The pitch, properly speaking, as originally defined, is distinguished as the axial intch, and is the same for all parts of the same screw : the normal and circular pitch depend on the radius of the pitch cylinder. Let r denote the radius of the pitch cylinder; n, the number of threads; z, the obliquity of the threads to the pitch circles, and of the normal helix to the axis; &=^°|"^°^'''^^'{dlridid pitch; ^ P„ V the normal < ?• ^ i \ • i -^ =p^ I I divided pitch; j^ci the circular pitch ; Then o Pc =Pa ' cotan i=pn' cosec i = ; 71/ 2 -Trr ' tan i Pa=Pn' sec l=p^'t'driil = Pn=Pc n '^'ttv sin Fig. 59 will make these formulae clear, in which the several lines are lettered to represent the pitches : the hypotenuse of the larger triangle is the linear development on the plane of the paper of one coil of the screw which, it will be remarked, = \/{pJ +Pc^)) Pn t^^e normal pitch is normal to this: it is also evident from the figure that with a constant axial pitch, the normal and radial or circumferential pitch, as well as the angle of obliquity of the threads to the pitch cylinders, vary with the radii of those cylinders. 161. Screw Gearing. — A pair of convex screws, each rotating about its axis, are used as an elementary combination, to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the screws, their threads must be parallel; and their line of connection is the common perpendicular to the acting surfaces of the threads at their point of contact. Hence the following principles: — I. If the screws are both right-handed or both left-handed, the angle between the directions of their axes is the sum of their obli- quities: — if one is right-handed and the other left-handed, that angle is the difierence of their obliquities, II. The normal pitch, for a screw of one thread, and the normal THE RELATIVE SLIDING OF A PAIR OF SCREWS. 95 divided pitch, for a screw of more than one thread, must be the same in each screw. III. The angular velocities of the screws are inversely as their number of threads. 162. The Wheel and Screw is an elementary combination of two screws, whose axes are at right angles to each other, both being right-handed or both left-handed. As the usual object of this combination is to produce a change of angular velocity in a ratio greater than can be obtained by any single pair of ordinary wheels, one of the screws is commonly wheel-like, being of large diameter and many- threaded, while the other is short and of few threads; and the angular velocities are inversely as the number of threads. Fis. GO. Fi^. 61. Fig. 60, represents a side view of this combination, and fig. 61 a cross section at right angles to the axis of the smaller screw. It has been shewn by Prof. Willis, that if each section of both screws be made by a plane perpendicular to the axis of the large screw or wheel, the outlines of the threads of the larger and smaller screw should be those of the teeth of a wheel and rack respectively : B^ Bj, in fig. 60 for example, being the pitch circle of the wheel, and B^ Bg the pitch line of the rack. The periphery and teeth of the wheel are usually hollowed to fit the screw, as shewn at T, fig. 61. To make the teeth or threads of a pair of screws fit correctly and work smoothly, a hardened steel screw is made of the figure of the smaller screw, with its thread or threads notched so as to form a cutting tool ; the larger screw, or wheel, is cast approximately of the required figure ; the larger screw and the steel screw are fitted lip in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure. 163. The Relative Sliding of a Pair of Screws at their point of contact is found thus : — Let r-^, r^, be the radii of their pitch cylin- ders, and «i, u, the obliquities of their threa'^s to their pitch circles, one of w^hich is to be considered as negative if the screws are con- trary-handed. Let u be the common component of the velocities of a pair of points of contact along a line touching the pitch sur- 96 THEORY OF MECHANISM. faces and perpendicular to the threads at the pitch point, and v the velocity of sliding of the threads over each other, where v may be considered to be made up of the algebraical sum of two quantities, Vi and V2, which act perpendicularly to u, and whose values are ^1 = % ^1 cos ii, and V2 = a^ r^ cos 1*2 the sum or difference being taken as the screws are similar or contrary-handed. Then so that and w = «! r^ • sin ii = ^2 ^2 * sin h l u u tti = -. — - : CTo = : — - ; ri • sin ^1 7-2 • sm i.^ ' .(1.) v = a^ rj • cos i-^ + ag ^2 * cos i^ = u (cotan i^ + cotan i^ (2.) ' When the screws are contrary-handed, the difference instead of the sum of the terms in Equation 2 is to be taken. 164. Oldham's Coupling.— A coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction, with the same mean angular velocity If the axes of the shafts are in the same straight line, the coupling consists in so connecting their contiguous ends that they shall rotate as one piece; but if the axes are not in the same straight line, combinations of mechanism are re- quired. A coupling for parallel shafts ■pj^ 02 which acts by sliding contact was invented "'* * by Oldham, and is represented in fig. 62. Cj, C2, are the axes of the two parallel shafts; Dj, Dg, two cross- heads, facing each other, fixed on the ends of the two shafts respectively; E^, E^, a bar, sliding in a diametral groove in the face of Di; Eo, Eg, a bar, sliding in a diametral groove in the face of T>^; those bars are fixed together at A, so as to form a rigid cross. The angular velocities of the two shafts and of the cross are all equal at every instant. The middle point of the cross, at A, revolves in the dotted circle described upon the line of centres Cj, Cg, as a diameter, twice for each turn of the shafts and cross; the instantaneous axis of rotation of the cross, at any instant, is at I, the point in the circle Cj O2, diametrically oppo- site to A. Oldham's coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practicability or jjermanency of their exact continuity. PRINCIPLE OF CONNECTION BY BANDS. 97 Section 3. — Connection by Bands. 165. Bands Classed. — Bands, or wrapping connectors, for com- municating motion between pulleys or drums rotating about fixed axes, or between rotating pulleys and drums and shifting pieces, may be thus classed : — I. Belts, which are made of leather or of gutta percha, are flat and thin, and require nearly cylindrical pulleys. A belt tends to move towards that part of a pulley whose radius is greatest; pulleys for belts therefore, are slightly swelled in the middle, in order that the belt may remain on the pulley unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one pulley on to another of equal size alongside of it, by pressing against that part of the belt which is moving towards the pulley. II. Cords, made of catgut, hempen or other fibres, or wire, are nearly cylindrical in section, and require either drums with ledges, or grooved pulleys. , III. Chains, which are composed of links or bars jointed together, require pulleys or drums, grooved, notched, and toothed, so as to fit the links of the chains. Bands for communicating continuous motion are endless. Bands for communicating reciprocating motion have usually their ends made fast to the pulleys or drums which they connect, and which in this case may be sectors. 166. Principle of Connection by Bands. — The line of connection of a pair of pulleys or drums connected by means of a band, is the central line or axis of that part of the band whose tension transmits the motion. The principle of Article 129 being applied to this case, leads to the following consequences : — I. For a pair of rotating pieces, let rj, rg, be the perpendiculars let fall from their axes on the centre line of the band, ctj, a^, their angular velocities, and i-^, i^, the angles which the centre line of the band makes with the two axes respectively. Then the longitudinal velocity of the band, that is, its component velocity in the direction of its own centre line, is u = r^ Oj sin ir^ — r^a^ sin i^; (1.) whence the angular velocity-ratio is O'^ _ ^1 sin ^l % ra sin 4 When the axes are parallel (which is almost always the case), {j and the longitudinal velocity of the band, j the angle between the direction of the centre line of the band and that of the motion of the sliding piece, and v the velocity of the sliding piece; then u = r a sin i — v cos j; (4) for r sin i is the projection on the plane of motion of r, and u the longitudinal velocity of the band must necessarily be equal to V cos J, the longitudinal velocity of the sliding piece owing to the rigidity of the band; and (5.) COS J "When the centre line of the band is parallel to the direction of motion of the sliding piece, and perpendicular to the direction of the axis of the rotating piece, sin i (90°) = cos^' (0°) = 1, and v = u = T a (6.) 167. The Pitch Surface of a Pulley or Drum is a surface to which the line of connection is always a tangent; that is to say, it is a surface parallel to the acting surface of the pulley or drum, and distant from it by half the thickness of the band. 168. Circular Pulleys and Drums are used to commuicate a Fig. 64. constant velocity-ratio. In each of them, the length denoted by r in the equations of Article 166 is constant, and is called the effec- tive radivs, being equal to the real radius of the pulley or drum added to half the thickness of the band. A crossed belt connecting a pair of circular pulleys, as in fig. 63, reverses the direction of rotation; an open belt, as in fig. 64, pre- serves that direction. THE LENGTH OF AN ENDLESS BELT. 99 169. The length of an Endless Belt^onnectiQg a pair of pulleys whose effective radii are C^ j_ = r^, O^T^^r^, with parallel axes whose distance apart is Ci C^ = c, is given by formulae founded on Equation 1 of Article 113, viz., L = 2•s+2•r^. Each of the two equal straight parts of the belt is evidently of the length s = Ti Ta = /c2 _ (r, + r.Y ^^^ ^ crossed belt j " ,(1.) s = Ti Ta = Jc^ - ir^ - r^Y for an open belt ■] 7*1 being the greater radius, and r^ the less. Let iy be the arc to radius unity of the greater pulley, and 4 that of the less pulley, with which the belt is in contact; then for a crossed belt ^ + 2 arc • sm -\ ; for the angle Vj C^ Wj at the centre is double of the angle at the circumference Cj Tj Wj, and this is equal to the angle Si Cg Ci as they both differ from a right angle by the same angle Ti C^ V^; and for an open belt, K2.) 1 = I TT + 2 arc • sin 1 ; i^=\is-'2 arc * sm I ; and the introduction of those values into Equation 1 of Article 113 gives the following results : — For a crossed belt L = 2 jG^-ir^ + r,f+{r^ + r,) • (x + 2 arc -sin ''i^) ; " and if similar reasoning be applied, it may be shewn that i /q \ for an open belt, f \ v L = 2 ^0^ - (rj - r^)- + ^(r^ + r^ + 2{r^ - r^ • arc • sin • ~ c As the last of these equations would be troublesome to employ in a practical application to be mentioned in the next Article, an approximation to it, sufficiently close for practical purposes, is obtained by considering, that if r-^ - r^ is small compared with c, /"9 7 \? (^1 - '^2)^ 1 1 . '''] - ^2 ^1 - '^'2 ^c- - (r^ - r^Y = c- - — ^ — - nearly, and arc • sin • =■ -^— — - nearly; whence, for an open belt, L nearly = 2c + ^(r, + r,) + ^^^^^ (3a.) 100 THEORY OF MECHANISM. 170. Speed-Cones (figs. 65, 66, 67, 68) are a contrivance for Fig. 66. Ficr. 67. Ficr. 68. varying and adjusting the velocity-ratio communicated between a pair of parallel shafts by means of a belt, and may be either conti- nuous cones or conoids, as in figs. 65, 66, whose velocity-ratio can be varied gradually while they are in motion by shifting the belt ; or sets of pulleys whose radii vary by stejis, as in figs. 67, 68, in which case the velocity-ratio can be changed by shifting the belt from one pair of pulleys to another. In order that the belt may be equally tight in every possible position on a pair of speed-cones, the quantity L in the equations of Article 169 must be constant. For a crossed belt, as in figs. 65 and 66, L depends solely on c and on r^ -f- r^. Now c is constant, because the axes are parallel, therefore the sum of the radii of the pitch circles connected in every position of the belt is to be constant. That condition is fulfilled by a pair of continuous cones generated by the revolution of two straight lines inclined opposite ways to their respective axes at equal angles, and by a set of pairs of pulleys in which the sum of the radii is the same for each pair. For an open belt, the following practical rule is deduced from the approximate Equation 3a of Article 169 : — Let the speed-cones be equal and similar conoids, as in fig. 66, but with their large and small ends turned opposite ways. Let 7\ be the radius of the large end of each, r^ that of the small end, rQ that of the middle ; and let y be the sagitta, measured perpendi- cular to the axis, of the arc by whose revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length ; then y rj + ^2 _ (rj - r^y •(!•) where the second value is obtained from the first by considering that in Equation 3a, 2 ^r ^q = ^ (r^ -f- rg) + {r,-r,f ; 2 ^-6-2832; but 6 may be used in most practical cases without sensible error. COUPLING OF PARALLEL AXES. 101 The radii at the middle and ends being thus determin-ed, make the generating curve an arc either of a circle or of a parabola. For a pair of stepped cones, as in fig. 68, let a series of differ- ences of the radii, or values of r-y — r.2, be assumed; then for each pair of pulleys, the sum of the radii is to be computed from the difference by the formula ^^ + ,., = 2r.-fc^'; (2.) 2 7*0 being that sum when the radii are equal. Section 4. — Linkwork. 171. Definitions. — The pieces which are connected by linkwork, if they rotate or oscillate, are usually called cranks, beams, and levers. The link by which they are connected is a rigid bar, which may be straight or of any other figure; the straight figure being the most favourable to strength, is used when there is no special reason to the contrary. The link is known by various names under various circumstances, such as couj)ling rod, connecting rod, crank rod, eccentric rod, &c. It is attached to the pieces which it con- nects by two pins, about which it is free to turn. The eflfect of the link is to maintain the distance between the centres of those pins invariable; hence the line joining the centres of the pins is the line of connection ; and those centres may be called the connected points. In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is the arm or crank arm. 172. Principles of Connection. — The whole of the equations already given in Article 166 for bands, are applicable to linkwork. The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connec- tion; in which case the angular velocity-ratio at any instant is the reciprocal of the ratio of the common perpendiculars let fall from the line of connection upon the respective axes of rotation (Article 166, Equation 3). 173. Dead Points. — If at any instant the direction of one of the crank arms coincides with the line of connection, the common perpendicular of the line of connection and the axis of that crank arm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crank arm in question at such an instant is called a dead point. The velocity of the other connected point at such an instant is null, unless it also reaches a dead point at the same instant, so that the line of connection is in the plane of the two axes of rotation, in which case the velocity-ratio is indeterminate. 174. Coupling of Parallel Axes.~The only case in which an nni- 102 THEORY OF MECHANISM^. form angular velocity-ratio (being that of equality) is communicated by linkwork, is that in which two or more parallel shafts (such as those of the driving wheels of a locomotive engine) are made to rotate with constantly equal angular velocities, by having equal cranks, which are maintained parallel by a coupling rod of such a length that the line of connection is equal to the distance between the axes. The cranks pass their dead points simultaneously. To obviate the unsteadiness of motion which this tends to cause, the shafts are provided with a second set of cranks at right angles to the first, connected by means of a similar coupling rod, so that one set of cranks pass their dead points at the instant when the other set are farthest from theirs. 175. The Comparative Motion of the Connected Points in a piece of linkwork at a given instant is capable of determination by the method explained in Article 98; that is, by finding the instantan- eous axis of the link ; for the two connected points move in the same manner with two points in the link, considered as a rigid body. If a connected point belongs to a turning piece, the direction of its motion at a given instant is perpendicular to the plane contain- ing the axis and crank arm of the piece. If a connected point belongs to a shifting piece, the direction of its motion at any instant is given, and a plane can be drawn perpendicular to that direction. The line of intersection of the planes perpendicular to the paths of the two connected points at a given instant, is the instantaneous axis of the link at that instant ; and the velocities of the . connected points are directly as their distances from that axis. In drawing on a plane surface, the two planes perpendicular to the paths of the connected points are represented by two lines (being their sections by a plane normal to them), and the instanta- neous axis by a point; and should the length of the two lines render it impracticable to produce them until they actually inter- sect, the velocity-ratio of the connected points may be found by the principle, that it is equal to the ratio of the segments which a line parallel to the line of connection cuts ofl[' from any two lines drawn from a given point, perpendicular respectively to the paths of the connected points. Example I. Two Rotating Pieces with Parallel Axes (fig. 69) — • Let Ci, C2, be the parallel axes of the pieces; Ti, Tj, their con- nected points; Ci Tj, Cg T3, their crank arms ; Ti Tg, the link. At a given instant, let Vi be the velocity of Ti; V2 that of Tg. To find the ratio of those velocities, produce Cj Ti, C2 T^, till they intersect in K ; K is the instantaneous axis of the link or connecting rod, and the velocity ratio is vi:v : : KTi ."KT^ (1.) AN ECCENTRIC. 103 Should K be inconveniently far off, draw any triangle with its sides respectively parallel to Ci Tj, Cg T^, and Ti Ta; the ratio of the two sides first mentioned will be the velocity-ratio required. For example, draw Ca A parallel to Ci Tj, cutting Ti T2 in A, then Vi.v^: : CaA rCgTa- .(2.) Fig. 70. Fig. 69. Example II. Rotating piece and sliding piece (fig. 70). — Let Ca be the axis of a rotating piece, and Ti R the straight line along which a sliding piece moves. Let Ti, Ta, be the connected points, Ca Ta the crank arm of the rotating piece, and Tj Ta the link or connecting rod. The point Ti, Tg, and the line Ti R, are supposed to be in one plane, perpendicular to the axis C. Draw Ti K per- pendicular to Ti K, intersecting Ca Ta in K ; K is the instantaneous axis of the link ; and ^•i : va : : K Ti : K Ta Or otherwise draw from a point Cg, Cg A perpendicular to Tj II the direction of motion of the sliding piece, Ca Tg perpendicular to the direction of motion of the rotating piece, then the line Tj Ta, or a line parallel thereto cuts off the segments Ca A, Ca Tg, or segments proportional thereto, and the velocity-ratio of the rotating piece to the sliding piece is as Cj Tg to Ca A. 176. An Eccentric (fig. 71) being a circular disc keyed on a shaft, with whose axis its centre does not coin- cide, and used to give a reciprocating motion to a rod, is equivalent to a crank whose con- nected point is T, the centre of the eccentric disc, and whose crank arm is C T, the distance of that point from the axis of the shaft, called the eccentricity. Fig. 71. 104 THEORY OF MECHANISM. 177. The Length of Stroke of a point in a reciprocating piece is the distance between the two ends of the path in which that point moves. When it is connected by a link with OHJ a point in a continuously rotating piece, the ends of the stroke of the reciprocating point correspond with the dead points of the continu- ously revolving piece (Article 173). Let S = B C be the length of stroke of the reciprocating piece, L = E C = D B the length of the line of connection, and B = A E = A I) the crank arm of the continuously turning piece. Then if the two ends of the stroke be in one straight line with the axis of the crank, S = 2B; (1.) and if their ends be not in one straight line with that axis, then S, L - B, and L + B, are the three sides of a triangle, having the angle opposite S at that axis; so that if be the supplement of the arc between the dead points, D and E, S2 = (L - B)2 + (L + B)2 - 2 (L - B) (L + B) cos 6; >i S2 = 2(L2 + B2)-2(L2-B^)cos 0: ' Fig. 72. cos 2 L^ + 2 B2— S2 2(L2 - B2) r .(2.) 178. Hooke's Universal Joint (fig. 73) is a contrivance for coupling shafts whose axes intersect each other in a point. Let O be the point of intersection of the axes O Ci, O Cg, and i their angle of inclination to each other. The pair of shafts C,, Cg, terminate in a pair of forks, Fi, Fg, in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross, having its centre at O. This cross is the link; the connected points are the centres of the bear- ings Fi, Fg. At each instant each of those points moves at right angles to the central plane of its shaft and fork, tlierefore the line of intersection of the central planes of the two forks, at any instant, is the instantaneous axis of the cross, and the velocity-ratio of the DEFINITIONS. 105 points Fi, Fa (which, as the forks are equal, is also the angular velocity-ratio of the shafts), is equal to the ratio of the distances of those points from that instantaneous axis. The r}iean value of that velocity-ratio is that of equality; for each successive quarter turn is made by both shafts in the same time; but its actual value fluctuates between the limits, - - = -. — 77^7^5 — ;r = - — -. when Fi is in the plane ] «! r ' sm (90^ - ^) cos ^ ^ | of the axis; a a = cos i when Fa is in that plane. ■(!•) 179. The Double Hooke's Joint (fig. 74) is used to obviate the vibratory and unsteady motion caused by the fluctuation of the velocity-ratio indicated in the equa- tion of Article 178. Between the two shafts to be connected, Ci, Cg, there is introduced a short interme- diate shaft Cg, making equal angles with Gi and Cg, connected with each of them by a Hooke's joint, and ^^S- *'*- having both its own forks in the same plane. By this arrangement the angular velocities of the first and third shafts are equal to each other at every instant. 180. A Click, being a reciprocating bar, acting upon a rachet wheel or rack, which it pushes or pulls through a certain arc at each forward stroke, and leaves at rest at each backward stroke, is an example of intermittent linkwork. During the forward stroke, the action of the click is governed by the principles of linkwork ; during the backward stroke, that action ceases. A catch or pally turning on a fixed axis, prevents the ratchet wheel or rack from reversing its motion. Section 5. — Reduplication of Cords. 181. Definitions — The combination of pieces connected by the several plies of a cord or rope consists of a pair of cases or frames called blocks, each containing one or more pulleys called sheaves. One of the blocks called the /all-block, Bi, is fixed ; the other, or running-block, Bg, is movable to or from the fall-block, with which 106 THEORY OF MECHANISM. it is connected by means of a rope of which one end is attached either to the fall-block or to the running-block, while the other end, Tj, called the fall^ or tackle-folly is free j while the intermediate, portion ol the rope passes alternately round the pulleys in the fall- block and running-block. The whole combination is called a tackle or purchase. 182. The Velocity-Ratio chief- ly considered in a tackle is that between the velocities of the running-block, u, and of the tackle-fall, v. That ratio is given by Equation 3 of Article Fig. 75a. 113 (which see), viz. nu;.. (1.) where n is the number of plies of rope by which the running-block is connected with the fall-block. Thus, in fig. 75 w = 7 ; and in fig. 75a w = 6. 182a. The Velocity of any Ply of the rope is found in the follow- ing manner : — I. For a ply on the side of the fall-block next the tackle-fall, such as 2, 4, 6, fig. 75, and 3, 5, fig. 75a, it is to be considered what would be the velocity of that ply if it were itself the tackle- fall. Let that velocity be denoted by v', and let n' be the number of plies between the ply in question and the point of attachment by which the first ply (marked 1 in the figures) is fixed to one or other block. Then ,(1.) II. For a ply on the side of the fall-block farthest from the tackle-fall, the velocity is equal and contrary to that of the next succeeding ply, with which it is directly connected over one of the .sheaves of the fall-block. III. If the first ply, as in fig. 75a, is attached to the fall-block, its velocity is nothing; if to the running-block, its velocity is equal to that of the block. 183. White's Tackle. — The sheaves in a block are usually made all of the same diameter, and turn on a fixed pin; and they have, consequently, different angular velocities. But by making the diameter of each sheaf proportional to the velocity, relatively to the CLASSIFICATION OF THE MECHANICAL POWERS. 107 hloch, of the ply of rope which it is to carry, the angular velocities of the sheaves in one block may be rendered equal, so that the sheaves may be made all in one piece, and may have journals turning in fixed bearings. This is called White's Tackle, from the inventor, and is represented in figs. 75 and 75a. Section 6. — Comparative Motion in the " Mechanical Powers." 184. Classification of the Mechanical Powers. — "Mechanical Powers" is a name given to certain simple or elementary machines, all of which, with the single exception of the pulley, are more simple than even an elementary combination of a driver and fol- lower; for, with that exception, a mechanical power consists essentially of only one primary n)oving piece; and the comparative motion taken into consideration is simply the velocity-ratio either of a pair of points in that piece, or of two components of the velocity of one point. There are two established classifications of the mechanical powers; an older classification, which enumerates six; and a newer classification, which ranges the six mechanical powers of the older system under three heads. The following table shews both these classifications : — Nkwer Classification Older Classification. The Lfvfp / ^^® ^^^®^*' IHE 1.EVER, I r^^^ y^^^^^ ^^^ ^^^^^ ( The Inclined Plane. The Inclined Plane, < The Wedge. (The Screw. The Pulley, The Pulley. In the present section the comparative motions in the mechanical powers are considered alone. The relations amongst the forces which act in those machines will be treated of in the kinetic division of this Treatise. In the lever and the wheel and axle of the older classification, which are both comprehended under the lever of the newer classi- fication, the primary moving piece turns about a fixed axis; and the comparative motion taken into consideration is the velocity- ratio of two points in that piece, which may be called respectively the driving point and the following point. The principle upon which that velocity-ratio depends has already been stated in Article 93, page 50 — viz., that the velocity of each point is proportional to the radius of the circular path which it describes; that is, to its perpendicular distance from the axis of motion. The distinction between the lever and the wheel and axle is 108 THEORY OF MECHANISM. this: that in the lever , the driving point, D, and the following point, F, are a pair of determinate points in the moving piece, as in iigs. 7Ga to 76d; whereas in the wheel and axle they may be any pair of points which are situated respectively in a pair of cylindrical pitch-surfaces, D and F, described about the axis A, fig. 76. In each of these figures the plane of projection is normal to the axis, and A is the trace of the axis. In fig. 76, D and F are the traces of two cylindrical pitch-surfaces. In figs. 76a to 76d, D and F are the projections of the driving and following points respectively. The axis of a lever is often called the fulcrum. A lever is said to be straight, when the driving point, D, and following point, F, are in one plane traversing the axis A, as in figs. 76a, 76b, and 76c. In other cases the lever is said to be hent, as in fig. 76d. Fig. 76. Ti%. 76a. Fig 7Cc. \ Fig. 76c. The straight lever is said to be of one or other of three kinds, according to the following classification : — In a lever of the first kind, fig. 76a, the driving and following points are at opposite sides of the fulcrum A. In a lever of the second kind, fig. 76b, the driving and following points are at the same side of the fulcrum, and the driving point is the further from the fulcrum. In Si, lever of the third kind, fig. 76c, the driving and following points are at the same side of the fulcrum, and the following point is the further from the fulcrum. In the inclined plane, and in the wedge, the comj)arative motion considered is the velocity-ratio of the entire motion of a straight- sliding primary piece and one of the components of that motion ; the principles of which velocity-ratio have been stated in Article 70, pages 38, 39. Fior. 76d. THE INCLINED PLANE. 109 III the inclined plane, fig. 76e, A A is the trace of a fixed plane; B, a block sliding on that plane in the direction BC; the plane of projection being perpendicular to the plane A A, and parallel to the direction of motion of B> B D is some direction oblique to B C. From any convenient point, C, in B 0, let fall C D perpendicular to B D; then B D -f- B C is the ratio of ^^S- 76e. the component velocity in the direction B D to the entire velocity ofB. In fig. 76f, a a is the trace of a fixed plane ; B C D, the trace of a wedge which slides on that plane. While the wedge advances through the distance C c, its oblique face advances from the posi- tion C D to the position c d ; and if C e be drawn normal to the plane C D, the ratio borne by the component velocity of the wedge Fig, 76r. in a direction normal to its oblique face to its entire velocity will be expressed by C e : C c. In the screw the comparative motion considered is the ratio borne by the entire velocity of some point in, or rigidly connected with, the screw, to the velocity of advance of the screw. The helical path of motion of a point in, or rigidly attached to, a screw may be developed (as has been already explained in Article 160, page 94) into a straight line : being the hypotenuse of a right angled triangle whose height is equal to the pitch of the screw, and its base to the circumference of a circle whose radius is the distance of the given point from the axis of the screw. Then if B D in fig. 76e be taken to represent the pitch of the screw, and D C, perpendicular to B D, the circumference of the circle described by the point in question about the axis, B C will be the develop- ment of one turn of the screw-line described by that point as it revolves and advances along with the screw; and B C -^- B D will be the ratio of its entire velocity to the velocity of advance; just as in the case of a body sliding on an inclined plane, A A, parallel to B C. This shews why the screw is comprehended under the 110 THEORY OF MECHANISM. general head of the inclined plane, in the newer classification of tlie mechanical powers. The terra pulley^ in treating of the mechanical powers, means any purchase or tackle of the class already described in Section 5 of this Chapter, pages 105 to 107. Section 7. — Hydraulic Connection. 185. The General Principle of the communication of motion between two pistons by means of an intervening fluid of constant density has already been stated in Article 119, viz., that the velo- cities of the pistons are inversely as their areas, measured on planes normal to their directions of motion. Should the density of the fl.uid vary, the problem is no longer- one of pure mechanism; because in that case, besides the communi- cation of motion from one piston to the other, there is an additional motion of one or other, or both pistons, due to the change of volume of the fluid. 186. Valves are used to regulate the communication of motion through a fluid, by opening and shutting passages through which the fluid flows; for example, a cylinder may be provided with valves which shall cause the fluid to flow in through one passage, and out through another. Of this use of valves, two cases may be distinguished. I. When the piston moves the fluid, the valves may be what is called self-acting; that is, moved by the fluid. If there be two passages into the cylinder, one provided with a valve opening inwards, and the other with a valve opening outwards; then during the outward stroke of the piston the former valve is opened and the latter shut by the inward pressure of the fluid, which flows in through the former passage; and during the inward stroke of the piston, the former valve is shut and the latter opened by the outward pressure of the fluid, which flows out through the latter passage. This combination of cylinder, piston, and valves, consti- tutes a pump. II. When the fluid moves the piston, the valves must be opened and shut by mechanism, or by hand. In this case the cylinder is a working cylinder. 187. In the Hydraulic Press, the rapid motion of a small piston 'n a pump causes the slow motion of a large piston in a working cylinder. The pump draws water from a reservoir, and forces it into the working cylinder: during the outward stroke of the pump jjiston, the piston of the working cylinder stands still; during the inward stroke of the pump piston, the piston of the working cylinder moves outward with a velocity as much less than that ot the pump piston as its area is greater. When the piston of the TRAINS OF ELEMENTARY COMBINATIONS. Ill working cylinder has finished its outward stroke, which may be of any length, it is permitted to be moved inwards again by opening a valve by hand and allowing the water to escape. 188. In the Hydraulic Hoist, the slow inward motion of a large piston drives water from a large cylinder into a smaller cylinder, and causes a more rapid outward motion of the piston of the smaller cylinder. When the latter piston is to be moved inward, a valve between the two cylinders is closed, and the valve of an outlet from the smaller cylinder opened, by hand, so as to allow the water to escape from the smaller cylinder. The larger cylinder is filled and its piston moved outward, when required, by means of a pump, in a manner resembling the action of a hydraulic press. Section 8. — Trains of Mechanism. 189. Trains of Elementary Combinations have been defined in Article 131, and illustrated in the case of wheel work, in Article 145, and in the case of a double Hooke's joint, in Article 179. The general principle of their action is that the comparative motion of the first driver and last follower is expressed by a ratio, which is found by multiplying together the several velocity-ratios of the series of elementary combinations of which the train consists, each with the sign denoting the directional relation. Two or more trains of mechanism may converge into one; as when the two pistons of a pair of steam engines, each through its own connecting rod, act upon one crank shaft. One train of mechanism may diverge into two or more; as when a single shaft, driven by a prime mover, carries several pulleys, each of which drives a diflferent machine. The principles of comparative motion in such converging and diverging trains are the same as in simple trains. LIBRARY ^^ UNIVEKSITY OF OALlFOilNlA. 112 CHAPTER III. ON AGGREGATE COMBINATIONS. 190. The General Principles of aggregate combinations have already been given in Part L, Chapter IL, Section 3. The pro- blems to which those principles are to be applied may be divided into two classes. I. Where a secondary moving piece is connected at three, or at two points, as the case may be, with three or with two other pieces whose motions are given ; so tliat the problem is, from the motions of three or of two points in the secondary piece, to find its motion as a whole, and the motion of any point in it. The solution of this problem is given in Articles 97 and 98. IT. Where a secondary piece, C, is carried by another piece, B ; and denoting the frame of the machine by A, there are given two out of the three motions of A, B, and C, relatively to each other, and the third is required. The motion of C relatively to A is the resultant of the motion of C relatively to B, and of B relatively to Aj and the problem is solved by the methods already explained in Articles 99 to 107, inclusive. Professor Willis distinguishes the effects of aggregate combina- tions into aggregate velocities, whether linear or angular, produced in secondary pieces by the combined action of different drivers, and aggregate paths, being the curves, such as cycloids and trochoids, epicycloids and ei)itrochoids, described by given points in such secondary pieces. A/ The following Articles give examples of two simple aggregate combinations. 191. Differential Windlass. — In fig. 77, the axis Aj carries two barrels of different radii, r-^ being the greater, and r^^ the less. A running block containing a single pulley is hung by a rope which passes below the pulley, and has one end wound round the larger barrel, and the other wound the contrary way round the smaller barrel. When the two barrels rotate together with the common angular velocity a, the division of the rope which hangs from the larger barrel moves with the COMPOUND SCREWS. 113 velocity a r-^, and the division wliicli hangs from the smaller barrel moves in the contrary direction with the velocity -ar^ (whose direction is denoted by the negative sign). These are also the velocities of the two points at opposite extremities of a diameter of the pulley, where it is touched by the two vertical divisions of the rope. The velocity of the centre of the pulley is a mean between those two velocities; that is, their half- difference, because their signs are opposite; or denoting it by v, a{r^-r^) .(!•) The instantaneous axis of the pulley may be found by the method of Article 98, as follows: — In fig. 35c, let A and B be t he two ends of the horizontal diameter of the pulley, and le t A.Y^ = a r^, and B Vj = (xr2 represent their velocities; join Y^Yj, cutting AB in O ; this is the instantaneous axis. Now A0-0B = AC + C0-0B = BC + C0-0B = 20C, AO + OB : AO-OB::AV„ + BY,:AY,-BY„ AB: 20C:: a{r^ + T^: a{o\-r^); and hence the distance of the instantaneous axis from the centre or moving axis of the pulley is obviously AB .(2.) 2(^1 + ^2)' ' The motion of the centre of the pulley is the same with that of a point in a rope wound on a barrel of the radius — ^ — . The use of the contrivance is to obtain a slow motion of the pulley without using a small, and therefore a weak, barrel. 192. Compound Screws. — (Fig. 78). On the same axis let there be two screws Sj Sj, and 85^83, of the respective pitches o N/ Fig. 78. Pi and P2, Pi being the greater, and let the screws in the first instance be both right-handed or both left-handed. Let Nj and N2 be two nuts, fitted on the two screws respectively. When the com- pound screw rotates with the angular velocity a, the nuts ap roach towards or recede from each other with the relative velocity I 114 THEORY OF MECHANISM. -"-%^; (1.) being that due to a screw whose pitch is the difference of the two pitches of the compound screw. (See Article 96, Equation 1.) The object of this contrivance is to obtain the slow advance due to a fine pitch, together with the strength of large threads. rig. 79 represents a compound screw in which the two screws are contrary-handed, and the relative velocity of the nuts Ni Ng is that due to the sum of the two pitches ; or as they are usually equal, to double the pitch of each screw. This combination is used in coupling railway-carriages. PAKT 111. PRINCIPLES OF STATICS. CHAPTER I. SUMMARY OF GENERAL PRINCIPLES. Nature and Division of the Subject. The present Chapter contains a summary of the Principles of Statics. 193. Forces — Action and Re-action. — Every force is an action exerted between a pair of bodies, tending to alter their condition as to relative rest and motion; it is exerted equally, and in con- trary directions, upon each body of the pair. That is to say, if A and B be a pair of bodies acting mechanically on each other, the force exerted by A upon B is equal in magnitude and contrary in direction to the force exerted by B upon A. This principle is sometimes called the equality of action and re-action. It is ana- logous to that of relative motion, explained in Article 61, page 34. 194. Forces, how Determined and Expressed. — A force, as respects one of the two bodies between which it acts, is deter- mined, or made known, when the following three things are known respecting it :— first, the place, or part of the body to which it is applied; secondly, the direction of its action; thirdly, its magnitude. The Place of the application of a force to a body may be the ■whole of its volume, as in the case of gravity; or the surface at which two bodies touch each other, or the bounding surface between two parts of the same body, as in the case of pressure, tension, shearing stress, and friction. Thus every force has its action distributed over a certain space, either a volume or a surface; and a force concentrated at a single point has no real existence. Nevertheless, in investigations respect- ing the action of a distributed force upon the position and move- ments, as a w^hole, of a rigid body, or of a body which without error may be treated as rigid, like the solid parts of a machine, fixed or moving, that force may be treated as if it were concen- trated at a point or points, determined by suitable processes ; and 116 PRINCIPLES OF STATICS. such is the use of those numerous propositions in statics which relate to forces concentrated at points; or single forces, as they are called. The Direction of a force is that of the motion which it tends to produce. A straight line drawn through the points of applica- tion of a single force, and along its direction, is the line op action of that force. The Magnitudes of two forces are equal when, being applied to the same body in opposite directions along the same line of action, they balance each other. The magnitude of a force is expressed arithmetically by stating in numbers its ratio to a certain unit or standard of force, which, for practical purposes, is usually the weight (or attraction towards the earth), at a certain latitude, and at a certain level, of a known mass of a certain material. Thus the British unit of force is the standard pound avoirdupois ; which is the weight, in the latitude of London, of a certain piece of platinum kept in a public office. For the sake of convenience, or of compliance with custom, other units of weight are occasionally employed in Britain, bearing certain ratios to the standard pound ; such The grain = -j^^q of a pound avoirdupois. The troy pound = 5,760 grains = 0-82285714 pound avoirdupois. The hundredweight = 112 pounds avoirdupois. The ton = 2,240 pounds avoirdupois. The French standard of weight is the kilogramme, which is the weight, in the latitude of Paris, of a certain piece of platinum kept in a public office. It was originally intended to be the weight of a cubic decimetre of pure water, measured at the temperature at which the density of water is greatest — viz., 4°'l Cent., or 39°-4 Fahr., and under the pressure which supports a barometric column of 760 millimetres of mercury; but it is in reality a little heavier. A kilogramme is 2-20462125 lbs. avoirdupois. A pound avoirdupois is 0-4535926525 of a kilogramme. For scientific purposes, forces are sometimes expressed in Absolute Units. The "Absolute Unit of Force" is a term used to denote the force which, acting on an unit of mass for an unit of time, produces an unit of velocity. The unit of time employed is always a second. The unity of velocity is in Britain one foot per second ; in France one metre per second. The unit of mass is the mass of so touch matter as weighs one eeRiesentation of forces by lines. 117 unit of weiglit near the level of the sea, and in some definite latitude. In Britain the latitude chosen is that of London ; in France, that of Paris. In Britain the unit of weight chosen is sometimes a grain, sometimes a pound avoirdupois; and it is equal to 32-187 of the corresponding absolute units of force. In France the unit of weight chosen is either a gramme or a kilogramme, and it is equal to 9-8087 of the corresponding absolute units of force. Each of those coefficients is denoted by the letter g. 195. Measures of Force and Mass. — If by the unit of force is understood the weight of a certain standard, such as the avoirdupois pound, then the mass of that standard is 1-^g', and the unit of mass is g times the mass of the standard; and this is the most convenient system for calculations connected with mechanical engineering, and is therefore followed in the present work. But if we take for the unit of mas?, the mass of the standard itself, then the unit of force is the absolute unit; and the weight of the standard in such units is expressed by g; for g is the velocity which a body's own weight, acting unbalanced, impresses on it in a second. This will be specially treated of in Part V. This is the system employed in many scientific writings, and in particular, in Thomson and Tait's Natural Philosophy. It has great advan- tages in a scientific point of view; but its use in calculations for practical purposes would be inconvenient, because of the prevailing custom of expressing forces in terms of the standard of weight. 196. Representation of Forces by Lines. — A single force may be represented in a drawing by a straight line; an extremity of the line indicating the point of application of the force, — the lirection of the line, the direc- tion of the force, — and the length of the line, the magnitude of the force, according to an arbitrary 3 ' "^"--3& scale. Yig. 80. For example, in fig. 80, the fact that the body B B B B is acted upon at the point Oi by a given force, may be expressed by drawing from Oj a straight line Oi Fi in the direction of the force, and of a length representing the magnitude of the force. If the force represented by Oi F^ is balanced by a force applied either at the same point, or at another point O2 (which must be in the line of action L L of the force to be balanced), then the second force will be represented by a straight line Og Fg, opposite in direc- 118 PRINCIPLES OF STATlCy. tion, and equal in length to Oi h\, and lying in the same line of action L L. If the body B B B B (fig. 81), be balanced by several forces acting in the same straight line LL, applied at points Oi O2, &c., and represented by lines O^ \^\, O^Fa, &c. j then either direction in the line L L (such as the direc- tion towards + L) is to be ^' considered as positive, and the opposite direction (such as the direction towards — L) as negative ; and if the sum of all the lines repre- senting forces which point pjg gj; positively be equal to the sum of all those which point negatively, the algebraical sum of all the forces is nothing, and the body is balanced. 197. Resultant and Component Forces — Their Magnitude. — The Besultant of any combination of forces applied to one body is a single force capable of balancing that single force which balances the combined forces; that is to say, the resultant of the combined forces is equal and directly opposed to the force which balances the combined forces, and is equivalent to the combined forces so far as the balance of the body is concerned. The com- bined forces are called components of their resultant. The resultant of a set of mutually balanced forces is nothing. The magnitudes and directions of a resultant force and of its components are related to each other exactly in the same manner with the velocities and directions of resultant and component motions. As to the position of the resultant, if the components act through one point, the resultant acts through that point also; but if the components do not act through one point, the position of the re- sultant is to be found by methods which will be stated further on. 198. Equilibrium or Balance is the condition of two or more forces which are so opposed that their combined action on a body produces no change in its rest or motion, and that each force merely tends to cause such change, without actually causing it. In treatises on statics, the word pressure is often used to denote any balanced force; although in the pojmlar sense that word is used to denote a force, of the nature of a thrust or push, distributed over a surface. 199. Parallel Forces are forces whose directions of motion are parallel, excepting couples and directly opposed forces. 200. Couples. — Two forces of equal magnitude applied to the same body in parallel and opposite directions, but not in the same DISTRIBUTED FORCES IN GENERAL. 119 line of action (such as F, F, in fig. 82), constitute what is called a " GowpUy The arm or leverage of a couple (L, fig. 82) is the perpendicular distance between the lines of action of the two equal forces. The tendency of a couple is to turn the body to which it is applied in the plane of the couple — that is, the plane which con- tains the lines of action of the two forces. (The plane in which a body turns is any plane parallel to those planes in the body whose position is not altered by the turning). The turning of a body is said to be right-handed when it appears to a spectator to take place in the same direction with that of °* the hands of a watch, and left-handed when in the opposite direc- tion; and couples are designated as right-handed or left-handed according to the direction of the turning which they tend to pro- duce. The couple represented in fig. 82 appears right-handed to the reader. The Moment of a couple means the product of the magnitude of its force by the length of its arm (F L) ; and may be represented by the area of a rectangle whose sides are F and L. If the force be a certain number of pounds, and the arm a certain number of feet, the product of those two numbers is called the moment in foot-pou7ids, and similarly for other measures. The moment of a couple may also be represented by a single line on paper, by setting off upon its axis (that is, upon any line perpendicular to the plane of the couple) a length proportional to that moment (O M, fig. 82) in such a direction, that to an observer looking from O towards M the couple shall seem right-handed. 201. The Centre of Parallel Forces is the single point referred to in the following principle. The forces to which that principle is applied are in general either weights or pressures; and the point in question is then called the Cent/re of Gravity or the Cent7'e of Pressure, as the case may be. If there be given a system of points, and the mutual ratios of a system of parallel forces applied to those points, which forces have a single resultant, then there is one point, and one only, which is tra- versed hy the line of action of the resultant of every system of parallel forces haviyig the given mutual ratios and applied to the given system of points, whatsoever may he the absolute magnitudes of those forces and the angular position of their lines of action. 202. Distributed Forces in General. — In Article 194, page 115, it has already been explained, that the action of every real force is distributed throughout some volume, or over some surface. It is always possible, however, to find either a single resultant, or a 120 PRINCIPLES OF STATICS. resultant couple, or a combination of a single force with a couple, to which a given distributed force is equivalent, so far as it affects the equilibrium of the body, or part of a body, to which it is applied. In the application of Mechanics to Structures, the only force dis- tributed throughout the volume of a body which it is necessary to consider, is its weight, or attraction towards the earth ; and the bodies considered are in every instance so small as compared with the earth, that this attraction may, without appreciable error, be held to act in parallel directions at each point in each body. More- over, the forces distributed over surfaces are either parallel al/ each point of their surfaces of application, or capable of being resolved into sets of parallel forces ; hence, parallel distributed forces have alone to be considered ; and every such force is statically equivalent . either to a single resultant, or to a resultant couple. The intensity of a distributed force is the ratio which the magni- tude of that force, expressed in units of weight, bears to the space over which it is distributed, expressed in units of volume, or in units of surface, as the case may be. An unit of intensity is an unit of force distributed over an unit of volume or of surface, as the case may be; so that there are two kinds of units of intensity. For example, one pound per cubic foot is an unit of intensity for a force distributed throughout a volume, such as weight; and one pound per square foot is an unit of intensity for a force distributed ©ver a surface, such as pressure or friction. 203. Specific Gravity — Heaviness — Density — Bulkiness. — I. Specific Gravity is the ratio of the weight of a given bulk of a given substance to the weight of the same bulk of pure water at a standard temperature. In Britain the standard temperature is 62° Fahr. = 16° -67 Cent. In France it is the temperature of the maximum density of water = 3°-94 Cent. = 39°-l Fahr. In rising from 39°-l Fahr. to 62° Fahr., pure water expands in the ratio of 1*001118 to 1 ; but that difference is of no consequence in calculations of specific gravity for engineering purposes. II. The heaviness of any substance is the weight of an unit of volume of it in units of weight. In British measures heaviness is most conveniently expressed in lbs. avoirdupois to the cubic foot ; in French measures, in hilogrammes to the cubic decimetre (or to the litre). The values of the heaviness of water at 39°-l Fahr., and at 62° Fahr., are respectively 62425 and 62-355 lbs. to the cubic foot. III. The density of a substance is either the number of units of mass in an unit of volume, in which case it is equal to the heavi- ness, — or the ratio of the mass of a given volume of the substance to the mass of an equal volume of water, in which case it is equal to the specific gravity. In its application to gaseSy the term THE INTENSITY OF PRESSURE. 121 " Density" is often used to denote the ratio of the heaviness of a given gas to that of air, at the same temperature and pressure. IV. The bulkiness of a substance is the number of units of volume which an unit of weight fills ; and is the reciprocal of the heaviness. In British measures bulkiness is most conveniently expressed in cubic feet to the lb. avoirdupois ; in French measures, in cubic decimetres (or in litres) to the Mlogramme. Rise of temper- ature produces (with certain exceptions) increase of bulkiness. The linear expansion of a solid body is one-third of its expansion in bulk. 204. The Centre of Gravity of a body or of a system of bodies, is the point always traversed hj the resultant of the weight of the body or system of bodies, — in other words, the centre of parallel forces for the weight of the body or system of bodies. To support a body, that is, to balance its weight, the resultant of the supporting force must act through the centre of gravity. When the centre of gravity of a geometrical figure is spoken of, it is to be understood to mean the point where the centre of gravity would be, if the figure were formed of a substance of uniform heaviness. 205. The Centre of Pressure in a plane surface is the point traversed by the resultant of a pressure that is exerted at that surface. When the intensity is uniform, the centre of pressure is at the centre of magnitude of the pressed surface. 206. The Centre of Buoyancy of a solid wholly or partly im- mersed in a liquid is the centre of gravity of the mass of liquid displaced. The resultant pressure of the liquid on the solid is equal to the weight of liquid displaced, and is exerted vertically upwards through the centre of buoyancy. 207. The Intensity of Pressure is expressed in units of weight on the unit of area; as pounds on the square inch, or kilogrammes on the square metre ; or by the height of a column of some fluid ; or in atmospheres, the unit in this case being the average pressure of the atmosphere at the level of the sea. 122 CHAPTER II. COMPOSITION, RESOLUTION, AND BALANCE OF FORCES. Section 1. — Forces Acting Through One Point. 208. Resultant of Forces Acting in One Straight Line.— The resultant of any number of forces acting on one body in the same straight line of action, acts along that line, and is equal in magni- tude to the sum of the component forces; it being understood, that when some of the component forces are opposed to the others, the word "sz*m" is to be taken in the algebraical sense; that is to say, that forces acting in the same direction are to be added to, and forces acting in opposite directions subtracted from each other. When a system of forces acting along one straight line are balanced, the sum of the forces acting in one direction is equal to the sum of the forces acting in the opposite direction. 209. Resultant and Balance of Inclined Forces — Parallelogram of Forces. — The smallest number of inclined forces which can balance each other is three. Those three forces must act through one point, and in one plane. Their relation to each other depends on the following theorem, called the " Parallelogram of Forces," from which the whole science of statics may be deduced. If two forces whose lines of action traverse one point he repre- sented in direction and magnitude hy the sides of a parallelogram, their resultant is represented hy the diagonal. For example, through the point O (fig. 83) let two forces act, represented in direction and magnitude by O A and O B. The re- sultant or equivalent single force of those two forces is represented in direction and magnitude by the diagonal O C of the parallelogram O A C B. Its magnitude is given algebraically by the equation. 0C=\/ |0A2 + 0B2 ^ ^ . h (1.) 20A-OBcosAOB Fig. 83. 210. Triangle of Forces.— To balance the forces O A and OB, a force is required equal and directly opposed to their resultant C. This may be expressed by saying, that if the directions and RESOLUTION OF A FORCE INTO TWO COMPONENTS. 123 angle magnitudes of three forces he reirresented hy the three sides of a triangle, taJcen in the same order (sucli as O A, A C, C O), then those three forces, acting through one point, balance each other, or in other words, that three forces in the same plane balance each other at one point, -when each is proportional to the sine of the between the other two. 211. Polygon of Forces. — If a numher of forces acting through the same point he represented hy lines equal and parallel to the sides of a closed polygon, taken in the same order, those forces halance each other. To fix the ideas, let there be five forces acting through the point O (fig. 84), and represented in direction and magnitude by the lines Fj, Fg, Fg, F^, Fg, which are equal and parallel to the sides of the closed poly- gon O ABC DO; viz.:— Fi^and II O A; Fo^and ii AB; Fg = and BC; F4 = and CD; Fg^andllDO. Then, by the principle of the parallelogram of forces, the resultant of F^ and Fg is O B; the resultant of F^, Fg, and F3 is O C ; the resultant of F^, Fg, Fg, and F^ is O D, equal and opposite to F^, gauche" — that is, so that the final resultant is nothing. The closed polygon may be either plane or not in one plane. 212. Principles of the Parallelepiped of Forces. — The simplest gauclie polygon is one of four sides. Let AOBCEFGrH (fig. 85), be a parallelopiped whose diagonal is O H. Then any three successive edges so placed as to begin at O and end at H, form, together with the diagonal H O, a closed quadrilateral; consequently, if three forces Fj, Fg, Fg, acting through 0, be represented by the three edges A O, O B, O C, of a parallelopiped, the dia- gonal O H represents their resultant, and a fourth force F^ equal and opposite to O H balances them. /% 213. Resolution of a Force into two / Components. — In order that a given V single force may be resolvable into two '^ components acting in given lines in- clined to each other, it is necessary, first, that the lines of action of those components should intersect the line of action of the given force in one point ; and secondly, that those three lines of action should be in one plane. Fis. 85. .124 PRINCIPLES OF STATICS. Returning then to fig. 83, let O represent the given force, which it is required to resolve into two component forces, acting in the lines O X, O Y, which lie in one plane with O C, and intei-sect it in one point 0. Though C draw C A li O Y, cutting O X in A, and C B li O X, cutting O Y in B. Then will O A and O B represent the com- ponent forces required. Two_ forces respectively equal to and directly opposed to O A and O B will balance 0. The magnitudes of the forces are in the following proportions: — 00:OA:OB ::sinAOB:sinBO0 :sinAOO (1.) 214. Resolution of a Force into three Components. — In order that a given single force may be resolvable into three components acting in given lines inclined to each other*, it is necessary that the lines of action of the components should intersect the line of action of the given force in one point. Returning to fig. 85, let O H represent the given force which it is required to resolve into three component forces, acting in the lines O X, Y, O Z, which intersect O H in one point O. Through H draw three planes parallel respectively to the planes Y O Z, Z O Y, X O Y, and cutting respectively O X in A, O Y in B, O Z in 0. Then will O A, O B, O 0, represent the component forces required. Three forces respectively equal to, and directly opposed to O A, OB, and O "0, will balance Oil. 215. Resolution of a Force. Rectangular Components.— The rectangular components of a force are those into which it is resolved when the directions of their lines of action are at right angles to each other. For example, in fig. 85, suppose O X, O Y, O Z, to be three axes of co-ordinates at right angles to each other. Then O H is resolved into three rectangular components, A O, O B, O C, simply by letting fall from H perpendiculars on O X, O Y, O Z, cutting them at A, B, C, respectively. Let the three rectangular components be denoted respectively by X, Y, Z, the resultant by R, and the angles which it makes with the components by a, /3, y, respectively ; then the relations between the three rectangular components and their resultant are expressed by the following equations : — X = Rcosa; Y = Rcos/3; Z = Rcosy; (2.) R2 = X2 + Y2 + Z2 (3.) RESULTANT OF COUPLES. 125 When the resultant is in the same plane with two of its com- ponents (as X and Y), the third component is null, and the Equations 2 and 3 take the following form : — X = R cos a = R sin/3; Y = R cos /3=:E- sin «j Z = 0j...(4.) R2 = X2 + Y2 (5.) In using Equations 2, 3, 4, and 5, it is to be remembered that cosines of obtuse angles are negative. 216. Resultant and Balance of any number of inclined Forces acting through one Point. — To find this resultant by calculation, assume any three directions at right angles to each other as axes; resolve each force into three components (X, Y, Z) along those axes, and consider the components along a given axis which act in one direction as positive, and those which act in the opposite direc- tion as negative; take the algebraical sums of the components along the three axes respectively (S • X, E • Y, 2 • Z); these will be the rectangular components of the resultant of all the forces; and its magnitude and direction will be given by the following equations : — E-2 = (2-X)2 + (2-Y)^ + (2-Z)2; (1.) 2X ^2Y 2Z ,_ cos tt =r ; cos, (i = -—- -^ COSy = — :p- (2.) If the forces all act in one plane, two rectangular axes in that plane are sufficient, and the terms containing Z disappear from the equations. If the forces balance each other, the components parallel to each axis balance each other independently; that is to say, the three follow^ing conditions are fulfilled : — 2-X = 0; S-Y = 0; 2-Z = (3.)^ If the forces all act in one plane, these conditions of equilibrium are reduced to two. Section 2. — Resultant and Balance of Couples. 217. Equivalent Couples. — If the moments of two couples acting in the same direction and in the same or parallel planes are equal, those couples are equivalent : that is, their tendencies to turn the body to which they are applied are the same. The following propositions are the chief consequences of the principle just stated : — 218. Resultant of Couples. — -The resultant of any number of couples acting in the same or parallel planes is equivalent to a couple whose moment is the algebraical sum of the moments of the combined couples. 126 PRINCIPLES OF STATICS. 219. Equilibrium of Couples with same Axis. — Two opposite couples of equal moment in the same or parallel planes balance each other. Any number of couples in the same or parallel planes balance each other when the moments of the right-handed couples are together equal to the moments of the left-handed couples; in other words, when the resultant moment is nothing — a condition expressed algebraically by 2-FL = (1.) 220. Parallelogram of Couples. — If the two sides of a parallelo- gram represent the axes and moments of two couples acting on the same body in planes inclined to each other, the diagonal of the parallelogram will represent the axis and moment of the resultant couple, which is equivalent to those two. In other words, three couples represented by the three sides of a triangle, taken in the same order, balance each other. 221. Polygon of Couples. — If any number of couples acting on the same body be represented by a series of lines joined end to end, and taken in the same order so as to form sides of a polygon, and if the polygon is closed, those couples balance each other. These propositions are analogous to corresponding propositions relating to single forces; and couples, like single forces, can be resolved into components acting about two or three given axes. 222. Resultant of a Couple and Single Force in Parallel Planes. — Let M denote the moment of a couple applied to a body (fig. 86); and at a point O let a single force F be applied, in a plane parallel to that of the couple. For the given couple substitute an equi- valent couple, consisting of a force - F equal and directly op- posed to F at O, and a force F acting through the point A, the arm A O perpendicular to F M being = ^, and parallel to the Fig. S6. plane of the couple M. Then the forces at O balance each other, and F acting through A is the resultant of the single force F applied at O, and the couple M ; that is to say, that if with a single force F there be combined a couple M whose plane is parallel to the force, the efifect of that combination is to shift the line of M action of the force parallel to itself through a distance O A = ^ ; — to the left if M is right-handed — to the right if M is left-handed. MAGNITUDE OF RESULTANT OF PARALLEL FORCES. 127 223. Moment of Force with respect to an Axis. let the straight line F represent a force. Let O X be any straight line perpendicular in direc- tion to the line of action of the force, and not intersecting it, and let A B be the common per- pendicular of those two lines. At B conceive a pair of equal and directly opposed forces to be applied in a line of action parallel to F, viz.: — F' = F, and - F' = - F. The supposed application of such a pair of balanced forces does not alter the statical condition of the body. Then the original ^ single force F, applied in a line traversing A, is ^ equivalent to the force F' applied in a line travers- ing B, the point in O X which is nearest to A, combined with the couple composed of F and - F', whose moment is F * A B. This is called the moment of the force F relatively to the axis O X, and sometimes also, the moment of the force F relatively/ to the plane traversing O X, parallel to the line of action of the force. If from the point B there be drawn two straight lines B D and B E, to the extremities of the line F representing the force, the area of the triangle B D E being = J F • A B, represents one-half of the moment of F relatively to O X. Fia. 87. Section 3. — Resultant and Balance of Parallel Forces. 224. Magnitude of Resultant of Parallel Forces. — A balanced system of parallel forces consists either of pairs of directly opposed equal forces, or of couples of equal forces, or of combinations of such pairs and couples. Hence the following propositions as to the relations amongst the magnitudes of systems of parallel forces. I. In a balanced system of parallel forces the sums of the forces acting in opposite directions are equal ; in other words, the alge- braical sum of the magnitudes of all the forces taken with their proper signs is nothing. II. The magnitude of the resultant of any combination of parallel forces is the algebraical sum of the magnitudes of the forces. The relations amongst the positions of the lines of action of balanced parallel forces remain to be shewn; and in this inquiry all pairs of directly opposed equal forces may be left out of con- sideration; for each such pair is independently balanced whatso- ever its position may be ; so that the question in each case is to be solved by means of the theory of couples. The following is the simplest case : — 128 PRINCIPLES OF STATICS. •225. Direction of Eesultant of Parallel Forces — Principle of the Lever. — If three parallel forces applied to one body balance each other, they must be in one plane; the two extreme forces Tnust act in the same direction; the middle force must act in the opposite direction; and the magnitude of each force TRUst be proportional to the distance between the lines of action of the other two. Let a body (fig. 88) be maintained in eqnilibrio by two opposite couples acting in the same plane, and of equal moments, and let those couples be so applied to the body that the lines of action of two of those forces, - F^ - F^, which act in the same direction, shall coincide. Then those two forces are equivalent to the single middle force F^ = - (F^ + F^), equal and opposite to the sum of the extreme forces + F^, + F^, and in the same plane with them ; and if the straight line A C B be drawn perpendicular to the lines of action of the forces, then AC and consequently L,; CB = L3; AB==L, + L, F, :F3 :F, ::CB : AC : AB;. .(!•) This proposition holds also when the straight line A C B crosses the lines of action of the three forces obliquely. 226. To find the Resultant of Two Parallel Forces.— The resultant is in the same jDlane with, and parallel to, the com- ponents. It is their sum or difference, according as they act in the same or contrary directions; and in the latter case its direction is that of the greater component. To find its line of action by construction, proceed as follows : — Fig. 89 representing the case in which the components act in the same direction, fig. 90 that in which they act in contrary directions. Let A D and B E be the components. Join A E and B D, cutting each other in F. In BD (produced in fig. 90) take BG = DF. Through G draw a line parallel to the components; this will be the line of action of the resultant. To find its magnitude by construction : parallel to A E, draw B C and D H, cutting the line of action of the result- ant in C and H; C H will represent the resultant required; and a force equal and opposite to C H will balance A D and B E. To find the line of action of the resultant by calculation ; make either KELATIVE PROPORTIONS OF FOUR PARALLEL FORCES. 129 BG = ADDB DG Fig. 90. BED B Fig. 91. H ^ C H When the two given parallel forces are opposite and equal, they form a couple, and have no single resultant. 227. To find the Relative Proportions of Three Parallel Forces which Balance each other, Acting in One Plane: their Lines of Action being given. — Across the three lines of action, in any convenient position, draw a straight line A C B, lig. 91, and measure the distances between the points where it cuts the lines of action. Then each force will be proportional to the distance between the lines of action of the other two. The direction of the middle force, C, is contrary to that of the other two forces, A and B. In symbols, let A, B, and C be the forces; then, A + B + C = 0; AB:BC:CA::C:A:B. Each of the three forces is equal and opposite to the resultant of the other two; and each pair of forces are equal and opposite to the components of the third. Hence this rule serves to resolve a given force into two parallel components acting in given lines in the same plane. 228. To find the Relative Proportions of Four Parallel Forces which Balance each other, not Acting in One Plane: their Lines of Action being given. — Conceive a plane to cross the lines of action in any convenient position; and in fig. 92 or fig. 93, let A, B, Fig. 92. C, J) reprc' 130 PRINCIPLES OF STATICS. sent the points where the four lines of action cut the plane. Draw the six straight lines joining those four points by pairs. Then the force which acts through each point will be proportional to the area of the triangle formed by the other three points. In fig. 92 the directions of the forces at A, B, and C are the same, and are contrary to that of the force at D. In fig. 93 the forces at A and D act in one direction, and those at B and C in the contrary direction. In symbols, A + B + C + D = 0; BCD:CD A:D AB:ABC C D. Each of the four forces is equal and opposite to the resultant of the other three ; and each set of three forces are equal and oppo- site to the components of the fourth. Hence the rule serves to resolve a force into three parallel components not acting in one plane. 229. Moments of a Force with respect to a Pair of Rectangular Axes. — In fig. 94, let F be any single force; O an arbitrarily- assumed point, called the " origin of co-ordinates;" - X O + X, - Y O + Y, a pair of axes travers- ing 0, at right angles to each other and to the line of action of F. Let A B = ?/, be the com- mon perpendicular of F and O X; let A C = x, be the common perpendicular of F and O Y. x and y are the "rectangular co- ordinates" of the line of action of F relatively to the axes - X O + X, - Y O + Y, respec- tively. According to the arrange- ment of the axes in the figure, X is to be considered as positive to the right, and negative to the Y O + Y; and y is to be considered as positive to the left, and negative to the right, of - X O + X; right and left referring to the spectator's right and left hand. In the particular case represented, x and y are both positive. Forces, in the figure, are considered as positive upwards, and negative downwards ; and in the particular case represented, F is positive. At B conceive a pair of equal and opposite forces, F' and - F', to be applied ; F' being equal and parallel to F, and in the same direction. Then, as in Article 223, F is equivalent to the single force Fig. 94. left, of RESULTANT OF ANY SYSTEM OF PARALLEL FORCES IN ONE PLANE. 131 F' = F applied at B, combined with the couple constituted by F and - F' with the arm y, whose moment is ?/ F ; being positive in the case represented, because the couple is right-handed. Next, at the origin O, conceive a pair of equal and opposite forces, F" and - F", to be applied, F" being equal and parallel to F and F', and in the same direction. Then the single force F' is equivalent to the single force F" = F' = F applied at O, combined with the couple constituted by F' and - F" with the arm OB = £c, whose moment is - a? F; being negative in the case represented, because the couple is left-handed. Hence, it appears finally, that a force F acting in a line whose co-ordinates with respect to a pair of rectangular axes perpendicular to that line are x and y, is equivalent to an equal and parallel force acting through the origin, combined with two couples whose moments are, y F relatively to the axis O X, and - x'¥ relatively to the axis O Y right-handed couples being considered positive ; and + Y lying to the left of + X, as viewed by a spectator looking from 4- X towards O, with his head in the direction of positive forces. 230. Balance of any System of Parallel Forces in one Plane. — In order that any system of parallel forces whose lines of action are in one plane may balance each other, it is necessary and sufficient that the following conditions should be fulfilled : — First — (As already stated) that the algebraical sum of the forces shall be nothing. Secondly — That the algebraical sum of the moments of the forces relatively to any axis perpendicular to the plane in which they act shall be nothing, two conditions which are expressed symbolically as follows : — Let F denote any one of the forces, considered as positive or negative, according to the direction in which it acts; let y be the perpendicular distance of the line of action of this force from an arbitrarily assumed axis O X, 3/ also being considered as positive or negative, according to its direction ; then, 2-F^ 0; 2-^F = 0. In summing moments, right-handed couples are usually con- sidered as positive, and left-handed couples as negative. 231. Let II denote the Resultant of any System of Parallel Forces in one Plane, and 3/^, the distance of the line of action of that resultant from the assumed axis O X to which the positiona of forces are referred; then, B = 2-F; Vt 2 • F 132 PRINCIPLES OF STATICS. In some cases the forces may have no single resultant, 2 • F being = ; and then, unless the forces balance each other com- pletely, their resultant is a couple of the moment 2 • ?/ F. 232. Balance of any System of Parallel Forces. — In order that any system of parallel forces, whether in one plane or not, may balance each other, it is necessary and sufficient that the three following conditions shall be fulfilled : — First — (As already stated) that the algebraical sum of the forces shall be nothing. Secondly and Thirdly — That the algebraical sums of the moments of the forces, relatively to a pair of axes at right angles to each other, and to the lines of action of the forces, shall each be nothing, two conditions which are expressed symbolically as follows : — Let O X and O Y denote the pair of axes ; let F be the magnitude of any one of the forces; y its perpendicular distance from O X, and X its perpendicular distance from O Y ; then, 2-F = 0; S-2/F = 0; 2-ajF = 0; 233. Let R denote the Resultant of any System of Parallel Forces, and x^ and y^ the distances of its line of action from two rectangular axes; then, In some cases the forces may have no single resultant, 2 • F being = ; and then, unless the forces balance each other com- pletely, their resultant is a couple, whose axis, direction, and moment, are found as follows : — Let M^ = 2'2/F; Mj, = -2-^F; be the moments of the pair of partial resultant couples about the axes O X and O Y respectively. From O, along those axes, set off two lines representing respectively M^. and M^, ; that is to say, pro- portional to those moments in length, and pointing in the direction from which those couples must respectively be viewed in order that they may appear right-handed. Complete the rectangle whose sides are those lines; its diagonal will represent the axis, direction, and moment of tlie final resultant couple. Let M,. be the moment of this couple ; then and if d be the angle which its axis makes with X, TO FIND THE CENTRE OF PARALLEL FORCES. 133 234. To find the Centre of Parallel Forces.— Let O in fig. 95 be any convenient point, taken as the origin of co-ordinates, and O X, O Y, O Z, three axes of co-ordinates at right angles to each other. Let A be any one of the points to which the system of parallel forces in question is applied. From A draw x parallel to X, and perpendicular to the plane Y Z, y parallel to O Y, and perpendicular to the plane Z X, and z parallel to O Z, and perpendicular to the plane XY. .cc, 2/, and 2; are the rectangu- lar co-ordinates of A, which, being known, the position of A is deter- mined. Let F denote either the magnitude of the force applied at A, or any magnitude proportional to that magnitude, x, y, z, and F are supposed to be known for every point of the given system of points. The position of the centre of parallel forces depends solely on the proportionate magnitudes of the parallel forces, not on their absolute magnitudes, nor on the angular positions of their lines of actions; so that for any system of parallel forces another may be substituted in any angular position : this is the statement of the principle of the centre of parallel forces given at Article 201, page 119. This is evident since, in considering the relations of parallel forces, they are not considered with reference to any parti- cular plane, and hence these relations must hold for any plane. First, conceive all the parallel forces to act in lines parallel to the plane Y Z. Then the distance of their resultant, and of the centre of parallel forces from that plane is Fig. 95. 2F * ■CO Secondly, conceive all the parallel forces to act in lines parallel to the plane Z X. Then the distance of their resultant, and of the centre of parallel forces from that plane is 2-2/F .(2.) Thirdly, conceive all the parallel forces to act in lines parallel to the plane X Y. Then the distance of their resultant, and of the centre of parallel forces from that plane is z,= 2-F .(3.) 134 PRINCIPLES OF STATICS. If the forces have no single resultant, so that S • F = 0, there is no centre of parallel forces. This may be the case with pressures, but not with weights. If the parallel forces applied to a system of points are all equal and in the same direction, it is obvious that the distance of the centre of parallel forces from any given plane is simply the mean of the distances of the points of the system from that plane. Section 4. — Op any System op Forces. 235. Resultant and Balance of any BfsUm of Forces in One Plane. — Let the plane be that of the axes O X and O Y in fig. 95; and in looking from Z towards O, let Y lie to the right of X, so that rotation from X towards Y shall be right-handed. Let a? and y be the co-ordinates of the point of application of one of the forces, or of any point in its line of action, relatively to the assumed origin and axes. Resolve each force into two rectangular com- ponents X and Y, as in Article 215, page 125; then the rectangular components of the resultant are S * X and; S • Y ; its magnitude is given by the equation R2 = (S-X)U(2-Y)2, (1.) and the angle »^ which it makes with X is found by the equations 2-X . 2-Y .,, cosa, = -^— ; sin«,= -^ (2.) This angle is acute or obtuse according as 2 • X is positive or nega- tive; and it lies to the right or left of O X according as 2 • Y is positive or negative. • The perpendicular distance from of the line of action of any force is X sin a. -y cos «, and hence the resultant moment of the system of forces about the axis O Z is M = 2(ajY-2/X), (3.) and is right or left-handed according as M is positive or negative. The perpendicular distance of the resultant force R from is -4- (^•) Let x^ and y^ be the co-ordinates of any point in the line of action of that resultant; then the equation of that line is* x,^'Y-y,^-X = M (5.) * The method of obtaining this result by Co-ordinate Geometry is the RESULTANT AND BALANCE OF ANY SYSTEM OF FORCES. 135 If M = the resultant acts througli the origin O; if M has magnitude, and R = (in which case 2 • X = 0, 2 • Y = 0) the resultant is a couple. The conditions of equilibrium of the system of forces are S-X = Oj 2-Y = 0; M = (6.) 236. Resultant and Balance of any System of Forces. — To find the resultant and the conditions of equilibrium of any system of forces acting through any system of points, the forces and points are to be referred to three rectangular axes tof co- ordinates. As before, let O in fig. 95, p. 133, denote the origin ^f co-or- dinates, and O X, O Y, O Z, the three rectangular axes : and let then* be arranged so that in looking from X] (Y towards 211 Y > towards 0, rotation from < Z towards X y Z } shall appear right-handed. ( X towards Y } Let X, Y, Z, denote the rectangular components of any one of the forces; x, y, z, the co-ordinates of a point in its line of action. Taking the algebraical sums of all the forces which act along the same axes, and of all the couples which act round the same axes, following :-Let C=L, A B=E, ZXAB ==«,.; and let B G=a> and 0G=2^rhe the co-ordinates of the point E. Then by Trigonometry x sin or-sinO AC=cos CO A = sin X DOG=cosDGO=sinEGF and - cos ar=cos A C = sin C A = cosDOG. L=DC+OD=FE+OD =EG-sinEGF + O G • cos D G = Xr'Sm.Uf, - llr COSa, multiplying by R L • E,=M=a;r 'B 'sin a^.- ^/y • B * cos or = Xr •'2,Y-yr"2,X. Fig. 96. by substituting the values in Equation 2 supra. 136 PRINCIPLES OF STATICS. the six following quantities are found, which compose the resultant of the given system of forces : — Forces. 2-X; 2'Y; S'Z; (!•) Couples. . about OX; M, =2 (2/Z-;s Y); ) „ OY; M,=^^{zX-xZ);\ .(2.). found as already explained in Article 235. The three forces are equivalent to a single force R=a/ |(2-X)2 + (2-Y)2 + (2-Z)2l -.(3.) acting through O in a line which makes with the axes the angles given by the equations 2-X ^ 2-Y S-Z ,,, cos cc = -^--; cos /3=— ^; COS y= -J, (4.) The three couples, M^, Mg, Mg, are equivalent to one couple, whose magnitude is given by the equation M= J{Ml + M.l + M^, (5.) and whose axis makes with the axes of co-ordinates the angles given by the equations . ^1 Mo Mo ,-, 1 • T- I I denote respectively the angles I /-^ ^^ I m winch I ^ j ^^^^ ^^y J^ ^^.^ y jj ^j^^= I Y I The conditions of equilibrium of the system of forces may be expressed in either of the two following forms : — 2-X = 0; 2- Y = 0; 2-Z=0; Mi=:0;M2 = 0; M3-O; (7.) or E = 0; M = (8.) When the system is not balanced, its resultant may fall under one or other of the following cases : — Case I. — When M = 0, the resultant is the single force E, acting through O. RESULTANT AND BALANCE OF ANY SYSTEM OF FORCES. 137 Case II. — When the axis of M is at right angles to the direction of R, — a case expressed by the following equation : — cos ec cos "^ + cos /3 COS /^ + COS y cos ^ = j (9.) (an equation of Co-ordinate Geometry) the resultant of M and H is a single force eqiial and parallel to R, acting in a plane perpendicular to the axis of M, and at a perpen- dicular distance from O given by the equation M L = f (10.) Case III. Wheii E = 0, there is no single resultant; and the only resultant is the couple M. Case IY. When the axis of ]M is parallel to the line of action of R, that is, when either A = «; ^ = /3; p^y, (11.) or X= -u; f^= -/3; u= -y; (12.) there is no single resultant; and the system of forces is equivalent to the force E, and the couple M, being incapable of being farther simplified. Case V. — When the axis of M is oblique to the direction of R, making with it the angle given by the equation cos ^ = COS A cos et + COS fC COS /3 + COS j, COS y,....(13.) the couple M is to be resolved into two rectangular components, viz : — M sin 6 round an axis perpendicular to R, and in the plane containing the direction of R and of the axis of M; M cos 6 round an axis parallel to R. (14.) The force R and the couple M sin ^ are equivalent, as in Case II., to a single force equal and parallel to R, whose line of action is in a plane perpendicular to that containing R and axis of M, and whose perpendicular distaflce from O is L = ^S^ (15.) JX The couple M cos ff, whose axis is parallel to the line of action of R, is incapable of further combination. Hence it appears finally, that every system of forces which is not self-balanced, is equivalent either, (A); to a single force, as in 138 PRINCIPLES OF STATICS. Cases I. and II. (B); to a couple, as in Case III. (C); tea force, combined with a couple whose axis is parallel to the line of action of the force, as in Cases IV. and Y. This can occur with inclined forces only; for the resultant of any number of parallel forces is either a single force or a couple. 237, Parallel Projections or Transformations in Statics. — If two figures be so related, that for each point in one there is a corre- sponding point in the other, and that to each pair of equal and parallel lines in the one, there corresponds a pair of equal and parallel lines in the other, those figures are said to be parallel PROJEiCEriONS of each other. The lotions between such a pair of figures is expressed alge- Jbf ai^caJIy as follows : — Let any figure be referred to axes of co- otdinalRB, whether rectangular or oblique ; let x, y, z, denote the (?o-ordinates of any point in it, which may be denoted by A : let a second figure be constructed from a second set of axes of co-ordinates, either agreeing with, or difiering from, the first set as to rectangu- larity or obliquity; let x', y\ z, be the co-ordinates in the second ^[giir^j, of the point A' which corresponds to any point A in the firsti figure, and let those co-ordinates be so related to the co-ordi- nates of A, that for each pair of corresponding points, A, A', in the two figures, the three pairs of corresponding co-ordinates shall bear to each other three constant ratios, such as x' y' J z' - =a; - =6; - =c: X y z then are those two figures parallel projections of each other. For example, all circles and ellipses are parallel projections of each other; so are all spheres, spheroids, and ellipsoids; so are all triangles; so are all triangular pyramids; so are all cylinders; so are all cones. The following are the geometrical properties of parallel projec- tions which are of most importance in statics : — I. A parallel projection of a system of three points, lying in one straight line and dividing it in a given proportion, is also a system of three points, lying in one straight line and dividing it in the same proportion. II. A parallel projection of a system of parallel lines, whose lengths bear given ratios to each other, is also a system of parallel lines whose lengths bear the same ratios to each other. III. A parallel projection of a closed polygon is a closed polygon. IV. A parallel projection of a parallelogram is a parallelogram. V. A parallel projection of a parallelepiped is a parallelopiped. ' VI. A parallel projection of a pair of parallel plane surfaces, PARALLEL PROJECTIONS OR TRANSFORMATIONS IN STATICS. 139 whose areas are in a given ratio, is also a pair of parallel plane surfaces, whose areas are in the same ratio. VII. A parallel projection of a pair of volumes having a given ratio, is a pair of volumes having the same ratio. The following are the mechanical properties of parallel projec- tions in connection with the principles set forth in this section : — YIII. If two systems of points be parallel projections of each other; and if to each of those systems there be applied a system of parallel forces bearing to each other the same system of ratios, then the centres of parallel forces for those two systems of points will be parallel projections of each other, mutually related in the eame manner with the other pairs of corresponding points in the two systems. IX. If a balanced system of forces acting through any .system of points be represented by a system of lines, then will any parallel projection of that system of lines represent a balanced system of forces; and if any two systems of forces be represented by lines which are parallel projections of each other, the lines, or sets of lines, representing their resultants, are corresponding parallel pro- jections of each other, — it being observed that couples are to be represented by pairs of lines, as pairs of opposite forces, or by areas, and not by single lines along their axes. uo CHAPTER III. DISTRIBUTED FORCES. Section 1. — Centres of Gravity. 238. Centre of Gravity of a Symmetrical Homogeneous Body. — If a body is homogeneous, or of equal specific gravity througliout, and so far symmetrical as to have a centre of figure ; that is, a point within the body, which bisects every diameter of the body drawn through it, that point is also the centre of gravity of the body. Amongst the bodies which answer this description, are the sphere, the ellipsoid, the circular cylinder, the elliptic cylinder, prisms whose bases have centres of figure, and parallelopipeds, whether right or oblique. 239. The Common Centre of Gravity of a Set of Bodies whose several centres of gravity are known, is the centre of parallel forces for the weights of the several bodies, each considered as acting through its centre of gravity. (See Article 234, p. 133.) 240. Planes of Symmetry —Axes of Symmetry. — If a homogeneous body be of a figure which is symmetrical on either side of a given plane, the centre of gravity is in that plane. If two or more such planes of symmetry intersect in one line, or axis of symmetry, the centre of gravity is in that axis. If three or more planes of symmetry intersect each other in a point, that point is the centre of gravity. 241. To find the Centre of Gravity of a Homogeneous Body of any Figure, assume three rectangular co-ordinate planes in any convenient position, as in fig. 95, p. 133. To find the distance of the centre of gravity of the body from one of those planes (for example, that of Y Z), conceive the body to be divided into indefinitely thin plane layers parallel to that plane. Let s denote the area of any one of those layers, and d x its thickness, so that sclxi^ the volume of the layer, and = \sd. the volume of the whole body, being the sura of the volumes of CENTRE OF GRAVITY FOUND BY SUBTRACTION. 141 the layers. Let x be the perpendicular distance of the centre of the layer sdx from the plane of Y Z. Then the perpendicular distance x^ of the centre of gravity of the body from that plane is giveii by the equation J~ ■(!•) Find, by a similar process, the distances 2/oj ^ot of the centre of gravity from the other two co-ordinate planes, and its position will be completely determined. If the centre of gravity is previously known to be in a particular plane, it is sufficient to find by the above process its distances from two planes perpendicular to that plane and to each other. If the centre of gravity is previously known to be in a particular line, it is sufficient to find its distance from one plane, perpendicular to that line. 242. If the Specific Gravity of the Body Varies, let w be the mean heaviness of the layer sdx, so that W = j wsdx, is the weight of the body. Then xwsdx f- w (2-) 243. Centre of Gravity found by Addition. — When the figure of a body consists of parts, whose resj)ective centres of gravity are known, the centre of gravity of the whole is to be found as in Article 239. 244. Centre of Gravity found by Subtraction. — When the figure of a homogeneous body, whose centre of gravity is sought, can be made by taking away a figure whose centre of gravity is known from a larger figure whose centre of gravity is known also, the following method may be used : — Let A C D be the larger figure, Gj its known centre of gravity, Wj its weight. Let A B E be the smaller figure, whose centre of gravity Gg is known, Wg its weight. Let E B C D be the figure whose centre of gravity Gg is sought, made by taking away ABE from A C D, so that its weight is W3 = W,-W2. 142 PRINCIPLES OF STATICS. Join Gi G2'} Gts ^^^^ be in tlie prolongation of that straight line beyond G^. In the same straight line produced, take any point O as origin of co-ordinates. Make O G^ unknown quantity) = ^a- Then h'j ^ ^2 = ^2) O G3 (the X, x,W, Wj - w, ,(3.) 245. Centre of Gravity Altered by Transposition.— In fig. 98, let A B C D be a body of the weight Wo, whose centre of gravity Gq is known. Let the figure of this body be altered, by trans- posing a part whose weight is Wj, from the position E C F to the position ¥ D H, so that the new figure of the body is A B H E. Let Gi be the original, and Gg the new position of the centre of gravity of the transposed part. Then the centre of gravity of the whole body will be shifted to G3, in a direction Go G3 parallel to G2 Gi, and through a distance given by the formula. Fm. 98. Go Gg = Gi Gj W, .(4.) 246. Centre of Gravity found by Projection or Transformation. — If the figures of two homogeneous bodies are parallel projections of each other, the centres of gravity of those two bodies are corres- ponding points in those parallel projections. To express this symbolically, — as in Article 237, let x, y, z, be the co-ordinates, rectangular or oblique, of any point in the figure of the first body; x', y', z', those of the corresponding point in the second body ; x^, yo, ^o) ^^^ co-ordinates of the centre of gravity of the first body; x\ second body, then y^ those of the centre of gravity of the Xo ^' Vo y' ^0 ^' .(5.) This theorem facilitates much the finding of the centres of gravity of figures which are parallel projections of more simple or more symmetrical figures. STRESS — ITS INTENSITY. 143 For example, let it be supposed that tlie centre of gravity of a sector of a circular disc has been found (Case IX. Article 44), and let it be required to find the centre of gravity of a sector of an elliptic disc. In fig. 99, let A B' A B' be the ellipse, A O A = 2 a, and B' B' = 2 6, its axes, and C O D' the sector whose centre of gravity is required. About the centre of the ellipse, O, describe the circle, A B A B, whose radius is the semi- axis major a. Through C and D' respectively draw E C C and FD'D, parallel to O B, and cutting the circle in C and D respectively ; the circular sector C O D is the parallel projection of the elliptic sector C D'. Let G- be the centre of gravity of the sector of the circular disc, its co-ordinates being Then the co-ordinates of the centre of gravity G' of the sector of the elliptic disc are (6. 247. Centre of Gravity found Experimentally.— The centre of gravity of a body of moderate size may be found approximately by experiment, by hanging it up successively by a single cord in two different positions, and finding the single point in the body which in both positions is intersected by the axes of the cord. Section 2. — Of Stress. 248. Stress — its Intensity. — The word Stress has been adopted as a general term to comprehend various forces which are exerted between contiguous bodies, or parts of bodies, and which are dis- tributed over the surface of contact of the masses between which they act. The Intensity of a stress is its amount in units of weight, divided by the extent of the surface over which it acts, in units of area. 144 TRINCIPLES OF STATICS. The following taLle gives a comparison of various units in which the intensity of stress is expressed : — , Pounds on the Pounds on the square foot. square inch. One pound on the square inch,.... 144 1 One pound on the square foot, 1 yij- One inch of mercury (that is, weight of a column of mercury at 32** Fahr., one inch high), 70-73 0-4912 One foot of water (at 39^-1 Fahr.), 62-425 0-4335 One inch of water (at 39°-l Fahr.), 5-2021 0-036125 One foot of water (at 62° Fahr.),... 62-355 0-43302 One inch of water (at 62° Fahr. ), . . . 5-19625 0036085 One atmosphere, of 29*922 inches of mercury, or 760 millimetres, 2116-4 14-7 One foot of air, at 32° Fahr., and under the pressure of one atmo- sphere, 0-080728 0-0005606 One kilogramme on the square metre, 0-20481 0-00142228 One kilogramme on the square millimetre, 204810 1422-28 One millimetre of mercury, 2-7847 0-01934 249. Classes of Stress. — The various kinds of stress may be thus classed : — I. Thrust, or Pressure, is the force which, acts between two con- tiguous bodies, or parts of a body, when each pushes the other from itself. II. Pull, or Tension, is the force which acts between two con- tiguous bodies, or parts of a body, when each draws the other towards itself. Pressure and tension may be either normal or oblique, relatively to the surface at which they act. III. Shear, or Tangential Stress, is the force which acts between two contiguous bodies, or parts of a body, when each draws the other sideways, in a direction parallel to their surface of contact. In expressing a Thrust and a Pull in parallel directions alge- braically, if one is treated as positive, the other must be treated as negative. The choice of the positive or negative sign for either is a matter of convenience. The word "Pressure," although, strictly speaking, equivalent to "thrust,'^ is sometimes applied to stress in general; and when this is the case, it is to be understood that thrust is treated as positive. The following are the processes for finding the magnitude of the resultant of a stress distributed over a plane surface, and the centre IN STKESS OF VARYING INTENSITY. 145 of stress; that is, the point where the line of action of that resultant cuts the plane surface : — 250. In Stress of Uniform Intensity, the magnitude of the re- sultant is the product of that intensity and the area of the surface ; and the centre of stress is at the centre of magnitude of the surface. Or in symbols, let S be the area of the surface, p the intensity of the stress, P its resultant, then — P=pS. 25 L In Stress of Varying Intensity, but of One Sign, there is all tension, or all pressure, or all shear in one direction. In fig. 100, let A A be the given plane surface at which the stress acts; X, O Y, two rectangular axes of co-ordinates in its plane; ^ O Z, a third axis perpendicular to that plane. Conceive a solid to exist, bounded at one end by the given plane surface A A, laterally by a cylindrical or prismatic surface generated by the motion of a straight line parallel to O Z round the outline of A A, and at the other end by a surface B B, of such a figure, that its ^^' ordinate z at any point shall be proportional to the intensity of the stress at the point a of the surface A A from which that ordinate proceeds, as shewn by the equation z = P^ (1.) where p represents the intensity of the stress and w the heaviness, or weight per unit. Conceive the surface A A to be divided into an indefinite number of small rectangular areas, each denoted hy dxdy, and so small that the stress on each is sensibly uniform; the entire area being B= I I dxdy. The volume of the ideal solid will be Y= f fz'dxdy (2.) So that if it be conceived to consist of a material whose heaviness isw^~, the amount of the stress will be equal to the weight of the z solid ; that is to say, -£ = j jpdxdy = wY (3.) 146 / PRINCIPLES OF STATICS. ; The centre of stress is the point on the surface A A perpendicu- larly opposite the centre of gravity of the ideal solid. The simplest, and at the same time the commonest, case of this kind is where the stress is uniformly-varying; that is, where its intensity at a given point is simply proportional to the per- pendicular distance of that point from a given straight line in the plane of the surface A A. To express this symbollically, take the straight line in question for the axis Y; conceive the substance to be divided into bands by lines jDarallel toO Y; let y denote the length of one of these bands, and d x its breadth, so that y dx h its area, and S = ydx the area of the whole surface. Let x be the perpendicular distance of the centre of a band from the line of no stress Y, and let the intensity of the stress there be P = ax; (4.) a being a constant coefficient; then the amount or resultant of the stress is 'P = jpydx = a fxydx', (5.) and the perpendicular distance of the centre of stress from O Y is I pxydx j x^ydx; j pydx .(6.) 252. In Stress of Contrary Signs, for example, pressure at one part of the surface and tension at another, the resultants and centres of stress of the pressure and tension are to be found separately. Those partial resultants are then to be treated as a pair of parallel forces acting through the two respective centres of stress; their final resultant will be equal to their difference, if any, acting through a point found as in Article 226, page 128. If the total pressure and total tension are equal to each other, they have no single resultant and no single centre of stress : their resultant being a couple, whose moment is equal to the total stress of either kind multiplied by the perpendicular distance between the resultant of the pressure and the resultant of the tension. principles of hydrostatics. 147 Section 3. — Principles of Hydrostatics and Internal Stress of Solids. 253. Pressure and Balance of Fluids: Principles of Hydro- statics. — Fluid is a term opposed to solid, and comprehending the liquid and gaseous conditions of bodies. The property common to the liquid and the gaseous conditions is that oinot tending to preserve a definite shape, and the possession of this property by a body in perfection throughout all its parts, constitutes that body a perfect fiuid. A necessary consequence of that property is the following prin- ciple, which is the foundation of the whole science of hydro- statics: — I. In a perfect fiuid, when still, the pressure exerted at a given point is normal to the surface on which it acts, and of equal intensity/ for all positions of tJiat surface. The following are some of the most useful consequences of that principle : — II. A surface of equal pressure in a still fiuid mass is everywhere perpendicular to the direction of gravity ; that is, horizontal through- out. In other words, the pressure at all points at the same level is of equal intensity. III. The intensity of the pressure at the lower of tvm points in a still fiuid mass is greater than the intensity at the higher point, hy an amount equal to the weight of a vertical column of the fiuid whose height is the difference of elevation of the points, and base an unit of area. To express this symbolically, let p^ denote the intensity of the pressure at the higher of two points in a fluid mass, and p^ the intensity at a point whose vertical depth below the former point is X. Let w be the mean heaviness of the layer of fluid between those two points; then Pl=P(i+V)X (1.) In a gas, such as air, w varies, being nearly proportional to p ; but in a liquid, such as water, the variations of w are too small to be considered in practical cases. For example, let the upper of the two points be the surface of a mass of water where it is exposed to the air j then p^ is the atmos- pheric pressure; let the depth x of the second point below the surface be given in feet, and let the temperature be 39°-l ; then j?i in lbs. on the square foot=j(?o + 62*425 x (2.) In many questions relating to engineering, the pressure of the atmosphere may be left out of consideration, as it acts with sensibly equal intensity on all sides of the bodies exposed to it, and so balances its own action. The pressure calculated, in such cases, is 148 PRINCIPLES OF STATICS. the excess of the pressure of the water above the atmospheric pressure, which may be thus expressed, — p'=/?i -^0 = 62-425 a; nearly (3.) ly. The pressure of a liquid on a floating or immersed body, is equal to the weight of the volume of fluid displaced by that body; and the resultant of that pressure acts vertically upwards through the centre of gravity of that volume; which centre of gravity is called the ^^ centre of buoyancy " V. The pressure of a liquid against a plane surface immersed in it is perpendicular to that surface in direction '. its magnitude is equal to the weight of a volume of the liquid, found by multiplying the area of the surface by the depth to which its centre of gravity is immersed. VI. The centre of pressure on such a surface, if the surface is horizontal, coincides with its centre of gravity; if the surface is vertical or sloping, the centre of pressure is always below the centre of gravity of the surface, and is found by considering that the pressure is an uniformly-varying stress, whose intensity at a given point varies as the distance of that point from the line where the given plane surface (produced if necessary) intersects the upper surface of the liquid. To express the last two principles by symbols in the case in which the pressed surface is vertical or sloping, let the line where the plane of that surface cuts the upper surface of the liquid be taken as the axis Y. Let 6 denote the angle of inclination of the pressed surface to the horizon. Conceive that surface to be divided by parallel horizontal lines into an indefinite number of narrow bands. Let y be the length of any one of those bands, d x its breadth, x the distance of its centre from O Y ; then ydx is its area, x sin $ the depth at which it is immersed; and if w be the weight of unity of volume of the fluid, the intensity of the pressure on that band is p = wxBmff (4.) The whole area of the pressed surface, being the sum of the areas of all the bands, is S = jydx; the whole pressure upon it is P= jpydx = W8m filxydx; (5.) the mean intensity of the pressure is p \ py ^^ I ^y dx - z^wBinS ; (6.) dx j ydx \y COMPOUND INTERNAL STRESS OF SOLIDS. 149 and the distance of the centre of pressure from Y is Ixpydx ix^ydx Xf\ — j xydx .(7.) For example, let the sloping pressed surface be rectangular, like a sluice, or the back of a reservoir- wall ; and in the first instance, let it extend from the surface of a mass of water down to a distance x-^, measured along the slope, so that its lower edge is immersed to the depth x-^^ sin 6. Then its centre of gravity is immersed to the depth x^ sin ^ -^ 2, and the mean intensity of the pressure in lbs. on the square foot, is P 62-4 X. sin 4 ,_ ^ s= 2 ; (^> The breadth y is constant; so that the area of the surface is ^ = x-^y; and the total pressure is ■p _ 62-4 a?fy sin <> ^ The distance of the centre of pressure from the upper edge is 2 a^o = 3 ^1" (1^) Next, let the upper edge, instead of being at the surface of the water, be at the distance x^ from it, so as to be immersed to the depth x^ sin 6. Then the centre of gravity of the pressed surface is immersed to the depth {x^ + x^ sin ^-r2, and the mean intensity of the pressure upon it, in lbs. on the square foot, is P _ 62-4 {x^ + x^ sin 6 . S" 2 ^ ^^'\ the area of the surface is {x^ ~ x.^ y, and the total pressure on it ■p ^ 62-4 (a^-apy sin xn. The locus of the point M is a circle of the radius , and that of the point R, an ellipse whose semi-axes are jOj and p^, and which may be called the Ellipse of Stress, because its semi- diameter in any direction represents the intensity of the stress in that direction. 259. Deviation of Principal Stresses by a Shearing Stress. — Problem. Let p^ and p^ denote the original intensities of a pair of principal stresses acting at right angles to each other through one particle of a solid. Suppose that with these there is combined a shearing stress of the intensity q, acting in the same plane with the original pulls or thrusts ; it is required to find the new inten- sities and new directions of the principal stresses. To assist the conception of this problem, the original stresses referred to are represented in fig. 104, as acting through a particle ^-E, of the form of a square prism. The principal stresses, both original and new, are represented as tensions, although any or all of them might be pressures. In the formulae annexed, tensions are considered positive, pressures negative ; angles lying to the right of A A are considered as positive, to the left as negative; and a shear- ing stress is considered as positive or negative according as it tends to make the upper right- hand and lower left-hand corner of the square particle acute or obtuse. The arrows A A represent the greater original Fig. 104. tension p^; the arrows B B, the less original tension;?^; C, C, D, D, represent the positive shear of the inten- sity q, as acting at the four faces of the particle. The combination of this shear with the original tensions is equivalent to a new pair of principal tensions, oblique to the original pair. The greater new (3.) FRICTION. 153 principal tension, p^, is represented by the arrows E, E ; it deviates to the right of p^ through an angle which will be denoted by 6. The less new principal tension p^ is represented by the arrows F, F ; it deviates through the same angle to the right of p^,. Tljen the intensities of the new principal stresses are given by the equations, and the double of the angle of deviation by either of the following, tan2<>= -^^ ', orcotan2^=^"„^ (4.) The greatest value of ^ is 45°, when p^ = Py. The new principal stresses are to be conceived as acting normally on the faces of a new square prism. 260. Parallel Projection of Distributed Forces. — In apjjlying the principles of parallel projection to distributed forces, it is to be borne in mind that those principles, as stated in Article 237, are applicable to lines representing the amounts or resultants of distri- buted forces, and not their intensities. The relations amongst the intensities of a system of distributed forces, whose resultants have been obtained by the method of projection, are to be arrived at by a subsequent process of dividing each projected resultant by the projected space over which it is distributed. 261. Friction is that force which acts between two bodies at their surface of contact so as to resist their sliding on each other, and which depends on the force with which the bodies are pressed together. It is a kind of shearing stress. The following law respecting the friction of solid bodies has been ascertained by experiment : — The friction which a given pair of solid bodies, with their surfaces in a given condition, are capable of exerting, is simply proportional to the force with which they are pressed together. If a body be acted upon by a force tending to make it slide on another, then so long as that force does not exceed the amount fixed by this law, the friction will be equal and opposite to it, and will balance it. There is a limit to the exactness of the above law, when the pressure becomes so intense as to crush or indent the parts of the bodies at and near their surface of contact. At and beyond that limit the friction increases more rapidly than the pressure; but 154 PRINCIPLES OF STATICS. that limit ought never to be attained in any structure. For some substances, especially those whose surfaces are sensibly indented by a moderate pressure, such as timber, the friction between a pair of surfaces which have remained for some time at rest relatively to each other, is somewhat greater than that between the same pair of surfaces when sliding on each other. That excess, how- ever, of the friction of rest over the friction of motion, is instantly destroyed by a slight vibration; so that the friction of motion is alone to be taken into account, as contributing to the stability of a structure. The friction between a pair of surfaces is calculated by multiply- ing the force with which, they are directly pressed together, by a factor called the coefficient of friction, which has a special value depending on the nature of the materials and the state of the , surfaces. Let F denote the friction between a pair of sur- faces; N, the force, in a direction perpendicular to the surfaces, with which they are pressed together ; and / the coeflB.cient of friction; then F=/N (1.) The coefficient of friction of a given pair of surfaces is the tangent of an angle called the angle of repose, being the greatest angle which an oblique pressure between the surfaces can make with a perpendicular to them, without making them slide on aceh other. Let P denote the amount of an oblique pressure between two plane surfaces, inclined to their common normal at the angle of repose of the tanf?ents of the inclinations is to be ( difterence J ^ used, according as the inclinations are < • -i . r 272. Open Polygonal Frame.— When the frame, instead of being closed, as in fig. 114, is converted into an open frame, by the omis- sion of one bar, such as E, the corresponding modification is made in the diagram of "inclined forces, fig. 115, by omitting the lines O E, D E, E A, and in the diagram of parallel forces, fig. 116, by omitting the line E. Then, in both diagrams, D O and O A represent the supporting forces respectively, equal and directly opposed to the stresses along the extreme bars of the frame, D and A, which must be exerted by the supports (called in this case abutments), at the points 4 and 5, against the ends of those bars, in order to maintain the equilibrium. In the case of parallel loads, the following formulae give the horizontal stress and supporting pressures. Let ia and i^ denote the angles of inclination of the bars D and A respectively. Let R^ = O D and P„ = O A be the stresses along them. Let 2 • P = A D denote the total load on the frame; then, jj_ 2 • P . tan i^ + tan'i^ ' ^ '' E^ = H -sec 4; Pa = H -seci^ (2.) 273. Polygonal Frame— Stability.— The stability or instability of a polygonal frame depends on the principles stated in Article 207, page 159, viz., that if a bar be free to change its angular position, then if it is a tie it is stable, and if a strut, unstable; and that a strut may be rendered stable by fixing its ends. For example, in the frame of fig. 114, E is a tie, and stable; A, B, C, and D, are struts, free to change their angular position, and therefore unstable. But these struts may be rendered stable in the plane of the frame by means of stays; for example, let two stay-bars connect the joints 1 with 4, and 3 with 5; then the points 1, 2, and 3, are all fixed, so that none of the struts can change their angular posi- 166 THEORY OF STRUCTURES. tions. The same effect might be produced by two stay -bars con- necting the joint 2 with 5 and 4. The frame, as a whole, is unstable, as being liable to overturn laterally, unless provided with lateral stays, connecting its joints with fixed points. Now, suppose the frame to be exactly inverted, the loads at 1,2, and 3, and the supporting forces at 4 and 5, being the same as before. Then E becomes a strut; but it is stable, because its ends are fixed in position ; and A, B, C, and D becomes ties, and are stable without being stayed. An open polygon consisting of ties, such as is formed by A, B, C, and D, when inverted, is called by mathematicians, a funicular polygon, because it may be made of ropes. It is to be observed, that the stability of an unstayed polygon of ties is of the kind which admits of oscillation to and fro about the position of equilibrium. That oscillation may be injurious in practice, and stays may be required to prevent it. 274. Bracing of Frames. — A brace is a stay-bar on which there is a permanent stress. If the distribution of the loads on the joints of a polygonal frame, though consistent with its equilibrium as a whole, be not consistent with the equilibrium of each bar, then, in the diagram of forces, when converging lines respectively parallel to the lines of resistance are drawn from the angles of the polygon of external forces, those converging lines, instead of meet- ing in one point, will be found to have gaps between them. The lines necessary to fill up those gaps will indicate the forces to be supplied by means of the resistance of braces.* The resistance of a brace introduces a pair of equal and opposite forces, acting along the line of resistance of the brace, upon the pair of joints which it connects. It therefore does not alter the resultant of the forces applied to that pair of joints in amount nor in position, but only the distribution of the components of that resultant on the pair of joints. To exemplify the use of braces, and the mode of determining the stresses on them, let fig. 117 represent a frame such as frequently * This method of treating braced frames contains an improvement sug- gested by Prof. Clerk Maxwell in 1867. BRACING OF FRAMES. 167 TJ.C.D occurs in iron roofs, consisting of two struts or rafters, A and E, and three tie-bars, B, C, and T>, form- ing a polygon of five sides, jointed at 1, 2, 3, 4, 5, loaded vertically at 1, and supported by the vertical resistance of a pair of walls at 2 and 5. The joints 3 and 4 having no loads applied to them, are connected with 1 by the braces 1 4 and 1 3. To make the diagram of forces (fig. 118), draw the vertical line E, A, as in Article 271, to represent the direction of the load and of the supporting forces. The two segments of that line, A B and D E, are to be taken to represent the supporting forces at 2 and 5 ; and the whole line E A will represent the load at 1. From the ends, and from the point of division of the scale of external forces, E A, draw straight lines parallel respectively to the lines of resistance of the frame, each line being drawn from the point in E A that is marked with the corresponding letter. Then A a and B h, meeting at a, b, will represent the stresses along A and B respectively ; and E e and D d, meeting in D e, will represent the stresses along D and E respectively ; but those four lines, instead of meeting each other and C c parallel to C in one point, leave gaps, which are to be filled up by drawing straight lines parallel to the braces: that is to say, from a, b, to c, parallel to 1 3; and from d, e, to c parallel to 4 1. Then those straight lines will represent the stresses along the braces to which they are respectively parallel; and C c will represent the tension along C. To each joint in the frame, fig. 117, there corre- sponds, in fig. 118, a triangle, or other closed polygon, having its sides respectively parallel, and therefore proportional, to the forces that act at that joint. For example, Joints, 1, 2, 3, 4, 5, Polygons, EAaceE; ABbA; BcbB; Bdcl); D E eD. The order of the letters indicates the directions in which the forces act relatively to the joints. Another method of treating simple cases of bracing is illu^Cr9«<;ed by fig. 119. A and B are two struts, forming 168 THEORY OF STRUCTURES. one straight bar; C and D are two equal tie-rods ; E, a stmt brace. A vertical rod P is applied at the joint 1, between A and B; two vertical supporting pressures, eacli denoted by R = P h- 2, act at the joints 4 and 2. The joint 3 has no external load. Fig. 120 is the diagram of forces, constructed as follows: — Through a point O draw O B A parallel to A and B, O C parallel to C, and O D parallel to D. Make O D = C; join CD; this line will be parallel to the brace E, and perpendicular to O A. Through D and C draw vertical lines I) B, C A; these, being equal to each other, are to be taken to represent the two sup- porting pressures E; and their sum D B + A C will represent the load P. The equal tensions on C and D will be represented by C and O B, and the thrusts along A, B, and E, by O A, O B, and C D. The polygon of external forces in this case is the crossed quad- rilateral A C D B, in which C A and B D represent (as already stated) the supporting pressures, and D C and A B the components of the load P respectively parallel and perpendicular to the brace E. When A and B are horizontal, and E vertical, A B in fig. 120 vanishes, and B D and C A coincide with the two halves of C D. 275. Rigidity of a Truss. — The word truss is applied in car- pentry to a triangular frame, and to • a polygonal frame to which rigidity is given by staying and bracing, so that its figure shall be incapable of alteration by turning of the bars about their joints. If each joint were like a hinge, incapable of offering any resistance to alteration of the relative angular position of the bars connected by it, it would be necessary, in order to fulfil the condition of rigidity, that every polygonal frame should be divided by the lines of resistance of stays and braces into triangles and other polygons, so arranged that every polygon of four or more sides should be surrounded by triangles on all but two sides and the included angle at farthest : for every unstayed polygon of four sides or more, with flexible joints, is flexible, unless all the angles except one be fixed by being connected with triangles. Sometimes, however, a certain amount of stiffness in the joints of a frame, and sometimes the resistance of its bars to bending, is relied upon to give rigidity to the frame, when the load upon it is subject to small variations only in its mode of distribution. For example, in the truss of fig. 121, the tie-beam A A is made in one piece, or in two or more pieces so connected together as to act like one piece; and part of its weight is suspended from the joints C, C, by the rods C B, C B, ^^^' ^^^' These rods also serve to make the re- sistance of the tie-beam A A to being bent act so as to prevent the /\ 7\ ^ y/ ^ % ' L \ ■4M SECONDARY AND COMPOUND TRUSSING. 169 struts A C, C C, C A, from deviating from their proper angular positions, by turning on the joints A, C, C, A. If A B, B B, and B A, were three distinct pieces, with flexible joints at B B, it is evident that the frame might be disfigured by distortion of the quadrangle B C C B. The object of stiffening a truss by braces is to enable it to sustain loads variously distributed; for were the load always distributed in one way, a frame might be designed of a figure exactly suited to that load, so that there should be no need of bracing. The variations of load produce variations of stress on all the pieces of the frame, but especially on the braces; and each piece must be suited to withstand the greatest stress to which it is liable. Some pieces, and especially braces, may have to act sometimes as struts and sometimes as ties, according to the mode of distribution of the load. 276. Secondary and Compound Trussing. — A secondary truss is a truss which is supported by another truss. When a load is distributed over a great number of centres of resistance, it may be advantageous, instead of connecting all those centres by one polygonal frame, to sustain them by means of several small trusses, which are supported by larger trusses, and so on, the whole structure of secondary trusses resting finally on one large truss, which may be called the primary truss. In such a combina- tion the same piece may often form part of different trusses ; and then the stress upon it is to be determined according to the follow- ing principle : — When the same bar forms at the same time part of two or more different frames, the stress on it is the resultant of the several stresses to which it is subject by reason of its position in the several frames. In a Compound Truss, several frames, without being distinguish- able into primary and secondary, are combined and connected in such a manner that certain pieces are common to two or more of them, and require to have their stresses determined by the principle above stated. Example. — Fig. 122, represents a kind of secondary trussing common in the framework of iron roofs. Fis. 122. The entire frame is supported by pillars at 2 and 3, each of which sustains in all, half the weight. 170 THEORY OF STRUCTURES. 1 2 3 is the primary truss, consisting of two rafters 1 3, 1 2, and a tie-rod 2 3. The weight of a division of the roof is distributed over the rafters. The middle point of each rafter is supported by a secondary truss; one of those is marked 14 3; it consists of a strut, 1 3 (the rafter itselfj, two ties 4 1, 4 3, and a strut-brace, 5 4, for transmitting the load, applied at 5, to the point where the ties meet. Each of the two larger secondary trusses just described supports two smaller secondary trusses of similar form and construction to itself; two of those are marked 1 7 5, 5 6 3; and the subdivision of the load might be carried still farther. In determining the stresses on the pieces of this structure, it is indifferent, so far as mathematical accuracy is concerned, whether we commence with the primary truss or with the secondary trusses; but by commencing with the primary truss, the process is rendered more simple. (1.) Primary Truss 12 3. Let W denote the weight of the roof; then I" W is distributed over each rafter, the resultants acting through the middle points of the rafters. Divide each of those resultants into two equal and parallel components, each equal to \ W, acting through the ends of the rafter; then ;| W is to be considered as directly supported at 3, J W at 2, and :j W + ;^ W = ^ W at 1 ; therefore the load at the joint 1 is P = J W. Let i be the inclination of the rafters to the horizon ; then by the equations of Article 270. H- ^ = vv , .J. 2 tan ?^ 4 tan i' ^ ' This is the pull upon the horizontal tie- rod of the primary truss, 2 3 ; and the thrust on each of the rafters 1 3, 1 2, is given by the equation . W cosec i 1^ R = H sec 1 = , (2.) (2.) Secondary Truss 14 3 5. The rafter 1 3 has the load J W dsstributed over it; and reasoning as before, we are to leave two quarters of this out of the calculation, as being directly supported at 1 and 3, and to consider one-half, or \ W, as being the vertical load at the point 5. The truss is to be considered as consisting of a polygon of four pieces, 5 1, 1 4, 4 3, 3 5, two of which happen to be in the same straight line, and of the strut-brace, 5 4, which exerts obliquely upwards against 5, and obliquely downward? RESISTANCE OF A FRAME AT A SECTION. 171 against 4, a thrust equal to the component perpendicular to the rafter of the load J W j which thrust is given by the equation I^54 = i "W COS i, (3.) Then we easily obtain the following values of the stresses on the rafter and ties, in which each stress is distinguished by having affixed to the letter R the numbers denoting the two joints between which it acts. Pulls / R,. 1 on ties I ^43 = ^41 = 2^ i " 8 ^ ^^*^^ ^' Thrusts I Ro, = o-^^ + I W sin ^ = ^, W cosec i, V (4.) ^^ ^ 30 2tan* & 8 ^ ^ rafter j ^ ^ _R,,_ _ ^ w sin z = J W (cosec i - 2 sin i) ; [ ^^ 2 tan ^ 8 8 ^ ^ J The difference between the thrusts on the two divisions of the rafter, 1^35 - 1^51 = i^^ sin i, is the coniponent along the rafter of the load at the point 5. (3.) Smaller Secondare/ Trusses, 1 7 5, 5 6 3. — These trusses are similar in every respect to the larger secondary trusses, except that the load on each point is one-half, and consequently each of the stresses is reduced to one-half of the corresponding stress in the Equations 3 and 4. (4.) Resultant Stresses. The pull on the middle division of the great tie-rod 2 3 is simply that due to the primary truss, 12 3. The pull on the tie 4 7 is simply that due to the secondary truss 14 3. The pulls on the ties 5 7, 5 6, are sitoply those due to the smaller secondary trusses, 1 5 7, 5 6 3. But agreeably to the Theorem stated at the commencement of this article, the pull on the tie 1 7 is the sum of those due to the larger secondary truss 14 3, and the smaller secondary truss 17 5. The pull on 6 4 is the sum of those due to the primary truss 12 3, and to the larger secondary truss 14 3. The pull on 6 3 is the sum of those due to the primary truss 1 2 3, to the larger secondary truss 14 3, and to the smaller secondary truss 5 6 3. The thrust on each of the four divisions of the rafter 1 3, is the sum of three thrusts, due respectively to the primary truss, the larger secondary truss, and one or other of the smaller secondary trusses. 277. Resistance of a Frame at a Section The labour of calcu- lating the stress on the bars of a frame may sometimes be abridged by the application of the following principle : — If a frame be acted upon by any system, of external forces, and if that frame be conceived to be coinpletely divided into two parts by an ideal surface, the stresses along the bars which are intersected by that 172 THEORY OF STRUCTURES. surface, hala?ice the external forces which act on each of the two i^arts of the frame. In most cases which occur in practice, the lines of resistance of the bars, and the lines of action of the external forces, are all in one vertical plane, and the external forces are vertical. In such cases the most convenient position for an assumed plane of section is vertical, and perpendicular to the i)lane of the frame. Take the vertical line of intersection of these two planes for an axis of co- ordinates, — say for the axis of y, and any convenient point in it for the origin O; let the axis of x be horizontal, and in the plane of the frame, and the axis of z horizontal, and in the plane of section. The external forces applied to the part of the frame at one side of the plane of section (either may be chosen), being combined, as in Article 235, page 134, give three data — viz., the total force along 57 = 2 • X; the total force along 2/ = 2 • Y j and the moment of the couple acting round 2; = M; and the bars which are cut by the plane of section must exert resistances capable of balancing those two forces and that couple. If not more than three bars are cut by the plane of section, there are not more than three unknown quantities, and three relations between them and given quantities, so that the problem is determinate; if more than three bars are cut by the plane of section, the jDroblem is or may be indeterminate. The formulre to which this reasoning leads are as follows : — Let X be positive in a direction from the plane of section towards the part of the structure which is considered in determining S • X, 2 • Y, and Mj let ■\-y\)% measured upwards; let angles measured from Ox towards + y, that is, upwards, be positive; and let the lines of resistance of the three bars cut by the jjlane of section make the angles ?j, ^2, ?3, with x. Let w^, n^, n^, be the perpendicular dis- tances of those three lines of resistance from O, distances lying f upwards ) n r\ ^ - • i j f positive ) < , ^ 1 > from O X bemff considered as < ^ , . V \ downwards j i negative. J Let E^, Rgj 1^3' ^® ^^^® resistances, or total stresses, along the three bars, pulls being positive, and thrusts negative. Then we have the following three equations : — 2 • X = Rj^ cos i^ + Rg ^^^^ h + -^3 ^^^ h '} ) 2 • Y=:Rj sin i^ + ^2. ^i" ^2"^ ^3 ^i^^ '''3; f (!•) - M = itj TZj + Rg ^^2 "^ -^^3 '^h } / from which the three quantities sought, 11^, Rg' ^^3 ^^^ ^® found. Si)eaking with reference to the given plane of section, 2 • X may be called the normal stress, 2 • Y, the shearing stress, and M, the RESISTANCE OF A FRAME AT A SECTION. 173 moment of flexure, or lending stress; for it tends to bend the frame at the section under consideration. M is to be considered as < P ' . > according; as it tends to make the frame become con- ( negative j ° J upwards ) ( downwards. J The following is one of the simplest examples of the solution of a problem by the method of polygons, and the method of sections. Fig. 121 represents a truss of a form very common in carpentry (already referred to in Article 275), and consisting of three struts, A C, C C, C A, a tie-beam A A, and two suspension-rods, C B, C B, which serve to suspend part of the weight of the tie-beam from the joints C C, and also to stiffen the truss in the manner men- tioned in Article 275. Let i denote the equal and opposite inclinations of the rafters AC, C A, to the horizontal tie-beam A A; and leaving out of consideration the portions of the load directly supported at A, A, let P, P, denote equal vertical loads applied at C, C, and - P, - P, equal upward vertical supporting forces applied at A, A, by the resistance of the props. Let H denote the pull on the tie-beam, K the thrust on each of the sloping rafters, and T the thrust on the horizontal strut C C Proceeding by the method of polygons, as in Article 271, we find at once, H = - T = P cotan i ; ) \ (2.) E, = - P cosec i. ) (Thrusts being considered as negative.) To solve the same question by the method of sections, suppose a vertical section to be made by a plane traversing the centre of the right hand joint C ; take that centre for the origin of co-ordinates; let X be positive towards the right, and y positive downwards; let £C]^, 2/i, be the co-ordinates of the centre of resistance at the right hand point of support A, When the plane of section traverses the centre of resistance of a joint, we are at liberty to suppose either of the two bars which meet at that joint on opposite sides of the plane of section to be cut by it at an insensible distance from the joint. First, consider the plane of section as cutting C A. The forces and couple acting on the part of the frame to the right of the section are r, = 0;F,= -P M = - Vx,. 174 THEORY OF STRUCTURES. Then, observing that for the strut AC, n-O, and that for the tie A A, w = 1/^, we have, by the equations 1 of this Article Ecos^ + H = ^, = 0; E/ sin i = - P ; H?/i= -M= +Fxj^; whence we obtain, from the last equation, H==?^ = Pcotan^ 1 2/1 I from the first, or from the second J> (3.) E. = -. = - P cosec i cos I J Next, conceive the section to cut C C at an insensible distance to the left of C. Then the equal and opposite applied forces + P at C, and - P at A, have to be taken, into account ; so that from the first of which equations we obtain H + T=:P^ = 0, and T= -H= -Pcotant (4.) In the example just given, the method of sections is tedious and complex as compared with the method of polygons, and is intro- duced for the sake of illustration only. 278. Balance of a Chain or Cord.— A loaded chain may be looked upon as a polygonal frame whose pieces and joints are so numerous that its figure may without sensible error be treated as a continuous curve. Q'he following are the princijjles respecting the equilibrium of loaded chains and cords which are of most importance in practice. I. Balance of a Chain in general. — Let D A C, in fig. 123, repre- sent a flexible cord or chain supported at the points C and D, and loaded by forces in any direction, constant or vary- ing, distributed over its whole length with constant or varying intensity. Let A and B be any _ two points in this chain ; lig. 123. from those points draw tangents to the chain, A P and B P, meeting in P. The load acting on the chain between the points A and B is balanced by the pulls along the chain at those two points respectively; those pulls must respectively act along the tangents A P, B P; hence the resultant of the load between A and B acts through the point of intersection of the tangents at A and B; and that load, and the tensions on the STABILITY OF BLOCKS. 175 Fi^. 121 cliain at A and B, are respectively proportional to the sides of a triangle parallel to their directions. II. Chain under Vertical Load. — Curve of Equilibrium. — If the direction of the load be everywhere parallel and vertical, draw a vertical straight line, CD, fig. 124, to represent the total load, amd from its ends draw C O and D O, parallel to two tangents at the points of support of the chain, and meeting in O ; those lines will represent the tensions on the chain at its points of support. Let A, in fig. 123, be the lowest point of the chain. In fig. 124, draw the horizontal line O A; this will represent the horizontal component of the tension of the chain at every point, and if O B be parallel to a tangent to the chain ^t B (fig. 123), A B will represent the portion of the load sup- ported between A and B, and O B the tension at B. To express this algebraically, let H = A = horizontal tension along the chain at A; E. = O B = pull along the chain at B; P = A B = load on the chain between A and B; i= ZX P B (fig. 123) = Z A O B fig. 124) = inclination of chain at B; then, P-Htan^;B= J {F^ + 11^)=IL seci (1.) To deduce from these formulae an equation by which the form of the curve assumed by the chain can be determined when the dis- tribution of the load is known, let that curve be referred to rect- angular, horizontal, and vertical co-ordinates, measured from the lowest point A, fig. 123, the co-ordinates of B being, A^ = x, X B = ?/, then tan i =-—= — , a differential equation, which enables ClX JlL the form assumed by the cord (or " curve of equilibrium") to be determined when the distribution of the load is known. 279. Stability of Blocks. — The conditions of stability of a single block supported upon another body at a plane joint may be thus summed up : — In fig. 125, let A A represent the upper block, B B part of the supporting body, e E the joint, C its centre of pressure, P C the resultant of the whole pressure distributed over the joint, N C, T C, its components perpendicular and parallel to the joints respectively. Then the conditions of stability are the following : — I. In order that the block may not slide, the obliquity of the Fig. 125. 176 THEORY OF STRUCTURES. pressure must not exceed the angle of repose (Article 231, page 154), that is to say, ZPCN^^ (1.) II. In order that the block may he in no danger of overturning, the ratio which the deviation of the centre of pressure from the centre of figure of the joint hears to the length of the diameter of the joint traversing those two centres, must not exceed a certain fraction. The value of that fraction varies, according to circumstances, from one- eighth to three-eighths. The first of these conditions is called that oi stahility of Jriction^ the second, that of stability of position. In a structure composed of a series of blocks, or of a series of courses so bonded that each may be considered as one block, which blocks or courses press against each other , '^'^ at plane joints, the two conditions of stability must be fulfilled at each joint. Let fig. 126 represent part of such a structure, 1, 1, 2, 2, 3, 3, 4, 4, being some of its plane joints. Suppose the centre of pressure Cj of the joint 1, 1, to be known, and also the amount and direction of the pressure, as indicated by the arrow traversing Cj. "With that pressure combine the weight of the block 1, 2, 2, 1, together with any other external force which may act on that block; the resultant will be the total pressure to be resisted, at the joint 2, 2, which will be given in magnitude, direction, and position, and will intersect that joint in the centre of pressure Cg. By continu- ing this process there are found the centres of pressure Cg, C4, &c., of any number of successive joints, and the directions and magni- tudes of the resultant pressures acting at those joints. The magnitude and position of the resultant pressure at any joint whatsoever, and consequently the centre of pressure at that joint, may also be found simply by taking the resultant of all the forces which act on one of the parts into which that joint divides the structure. The centres of pressure at the joints are sometimes called centres of resistance. A line traversing all those centres of resistance, such as the dotted line R R, in fig. 126, has received from Mr., Moseley the name of the " line of resistance ; " and that author has also shewn how in many cases the equation which expresses the form of that line may be determined, and applied to the solution of useful problems. The straight lines representing the resultant pressures may be all parallel, or may all lie in the same straight line, or may all STABILITY OF BLOCKS. 177 intersect in one point. The more common case, however, is that in which those straight lines intersect each other in a series of points, so as to form a polygon. A curve, such as P P, in fig. 126 touching all the sides of that polygon, is called by Mr. Moseley the " line of prtssuresy The properties which the line of resistance and line of pressures must have, in order that the conditions of stability may be fulfilled, are, as already stated, the' following : — To insure stability of position, the line of resistance must not deviate from the centre of figure of any joi7it hy mo7'e than a certain • fraction of the diameter of the joint, measured in the direction of deviation. To insure stability of friction, the normal to each joint must not make an angle greater than the angle of repose with a tangent to the line of pressures drawn through the centre of resistance ofthatjoint. Conceive a line to pass tlirough all the limiting positions of the centre of resistance of the joint, so as to enclose a space beyond which that centre must not be found. The product of the weight of the structure into the horizontal dis- tance of a point in this line from a vertical line traversing the centre of gravity of the structure is the moment of stability of the struc- ture, when the applied thrust acts in a vertical plane parallel to that horizontal, distance, and tends to overturn the structure in the direc- tion of the given point in the line limiting the position of tJie centre of resistance; for that, according to Article 222, is the moment of the couple, which, being combined with a single force equal to the weight of the structure, transfers the line of action of that force parallel to itself through a distance equal to the given horizontal distance of the centre of resistance from the centre of gravity of the structure. The applied couple usually consists of the thrust of a frame, or an arch, or the pressure of a fluid, or of a mass of earth, against the structure, together with the equal, opposite, and parallel, but not directly opposed, resistance of the joint to that lateral force. To express this symbolically, let t be the length of the diameter of the joint where it is cut by the vertical plane traversing the centre of gravity of the structure and parallel to the applied thrust; let J be the inclination of that diameter to the horizon; let qthQ the distance of the given limiting centre of resistance from the middle point of that diameter, and q' t the distance from the same middle point to the point where the diameter is cut by the vertical line through the centre of gravity of the structure, and let W be the weight of the structure. Then the moment of stability is W {q± q') t cos;; (1.) the sign \ _ > being used according as the centre of resistance. 178 THEORY OF STRUCTURES. and the vertical line through the centre of gravity, lie towards I xf -J !- of the middle of the diameter. ( the same side J Let h denote the height of the structure above the middle of the plane joint which is its base, b the breadth of that joint in a direc- tion perpendicular or conjugate to the diameter t, and w the weight of an unit of volume of the material. Then we shall have W = n'whhf (2.) where w is a numerical factor depending on the figure of the structure, and on the angles which the dimensions, h, b, t, make with each other; that is, the angles of obliquity of the co-ordinates to which the figure of the structure is referred. Introducing this value of the weight of the structure into the formula 1, we find the following value for the moment of stability : — n (q ± q') cos J • w hb fi (3.) This quantity is divided by points into three factors, viz. : — '(1.) n{q ± q) cos J, a numericol factor, depending on the figure of the structure, the obliquities of its co-ordinates, and the direction in which the applied force tends to overturn it. (2.) ID, the specific gravity of the material. (3.) hb t^, a geometrical factor, depending on the dimensions of the structure. Now the first factor is the same in all structures having figures of the same class, with co-ordinates of equal obliquity, and exposed to similarly applied external forces; that is say, to all structures whose figures, together with the lines of action of the applied forces, are ^;a?'«^/e^ j^'^'^j^^^'^ons of each other, with co-ordinates of equal obli- quity; hence for any set of structures which fulfil that condition, the moments of stability are proportional to — I. The specific gravity of the material; II. The height; III. The breadth; IV. The square of the thickness; that is, of the dimension of the base which is parallel to the vertical plane of the applied force. 280. Transformation of Blockwork Structures. — If a structure composed of blocks have stability of position when acted on by forces represented by a given system of lines, then will a structure whose figure is a parallel projection of the original structure have stability of position when acted on by forces represented by the corresponding parallel projection of the original system of lines; also, the centres of pressure in the new structure will be the corresponding projections of the centres of pressure in the original structure. The question, whether the new structure obtained by transfor- mation will possess stability of friction is an independent problem. 179 CHAPTEE II. PRINCIPLES AND RULES RELATING TO STRENGTH AND STIFFNESS. 281. The Object of this Chapter is to give a summary of the principles, and of the general rules of calculation, which are applicable to problems of strength and stiffness, whatsoever the particular material may be. Section I. — Of Strength and Stiffness in General. 282. Load, Stress, Strain, Strength. — The load, or combination of external forces, which is applied to any piece, moving or fixed, in a structure or machine, produces stress amongst the particles of that piece, being the combination of forces which they exert in resisting the tendency of the load to disfigure and break the piece, accompanied by strain, or alteration of the volumes and figures of the whole piece, and of each of its particles. If the load is continually increased, it at length produces either fracture or (if the material is very tough and ductile) such a disfigurement as is practically equivalent to fracture, by rendering the piece useless. The Ultimate Strength of a body is the load required to produce fracture in some specified way. The Proof Strength is the load required to produce the greatest strain of a specific kind con- sistent with safety; that is, with the retention of the strength of the material unimpaired. A load exceeding the proof strength of the body, although it may not produce instant fracture, produces fracture eventually by long-continued application and frequent repetition. The Working Load on each piece of a machine is made less than the ultimate strength, and less than the proof strength, in certain ratios determined partly by experiment and partly by practical experience, in order to provide for unforeseen contingencies. Each solid has as many different kinds of strength as there are different ways in which it can be strained or broken, as shewn in the following classification :— Strain. Fracture. Elementary { Extension Tearing. •' ( Compression (Jrushing. i Distortion Shearing. Compound -^ Twisting Wrenching. ( Bending Breaking across 180 THEORY OF STRUCTURES. 283. Coefficients or Moduli of Strength are quantities expressing the intensity of the stress under which a piece of a given material gives way when strained in a given manner; such intensity being expressed in units of weight for each unit of sectional area of the layer of particles at which the body first begins to yield. In Britain, the ordinary unit of intensity employed in expressing the strength of materials is the pound avoirdupois on the square inch. Coefficients of strength are of as many difierent kinds as there are diflferent ways of breaking a body. Their use will be explained in the sequel. Coefficients of strength, when of the same kind, may still vary according to the direction in which the stress is applied to the body. Thus the tenacity, or resistance to tearing, of most kinds of wood is much greater against tension exerted along than across the grain. 284. Factors of Safety. — A factor of safety, in the ordinary sense, is the ratio in which the load that is just sufficient to overcome instantly the strength of a piece of material is greater than the greatest safe ordinary working load. The proper value for the factor of safety depends on the nature of the material; it also depends upon how the load is applied. The load upon any piece in a structure or in a machine is distin- guished into dead load and live load. A dead load is a load which is put on by imperceptible degrees, and which remains steady; such as the weight of a structure, or of the fixed framing in a machine. A live load is one that is or may be put on suddenly, or accom- panied with vibration ; like a swift train travelling over a railway bridge; or like most of the forces exerted by and upon the moving pieces in a machine. It can be shewn that in most cases which occur in practice a live load produces, or is liable to produce, twice, or very nearly twice, the effect, in the shape of stress and strain, which an equal dead load would produce. The inean intensity of the stress pro- duced by a suddenly applied load is no greater than that produced by the same load acting steadily; but in the case of the suddenly applied load, the stress begins by being insensible, increases to double its mean intensity, and then goes through a series of fluctuations, alternately below and above the mean, accompanied by vibration of the strained body. Hence the ordinary practice is to make the factor of safety for a live load double of the factor of safety for a dead load. A distinction is to be drawn between real and apparent factors of safety. A real factor of safety is the ratio in which the ultimate or breaking stress is greater than the real working stress at the time when the straining action of the load is greatest. The apparent factor of safety has to be made greater than the real FACTORS OF SAFETY. 181 factor of safety in those cases in winch the calculation of strength is based, not upon the greatest straining action of the load, but upon a mean straining action, which is exceeded by the greatest straining action in a certain proportion. In such cases the apparent factor of safety is the product obtained by multiplying the real factor of safety by the ratio in which the greatest straining action exceeds the mean. Another class of cases in which the apparent exceeds the real factor of safety is when there are additional straining actions besides that due to the transmission of motive power, and when those additional actions, instead of being taken into account in detail, are allowed for in a rough way by means of an increase of the factor of safety. A third class of cases is when there is a possibility of an increased load coming by accident to act upon the piece under consideration. For example, a steam engine may drive two lines of shafting, exerting half its power on each; one may suddenly break down, or be thrown out of gear, and the engine may for a short time exert its whole power on the other. The following table shews the ordinary values of real factors of safety : — Eeal Factors op Safett. Dead Load. Live Load Perfect materials and workmanship, — 2 4 Ordinary materials and workmanship- Metals, 3 6 Wood, Hempen Ropes, ,from 3 to 5 10 Masonry and Brickwork, 4 8 The following are examples of apparent factors of safety ; — Ratio in which . ^„„ * Greatest Effort ^^Pflf"? Real Factor of Safety, 6 exceeds Mean eiV^A,^^ Effort, nearly. ^^^^^y- Steam engines acting against a constant resistance — Single engine, 1'6 9"6 Pair of engines driving cranks at right ) -. ^ ^,p aiagles,... j Three engines driving equiangular ) -j ^^ P o cranks, j Ordinary cases of varying effort and ) ^.q -lij.r^ resistance, Lines of shafting in mill work; apparent factor of safety for twisting stress due to motive power, to cover allow- y from 18 to 36 ances for bending actions, accidental | extra load, &c., J 182 THEORY OP STRUCTURES. Almost all tlie experiments hitherto made on the strength of materials give coefficients or moduli of ultimate strength; that is, coefficients expressing the intensity of the stress exerted by the most sev^erely strained particles of the material just before it gives way. In calculations for the purpose of designing framework or machinery to bear a given working load, there are two ways of using the factor of safety, — one is, to multiply the working load by the factor of safety, so as to determine the breaking load, and ■use this load in the calculation, along with the modulus of ultimate strength : the other is, to divide the modulus of ultimate strength by the factor of safety, and thus to find a modulus or coefficient of working stress, which is to be used in the calculation, along with the working load. It is obvious that the two methods are mathematically equivalent, and must lead to the same result; but the latter is on the whole the more convenient in designing machines, 285. The Proof or Testing by experiment of the strength of a piece of material is conducted in two different ways, according to the object in view. I. If the piece is to be afterwards used, the testing load must be so limited that there shall be no possibility of its impairing the strength of the piece; that is, it must not exceed the proof strength, being from one-third to one-half of the ultimate strength. About double or treble of the working load is in general sufficient. Care should be taken to avoid vibrations and shocks when the testing load approaches near to the proof strength. II. If the piece is to be sacrificed for the sake of ascertaining the strength of the material, the load is to be increased by degrees imtil the piece breaks, care being taken, especially when the breaking point is approached, to increase the load by small quantities at a time, so as to get a sufficiently precise result. The proof strength requires much more time and trouble for its determination than the ultimate strength. One mode of approxi- mating to the proof strength of a piece is to apply a moderate load and remove it, apply the same load again and remove it, two or three times in succession, observing at each time of application of the load the strain or alteration of figure of the piece when loaded, by stretching, compression, bending, distortion, or twisting, as the case may be. If that alteration does not sensibly increase by re- peated applications of the same load, the load is within the limit of proof strength. The effects of a greater and a greater load being successively tested in the same way, a load will at length be reached whose successive applications produce increasing disfigurements of the piece; and this load will be greater than the proof strength, which will lie between the last load and the last load but one in the series of experiments. ELASTICITY OF A SOLID. 183 It was formerly supposed tliafc the production of a set — that is, a disfigurement which continues after the removal of the load — was a test of the proof strength being exceeded ; but Mr. Hodgkinson shewed that supposition to be erroneous, by proving that in most materials a set is produced by almost any load, how small soever. The strength of bars and beams to resist breaking across, and of axles to resist twisting, can be tested by the application of known weights either directly or through a lever. To test the tenacity of rods, chains, and ropes, and the resist- ance of pillars to crushing, more powerful and complex mechanism is required. The apparatus most commonly employed is the hydraulic press. In computing the stress which it produces, no reliance ought to be placed on the load on the safety valve, or on a weight hung to the pump handle, as indicating the intensity of the pressure, which should be ascertained by means of a pressure gauge. This remark applies also to the proving of boilers by water pressure. From experiments by Messrs. Hick and Liithy it appears that, in calculating the stress produced on a bar by means of a hydraulic press, the friction of the collar may be allowed for by deducting a force equivalent to the pressure of the water upon an area of a length equal to the circumference of the collar, and one- eightieth of an inch broad. For the exact determination of general laws, although the load may be applied at one end of the piece to be tested by means of a hydraulic press, it ought to be resisted and measured at the other end by means of a combination of levers. 286. Stiffness or Rigidity, Pliability, their Moduli or Coefficients. — Rigidity or stiffness is the property which a solid body possesses of resisting forces tending to change its figure. It may be expressed as a quantity, called a modulus or coefficient of stiffness, by taking the ratio of the intensity of a given stress of a given kind to the strain, or alteration of figure, with which that stress is accom- panied — that strain being expressed as a quantity by dividing the alteration of some dimension of the body by the original length of that dimension. In most materials which are used in machinery, the moduli of stiffness, though not exactly constant, are nearly constant for stresses not exceeding the proof strength. The reciprocal of a modulus of stiffness may be called a " modulus of pliability " that is to say, -,/r 1 1 fci.-,-,' Intensity of Stress • Modulus of Stiffness = -^^^^ ; Strain Modulus of Pliability = ^ . ^^^" ., . •^ Intensity of fetress 287. The Elasticity of a Solid consists of stiffness, or resistance to change of figure, combined with the T)ower of recovering the 184 THEORY OF STRUCTURES. original figure when the straining force is withdrawn. If that recovery is complete and immediate, the body is perfectly elastic; if there is a set, or permanent change of figure, after the removal of the straining force, the body is imperfectly elastic. The elasticity of no solid substance is absolutely perfect, but that of many sub- stances is nearly perfect when the stress does not exceed the proof strength, and may be made sensibly perfect by restricting the stress within small enough limits. Moduli or Coefficients of Elasticity are the values of moduli of stifi'ness when the stress is so limited that the value of each of those moduli is sensibly constant, and the elasticity of the body sensibly perfect. 288. Resilience or Spring is the quantity of mechanical work* required to produce the proof stress on a given piece of material, and is equal to the product of the proof strain, or alteration of figure, into the mean load which acts during the production of that strain ; that is to say, in general, very nearly one-half of the proof load. 289. Heights or Lengths of Moduli of Stiffness and Strength. — The term height or length, as applied to a modulus or coefficient of strength or of stiflfness, means the length of an imaginary vertical column of the material to which the modulus belongs, whose weight would cause a pressure on its base equal in intensity to the stress expressed by the given modulus. Hence Height of a modulus in feet '&" Modulus in lbs, on the square foot Heaviness of material in lbs. to the cubic foot' Modulus in lbs. on the square inch Weight of 12 cubic inches of the material' Height of a modulus in inches Modulus in lbs. on the square inch Heaviness of material in lbs. to the cubic inch' Height of a modulus in metres Modulus in kilogrammes on the square metre Heaviness of material in kilogrammes to the cubic metre * Section 2. — Of Resistance to Direct Tension. 290. Strength, Stiffness, and Resilience of a Tie. — The word tie is here used to denote any piece in framing or in mechanism, such * Mechanical Work, which will be fully treated of in Part VI., may be defined as the product of o. force into the space through which it acts. RESILIENCE OR SPRING. 185 as a rod, bar, band, cord, or chain, which is under the action of a pair of equal and opposite longitudinal forces tending to stretch it, and to tear it asunder. The common magnitude of those two forces is the load ; and it is equal to the product of the sectional area of the piece into the intensity of the tensile stress. The values of that intensity, corresponding to the immediate breaking load, the proof load, and the working load, are called respectively the moduli or coefficients of ultimate tenacity/, of proof tension, and of working tension. In symbols, let P be the load, S the sectional area, and p the intensity of the tensile stress; then i'=i'S (1.) If the sectional area varies at different points, the least area is to be taken into account in calculations of strength. The elongation of a tie produced by any load, P, not exceeding the proof load, is found as follows, provided the sectional area is uniform : — Let X denote the original length of the tie, A x the elongation, A ^ and a = the extension ; that is, the proportion which that elongation bears to the original length of the bar, being the numerical measure of the strain. Let E denote the modulus of direct elasticity, or resistance to stretching. Then « = |^ A^ = «^ = | ^' (2-) Let/' denote the proof tension of the material, so that/' S is the proof load of the tie; then the proof extension is/' -f- E. The Resilience or Spring of the tie, or the work done in stretch- ing it to the limit of proof strain, is computed as follows. The length, as before, being x, the elongation of the tie produced by the proof load is / aj -=- E. The force which acts through this space has for its least value 0, for its greatest value P =/ S, and for its mean value/' S-^2; so that the work done in stretching the tie to the proof strain, tliat is, its resilience or spring , is /S fx /2 ^X .^. "2"-E -"E-F ^""'^ The coefficient /2-^E, by which one-half of the volume of the tie is multiplied in the above formula, is called the Modulus of Resilience. A sudden pull of / S -r- 2, or one-half of the proof load, being applied to the bar, will produce the entire proof strain of /'-r-E, which is produced by the gradual application of the proof load itself; for the work performed bv the action of the constant force 186 THEORY OF STRUCTURES. /' S -^ 2, tLrongli a given space, is the same with the work per- I'ormed by the action, through the same space, of a force increasing at an uniform rate from up to/' S. Hence a tie, to resist with safety the sudden application of a given pull, requires to have twice the strength that is necessary to resist the gradual application and steady action of the same pull. This is an illustration of the principle, that the factor of safety for a live load is twice that for a dead load. 291. Thin Cylindrical and Spherical Shells. — Let r denote the radius of a thin hollow cylinder, such as the shell of a liigh -pressure boiler ; t, the thickness of the shell ; /, the ultimate tenacity of the material, in pounds per square inch ; p, the intensity of the pressure, in pounds per square inch, re- quired to burst the shell. This ought to be taken at six times the effective working pressure — effective pi^essure meaning the excess of the jjressure from within above the pressure from without, which last is usually the atmospheric pressure, of 14-7 lbs. on the square inch or thereabouts. Then P = l,> (!•) and the proper proportion of thickness to radius is given by the formula, — i^p „..(2.) *• / ^^ Thin spherical shells are twice as strong as cylindrical shells of the same radius and thickness. The tenacity of good wrought-iron boiler-plates is about 50,000 lbs. Section 3. — Of Besistance to Distortion and Shearing. 292. Distortion and Shearing Stress in General.— In framework and mechanism many cases occur in which the principal pieces, such as plates, links, bars, or beams, being themselves subjected to ten- sion, pressure, twisting, or bending, are connected with each other at their joints by rivets, bolts, pins, keys, or screws, which are under the action of a shearing force, tending to make them give way by the sliding of one part over another. Every shearing stress is equivalent to a pair of direct stresses of the same intensity, one tensile and the other compressive, exerted STRENGTH OF A CYLINDRICAL SHAFT. 187 in directions making angles of 45° with the shearing stress. Hence it follows that a body may give way to a shearing stress either by actual shearing, at a plane parallel to the direction of the shearing force, or by tearing, in a direction making an angle of 45° with that force. The manner of breaking depends on the structure of the material, hard and brittle materials giving way by tension, and soft and tough materials by shearing. When a shearing force does not exceed the limit within which moduli of stiffness are sensibly constant, it produces distortion of the body on which it acts. Let q denote the intensity of shearing stress applied to the four lateral faces of an originally square prismatic particle, so as to distort it; and let » be the distortion, expressed by the tangent of the difference between each of the distorted angles of the prisjn and a right Ojngle; then q = c, (1.) is the modulus of transverse elasticity, or resistance to distortion. One mode of expressing the distortion of an originally square prism is as follows : — Let « denote the proportionate elongation of one of the diagonals of its end, and — a the proportionate shorten- ing of the other J then the distortion is V=2 Ot. C The ratio -^ of the modulus of transverse elasticity to the modulus of direct elasticity defined in Article 287, page 184, has different values for different materials, ranging from to 77. For wrought- . . 1 ^ iron and steel it is about ^. Section 4. — Of Resistance to Twisting and Wrenching. 293. Twisting or Torsion in General.— Torsion is the condition of strain into which a cylindrical or prismatic body is put when a pair of couples of equal and opposite moment, tending to make it rotate about its axis in contrary directions, are applied to its two ends. Such is the condition of t^hafts which transmit motive power. The moment is called the tivisting moment, and at each cross- section of the bar it is resisted by an equal and opposite moment of stress. Each particle of the shaft is in a state of distortion, and exerts shearing stress. In British measures, twisting moments are expressed in inch-lbs. 294. Strength of a Cylindrical Shaft.~A cylindrical shaft, A B, l'^8 THEORY OF STEUCTURES. fig. 127, being subjected to the twisting raoment of a pair of eqnal and opposite couples applied to the cross-sections, A and B, it is required to find the condition of stress and strain at any intermediate cross-sec- tion, such as S, and also the angular displacement of any cross-section rela- ■^^S* 127. tively to any other. From the uniformity of the figure of the bar, and the uniformity of the twisting moment, it is evident that the condition of stress and strain of all cross-sections is the same; also, because of the circular figure of each cross-section, the condition of stress and strain of all particles at the same distance from the axis of the cylinder must be alike. Suppose a circular layer to be included between the cross-section S, and another cross-section at the longitudinal distance d x from it. The twisting moment causes one of those cross-sections to rotate relatively to the other, about the axis of the cylinder, through an angle which may be denoted by d 6. Then if there be two points at the same distance, r, from the axis of the cylinder, one in the one cross-section and the other in the other, which points were originally in one straight line parallel to the axis of the cylinder, the twisting moment shifts one of those points laterally, relatively to the other, through the distance r d 6. Consequently, the part of the layer which lies between those points is in a con- dition oi distortion, in a plane perpendicular to the radius r; and the distortion is expressed by the ratio '^''d-.' (^-^ which varies proportionally to the distance from the axis. There is therefore a shearing stress at each point of the cross-section, whose direction is perpendicular to the radius drawn from the axis to that point, and whose intensity is proportional to that radius, being represented by ?=<^'=«:-£ (2-) The STRENGTH of the shaft is determined in the following man- ner: — Let q^ be the limit of the shearing stress to which the material is to be exposed, being the ultimate resistance to wrench- ing if it is to be broken, the pi^oof resistance if it is to be tested, and the working resistance if the working moment of torsion is to be determined. Let r^ be the external radius of the axle. Then RESISTANCE TO BENDING IN GENERAL. 189 ^1 is the value of q at the distance r-^ from the axis ; and at any other distance, r, the intensity of the shearing stress is 2=-^2ii: (3.) Conceive the cross-section to be divided into narrow concentric rings, each of the breadth d r. Let r be the mean radius of one of these rings. Then its area is 2 -r r cZr; the intensity of the shear- ing stress on it is that given by Equation 3, and the leverage of that stress relatively to the axis of the cylinder is r; consequently the moment of the shearing stress of the ring in question, being the product of the three quantities, ^^, r, and ^ -^ r dr is '^'^ • r^ d which being integrated for all the rings from the centre to the circumference of the cross-section, gives for the moment of torsion, and of resistance to torsion, M=^gir^ = y^giA!; (4.) ii h = 2 r-^ be the diameter of the shaft, (^^ 1-5708 ; ^ = 0-196 nearly). If the axle is hollow, h^ being the diameter of the hollow, the moment of torsion becomes ^^=i-G-*'-Ar ('-^ The following formulae serve to calculate the diameters of shafts when the twisting moment and stress are given ; solid shafts : — '''=i^ ' ^^-^ hollow shafts— h. = \ ( 5-i iVi_ \ 1 (7.) Section 5. — Of Resistance to Bending and Cross-Breaking. 295. Resistance to Bending in General. — In explaining the prin- ciples of the resistance which bodies oppose to bending and cross- breaking, it is convenient to use the word beam as a general term 190 THEORY OF STRUCTURES. to denote the body under consideration ; but those principles are applicable, not only to beams for supporting weights, but to levers, cross-heads, cross-tails, shafts, journals, cranks, and all pieces in machinery or framework to which forces are applied tending to bend them and to break them across ; that is to say, forces trans- verse to the axis of the piece. Conceive a beam which is acted upon by a combination of parallel transverse forces that balance each other, to be divided into two parts by an imaginary transverse section ; and consider separately the conditions of equilibrium of one of those parts. The external transverse forces which act on that part, and constitute the load on it, do not necessarily balance each other. Their result- ant may be found by the rule of Article 233, \iSLge 132. That resultant is called the Shearing Load at the cross-section under con- sideration, and it is balanced by the Shearing Stress exerted by the particles which that cross-section traverses. The resultant moment of the same set of forces, relatively to the same cross-section, may be found by the same rule ; it is called the Bending Moment at that cross-section, and it is balanced (if the beam is stroug enough) by the Moment of Stress exerted by the particles which the cross-section traverses, called also the Moment of Resistance. That moment of stress is due wholly to longitudinal stress, and it is exerted in the following way: — The bending of the beam causes the originally straight layers of particles to become curved; those near the concave side of the beam become shortened ; those near the convex side, lengthened ; the shortened layers exert longitudinal thrust ; the lengthened layers, longitudinal tension; the resultant thrust and the resultant tension are equal and ojDposite, and compose a couple, whose moment is the moment of stress, equal and opposite to the bending moment. In the solution of problems respecting the transverse strength of beams, it is necessary to determine the shearing load and bending moment produced by the transverse external forces at different cross-sections, and especially at those cross-sections at which they act most intensely, and the relations between the dimensions and figure of a cross-section of the beam, and the moment of stress which that cross-section is capable of exerting, so that each cross- section, and especially that at which the bending moment is greatest, may have sufficient strength. 296. Calculation of Shearing Loads and Bending Moments. — In the formulae which follow, the shearing load at a given cross- section will be denoted by F, and the bending moment by M. In British measures it is most convenient to express the bending moment in inch-lbs., because of the transverse dimensions of pieces in machines being expressed in inches. The mathematical process for finding F and M at any given CALCULA.TION OF SHEARING LOADS. 191 cross-section of a beam, though always the same in principle, may be varied considerably in detail. The following is on the whole the most convenient way of conducting it : — Fig. 128 represents a beam supported at both ends, and loaded between them. Fig. 129 represents a bracket; that is, a beam supported 'dnd^fixed at one end, and loaded on a projecting portion. P, Q, represent in each case the supporting forces; in fig. 128, Wj, Fig. 128. Fig. 129. W^, W3, (fee, represent portions of the load; in fig. 129, Wq re- presents the endmost portion of the load, and W^, W5, Wg, other portions; in both figures, Aooi, A0C2, Ax^, &c., denote the lengths of the intervals into which the lines of action of the portions of the load divide the longitudinal axis of the beam. The forces marked W may be the weights of parts of the beam itself, or of bodies carried by it ; or they may be forces exerted by moving pieces in a machine on each other; or, in short, they may be any external transverse forces. If the body called the beam is a shaft, P and Q will be the bearing pressures. The figures represent the load as applied at detached points ; but when it is continuously distributed, the length of any inde- finitely short portion of the beam may be denoted by d x, the intensity of the load upon it per unit of length by w, and the amount of the load upon it by lo d x. The process to be gone through will then consist of the follow- ing steps : — Step I. To find the Supporting Forces or Bearing Pressures, P and Q. — Assume any convenient point in the longitudinal axis as origin of co-ordinates, and find the distance x^ of the resultant of the load from it, by the rule of Article 233, page 132 ; that is to say, 2W or fxwdx j wdx .(2.) 192 THEORY OF STRUCTURES. Then, by the rule of Article 227, page 129, find the two sup- porting forces or bearing pressures, P and Q; that is to say, let R be the resultant load, and P R and E, Q its distances from the points of support; and make PQ:PR:QR :R : Q : P ^} (3.) Step II. To find the shearing loads at a series of sections. — In what position soever the origin of co-ordinates may have been during the previous step, assume it now, in a beam supported at both ends, to be at one of the points of support (as A, fig. 128), and in a bracket to be at the loaded point farthest from the fixed end (as A, fig. 129). Consider P as positive and W as negative. Then the shearing load in any given interval of the length of the beam is the resultant of all the forces acting on the beam from the origin to that interval ; so that it has the series of values, In Fig. 128. r,,^p-w,; F.23 = P-W,-W,; F3^ = P_Wi-W2-W3; &c. and generally, F = P-2-W; (4.) In Fig. 129. Foi = Wo,; -F,3 = Wo + Wl + ^Y,; -F3, = Wo + Wi + W, + W3,&c.; and generally, -F-2-W; (5.) so that the shearing loads which act in a series of intervals of the length of the beam can be computed by successive subtractions or successive additions, as the case may be. For a continuously distributed load, these equations become respectively, In a beam supported at both ends, F = P - / w dx',...{^.) In a bracket, - F = I wdx; (7.) in which expressions, x denotes the distance from the origin. A, to the plane of section under consideration. The positive and negative signs distinguish the two contrary directions of the distortion which the shearing load tends to produce. The Greatest Shearing Load acts in a beam supported at both ends, close to one or other of the points of support, and its value is either P or Q. In a bracket, the greatest shearing load on the projecting part acts close to the outer point of support, and its value is equal to the entire load. In a beam supported at both ends the Shearing Load vanishes, or changes from positive to negative at some intermediate section, GREATEST SHEARING LOAD. 193 "whose position may be found from Equation 4 or Equation 6, by making F = 0. At the second point of support, F = - Q. Step III. To find the bending moments at a series of sections. — At the origin A there is no bending moment. Multiply the length of each of the intervals A a; of the longitudinal axis of the beam by the sliearing load F, which acts throughout that interval; the first of the products so obtained is the bending moment at the inner end of the first interval ; and by adding to it the other j)roducts successively, there are obtained the bending moments at the inner ends of the other intervals in succession.* That is to say, — bending moment at the origin A ; Mq = ; at the line of action of W^; M-^ = Fqi • £\x^, W^; Mg = Foi • A^i + Fi2 A ^gJ „ „ „ Wg; M3 = Foi-A^i + -Fi2Aa;2 + F23-Aa73; &c. &c. and generally, M = S • ¥ Ax (8-) If the divisions A x are of equal length, this becomes M = Aa;-2F; (9.) and for a continuously distributed load, M = f^Fdx (10.) -' The three preceding Equations 8, 9, and 10, are applicable to beams whether supported at both ends or fixed at one end. By substituting for F in Equation 10 its values as given by Equations 6 and 7 respectively, we obtain the following results : — For a beam supported at both ends, M = ?!«;'- f' [''wdx^ •' •' = Tix'- I (x -x)wdx) (11.) J For a beam fixed at one end, -M= ]'' r wdx^= i" {x'-x)wdx; (12.) in the latter of which equations the symbols — M denotes that the bending moment acts downwards. * This process is substantially the same with that employed by Mr. Herbert Latham, in his work On Iron Bridges, to compute the stress in a half-lattice girder. O 194 THEORY OF STRUCTURES. The Greatest Bending Moment acts, in a bracket, at the outer point of support; and in a beam supported at both ends, at the section where the shearing load vanishes; found, as already stated in Step IL, from the Equation F = 0. "When the transverse forces applied to a beam supported at both ends are symmetrically distributed relatively to its middle section, the Greatest Bending Moment acts at that section ; and it is some- times convenient to assume a point in that section as the origin of co-ordinates. Step IV. To deduce the shearing load and bending moment in one beam from those m another beam similarly supported and loaded. — This is done by the aid of the following principle : — When beams differing in length and in the amounts of the loads upon them are similarly supported, and have their loads similarly distributed, the shearing loads at corresponding sections in tliem vary as the total loads, and the bending moments as the products of the loads and lengths. This principle may be expressed by symbols in either of the two following ways: — First, Let I, V, denote the lengths of two beams, similarly sup- ported ; let "W, W, denote their total loads, similarly distributed ; let F, F', be the shearing forces, and M, M', the bending moments, at sections similarly situated in the two beams ; then W: W : :F :F'; (13.) ZW : ^' W : : M : M'.... (14.) Secondly, Let k and m be two numerical factors, depending on the way in which a beam is supported, the mode of distribution of its load, and the position of the cross-section under consideration; then F = A;W; (15.) M = 7?zWZ (16.) The length between the points of support of a beam supported at the ends, as in fig. 128, is often called the span. 297. Examples. — In the following formulae, which are examples of the application of the principles of the preceding Article to the cases which occur most frequently in practice, W denotes the total load; w, when the load is distributed, the load per unit of length of the beam ; c, in brackets, the length of the free part of the bracket ; c, in beams either loaded or supported at both ends, the half span, between the extreme points of load or support and the middle ; M, the greatest bending moment. EXAMPLES. 195 I. Bracket fixed at on end and loaded) M = cW (1.) at the other, j II. Bracket fixed at one end and imi- ) „ ^ cW^ _ -w^^ formerly loaded, / 2 2 ""^'^'^ III. Beam supported at both ends and \ loaded at an intermediate point, V j^ _ (<^^ - ^^) ^ /^ ^ whose distance from the middle of r 2c ""^ '' the span is £c, / lY. Beam supported at both ends and) -iv/r ^ ^ />< \r^ loaded in the middle, / ^= ~2~ ^^^ V. Beam supported at both ends and |Tiyi-_cW w c^ uniformly loaded, J "T""'2~ ^^ di X In Example III. the greatest force exerted is — — W, and the ^ c leverage with which it acts is c + x; and Examples IV. and Y. follow from it by making x = o. YI. If a beam has equal and opposite couples applied to its two ends; for example, if the beam in fig. 130 has the couple of equal and opposite forces Pj applied at A and B, and the couple of equal and opposite forces Pg at C and D, and if the opposite moments Pj'A B * '' jp^ = Pa * C D = M are equal, then each of the endmost divisions, A B and C D, is ^= ]^ ^r —^ in the condition of a bracket fixed at one end and loaded at the other (Example 1.); '^'^^ ^^s and the middle division B C is acted upon ^^' by the uniform bending moment M, and by no shearing load. VII. Let a beam of the half span c be loaded with an uniformly distributed load of w units of weight per unit of span; and at a point whose distance from the middle of the span is a, let there be applied an additional load W. It is required to find x, the dis- tance from the middle of the span at which the greatest bending moment is exerted, and M, that greatest moment. Make W z = m ; 2 cm; then the solutions are as follows : — Case 1. — When - = or^^, : x-m (c - a) ; and 196 THEORY OP STRUCTURES. TIT ^ ^ 2 Case 2. — When = or ^i^:. : x = a : and c 1 +m' M=?^'(lH-2»)(l-J) (7.) In the following case both sets of formulae give the same result; when -^ = = : x = a = m (c — d) ; and Vtt^J (^^ M=^ 298. Bending Moments produced by Longitudinal and Oblique Forces. — When a bar is acted upon at a given cross-section by any- external force, whose line of action, whether transverse, oblique, or parallel to the axis of the bar, does not traverse the centre of magnitude of that cro'ss-section, that force exerts a moment upon that cross-section equal to the product of the force into the perpen- dicular distance of its line of action from the centre of the cross- section, and that moment is to be balanced by the moment of longitudinal stress at the cross-section. The external force may be resolved into a longitudinal and a transverse component. The longitudinal component is balanced by an uniform longitudinal tension or pressure, as the case may be, exerted at the cross-section, and combined with the stress which resists the bending moment; and the transverse component is re- sisted by shearing stress. 299. Moment of Stress— Transverse Strength. — The bending moment at each cross-section of a beam bends the beam so as to make any originally plane longitudinal layer of the beam perpendicular to the plane in which the load acts, become concave in the direction towards which the moment acts, and convex in the opposite direction. Thus, fig. 131 represents a side view of a short portion of a bent beam ; C C is a layer, origin- ally plane, which is now bent so as to become concave at one side and convex at the other. The layers at and near the concave side of the beam, A A', are shortened, Fig. 131. and the layers near the convex side, B B' lengthened, by the bending action MOMENT OF STRESS. 197 of the load. There is one intermediate surface, 0', which is neither lengthened nor shortened ; it is called the " neutral surface'^ The particles at that surface are not necessarily, however, in a state devoid of strain ; for, in common with the other particles of the beam, they are compressed and extended in a pair of diagonal directions, making angles of 45° with the neutral surface, by the shearing action of the load, when such action exists. The condition of the particles of a beam, produced by the com- bined bending and shearing actions of the load, is illustrated by fig. 132, which represents a vertical longitudinal section of a rectangular beam, supported at the ends, and loaded at intermediate points. It is covered with a network consist- ing of two sets of curves cutting each other at right angles. The curves convex upwards are lines of direct thrust ; those convex downwards are Fig. 132. lines of direct tension. A pair of tangents to the pair of curves which traverse any particle are the axes of stress of that particle. The neutral surface is cut by both sets of curves at angles of 45°. At that vertical section of the beam where the shearing load vanishes, and the bending moment is greatest, both sets of curves become parallel to the neutral surface. When a beam breaks under the bending action of its load, it gives way, either by the crushing of the compressed side, A A', or by the tearing of the stretched side, B B'. In fig. 133, A represents a beam of a granular material, like cast iron, giving way by the crushing of the compressed side, i?- i qo ^ out of which a sort of wedge is ^°* forced. B represents a beam giving way by the tearing asunder of the stretched si^e. The resistance of a beam to bending and cross-breaking at any given cross-section is the moment of a couple, consisting of the thrust along the longitudinally-compressed layers, and the equal and opposite tension along the longitudinally-stretched layers. It has been found by experiment, that in most cases which occur in practice, the longitudinal stress of the layers of a beam may, without material error, be assumed to be uniformly varying, its intensity being simply proportional to the distance of the layer from the neutral surface. Let fig. 134 represent a cross-section of a beam (such as that represented in fig. 131), A the compressed side, B the extended side, C any layer, and O O the neutral axis of the section, being the line in which it is cut by the neutral surface. Let p denote 198 THEORY OF STRUCTURES. the intensity of the stress along the layer C, and y the distance of that layer from the neutral axis. Because the A . '' .f, ■, stress IS unitormiy varying, p -^ y \s a constant quantity. Let that constant be denoted for the present by a. Let z be the breadth of the layer C, and d y its thickness; Then the amount of stress along it is pzdy = ayzdy] the amount of the stress along all the layers at the given cross- section is ijyzdy] and this amount must be nothing, — in other words, the total thrust and total tension at the cross-section must be equal, — because the forces applied to the beam are wholly transverse ; from which it follows that j yzdy = 0, (1.) and the neutral axis traverses the centre of magnitude of the cross- section. This principle enables the neutral axis to be found by the aid o^the methods explained in Section 1, Chapter III, Part IIL To find the greatest value of the constant p -^ y consistent with the strength of the beam at the given cross-section, let y^ be the distance of the compressed side, and ^/^ that of the extended side from the neutral axis; /„ the greatest thrust, and /j the greatest tension which the material can bear in the form of a beam ; com- pute/^ -f- 2/a) and/j-^^/j, and adopt the less of those two quantities as the value of p -^ y, which may now be denoted hyf-^y^^; f being f^ or f, and 3/1 being y^ or y^,, according as the beam is liable to give way by crushing or by tearing. For the best economy of material, the two quotients ought to be equal; that is to say, — =z~ = '~^= -^ "^ "^ . n A ^ yi Va 2/6 h ' ^ and this gives what is called a cross-section of equal strength. The moment relatively to the neutral axis, of the stress exerted along any given layer of the cross-section, is y pzdy = ^y'^zdy; MOMENT OP STRESS. 199 and the sum of all such moments, being the moment of stress, or MOMENT OP RESISTANCE of the given cross-section of the beam to breaking across, is given by the formula, = \pyzdy=—Jy^zdy] (2.) M or making j y'^zdy = lf M=f^ (2 a.) "When the breaking load is in question, the coefficient / is what is called the modulus op rupture of the material. When the proof load or working load is in question, the co- efficient / is the modulus of rupture divided by a suitable factor of safety, which, for the working stress in parts of machinery that are made of metal, is usually 6, and for the parts made of wood, 10. Thus, the working modulus f is usually 9,000 lbs. on the square inch for wrought iron, 4,500 for cast iron, and from 1,000 to 1,200 for wood. The factor denoted by I in the preceding equation is what is called the ^'geometrical moment of inertia" of the cross-section of the beam. For sections whose figures are similar, or are parallel projections of each other, the moments of inertia are to each other as the breadths, and as the cubes of the depths of the sections, and the values of 2/i are as the depths. If, therefore, b be the breadth and h the depth of the rectangle circumscribing the cross-section of a given beam at the point where the moment of stress is greatest, we may put I=n'bh^, (3.) yi = 'm'h, (4.) n and m' being numerical factors depending on the form of section, and making n'-i-m'=^n, the moment of resistance may be thus expressed, — M. = nfbh^ (5.) Hence it appears that the resistances of similar cross-sections to cross-breaking are as their breadths and as the squares of their depths. The relation between the load and the dimensions of a beam is found by equating the value of the greatest bending moment in terms of the load and span of the beam, as given in Article 296, Equations 10, 11, 12, 16, to the value of the moment of resist- ance of the beam, at the cross-section where that greatest bending moment acts, as given in Equation 5 of this Article. The depth h is usually fixed by considerations of stiffioess, and then the unknown quantity is the breadth, b. Sometimes, as when 200 THEORY OF STRUCTURES. the cross-section is circular or square, we have h — h] and then we have h^, instead of b h^ in Equation 5, which is solved so as to give h by extraction of the cube root, formulae for these calculations: — The following are the and when h = b, b = M nfh 2^' m- ...(6.) .(6 a.) Examples op the Numerical Factors in Equations 3, 4, 5 AND 6. Form of Cross-Sections. I ™-=^i 11 I Rectal) (^le 6 /i ) 1 12 1 2 1 6 (including square) ) IL Ellipse- Vertical axis h, ) Horizontal axis b, > (including circle) ) III. Hollow rectangle, 6 h—b' h'; . also I - formed section, j where b' is the sum of the > breadths of the lateral \ hollows, ' TT 1 64 20-4 = 0491 1 2 •TT 1 32 10-2 = 0-0982 1 / b'h-x 12 V 6AV 1 2 U^~W) IV. Hollow square — ) f^'-h'\ i V. Hollow ellipse 12 V h^J 1 2 K-t^) 64 V bh'J 1 2 32 V bk'/ VI. Hollow circle, iO-^) 1 2 U^-'S) VII. Isosceles triangle; base 6, ) height /i ; i/i measured > from summit, ) 1 36 2 3 1 24 300. Allowance for Weight of Beam — Limiting Length of Beam. — When a beam is of great span, its own weight may bear a proportion to the load which it has to carry, sufficiently great to require to be taken into account in determining the dimensions of the beam. The following is the process to be performed for that purpose, when the load is uniformly distributed, and the ALLOWANCE FOR WEIGHT OF BEAM. 201 heain. of uniform cross-section. Let W' be the external working load, Si its factor of safety, ^g a factor of safety suited to a steady load, like the weight of the beam. Let b' denote the breadth of any part of the beam, as computed by considering the external breaking load alone, s^ W. Compute the weight of the beam from that provisional breadth, and let it be s W denoted by B/ Then — ^, ^ is the proportion in which the S^ W — ^2 1j gross breaking load exceeds the external part of that load. Conse- quently, if for the provisional breadth 6' there be substituted the exact breadth, b' Sj W * = «,W'-.,B" (!•) the beam will now be strong enough to bear both the proposed external load W, and its own weight, which will now be B- B'%W- , ^-s,W'-s,B" ^^-I and the true gross breaking load will be w=^^w'^'^^=vI5f (3) As the factor of safety for a steady load is in general one-half of that for a moving load, s^ may be made = 2 Sgi ia which case the preceding formulae become 2b'W_^ ^~9 W' Ti" \^-) 2 W - B' 2B W^ 2W'-B'^ 2B'W' In all these formulae, both the external load and the weight of the beam are treated as if uniformly distributed — a supposition which is sometimes exact, and always sufficiently near the truth for the purposes of the present Article. The gross load of beams of similar figures and proportions, vary- ing as the breadth and square of the depth directly, and inversely as the length, is proportional to the square of a given linear dimension. The weights of such beams are proportional to the cubes of corresponding linear dimensions. Hence the weight increases at a faster rate than the gross load ; and for each parti- 202 THEORY OF STRUCTURES. cular figure of a beam of a given material and proportion of its dimensions, there must be a certain size at which the beam will bear its own weight only, without any additional load. To reduce this to calculation, let the uniformly distributed gross breaking load of a beam of a given figure be expressed as follows: — ^^ _^^ ^ M SnfhA .(7. 7n I I the value of m for an uniformly distributed load and rectangular cross-section being ^ ; and n/h A being = n/b It^, Equation 6, Article 299; I, h said A being the length, depth, and sectional area of the beam, / the modulus of rupture, and n a factor depend- ing on the form of cross-section. The weight of the beam will be expressed by B = kw'lA; (8.) W being the weight of an unit of volume of the material, and k a factor depending on the figure of the beam. Then the ratio of the weight of the beam multiplied by its proper factor of safety to the gross breaking load is SjB _ S2 k w P' W ~ 8 nfh ' ^ -^ which increases in the simple ratio of the length, if the proportion Ixh is fixed. When this is the case, the length L of a beam, whose weight (treated as uniformly distributed) is its working load, is given by the condition §2 B = W ; that is, L= ., -, (10.) This limiting length having once been determined for a given class of beams, may be used to compute the ratios of the gross breaking load, weight of the beam, and external working load to each other, for a beam of the given class, and of any smaller length, I, according to the following proportional equation : — L:- :— ::W:B:W'; (11.) ^2 *i Section 6. — Of Resistance to Thrust or Pressure. 301. Resistance to Compression and Direct Crushing. — Resist- ance to longitudinal compression^ when the proof stress is not RESISTANCE TO COMPRESSION-. 203 exceeded, is sensibly equal to the resistance to stretching, and is expressed by the same modulus of elasticity, denoted by E. When that limit is exceeded, it becomes irregular. The present Article has reference to direct and simple crushing only, and is limited to those cases in which the pillars, blocks, struts, or rods along which the thrust acts are not so long in pro- portion to their diameter as to have a sensible tendency to give way by bending sideways. Those cases comprehend — Stone and brick pillars and blocks of ordinary proportions ; Pillars, rods, and struts of cast iron, in which the length is not more than five times the diameter, approximately; Pillars, rods, and struts of wrought iron, in which the length is not more than ten times the diameter, approximately; Pillars, rods, and struts of dry timber, in which the length is not more than about five times the diameter. In such cases the rules for the strength of ties (Article 290) are approximately applicable, substituting thrust for tension, and using the proper modulus of resistance to direct crushing instead of the tenacity. Blocks whose lengths are less than about once-and-a-half their diameters ofier greater resistance to crushing than that given by the rules; but in what proportion is uncertain. The modulus of resistance to direct crushing often differs con- siderably from the tenacity. The nature and amount of those differences depend mainly on the modes in which the crushing takes place. These may be classed as follows : — I. Crushing hy splitting (fig. 135) into a number of nearly pris- matic fragments, separated by smooth surfaces whose general direction is nearly parallel to the direction of the load, is character- istic of very hard homogeneous substances, in which the resistance to direct crushing is greater than the tenacity; being in many examples about double. Fig. 135. Fig. 136. Fig. 137. Fig. 138. II. Crushing hy shearing or sliding of portions of the block along oblique surfaces of separation is characteristic of substances of a granular texture, like cast iron, and most kinds of stone and brick. Sometimes the sliding takes place at a single plane surface, like A B in fig. 136; sometimes two cones or pyramids are formed, like 'c, c in fig, 137, which are forced towards each other, and split or drive outwards a number of wedges surrounding them, like w, w, 204 THEORY OF STRUCTURES. in the same figure. Sometimes the block splits into four wedges, as in fig. 138. In substances which are crushed by shearing, the resistance to crushing is always much greater than the tenacity; for example, in cast iron it is from four times to six times. Ill, Crushing hy bulging, or lateral swelling and spreading of the block which is crushed, is characteristic of ductile and tough materials, such as wrought iron. Owing to the gradual manner in which materials of this nature give way to a crushing^ load, it is difficult to determine their resistance to that load exactlye That resistance is in general less, and sometimes considerably less, than the tenacity. In wrought iron, the resistance to the direct crush- ing of pillars or struts of moderate length, as nearly as it can be 2 4 ascertained, is from ,, to - of the tenacity, i O TV. Crushing by buckling or crippling is characteristic of fibrous subtances, such as wood, under the action of a thrust along the fibres. It consists in a lateral bending and wrinkling of the fibres, sometimes accompanied by a splitting of them asunder- V. Crushing by cross-breaking is the mode of fracture of columns and struts in which the length greatly exceeds the diameter, under the breaking load they yield sideways, and are broken across like beams under a transverse load. LIBRARY ^ UNIYEKSITV OF CALIFORNIA. PART V. PRINCIPLES OF KINETICS. CHAPTER I. SUMMARY OF GENERAL PRINCIPLES. NATURE AND DIVISION OF THE SUBJECT. The present Chapter contains a siimmaiy of the Principles of Kinetics. 302. Effort ; Resistance ; Lateral Force. — Let F denote a force applied to a moving point, and 6 the angle made by the direction of that force with the direction of the motion of the point. Then, by the principles of Article 215, the force F may be resolved into two rectangular components, one along, and the other across, the direction of motion of the point, viz : — The direct force, F cos 6. The lateral force, F sin 6. A direct force is further distinguished, according as its acts vnth or against the motion of the point (that is, according as 6 is acute or obtuse), by the name of effort, or of resistance, as the case may be. Hence, each force applied to a moving point may be thus decom- posed : — Effort, P = F cos 6, if & is acute ; Resistance, R = F cos (t - e) if ^ is obtuse ; Lateral Force, Q = F sin 6. 303. The Conditions of Uniform Motion of a pair of points are, that the forces applied to each of them shall balance each other ; that is to say, that the lateral forces applied to each point shall balance each other, and that the efforts applied to each point shall balance the resistances. The direction of a force being, as stated in Article 194, that of the motion which it tends to produce, it is evident that the balance of lateral forces is the condition of uniformity of direction of motion, that is, of motion in a straight line ; and that the balance of efforts and resistances is the condition of uniformity of velocity. 206 PRINCIPLES or KINETICS. 304. Work consists in moving against resistance. The work is said to be performed^ and the resistance overcome. Work is mea- sured by the product of the resistance into the distance through which its point of application is moved. The unit of work com- monly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called a foot-pound. 305. Energy means capacity for performing work. The energy of an effort, or potential energy, is measured by the product of the effort into the distance through which its point of application is capable of being moved. The unit of energy is the same with the unit of work. When the point of application of an effort has been moved through a given distance, energy is said to have been exerted to an amount expressed by the product of the effort into the distance through which its point of application has been moved. 306. The Conservation of Energy, in the case of uniform motion, means the fact, that the energy exerted is equal to the work per- formed. 307. The Principle of Virtual Velocities is the name given to the application of the principle of the conservation of energy to the determination of the conditions of equilibrium amongst the forces externally applied to any connected system of points. 308. The Mass, or Inertia, of a body, is a quantity proportional to the unbalanced force which is required in order to produce a given definite change in the motion of the body in a given interval of time. It is known that the weight of a body, that is, the attraction between it and the earth, at a fixed locality on the earth's surface, acting unbalanced on the body for a fixed interval of time (e. g., for a second), produces a change in the body's motion, which is the same for all^ bodies whatsoever. Hence it follows, that the masses of all bodies are proportional to their loeights at a given locality on the eartNs surface. This fact has been learned by experiment; but it can also be shewn that it is necessary to the permanent existence of the uni- verse ; for if the gravity of all bodies whatsoever were not propor- tional to their respective masses, it would not produce similar and equal changes of motion in all bodies which arrive at similar posi- tions with respect to other bodies, and the different parts which make up stars and systems would not accompany each other in their motions, never departing beyond certain limits, but would be dis- persed and reduced to chaos. Neither an imponderable body, nor a body whose gravity, as compared with its mass, differs in the slightest conceivable degree from that of other bodies, can belong to the system of the universe.* • See the Rev. Dr. Whewell's demonstration ** that all matter gravitates." THE ACTUAL ENERGY. 207 309. The Centre of Mass of a body is its centre of gravity, found in the manner explained in Part III., Chapter III., Section 1. 310. The Momentum of a body means, the product of its mass into its velocity relatively to some point assumed as fixed. The momentum of a body, like its velocity, can be resolved into com- ponents, rectangular or otherwise, in the manner already explained for motions in Part I., Chapter I. 311. The Resultant Momentum of a system of bodies is the resultant of their separate momenta, compounded as if they were motions or statical couples. 312. Variations and Deviations of Momentum are the products of the mass of a body into the rates of variation of its velocity and deviation of its direction, found as explained in Part I., Chapter I., Section 3. 313. Impulse is the product of an unbalanced force into the time during which it acts unbalanced, and can be resolved and com- pounded exactly like force. If F be a force, and d t an interval of time during which it acts unbalanced, F c? ^ is the impulse exerted by the force during that time. The impulse of an unbalanced force in an unit of time is the magnitude of the force itself. 314. Impulse, Accelerating, Retarding, Deflecting.— Correspond- ing to the resolution of a force applied to a moving body into effort or resistance, as the case may be, and lateral force as explained in Article 302, there is a resolution of impulse into accelerating or retarding impulse, which acts with or against the body's motion, and deflecting impulse, which acts across the direction of the body's motion. Thus, if ^, as before, be the angle which the unbalanced force F makes with the body's path during an indefinitely short interval, d t. V dt = F cos 6 • dt\% accelerating impulse if & is acute; Re? ^ = F cos {-TT- 6) ' dti^ retarding impulse if 6 is obtuse ; Q,dt = 'F &m 6 ' d t\^ defiecting impulse. 315. A Deviating Force is one which acts unbalanced in a direc- tion perpendicular to that of a body's motion, and changes that direction without changing the velocity of the body. 316. Centrifugal Force is the force with which a revolving body reacts on the body that guides it, and is equal and opposite to the deviating force with which the guiding body acts on the revolving body. In fact, as has been stated in Article 193, every force is an action between two bodies ; and deviating force and centrifugal force are but two different names for the same force, applied to it according as its action on the revolving body or on the guiding body is under consideration at the time. 317. The Actual Energy of a moving body relatively to a fixed 208 PRINCIPLES OP KINETICS. point is the product of the mass of the body into one-half oi the square of its velocity, that is to say, it is represented by 2 2g The product m v'^, the double of the actual energy of a body, was formerly called its vis-viva. Actual energy, being the product of a weight into a height, is expressed, like potential energy and work, in foot-pounds (Articles 304, 305.) 318. Energy Stored and Restored. — A body alternately acceler- ated and retarded, so as to be brought back to its original speed, performs work by means of its retardation exactly equal in amount to the potential energy exerted in producing its acceleration; and that amount of energy may be considered as stored during the acceleration, and restored during the retardation. 319. The Transformation of Energy is a term applied to such processes as the expenditure of potential energy in the production of an equal amount of actual energy, and vice versa. 320. Periodical Motion. — If a body moves in such a manner that it periodically returns to its original velocity, then at the end of each period, the entire variation of its actual energy is nothing; and in each such period the whole potential energy exerted is equal to the whole work performed, exactly as in the case of a body moving uniformly (Article 306.) 321. A Reciprocating Force is a force which acts alternately as an effort and as an equal and opposite resistance, according to the direction of motion of the body. The work which a body performs in moving against a reciprocating force is employed in increasing its own potential energy, and is not lost by the body. 322. Collision is a pressure of inappreciably short duration be- tween two bodies. 323. The Moment of Inertia of an indefinitely small body, or " physical point," relatively to a given axis, is the product of the mass of the body, or of some quantity proportional to the mass, such as the weight, into the square of its perpendicular distance from the axis. 324. The Radius of Gyration of a body about a given axis is that length whose square is the mean of all the squares of the dis- tances of the indefinitely small equal particles of the body from the axis, and is found by dividing the moment of inertia by the mass. 325. The Centre of Percussion of a body, for a given axis, is a point so situated, that if part of the mass of the body were con- centrated at that point, and the remainder at the point directly opposite in the given axis, the statical moment of the weight so distributed, and its moment of inertia about the given axis, would THE CENTRE OF PERCUSSION. 209 be the same as those of the actual body in every position of the body. 326. The subjects to which the principles of kinetics relate will be classed in the following manner : — I. Uniform Motion. 'I. Varied Translatio [I. Rotations of Rigi IV. Motions of Pluid». II. Varied Translation of Points and Rigid Bodies. III. Rotations of Rigid Bodies. 210 CHAPTER II. ON UNIFORM MOTION UNDER BALANCED FORCES. 327. First Law of Motion. — A body under the action of no force, or of balanced forces, is either at rest, or moves uniformly. (Uni- form motion has been defined in Article 66.) Such is the first law of motion as usually stated ; but in that statement is implied something more than the literal meaning of the words; for it is understood, that the rest or motion of the body to which the law refers, is its rest or motion relatively to another body which is also under the action of no force or of balanced forces. Unless this implied condition be fulfilled, the law is not true. Therefore the complete and explicit statement of the first law of motion is as follows : — If a pair of bodies be each under the action of no force, or of balanced forces, the motion of each of those bodies relatively to the other is either none or uniform. The first law of motion has been learned by experience and observation: not directly, for the circumstances supposed in it never occur; but indirectly, from the fact that its consequences, when it is taken in conjunction with other laws, are in accordance with all the phenomena of the motions of bodies. The first law of motion may be regarded as a consequence of the definitions of /or ce and of 5a^a72ce (Articles 55, 56); at the same time it is to be observed, that the framing of those definitions has been guided by experimental knowledge. 211 CHAPTER III. ON THE VARIED TRANSLATION OF POINTS AND RIGID BODIES. Section 1. — Law of Varied Translation. 328. Second Law of Motion. — Change of momentum is propor- tional to the impulse 'producing it. In this statement, as in that of the first law of motion, Article 327, it is implied that the motion of the moving body under consideration is referred to a fixed point or body whose motion is uniform. In questions of applied me- chanics, the motion of any part of the earth's surface may be treated as uniform without sensible error in practice. The units of mass and of force may be so adapted to each other as to make change of momentuin equal to the impulse producing it. (See Articles 330, 331.) 329. General Equations of Dynamics. — To express the second law of motion algebraically, two methods may be followed : the first method being to resolve the change of momentum into direct variation and deviation, and the impulse into direct and deflecting impulse; and the second method being to resolve both the change of momentum and the impulse into components parallel to three rectangular axes. First method, m being the mass of the body, v its velocity, and r the radius of curvature of its path, it follows from Articles 73 and 75 that the rate of direct variation of its momentum is dv d^s and from Articles 77 and 78, that the rate of deviation of its momentum is m— . r Equating these respectively to the direct and lateral impulse per unit of time, exerted by an unbalanced force F, making an angle 6 with the direction of the body's motion, we find the two following equations (see Article 314): — T» T-.-rn/i dv d^s .^ . P or -E = r (OS 6 = m • y-. = w^-2i (1.) at dv ^ 2 Q = P sin Q^~ (2.) 212 PRINCIPLES OF KINETICS. The radius of curvature r is in the direction of the deviating force Q. Second metJiod. As in Article 80, let the velocity of the body (L QC (J IJ d ^ be resolved into three rectangular components, -^-, -~, -7- ; so that the three component rates of variation of its momentum are d'^ X d^ y d^ z .(3.) Also let the unbalanced force F, making the angles a, /3, y, with the axes of co-ordinates, and its impulse per unit of time, be resolved into three components, Y^, F^, F^. Then we obtain -n • •!-! d^X -n. -r. ^^ ^ F, =: JB COS y = m -7-^ : three equations, which are substantially identical with the Equa- tions 1 and 2. 330. Mass in Terms of Weight. — A body's own weight, acting unbalanced on the body, produces velocity towards the earth, increasing at a rate per second denoted by the symbol g, whose numerical value is as follows: — Let x denote the latitude of the place, h its elevation above the mean level of the sea, ^j = 32*1695 feet, or 9-8051 metres, per second; being the value of ^ for A = 45° and h = 0, and R = 20900000 feet, or 6370000 metres, nearly, being the earth's mean radius; then ^ = ^^ .(1 _ 0-00284 cos 2 ?i) • (l " ^) (!•) For latitudes exceeding 45°, it is to be borne in mind that cos 2 x is negative, and the terms containing it as a factor have their signs reversed. For practical purposes connected with ordinary machines, it is sufficiently accurate to assume g= 32*2 feet, or 9-81 metres, per second nearly (2.) If, then, a body of the weight W be acted upon by an unbalanced THE MOTION OF A FALLING BODY. 213 force F, the change of velocity in the direction of F produced in a second will be F_F^ whence m = — (3.) 9 is the expression for the mass of a body in terms of its weight, suited to make a change of momentum equal to the impulse pro- ducing it. m being absolutely constant for the same body, g and W vary in the same proportion at different elevations and in different latitudes. 331. An Absolute Unit of Force is the force which, acting during an unit of time on an arbitrary unit of mass, produces an unit of velocity. In Britain, the unit of time being a second (as it is else- where), and the unit of velocity one foot per second, the unit of mass employed is the mass whose weight in vacuo at London and at the level of the sea is a standard avoirdupois pound. The weight of an unit of mass, in any given locality, has for its value, in absolute units of force, the coefficient g. When the unit of vjeight is employed as the unit of force, instead of the absolute unit, the corresponding unit of mass becomes g times the unit just mentioned: that is to say, in British measures, the mass of 32-2 lbs.; or in French measures, the mass of 9*81 kilogrammes. 332. The Motion of a Falling Body, under the unbalanced action of its own weight, a sensibly uniform force, is a case of the uni- formly varied velocity described in Article 73. In the equations of that Article, for the rate of variation of velocity a, is to be sub- stituted the coefficient g, mentioned in the last Article. Then if Vq be the velocity of the body at the beginning of an interval of time tf its velocity at the end of that time is v = VQ + gt, .(1.) the mean velocity during that time is and the vertical height fallen through is ^-^0^ + ^/ (3.) The preceding equations give the final velocity of the body, and the height fallen through, each in terms of the initial velocity and the time. To obtain the height in terms of the initial and final velo- cities, or vice versa, Equation 2 is to be multiplied hy v-VQ = gt, 2U PRINCIPLES OF KINETICS. the acceleration, and compared witli Equation 3 j giving the follow- ing results : — ^-^" ^''' = gh^ ^9 When the body falls from a state of rest, that the following equations are obtained \ (4.) J is to be made = ; so v = gt; h,- 2 2^' ,(3. The height h in the last equation is called the height or fall due to the velocity v; and that velocity is called the velocity due to the height or fall h. Should the body be at first projected vertically upwards, the initial velocity Vq is to be made negative. To find the height to which it will rise before reversing its motion and beginning to fall, V is to be made =0 in the last of the Equations 4; then h- ^ .(6.) being a rise equal to the fall due to the initial velocity Vq. 333. An Unresisted Projectile, or a projectile to whose motion there is no sensible resistance, has a motion compounded of the vertical motion of a falling borly, and of the horizontal motion due to the horizontal component of its velocity of projection. In fig. 139, let represent the point from which the projectile is originally projected in the direction O A, making the angle X O A = 6 with a horizontal line O X in the same vertical plane with O A. Let horizontal distances parallel to O X be denoted by cc, and verti- cal ordinates parallel to Z by z, positive upwards, and negative downwards. In the equations of vertical motion, the symbol h of the equations of Article 332 is to be replaced by - z, because of h and z being measured in opposite directions. Let Vq be the velocity of projection. Then at the instant of pro- jection, the components of that velocity are, horizontal, -j- = Vq cos d; vertical, -y- = Vq sin ff; Fig. 139. AN UNIFORM EFFORT OR RESISTANCE. 215 and after the lapse of a given time t, tliose components have become dx .(1.) Hence the co-ordinates of the body at the end of the time t are horizontal, x = Vq cos Q ' t, \ a t^ > (2) vertical, z^^v^sin 6 't- —^ ; i ^ ■' the Equations 2 being those of which the differential coefficients X are Equations 1, and because t = -, those co-ordinates are ^ ' Vq cos 6' thus related, S! = x'taiii3-^r-^^-^'x^; (3.) 2vl cos^ & ^ ^ an equation which shews the path O B C of the projectile to be a parabola with a vertical axis, touching O A in O. The total velocity of the projectile at a given instant, being the resultant of the components given by Equation 1, has for the value of its square (remembering that sin^ + cos^ ^ = 1)> v^ = ^^ + ^^ = vl-2v,sme'gt + gU^ = vl-2gz', (4.) from the last form of which is obtained the equation -^^ (^o which, being compared with Equation 4 of Article 332, shews that the relation between the variation of vertical elevation, and the varia- tion of the square of the resultant velocity, is the same, whether the velocity is in a vertical, inclined, or horizontal direction. The resistance of the air prevents any actual projectile near the earth's surface from moving exactly as an unresisted projectile. The approximation of the motion of an actual projectile to that of an unresisted projectile is the closer, the slower is the motion, and the heavier the body, because of the resistance of the air increasing with the velocity, and because of its proportion to the body's weight being dependent upon that of the body's surface to its weight. 334. An Uniform Effort or Resistance, unbalanced, causes the velocity of a body to vary according to the law expressed by this equation, Trfr> (1.) 216 PRINCIPLES OF KINETICS. where f is the constant ratio which the unbalanced force bears to the weight of the moving body, positive or negative according to the direction of the force; so that by substituting/^ for g in the equations of Article 332, those equations are transformed into the equations of motion of the body in question, h being taken to represent the distance traversed by it in a positive direction. In the apparatus known by the name of its int'-entor, Attwood, for illustrating the effect of uniform moving forces, this principle is applied in order to produce motions following the same law with those of falling bodies. Two weights, P and E, of which P is the greater, are hung to the opposite ends of a cord passing over a finely constructed pulley. Considering the masses of the cord and pulley to be insensible, the weight of the mass to be moved is P + R, and the moving force P - E, being less than the weight in the ratio, P-E -^"P + E* consequently the two weights move according to the same law with a falling body, but more slowly in the ratio of/ to 1. 335. Deviating Force of a Single Body. — It is part of the first law of motion, that if a body moves under no force, or balanced forces, it moves in a straight line. It is one consequence of the second law of motion, that in order that a body may move in a curved path, it must be continually acted upon by an unbalanced force at right angles to the direction of its motion, the direction of the force being that towards which the path of the body is curved, and its magnitude bearing the same ratio to the weight of the body that the height due to the body's velocity bears to half the radius of curvature of its path. This principle is expressed symbolically as follows : — Half radius of Height due Body's Deviating curvature. to velocity. weight. force. \ : t- W = Q = " (!■) 2 2^ gr ^ ' or otherwise that the acceleration produced by gravity, bears the same ratio to the rate of deviation, that the weight bears to the magnitude of the deviating force, which may be symbolically expressed g \ - : : W ; Q = . In the case of projectiles, just described, and of the heavenly bodies, deviating force is supplied by that component of the mutual attraction of two masses which acts perpendicular to the direction of their relative motion. In machines, deviating force is supplied by the strength or rigidity of some body, which guides the deviating mass, making it move in a curve. DEVIATING FORCE IN TERMS OF ANGULAR VELOCITY. .217 A pair of free bodies attracting each other have both deviated motions, the attraction of each guiding the other; and their devia- tions of momentum are equal in equal times; that is, their devia- tions of motion are inversely as their masses. In a machine, each revolving body tends to press or draw the body which guides it away from its position, in a direction from the centre of curvature of the path of the revolving body; and that tendency is resisted by the strength and stiffness of the guiding body, and of the frame with which it is connected. 336. A Revolving Simple Pendulum consists of a small mass A, suspended from a point by a rod or cord C A of insensibly small weight as compared with the mass A, and revolving in a circle about a vertical axis C B. The tension of the rod is the resultant of the weight of the mass A, acting vertically, and of its centrifugal force, acting horizontally; and therefore the rod will assume such an inclination that Fi.cr. 140. height B C _ radius ATB centrifugal force ~ v^ weight gr •(!•) where r = A B. Let n be the number of turns per second of the pendulum; then = z 'Trnr: and therefore, making B C = 7i, = (in the latitude of London) '^ = 9-7848 inches ....(2.; When the speed of revolution varies, the inclination of the pendu- lum varies so as to adjust the height to the varying speed. 337. Deviating Force in Terms of Angular Velocity.— If the radius of curvature of the path of a revolving body be regarded as a sort of arm of constant or variable length at the end of which the body is carried, the angular velocity of that arm is given by the expression, a= - T •(!•) Let ar be substituted for v in the value of deviating force of Article 335, and that value becomes Q ■W«2 •(2-) 218 PRINCIPLES OF KINETICS. In the case of a body revolving with tiniform velocity in a circle, like the bob A of the revolving pendulum of Article 336, a = 2 «• n, where n is the number of revolutions per second, so that ^- g y W from which equation the height of a revolving pendulum might be deduced with the same result as in the last Article. 338. A Simple Oscillating Pendulum consists of an indefinitely small weight A, fig. 141, hung by a cord or rod of in- sensible weight A C from a point C, and swinging in a vertical plane to and fro on either side of a central point D vertically below C. The path of the weight or bob is a circular arc, A D E. The weight W of the bob, acting vertically, may be resolved at any instant into two components, viz. : — W-cosZDCA = W-2J2, CA' acting along C A, and balanced by the tension of the Fig. 141. rod or cord, and W-sinZDCA = W-4£, c a' acting" in the direction of a tangent to the arc, towards D, and un- balanced. The motion of A depends on the latter force. When the arc A D E is small compared with the length of the pendulum A C, it very nearly coincides with the chord ABE; and the horizontal distance A B, to which the moving force is propor- tional, is very nearly equal to the distance of the bob from D, the central point of its oscillations. Then if the length of the pendu- lum, C A, be denoted by I, we have approximately, for small arcs of oscillation, and (1) 1 = 2.a/ n V and the following statement shews the connection between a simple oscillating and revolving pendulum, viz., that the leiigth of a simple oscillating pendulum, making a given number of small double oscilla- tions in a second, is sensibly equal to the height of a revolving pendu- lum^ making the same number of revolutions in a second. THE ANGULAR MOMENTUM. 219 Section 2. — Varied Translation of a System of Bodies. 339. Conservation of Momentum. — Theorem. The mutual actions of a system, of bodies cannot change their resultant momentum. (Resultant momentum has been defined in Article 311.) Every force is a pair of equal and opposite actions between a pair of bodies ; in any given interval of time it constitutes a pair of equal and opposite impulses on tliose bodies, and produces equal and opposite momenta. Therefore the momenta produced in a system of bodies by their mutual actions neutralize each other, and have no resultant, and cannot change the resultant momentum of the system. 340. Motion of Centre of Gravity. — Corollary. The variations of the motion of the centre of gravity of a system of bodies are wholly produced by forces exerted by bodies external to the system; for the motion of the centre of gravity is that which, being multiplied by the total mass of the system, gives the resultant momentum, and this can be varied by external forces only. It follows that in all dynamical questions in which the mutual actions of a certain system of bodies are alone considered, the centre of gravity of that system of bodies may be correctly treated as a point whose motion is none or uniform ; because its motion cannot be changed by the forces under consideration. 341. The Angular Momentum, relatively to a fixed point, of a body having a motion of translation, is the product of the momen- tum of the body into the perpendicular distance of the fixed point from the line of direction of the motion of the body's centre of gravity at the instant in question. Let m be the mass of the body, V its velocity, I the length of the before-mentioned perpendicular; then , Wvl mv 1 = 9 is the angular momentum relatively to the given point. Angular momenta are compounded and resolved like forces, each angular momentum being represented by a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of the motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius vector of the body seems to have right-handed rotation. The direction of such a line is called the axis of the angular momentum which it represents. The resultant angular momentum of a system of bodies is the resultant of all their angular momenta relatively to their common centre of gravity; and the axis of that resultant angular 220 PRINCIPLES OF KINETICS. momentum is called the axis of angular momentum of tlie system. The term angular momentum was introduced by Mr. Hayward. 342. Angular Impulse is the product of the moment of a couple of forces (Article 200) into the time during which it acts. Let F be the force of a couple, I its leverage, and d t the time during which it acts, then ^Idt is the angular impulse. Angular impulses are compounded and resolved like the moments of couples. 343. Relations of Angular Impulse and Angular Momentum. — Theorem. The variation, in a given time, of the angular momentum of a body, is equal to the angular impulse producing that variation, and has the same axis. This is a consequence which is deduced from the second law of motion in the following manner : — Conceive an unbalanced force F to be applied to a body m, and an equal, opposite, and parallel force, to a fixed point, during the interval d t ; and let I be the perpendicular distance from the fixed point to the line of action of the first force. Then the couple in question exerts the angular impulse 'Fldt At the same time, the body m acquires a variation of momentum in the direction of the force applied to it, of the. amount mdv^'F dt-j so that relatively to the fixed point, the variation of the body's angular momentum is m ldv = F Idt; being equal to the angular impulse, and having the same axis. — Q. E. D. 344. Conservation of Angular Momentum. — Theorem. The resultant angular momentum of a system of bodies cannot be changed in magnitude, nor in Hie direction of its axis, by the mutual actions of the bodies. Considering the common centre of gravity of the system of bodies as a fixed point, conceive that for each force with which one of the bodies of the system is urged in virtue of the combined action of all the other bodies upon it, there is an equal, opposite, and parallel force applied to the common centre of gravity, so as to form a couple. The forces with which the bodies act on each other are equal and opposite in pairs, and their resultant is nothing; there- fore, the resultant of the ideal forces conceived to act at the common centre of gravity is nothing, and the supposition of these forces does not efiect the equilibrium or motion of the system. Also, the resultant of all the couplts thus formed is nothing; therefore, the COLLISION. 221 resultant of their angular impulses is nothing; therefore, the resultant of the several variations of angular momentum produced by those angular impulses is nothing; therefore, the resultant angular momentum of the system is invariable in amount and in the direction of its axis. — Q. E. D. 345. Collision. — The most useful problem in cases of collision is, when two bodies whose masses are given move before the collision in one straight line with given velocities, and it is required to find their velocities after the collision. The two bodies form a system whose resultant momentum and internal energy are each unaltered by the collision ; but a certain fraction of the internal energy disappears as visible motion, and appears as vibration and heat. If the bodies are equal, similar, and perfectly elastic, that fraction is nothing. Let m^, mg, be the masses of the two bodies, and w^, u<^, their velocities before the collision, whose directions should be indicated by their signs. Then the velocity of their common centre of gravity is UiTni + u^m^ ,- . Uq = ; (1.) and this is not altered by the collision. 222 CHAPTER IV. KOTATIONS OF RIGID BODIES. 346. The Motion of a Rigid Body, or of a body wliicH sensibly preserves the same figure, has already been shewn in Part I., Chapter II., to be always capable of being resolved at each instant into a translation and a rotation; and by the aid of the principles explained in Section 3 of that chapter, the component rotation can always be conceived to take place about an axis traversing the centre of gravity of the body, and to be combined, if necessary, with a translation of the whole body in a curved or straight path along with its centre of gravity. The variations of the momentum of the translation, whether in amount or in direction, are due to the resultant force acting through the centre of gravity of the body, and are exactly the same with those of the momentum of the entire mass if it were concentrated at that centre; the variations of the angular momentum of the rotation are due to the resultant couple which is combined with that resultant force. The varia- tions of actual energy are due to both causes. When the translation of the centre of gravity of a rotating body, and its rotation about an axis traversing that centre, are known, the motion of every point in the body is determined by cinematical principles, which have been explained in Part I., Chapter IL, Section 3. Section 1. — On Moments of Inertia, Radii of Gyration, AND Centres of Percussion. 347. The Moment of Inertia of an indefinitely small body, or " physical point," relatively to a given axis, is the product of the mass of the body, or of some quantity proportional to the mass, such as the weight, into the square of its perpendicular distance from the axis : thus in the following equation : — = m r^ = , (1.) 9 9 ^ ' r is the perpendicular distance of the mass m, whose weight is W, from a given axis; and the moment of inertia, according to the unit employed, is either I, or I-^-<7; the former, when the unit is the moment of inertia of an unit of weight at the end of an arm whose length is unity; and the latter, when the unit is the moment THE RADIUS OF GYRATION. 223 of inertia of an unit of mass at the end of the same arm. The former is the more convenient unit, and will be employed in this treatise. By an extension of the term " moment of inertia," it is applied to the product of any quantity, such as a volume, or an area, into the square of the distance of the point to which that quantity relates from a given axis; but in the remainder of this treatise the term will be used in its strict sense, and according to the unit of measure already specified; that is, in British measures, moment of inertia will be expressed by the product of a certain number of pounds avoirdupois into the square of a certain number oifeet. The geometrical relations amongst moments of inertia, to which the present section refers, are independent of the unit of measure. 348. The Moment of Inertia of a System of Physical Points, relatively to a given axis, is the sum of the moments of inertia of the several points; that is, I-2-W-2 (1.) 349. The Moment of Inertia of a Rigid Body is the sum of the moments of inertia of all its parts, and is found by integration; that is, by conceiving the body to be divided into small parts of regular figure, multiplying the mass of each of those parts into the square of the distance of its centre of gravity from the axis, adding the products together, and finding the value towards which their sum converges when the size of the small parts is indefinitely diminished. For example, let the body be conceived to be built up of rectangular molecules, whose dimensions are d x, d y, and d z, the volume of each d X d y d Zy and the mass of unity of volume w. Then 1= / / ir^w'dxdydz (1.) Hence follows the general principle that propositions relative to the geometrical relations amongst the moments of inertia of systems of points are made applicable to continuous bodies by substituting integration for ordinary summation ; that is, for example, by putting / / / for 2, and w d x d y d ziov W. 350. The Radius of Gyration of a body about a given axis is that length whose square is the mean of all the squares of the distances of the indefinitely small equal particles of the body from the axis, and is found by dividing the moment of inertia by the mass, thus, * 2-W~ 2W '^ ■' 224 PRINCIPLES OF KINETICS. When symbols of integration are used, this becomes / / li^wdxdydz e^ = — T-f-r (^'^ j I Iw'dxdydz 351. Components of Moment of Inertia. — Let the positions of the particles of a body be referred to three rectangular axes, one of which, O X, is that about which the moment of inertia is to be taken. Then the square of the radius vector of any particle is so that the moment of inertia round the axis of aj is I, = 2 -W 2/2 + 2 -W^s^; (1.) that is to say, the moment of inertia of a body round a given axis Tnay be found by adding together the sum of the products of tJie masses of the particles, each multiplied by the square of each of its distances from a pair of plaries cutting each other at right angles in the given axis. In the same manner it may be shewn that the moments of inertia of the same body round the other two axes are given by the equations I^ = 2-W;s2 + 2-Wai2; 1, = ^- W ay' + ^'W y"" (2.) 352. Moments of Inertia Round Parallel Axes Compared.— Theorem. The moment of inertia of a body about any given axis is equal to its moment of inertia about an axis traversing its centre of gravity parallel to the given axis, added to the moment of inertia about the given axis due to the whole mass of the body concentrated at its centre of gravity. This theorem may be expressed as follows: — Let lo be the moment of inertia of a body about an axis traversing its centre of gravity in any given direction, and I the moment of inertia of the same body about an axis parallel to the former at the perpendicular distance r^] then I = r^2W + Io (1.) Corollary I. The radius of gyration (?) of a body about any axis is equal to the hypotenuse of a right-angled triangle, of which the two sides are respectively equal to the radius of gyration of the body about an axis traversing the centre of gravity parallel to the MOMENTS OF INERTIA AND RADII OF GYRATION. 225 given axis {pq), and to tlie perpendicular distance between these axes (vq). That is to say, i' = rl + il (2.) Corollary II. The moment of inertia of a body about an axia traversing its centre of gravity in a given direction, is less than the moment of inertia of the same body about any other axis parallel to the first. Corollary III. The moments of inertia of a body about all axes parallel to each other, which lie at equal distances frora its centre of gravity, are equal. 353. Combined Moments of Inertia. — Theorem. The combined moment of inertia of a rigidly connected system of bodies about a given axis, is equal to the combined moment of inertia which the sys- tem would have about the given axis, if each body were concentrated at its own centre of gravity^ added to the sum of the several moments of inertia of the bodies, about axes traversing their respective centres of gravity, parallel to the given axis. Let W now denote the mass of one of the bodies, Iq its moment of inertia about an axis traversing its own centre of gravity parallel to the given common axis, and ^othe distance of its centre of gravity from that common axis. Then the moment of inertia of that body about the common axis, according to Article 352, Equation 1, is I=:Wr^ + Io. Consequently, the combined moment of inertia of the system of bodies is 3I = 2-Wr^4-2Io; .' (1). — Q. E. D. 354. Examples of Moments of Inertia and Radii of Gyration of homogeneous bodies of some of the more simple and ordinary figures, are given in the following tables. In each case, the axis is supposed to traverse the centre of gravity of the body; for the principles of Article 352 enable any other case to be easily solved. The axes are also supposed, in each case, to be axes of symmetry of the figure of the body. The column headed W gives the mass of the body; that headed lo gives the moment of inertia ; that headed ^l, the square of the radius of gyration. The mass of an unit of volume is in each case denoted by w. 226 PRINCIPLES OF KINETICS. Body. I. Sphere of radius r, II. Spheroid of revolution — polar semi-axis a, equa- torial radius r...... III. IV. V. VI. VII. VIII. IX. X. XI. XII. Ellipsoid — semi-axes, a, h, c Spherical shell — external radius r, internal /,.... Spherical shell, insensibly thin — radius r, thick- ness dr, Circular cylinder — length 2a, radius r, Elliptic cylinder — length 2a, transverse semi- axes b,c, Hollow circular cylinder- length 2a, external ra- dius r, internal r', Hollow circular cylinder, insensibly thin — length 2a, radius r, thickness Jr, Circular cylinder — length 2a, radius r, Elliptic cylinder — length 2a, transverse semi-axes &. c, Hollow circular cylinder- length 2a, external ra- dius r, internal r', XIII. Hollow circular cylinder, insensibly thin — radius r, thickness dr, XIV. Rectangular prism — di- mensions 2a, 2b, 2c, XV. Rhombic prism — length 2a, diagonals 25, 2c,.... XVI. Rhombic prism, as above. Diameter Polar axis Axis, 2a Diameter Diameter Longitudinal axis, 2a Longitudinal axis, 2a Longitudinal axis, 2a Longitudinal axis, 2a Transverse diameter Transverse axis, 2b Transverse diameter Transverse diameter Axis, 2a Axis, 2 a Diagonal, 2b 4:i be the atmospheric pressure, Zi the vertical ordinate, mea- sured positively upwards from a given horizontal plane, of any point in the free surface of the liquid, and h^ the dynamic head at the same point; then it appears from Article 360, Equation 2, that for that surface, h^-z^ = — = constant (1.) e 364. A Surface of Equal Pressure is characterized by an ana- logous equation, ^-;2;== — = constant; (1.) 232 PRINCIPLES OF KINETICS. and all surfaces of equal pressure fulfil the differential equation, dh^dz; (2.) for the differential coefficient of a constant being equal to dh - dz — Equation 1, and .-. d h = dz which, for steady motioji, becomes dz = dh= -d-~; (3.) found by differentiating the equation of Article 361, expressing that the variations of actual energy are those due to the variations of level simply. 365. Motion in Plane Layers is a state which is either exactly or approximately realized in many ordinary cases of liquid motion; Fi^. 143. Fig. 144. and the assumption of which is often used as a first approximation to the solution of various questions in hydraulics. It consists in the motions of all the particles in one plane being parallel to each other, per- pendicular to the plane, and equal in velocity. It is illustrated by the three figures 143, 144, and 145, each of which represents a reservoir containing liquid up to the elevation OZ^ = z-^ above a given datum, and discharging the liquid from an orifice Ao at the smaller elevation O Z^ = Zq. The liquid moves exactly or nearly in plane layers at the upper surfuce Ai and at the orifice A^. Let these symbols denote the areas of the upper surface and of the issuing stream respectively. Let Q denote the rate of flow per second, v-^ the velocity of descent of the liquid at the upper surface, v^ its velocity of outflow from the Fis. 145. GENERAL LAWS OF FLUID miCTICM/ >^ 5331 orifice; then, according to Article IIG, tlie equaCioxrfof contiiiuk;^ i3 ^iAi = 2;oAo-Q; "1 -V^Vi. ^" Q Q "^'^^ = A,^^«=^Ao' The pressures at the upper surface and at the orifice respectively are each equal to the atmospheric pressure; hence the difference of dynamic head is simply the difference of elevation j that is to say, "^1 ~ "o — ^1 ~ ^0 ^ therefore, according to Article 361 and Article 364, Equations 2 and 3, 25^ A0-®--^o ^^\ This gives for the velocity of outflow, .) from which can be computed the rate of flow or discharge by means of Equation 1. 366. The Contracted Vein is the name given to a portion of a jet of fluid at a short distance from an orifice in a plate, which is smaller in diameter and in area than the orifice, owing to a spon- taneous contraction which the jet undergoes after leaving the orifice. The area of the. narrowed part of the contracted vein is in every case to be considered as the virtual or effective outlet^ and used for Aq in the equations of the last Article. The ratio of the area of the contracted vein, or effective orifice, to that of the actual orifice, is called the coefficient of contraction. For sharp edged orifices in thin plates, it has different values for different figures and proportions of the orifice, ranging from about 0-58 to 0-7, and being on an average about f. It diminishes some- what for great pressures, and for dynamic heads of six feet and upwards may be taken at about 0-6. The mo,5t elaborate table of those coefficients is that of Poncelet and Lesbros. For orifices with edges that are not sharp and thin, the discharge is modified sensibly by friction. Section 2. — Motions op Liquids with Friction. 367. General Laws of Fluid Friction. — It is known by experi- m(3nt, that between a fluid, and a solid surface over which it glides, 234 PRINCIPLES OF KINETICS. there is exerted a resistance to tlieir relative motion which is pro- portional to their surface of contact, and to the density of the fluid, and is approximately proportional to the square of the velocity of the relative motion; that is, the resistance is approximately pro- portional to the weight of a prism of the fluid, whose base is the sur- face of contact, and its height the height due to the relative velocity. Let S be the surface of contact, v the velocity, ^ the weight of an unit of volume of the fluid, and/ a factor called the coefficient of friction; then R=/eS|^. (1-) is the amount of the friction at the surface S. The coefficient /is not absolutely constant at different velocities. The mode of calculation employed in practice, where the velocity is one of the unknown quantities to be determined, is to find an approximate value of the velocity from the mean value of /; then to compute the value of / corresponding to that approximate velocity, and use it to compute the velocity more exactly. The following are some of the values of the coefficients of friction, according to different authorities, for streams of water, gliding over various surfaces; v being the mean velocity of the stream, in feet per second : — Iron pipes (Darcy). Let c? = diameter of pipe in feet; then, or for velocities that are not very small. Iron pipes, value of/ for first approximation, 0-0064 Beds of rivers (Weisbach), f — a + -; a = 0-0074. h = 0-00023 foot. Beds of rivers, value of / for first ) o-DOTfi approximation, j A collection of numerous formulae for fluid friction, proposed by different authors, together with tables of the results of the best formulae, is contained in Mr. Neville's work on hydraulics. The formulae of many authors, though differing in appearance, are founded on the same, or nearly the same, experimental data, being chiefly those of Du Buat, with additions by subsequent inquirers; and their practical results do not materially differ. The two formulae given above, on the authority of Darcy, for iron pipes, FRICTION IN AN UNIFORM STREAM. 235 are based on his experiments as recorded in his treatise du Mouve- ment de VEau dans les Tuyaux. 368. Internal Fluid Friction. — Although the particles of fluids have no transverse elasticity — that is, no tendency to recover a certain figure after having been distorted — it is certain that they resist being made to slide over each other, and that there is a lateral communication of motion amongst them; that is, that there is a tendency of particles which move side by side in parallel lines to assume the same velocity. The laws of this lateral communica- tion of motion, or internal friction of fluids, are not known exactly; but its eflTects are known thus far : — that the energy due to differ- ences of velocity, which it causes to disappear, is replaced by heat in the proportion of one thermal unit of Fahrenheit's scale for 772 foot pounds of energy, and that it causes the friction of a stream against its channel to take effect, not merely in retarding the film of fluid which is immediately in contact with the sides of the channel, but in retarding the whole stream, so as to reduce its motion to one approximating to a motion in plane layers perpen- dicular to the axis of the channel (Article 365). 369. Friction in an Uniform Stream. — It is this last fact which renders possible the existence of an open stream of uniform section, velocity, and declivity. In hydraulic calculations respecting the resistance of this, or any other stream, the value given to the velocity is its mean value throughout a given cross-section of the stream A, "1 <'■) The greatest velocity in each cross-section of a stream takes place at the point most distant from the rubbing surface of the channel. Its ratio to the mean velocity is given by the following empirical formula of Prony, where V is the greatest velocity in feet per second : — V 10-25 + V ^ ' In an uniform stream, the dynamic head which would otherwise have been expended in producing increase of actual energy, is wholly expended in overcoming friction. Consider a portion of the stream whose length is I, and fall z. The loss of head is equal to the fall of the surface of the stream, according to Article 363; and the expenditure of potential energy in a second is accordingly zzQ, = z z'v A.. Equating this to the work performed in a second in overcoming fiiction, viz., v E, Equation 1, Article 367, we find 236 PRINCIPLES OF KINETICS. or dividing by common factors, and by the area of section A, we find for the value of the fall in terms of the velocity -f' S .(3.) A 2^" Let s be what is called the wetted perimeter of the cross-section of the stream; that is, the cross-section of the rubbing surface of the stream and channel; then and dividing both sides of Equation 3 by I, we find for the relation between the rate of declivity and the velocity, ,(4.) sin ^^-^ = /— ^ — - I A. 2 g — is what is called the "hydraulic mean depth" of the stream; s and as the friction is inversely proportional to it, it is evident that the figure of cross-section of channel which gives the least friction is that whose hydraulic mean depth is greatest, viz., a semicircle. When the stability of the material limits the side-slope of the channel to a certain angle, Mr. Neville has shewn that the figure of least friction consists of a pair of straight side-slopes of the given inclination connected at the bottom by an arc of a circle whose radius is the depth of liquid in the middle of the channel; or, if a flat bottom be necessary, by a horizontal line touching that arc. Por such a channel, the hydraulic mean depth is half of the depth of liquid in the middle of the channel. 370. Varying Stream. — In a stream whose area of cross-section varies, and in which, consequently, the mean velocity varies at different cross-sections, the loss of dynamic head is the sum of that expended in overcoming friction, and of that expended in producing increased velocity, when the velocity increases, or the difference of those two quantities when the velocity diminishes, which difference may be positive or negative, and may represent either a loss or a gain of head. The following method of representing this principle sym- bolically is the most convenient for practical purposes. In fig. 146, let the origin of co-or- dinates be taken at a point O completely below the part of the stream to be considered; let ho- rizontal abscissae x be measured against the direction of flow, and vertical ordinates to the Fig 146. surface of the stream, z, up- wards. Consider any indefinitely short portion of the stream whose THE FRICTION OF A PIPE RUNNING FULL. 2o7 horizontal length is d x; in practice this may almost always be con- sidered as equal to the actual length. The fall in that portion of the stream is d z, and the acceleration -d v, because of v being opposite to X. Then modifying the expression for the loss of head due to friction in Equation 3 of Article 369 to meet the present case, and adding the loss of head due to acceleration, we find ^ sdx v^ vdv ,- . dz - /•— T" * o (1-) In applying this differential equation to the solution of any parti- cular problem, for v is to be put Q -f- A, and for A and s are to be put their values in terms of x and z. Thus is obtained a differential equation between z and x, and the constant quantity Q, the flow per second. If Q is known, then it is sufficient to know the value of z for one particular value of x, in order to be able to determine the integral equation between z and x If Q is unknown, the dz values of z for two particular values of x, or of z and -j~~ (the declivity), for one particular value of x, are required for the solu- tion, which comprehends the determination of the value of O. 371. The Friction in a Pipe Running Full produces loss of dynamic head according to the same law with the friction in a channel, except that the dynamic head is now the sum of the ele- vation of the pipe above a given level, and of the height due to the pressure within it. The differential equation which expresses this is as follows : — Let d I he the length of an indefinitely short portion of a pipe measured in the direction of flow, s its internal circumference, A its area of section, z its elevation above a given level, p the pressure within it, h the dynamic head. Then the loss of head is ,, , dp vdv . sdl v^ ._. -dh= -dz--^ = +/• • -— (1.) The ratio -y-, is called the virtual or hydraulic declivity, being the rate of declivity of an open channel of the same flow, area, and hydraulic mean depth. This may differ to any extent from the dz actual declivity of the pipe, -j-j. CO When the pipe is of uniform section, dv-0, and the first term of the right-hand side of Equation 1 vanishes. When the section of the pipe varies, s and A are given functions of L If Q is given, v^Q^A is also a given function of I; and to solve the equation completely, there is only required in addition 238 PRINCIPLES OF KINETICS. the value of h for one particular value of I. If Q is unknown, the values of h for two particular values of I, or of h and -— - for one particular value of I, are required for the solution, which compre- hends the determination of Q. 372. Resistance of Mouthpieces. — A mouthpiece is the part of a channel or pipe immediately adjoining a reservoir. The internal friction of the fluid on entering a mouthpiece causes a loss of head equal to the height due to the velocity multiplied by a constant depending on the figure of the mouthpiece, whose values for certain figures have been found empirically; that is to say, let - A ^* he the loss of head ; then -^'-V ^'-^ /' being a constant. For the mouthpiece of a cylindrical pipe, issuing from the flat side of a reservoir, and making the angle i with a normal to the side of the reservoir, according to Weisbach, /' = 0-505 + 0-303 sin ^ + 0-226 sin2i (2) 373. The Resistance of Curves and Knees in pipes causes a loss of head equal to the height due to the velocity multiplied by a coefficient, whose values, according to Weisbach, are given by the following formulae : — For cui'ves, let i be the arc to radius unity, r the radius of curvature of the centre line of the pipe, and d its diameter. Then for a circular pipe, /" = i{0-131.1-847(^J}; and for a rectangular pipe, |" (!•) for hnees, or sudden bends, let i be the angle made by the two por- tions of the pipe at either side of the knee with each other,; then /" = 0-9457 sin2 ^ + 2-047 sin4 \ (2.) 374. A Sudden Enlargement of the channel in which a slream of liquid flows, causes a sudden diminution of the mean velocity in the same proportion as that in w^hich the area of section is in- THE GENERAL PROBLEM. 239 creased. Thus, let v^ be tlie velocity in the narrower portion of the channel, and let m be the number expressing the ratio in which the channel is suddenly enlarged: the velocity in the enlarged part is A Now it appears from experiment, that the actual energy due to the velocity of the narrow stream relatively to the wide stream, that is, to the difference -y^ f 1 j, is expended in over- coming the internal fluid friction of eddies, and so producing heat; so that there is a loss of total head, represented by ftO-D- 0) 375. The General Problem of the flow of a stream with friction vl v^ is thus expressed : — Let h. + -r—, and Ao + ~, be the total heads at ^g ^9 the beginning and end of the stream respectively; then the loss of total head is represented by ;,-4+!|--^ = 2-Fi^ (1-) where the right-hand side of the equation represents the sum of all the losses of head due to the friction in various parts of the channel. 210 PART VI. THEORY OF MACHINES. CHAPTER I. DEFINITIONS AND GENERAL PRINCIPLES. 376. Nature and Division of the Subject. — In the present Part of this work, machines are to be considered not merely as modify- ing motion, but also as modifying force, and transmitting energy from one body to another. The theory of machines consists chiefly in the application of the principles of dynamics to trains of mechanism; and therefore much of the present Part of this treatise will consist of references back to Parts 11. and V. There are two fundamentally different ways of considering a machine, each of which must be employed in succession, in order to obtain a complete knowledge of its working, I. In the first place is considered the action of the machine during a certain period of time, with a view to the determination of its efficiency; that is, the ratio which the useful part of its work bears to the whole expenditure of energy. The motion of every ordinary machine is either uniform or periodical; and there- fore the principle of the equality of energy and work is fulfilled, either constantly, or periodically at the end of each period or cycle of changes in the motion of the machine. II. In the second place is to be considered the action of the machine during intervals of time less than its period or cycle, if its motion is periodic, in order to determine the law of the periodic changes in the motions of the pieces of which the machine con- sists, and of the periodic or reciprocating forces by which such changes are produced. 377. A Prime Mover is an engine, or combination of moving pieces, which serves to transfer energy from those bodies which naturally develop it, to those by means of which it is to be enj- ployed, and to transform energy from the various forms in which POWER AND effect; HORSE POWER. 24:1 it may occur, such as cliemical affinity, heat, or electricity, into the form of mechanical energy, or energy of force and motion. The mechanism of a prime mover compreliends all those parts by means of which it regulates its own operations. The useful work of a prime mover is the energy which it trans- mits to any machine driven by it; and its efficiency is the ratio of that useful work to the whole energy received by it from a natural source of energy. The effect or available power of a prime mover is its useful work in some given unit of time, such as a second, a minute, an hour, or a day. 378. The Regulator of a prime mover is some piece of apparatus by which the rate at which it receives energy from the source of energy can be varied. 379. A Governor is a self-acting adjusting apparatus, usually consisting of a pair of rotating pendulums, whose angle of devia- tion from their axis depends upon the speed. 380. Fluctuations of Speed in a machine are caused by the alternate excess of the energy received above the work performed, and of the work performed above the energy received, which pro- duce an alternate increase and diminution of actual energy. 381. A Fly -Wheel is a wheel with a heavy rim, wliose great moment of inertia reduces the coefficient of fluctuation of speed to a certain fixed amount. 382. A Brake is employed to stop a machine in a shorter time than can be done by simply suspending the efibrt of the prime mover. 383. Useful and Lost Work. — The whole work performed by a machine is distinguished into useful work^ being that performed in producing the effect for which the machine is designed, and lost work being that performed in producing other effects. 384. Useful and Prejudicial Resistance are overcome in per- forming useful work and lost work respectively. 385. The Efficiency of a machine is a fraction expressing the ratio of the useful work to the whole work performed, which is equal to the energy expended. The limit to the efficiency of a machine is unity, denoting the efficiency of a perfect machine in which no work is lost. The object of improvements in machines is to bring their efficiency as near to unity as possible. 386. Power and Effect; Horse Power. — The power of a machine is the energy exerted, and the effect, the useful work performed, in some interval of time of definite length. The unit of power called conventionally a horse power, is 550 foot pounds per second, or 33,000 foot pounds per minute, or 1,980,000 foot pounds per hour. The effect is equal to the power multiplied by the efficiency. 242 THEORY OF MACHINES. 387. Driving Point; Train; Working Point.— The driving point is that througli which the resultant effort of the prime mover acts. The train is the series of pieces which transmit motion and force from the driving point to the working point, through which acts the resultant of the resistance of the useful work. 388. Points of Resistance are points in the train of mechanism through which the resultants of prejudicial resistances act. 389. Efficiencies of Pieces of a Train. — The useful work of an intermediate piece in a train of mechanism consists in driving the piece which follows it, and is less than the energy exerted iipon it by the amount of the work lost in overcoming its own friction. Hence the efficiency of such an intermediate piece is the ratio of the work performed by it in driving the following piece, to the energy exerted on it by the preceding piece ; and it is evident that the efficiency of a machine is the product of the efficiencies of the series of moving pieces which transmit energy from the driving point to the working point. The same principles apply to a train of successive machines^ each driving that which, follows it. 243 CHAPTER II. OF THE PERFORMANCE OF WORK BY MACHINES. Section 1. — Of Work. 390. The Action of a Machine is to produce motion against Resistance. For example, if the machine is one for lifting solid bodies, such as a crane, or fluid bodies, such as a pump, its action is to produce upward motion of the lifted body against the resist- ance arising from gravity ; that is, against its own weight : if the machine is one for propulsion, such as a locomotive engine, its action is to produce horizontal or inclined motion of a load against the resistance arising from friction, or from friction and gravity combined : if it is one for shaping materials, such as a planing machine, its action is to produce relative motion of the tool and of the piece of material shaped by it, against the resistance which that material ofiers to having part of its surface removed; and so of other machines. 391. Work. — The action of a machine is measured, or expressed as a definite quantity, by multiplying the motion which it pro- duces into the resistance, or force directly opposed to that motion, which it overcomes; the product resulting from that multiplication being called work. In Britain, the distances moved through by pieces of mechanism are usually expressed in feet ; the resistances overcome, in pounds avoirdupois; and quantities of work, found by multiplying dis- tances in feet by resistances in pounds, are said to consist of so many foot-pourwis. Thus the work done iij lifting a weight of one pound, through a height of one foot, is one foot-pound; the work done in lifting a weight of twenty pounds, through a height of one hundred feet, is 20 x 100 = 2,000 foot-pounds. In France, distances are expressed in metres, resistances over- come in kilogrammes, and quantities of work in what are called hilogrammetres, one kilogrammetre being the work performed in lifting a weight of one killogramme through a height of one m6tre. 392. The Rate of Work of a machine means, the quantity of work which it performs in some given interval of time, such as a second, a minute, or an hour. It may be expressed in units of work (such as foot-pounds) per second, per minute, or per hour, as the case 244 THEORY OF MACHINES. may be; but tliere is a peculiar unit of power appropriated to its expression, called a horse-power, which is, in Britain, 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour. in France, the term Force de Cheval is applied to the following rate of work: — Foot-ponnds. 75 kilogramm^tres per second = 542J or 4,500 kilogram metres per minute = 32,549 or 270,000 kilogram metres per hour = 1,952,932 being about one-seventieth part less than the British horse-power. 393. Velocity. — If the velocity of the motion which a machine causes to be performed against a given resistance be given, then the j)roduct of that velocity into the resistance obviously gives tbe rate of work, or effective power. If the velocity is given in feet per second, and the resistance in pounds, then their product is the rate of work in foot-pounds per second, and so of minutes, or hours, or other units of time. It is usually most convenient, for purposes of calculation, to express the velocities of the parts of machines either in feet per second or in feet per minute. For certain kinetic calculations the second is the more convenient unit of time • in stating the performance of machines for practical purposes^ the minute is the unit most commonly employed. 394. Work in Terms of Angular Motion. — When a resisting force opposes the motion of a part of a machine which moves round a fixed axis, such as a wheel, an axis, or a crank, the product of the amount of that resistance into its leverage (that is, the perpen- dicular distance of the line along which it acts from the fixed axis) is called the moment, or statical moment, of the resistance. If the resistance is expressed in pounds, and its leverage in feet, then its moment is expressed in terms of a measure which may be called 2i foot-pound, but which, nevertheless, is a quantity of an entirely difierent kind from a foot-pound of work. Suppose now that the body to whose motion the resistance is opposed turns through any number of revolutions, or parts of a revolution; and let T denote the angle through which it turns, expressed in revolutions, and parts of a revolution , also, let 710 2 x = 6-2832 = |i^ denote, as is customary, the ratio of the circumference of a circle to WORK IN TERMS OF PRESSURE AND VOLUME 245 its radius. Then the distance through which the given resistance is overcome is expressed by the leverage x 2 ^ x T; that is, by the product of the circumference of a circle whose radius is the leverage, into the number of turns and fractions of a turn made by the rotating body. The distance thus found being multiplied by the resistance over^ come, gives the work performed ; that is to say, The work 'performed =ihe resistance x the leverage x 2 ^ x T: But the product of the resistance into the leverage is what is called the inoment of the resistance, and the product 2 «■ T is called the angular motion of the rotating body ; consequently, The work performed = the moment of the resistance x the angular motion. The mode of computing the work indicated by this last equation is often more convenient than the direct mode already explained in Article 391. The angular motion 2 ^r T of a body during some definite unit of time, as a second or a minute, is called its angular velocity; that is to say, angular velocity is the product of the turns and fractions of a turn made in an unit of time into the ratio of the circumference of a circle to its radius. Hence it appears that The rate of work = the momeiit of the resistance x the angular velocity. 395. Work in Terms of Pressure and Volume. — If the resistance overcome be a pressure uniformly distributed over an area, as when a piston drives a fluid before it, then the amount of that resistance is equal to the intensity of the pressure, expressed in units of force on each unit of area (for example, in pounds on the square inch, or pounds on the square foot) multiplied by the area of the sur- face at which the pressure acts, if that area is perpendicular to the direction of the motion; or, if not, then by the projection of that area on a plane perpendicular to the direction of motion. In practice, when the area of a piston is spoken of, it is always imderstood to mean the projection above mentioned. Now, when a plane area is multiplied into the distance through which it moves in a direction perpendicular to itself, if its motion is straight, or into the distance through which its centre of gravity moves, if its motion is curved, the product is the volume of the space traversed by the piston. Hence the work performed by a piston in driving a fluid before 246 THEORY OF MACHINES. it, or by a fluid in driving a piston before it, may be expressed in either of the following ways : — Resistcmce x distance traversed = intensity of pressure x area x distance travefsed; =:: intensity of pressure x volume traversed. In order to compute the work in foot-pounds, if the pressure is stated in pounds on the square foot, the area should be stated in square feet, and the volume in cubic feet ; if the pressure is stated in pounds on the square inch, the area should be stated in square inches, and the volume in units, each of which is a prism of one foot in length, and one square inch in area ; that is, of j^j ^^ ^ cubic fpot in volume. 396. Algebraical Expressions for Work. — To express the results of the preceding articles in algebraical symbols, let s denote the distance in feet through which a resistance is over- come in a given time ; E, the amount of the resistance overcome in pounds. Also, supposing the resistance to be overcome by a piece which turns about an axis, let T be the number of turns and fractions of a turn made in the given time, and i = 2 cr T = 6'2832 T the angular motion in the given time ; and let ? be the leverage of the resistance ; that is, the perpendicular distance of the line along which it acts from the axis of motion ; so that s = il, and R Z is the statical moment of the resistance. Supposing the resistance to be a pressure, exerted between a piston and a fluid, let A be the area or projected area of a piston, and j3 the intensity of the pressure in pounds per unit of area. Then the following expressions all give quantities of work in the given time in foot-pounds : — Rs; i'Rl; p As ; ipA.1. The last of these expressions is applicable to a piston turning on an axis, for which I denotes the distance from the axis to the centre of gravity of the area A. 397. Work against an Oblique Force.— The resistance directly due to a force which acts against a moving body in a direction oblique to that in which the body moves, is found by resolving that force into two components, one at right angles to the direction of motion, which may be called a lateral force, and which must be balanced by an equal and opposite lateral force, unless it takes efiect by altering the direction of the body's motion, and the other component directly opposed to the body's motion, which is the SUMMATION OF QUANTITIES OF WORK. 247 resistance required. That resolution is effected by means of the well-known principle of the parallelogram of forces as follows : — In fig. 147, let A represent the point at which a resistance is over- come, A B the direction in which ^ ^ ^ ^ 33 that point is moving, and let A F be a line whose direction and length represent the direction and magnitude of a force obliquely Yw. 147. opposed to the motion of A. From F upon B A produced, let fall the perpendicular F K; the length of that perpendicular will represent the magnitude of the lateral component of the oblique force, and the length A R will represent the direct component or resistance. The w^ork done against an oblique resisting force may also be calculated by resolving the motion into a direct component in the line of action of the force, and a transverse component, and multiplying the whole force by the direct component of the motion. 398. Summation of Quantities of Work. — In every machine, resistances are overcome during the same interval of time, by different moving pieces, and at different points in the same moving piece ; and the whole work performed during the given interval is found by adding together the several products of the resistances into the respective distances through which they are simultaneously overcome. It is convenient, in algebraical symbols, to denote the result of that summation by the symbol — 2-Rsj..., , (1.) in which 2 denotes the operation of taking the sum of a set of quantities of the kind denoted by the symbols to which it is pre- fixed. When the resistances are overcome by pieces turning upon axes, the above sum may be expressed in the form — 2 iRZj (2.) and so of other modes of expressing quantities of work. The following are particular cases of the summation of quantities of work performed at different points : — I. In a shifting piece, or one which has the kind of movement called translation only,' the velocities of every point at a given instant are equal and parallel ; hence, in a given interval of time, the motions of all the points are equal ; and the work performed is to be found by multiplying the sum of the resistances into the motion as a common factor ; an operation expressed algebraically thus — s2E; (3.) 248 THEORY OF MACHINES. II. For a turning piece, the angular motions of all the points during a given interval of time are equal; and the work performed is to be found by multiplying the sum of the moments of the resist- ances relatively to the axis into the angular motion as a common factor — an operation expressed algebraically thus — iS-R/; (4.) The sum denoted by 2 • R Hs the total moment of resistance of the piece in question. III. In every tnrain of mechanism, the proportions amongst the motions performed during a given interval of time by the several moving pieces, can be determined from the mode of connection of those pieces, independently of the absolute magnitudes of those motions, by the aid of the Theory of Pure Mechanism, Part II. This enables a calculation to be performed which is called reducing the resistances to the driving point; that is to say, determining the resistances, which, if they acted directly at the point where the motive power is applied to the machine, would require the same quantity of work to overcome them with the actual resistances. Suppose, for example, that by the principles of pure mechanism it is found, that a certain point in a machine, where a resistance R is to be overcome, moves with a velocity bearing the ratio w : 1 to the velocity of the driving point. Then the work performed in overcoming that resistance will be the same as if a resistance n R were overcome directly at the driving point. If a similar calcula- tion be made for each point in the machine where resistance is overcome, and the results added together, as the following symbol denotes : — 2-nR, (5.) that sum is the equivalent resistance at the driving point ; and if in a given interval of time the driving point moves through the dis- tance s, then the work performed in that time is — s2-nR (6.) The process above described is often applied to the steam engine, by reducing all the resistances overcome to equivalent resistances acting directly against the motion of the piston. A similar method may be applied to the moments of resistances overcome by rotating pieces, so as to reduce them to equivalent moments at the driving axle. Thus, let a resistance R, with the leverage I, be overcome by a piece whose angular velocity of rota- tion bears the ratio n :! to that of the driving axle. Then the equivalent moment of resistance at the driving axle is n R Z ; and if a similar calculation be made for each rotating piece in the machine which overcomes resistance, and the results added to- gether, the sum — ^-n'Rl (7.) WORK AGAINST VARYING RESISTANCE, 249 is the total equivalent moment of resistance at the driving axle; and if in a given interval of time the driving axle turns through the arc 2) to radius unity, the work performed in that time is — i^ • n'Rl. .(8.) IV. Ce7itre of gravity. — The work performed in lifting a body is the product of the weight of the body into the height through which its centre of gravity is lifted. If a machine lifts the centres of gravity of several bodies at once to heights either the same or different, the whole quantity of work performed in so doing is the sum of the several products of the weights and heights; but that quantity can also be computed by multiplying the sum of all the weights into the height through which their common centre of gravity is lifted. 399. Representation of Work by an Area. — As a quantity of work is the product of two quantities, a force and a motion, it may be represented by the -^^S- 148. area of a plane figure, wliich is the product of two dimensions. Let the base of the rectangle A, fig. 148, represent one foot of motion, and its height one pound of resistance; then will its area represent one foot-pound of work. In the larger rectangle, let the base O S represent a certain motion s, on the same scale w^ith the base of the tinit-area A; and let the height O K represent a certain resistance R, on the same scale with the height of the unit-area A ; then will the number of times that the rectangle O S • O R contains the unit-x-ectangle A, express the number of foot-pounds in the quantity of work E, s, which is performed in overcoming the resistance R through the distance s. 400. Work against Varying Resistance.— In fig. 149, let dis- tances, as before, be represented Y by lengths measured along the base line O X of the figure; and let the magnitudes of the resist- ance overcome at each instant be represented by the lengths of ordinates drawn perpendicular to O X, and parallel to Y :— For example, when the working body ^^' has moved through the distance represented by OS, let the resiirt- ance be represented by the ordinate S ii. 7^ A R 250 THEORY OP MACHINES. If the resistance were constant, the summits of those ordinates would lie in a straight line parallel to O X, like E, B in fig. 148; but if the resistance varies continuously as the motion goes on, the summits of the ordinates will lie in a line, straight or curved, such as that marked ERG, fig. 149, which is not parallel to O X. The values of the resistance at each instant being represented by the ordinates of a given line ERG, let it now be required to deter- mine the work performed against that resistance during a motion represented by O F = s. Suppose the area O E G F to be divided into bands by a series of parallel ordinates, such as A C and B D, and between the upper ends of those ordinates let a series of short lines, such as C D, be drawn parallel to O X, so as to form a stepped or serrated outline, consisting of lines parallel to O X and O Y alternately, and approxi- mating to the given continuous line E G. Now conceive the resistance, instead of varying continuously, to remain constant during each of the series of divisions into which the motion is divided by the parallel ordinates, and to change abruptly at the instants between those divisions, being represented for each division by the height of the rectangle which stands on that division : for example, during the division of the motion represented by A B, let the resistance be represented by A C, and so for other divisions. Then the work performed during the division of the motion re- presented by A B, on the supposition of alternate constancy and abrupt variation of the resistance, is represented by the rectangle A B • A C j and the whole work performed, on the same supposition during the whole motion O F, is represented by the sum of all the rectangles lying between the parallel ordinates; and inasmuch as the supposed mode of variation of the resistance represented by the stepped outline of those rectangles is an approximation to the real mode of variation rej)resented by the continuous line E G, and is a closer approximation the closer and the more numerous the parallel ordinates are, so the sum of the rectangles is an approximation to the exact representation of the work performed against the conti- nuously varying resistance, and is a closer approximation the closer and more numerous the ordinates are, and by making the ordinates numerous and close enough, can be made to difier from the exact representation by an amount less than any given difference. But the sum of those rectangles is also an approximation to the area O E G F, bounded above by the continuous line E G, and is a closer approximation the closer and the more numerous the ordi- nates are, and by making the ordinates numerous and close enough, can be made to dififer from the area O E G F by an amount less than any given difference. Therefore the area O E G F, hounded hy the straight line F, which THE WOEK PERFORMED AGAINST FRICTION. 251 represents the motion, hy the line E G, whose ordinates represent the values of the resistance, and hy the two ordinates O E and F G, repre- sents exactly the work performed. (See Article 34, page 17). The MEAN RESISTANCE during the motion is found by dividing the area O E G F by the motion CTF. 401. Useful Work and Lost Work. — The useful work of a ma- chine is that which is performed in effecting the purpose for which the machine is designed. The lost work is that which is performed in producing effects foreign to that purpose. The resistances over- come in performing those two kinds of work are called respectively useful resistance and prejudicial resistance. The useful work and the lost work of a machine together make up its total or gross work. In a pumping engine, for example, the useful work in a given time is the product of the weight of water lifted in that time into the height to which it is lifted : the lost work is that performed in overcoming the friction of the water in the pumps and pipes, the friction of the plungers, pistons, valves, and mechanism, and the resistance of the air pump and other parts of the engine. For example, the useful work of a marine steam engine in a given time is the product of the resistance opposed by the water to the motion of the ship, into the distance through which she moves : the lost work is that performed in overcoming the resist- ance of the water to the motion of the propeller through it, the friction of the mechanism, and the other resistances of the engine, and in raising the temperature of the condensation water, of the gases which escape by the chimney, and of adjoining bodies. There are some cases, such as those of muscular power and of windmills, in which the useful work of a prime mover can be determined, but not the lost work, 402. The Work Performed against Friction in a given time, between a pair of rubbing surfaces, is the product of that friction into the distance through which one surface slides over the other. When the motion of one surface relatively to the other consists in rotation about an axis, the work performed may also be cal- culated by multiplying the relative angular motion of the surfaces to radius unity into the moment of friction; that is, the product of the friction into its leverage, which is the mean distance of the rubbing surfaces from the axis. For a cylindrical journal, the leverage of the friction is simply the radius of the journal. For a flat pivot, the leverage is two-thirds of the radius of the pivot. For a collar, let r and r be the inner and outer radii; then the leverage of the friction is 3 • ^2 _ ^'2 \^') 252 THEORY OF MACHINES. In the cup and hall pivot, the end of the shaft, and the step on which it presses, present two recesses facing each other, into which are fitted two shaljow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere, or a lens having convex surfaces of a some- what less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable, from the extreme smallness of the radius of the circles of contact of the ball and cups; but as they wear, that radius and the moment of friction increase. By the rolling of two surfaces over each other without sliding, a resistance is caused, which is called sometimes " rolling friction," but more correctly rolling resistance. It is of the nature of a couple resisting rotation ; its moment is found by multiplying the normal pressure between the rolling surfaces by an arm whose length depends on the nature of the rolling surfaces; and the work lost in an unit of time in overcoming it is the product of its moment, by the angular velocity of the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot: — Oak upon oak, 0*006 (Coulomb). Lignum-vitse on oak, 0-004 „ Cast-iron on cast-iron, 0-002 (Tredgold). The work lost in friction produces heat in the proportion of one British thermal unit, being so much heat as raises the temperature of a pound of water 1° of Fahr., for every 772 foot-pounds of lost work. The heat produced by friction, when moderate in amount, is useful in softening and liquefying unguents; but when excessive it is prejudicial by decomposing the unguents, and sometimes even by softening the metal of the bearings, and raising their tempera- ture so high as to set fire to neighbouring combustible matters. Excessive heating is prevented by a constant and copious supply of a good unguent. When the velocity of rubbing is about four or five feet per second, the elevation of temperature is found to be, with good fatty and soapy unguents, 40° to 50° Fahr., witli good mineral unguents, about 30°. The effect of friction upon the efficiency of machines will be considered at the end of this Part. 403. Work of Acceleration. — In order that the velocity of a body's motion may be changed, it must be acted upon by some other body with a force in the direction of the change of velocity, which force is proportional directly to the change of velocity, and to the mass of the body acted upon, and inversely to the time^ occupied in pro- ducing the change. If the change is an acceleration or increase of velocity, let the first body be called the driven body, and the second WORK OF ACCELERATION. 253 the driving hody. Then the force miTst act upon the driven body in the direction of its motion. Every force being a pair of equal and opposite actions between a pair of bodies, the same force which accelerates the driven body is a resistance as respects the driving body. For example, during the commencement of the stroke of the piston of a steam engine, the velocity of the piston and of its rod is accelerated; and that acceleration is produced by a certain part of the pressure between the steam and the piston, being the excess of that pressure above the whole resistance which the piston has to overcome. The piston and its rod constitute the driven body; the steam is the driving body ; and the same part of the pressure which accelerates the piston, acts as a resistance to the motion of the steam, in addition to the resistance which would have to be over- come if the velocity of the piston were uniform. The resistance due to acceleration is computed in the following manner : — It is known by experiment, that if a body near the earth's surface is accelerated by the attraction of the earth, — that i3, by its own weight, or by a force equal to its own weight, its velocity goes on continually increasing very nearly at the rate of 32'2 feet per second of additional velocity, for each second during which the force acts. This quantity varies in different latitudes, and at different elevations, but the value just given is near enough to the truth for purposes of mechanical engineering. For brevity's sake, it is usually denoted by the symbol g', so that, if at a given instant the velocity of a body is v^ feet per second, and if its own weight, or an equal force, acts freely on it in the direction of its motion for t seconds, its velocity at the end of that time will have increased to V2 = v^ + gt (1.) If the acceleration be at any different rate per second, the force necessary to produce that acceleration, leing the resistance on the driving body due to the acceleration of the driven body, bears the same proportion to the driven body's weight which the actual rate of acceleration bears to the rate of acceleration produced by gravity acting freely. (In metres per second, ^ = 9-81 nearly.) To express this by symbols, let the weight of the driven body be denoted by W. Let its velocity at a given instant be v-^^ feet per second; and let that velocity increase at an uniform rate, so that at an instant t seconds later, it is v^ feet per second. Let /denote the rate of acceleration ; then /=^^; ..........(2.) 254 THEORY OP MACHINES. and the force R necessary to produce it will be given by the pro- portion, that is to say, ^^/W^W (..-..) _ g gt ^ ^ w The factor — , in the above expression, is called the mass of the driven body; and being the same for the same body, in what place soever it may be, is held to represent the quantity of matter in the body. (See Article 195, page 117.) W V The product of the mass of a body into its velocity at any instant, is called its momentum ; so that the resistance due to a given acceleration is equal to the increase of momentum divided hy the time which that increase occupies. If the product of a force by which a body is accelerated, equal and opposite to the resistance due to acceleration, into the time during which it acts, be called impulse, the same principle may be otherwise stated by saying, that the increase of momentum is equal to the impulse by which it is caused. If the rate of acceleration is not constant, but variable, the force K varies along with it. In this case, the value, at a given instant of the rate of acceleration, is represented by / = -j— , and the cr- ct t responding value of the force is K=^ = ^-.J4.. (4.) g g d t ^ ' The WORK performed in accelerating a body is the product of the resistance due to the rate of acceleration into the distance moved through by the driven body while the acceleration is going on. The resistance is equal to the mass of the body, multiplied by the increase of velocity, and divided by the time which that increase occupies. The distance moved through is the product of the mean velocity into the same time. Therefore, the work per- formed is equal to the mass of the body multiplied by the increase of the velocity, and by the mean velocity ; that is, to the mass of the body, multiplied hy the increase of the half-square of its velocity. To express this by symbols, in the case of an uniform rate of acceleration, let s denote the distance moved through by the driven body during the acceleration j then -''-^:"''; (5.) WORK OP ACCELERATION. 255 whicli being multiplied by Equation 3, gives for the work of accele- ration, xis = — — - — — ^ — t = — — ^ — (0.) In the case of a variable rate of acceleration, let v denote the mean velocity, and d s the distance moved through, in an interval of time dt &o short that the increase of velocity dvi^ indefinitely small compared with the mean velocity. Then ds = vdt; (7,) which being multiplied by Equation 4, gives for the work of accele- ration during the interval d t, ^. W dv . 9 dt W = — ' vdv \ (8.) g ' ^ and the integration of this expression (see Article 29) gives for the work of acceleration during a finite interval. \^d, = -\vdv^ '-^^ (9.) g J g being the same with the result already arrived at in Equation 6. From Equation 9 it appears that the work performed in producing a given acceleration depends on the initial and final velocities, Vj and Va, and not on the intermediate changes of velocity. If a body falls freely under the action of gravity from a state of rest through a height k, so that its initial velocity is 0, and its final velocity v, the work of acceleration performed by the earth on the body is simply the product W A of the weight of the body into the height of fall. Comparing this with Equation 6, we find — ^=1^ (!«•) This quantity is called the height, or fall, due to the velocity v ; and from Equations 6 and 9 it appears that the work performed in j)roducing a given acceleration is the same with that performed in lifting the driven body through the difference of the heights due to its initial and final velocities. If work of acceleration is performed by a prime mover upon bodies which neither form part of the prime mover itself, nor of the machines which it is intended to drive, that work is lost; as when a marine engine performs work of acceleration on the water that is struck by the propeller. 25G THEORY OF MACHINES. Work of acceleration performed on the moving pieces of the prime mover itself, or of the machinery driven by it, is not neces- sarily lost, as will afterwards appear. (Article 413.) 404. Summation of Work of Acceleration.— If several pieces of a machine have their velocities increased at the same time, the work performed in accelerating them is the sum of the several quantities of work due to the acceleration of the respective pieces; a result expressed in symbols by If-*-?"}- <■■) The process of finding that sum is facilitated and abridged in certain cases by special methods. I. Accelerated Rotation. — Let a denote the angular velocity of a solid body rotating about a fixed axis; — that is, as explained in Article 87, the velocity of a point in the body whose radius- vector, or distance from the axis, is unity. Then the velocity of a particle whose distance from the axis is r is v = a r; (2.) and if in a given interval of time the angular velocity is accelerated from the value CTj, to the value a^, the increase of the velocity of the particle in question is v^-v^ = r{a^-a;) (3.) Let w denote the weight, and — the mass of the particle in ques- tion. Then the work performed in accelerating it, being equal to the product of its mass into the increase of the half-square of its velocity, is also equal to the product of its mass into the square of its radius-vector, and into the increase of the half-square of the angular velocity; that is to say, in symbols, w vl - vl w r-2 ^ al - of g'~^' '~~gi '"'2" ^^ To find the work of acceleration for the whole body, it is to be con- ceived to be divided into small particles, whose velocities at any given instant, and also their accelerations, are proportional to their distances from the axis; then the work of acceleration is to be found for each particle, and the results added together. But in the sum so obtained, the increase of the half-square of the angular velocity is a common factor, having the same value for each particle of the body; and the rate of acceleration produced by gravity, g = 32*2 is a common divisor. It is therefore sufficient to add together the E EDUCED INERTIA. 257 jyroducts of the weight of each particle (w) into the square of its radius-vector (v^), and to multiply the sum so obtained (D • w r^) by the increase of the half-square of the angular velocity Lj(al - al)!, and divide by the rate of acceleration due to gravity (g). The result, viz.: — 2{^.^^iU^4z^.2^r2 (5.) is the woi'k of acceleration sought. In fact, the sum 2 w r^ is the iveight of a body, which, if concentrated at the distance unity from the axis of rotation, would require the same work to produce a given increase of angular velocity which the actual body requires. 405. Reduced Inertia. — If in a certain machine, a moving piece whose weight is W has a velocity always bearing the ratio n : 1 to the velocity of the driving point, it is evident that when the driving point undergoes a given acceleration, the work performed in pro- ducing the corresponding acceleration in the piece in question is the same with that which would have been required if a weight 9^2 W had been concentrated at the driving point, the work per- formed in producing the acceleration depending on the square of the velocity. If a similar calculation be performed for each moving piece in the machine, and the results added together, the sum 2-w2W (1.) gives the weight which, being concentrated at the driving point, would require the same work for a given acceleration of the driving point that the actual machine requires; so that if v-^ is the initial, and v^ the final velocity of the driving point, the work of accelera- tion of the whole machine is -^—^ -^-n^W (2.) 2g ^ ^ This operation may be called the reduction of the inertia to tJie driving point. Mr. Moseley, by whom it was first introduced into the theory of machines, calls the expression (1.) rhe ^^coefficient of In finding the reduced inertia of a machine, the mass of each rotating piece is to be treated as if concentrated at a distance from its axis equal to its radius of gyration j; so that if v represents the velocity of the driving point at any instant, and a the corresponding angular velocity of the rotating piece in question, we are to make »^="y (3.) in performing the calculation expressed by the formula (1.) 258 THEORY OF MACHINES. 406. Summary of Various Kinds of Work. — In order to present at one view the symbolical expression of the various modes of per- forming work described in the preceding articles, let it be supposed that in a certain interval of time d t the driving point of a machine moves through the distance ds; that during the same time its centre of gravity is elevated through the height dh', that resist- ances, any one of which is represented by E,, are overcome at points, the respective ratios of whose velocities to that of the driving point are denoted by n ; that the weight of any piece of the mechanism is W, and that n' denotes the ratio of its velocity (or if it rotates, the ratio of the velocity of the end of its radius of gyration) to the velocity of the driving point ; and that the driving point, whose mean velocity is v = -y , undergoes the acceleration d V. Then the wJioU work performed during the interval in ques- tion is dA-2 W + c^s•27^R + '^•2 7^,'2W (1.) 9 The mean total resistance, reduced to the driving point, may be computed by dividing the above expression by the motion of the driving point ds = vdt, giving the following result : — ^•■2W + 27^R + ^•S7^'2W (2.) a 8 gdt . ^ ' Section 2. —Of Energy, Power, and Efj'iciency. 407. Condition of Uniform Speed.— According to the first law of motion, in order that a body may move uniformly, the forces applied to it, if any, must balance each other; and the same j)rinciple holds for a machine consisting of any number of bodies. When the direction of a body's motion varies, but not the velocity ,, the lateral force required to produce the change of direction depends on the principles set forth in Article 335; but the condition of balance still holds for the forces which act along the direction of the body's motion, that is, for the efforts and resistances ; so that, whether for a single body or for a machine, the condition of uniform velocity is, that the efforts shall balance the resistances. In a machine, this condition must be fulfilled for each of the single moving pieces of which it consists. It also follows, from the principles of statics, that in any body, system, or machine, that condition is fulfilled when the sum of the products of the efforts into the velocities of their respective points of action is equal to the sum of the products of the resistances into the velocities of the points where they are overcome. ENERGY — POTENTIAL ENERGY. 25^ Thus, let V be the velocity of a driving point, or point where an. effort P is applied ; v' the velocity of a working point, or point where a resistance R is overcome ; the condition of uniform velocity for any body, system, or machine is 2 • Pi; = 2 • R V'. (1.) If there be only one driving point, or if the velocities of all the driving points be alike, then P being the total effort, the single product P V may be put in in place of the sum 2 • P i; ; reducing the above equation to Vv = ^'^v'.... (2.) Referring now to Article 398, let the machine be one in which the comparative or proportionate velocities of all the points at a given instant are known independently of their absolute velocities, from the construction of the machine ; so that, for example, the velocity of the point where the resistance R is overcome bears to that of the driving point the ratio V V then the condition of uniform speed may be thus expressed : — - P = 2 -TiR; (3.) that is, the total effort is equal to the sum of the resistances reduced to the drivi7ig point. 408 Energy — Potential Energy. — Energy means capacity for performing work, and is expressed, like work, by the product of a force into a space. The energy of an effort, sometimes called ^^ potential energy" (to distinguish it from another form of energy to be referred to in Article 414), is the product of the effort into the distance through which it is capable of acting. Thus, if a weight of 100 pounds be placed at an elevation of 20 feet above the ground, or above the lowest plane to which the circumstances of the ease admit of its descending, that weight is said to possess potential energy to the amount of 100 X 20 = 2,000 /oo^/?OMnc?s; which means, that in descending from its actual elevation to the lowest point of its course, the weight is capable of performing work to that amount. To take another example, let there be a reservoir containing 10,000,000 gallons of water, in such a position that the centre of gravity of the mass of water in the reservoir is 100 feet above the lowest point to which it can be made to descend while overcoming resistance. Then as a gallon of water weighs 10 lbs., the weight of the store of water is 100,000,000 lbs., which being multiplied by the height through which that weight is capable of acting, 100 feet, gives 10,000,000,000 foot-pounds for the potential energy of the weight of the store of water. 260 THEORY OF MACHINES. 409. Equality of Energy Exerted and Work Performed, or the Conservation of Energy. — When an efFort actually does drive its point of application tbrongli a certain distance, energy to the amount of the product of the effort into that distance is said to be exerted ; and the potential energy, or energy which remains capable of being exerted, is to tliat amount diminished. When the energy is exerted in driving a machine at an uniform speed, it is equal to the work performed. To express this algebraically, let t denote the time during which the energy is exerted, v the velocity of a driving point at which an effort P is applied, s the distance through which it is driven, v' the velocity of any working point at which a resistance K is overcome, s' the distance through which it is driven; then 8 = vt; s' = v' t; and multiplying Equation 1 of Article 407 by the time t, we obtain the following equation : — 2'Pt?^ = 2-Ilv'j5 = 2-Ps = 2Ils'; (1.) which expresses the equality of energy exerted, and work per- formed, for constant efforts and resistances. When the efforts and resistances vary, it is sufficient to refer to Articles 400 and 29, to shew that the same principle is expressed as follows : — :fVds = 2 fRds'; (2.) where the symbol f expresses the operation of finding the work performed against a varying resistance, or the energy exerted by a varying effort, as the case may be; and the symbol 2 expresses the operation of adding together the quantities of energy exerted, or work performed, as the case may be, at different points of the machine. 410. Various Factors of Energy. — A quantity of energy, like a quantity of work, may be computed by multiplying either a force into a distance, or a statical moment into an angular motion, or the intensity of a pressure into a volume. These processes have already been explained in detail in Articles 394 and 395, pages 244 to 246. 411. The Energy Exerted in Producing Acceleration is equal to the work of acceleration, whose amount has been investigated in Articles 403 and 404, pages 252 to 257. 412. The Accelerating Effort by which a given increase of velocity in a given mass is produced, and which is exerted by the driving body against the driven body, is equal and opposite to the resistance due to acceleration which the driven body exerts against the driving body, and whose amount has been given in Articles THE ACCELEKATING EFFORT. 261 403 and 404. Referring, therefore, to Equations 4 and 8 of Article 403, we find the two following expressions, the first of which gives the accelerating efibrt required to produce a given acceleration d v in a body whose weight is W, when the time dt in. which that acceleration is to be produced is given, and the second, the same accelerating effort, when the distance ds = vdt in which the ac- celeration is to be produced is given : — p d t w vdv W d(v') V -ds' z — • 9 2ds' •(!■) .(2.) Referring next to Article 404, page 257, we find, from Equation 5, that the work of acceleration corresponding to an increase da in the angular velocity of a rotating body whose moment of inertia is I, is I • d (a^) _I a d a Let d t be the time, and di — adt the angular motion in which that acceleration is to be produced ; let P be the accelerating effort, and I its leverage, or the perpendicular distance of its line of action from the axis ; then, according as the time d t, or the angle d i, is given, we have the two following expressions for the accelerating couple: — P^ = -^ -^^ (3.) g dt ^ ' _I ^ada _ I ^d {a-) . , g di g 2di ^ '' Lastly, referring to Article 405, page 257, Equation 2, we find, that if a train of mechanism consists of various parts, and if W be the weight of any one of those parts, whose velocity v' bears to that \ v' of the driving point v the ratio — = n, then the accelerating effort which must be applied to the driving point, in order that, during the interval d t, in which the driving point moves through the distance d s — v dt, that point may undergo the acceleration d v, and each weight W the corresponding acceleration ndv, is given by one or other of the two formulae — J^n^W dv ^- n 'Z7 V^) dt _ ^n^W ^ vdv _ ^n^^ d{v^) g ds " g 2ds .(6.) 262 THEORY OP MACHINES. both of which are derived from the equation 'P ds =Tv'dt = vdv 2 9 413. Work During Retardatioil— Energy Stored and Restored.— In order to cause a given retardation, or diminution of the velocity of a given body, in a given time, or while it traverses a given dis- tance, resistance must be opposed to its motion equal to the effort which would be required to produce in the same time, or in the same distance, an acceleration equal to the retardation. A moving body, therefore, while being retarded, overcomes re- sistance and performs work; and that work is equal to the energy exerted in producing an acceleration of the same body equal to the retardation. It is for this reason that it has been stated, in Article 403, that the work performed in accelerating the speed of the moving pieces of a machine is not necessarily lost ; for those moving pieces, by returning to their original speed, are capable of performing an equal amount of work in overcoming resistance ; so that the per- formance of such work is not prevented, but only deferred. Hence energy exerted in acceleration is said to be stored; and when by a subsequent and equal retardation an equal amount of work is per- formed, that energy is said to be restored. The algebraical expressions for the relations between a retarding resistance, and the retardation which it produces in a given body by acting during a given time or through a given space, are ob- tained from the equations of Article 412 simply by putting K, the symbol for a resistance, instead of P, the symbol for an effort, and -dv, the symbol for a retardation, instead of dv, the symbol for an acceleration. 414. The Actual Energy of a moving body is the work which it is capable of performing against a retarding resistance before being brought to rest, and is equal to the energy which must be exerted on the body to bring it from a state of rest to its actual velocity. The value of that quantity is the product of the weight of the body into the height from which it must fall to acquire its actual velocity ; that is to say, Jl^ (1.) The total actual energy of a system of bodies, each moving with its own velocity, is denoted by ~~^^' ^^^ and when those bodies are the pieces of a machine, whose velocities A RECIPROCATING FORCE. 263 bear definite ratios (any one of whicli is denoted by n) to tbe velo- city of the driving point v, their total actual energy is |^-2«2W, (3.) being the product of the reduced inertia (or coefficient of steadiness, as Mr. Moseley calls it) into the height due to the velocity of the driving point. The actual energy of a rotating body whose angular velocity is a, and moment of inertia 2 W r^ = I, is «'i. (4) that is, the product of the moment of inertia into the height due to the velocity y a, of a pointy whose distance from the axis of rotation is unity. When a given amount of energy is alternately stored and restored by alternate increase and diminution in the speed of a machine, the actual energy of the machine is alternately increased and diminished by that amount. Actual energy, like motion, is relative only. That is to say, in computing the actual energy of a body, which is the capacity it possesses of performing work upon certain other bodies by reason of its motion, it is the motion relatively to those other bodies that is to be taken into account. For example, if it be wished to determine how many turns a wheel of a locomotive engine, rotating with a given velocity, would make, before being stopped by the friction of its bearings only, sup- posing it lifted out of contact with the rails, — the actual energy of that wheel is to be taken relatively to the frame of the engine to which those bearings are fixed, and is simply the actual energy due to the rotation. But if the wheel be supposed to be detached from the engiee, and it is inquired how high it will ascend up a perfectly smooth inclined plane before being stopped by the attraction of the earth, then its actual energy is to be taken relatively to the earth; that is to say, to the energy of rotation already mentioned, is to be added the energy due to the translation or forward motion of the wheel along with its axis. 415. A Reciprocating Force is a force which acts alternately as an efibrt and as an equal and opposite resistance, according to the direction of motion of the body. Such a force is the weight of a moving piece whose centre of gravity alternately rises and falls ; and such is the elasticity of a perfectly elastic body. The work which a body performs in moving against a reciprocating force is employed in increasing its own potential energy, and is not lost by 264 THEORY OF MACHINES. the body; so that by the motion of a body alternately against and with a reciprocating force, energy is stored and restored, as well as by alternate acceleration and retardation. Let 2 W denote the weight of the whole of the moving pieces of any machine, and h a height through which the common centre of gravity of them all is alternately raised and lowered. Then the quantity of energy — is stored while the centre of gravity is rising, and restored while it is falling. These principles are illustrated by the action of the plungers of a single-acting pumping steam engine. The weight of those plungers acts as a resistance while they are being lifted by the pressure of the steam on the piston: and the same weight acts as effort when the plungers descend and drive before them the water with which the pump barrels have been filled. Thus the energy exerted by the steam on the piston is stored during the up-stroke of the plungers; and during their down-stroke the same amount of energy is restored, and employed in performing the work of raising water and overcoming its friction. 416. Periodical Motion. — If a body moves in such a manner that it periodically returns to its original velocity, then at the end of each period, the entire variation of its actual energy is nothing; and if, during any part of the period of motion, energy has been stored by acceleration of the body, the same quantity of energy exactly must have been during another part of the period restored by retardation of the body. If the body also returns in the course of the same period to the same position relatively to all bodies which exert reciprocating forces on it — for example, if it returns periodically to the same elevation relatively to the earth's surface — any quantity of energy which has been stored during one part of the period by moving against reciprocating forces must have been exactly restored during another part of the period. Hence at the end of each period ^ the equality of energy and work, and the balance of mean effort and mean resistance, holds with respect to the driving effort and tJie resistances, exactly as if the sp>eed were uniform and the reciprocating forces null; and all the equa- tions of Articles 407 and 409 are applicable to periodic motion, pro- vided that in the equations of Article 407, and Equation 1 of Article 409, P, R, and v are held to denote the mean values of the efforts, resistances, and velocities, — that s and s are held to denote spaces moved through in one or more entire periods, — and that in Equation 2 of Article 409, the integrations denoted by \ be held to extend to one or more entire periods. THE EFFICIENCY OF A MACHINE. 265 These principles are illustrated by the steam engine. The velo- cities of its moving parts are continually varying, and those of 8ome of them, such as the piston, are periodically reversed in direc- tion. But at the end of each period, called a revolution, or double- stroke, every part returns to its original position and velocity; so that the equality/ of energy and worh, and the equality of the mean effort to the w,ean I'esistance reduced to the driving point, — that is, the equality of the mean effective pressure of the steam on the piston to the mean total resistance reduced to the piston — hold for one or any whole number of complete revolutions, exactly as for uniform speed. It thus appears that (as stated at the commencement of this Part) there are two fundamentally different ways of considering a periodically moving machine, each of which must be employed in succession, in order to obtain a complete knowledge of its working. " I. In the first place is considered the action of the machine during one or more whole periods, with a view to the determination of the relation between the mean resistances and mean efforts, and of the EFFICIENCY; that is the ratio which the useful part of its work bears to the whole expenditure of energy. The motion of every ordinary machine is either uniform or periodical. " II. In the second place is to be considered the action of the machine during intervals of time less than its period, in order to determine the law of the periodic changes in the motions of the pieces of which the machine consists, and of the periodic or recip- rocating forces by which such changes are produced." 417. Starting and Stopping. — The starting of a machine consists in setting it in motion from a state of rest, and bringing it up to its proper mean velocity. This operation requires the exertion, besides the energy required to overcome the mean resistance, of an additional quantity of energy equal to the actual energy of the machine when moving with its mean velocity, as found according to the principles of Article 414, page 262. If, in order to stop a machine, the effort of the prime mover is simply suspended, the machine will continue to go until work has been performed in overcoming resistances equal to the actual energy due to the speed of the machine at the time of suspending the effort of the prime mover. In order to diminish the time required by this operation, the resistance may be increased by means of the friction of a brake. Brakes will be further described in the sequel. 418. The Efl&ciency of a machine is a fraction expressing the ratio of the useful work to the whole work, which is equal to the energy expended. The Counter-efficiency is the reciprocal of the efficiency, and is the ratio in which the energy expended is greater than the useful work. The object of improvements in 2G6 THEORY OF MACHINES. machines is to bring their efficiency and counter-efficiency as near to unity as possible. As to useful and lost work, see Article 401. The algebraical expression of the efficiency of a machine having uniform or perio- dical motion, is obtained by introducing the distinction between useful and lost work into the equations of the conservation of energy, Article 409. Thus, let P denote the mean effort at the driving point; s, the space described by it in a given interval of time, being a whole number of periods of revolutions; E.^, the mean useful resist- ance; Sj, the space through which it is overcome in the same interval; Eg, any one of the wasteful resistances; s.2, the space through which it is overcome; then Fs = Bs, + 2':R^s^;... (1.) and the efficiency of the machine is expressed by 5lii^__J?i_fi__ (o\ Fs Ri^i-hS • RgSg ^"'^ In many cases the lost work of a machine, 'Re, Sg' consists of a con- stant part, and of a part bearing to the useful work a proportion depending in some definite manner on the sizes, figures, arrange- ment, and connection of the pieces of the train, on which also depends the constant part of the lost work. In such cases the whole energy expended and the efficiency of the machine are expressed by the equations Ps = (l + A)Ri5i + B; ] ^= — '—^ \ (3) 1 + A-I-- — and the first of these is the mathematical expression of what Mr. Moseley calls the " modulus" of a machine. The useful work of an intermediate piece in a train of mechanism consists in driving the piece which follows it, and is less than the energy exerted upon it by the amount of the work lost in over- coming its own friction. Hence the efficiency of such an inter- mediate piece is the ratio of the work performed by it in driving the following piece, to the energy exerted on it by the preceding piece; and it is evident that the effi^ciency of a machine is the product of the efficiencies of the series of 7noving pieces which transmit energy from the driving point to the working point. The same principle applies to a train of successive machines, each driving that M'^hich follows it ; and to counter-efficiency as well as to efficiency. 419. Power and Effect— Horse Power. — Th^ power of a machine THE PRINCIPLE OF VIRTUAL VELOCITIES. 267 is the energy exerted, and the effect^ the useful work performed, in some interval of time of definite length, such as a second, a minute, an hour, or a day. The unit of power called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour. The eJBfect is equal to the power multiplied by the efficiency; and the power is equal to the eflfect multiplied by the counter-efficiency. The loss of power is the dif- ference between the efifect and the power. As to the French " Force de Cheval," see Article 392, page 244. It is equal to 0*9863 of a British horse-power; arid a British horse-power is 1*0139 of a French force de cheval. 420. General Equation. — The following general equation pre- sents at one view the principles of the action of machines, whether moving uniformly, periodically, or otherwise : — where W is the weight of any moving piece of the machine ; h, when positive, the elevation, and when negative, the depres- sion, which the common centre of gravity of all the moving pieces undergoes in the interval of time under consideration; v-^ the velocity at the beginning, and v^ the velocity at the end, of the interval in question, with which a given particle of the machine of the weight W is moving; g, the acceleration which gravity causes in a second, or 32*2 feet per second, or 9*81 metres per second. j^ds', the work performed in overcoming any resistance during the interval in question ; I P ds, the energy exerted during the interval in question. The second and third terms of the right-hand side, when positive, are energy stored; when negative, energy restored. The principle represented by the equation is expressed in words as follows : — The energy exerted, added to the energy restored, is equal to the energy stored added to the work performed. 421. The Principle of Virtual Velocities, when applied to the uniform motion of a machine, is expressed by Equation 3 of Article 407, already given in page 259; or in words as follows : — The effort is equal to the sum of the resistances reduced to the driving point ; that is, each multiplied by the ratio of the velocity of its working point to the velocity of the driving point. The same principle, when applied to reciprocating forces and to re-actions due to varying speed, as well as to passive resistances, is expressed by 268 THEORY OF MACHINES. means of a modified form of tbe general equation of Article 420, obtained in the following manner: — Let n denote either the ratio torne at a given instant by the velocity of a given working point, where the resistance R is overcome, to the velocity of the driving point, or the mean value of that ratio during a given interval of time; let n" denote the corresponding ratio for the vertical ascent or descent (according as it is positive or negative) of a moving piece whose weight is W; let tl denote the corresponding ratio for the mean velocity of a mass whose weight is W, undergoing acceleration or retardation, and —7— either the rate of acceleration at of that mass, if the calculation relates to an instant, or the mean value of that rate, if to a finite interval of time. Then the effort at the instant, or the mean effort during the given interval, as the case may be, is given by the following equation : — gdt • If the ratio 71, which the velocity of the mass W bears to that of the drivin Counter-efficiency = p- = ^-—^^^^^. (2.). It is to be remarked, that the efficiency diminishes to nothing when cotan /3 =y ; that is to say, when /3 is the complement of the ,angle of repose, (p. In other words, if the oblique effort is applied in the direction C Q, no force, how great soever, will be sufficient to keep the piece B in motion. 438. Efficiency of an Axle. — In fig. 156, let the circle A A A represent the trace of the bearing-surface of an axle on a plane perpendicular to its axis of rotation, — in other words, the trans- u 290 THEORY OF MACHINES. verse section of that surface. Let the arrow near the letter N represent the direction of rotation. Let C D be the given force ; that is, as before, the resultant of the weight of the whole piece that rotates with the axle, and of the useful resistance or re-action exerted on that piece by the piece which it drives ; C J, the line of action of the effort by which the rotating piece is driven. m-— Fig. 156. Let r denote the radius of the bearing-surface. About O describe the small circle B B, with a radius = r sin

H the effort; the actual bear- ing-pressure is represented by E C, and the actual effort by C Pi = D E. Hence the eflSciency and counter-efficiency are as follows : — Po_DH.Pi_DE Pi~DE^ Po"I>H ^^ Another method, applicable whether the forces are inclined or parallel, is as follows : — From the axis of rotation O, let fall O L^ and O Mq perpendicular respectively to the lines of action of the given force and of the effort. Then, by the balance of moments, the effort in the absence of friction is From a convenient point in the actual line of action, C Q, of the bearing-pressure (such, for example, as T, where it touches the small circle B B), let fall T L^ and T Mj perpendicular respec- tively to the same pair of lines of action; then the actual effort will be ^^-^ tm; Hence the efficiency and the counter-efficiency have the following value : — .(2.) P„_OL<, •TMi ] ■pro Mo Pi OMo Po OLo" •TL/ •TLi ^T M,' The same results are expressed, to a degree of approximation sufficient for practical purposes, by the following trigonometrical formula:— Let O Lq = 1; O Mo = m; ZCO Lo = a; Z.GOMq-0. Then we have, very nearly, fr Pq_ ^ m-/rsm/3_ m /« \ Pi~ m I +fr sin ec ^ fr . 1 +"'-y-'Sin06 c In making use of the preceding formula, it is to be observed tliat the contrary algebraical signs of sin a and sin /3 apply to those cases in which the two angles a and i3 lie at contrary sides of O C. In the cases in which those angles lie at the same side of C, their algebraical signs are the same; and in the formula they are to be 292 THEORY OF MACHINES. made both positive or both negative, according as /3 is less or greater than a ; so that the efficiency may be always expressed by a frac- tion less than unity. That is to say, fr p 1 -*^ - sin /3 If/3>.;^o^ ^ (3 a.) 1 --J- sm » L -n 1 + — sm /3 If/3<«;p^= ;> . ■' (3b.) 1 + -, sin ct When the lines of action intersect, let O C be denoted by c; then I = c cos a, and m = c cos /3; and consequently the three preceding equations take the following form ; — fr 1 -"^-' tan /3 ^0--^ • (4.) /8 and <« of contrary signs; p" = ^ ; ^ 1 + '- tan « c fi and « of the same sign; 1 --^ tan /3 ''>''^r:=^-^^ (^-) 1 tan a c fr 1 ^--^ tan /3 ^<'''^T=''—fr ^ (^^-^ ^ 1 + — tan a c When the lines of action of the forces are parallel, we have sin fi and sin a= +1 or— 1, as the case may be; and the formulae take the following shape : — When I and m lie at contrary sides of O, the piece is a " lever of the first kind; " and i^rrz^ ^^-^ When I and m lie at the same side of 0; EFFICIENCY OF MODES OF CONNECTION IN GENERAL. 293 If w > /, the piece is a " lever of the second kind;" and ^" "" (5a.) Pi-i_yv I l{m<_l, the piece is a " lever of the third kind;" and (As to levers of the first, second, and third kinds, see Article 184, page 108.) The following method is applicable whether the forces are inclined or parallel; in the former case it is approximate, in the latter exact. Through O, perpendicular to O C, draw XJ O Y, cutting the lines of action of the given force and of the effort in XJ and V respectively. The point where this transverse line cuts the small circle B B coincides exactly with T when the forces are parallel, and is very near T when they are inclined ; and in either case the letter T will be used to denote that point. Then OJJ TY O Y T XJ* .(6.) It is evident that with a given radius and a given coefficient of friction, the efficiency of an axle is the greater the more nearly the effort and the given force are brought into direct opposition to each other, and also the more distant their lines of action are from the axis of rotation. 439. Efficiency of a Screw. — The efficiency of a screw acting as a primary piece is nearly the same with that of a block sliding on a straight guide, which represents the development of a helix situated midway between the outer and inner edges of the screw-thread ; the block being acted upon by forces making the same angles with the straight guide that the actual forces do with that helix. . As to the development of a helix, see Article 160, page 94; and as to the efficiency of a piece sliding along a straight guide, see Article 437, page 288. Section 2. — Efficiency and Counter-efficiency of Modes of Connection in Mechanism. 440. Efficiency of Modes of Connection in General. — In an ele- mentary combination consisting of two pieces, a driver and a 294 THEORY OF MACHINES. follower, there is always some work lost in overcoming wasteful resistance occasioned by the mode of connection j the result being that the work done by the driver at its working-point is greater than the work done upon the follower at its driving-point, in a proportion which is the counter-efficiency of the connection ; and the reciprocal of that proportion is the efficiency of the connection. In calculating the efficiency or the counter-efficiency of a train of mechanism, therefore, the factors to be multiplied together comprise not only the efficiencies, or the counter-efficiencies, of the several primary pieces considered separately, but also those of the several modes of connection by which they communicate motion to each other. 441. Efficiency of Rolling Contact. — The work lost when one primary piece drives another by rolling contact is expended in overcoming the rolling resistance of the pitch-surfaces, a kind of resistance whose mode of action has been explained in Article 402, page 251; and the value of that work in units of work per second is given by the expression a 6 N ; in which N is the normal pressure exerted by the pitch-surfaces on each other; 6, a constant arm, of a length depending on the nature of the surfaces (for example 0*002 of a foot = 0*6 millimetre for cast iron on cast iron, see page 252); and a the relative angular velocity of the surfaces. The useful work per second is expressed by ^^/JS", in which y* is the coefficient of friction of the surfaces, and u the common velocity of the pitch lines. Hence the counter-efficiency is -ah .- . c = l+— -> .(1.) Let />! and 'p^ be the lengths of two perpendiculars let fall from the two axes of rotation on the common tangent of the two pitch- lines; if the pieces are circular wheels, those perpendiculars will be the radii. Then the absolute angular velocities of the pieces are respectively - and ; and their relative angular velocity is therefore which value being substituted in Equation 1, gives for the counter- efficiency the following value ; — e.l^l(A^l\ (2.) It is assumed that the normal pressure is not greater than is EFFICIENCY OF SLIDING CONTACT IN GENERAL. 295 necessary in order to give sufficient friction to communicate the motion. It is evident, from the smallness of h, that the lost work in this case must be almost always a very small fraction of the whole. 442. Efficiency of Sliding Contact in General.— In fig. 157, let T be the point of contact of a pair of moving pieces connected by sliding contact. Let the plane of the figure be that containing the directions of motion of the two particles which touch each other at the point T ; and let T V be the velocity of the driving- particle, and T W the velocity of the following particle ; whence V W will represent the velocity of sliding, and T U, perpendicular to Y W, the common component of the velocities of the two particles along their line of connection RTF. C T C, parallel to Y "W, and perpendicular to R T P, is a common tangent to the two acting surfaces at the point T j the arrow A represents the direction in which the driver slides relatively to the follower; and the arrow B, the direction in which the follower slides relatively to the driver. ^ig- 1^7. Along the line of connection, that is, normal to the acting sur- faces at T, lay off T P to represent the effort exerted by the driver on the follower, and T K ( = - T P) to represent the equal and opposite useful resistance exerted by the follower against the driver. Draw S T Q, making with R T P an angle equal to the angle of repose of the rubbing surfaces, (see Article 261, page 154), and inclined in the proper direction to represent forces opposing the sliding motion; draw P Q and II S parallel to C C. Then T Q will represent the resultant pressure exerted by the driver on the follower, and T S ( = — T Q), the equal and opposite resultant pressure exerted by the follower against the driver, and P Q = - R S will represent the friction which is overcome, through the dis- tance Y W, in each second ; while the useful resistance, T R, is overcome through the distance T U. Hence the useful work per second is T U counter-efficiency is T R; the lost work is Y W • R S; and the YW-RS T U • T R* .(1.) 296 THEORY OF MACHINES. Let the angle U T V = a, the angle U T W = /3, and let/ be the coefficient of friction. Then we have — -^p^=tan« + tan^; TR"-^^ and consequently c = l +/(tan a + tan /3) (2.) 443. Efficiency of Teeth. — It has already been shewn, in Article 148, page 87, that the relative velocity of sliding of a pair of teeth in outside gearing is expressed at a given instant by (aj + a^ t; where t denotes the distance at that instant of the point of contact from the pitch-point. (In inside gearing the angular velocity of the greater wheel is to be taken with the negative sign.) The distance t is continually varying from a maximum at the beginning and end of the contact, to nothing at the instant of passing the pitch-point. Its mean value may be assumed, with sufficient accuracy for practical purposes, to be sensibly equal to one-half of its greatest value ; and in the formulae which follow, the symbol t stands for that mean value. Let P be the mutual pressure exerted by the teeth; f, the coefficient of friction ; then the work lost per second through the friction of the teeth is (% + a^) tfY. Let u be the common velocity of the two pitch-circles ; , the mean obliquity of the line of connection to the common tangent of the pitch-circles ; then u cos ^ is the mean value of the common component of the velocities of the acting surfaces of the teeth along the line of connection ; and the useful work done per second is expressed by P u cos 6. St) that the counter-efficiency is ,^l+K±«2)i/. (1.) u cos ^ ' Let r^ and rg be the radii of the two pitch circles; then we have _ u _u and consequently l+pseca{i + i} (2.) EFFICIENCY OF BANDS. 297 If two pairs of teeth at least are to be in action at each instant (as in the case of involute teeth, and of some epicycloidal teeth), and if the pitch be denoted by^, we have ^ see e = «; and therefore where Wj and n^ are the number of teeth in the two wheels. In many examples of epicycloidal teeth, especially where small 2 3 pinions are used, the duration of the contact is only ^ or r of that assumed in Equation 3 j and the work lost in friction is less in the same proportion. 444. EflBciency of Bands. — A band, such as a leather belt or a hempen rope, which is not perfectly elastic, requires the expenditure of a certain quantity of work — first to bend it to the curvature of a pulley, and then to straighten it again; and the quantity of work so lost has been found by experiment to be nearly the same as would be required in order to overcome an additional resistance, varying directly as the sectional area of the band, directly as its tension, and inversely as the radius of the pulley. In the follow- ing formulae for leather belts, the stiffness is given as estimated by Keuleaux {Constructionslehre filr Maschinenhau, § 307). Let T be the mean tension of the belt; S, its sectional area; r, the radius of the pulley; h, a constant divisor determined by experiment; R', the resistance due to stiffness; then E' = |^ (1-) r h (for leather) = 3-4 inch = 87 millimetres. To apply this to an endless belt connecting a pair of pulleys of the respective radii r^ and r^, let Tj and Tg be the tensions of the two sides of the belt. Then the useful resistance is Tj - Tg,. the T +T mean tension is -^-^ — ^; and the additio»al resistance due to stifiness is T, + T,Sri 1 \. 2 bir^^rj' consequently the counter-efficiency is Ti + Tg S f 1 Hi 1 -JLlL?/1 _h .(2.) 298 THEORY OF MACHINES. T N denoting ~. The sectional area, S, of a leather belt is given by the formula ^-p' (3.) where p denotes the safe working tension of leather belts, in units of weight per unit of area; its value being, according to Morin, 0'2 kilogi'arame on the square millimetre, or 285 lbs. on the square inch. The ordinary thickness of the leather of which belts are made is about 0-16 of an inch, or 4 millimetres; and from this and from the area the breadth may be calculated. A double belt is of double thickness, and gives the same area with half the breadth of a single belt. When a band runs at a high velocity, the centrifugal tension, or tension produced by centrifugal force, must be added to the tension required for producing friction on the pulleys, in order to find the total tension at either side of the band, with a view to determining its sectional area and its stiffness. The centrifugal tension is given by the following expression : — -^'' <'•) in which w is the heaviness (being, for leather belts, nearly equal to that of water); S, the sectional area; v, the velocity; and g, gravity ( = 32-2 feet, or 9-81 metres per second). When centrifugal force is taken into consideration, the following formula is to be used for calculating the sectional area; T^ being the tension at the driving-side of the belt, exclusive of centrifugal fension : — S ^"T .(5. and the following fomiula for the counter-efficiency : — 2 wv^ . T, + T c = l + •|.ji.l| (6.) 2(T,-T,) For calculating the efficiency of hempen ropes used as bands, it is unnecessary in such questions as that of the present article to use a more complex formula than that of Eytelwein — viz., D2T «' = W' (^•) EFFICIENCY OF LINKWORK. 299 where D is the diameter of the rope, and h' - 54 millimetres = 2-125 inches. D2 g In all the formulae, -^r is to be substituted for -^ . 6 6 value of D^ is given by the formula T The proper (8.) ■where p' = i,000 for measures in inches and lbs. ; and j9'= 0-7 for measures in millimetres and killogrammes. 445. Efficiency of Linkwork.— In fig. 158, let C^ T^, Cg T2 be two levers, turning about parallel axes at Cj and Gg, and connected with each other by the link Tj Tgj Ti and Tg being the connected points. Ficr. 158. The pins, which are connected with each other by means of the link, are exaggerated in diameter, for the sake of distinctness.' Let C^ Ti be the driver, and Cg Tg the follower, the motion being as shewn by the arrows. From the axes let fall the perpendiculars ^1 1*1) p2 1*2) upon the, line of connection. Then the angular, velocities of the driver and follower are inversely as those perpen- diculars ;V' and, in the absence of friction, the driving moment of the first leve'rland the working moment of the second' are ^ directly as' those perpendiculars; the driving pressure being exerted along the' line of connection T^ Tg. Let Mg be the working moment; and let Mq be the driving moment in the absence of friction; then we have M„ = M„ CiP, Co Pq 300 THEORY OF MACHINES. To allow for the friction of the pins, multiply the radius of each pin by the sine of the angle of repose; that is, very nearly by the coefficient of friction ; and with the small radii thus computed, Tj Ai and Tg Ag, draw small circles about the connected points. Then draw a straight line, Qi Aj Bj Q2 Ag Bg, touching both the small circles, and in such a position as to represent the line of action of a force that resists the motion of both pins in the eyes of the link. This will be the line of action of the resultant force exerted through the link. Let fall upon it the perpendiculars Cj Qi, C2 Q2; these will be proportional to the actual driving moment and working moment respectively; that is to say, let JVI^ be the driving moment, including friction; then M,=^ Mo • Ci Qi C2Q2 • Comparing this with the value of the d,riving moment without friction, we find for the counter-efficiency M,_ C,Qi'C,P2 . , . '-Mo~C,Q,-C,F./ ^"-'^ and for the efficiency c~Mi CiQi-C^Pa* ^^ 446. Efficiency of Blocks and Tackle.— (See Articles 181, 182, pages 105 and. 106.) — In a tackle composed of a fixed and a running block containing sheaves connected together by means of a rope, let the number of plies of rope by which the blocks are connected with each other be n. This is also the collective number of sheaves in. the two blocks taken together, and is the number expressing the jmrchase, when friction is neglected. Let c denote the counter-efficiency of a single sheave, as depend- ing on its friction on the pin, according to the principles of Article 373, page 290. Let c denote the counter-efficiency of the rope, when passing over a single sheave, determined by the principles of Article 444, the tension being taken as nearly equal to — ; where R is the useful load, or resistance opposed to the motion of the running block. B -f- n is also the effi^rt to be exerted on the hauling part of the rope, in the absence of friction. Then the counter-efficiency of the tackle will V)e expressed approximately by (ccV; ' (1.) so that the actual or effective purchase, instead of being expressed by w, will be expressed by «(<'«')-" (2.) EFFICIENCY OF CONNECTION BY MEANS OP A FLUID. 301 447. Efficiency of Connection by means of a Fluid. — Whea motion is communicated from one piston to another by means of an intervening mass of fluid, as described in Articles 185 to 188, pages 110 and 111, the efficiencies and counter-efficiencies of the two pistons have in the first place to be taken into account; that is to say, with ordinary workmanship and packing, the efficiency of each piston may be taken at 0-9 nearly; while with a carefully made cupped leather collar the counter-efficiency of a plunger may be taken at the following value : — ■-'^^ (..1 in which d is the diameter of the plunger; and h a constant, whose value is from 0*01 to 0015 of an inch, or from 0-25 to 0*38 of a millimetre. For if c be the circumference of the plunger, and ^ the effective pressure of the liquid, the whole amount of the pres- sure on the plunger is —j^- i and the pressure required to overcome the friction is pcb. The efficiency and counter-efficiency of the intervening mass of fluid remain to be considered; and if that fluid is a liquid, and may therefore be regarded as sensibly incompressible, these quan- tities depend on the work which is lost in overcoming the resist- ance of the passage which the liquid has to traverse. To prevent unnecessary loss of work, that passage should be as wide as possible, and as nearly as possible of uniform transverse section; and it should be free from sudden enlarge- ments and contractions, and from sharp bends, all necessary enlargements and contractions which may be required being made by means of gradually tapering conoidal parts of the passage, and all bends by means of gentle curves. When those conditions are fulfilled, let Q be the volume of liquid which is forced through the passage in a second; S, the sectional area of the passage; then, "I <^> is the velocity of the stream of fluid. Let b denote the wetted border or circumference of the passage ; then, m = ^, (3-) is what is called the hydraulic mean depth of the passage. In a cylindiical pipe, m = ^ diameter. Let I be the length of the 302 THEORY OF MACHINES. passage, and w tlie heaviness of the liquid. Then the loss of pres- sure in overcoming the friction of the passage is ^-^•-i-g' (*•) in which g denotes gravity, and f a coefficient of friction whose value, for water in cylindrical cast-iron pipes, according to the experiments of Darcy, is /=0005(l+^^);* (5.) d being the diameter of the pipe in feet. Let p be the pressure on the driven or following piston ; then the pressure on the driving piston is p + p'y and the counter- efficiency of the fluid is i-f; («•) which, being multiplied by the product of the counter-efficiencies of the two pistons, gives the counter-efficiency of the ' intervening liquid. When the intervening fluid is air, there is a loss of work through friction of the passage, depending on principles similar to those of the friction of liquids; and there is a further loss through the escape by conduction of the heat produced by the compression of the air. The friction which has to be overcome by the air, and which causes a certain loss of pressure between the compressing pumps and the working machinery, consists of two parts, one occasioned by the resistance of the valves, and the other by the friction along the internal surface of pipes. , To overcome the resistance of valves, about five per cent, of the effective pressure may be allowed. The friction in the pipes depends on their length and diameter, and on the velocity of the current of air through them. It is nearly proportional to the square of the velocity of the air. A velocity of about forty feet per second for the air in its com- pressed state has been found to answer in practice. The diameter of pipe required in order to give that velocity can easily be com- puted, when the dimensions of the cylinders of the machinery to be driven, and the number of strokes per minute, are given. When the diameter of a pipe is so adjusted that the velocity of the air is 40 feet per second, the pressure expended in overcoming its friction may be estimated at one per cent, of the total or absolute 1 25*4 • When the diameter is expressed in millimetres, forr^ substitute -t-* EFFICIENCY OP CONNECTION BY MEANS OF A FLUID. 303 pressure of the air, for every Jive hundred diameters of the pipe that its length contains. Although the abstraction from the air of the heat produced by the compression involves a certain sacrifice of motive power (say from 30 to 35 per cent.) still the effects of the heated air are so inconvenient in practice, that it is desirable to cool it to a certain extent during or immediately after the compression. This may be effected by injecting water in the form of spray into the com- pressing pumps j and for that purpose a small forcing pump of about xJijth of the capacity of the compressing pumps has been found to answer in practice. The air may thus be cooled down to about 104'' Fahr. or 40° Cent. The factor in the counter-efficiency due to the loss of heat expresses the ratio in which the volume of air as discharged from the compressing pump at a high temperature is greater than the volume of the same air when it reaches the working machinery at a reduced temperature; which ratio may be calculated approxi- mately by taking two-sevenths of the logarithm of the absolute working pressure of the compressed air in atmospheres, and finding the corresponding natural number. That is to say, let p^ denote one atmosphere ( = at the level of the sea 14*7 lbs. on the square inch, or 10,333 kilogrammes on the square m^tre); let p^ be the absolute working pressure of the air, so that j^i-jOo is the effective pressure; then the counter-efficiency due to the escape of heat is, '© (7.) From examples of the practical working of compressed air, when used to transmit motive power to long distances, it appears that in order to provide for leakage and various other imperfec- tions in working, the capacity of the compressing pumps should be very nearly double of the net volume of uncompressed air required; and it has also been found necessary, in working the compressing pumps, to provide from three to four times the power of the machinery driven by the compressed air. INDEX, Absolute unit of force, 213. Acceleration, work of, 252. Accelerating effect of gravity, 213. force, 213. impulse, 207. Action and re-action, 115. Actual energy, 207. Addendum of a tooth, 81. Aggregate combinations, 73, 112. Angle of repose, 154. of rotation, 48. Angular impulse, 220. momentum, 219, 228. momentum, conservation of, 220. momentum and angular impulse relation of, 220. velocity, 48. velocity, variation of, 63. Arch, line of pressures in, 177. Arcs, measurement of, 23, 24. Areas, centre of, 26. mensuration of, 16, 17. Axis, instantaneous, 55. of rotation, 47, 48, Axle, strength of, 187 torsion of, 187. Axles and shafts, efficiency of, 289. friction of (see Efficiency). Balance, 31, 118. of any system of forces, 135, 136, 137. of any system of forces in one l^lane, 134. of chain or cord, 174. of couples, 126. of forces in one line, 118. of inclined forces, 122. of parallel forces, 131, 132. of structures, 157. Balanced forces, motion under, 210. Bands, classed, 97. connection by, 72, 97, 98. efficiency of, 297. length of, 99. motion of, 97. principle of connection by, 97. Bar, 158. Beam, 158. allowance for weight of, 200. limiting length of, 200. in link work, 101. Bearings, 71. friction of, 251. Belt, with speed cones, 100. Bending moment, at a series of sec- tions, 193. Bending moment, greatest, 194. Bending moments, calculation of, 190. Bending, resistance to, 189. moment of, 190. Bevel-wheels (see Wheels). Blocks and tackle, 105. efficiency of, 300. Blocks, stability of a series of, 158, 175. Bodies, 30. rigid, 47. Bracing of frames, 166, 167, 108. Brake, 241. Brakes, 276. block, 277. Bulkiness, 12L Buoyancy, centre o£, 121. Cam or Wiper, 92. Centre of area, 26. of a curved line, 27. of a plane area, 26. of buoyancy, 121. of gravity, 121, 140. of magnitude, 25, 26, 27, 28, 29. of mass, 207. of oscillation or percussion, 208, 227. of parallel forces, 119, 133. of pressure, 121. of resistance, 176. of special figures, 28. of volume, 27. Centrifugal force, 207 (see also De- viating Force). Chains, equilibrium of, 158, 174. Channel, 68. Cinematics, 31. 306 INDEX. Cinematics, principles of, 33. Circle, involute of (see Involute). area of, 21. Circular arcs, measurement of, 23. Circular measure, 8. sector, area of, 22, arcs, length of, 23, 24. Click, 105. Coefficient of stiffness, 183. of elasticity, 184. of pliability, 183. Cog, hunting, 83. Collar, friction of, 251. Collision, 208, 221. Combinations, aggregate (see Aggre- gate). elementary (see Elementary). Comparative motion, 38, 45, 50, 63. Components, 123. of motion, 35, of varied motion, 40. Compression, resistance to, 202. Cones, x>itch (see also Wheels, bevel). rolling, G3. speed, 100. Connected points, motion of, 102, Connecting-rod, 101 (see Linkwork). Connection, line of, 73. principle of, 73. Connectors, 71. Conservation of energy, 206, 260. of angular momentum, 220. of momentum, 219. Continued fractions, 2. Continuity, equations of, in liquids, 67, 69. Contracted vein, 233. Contraction, coefficient of, 233. Cord, equilibrium of, 158, 174. guided by surfaces of revolution, 66. motion of, 65. Counter-efficiency, (see Efficiency). Cou])led parallel shafts, 101. Couples, 118, 119. equivalent, 125. parallelogram of, 126. polygon of, 126. resultant of, 125. with parallel axes, 126. Coupling, double, Hooke's, 105. Hooke's, 104. Oldham's, 96. Coupling-rod, 101 (see Linkwork). Crank-rod, 101 (see Linkwork). Crosp-br-paJiiDcr. resistance to 389. Crushing, direct resistance to, 202. Curved lines, measurement of, 23. Curves, measurement of the length of, 23, 24, 25. Cycloid, 55. Cylinders, strength of, 186, 187. Dead points in linkwork, 101. Dead load, 180. Density, 120. Deviating force, 207, 216. in terms of angular velocity, 217. Deviation (of motion), uniform, 44. varying, 45. Differential and integral calculus, 10. coefficients, 11, 12. calculus, geometrical illustration of, 12. Direction, fixed and nearly fixed, 33. Directional relation, 38. Distributed forces, 119, 120, 140. loads, 160. Driving-point, 242, Dynamics, 32. general equations of, 211. Dynamometer, 271. Eccentric, 103. rod, 101. Effect and power, 241, 266. _ Efficiency'- and counter-efficiency, 241, 265, 286. of a machine, 265, 266. of a shaft or axle, 2S9. of a sliding piece, 288. of modes o^ connection in mechan- ism, 293. of primary pieces, 287. of bands, 297. of linkwork, 299. of blocks and tackle, 300. of fluid connection, 301. of a screw, 293. of rolling contact, 294. of sliding contact, 295. of teeth,'' 296. Effort, 205._ accelerating, 260. when speed is uniform, balances resistances, 215. Elasticity, 183. coefficients of, 184. modulus of, 184. 1 Elementary combinations, 72. ' classed generally, 72. INDEX. 307 Energy, 206, 259. actual (or kinetic), 207, 262. and work, general equation of, 267. exerted and work done, equality of, 260. potential, 259. stored and restored, 208, 262. conservation of, 206, 260. transformation of, 208. Epicycloid, 58. Epicycloidal teeth, 89, 90. Epitrochoid, 58. curtate, 60. prolate, 59. Equilibrium (see Balance). Face of a tooth, 81. Factors of safety, 180. prime, of a number, 1. Falling body (see Gravity). Fixed direction, 33. point, 31. Flank of a tooth, 81. Flow of liquid, 66, 67. in a stream, 67. Fluctuations of speed, 241. Fluid, motion of, 66, 68, 69, 230. pressure of, 147. steady motion of, 68. velocity and flow of, 66. Fluids, flow of volume of, 69. balance of, 147. flow of mass of, 69. Fly-wheels, 241, 278, 280. Foot-pound, 243. Force, 31, absolute unit of, 116, 213. centrifugal (see Deviating Force). deviating (see Deviating Force). direction of, 116. distributed, 119, 120, 140. magnitude of, 116. measure of, 117. moments of, 127, 130. rectangular components of, 124. representation of, 115, 116. reciprocating, 208, 263. Forces, action and reaction, 115. how determined and expressed, 115. inclined, resultant and balance of, 122, 125. parallel, 118. parallel, magnitude of resultant of, 127. direction of, 128. Forces, parallelogram of, 122. parallelepiped of, 123. polygon of, ] 23. representation of by line, 117. resolution of, 122, 123, 124. resultaftt and component of, llg, triangle of, 122. Fractions, continued, 2. Frames, 71. bracing of, 166. equilibrium and stability of, 15S. of two bars, 161. polygonal, 163, 164, 165. resistance of, at a section, 171* triangular, 162, 163. Friction, 153, 154. coefficient of, 154. moment of, 251. of liquid, 235. of solid bodies, law of, 153. tables of, 155. work done against, 251. Frictional stability, 176. Function, 6. Governors, 241, 282. pendulum, 283. loaded, 285. Gravity, accelerating effect of, 213. centre of, 121, 140. motion under, 213. specific, 120. Greatest common measure, 1. Gyration, radius of, 208, 223. table of radii of, 226. Head, dynamic, of liquid, 230, Heat of friction, 252. Heaviness, 120. Helical motion, 51, 52. Helix (see Screw-line). normal, 93. Horse-power, 241, 266. Hunting-cog, 83. Hydraulic connection, 110. efficiency of, 301. hoist. 111. Hydraulic press, 110. Hydrostatics, principles of, 147, 148, 149. Impulse, 207. and momentum, law of, 254. Inclined plane, 107. Indicator, 271. 308 INDEX. Indicator diagram, 273. Inertia, or mass, 206. moment of (see Moment). reduced, 257. Integrals, approximate computation of, 13, 14, 15. Intensity of distributed force, 120. of pressure, 121. of stress, 143. Intervening fluid, connection by, 73. Involute, 56. Joints, of a structure, 156. Journal, friction of, 251. Kinetics, 32, 205. general equations of, 211. Lateral force, 205. Length, measure of, 30, 31. Lever, 101, 107, 128. Line, 30. Link, ,101. Linkwork, connection by, 72, 101. comparative motion of the con nected points in, 102. efi&ciency of, 299. Liquid, dynamic head of, 230. equilibrium of, 147. free surface of, 231. motion of, 230, 233. motion of, in plane layers, 232. motion of, vv^ith friction, 233. surface of equal pressure in, 231. without friction, motion of, 230. Live load, 180. Load, 179. dead, 180. Hve, 180. working, 179. Logarithms, common, 4, 5, 6. Machine, eflaciency of (see Efl&- ciency). action of, 243. general equation of the action of, 267. moving pieces in, jjrimary and secondary, 72. Machines, 32. theory of, 240. Magnitude, centre of, 25. Mass, 206. centre of, 207. in terms of weight, 212. measure of, 117. Matter, 30. Measure, greatest common, 1. Measures offeree and mass, 117. of length, 30, Mechanical powers, comparative mo- tion in, 107. forces in, 268. Mechanics, 30. Mechanism, theory of, 70. aggregate combinations in, 73. elementary combinations in, 72. principle of connection in, 73. Mensuration of areas, 17. of curved lines, 23. of geometrical moments, 25 of volumes, 22. Merrifield's trapezoidal rule, 19, 20. Modulus of elasticity, 184. height or length of, 184. of i)liability, 183. of resilience, 185. of stiffness, 183. of transverse elasticity, 187. Moment, bending, 190. geometrical, 25. geometrical, of inertia, 199. greatest, 194. of a couple, 127. of a force, 127, 130. of inertia, 208, 222. of inertia, table of, 226. of stability, 177. of stress, 196. Momentum, 207. and impulse, law of, 254. angular (see Angular Momentum). conservation of, 219. of a rotating body, 228. resultant, 207. variation and deviation of, 207. Motion, 31. combination of uniform, and uni- formly accelerated, 43. comparative, 38, 39, 50, 63. component and resultant, 35. first law of, 210. graphical representation of, 42. of a falling body, 213. of fluid of constant density, 66. of pistons, 68. of points, 34, 37. of points, varied, 39, 40. of pliable bodies and fluids, 65. of rigid bodies, 47. of varying density, 69. periodical, 208, 264, 278. INDEX. 309 Motion, second law of, 21 1. uniform, 37, 205. uniform, dynamical principles of, 210. Neutral surface, 197. Parabolic curves, 16, 17. Parallel forces, 118, 127. centre of, 119, 133. forces, resultant of, 127, 128, 129, 131, 132. projection (see Projection, Parallel). Parallelogram, area of, 16. Parallelopiped of motions, 38. Pendulum, rotating, 217. simple oscillating, 218. simple revolving, 217. Percussion, centre of (see Centre). Periodic motion, 208, 264, 278. Periodical motion of machines, 208. Pieces, moving, 71. of a structure, 156. Pinion, smallest, with involute teeth, 89. Pipes, friction in, 237. resistance caused by sudden en- largement in, 238. resistance of curves and knees in, 238. resistance of mouthpieces of, 238. Piston, 110. action of a fluid upon, 110. motion of, 68. Piston-rod, 101. Pitch of a screw, axial, 94. divided, 93. normal, 93. of teeth, 81 (see Teeth). Pitch-circles, 81. Pitch-lines, 81. Pitch-point, 81. Pitch-surfaces, 74, 81 (see Wheels). Pivot, friction of, 251. Plane of rotation, 48. Pliability, 183. coefficients of, 183. Point, 30. fixed, 31, 35. motions of, 34. moving, 35. physical, 30. Power, 241. and effect, 241, 266. horse, 241, 266. Powers, mechanical (see Mechanical powers). Press, hydraulic (see Hydraulic press). Pressure, 144. centre of, 121. intensity of, 121. Primary moving pieces, efficiency of, 287. motions of, 72. Prime factors, 1. Prime movers, 240. Projection, paraUel, 138, 153, 178. Projectile, unresisted, 214. Proof strength, 182, 183. Pull (see Tension). Pulley-blocks (see Tackle). Pulley (mechanical power), 107. Racks, toothless, 74. smooth, 74. st^-aight and circular wheels, 75. Eadius, geometrical, 81. of gyration, 208. real, 81. Eatio, 2. approximation to, 2. Reaction and action, 115. Reciprocating force, 208, 263. Reduced inertia, 257. Reduction of forces and couples in machines to the driving point, 257. Reduplication (see Tackle). Regulating apparatus, 276. Regulator of a prime mover, 241. Repose, angle of (see Angle). Resilience, 184. Resistance, 205. centre of, 176. line of, 176. points of, 242. of curves and knees, 23S. of mouthpieces, 238. of rolling, 252. useful and prejudicial, 241. Resolution offerees, 122. Rest, 31. Resultant, 118. momentum, 207. of any system of forces, 135. of any system of forces in one plane, 134. of couples, 125. of inclined forces, 125. motions, 35. of parallel forces, 127, 128, 129. 131, 132. 310 INDEX. Kigid body, motion of, 47, 222 (see Rotation). .Rigidity or stiffness, 183. coefficients of, 183. Rod (see Crauk-rod, Coupling-rod, Connecting-rod, Eccentric -rod, Link, Piston-rod). Rolled curves (see Cycloid, Epicy- cloid, Epitrochoid, Involute, Spiral, Trochoid). Rollers, 74. Rolling contact, connection by, 72. cones, 63. efficiency of, 294. general conditions of, 74. of cylinder on plane, 55. of cylinder on cylinder, 58. of plane on cylinder, 55. resistance, 252. Rotating body, comparative motion of points in, 50, components of velocity of a point in, 50. relative motion of a pair of points in, 49. Rotation, 47. actual energy of, 229. angle of, 48. angular velocity of, 48. axis of, 47, 48. combined with translation, 51, 54. combined parallel, 56, 57, 62. components of, varied, 64. instantaneous axis of, 55. plane of, 48; right and left handed, 49. uniform, 48, 228. varied, 63, 64. Rotations about intersecting axes combined, 62. Safety, factors of, 180. Screw, 92. circular, pitch of, 93. efficiency of, 293. mechanical power, 107. pitch of, 92, 93. Screw-gearing, 94. axial pitch of, 94. development of, 94. divided pitch of, 93. Screw-like or helical motion, 51, 52. Screw-line, normal pitch of, 93. Screws, compound, 113. relative sliding of a pair of, 95. right and left handed, 93. Secondary moving pieces, 72. efficiency of, 289. Sections, method of, applied to frame- work, 171. Shaft, strength of (see Axle). Shear, 144. Shearing load, greatest, 192. at a series of sections, 192. Shearing loads, calculation of, 190. Shearing, resistance to, 186. Sheaves, 105. Shifting, or translation, 47. Simpson's Rules, 18, 19. Skew-bevel wheels (see Wheels). Sliding contact, connection by, 72. efficiency of, 295. principle of, 79, 80. Sliding piece, efficiency of, 288. Solid, 30. Solids, mensuration of, 22. Specific gravity (see Gravity, Specific). Speed (see Velocity). Speed, adjustments of, 73. cones, 100. fluctuations of, 241. periodic fluctuations of (see Periodic motion). uniform, condition of, 258. Spheres, strength of, 186. Spiral, 55, 56. Spring, 184. Stability, 156. frictional, 176. of position, 176. Standard measure of length, 30. measure of weight, 116. Starting a machine, 265. Statics, 32. principles of, 115. Stiffness, 157, 179. Stopping a machine, 265. Strain, 179. Stream of liquid, friction of, 235. hydraulic, mean depth o^ 236. varying, 236. Strength, 156, 179, coefficients or moduli of, 180. proof, 179. transverse, 196. ultimate, 179. Stress, 143, 179. classes of, 144. compound internal, 149. intensity of, 143. internal, 147. INDEX. 311 Stress, moment of, 196. shearing, 150. tangential, 144. uniform, 145. varying, 145. Stresses, conjugate, principal, 150. Stretching, resistance to, 184. Structures, 32. equilibrium of, 157. theory of, 156. Stroke, length of, in linkwork, 104. Struts, 158. Supports, 156. Surface, 30. System of parallel forces, 131. Tackle, 105. connection by, 73, 105. eflBciency of, 300. Tearing, resistance to, 184. Teeth, arc of contact of, 88. dimensions of, 91. efficiency of, 296. epicycloidal, 89. involute, for circular wheels, 88, 89. of mitre or bevel-wheels, 91, 92. of non-circular wheels, 92. of spur wheels and racks, 86. of wheels, 81. of wheel and trundle, 90. pitch and number of, 81. sliding of, 87. traced by rolling curves, 86. Tension, 144, 184. Testing, 182. Thrust, 144. Tie, 158. strength of, 1 84. Time, measure of, 35. Tooth, face of, 81. flank of, 81. Torsion (see Wrenching). Trains of mechanism, 73, 111. of wheelwork, 83, 84, 85. Transformation (see Projection). Transformation of energy, 208. Translation or shifting, 47. varied, 211, 219. Transverse strength, 196. table, 200. Trapezoid, area of, 16. Trapezoidal rule, Merrifield's, 19, 20. common, 21. Triangles, area of, 10, 16. solution of plane, 8, 9. Trigonometrical rules, 6, functions of one angle, 7. functions of two angles, 8. Trochoid, 55. Trundle, 90. Truss, 168. compound, 169. Trussing, secondary, 169, 170, 171- Turning (see Eotation). Twisting (see Wrenching). Unguents, 252. Uniform motion, 37, 205. deviation, 44. effort or resistance, effect of, 215. motion under balanced forces, 210. rotation, 48. stress, 145. velocity, 36. Universal joint, 104. double, 105. Valves, 110. Velocities, virtual, 206, 267. Velocity, 36, 244. angular, 48. angular, variation of, 63. ratio, 38. uniform, 36. uniformly -varied, 41. varied, 39. varied rate of variation of, 43. Virtual velocities, 206, 267. Volume, 30. Volumes, measurement of, 22. Wedge (mechanical power), 107. Weight, 116. mass in terms of, 212. Wheel and axle, 107. and rack, 75. and screw, 95. Wheels, bevel, 76, 81. circular, in general, 75. non-circular, 77. pitch-surfaces, pitch-lines, pitch- points of, 81. skew-bevel, 77, 78, 81. spur, 81. Wheelwork, train of, 83. White's tackle, 106. Windlass, differential, 112. Wooley's rule, 22. Work, 206, 243. against an oblique force, 246. against friction, 251. 312 INDEX Work, against varying resistance, 249, 250. algebraical expressions for, 246. and energy, general equation of, 267. done, and energy exerted, equality of, 260. done during retardation, 262. in terms of angular motion, 244. in terms of pressure and volume, 245. measures of, 243. of acceleration, 252. Work, of acceleration, summation of, 256. of machines, 243. rate of, 243. represented by an area, 249. summary of various kinds of, 258. summation of, 247, 248, 256. useful and lost, 241, 251. Working point, 242. Working stress, working load, 179. Wrenchmg, resistance to, 187. Yard, standard, 30. THE END. PKIMTED BT BKLL AUD BAIK, GLASGOW 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 2Way'6 ^TPf r3rr^r?03 *T8 FU„-£iVED juN 2 1 mh % CDS71EEE31