THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF John S.Prell A MECHANICAL TEXT-BOOK. STANDARD TECHNICAL WORKS. SECOND EDITION, Revised and Enlarged. Pocket-Size, Leather, 12s./ also for Office Use, Cloth, 12s. BOILERS: THEIR CONSTRUCTION AND STRENGTH. A Handbook of Rules, Formulae, Tables, &c., relative to Material, Scantlings, and Pressures, Safety Valves, &c. For the use of Engineers, Inspectors, Surveyors, Draughtsmen, Boilermakers, and Steam Users. By T. W. TRAILL, F.E.R.N., M. INST. C.E., Engineer-Surveyor-in-Chitfto the Board of Trade. " We can strongly rec9mmend Mr. Train's Book as being the most complete and eminently Pra Will pro r T k eT H VA e iu!^ tolto B&iSW*rtic.l Boilennaker.-ProcticaZ Engineer. NINTH EDITION, Revised and Enlarged. Price 18s. MARINE ENGINEERING (A MANUAL OF). The Designing, Construction, and Working of Marine Machinery. BY A. E. SEATON, M.I.N.A., &c., Lecturer on Marine Engineering, Ruyal Naval College, Greenwich. "The Student. Draughtsman, and Engineer will find this work the most valuable Handbook of Reference on the Marine Engine now in existence." Marine Engineer. In Crown 8vo, very fully Illustrated. Cloth, 3a. 6d. STEAM-BOILERS: THEIR DEFECTS, MANAGEMENT, AND CONSTRUCTION. BY R. D. MUNRO, Engineer of the Scottish Boiler Insurance and Engine Inspection Company. "A valuable companion for Workmen and Engineers engaged about Steam-Boil ers, oiiprlit, to be carefully studied, and always at hand." Colliery Guardian. " Presents a great deal of useful information in a succinct and practical manner." Ironnion/ier. BY W. J. MACQUORN RANKINE, C.E., LL.D., F.R.S., Late Regius Professor of Civil Engineering in the University of Glasgow, In Crown 8vo, Cloth, with Numerous Tables and Diagrams. 1. APPLIED MECHANICS. Twelfth Edition. 12s. 6d 2. CIVIL ENGINEERING. Seventeenth Edition. 16s. 3. THE STEAM ENGINE. Twelfth Edition. 12s. Qd. 4. MACHINERY AND MILL WORK. Sixth Edition. 12s. Qd. 5. USEFUL RULES AND TABLES. With Appendix for Electricians. Seventh Edition. 10s. Qd. 6. A MECHANICAL TEXT-BOOK. Third Edition. 9s. ** The "MECHANICAL TEXT-BOOK," by Prof. RANKINE and E. F. BAMBER, E., was designed as an INTRODUCTION to the above series of Manuals. SIXTH EDITION. Poclcet-Size, 628pp., Leather, 8s. Gd. A POCKET-BOOK OF ELECTRICAL RULES AND TABLES ffor tbe inse of Electricians an& Engineers. BY JOHN MUJSRO, C.E., ANDREW JAMIESON, F^.R.S.E., M.I.C.E., M.S.T.B. ifecw EEPDLLT PSEPECT - Worthy of the highest commendation we can give if- " The STKBUBO VALUE of Messrs. MUNBO & JAMIBSON'S ocKsa-BooK."-Electrical Review. LONDON: CHARLES GRIFFIN & CO., EXETER STREET, STRAND. JOHjV S. PRELL CM & Mechanical Engineer. SAN FBAHCISCO, OAL. MECHANICAL TEXT-BOOK; OR, INTRODUCTION TO THE STUDY OF MECHANICS, BY WILLIAM JOHN MACQUORN RANKINE, CIVIL ENGINEER; LL.D. TRIN. COLL. DUB.; F.R.SS. LOND. AND KDIN.; F.R.S.^.A.; LATE REGIUS PROFESSOR OF CIVIL BNGINEKIIING AND JIKCHANICS IN THE UNIVERSITY OF Q' ASOO\ EDWARD FISHER BAMBER, C.M With numerous Diagrams, FOURTH EDITION, REVISED. LONDON: CHARLES GRIFFIN AND COMPANY, EXETER ST11EET, STRAND. Engineering Library TA 350 ~K Ib PBEFACE. 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. I 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 JfflKgJto do hardly justified my 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 THE THIRD EDITION. THE text of this, the Third Edition, has been carefully revised, and where errata had occurred, they have been corrected. K F. B. LONDON, August, 1884. CONTENTS. MATHEMATICAL INTRODUCTION. ARITHMETICAL RULES. 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. ToExprcssthe Ratio of 'Two Num- bersin the Form of'a Continued Fraction, .... 2 5. To form a Series of Approxima- tions to a Given Ratio, . . 2 6. Logarithms Definitions,. . 4 6-1G. Logarithms, . . . 4, 5 17. Antilogaritluns, (J TRIGONOMETRICAL RULES. 18. Trigonometrical Functions De fined, 19. Relations amongst the Trigono- metrical Functions of One Angle, A, and of its Supple- ment, ..... 7 20. The Circular Measure of an Angle, ..... 8 21. Trigonometrical Functions of Two Angles, ... 8 22. Formulae for the Solution of Plane Triangles, ... 8 23. To Solve a Right-angled Triangle, 9 21. 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 2ij. Rules for finding Differential Co- efficients, . . . .11 27. Illustration of the Differential Calculus to Geometry, . . 12 23. The Integral Calculus the Inverse of the Differential, . . 13 29. Approximate Computation of Integrals, . . . .13 RULES FOR THE MENSURATION OF 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, . . . .2.5 41. Centre of'a Plane Area, . . 26 42. Centre of a Volume, . . 27 43. Centre of Magnitude of a Curved Line, 27 44. Special Figures, ... 28 ELEMENTARY MECHANICAL NOTIONS. Definition of General Terms and Division of the Subject. 45. Mechanics, . 30 46. Matter, 30 47. Bodies 30 48. A Material or Physical Volume, 30 49. A Material or Physical Surface, 30 50. Line, Point, Physical Point, Measure of Length, . . 30 51. Rest, 31 52. Motion, 31 53. Fixed Point, . . . .31 54. Cinematics, . . . .31 55. Force, . . . . .31 50. Equilibrium or Balance, . . 3L 57. Dynamics Statics and Kinetics. 32 68. Structures and Machines, . 82 CONTENTS. PART I. PRINCIPLES OF CINEMATICS, OR THE COMPARISON OF MOTIONS. Article 59. Division of the Subject, CHAPTER I. MOTIONS OF POINTS. SECTION I. 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 Parallelepiped of Motions, . 38 70. Comparative Motion, . . 38 SECTION 3 Varied Motion of Points. 71. Velocity and Direction of Varied Motion, .... 39 2. Components of Varied Motion, . 40 3. Uniformly Varied Velocity, 41 4. Graphical Representation of Mo- tions, . . . _ - .42 5. Varied Rate of Variation of Velocity, . . . .43 6. Combination of Uniform and Uni- formly Accelerated Motion, . 43 7. Uniform Deviation, . . .44 8. Varying Deviation. . . .45 79. The Resultant Rate of Variation, 45 80. The Rates of Variation of the Component Velocities, . . 45 81. The Comparison of the Varied Motions, . . . .45 CHAPTER II. MOTIONS o SECTION \.-Riijid 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 80. Plane of Rotation, ... 48 87. Angular Velocity, . . . 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 03. Comparative Motions of Two Points relatively to an Axis, . 50 94. Components of Velocity of a Point in a Rotating Body, . 50 SECTION 3. Combined Rotations and Translations. 95. Property of all Motions of Ri, o55 then we have the following series : Quotients, ................... 1 3 1 1 1 15 2 _ . .. 1 1 4 5 9 14 219 452 Fractl0ns ' ............... 1 1 3 I 7 n 172 355 O L G L G L G 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 - 98 1 7 4 28' 7 7 are alternately less and greater than the given ratio towards which they approximate, as indicated by the letters L and G- in the example; and, fourthly, the difference between any one of them and the given ratio is less than the difference between that one and the next fraction of the series. Fractions intermediate between the converging fractious may be MATHEMATICAL INTRODUCTION. found by means of the formula ri where and any two of the converging fractions, and h and Jc ai-e 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 i-aisecl to be equal to the given number. That number of which the indices of the powers are the logarithms, is called the base of the system. A suffix denotes the base of the logarithm ; if a? = 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 ! 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 figin*es; 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 Tj 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. EXAMPLES. Number. Logarithms. 377000 5-57634 37700 4-57634 3770 3-57634 377 2-57634 37-7 1-57634 3-77 0-57634 377 1-57634 0377 2-57634 00377 3-57634 and so on. 11. The logarithm 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 2, 7 instead of 3, and so on. In such cases, it is always to be understood that each such complementary index has 10 combined with it; and to prevent mistakes, it is useful to prefix 10 + to it; for example, N-nmhpr Logarithm with Logarithm with 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 100; the quotient will be a correction, to be added to the logarithm already found. 6 MATHEMATICAL INTRODUCTION. Example. Find the common logarithm of 37725. Locr. 377, 57634 Log. 378, 57749 Difference, 115 x 25-^100 Correction, 29 Add log. 377, 57634 Log. 37725, 57663 Answer. 17. To find the natural number, or 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 difference. 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 difference ; 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, ~Tl5 Given logarithm, 57663 Subtract log. 377, 57634 Divide by difference, 115)2900 Two additional figures, 25 o that the answer is 37725. Note. The last two rules refer particularly to the tables in Rankine'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. TEIGONOMETRICAL 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 a function of that number, as 2x, x 2 , 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 b sin A = - : cos A = -; c c c-b . c - a versin A = : coversm A = : c c a b tan A = f ; cotan A = - ; b' a The complement of A means the angle B, such that 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: ST-T tan A 1 sin A.= JL cos* A = j- = j- : sec A cosec A .-, r-s-j- cotan A 1 cos A = Jl - sin. 2 A = j- = T- ; cosec A sec A versin A = 1 - cos A ; coversin A = 1 - sin A; sin A 1 tan A- cotan A = cosec A = . -j- = Jl + cotan 2 A. sin A sin (180 -A) -sin A; cos (180- A) = -cos A; versin (1 80 A) = 1 + cos A = 2 versin A ; coversin (180 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 unit of circular measure is the angle whose arc is equal to the radius, that is, 360 -=- 2*- = (57 17' 45" = 206265"). To compute sines, &c., approximately by series ; reduce the angle to circular measure that is, to radius-lengths and fractions of a radius-length let it be denoted by A. Then A? 2.3.4.5.6.1 A 2 A 4 A 6 cos A = l--^- + 3-^-3 3456 + &c. 21. Trigonometrical Functions of Two Angles : sin (A B) - sin A cos B cos A sin B ; cos (A B) = cos A cos B + sin A sin B ; / A j. -r>\ tan A tan B tan (A B) --^ . ' I + tan A tan B 22. Formulas for the Solution of Plane Triangles. Let A, B, C be the angles, and a, b, c the sides respectively opposite them. I. delations amongst the Angles or if A and B are given, C = 180 - A - B. II. When the Angles and One Side are given, let a be the tten Area = 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 a function of x. A quantity, x, may be assumed to be made up of an infinite number of infinitesimal parts, dx, this expression meaning simply one of the small infinitesimal differences 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 dx ~~ * S ^ 6 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 that of the independent variable, the former being a function of the latter, is called the differential coefficient, thus -X is the differential co- dx efficient of y with respect to x, it being always borne in mind that -j- is one quantity, which cannot be divided into a numerator dy, and a denominator dx. 26. Rules for finding differential coefficients, If y = G (a constant); ^f = 0. The Differential Coefficient of the sum of functions is equal to the sum of the differential coefficients of the functions, or if v = w + y + z where all of these quantities are functions of x, then dv _ dw dy dz dx dx dx dx' In the same way to find the differential coefficient of the differ- ence, product, and quotient of functions of quantities. If v = y - z, then -^ = -j- - ~, where v, y, z, are functions of x. Al dv dw dy dz v, w, y, z, are functions of x. dy dz % J _ y _ -,- V , dv dx dx , m. -is * Ifv = , then = - - - , where v, y, z, are functions of x. If (t>x = nx, (j>'x = n; or otherwise let ^x-y-nx } then -~j- = n ) thus if $x = 7x, Vx = 7. If (j,x = x n , (f>'x = nx n ~ l , thus if 4>x = x 7 , 'x--= 7x 6 . If x = \og a x, tfx = , thus if 'x = a x log a. If * = *, 'x = * x . If x sin x, <$>'x = cos x. If Qx cos x, rx = sin x. If x = 5 . cos-x Definition. By sine ~ l x is meant the angle whose sine is x, thus if y = sine ~ l x, x = sine y. 12 MATHEMATICAL INTRODUCTION. If f x = sin ~ J x, = x, and V- v = x x, then x' x = 4' v'x, or otherwise ; If * x = log. sin x, then ? ' a; = cos x : here log. sin x is first sm a? differentiated with respect to sin x, and then sin x with respect to x. Definition. The differential coefficient of the differential coeffi- cient is called the second differential coefficient; the differential coefficient of the second is called the third; and so on : Thus let nearly; in which u' and u" denote the plain ordinates E F and G H, and u" the intermediate dotted ordinate ; and consequently, adding together the approximate areas of all the pairs of bands, we find, for the approximate value of the integral j u d x = (u a + u b + 2 2 u t (plain) + 42 >u t (dotted)) , It is obvious, that if the values of the ordinates u required in these computations can be calculated, it is unnecessary to draw the figure to a scale, although a sketch of it may be useful to assist the memory. When the symbol of integration is repeated, so as to make a double integral, such as f ju-dxdy, or a triple integral, such as j J ju-dxdydz, it is to be understood as follows : Let -/* be the value of this single integral for a given value of y. Con- 16 MATHEMATICAL INTRODUCTION. struct a curve whose abscissae are the various values of y within the prescribed limits, and its ordinates the corresponding values of v. Then the area of that curve is denoted by J V 'dy= J I u-dxdy. Next, let t=fvdy be the value of this double integral for a given value of z. Con- struct a curve whose abscissae are the various values of z within the prescribed limits, and its ordinates the corresponding values of t. Then the area of that curve is denoted by ffdz=j]vdydz=JJJu-dxdydi8; and so on for any number of successive integrations. RULES FOR THE MENSURATION OF FIGURES AND FINDING OF CENTRES OF MAGNITUDE. SECTION 1. AREAS OF PLANE SURFACES. 30. Parallelogram. Rule A. Multiply the length of one of the sides by the perpendicular distance between that side and the opposite side. Rule B. Multiply together the lengths of two adjacent sides and the sine of the angle which they make with each other. (When the parallelogram is right-angled, that sine is = 1.) 31. Trapezoid (or four-sided figure bounded by a pair of parallel straight lines, and a pair of straight lines not parallel). Multiply the half sum of the two parallel sides by the perpendicular distance between them. 32. Triangle. Rule A. Multiply the length of any one of the sides by one-half of its perpendicular distance from the opposite angle. ^ Rule B. Multiply one-half of the product of any two of the sides by the sine of the angle between them. Rule C. Multiply together the following four quantities : the half sum of the three sides, and the three remainders left after subtracting each of the three sides from that half sum ; extract the square root of the quotient ; that root will be the area required. Note. Any Polygon may be measured by dividing it into tri- angles, measuring those triangles, and adding their areas together. 33. Parabolic Figures of the Third Degree. The parabolic ANY PLANE AREA. 17 figures to which the following rules apply are of the following kind (see figs. 5 and 6.) One boundary is a straight line, A X, called the base or axis; two other boundaries are either points in that line, or straight lines at right angles to it, such as A B and X C, called ordinates; and the fourth boundary is a curve, B C, of the parabolic class, and of the third degree; that is, a curve whose ordinate Fig. 5. Fig. 6. (or perpendicular distance from the base A X) at any point is expressed by what is called an algebraical function of the third degree of the abscissa (or distance of that ordinate from a fixed point in the base). An algebraical function of the third degree of a quantity consists of terms not exceeding four in number, of which one may be constant, and the rest must be proportional to powers of that quantity not higher than the cube. Rule A. Divide the base, as in fig. 5, into two equal parts or intervals; measure the endmost ordinates, A B and X C, and the middle ordinate (which is dotted in the figure) at the point of divison ; add together the endmost ordinates and four times the middle ordinate, and divide the sum by six ; the quotient will be the mean breadth of the figure, which, being multiplied by the length of the base, A X, will give the area. Rule B. Divide the base, as in fig. 6, into three equal intervals; measure the endmost ordinates, A B and X C, and the two inter- mediate ordiuates (which are dotted) at the points of division; add together the endmost ordinates and three times each of the inter- mediate ordinates; divide the sum by eight; the quotient will be the mean breadth of the figure, which, being multiplied by the length of the base, A X, will give the area. In applying either of those rules to figures whose curved boundaries meet the base at one or both ends, the ordinate at each such point of meeting is to be made = 0. 34. Any Plane Area. Draw an axis or base-line, A X, in a convenient position. The most convenient position is usually parallel to the greatest length of the area to b measured. Divide the length of the figure into a convenient number of equal intervals, and measure breadths in a direction perpendicular to the axis at the two ends of that length, and at the points of division, which breadths will, of course, be one more in number than the intervals. (For example, in fig. 7, the length of the figure is divided into ten equal intervals, and eleven breadths are measured at 6 , 6 1 , &c.) Then the following rules are exact, if the sides of the figures are bounded by straight lines, and by 18 MATHEMATICAL INTRODUCTION. parabolic curves not exceeding the third degree, and are approxi- mate for boundaries of any other figures. Rule A. ("Simpson's First Rule" to be used when the number of intervals is even.) Add together the two endmost breadths, twice every second intermediate breadth, and four times each of the remaining intermediate breadths; multiply the sum by the common interval between the breadths, and divide by 3; the result will be the area required. For two intervals the multipliers for the breadths are 1, 4, 1 (as in Rule A of the preceding Article); for four intervals, 1, 4, 2, 4, 1 ; for six intervals, 1, 4, 2, 4, 2, 4, 1 ; and so on. These are called " Simpson's Multipliers."* Example. Length, 120 feet, divided into six intervals of 20 feet each. idths in Feet 1 Decimals. 17-28 Simpson's Multipliers. 1 Products. 17-28 16-40 4 65-60 14-08. . 2 28-16 10-80 4 ...43-20 14-08 7-04 2 3-28 4 .... 13-12 1 ... 0-00 Sum, x Common interval, 181-44 20 feet. -=- 3)3628-8 Area required, 1209-6 square feet. Rule B. ("Simpson's Second Rule" to be used when the number of intervals is a multiple of 3.) Add together the two endmost breadths, twice every third intermediate breadth, and thrice each of the remaining intermediate breadths; multiply the sum by the common interval between the breadths, and by 3; divide the product by 8; the result will be the area required. " Simpson's Multipliers " in this case are, for three intervals, 1, 3, 3, 1; for six intervals, 1, 3, 3, 2, 3, 3, 1; for nine intervals, 1, o, 6, 2, 3, 3, 2, 3, 3, 1 ; and so on. Example. Length, 120 feet, divided into six intervals of 20 feet each. * This rule has been given in symbols at page 15. TRAPEZOIDAL RULE. 19 Breadths in Feet Simpson's i>_ j and Decimals. Multipliers. Products. 17-28 1 17-28 16-40 3 49-20 14-08 3 42-24 10.80 2 21-60 7-04 3 21-12 3-28 3 9-84 .. ...1... .. 0-00 Sum, 161-28 x Common interval, 20 feet. 3225-6 x 3 -=- 8)9676-8 Area required, 1209-6 square feet.. Remarks. The preceding examples are taken from a parabolic figure of the third degree, for which, botli Simpson's Rules are exact; and the results of using them agree together precisely. For other figures, for which, the rules are approximate only, the first rule is in general somewhat more accurate than the second, and is therefore to be used unless there is some special reason for pre- ferring the second. The probable extent of error in applying Simpson's First .Rule to a given figure is, in most cases, nearly proportional to the fourth power of the length of an interval. The errors are greatest where the boundaries of the figure are most curved, and where they are nearly perpendicular to the axis. In such positions of a figure the errors may be diminished by sub- dividing the axis into smaller intervals. Rule G. (" Merrifield's Trapezoidal Rule," for calculating sepa- rately the areas of the parts into which a figure is subdivided by its equidistant ordinates or bi-eadths.) Write down the breadths in their order. Then take the differences of the successive breadths, distinguishing them into positive and negative, according as the breadths are increasing or diminishing, and write them oppo- site the intervals between the breadths. Then take the dif- ferences of those differences, or second differences, and write them opposite the intervals between the first differences, distinguishing them into positive and negative, according to the following principles : 20 MATHEMATICAL INTRODUCTION. Second Diffi First Differences. Positive increasing, or ) ...Positive, linishing, J Negative diminish! Negative increasing, or ) ^ .Negative. Positive diminishing, J In the column of second differences there will now be two blanks opposite the two endmost breadths; those blanks are to be filled up with numbers each forming an arithmetical progression with the two adjoining second differences, if these are unequal, or equal to them, if they are equal. Divide each second difference by 12; this gives a correction, which is to be subtracted from the breadth opposite it if the second difference is positive, and added to that breadth if the second difference is negative. Then to find the area of the division of the figure contained between a given pair of ordinates or breadths; multiply the half sum of the corrected breadths by the interval between them. The area of the whole figure may be formed either by adding together the areas of all its divisions, or by adding together the halves of the endmost corrected breadths, and the whole of the intermediate breadths, and multiplying the sum by the common interval. Example. Length, 120 feet, divided into six intervals of 20 feet each. Corrected Areas of Breadths. Divisions. Feet. Sq. Feet 17-44 . I 339-6 16-52 { I 306-8 Breadths in Feet and Decimals. First Differences. Second Differences. Corrections. 17-28 (-1-92) + 0-16 -0-88 16-40 -1-44 + 0-12 -2-32 14-08 -0-96 + 0-08 -3-28 10-80 -0-48 + 0-04 -3-76 7-04 -3-76 3-28 + 0-48 -0-04 -3.28 ( + 0-96) -0-08 14-16 250-0 10-84 ( I 178-8 7-04 J 3-24 -0-08 102-8 31-6 Total area, square feet, 1209-6 The second differences enclosed in parentheses at the top and bottom of the column are those filled in by making them form an arithmetical progression with the second differences adjoining them. CIRCLE. 21 The last corrected breadth in the present example is negative, and is therefore subtracted instead of added in the ensuing com- putation. Rule D. (" Common Trapezoidal Rule" to be used when a rough approximation is sufficient.) Add together the halves of the endmost breadths, and the whole of the intermediate breadths, and multiply the sum by the common interval. Example. The same as before. Feet. Half breadth at one end, 17-28 -H 2 = 8-64 f 16-40 | 14-08 Intermediate breadths, \ 10-80 7-04 [ 3-28 Half breadth at the other end, . . 60-24 x Common interval, . 20 Approximate area, .... 1204-8 square feet. True area as before computed, . . 1209-6 Error, 4-8 square feet. 35. Circle. The area of a circle is equal to its circumference multiplied by one-fourth of its diameter, and therefore to the square of the diameter multiplied by one-fourth of the ratio of the circum- ference to the diameter. The ratio of the area of a circle to the square of its diameter f which ratio is denoted by the symbol ~\ is incommensurable; that is, not expressible exactly in figures; but it can be found approximately, to any required degree of precision. Its value has been computed to 250 places of decimals; but the following approximations are close enough for mosb purposes, scientific or practical : Approxi.ateValueso^. l^taF^ o, the 7853981634 - ....... + one-300,000,000,000th. 785398+ ..................... - one-5,000,000th. 7854 - ......................... + one-400,000th. QKK .................. + one-13,000,000th. -f one-2,500th. The diameter of a circle equal in area to a given square is veiy neai'ly 1-12838 x the side of the square. The following table gives examples of this : 22 MATHEMATICAL INTRODUCTION. TABLE MULTIPLIERS FOR CONVERTING Sides of Squares into Diameters of Circles Diameters of into sides of Equal Circles. Equal Squares. 1 1-12838 0-88623 1 2 2-25676 1-77245 2 3 3-38514 2-65868 3 4 4-51352 3-54491 4 5 5-64190 4-43113 5 6 6-77028 5-31736 6 7 7-89866 6-20359 7 8 9-02704 7-08981 8 9 10-15542 7-97604 9 10 11-28380 8-86227 10 36. The area of a Circular Sector (0 A C B, fig. 8) is the same fraction of the whole circle that the angle A O B of the sector is of a whole revolution. In other words, multiply half the square of the radius, or one-eighth of the square of the diameter, by the circular measure (to radius unity) of the angle A O B; the product will be the area of the sector. SECTION 2. VOLUMES OF SOLID FIGURES. 37. To Measure the Volume of any Solid. Method I. By layers. Choose a straight axis in any convenient position. (The most convenient is usually parallel to the greatest length of the solid.) Divide the whole length of the solid, as marked on the axis, into a convenient number of equal intervals, and measure the sectional area of the solid upon a series of planes crossing the axis at right angles at the t\vo ends and at the points of division. Then treat those areas as if they were the breadths of a plane figure, applying to them Rule A, B, or C of Article 34, page 17; and the result of the calculation will be the volume required. If Rule C is used, the volume will be obtained in separate layers. Method II. By prisms or columns (" Wooley's Mule"). Assume a plane in a convenient position as a base, divide it into a network of equal rectangular divisions, and conceive the solid to be built of a set of rectangular prismatic columns, having those rectangular divisions for their sectional areas. Measure the thickness of the solid at the centre and at the middle of each of the sides of each of those rectangular columns; add together the doubles of all the thicknesses before-mentioned, which are in the interior of the solid, and the simple thicknesses which are at its boundaries ; divide the sum by six, and multiply by the area of one rectangular division of the base. TO CALCULATE THE LENGTHS OP CIRCULAR ARCS. 23 SECTION 3. LENGTHS OF CURVED LINES. 38. To Calculate the Lengths of Circular Arcs. When the proportion of the arc to an entire circumference is given, the length of the arc, in terms of the radius, is to be calculated by- multiplying that proportion by the well-known approximate value of the ratio of the circumference of a circle to its radius : viz., circumference 710 , /.nnoio^ 1^11 , r . = v^ nearly, = 6-283185 nearly: the above ratio is radius 113 commonly denoted by the symbol 2 ; the reciprocal of the above 113 ratio is very nearly ,-?- = 0-1 59 155 nearly; but it is often much more convenient in practice to proceed by drawing; and then the following rules are the most accurate yet known : * I. (Fig. 9). To draw a straight line approximately equal to a given circular arc, A B. Draw the straight chord B A; produce A to C, making A C = | BA; about C, with the radius C B = f B A, draw a circle; then draw the straight line A D, touching the given arc in A, and meeting the last-mentioned circle Fig. 9. in D ; AD will be the straight line required. The error of this rule consists in the straight line being a little shorter than the arc : in fractions of the length of the arc, it is about j-^Vo- f r an ai>c equal in length to its own radius ; and it varies as the fourth power of the angle subtended by the arc ; so that it may be diminished to any required extent by subdividing the arc to be measured by means of bisections. For example, in drawing a straight line approximately equal to an arc subtending 60, the error is about -^ of the length of the arc ; divide the arc into two arcs, each subtending 30 ; draw a straight line approxi- mately equal to one of these, and double it; the error will be reduced to one-sixteenth of its former amount ; that is, to about J^^Q of the length of the arc. The greatest angular extent of the arcs to which the rule is applied should be limited in each case according to the degree of pre- cision required in the drawing. II. (Fig. 10). To draw a straight line ap- proximately equal to a given circular arc, A B. (Another Method.) Let C be the centre of the arc. Bisect the arc A B in D, and the Fig. 10. arc A D in E; draw the straight secant * These rules are extracted from Papers read to the British Association in 1867, and published in the Philosophical Magazine for September and October of that year. 24 MATHEMATICAL INTRODUCTION. _ - C E F, and the straight tangent A F, meeting each other in F ; draw the straight line F B ; then a straight line of the length A F + F B will be approximately equal in length to the arc A B. The eiTor of this rule, in fractions of the length of the arc, is just one-fourth of the error of Rule I., but in the contrary direction; and it varies as the fourth power of the angle subtended by the arc. III. To lay off upon a given circle an arc approximately equal in length to a given straight line. In fig. 11, let A D be part of the circumference of the given circle, A one end of the required arc, and AB a straight line of the >\ gi ven length, drawn so as to touch the circle at the point A. In A B take A C = \ A B, and about C, with the radius C B = f A B draw a Fig. 11. circular arc B D, meeting the given circle in D. A D will be the arc required. The error of this rule, in fractions of the given length, is the same as that of Rule I., and follows the same law. IV. (Fig. 11.) To draw a circular arc which shall be approxi- mately equal in length to the straight line A B, shall with one of its ends touch that straight line at A, and shall subtend a given angle. In A B take A C = A B ; and about C, with the radius C B = | A B, draw a circle, B D. Draw the straight line A D, making the angle B A D = one-half of the given angle, and meeting the circle B D in D. Then D will be the other end of the required arc, which may be drawn by well-known rules. The error of this rule, in fractions of the given length, is the same with that of Rules I. and III., and follows the same law. V. To divide a circular arc, approximately, into any required number of equal parts. By Rule I. or II., draw a straight line approximately equal in length to the given arc; divide that straight line into the required number of equal parts, and then lay off upon the given arc, by Rule III., an arc approximately equal in length to one of the parts of the straight line. Rule V. becomes unnecessary when the number of parts is 2, 4, 8, or any other power of 2 ; for then the required division can be performed exactly by plane geometry. VI. To divide the whole circumference of a circle approximately into any required number of equal arcs. When the required number of equal arcs is any one of the following numbers, the division can be made exactly by plane geometry, and the present rule is not needed : any power of 2 ; 3 ; 3 x any power of 2 ; 5 ; 5 x any power of 2 ; 15 ; 15 x any power of 2.* In other cases It may be convenient here to state the methods of subdividing arcs and whole circles by plane geometry. (1.) To bisect any circular arc On the chord ot the arc as a base, construct any convenient isosceles triangle, with the summit pointing away from the centre of the arc; a straight line from. CENTRE OF MAGNITUDE. 25 proceed as follows : Divide the circumference exactly, by plane geometry, into such a number of equal arcs as may be required, in, order to give sufficient precision to the approximative part of the process. Let the number of equal arcs in that preliminary division be called n. Divide one of them, by means of Rule V., into the required number of equal parts; n times one of those parts will be one of the inquired equal arcs into which the whole circumfer- ence is to be divided. Rules L, III., and V., are applicable to arcs of other curves besides the circle, provided the changes of curvature in such arcs are small and gradual. 39. To Measure the Length of any Curve. Divide it into short arcs, and measure each of them by Rule I. of Article 38, page 23. SECTION 4. GEOMETRICAL CENTRES AND MOMENTS. 40. Centre of Magnitude General Principles. By the magni- tude of a figure is to be understood its length, area, or volume, according as it is a line, a surface, or a solid. The centre of magnitude of a figure is a point such that, if the figure be divided in any way into equal parts, the distance of the centre of magnitude of the whole figure from any given plane is the mean of the distances of the centres of magnitude of the several equal parts from that plane. The geometrical moment of any figure relatively to a given plane is the product of its magnitude into the perpendicular distance of its centre from that plane. I. Symmetrical figure. If a plane divides a figure into two symmetrical halves, the centre of magnitude of the figure is in that plane; if the figure is symmetrically divided in the like manner by two planes, the centre of magnitude is in the line where those planes cut each other; if the figure is symmetrically divided by three planes, the centre of magnitude is their point of intersection; and if a figure has a centre of figure (for example, a circle, a sphere, the centre of the arc to that summit will bisect the arc. (2.) To mark the sixth part of the circumference of a circle. Lay off a chord equal to the radius. (3.) To mark the tenth part of the circumference of a circle. In fig. 12, draw the straight line A B = the radius of the circle; and perpendicular to A B, draw B C = 4 A B. Join A C, and from it cut off C D = C B. AD will be the chord of one-tenth part of the circumference of the A. circle. (4.) For the fifteenth part, take the difference between one-sixth and one-tenth. It may be added that Gauss discovered a method of dividing the circumference of a circle by geometry exactly, when the number of equal parts is any prime number that is equal to 1 + a power of 2; such as 1 + 2*^17; 1 + 2 s 257, &c.; but the method is too laborious for use in designing mechanism. 26 MATHEMATICAL INTRODUCTION. an ellipse, an ellipsoid, a parallelogram, &c.), that point is its centre of magnitude. II. Compound figure. To find the perpendicular distance from a given plane of the centre of a compound figure made up of parts whose centres are known. Multiply the magnitude of each part by the perpendicular distance of its centre from the given plane; distinguish the products (or geometrical moments) into positive or negative, according as the centres of the parts lie to one side or to the other of the plane ; add together, separately, the positive moments and the negative moments : take the difference of the two sums, and call it positive or negative according as the positive or negative sum is the greater; this is the resultant moment of the compound figure relatively to the given plane ; and its being positive or nega- tive shews at which side of the plane the required centres lies. Divide the resultant moment by the magnitude of the compound figure; the quotient will be the distance required. The centre of a figure in three dimensions is determined by find- ing its distances from three planes that are not parallel to each other. The best position for those planes is perpendicular to each other; for example, one horizontal, and the other two cutting each other at right angles in a vertical line. To determine the centre of a plane figure, its distances from two planes perpendicular to the plane of the figure are sufficient. 41. Centre of a Plane Area. To find, approximately, the centre of any plane area. Rule A. Let the plane area be that represented in fig. 7 (of Article 34, page 17). Draw an axis, AX, in a convenient posi- tion, divide it into equal intervals, measure breadths at the ends and at the points of division, and calculate the area, as in Article 34. Then multiply each breadth by its distance from one end of the axis (as A) ; consider the products as if they were the breadths of a new figure, and proceed by the rules of Article 34 to calculate the area of that new figure. The result of the operation will be the geometrical moment of the original figure relatively to a plane perpendicular to A X at the point A. Divide the moment by the area of the original figure; the quotient will be the distance of the centre required from the plane perpendicular to A X at A. Draw a second axis intersecting A X (the most convenient posi- tion being in general perpendicular to A X), and by a similar pro- cess find the distance of the centre from a plane perpendicular to the second axis at one of its ends; the centre will then be completely determined. Rule B.If convenient, the distance of the required centre from a plane cutting an axis at one of the intermediate points of divi- CENTRE OF MAGNITUDE OF A CURVED LINE. 27 sion, instead of at one of its ends, may be computed as follows : Take separately the moments of the two pai'ts into which that plane divides the figure; the required centre will lie in the part which has the greater moment. Subtract the less moment from the greater; the remainder will be the resultant moment of the whole figure, which being divided by the whole area, the quotient will be the distance of the required centre from the plane of division. Remark. When the resultant moment is = 0, the centre is in the plane of division. Rule C. To find the perpendicular distance of the centre from the axis A X. Multiply each breadth by the distance of the middle point of that breadth from the axis, and by the proper "Simpson's Multiplier," Article 34, page 18; distinguish the pro- ducts into right-handed and left-handed, according as the middle points of the breadths lie to the right or left of the axis; take separately the sum of the right-handed products and the sum of the left-handed products; the required centre will lie to that side of the axis for which the sum is the greater; subtract the less sum from the greater, and multiply the remainder by i of the common interval if Simpson's first rule is used, or by f of the common interval if Simpson's second rule is used; the product will be the resultant moment relatively to the axis A X, which being divided by the area, the quotient will be the required distance of the centre from that axis.* 42. Centre of a Volume. To find the perpendicular distance of the centre of magnitude of any solid figure from a plane perpen- dicular to a given axis at a given point, proceed as in Rule A of the preceding Article to find the moment relatively to the plane, substituting sectional areas for breadths; then divide the moment by the volume (as found by Article 37) ; the quotient will be the required distance. To determine the centre completely, find its distances from three planes, no two of which are parallel. In general it is best that those planes should be perpendicular to each other. 43. Centre of Magnitude of a Curved Line. Rule A. To find approximately the centre of magni- tude of a very fiat curved line. In fig. 13, let A D B be the arc. Draw the straight chord A B, which bisect in 0; draw CD (the defiec- Fig. 13. tion of the arc) perpendicular to AB; from D lay off DE = ^ CD; E will be very nearly the centre required. * The rules of this Article are expressed in symbols, as follows : Let x and y be the perpendicular distances of any point in the plane area from two MATHEMATICAL INTRODUCTION. Fig. 14. This process is exact for a cycloidal arc whose chord, A B, is parallel to the base of the cycloid. For other curves it is approxi- mate. For example, in the case of a circular arc, it gives D E too small ; the error, for an arc subtending 60, being about ^^ of the deflection, and its proportion to the deflection varying nearly as the square of the angular extent of the arc. Rule B. When the curved line is not very flat, divide it into very flat arcs; find their several centres of magnitude by Rule A, and measure their lengths; then treat the whole curve as a com- pound figure, agreeably to Rule II. of Article 40, page 26. 44. Special Figures. I. Triangle (fig. 14). From any two of the angles draw straight lines to the middle points of the opposite sides; these lines will cut each other in the centre required; or otherwise, from any one of the angles draw a straight line to the middle of the opposite side, and cut off one-third part from that line commencing at the side. II. Quadrilateral (fig. 15). Draw the two diagonals A C and B D, cutting each other in E. If the quadrilateral is a parallelo- gram, E will divide each diagonal into two equal parts, and will itself be the centre. If not, one or both of the diagonals will be divided into unequal parts by the point E. Let B D be a diagonal that is unequally divided. From D lay off D F in that diagonal = B E. Then the centre of the triangle FAG, found as in the preceding rule, will be the centre required". III. Plane polygon. Divide it into tri- angles ; find their centres, and measure their areas ; then treat the polygon as a compound figure made up of the triangles, by Rule II. of Article 40, page 26. IV. Prism or cylinder with plane par- allel ends. Find the centres of the ends; a straight line joining them will be the axis of the prism or cylinder, and the middle point of that line will be the centre required. planes perpendicular to the area and to each other, and a; and ?/ the per- pendicular distances of th3 centre of magnitude of the area from the same planes; then _//xdxdy. _ffydxdy -ffdxdy ' y * ffdxdy ' See Article 29, page 16. SPECIAL FIGURES. Fig. 16. V. Tetrahedron, or triangular pyramid (fig. 16). Bisect any two opposite edges, as A D and B C, in E and F; join E F, and bisect it in G ; this point will be the centre required. VI. Any pyramid or cone with a plane base. Find the centre of the base, from which draw a straight line to the summit ; this will be the axis of the pyramid or cone. From the axis cut off one-fourth of its length, begin- ning at the base ; this will give the centre required. VII. Any polyhedron or plane-faced solid. Divide it into pyramids ; find their centres and measure their volumes ; then treat the whole solid as a compound figure by Kule II. of Article 22. VIII. Circular arc. In fig. 17, let A B be the arc, and C tho the centre of the circle of which it is part. Bisect the arc in D, and join C D and A B. Multiply the radius C D by the chord A B, and divide by the length of the arc A D B ; lay off the quotient C E upon C D ; E will be the centre of magnitude of the arc. IX. Circular sector, C A D B, fig. 17. Find C E as in the preceding rule, and make C F = f C E; F will be the centre re- quired. X. Sector of a flat riny. Let r be the external and r' the internal radius of the ring. Draw a circular arc of the same angular extent with the sector, and of the radius ^ -^ -^ and find its centre of magnitude by Rule VIII. MECHANICS. ELEMENTARY MECHANICAL NOTIONS. DEFINITION OF GENEKAL TERMS AND DIVISION OF THE SUBJECT. 45. Mechanics is the science of rest, motion, and force. The laws, or first principles of mechanics, are the same for all bodies, celestiaf and terrestrial, natural and artificial. The methods of applying the principles of mechanics to particular cases are more or less different, according to the circumstances of the case. Hence arise branches in the science of mechanics. 46. Matter (considered mechanically) is that which fills space. 47. Bodies are limited portions of matter. Bodies exist in three conditions the solid, the liquid, and the gaseous. Solid bodies tend to preserve a definite size and shape. Liquid bodies tend to preserve a definite size only. Gaseous bodies tend to expand inde- finitely. Bodies also exist in conditions intermediate between the solid and liquid, and possibly also between the liquid and the gaseous. 48. A Material or Physical Volume is the space occupied by a body or by a part of a body. 49. A Material or Physical Surface is the boundary of a body, or between two parts of a body. 50. Line, Point, Physical Point, Measure of Length. In mechanics, as in geometry, a LINE is the boundary of a surface, or between two parts of a surface ; and a POINT is the boundary of a line, or between two parts of a line; but the term "Physical Point" is sometimes used by mechanical writers to denote an immeasurably small body a sense inconsistent with the strict meaning of the word " point ;" but still not leading to error, so long as it is rightly understood. In measuring the dimensions of bodies, the standard British unit of length is the yard, being the length at the temperature of 62 Fahrenheit, and at the mean atmospheric pressure, between the two ends of a certain bar which is kept in the office of the Ex- chequer, at "Westminster. In computations respecting motion and force, and in expressing the dimensions of large structures, the unit of length commonly employed in Britain is the foot, being one-third of the yard. In expressing the dimensions of machinery, the unit of length commonly employed in Britain is the inch, being one-thirty-sixth part of the yard. Fractious of an inch are very commonly stated by mechanics and other artificers in halves, quarters, eighths, six- STRUCTURES AND MACHINES. 31 teenths, and thirty-second parts; but accoi'ding to a resolution of the Institution of Mechanical Engineers, passed at the meeting held at Manchester in June, 1857, the practice has been introduced of expressing fractions of an inch in decimals. The French unit of length is the metre, being about 4o0o 1 0()00 of the earth's circumference, measured round the poles. 51. Best is the relation between two points, when the straight line joining them does not change in length nor in direction. A body is at rest relatively to a point, when every point in the body is at rest relatively to the first mentioned point. 52. Motion is the relation between two points when the straight line joining them changes in length, or in direction, or in both. A body moves relatively to a point when any point in the body moves relatively to the first mentioned point. 53. Fixed Point. When a single point is spoken of as having motion or rest, some other point, either actual or ideal, is always either expressed or undei-stood, relatively to which the motion or rest of the first point takes place. Such a point is called a. fixed point. So far as the phenomena of motion a]one indicate, the choice of a fixed point with which to compare the positions of other points appears to be arbitrary, and a matter of convenience alone ; but when the laws of force, as affecting motion, come to be considered, it will be seen that thei'e are reasons for calling certain points fixed, in preference to others. In the mechanics of the solar system, the fixed point is what is called the common centre of mass of the bodies composing that system. In applied mechanics, the fixed point is either a point which is at rest relatively to the earth, or (if the structure or machine under consideration be movable from place to place on the earth), a point which is at i-est relatively to the structure, or to the frame of the machine, as the case may be. Points, lines, surfaces, and volumes, which ai*e at rest relatively to a fixed point, are fixed. 54. Cinematics. The comparision of motions with each other, without reference to their causes, is the subject of a branch of geometry called " Cinematics." 55. Force is an action between two bodies, either causing or tending to cause change in their relative rest or motion. The notion of force is first obtained directly by sensation; for the forces exerted by the voluntary muscles can be felt. The existence of forces other than muscular tension is inferred from their effects. 56. Equilibrium or Balance is the condition of two or more forces which ai-e so opposed that their combined action on a body produces no change in its rest or motion. 32 MECHANICS. The notion of balance is first obtained by sensation; for the forces exerted by voluntary muscles can be felt to balance some- times each other, and sometimes external pressures. 57. Dynamics Statics and Kinetics. Forces may take effect, either by balancing other forces, or by producing change of motion. The former of those effects is the subject of Statics; the latter that of Kinetics, and the Science which treats of both is by modern practice entitled Dynamics; these, together with Cinematics, already defined, form the three great divisions of pure, abstract, or general mechanics. 58. Structures and Machines. The works of human art to which the science of applied mechanics relates, are divided into two classes, according as the parts of which they consist are intended to rest or to move relatively to each other. In the former case they are called Structures; in the latter, Machines. Structures are subjects of Statics alone; Machines, when the motions of their parts are considered alone, are subjects of Cine- matics; when the forces acting on and between their parts are also considered, machines are subjects of Dynamics. PART I. PRINCIPLES OF CINEMATICS, OR THE COMPARISON OF MOTIONS. 59. Division of the Subject. The Science of Cinematics, and the fundamental notions of rest and motion to which it relates, having already been defined among the ELEMENTARY MECHANICAL NOTIONS, Articles 51, 52, 53, 54, it remains to be stated, that the principles of Cinematics, or the comparison of motions, will be divided and arranged in the present part of this treatise in the following manner : I. Motions of Points. II. Rigid Bodies or Systems. III. Pliable Bodies and Fluids. CHAPTER I. MOTIONS OF POINTS. SECTION 1. MOTIONS OF A PAIR OF POINTS. 60. Fixed and Nearly Fixed Directions. From the definition of motion given in Article 52, it follows, that in order to deter- mine the relative motion of a pair of points, which consists in the change of length and direction of the straight line joining them, that line must be compared, at the beginning and end of the motion considered, with some fixed or standard length, and with at least two fixed directions. Standard lengths have already been considered in Article 50. An absolutely fixed direction may be ascertained by means whose principles cannot be demonstrated until the subject of kinetics is considered. For the present it is sufficient to state, that when a solid body rotates free from the influence of any external force tending to change its rotation, there is an absolutely fixed direction called that of the axis of angular momentum, which bears certain relations to the successive positions of the body. 34 PRINCIPLES OF CINEMATICS. A nearly fixed direction is that of a straight line joining a pair of points 'in two bodies whose distance from each other is very great, such as the earth and a fixed star. A line fixed relatively to the earth changes its absolute direction (unless parallel to the earth's axis) in a manner depending on the earth's rotation, and returns periodically to its original absolute direction at the end of each sidereal day of 86,164 seconds. This rate of change of direction is so slow compared with that which takes place in almost all pieces of mechanism to which cinematical and kinetic principles are applied, that in almost all questions of applied mechanics, directions fixed relatively to the earth may be treated as sufficiently nearly fixed for practical purposes. When the motions of pieces of mechanism relatively to each other, or to the frame by which they are carried, are under con- sideration, directions fixed relatively to the frame, or to one of the pieces of the machine, may be considered provisionally as fixed for the purposes of the particular question. POSTULATE. Let it be granted that a line may represent a motion, where the term motion is employed to represent the path of motion, the direction and the velocity or length of motion in a unit of time. This is a self-evidently possible problem, for a line may be drawn to represent any path, in any direction to represent any direction of motion, and of any length to represent any length of motion, or velocity, limited always by the space within which motions can take place or lines be drawn. 61. Motion of a Pair of Points. In fig. 18, let A x B x repre- sent the relative situation of a pair of points at one instant, and A 2 B 2 the relative situation of the same pair of points at a later instant. Then the change of the straight line A B between those points, from the length and direc- Fig. 18. Fig. 19. Fig. 20. tion represented by Aj B x to the length and direction represented by A 2 B 2 , constitutes the relative motion of the pair of points A B, during the interval between the two instants of time considered. To represent that relative motion by one line, let there be drawn, from one point A, fig. 19, a pair of lines, A B x , A B 2 , equal and parallel to Aj B p A^B^ of fig. 18 ; then A represents one of the rof points whose relative motion is under consideration, and u , B 2 , represent the two successive positions of the other point B THE MEASUREMENT OF TIME. 35 relatively to A ; and the line B x B 2 represents the motion of B relatively to A, which, for the purposes of the representation, is assumed to be fixed. Or otherwise, as in fig. 20, from a single point B let there bo drawn a pair of lines, BA^ B A 2 , equal and parallel to Aj ~B V A 2 B 2 , of fig. 18 ; then AJ A 2 , represent the two successive positions of A relatively to B; and the line Aj^ A g , equal and parallel to Bj B 2 of fig. 19, but pointing in the contrary direction, represents th& motion of A relatively to B. 62. Fixed Point and Moving Point. In fig. 19, A is treated as the fixed point, and B as the moving point ; and in fig. 20, B is treated as the fixed point, and A as the moving point; and these are simply two different methods of representing to the mind the same relation between the points A and B (see Article 53). 63. Component and Resultant Motions. Let O be a point, assumed as fixed, and A and B two successive positions of a second point relatively to 0. In order to express mathematically the amount and direction of A B, the motion of the second point relatively to O, that line may be com- pai'ed with three axes, or lines in fixed directions, traversing the fixed point O, such as O X, O Y, O Z. Through A and B draw straight lines A C, B D, parallel to the plane of O Y and Z, and cutting the axis O X in C \ and D. Then CD is said to be the com- Flg> 2L ponent of the motion of the second point relatively to 0, along, or in the direction of the axis O X ; and by a similar process are found the components of the motion AB along Y and O Z. The entire motion A B is said to be the resultant of these components, and is- evidently the diagonal of a parallelepiped of which the components are the sides. The three axis are usually taken at right angles to each other ; in which case A C and B D are perpendiculars let fall from A and B upon O X ; and if be the angle made by the direction of the motion A B" with O X, C~I) = A~B cos *. 64. The Measurement of Time is effected by comparing the events, and especially the motions, which take place in intervals of time. Equal times are tho times occupied by the same body, or by equal and similar bodies, under precisely similar circumstances, in 36 PRINCIPLES OP CINEMATICS. performing equal and similar motions. The standard unit of time is the period of the earth's rotation, or sidereal day, which has been proved by Laplace, from the records of celestial phenomena, not to have changed by so much as one eight-millionth part of its length in the course of the last two thousand years. A subordinate unit is the second, being the time of one swing of a pendulum, so adjusted as to make 86,400 oscillations in 1-00273791 of a sidereal day; so that a sidereal day is 86164-09 seconds. The length of a solar day is variable; but the mean solar day, being the exact mean of all its different lengths, is the period already mentioned of 1-00273791 of a sidereal day, or 86,400 seconds. The divisions of the mean solar day into 24 hours, of each hour into 60 minutes, and of each minute into 60 seconds, are familiar to all. Fractions of a second are measured by the oscillations of small pendulums, or of springs, or by the rotations of bodies so contrived as to rotate through equal angles in equal times. 65. Velocity is the ratio of the number of units of length described by a point in its motion relatively to another point, to the number of units of time in the interval occupied in describing the length in question; and if that ratio is the same, whether it be computed for a longer or a shorter, an earlier or a later, part of the motion, the velocity is said to be UNIFORM. Velocity is expressed in units of distance per unit of time. For different purposes, there are employed various units of velocity, some of which, together with their proportions to each other, are given in the following table : Comparison of Different Measures of Velocity. Miles Feet Feet Feet per hour. per second. per minute, per hour. 1 = 1-46 =88 = 5280 0-68i8 =1 =60 = 3600 0-01136 = 0-016 = 1 =60 0-0001893 = 0-00027 = 0-016 = 1 1 nautical mile \ per hour, or } = 1-1507 = 1-6877 = 101-262 = 6075-74 "knot," J In treating of the general principles of mechanics, the foot per second is the unit of velocity commonly employed in Britain. The units of time being the same in all civilized countries, the propor- tions amongst their units of velocity are the same with those amongst their linear measui-es. 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. 37 way with those motions, which have already been treated of in Article 63. 66. Uniform Motion consists in the combination of uniform velocity with uniform direction; that is, with motion along a straight line whose direction is fixed. SECTION 2. UNIFORM MOTION OF SEVERAL POINTS. 67. Motion of Three Points. THEOREM. The relative motions of three points in a given interval of time are represented in direction and magni- tude by 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 A x and B x be the positions of A and B relatively to at the beginning of the given interval of time, and A 2 and B 2 their positions at the end of that interval. Then AJ A 2 and B x B 2 are the respective motions of A and B relatively to 0. Complete the parallelogram Aj B x b A 2 ; then because A 2 6 is parallel and equal to A l B : , b is the position which B would have at the end of the interval, if it had no motion relatively to A; but B 2 is the actual position of B at the end of the interval ; therefore, b B 2 is the motion of B relatively to A. Then in the triangle B x 6 B 2 , B x b = A! A 2 is the motion of A relatively to 0, b B 2 is the motion of B relatively to A, B x B 2 is the motion of B relatively to 0; so that those three motions are represented by the three sides of a triangle. Q. E. D. This Theorem might be otherwise expressed by saying, that if three moving points be considered in any order, the motion of the third relatively to the first is the resultant of the motion of the third relatively to the second, and of the motion of the second relatively to the first; the word "resultant" being understood as already ex- plained in Article 63. 68. Motions of a Series of Points. COROLLARY. If a series of points be considered in any order, and the motion of each point determined relatively to that which precedes it in the series, and if the relative motion of the last point and the first point be also deter- mined, then will those motions be represented by the sides of a closed gg PRINCIPLES OF CINEMATICS. polygon. Let be the first point, A, B, 0, &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 0, 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 (0, C), are the three sides of a triangle ; therefore (O, A), (A, B), (B, C), and (0, 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, (0, N), is the closing side of a polygon, of which (0, A), (A, B), (B, C), (C, D), &c., (M, N) are the other sides. Q. E. D. In other words, the motion 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 Parallelepiped of Motions. In iig. 23, let there be four points, 0, A, B, C, of which one, O, is assumed as fixed, and is traversed by three axes in fixed directions, X, Y, O 7i. In a given interval of time, let A have the motion A 1 A 2 along or parallel to O X; let B have, in the same interval, the motion b B^ parallel to Y, and rela- tively to A; then B x B 2 , the diagonal of the parallelogram whose sides are B x b = " 23 ' ATA^ and UB^ is the motion_of B rela- tively to O. Let C have, relatively to B, the motion c C 2 parallel to O Z ; then O x C 2 , the diagonal of the parallelepiped whose edges are A : A 2 , b B 2 , and c C 2 , is the motion of C relatively to O, being the resultant of the motions represented by those three edges. This is a mechanical explanation of the composition of motions, leading to results corresponding with the geometrical explanation of Article 63. 70. Comparative Motion is the relation which exists between the simultaneous motions of two points relatively to a third, which is assumed as fixed. The comparative motion of two points is expressed, in the most general case, by means of four quantities, viz. : (1.) The velocity ratio* or the proportion which their velocities bear to each other, that is, the proportion borne to each other by the distances moved through by the two points in the same interval of time. (2.) (3.) (4.) The directional relation* which is the relation be- tween the directions in which the two points are moving at the same instant, and which requires, for its complete expression, three * These terms are adopted from Prof. WILLIS'S wurii ua Al VELOCITY AND DIRECTION OF VARIED MOTION. 39 angles. Those three angles may be measured in different ways, and one of those ways is the following : (2.) The angle made by the directions of the compared motions with each other. (3.) The angle made by a plane parallel to those two directions with a fixed 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. TARIED 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 genei-al. In fig. 24, let O represent a point assumed as fixed, O X, O Y, Z, fixed directions, and A B part of the path or orbit traced by a second point in its ' -p io . ^ varied motion relatively to O. At the instant when the second point reaches a given position, such 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 As the dis- tance traced in that interval. Then As A is an approximation to the velocity at the instant in question, which will approach continually nearer and nearer to the exacfc velocity as the interval A t and the distance A s are made shorter 40 PRINCIPLES OF CINEMATICS. and shorter; and the limit towards which ^converges, as A and A t are indefinitely diminished, and which is denoted by ds 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 to the time, thus s = Q t and = is indefinitely diminished. In the fraction d t ~di~' ds is the limit of the difference of either of the spaces As in equa- tion (5), Article 73, and d d s, is the limit of the difference of that difference, viz., A# 2 As x ; that is, d in this fraction is represented by the minus sign ( ) in the other, and ds by the limit of either of the quantities AS I} A^- Here in the language of the differential calculus, the velocity (v) is a function of the time (), and the acceleration (a) is the differential coefficient of the velocity with respect to the time, thus v = t and a = 't, or = . 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 $'t, a = Vt. 76. Combination of Uniform and Uniformly Accelerated Motion. Assume a pair of rectangular axes of co-ordinates. Let the uniform motion be represented by abscissae along X, and the uniformly accelerated motion by ordinates parallel to O Y; let OB ( = x) = vt, represent the space described in the time t with a t z the velocity v, and let 00 (=y)= ~-, represent the space de- 44 PRINCIPLES OP CINEMATICS. scribed with a uniform rate of acceleration, a, in the same time t, see Article 73, then x* = v 2 t 2 and at* , ., = II - , where the y=-r' ' =y-a~> square of any abscissa bears a con- stant ratio to the corresponding ordi- nate, and the path of the point is Y known by Conic Sections to be a Fig- 23. Parabola. The same follows for any axes of co-ordinates; but if the direction of the uniformly accelerated motion be that of the uniform motion or directly opposed to it, the resultant direction will be the same as that of either motion, or will be that of the greater component. 77. Uniform Deviation is the change of motion of a point which moves with uniform velocity in a circular path. The rate at which uniform deviation takes place is determined in the following manner : Let C, fig. 29, be the centre of the cir- cular path described by a point A with an uniform velocity v, and let the radius C A be denoted by r. At the beginning and end of an interval of time A^, let A l and A 2 be the positions of the moving point. Then the arc A x A 2 = vAt; Fig. 29. ., , , . chord the chord A, A 9 = v At arc The velocities and directions at Aj and A 2 are represented by the equal lines A x V x = A 2 V 2 = y, touching the circle at A 1 and A 2 respec- tively. From A 2 draw A. 2 v equal and parallel to A^V^ and join V 2 v. Then the velocity A 2 V 2 may be considered as compounded of A z v and v V 2 ; so that v V 2 is the deviation of the motion dur- ing the interval A; and because the isosceles triangles A v ~V n , C A x Ag, are similar : A 2 V 2 A x A 2 v 2 - A t chord arc deduced by substituting the value of A l A 2 already found; and the approximate rate of that deviation being the deviation divided by the interval of time in which it occurs, is 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 remains 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 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: V/dv\ z v* On) + ;- the first term of the quantity under the first radical is the square of -j- in Article 73, and the second the square of - , Equation (1), Article 77. 80. The Rates of Variation of the Component Velocities of a point parallel to three rectangular axes, are represented as follows: and if a rectangular parallelepiped be constructed, of which the edges represent these quantities, its diagonal, whose length is V 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 PRINCIPLES OF 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 II. MOTIONS OF RIGID 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 figure undergoes no alteration during the motion which is under con- sideration. 83. Translation or Shifting is the motion of a rigid body rela- tively to a fixed point, when the points of the rigid body have no motion relatively to each other; that is to say, when they all move with the same velocity and in the same direction at the same instant, so that no line in the rigid body changes its direction. It is obvious that if three points in the rigid body, not in the same straight line, move in parallel directions with equal velocities at each instant, the body must have a motion of translation. The paths of the different points of the body, provided they are all equal and similar, and at each instant parallel, may have any figure whatsoever. SECTION 2. SIMPLE ROTATION. 84. Rotation or Turning is the motion of a rigid body when lines in it change their direction. Any point in or rigidly attached to the body may be assumed as a fixed point to which to refer the motions of the other points. Such a point is called a centre of rotation. 85. Axis of Rotation. THEOREM. In every possible change of position of a rigid 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, A x , B!, and after the turning, A 2 , B 2 . Join A x A 2 , Fio~30 ^ B! B 2 , forming the isosceles tri- angles O A! A 2 , B! B 2 . Bisect the bases of those triangles in C 48 PRINCIPLES OF CINEMATICS. and D respectively, and through the points of bisection draw two planes perpendicular to the respective bases, intersecting each other in the straight line O E, which must traverse O. Let E be any point in the line O E; then E Aj A 2 , and E Bj B 2 , 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 OC intersect, and the axis will be represented by a straight line perpendicular to the plane of the paper through O and E. COROLLARY. It is evident that every Fig. 31. line in the body, parallel to the axis, has its direction unchanged. 86. The Plane of 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 l O A 2 , or B x O B 2 , 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 j then a = 2 * T ; '2s- = 6-2831852 = 710 \* ii yj 88. Uniform Rotation consists in uniformity of the angular * The value of * may be easily remembered by taking the first three odd numbers twice each, and placing the six in a row, using; the first three as the denominator, and the last three as the numerator of a fraction: we thus 355 obtain 113 | 355 = ^; this is a nearer approximation than S'14159, and is generally much more easily employed in calculation. ROTATING BODY. 49 velocity of the turning body, and constancy of the direction of its axis of rotation. 89. Rotation common to all Parts of Body. Since the angu- lar motion of rotation consists in the change of direction of a line in a plane of rotation, and since that change of direction is the same how short soever the line may be, it is evident that the condition of 1'otation. 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 bee. 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 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 sui-face 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 lt r 2 , be the per- pendicular distances of A and B from the axis traversing O, and ?! and v z their velocities; then 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 .p '" point being v = a r (a representing the velocity of a point situated at the distance unity from the axis of rotation), let that velocity be represented by the line A V perpendicular to 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 ^/ Y A U = e be the angle made by that line with A V. From V let fall V U perpendicular HELICAL MOTION. 51 to B A; then A U represents the component in question; and de- noting it by u, u = v cos 6 ar' cos 6 (1.) From O let fall O B perpendicular to B A. Then Z A O B = Z V A U = 6 ; and the right-angled triangles O B A and A U V are similar; so that A~V : All : :OA~:OBW cos 6 (2.) Now the entire velocity of B relatively to the axis O is ar cos = M, (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 EOTATIONS AND TRANSLATIONS. 95. Property of all Motions of Rigid Bodies. The foregoing proposition may be regarded as a particular case of the following, which is true of all motions of a rigid body. The components, along a given straight line in a rigid body, of the velocities of the points in that line relatively to any point, whether in or attached to the body or otherwise, are all equal to each other; for otherwise, the distances between points in the given straight line must alter, which is inconsistent with the idea of rigidity. 96. Helical Motion. Rotation is the only movement which a rigid body as a whole can have relatively to a point belonging to it or attached to it. But if the motion of the body be determined relatively to a point not attached to it, a translation may be com- bined with the rotation. When that translation takes place in the direction of the axis of rotation, the motion of the rigid body is said to be helical, or screw-like, because each point in the rigid body describes a helix or screw, or a part of a helix or screw. Let v 1 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 Cb of one turn (2 if 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 PRINCIPLES OF CINEMATICS. a p vv , P p = J ; whence v : = ^ Let r as before, be the radius-vector of any point in the body, and kt ' % _., (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= *]v\ + v\ a ' A/ i -i 2 + *" 2 r (3-) The inclination of the helix described by that point to the plane of rotation is given by the equation i = arc tan = arc tan ~ ..(4.) that is, an angle whose tangent is equal to ^ divided by v 2 , or to p divided by 2 v r, the tangent of that angle beig the ratio of the pitch to the circumference of the circle described by the point rela- tively to the axis of rotation. 97. PROBLEM. To find the Motion of a Rigid Body from the Motions of Three of its Points. Let A, B, C, fig. 33, be three points in a rigid body, and at a given instant let them have mo- tions relatively to a point indepen- dent of the body, which motions are represented in velocity and direction by the three lines A V , B Y 6 , C V, It is required to find the motion of the entire rigid body relatively to the same fixed point. Through any point o, fig. 34, draw three lines oa, ob, oc, equal and parallel to the three lines C V c . Thi-ough a, b, 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 Y , Y 6 , Y c , draw lines Y^ TJ^ VTUl, Y7TJ C , equal and parallel to o n, but opposite in direction to it; and join SPECIAL CASES. 53 A U a , B U 6 , C U c , which will all be parallel to the same plane; that is, to the plane a b c. The last three lines will represent the component velocities which, along with the common velocity of translation parallel to o n, make up the resultant velocities of the three points. Through the point A draw a plane perpendicular to the component of its motion, which is parallel to a b c; that is, to A U a , and through B draw a plane perpendicular to B TJ 6 . 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 U c , will cut the other two planes in the same axis : the three revolving component velocities ATI;, BU;, cu; will be respectively proportional to the perpendicular distances, or radii-vectores, AD, B~E, 01", of the three points from that axis ; and the angular velocity will be equal to each of the three quotients made by dividing the revolving component velocities of the points by their respective radii-vectores. This rotation, combined with a translation parallel to the axis, with a velocity represented by o n, constitutes a helical or screw-like motion, being the required motion of the rigid body. Q. E. I. 98. Special Cases of the preceding problem occur, in which either a more simple method of solution is sufficient, or the general method fails, and a special method has to be employed. I. When the motions of the points of the body are knoion to be all parallel to one plane, it is sufficient to know the motions of two points, such as A, B, fig. 35. Let A O, BO, be two planes tra- versing A and B, and perpendicular to the respective directions of the simul- taneous velocities of those points ; if those planes cut each other, the entire motion is a rotation ; the line of intersection of the planes O, being the axis of rotation, and the angular velocity, are found as in the last Article. If tho 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. //' three points in the same plane move perpendicularly 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. 35 a. -V* Kg. 35c. mities of the three lines which represent their velocities viz., through the points, V a , V 6 , V c ; 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. 99 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 TL 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- Fig. 36. 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 _ Then the point T has, in virtue of translation along with the axis C, a fonvard motion with the velocity u; and in virtue of rotation about that axis, it has a backward motion with the velocity a C"T~= 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- tion above described, is at rest at the instant of its arriving at that position; that is, it has just ceased to move in one direction, and is about to move in another direction; and this is true of every SPIRAL PATHS. 55 point which arrives at a line traversing T parallel to 0. Conse- quently the resultant motion of the body, at any given instant, is the same as if it were rotating about the line which at the instant in question occupies the position T, parallel to 0, at the distance It - ; and that line is called THE INSTANTANEOUS AXIS. To find the motion of any point A in the body at a given instant, let fall the perpendicular A T from that point on the instantaneous axis; then the motion of A is in the direction 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-TT (2.) 100. Rolling Cylinder; Trochoid. Every straight line parallel to the moving axis 0, in a cylindrical surface described about G 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 FTP parallel to C and to C U, at the distance -. The path described by any point in the body, such as A, which is not in the moving axis C, is a curve well known by the name of trochoid. The particular form of trochoid called the cycloid, is described by each of the points in the rolling cylindrical surface ; 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 the 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 which the point T in the rotating plane has at a given instant, shall be in the contrary direction to the equal velocity of translation u, which the rigid body has relatively to the rotating 56 PRINCIPLES OP CINEMATICS. plane. Then each point in tJie rigid body which arrives at the position T, or at any position in a line traversing T parallel to the fixed axis O, is at rest at the instant of its occupying that position ; therefore the line traversing T parallel to the fixed axis O is the instantaneous axis; the motion at a given instant of any point in the 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- AT. All the lines in the rigid body which successively occupy the position of instantaneous axis are situated in a plane of that body, P T P, perpendicular to O C; and all the positions of the instan- taneous axis are situated in a cylinder described about O with the radius O T ; so that the motion of the rigid body is such as is pro- duced by the rolling of the plane P P on the cylinder whose radius is O T = . Each point in the rigid body, such as A, describes a plane spiral about the fixed axis O. For each point in the rolling plane, P P, that spiral is the involute of the circle whose radius is OT. 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 relatively to the plane OC; 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 b are similar in direction; fig. 39, that in which a and b are in opposite directions, COMBINED PARALLEL ROTATIONS. 57 and 6 is the greater; and fig. 40, that in, which a and b are in opposite directions, and a is the greater. Fig. 39. 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 : 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 a OT; and the velocity of the line T of the rigid body, relatively to the plane C, is b 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 + l; ........................... (2.) regard being had to the directions or signs of a and b ; that is to say, if we now take a and b to represent arithmetical magnitudes, and affix explicit signs to denote their directions, the dii'ection 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 = b-a-, c = a-b .................. (2 A.) The relative proportions of a, b, and c, and of the distances between the fixed, moving, and instantaneous axes, are given by the equation a:b:c: :C~T:OT:OT5 .................. (3.) The motion of any point, such as A, in the rigid body, is at each 58 PRINCIPLES OF CINEMATICS. instant at right angles to the radius-vector A T 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 with the radius CT; and all the positions of the instantaneous axis are contained in a cylindrical surface described about O with the radius O T; therefore the resultant motion of the rigid body is that which is produced by rolling the former cylindei-, 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 ti*aced by a point in the surface of the rolling cylinder is an epicycloid. In certain cases, the epitrochoids become curves of a more simple class. For example, each point in the moving axis G traces a circle. When a cylinder, as in fig. 39, rolls within a concave cylinder of double its radius, each point in the surface of the rolling cylinder moves backwards and forwards in a straight line, being a diameter of the fixed cylinder; each point in the axis of the rolling cylinder traces a circle of the same radius with that cylinder, and each other point in or attached to the rolling cylinder traces an ellipse of greater or less eccentricity, having its centre in the fixed axis 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 ihree-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 cusjis 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. EPITROCHOIDS. 59 Fig. 42. CO PRINCIPLES OP CINEMATICS. are outside the rolling-circles; and the curves traced are curtate epitrochoids, distinguished by their looped form. An important property of curves traced by rolling 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 = CD = AB-BC; also a point C' outside the rolling- EPITROCHOIDS. Gl circle, but rigidly attached to it, traces a quadrant, C' c', of an ellipse whose semiaxes are A C' = B C' + A B, and A c = C' D = B C' - A B. The former may be called an internal, and the latter an external, ellipse. The proportions of the axes of either of them 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. 6$ 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 in that plane parallel to 0, with an angular velocity -a, equal and opposite to that of the plane. Then the angular velocity of the rigid body is nothing; that is, its motion is one of translation only, all its points moving in equal circles of the radius O C, with the velocity a O 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 b re- latively to the rotating plane. Fi g _ 45. Because the point O in the rigid body is fixed, the instantaneous axis must traverse that point. The direction of that axis is determined, as before, by considering that each point which arrives at that line must have, in virtue of the rotation about C, a velocity relatively to the rotating plane, equal and directly opposed to that which the coincident point of the rotating plane has. Hence it follows, that the ratio of the perpendicular distances of each point in the instantaneous axis from the fixed and moving axes respectively that is, the ratio of the sines of the angles which the instantaneous axis makes with the fixed and moving axes must be the reciprocal of the ratio of the component angular velocities about those axes; or symbolically, if T be the instantaneous axis, sin AOT:sin COT : :b:a (1.) This determines the direction of the instantaneous axis, which may also be found by graphic construction as follows : On A take O a proportional to a ; and on O C take Ol> proportional to b. Let those lines be taken in such directions, that to an observer looking from their extremities towards 0, the component rotations seem both right-handed. Complete the parallelogram Obca; 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 O 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 A, O T, respectively; then a-~CD = c-CE; but CT3 : "CE : : sin Z A C : sin Z O T; and therefore sinZCOT :sinZAOC: :a:c; and, combining this proportion with that given in Equation 1, we obtain the following proportional equation: sinZCOT :sinZAOT:sinZAOC } : : _o_ : : _e_ > (2.) : : Oa : Ob : Oc ) 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 loth 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 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 lattsr. 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 O T become obtuse, that cone would be hollow, and would be touched internally by the rolling cone ; and similar changes may be made in the rolling cone. The path described by a point in or attached to the rolling cone is a spherical epitrochoid; but for the purposes of the present trea- tise, wb 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 in 64 PRINCIPLES OF CINEMATICS. the angular velocity of a rotating body, A a, during a given interval of time, with the length of that interval, A t, and the rate of variation is the value towards which the ratio of the change of angular velocity to the interval of time, - , converges, as the length of the interval is indefinitely diminished; being represented by -^, and found by the operation of differentiation. 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 x , a v , a 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 is the resultant angular velocity, and are the cosines of th.e angles which the instantaneous axis makes with the axis of x, y, and z respectively. CHAPTER III. MOTIONS OP PLIABLE BODIES, AND OP 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. II. The Motions of Fluids not altering in Volume. 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. 66 PRINCIPLES OP 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 coi'ds are those in which cords are the means of transmitting motion between two pieces in a train of mechanism. Such pro- blems will be considered in Part II. of this treatise. Next in point of frequency in practice 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 s be the length of any straight portion of the cord, the total length between two given. points fixed in the cord may be expressed thus : L = 2-s + 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 the -I difference r f ^e rac ^ being employed, according as the cord 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 1=180 in each case, while a certain number, n, of the guiding bodies or pulleys all move simultaneously in a direction parallel to the plies of the cord with the same velocity, u; where u represents the velocity of translation of the guiding surfaces, and v the longitudinal velocity of any point in the cord v = 2nu ................................. (3.) SECTION 2. MOTIONS 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 tempei'ature and pressure, or sensibly invariable, from the smallness of the alterations of volume which the actual altera- PIPES, CHANNELS, CURRENTS, JETS. 6T tions of pressure and temperature are capable of producing. The latter is the case in most problems of practical mechanics affecting liquids. Conceive an ideal surface of any figure, and of the area A, to be situated within a fluid mass, the parts of which have motion rela- tively to that surface; and let u denote, as the case may be, the? uniform velocity, or the mean value of the varying velocity,. resolved in a direction perpendicular to A, with which the particles of the fluid pass A. Then Q = "A ................................. (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 flow, or rate of flow, through A. When the particles 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 .............................. (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 fllled with the fluid, the flow into the space and the flow out of it, in any one given interval of time y must be equal a principle expressed symbolically by 2-Q = ............................. (3.) 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; u, id, 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, u A = u' A.' = Q = Q' = constant for all cross sections of the channel at the given instant; ..................... (1.) consequently, (2.) v ' 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 pipe or tube, the area of each cross section is given by the figure and dimensions of the pipe, and for similar forms of section varies as the square of the diameter. Hence the mean normal velocities of a stream flowing in a full pipe, at different cross sections of the pipe, are inversely as the squares of tho diameters of those sections. A channel partially encloses the stream flowing in it, leaving the upper surface free; and this description applies not only to channels commonly so called, but to pipes partially filled. In this case the area of a cross section of the stream depends not only on the figure and dimensions of the channel, but on the figure and elevation of the free upper surface of the stream. A current is a stream bounded by other portions of fluid whose motions are different. A jet is a stream whose surface is either free all round, or is touched by a solid body in a small portion of its extent only. 118. Steady Motion of a fluid relatively to a given space con- sidered as fixed is that in which the velocity and direction of the motion of the fluid at each fixed point is uniform at every instant of the time under consideration; so that although the velocity and direction of the motion of a given particle of the fluid may vary while it is transferred from one point to another, that particle assumes, at each fixed point at which it arrives, a certain definite velocity and direction depending on the position of that point alone; which velocity and direction ai'e 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), If u represents the normal velocity of a fluid moving steadily, at a given fixed point, expresses the condition of steady motion. 119. Motion of Pistons. 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 hydraulic press and hydraulic crane. The flow produced by a piston whose velocity is , 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 SECTION 3. MOTIONS OF FLUIDS OF VARYING DENSITY. 120. Flow of Volume and Flow of Mass. In the case of a fluid of varying density, the volume, which in an unit of time flows through a given area A, with a normal velocity u, is still repre- sented, as for a fluid of constant density, by (1.) but the absolute quantity, or mass of fluid which so flows, bears no longer a constant proportion to that volume, but is proportional to the volume multiplied by the density. The density may be expressed, either in units of weight per unit of volume, or in arbitrary units suited to the particular case. Let e be the density; then the flow of mass may be thus expressed : (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 ei to &. Then the mass of fluid which at the beginning of the interval occupied the volume V, occupies at the end of the interval the volume : and the difference of Pa those volumes is the volume which flows through the surface bounding the space, outward if j 2 is less than &, inward if ^ is greater than ft. Let t 2 - ^ be the length of the interval of time ; then the rate of flow of volume is expressed as follows : 70 PAKT II, THEORY OF MECHANISM. CHAPTER I. DEFINITIONS AND GENERAL PRINCIPLES. 122. Theory of Pure Mechanism Defined. 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 ^)he hands. A machine transmits and modifies force when it is the means of making a given kind of physical energy perform a given kind of work; as when the furnace, boiler, water, and mechanism of a marine steam engine are the means of making the energy of the chemical combination of fuel with oxygen perform the work of overcoming the resistance of water to the motion of a ship. The acts of transmitting and modifying motion, and of transmitting and modifying force, take place together, and are connected by a cer- tain law ; and until lately, they were always considered together in treatises on mechanics ; but recently great advantage in point of clearness has been gained by first considering separately the act of transmitting and modifying motion. The principles which re- gulate this function of machines constitute a branch of Cinematics, called the theory of pure mechanism. The principles of the theory of pure mechanism having been first established arid understood, ihose of the theory of the work of machines, which will form the subject of Part VI. of this work, which regulate the act of trans- mitting and modifying force, are much more readily demonstrated iind apprehended than when the two departments of the theory of machines are mingled. The establishment of the theory of pure mechanism as an independent subject has been mainly ac- MOVING PIECES; CONNECTORS; BEARINGS. 71 com pli shed by the labours of Professor Willis, whose nomenclature and methods are, to a great extent, followed in this treatise. 123. The General Problem of the theory of pure mechanism may be stated as follows : Given the mode of connection of two or more movable points or bodies with each other, and with certain fixed bodies; required the comparative motions of the movable points or bodies : and conversely, when 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 frame of a machine is a structure which supports the moving pieces, and regulates the path or kind of motion of most of them directly. In considering the movements of machines mathematically, the frame is considered as fixed, and the motions of the moving pieces are referred to it. The frame itself may have (as in the case of a ship or of a locomotive engine) a motion relatively to the earth, and in that case the motions of the moving pieces relatively to the earth are the resultants of their motions relatively to the frame, and of the motion of the frame relatively to the earth; but in all problems of pure mechanism, and in many problems of the work of machines, the motion of the frame relatively to the earth does not require to be considered. The moving pieces maybe distinguished 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 pieces. 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. Searings 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 turned to figures of revolu- tion, such as cylinders, spheres, conoids, and flat discs. The beariug of a piece whose motion is helical, must be an exact screw, of a pitch equal to that of the helical motion (Article 96). Those parts of moving pieces which touch the bearings, should have surfaces accurately fitting those of the bearings. They may be distinguished into slides, for pieces which move in straight lines, gudgeons, journals, lushes, 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 rotation, or turning about a fixed axis, which motion may be either continuous or reciprocating, being called in the latter case oscillation. (See Part I., Chapter II., Section 2.) III. Helical or screw-like motion, to which the same remarks apply as to straight translation. (See Part I., Chapter II., Section 3, Article 96.) 126. The Motions of Secondary Moving Pieces relatively to the pieces which carry them, are limited by the same principles which apply to the motions of primary pieces relatively to the frame. But the motions of secondary moving pieces relatively to the frame may be any motions which can be compounded of straight translations and simple rotations according to the principles already explained in Part I., Chapter II., 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 mav be I. By rolling contact of their surfaces, as in toothless wheels. II. By sliding contact of their surfaces, as in toothed xvheels, screws, wedges, cams, and escapements. III. By bands or wrapping connectors, such as belts, cords, and gearing-cJtains. IV. 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 primary division of the subject of elementary combinations in mechanism into classes, which are subdivided according to the mode of connection of the pieces. In the present treatise, elemen- tary combinations will be classed primarily according to the mode of connection; which is the classification employed by Professor Willis in the Edition of 1870. 128. Line of Connection. In every class of elementary combina- tions, except those in which the connection is made by reduplica- tion of cords, or by an intervening fluid, there is at each instant a certain straight line, called the 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 by the paths oj '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 secondaiy pieces. 74 CHAPTER II. ON ELEMENTARY COMBINATIONS AND TRAINS OP 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. 135. General Conditions of Rolling Contact. The whole of the principles which regulate the motions of a pair of pieces in rolling contact follow from the single principle, that each pair of points in the pitch surfaces, which are in contact at a given instant, must at that instant be moving in the same direction iviili 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 thei-e was motion among the particles, or in a particle at least, of the body, which is contrary to the hypothesis of rigidity. The direction of motion of a point in a rotating body being per- pendicular to a plane passing through its axis, the condition, that each pair of points in contact with each other must move in the same direction leads to the following consequences: I. That when both pieces rotate, their axes, and all their points of contact, lie in the same plane. II. That when one piece rotates and the other slides, the axis of the rotating piece, and all the points of contact, lie in a plane per- pendicular to the direction of motion of the sliding piece. The condition, that the velocities of each pair of points of con- tact must be equal, leads to the following consequences : III. That the angular velocities of a pair of wheels, in rolling contact, must be inversely as the perpendicular distances of any pair of points of contact from the respective axes. IV. That the linear velocity of a smooth rack in rolling contact with a wheel, is equal to the product of the angular velocity of the A STRAIGHT HACK 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). VI. That for a pair of wheels, with intersecting axes, the line of contact is also straight, and traverses the point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term " conical" including all surfaces generated by the motion of a straight line which traverses a fixed point). 136. Circular Cylindrical Wheels are employed when an uniform velocity-ratio is to be communicated between parallel axes. Figs. 38, 39, and 40, of Article 102, may be taken to represent pairs of such wheels ; 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 be their radii : let O C = c be the line of centres, or perpendicular distance between the axes, so that for Let ffj, a 2 , be the angular velocities of the wheels, and v the common linear velocity of their pitch surfaces; then c : 7'j : r 2 : : a 2 ax : a 2 : a a ; / the sign applying to { ? u ^jf e 1 gearing. 137. A Straight Back and Circular Wheel, which are used when an uniform velocity-ratio is to be communicated between a sliding piece and a turning piece, may be represented by fig. 36 of Article 99, C being the axis of the wheel, FTP the plane surface of the rack, and T a point in their line of contact. Let r be the radius 76 THEORY OF MECHANISM. of the wheel, a its angular velocity, and v the linear velocity of the rack; then v r a. 138. Bevel Wheels, -whose pitch surfaces are frustra of regular cones are used to transmit an uniform angular velocity-ratio between a pair of axes which intersect each other. Fig. 45 of Article 105 will serve to illustrate this case; O A and O 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 C respectively being the pitch surfaces, of which narrow zones or frustra are used in practice. Let !, 2 , be the angular velocities about the two axes respec- tively; and let ^ = ZAOT, 4-ZCOT, be the angles made by those axes respectively with the line of contact; then from the principle III. of Article 135 it follows, that the angular velocity- ratio is 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, be given, and also the ratio -? ; then the position of the line of Let ~ be the velocity-ratio required, reduced to its least terms, \j and let B be greater than C. T> If p is not greater than 6, and C lies between the prescribed minimum number of teeth (which may be called t), 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 T> ~-, some ratio approximating to that ratio, and capable of resolu- tion into convenient factors, is to be found by the method of continued fractions. See MATHEMATICAL INTRODUCTION, page 2, Article 4. T> 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, ml. 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 1' } or if B and C contain incon- veniently large prime factors, an approximate velocity-ratio, found -p> by the method of continued fractions, is to be substituted for , as o before. When the prime factors of either B or C are fewer in A TRAIN OF WHEELWORK. 85 number than m 1, the required number of factors is to be made tip 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 been observed in the choice of multipliers, this will be insured by so placing each driving wheel that it shall work with a following wheel whose number of teeth does not contain any of the same multipliers; for the original numbers B and C contain no common factor except 1. The following is an example of a case requiring the use of additional multipliers : Let the required velocity-ratio, in its least terms, be B 360 width of space = pitch. The difference of ^ of the pitch is called the back-lash. The clearance allowed between the points of teeth and the bot- toms of the spaces between the teeth of the other wheel, is about one-tenth of the pitch. The thickness of a tooth is fixed according to the principles of strength ; and the breadth is so adjusted, that when multiplied by pitch, the product shall contain one square inch for each 160 Ibs. 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 on a plane, substituting poles for centres, and great circles for straight lines. In consideration of the practical difficulty, especially in the case of la'-ge \vheels, 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 O, fig. 56, be the apex, and O C the axis of the pitch cone of a bevel-wheel; and let the largest pitch circle be that whose radius is C B. Perpendicular to O B draw B A cutting the axis produced in A, let the outer riin 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 sufficiently near to a zone of the sphere described about 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 fiat. 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 of 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, when viewed by an observer from whom the advance takes place. Fig. 57 re- presents a right-handed screw, and fig. 58 a left-handed screw. The pitch of a screw of one thread, and the total pitch of a screw of any number of threads, is the pitch of the helical motion of that screw, as ex- plained in Article 96, and is the dis- F[ S- 5 ?. 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. Normal and Circular Pitch. When the pitch of a screw is not otherwise specified, it is always understood to be measured parallel to the axis. But it is sometimes convenient for particular purposes to measure it in other directions; and for that purpose a cylindrical pitch surface is to be conceived as described about the axis of the screw, intermediate between the crests of the threads and the bottoms of the grooves between them. If a helix be now described upon the pitch cylinder, so as to cross each turn of each thread at right angles, the distance between two corresponding points on two successive turns of the same thread, measured along this normal helix, may be called the normal pitch; and when the screw has more than one thread, the normal pitch from thread to thread may be called the normal divided pitch. The distance from thread to thread measured on a circle described on the pitch cylinder, and called the pitch circle, may be called the circular pitch ; for a screw of one thread it is one circumference j 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 pitch, and is the same for all parts of the same screw : the normal and circular pitch depend on the radius of the pitch cylinder. Let r denote the radius of the pitch cylinder; n, the number of threads; i, the obliquity of the threads to the pitch circles, and of the noi'mal helix to the axis; pitch; ) P n P " 1 the normal / I jitch > " * , ,. } divided pitch; Fig p c , the circular pitch ; Then Pe=pa' cotan i = p n ' cosec i = ; 2 ic r tan i Pa-Pn ' sec 1'=Pe' tan * = * 77- . 2 r sin i Pn = Pc' Sm * = Pa ' COS * n 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, iJ(Pa 2+ P?)'} Pn the normal pitch is normal to this: it is also evident from the figure that with a constant axial pitch, the normal and radial or circumferential pitch, as well as the angle of obliquity of the threads to the pitch cylinders, vary with the radii of those cylinders. 161. Screw Gearing. A pair of convex screws, each rotating about its axis, are used as an elementary combination, to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the sci-ews, 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 ai-e 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 difference of their obliquities. II. The normal pitch, for a screw of one thread, and the normal THE RELATIVE SLIDING OP A PAIR OF SCREWS. 93 divided pitch, for a screw of more than one thread, must be the same in each screw. III. The angular velocities of the screws are inversely as their number of threads. 162. The Wheel and Screw is an elementary combination of two screws, whose axes are at right angles to each other, both being right-handed or both left-handed. As the usual object of this combination is to produce a change of angular velocity in a ratio greater than can be obtained by any single pair of ordinary wheels, one of the screws is commonly wheel-like, being of large diameter and many-threaded, while the other is short and of few- threads; and the angular velocities are inversely as the number of threads. Fig. CO. Fig. 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 : B 1? in fig. GO for example, being the pitch circle of the wheel, and B 2 B 2 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 up in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure. 163. The Relative Sliding of a Pair of Screws at their point of contact is found thus : Let r v r 2 , be the radii of their pitch cylin- ders, and i lt i 2 , the obliquities of their threads to their pitch circles, one of which is to be considered as negative if the screws are con- trary-handed. Let u be the common component of the velocities of a pair of points of contact along a line touching the pitch sur- 90 THEORY OP MECHANISM. faces and perpendicular to the threads at the pitch point, and v the velocity of sliding of the threads over each other, where v may be considered to be made up of the algebraical sum of two quantities, i\ and v 2 , which act perpendicularly to u, and whose values are i = i **i cos ?!, and v 2 = a 2 r 2 cos t 2 the sum or difference being taken as the screws are similar or contrary-handed. Then u = a 1 r 1 ' sin i : = a 2 r 2 sin i 2 ;"> v = a 1 r l ' cos i x + a 2 r 2 cos i 2 = M (cotan ^ + cotan ? 2 ) (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 which acts by sliding contact was invented by Oldham, and is represented in fig. 62. Cj, C 2 , are the axes of the two parallel shafts; D 19 D 2 , two cross- heads, facing each other, fixed on the ends of the two shafts respectively ; E 15 E 1; a bar, sliding in a diametral groove in the face of D x ; E 2 , E 2 , a bar, sliding in a diametral groove in the face of D 2 ; 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 C^ C 2 , 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. C 2 , diametrically oppo- site to A. Oldham's coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practicability or permanency of their exact continuity. PRINCIPLE OP CONNECTION BY BANDS. 97 SECTION 3. CONNECTION BY BANDS. 1G5. 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 baud, 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 r 1} r 2 , be the perpendiculars let fall from their axes on the centre line of the band, a^, a z , their angular velocities, and i 1} ?' 2 , 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-i Oi sin % = r 2 a 2 sin i z ; (1.) whence the angular velocity-ratio is .(2.) When the axes are parallel (which is almost always the case), i^ = i and 35 THEORY OF MECHANISM. The same equation holds when both axes, whether parallel or not, are perpendicular in direction to that part of the band which transmits the motion; for then sin i 1 = sin 4=1. II. For a rotating piece and a sliding piece, let r be the perpen- cular from the axis of the rotating piece on the centre line of the band, a the angular velocity, i the angle between the directions of the band and axis, u the longitudinal velocity of the baud, j the angle between the direction of the centre line of the baud and that of the motion of the sliding piece, and v the velocity of the sliding piece; then ur a sin i = v cosj; (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.;, the longitudinal velocity of the sliding piece owing to the rigidity of the band; and -'-*-'- ....(5-) When the centre line of the band is parallel to the direction of motion of the sliding piece, and perpendicular to the direction of the axis of the rotating piece, sin i (90) = cosj (0) = 1, and v = u = r 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 pitlley or drum, and distant from it by half the thickness of the band. 168. Circular Pulleys and Drums are used to commuicate a Fig. 63. 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 radius, being equal to the real radius of the pulley or drum added to half the thickness of the band. A crossed belt connecting a pair of circular 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, connecting a pair of pulleys whose effective radii are 0^ = ^, C 2 T 2 = r 2 , with parallel axes whose distance apart is G 1 2 = c, is given by formulae founded on Equation 1 of Article 113, viz., L = s-s+2"r^. Each of the two equal straight parts of the belt is evidently of the length a = Tj. T 2 = ^/c 2 - fa + r 2 ) 2 for a crossed belt ; ^ s = T! T 2 = J& - ( TI - r 2 )2 for an open belt ; j r-i being the greater radius, and r 2 the less. Let z a be the arc to radius unity of the greater pulley, and i 2 that of the less pulley, with which the belt is in contact; then for a crossed belt for the angle 'U 1 C x W x at the centre is double of the angle at the circumference C x T a W x , and this is equal to the angle S x C 2 G! as they both differ from a right angle by the same angle T x C x V x ; and for an open belt, / n r i~ r "\ ( n r i~ T 2\ tt= i,+ 2 arc sin - -J ; i 2 = I - 2 arc sin - ji - - ) ; and the introduction of those values into Equation 1 of Article 113 gives the following results : For a crossed belt L = 2 /c 2 -r 1 + r 2 2+r 1 + r 2 + 2 arc -sin . and if similar reasoning be applied, it may be shewn that for an open belt, L = 2 V 2 - (n - r a ) 2 + , (r, + r 2 ) + 2fa - r 2 ) arc sin 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 x - * 2 is small compared with c, 1 4 C nearly; whence, for an open belt, nearly, and arc sin L nearly = 2c + *(r x + r a ) + ^*- .......... (3 A .) 100 THEORY OF MECHANISM. 170. Speed-Cones (figs. 65, G6, 67, 68) are a contrivance for Fig. 65. Fig. 66. Fig. 67. Fig. 68. varying and adjusting the velocity-ratio communicated between a pair of parallel shafts by means of a belt, and may be either conti- nuous cones or conoids, as in figs. 65, 66, whose velocity-ratio can be varied gradually while they are in motion by shifting the belt ; or sets of pulleys whose radii vary by steps, as in figs. 67, 68, in which case the velocity-ratio can be changed by shifting the belt from one pair of pulleys to another. In order that the belt may be equally tight in every possible position on a pair of speed-cones, the quantity L in the equations of Article 169 must be constant. For a crossed belt, as in figs. 65 and 66, L depends solely on c and on r^ + r 2 . 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 involution 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 o\ be the radius of the large end of each, r 2 that of the small end, r that of the middle; and let y be the sagitta, measured perpendi- cular to the axis, of the arc by whoso revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length ; then '--^-^ <'> where the second value is obtained from the first by considering that in Equation 3A, 2 * r = (r + r 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 determined, make the generating curve an arc either of a circle or of a parabola. For a pair of stepped cones, as in fig. 68, let a series of differ- ences of the radii, or values of 1\ 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 M-,*-; ..................... (2.) 2 r Leing that sum when the radii are equal. SECTION 4. LINKWORK. 171. Definitions. The pieces which are connected by linkwork, if they rotate or oscillate, are usually called cranks, beams, and levers. The link by which they are connected is a rigid bar, which may be straight or of any other figure; the straight figure being the most favourable to strength, is used when there is no special reason to the contrary. The link is known by various names under various circumstances, such as coupling rod, connecting rod, 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 effect of the link is to maintain the distance between the centres of those pins invariable; hence the line joining the centres of the pins is the line of connection ; and those centres may be called the connected points- In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is the 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 arras coincides with the line of connection, the common perpendicular of the line of connection and the axis of that crank arm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crank arm in question at such an instant is called a dead point. The velocity of the other connected point at such an instant is null, unless it also reaches a dead point at the same instant, so that the line of connection is in the plane of the two axes of rotation, in which case the velocity-ratio is indeterminate. 174. Coupling of Parallel Axes. The only case in which an uni- 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 repi-esented 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 pai'allel to the line of connection cuts off from any two lines drawn from a given point, perpendicular respectively to the paths of the connected points. Example I. Two Rotating Pieces with Parallel Axes (fig. 69) Let C 1( C a , be the parallel axes of the pieces; T 17 T 2 , their con- nected points; C t T,, C a T 3 , their crank arms ; T! T 2 , the link. At a given instant, let v^ be the velocity of T x ; v 2 that of T 2 . To find the ratio of those velocities, produce d T lf C 2 T 2 , till they intersect in K ; K is the instantaneous axis of the link or connecting rod, and the velocity ratio is n :v 2 : : Kl^ . K T a (1.) AN ECCENTRIC. 103 Should K be inconveniently far off, draw any triangle with its sides respectively parallel to d T v C 2 T 2 , and T x T 2 ; the ratio of the two sides first mentioned will be the velocity-ratio required. For example, draw C a A parallel to C a T lf cutting TI T 2 in A, then Fig. 70. Fig. G9. Example II. Rotating piece and sliding piece (fig. 70). Let C 2 be the axis of a rotating piece, and T 1 E, the straight line along which a sliding piece moves. Let T 1? T 2 , be the connected points, C a T 2 the crank arm of the 1'otating piece, and 1\ T 2 the link or connecting rod. The point T x , T 2 , and the line T^ R, are supposed to be in one plane, perpendicular to the axis 0. Draw T l K per- pendicular to T! E, intersecting C 2 T 2 in K; K is the instantaneous axis of the link; and vi : v t : : K 3\ : K T, Or otherwise draw from a point C 2 , C 2 A perpendiciilar to T x R the direction of motion of the sliding piece, C 2 T 2 perpendicular to the direction of motion of the rotating piece, then the line T\ T 2 , or a line parallel thereto cuts off the segments C 2 A, C 2 T 2 , or segments proportional thereto, and the velocity-ratio of the rotating piece to the sliding piece is as C 2 T 2 to C 2 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 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 R = A E A D the crank arm of the continuously turning piece. Then if the two ends of the stroke be in one straight line with the axis of the crank, SO T> . / 1 \ = 2 R ; (i.) and if their ends be not in one straight line with that axis, then S, L - R, and L + R, are the three sides of a triangle, having the angle opposite S at that axis; so that if 6 be the supplement of the arc between the dead points, D and E, S 2 - (L - R) 2 + (L + R) 2 - 2 (L - R) Fig. 72. S 2 = 2 (L 2 + R 2 ) - 2 (L 2 2 L 2 + 2 R 2 S 2 2Jr).t;l ^ 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 G 1} O C 2 , and i their angle of inclination to each other. The pair of shafts C,, C 2 , terminate in a pair of forks, F 1; F 2 , in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross, having its centre at 0. This cross is the link; the connected points are the centres of the bear- ings Fj, F 2 . At each instant each of those points moves at right angles to the central plane of its shaft and fork, therefore the line of intersection of the central planes of the two forks, at any instant, is the instantaneous axis of the cross, and the velocity-ratio of the Fig. 73. DEFINITIONS. 105 points F 1; F s (which, as the forks are equal, is also the angular velocity-ratio of the shafts), is equal to the ratio of the distances of those points from that instantaneous axis. The mean value of that velocity-ratio is that of equality; for each successive quarter turn is made by both shafts in the same time; but its actual value fluctuates between the limits, r sin (90 - 1) cos ^ " - when Fj is in the plane of the axs = cos i when F 2 is in that plane. (1.) 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, C^, C s , there is introduced a short interme- diate shaft C 2 , making equal angles with d and C 3 , connected with each of them by a Hooke's joint, and 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 pall, turning on a fixed axis, prevents the ratchet wheel or rack from reversing its motion. SECTION 5. REDUPLICATION OP CORDS. 181. Definitions The combination of pieces connected by the several plies of a cord or 1'ope consists of a pair of cases or frames called blocks, each containing one or more pulleys called sheaves. One of the blocks called the fall-block, B,, is fixed ; the other, or running-block, B 2 , is movable to or from the fall-block, with which 100 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, T u called the fall, or tackle-fall, is free ; while the the intermediate rope passes Fig. 75. Fig. 75A. portion of 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 113 (which see), viz.: v = nu; .............................. (1.) where n is the number of plies of rope by which the running-block is connected with the fall-block. Thus, in fig. 75 n = l ; and in fig. 7oA n = 6. 182 A. 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 v' = n' u .............................. (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 tlie CLASSIFICATION OF THE MECHANICAL POWERS. 107 block, 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 rheaves may be made all in one piece, and may have journals turning in fixed bearings. This is called White's TacUe, 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 moving piece; and the comparative motion taken into consideration is simply the velocity-ratio either of a pair of points in that piece, or of two components of the velocity of one point. There are two established classifications of the mechanical powers; an older classification, which enumerates six; and a newer classification, which ranges the six mechanical 1 towers of the older system under three heads. The following table shews both these classifications: NKiVF.R CLASSIFICATION OLDER CLASSIFICATION. T WPP / The Lever LEVER, ........................ { The Whee ( The Inclined Plane. THE INCLINED PLANE, ........... 1 'I he Wedge. ( The Screw. THE PULLEY, ......................... The Pulley. In the present section the compai-ative 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. tliis: that in the lever, the driving point, D, and the following point, F, are a pair of determinate points in the moving piece, as in figs. 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 76o, D and F are the o- '" projections of the driving and following points respectively. The axis of a lever is often called the fulcrum. A lever is said to be straight, when the driving point, D, and following point, F, are in one plane traversing the axis A, as in figs. 76A, 76B, and 76c. In other cases the lever is said to be bent, as in fig. 7 60. A. Fig. 76A. Fig "6c. 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 j, sides of the fulcrum A. ^- In a lever of the second kind, fig. 76e, the 'J" driving and following points are at the same Fig. 76o. si^ 6 * tne fulcrum, and the driving point is the further from the fulcrum. In a. 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 comparative motion considered is the velocity-ratio of the entire motion of a straight- sliding primary piece and one of the components of that motion ; the principles of which velocity-ratio have been stated in Article 70, pages 38, 39. THE INCLINED PLANE. 109 In the inclined plane, fig. 7GE, 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 0. From any con venient point, 0, in B 0, let fall D perpendicular to B D; then B D -^- B C is the ratio of the component velocity in the direction B D to the entire velocity of B. In fig. 76p, A A is the trace of a fixed plane; BCD, the trace of a wedge which slides on that plane. While the wedge advances through the distance C c, its oblique face advances from the posi- tion 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. 7GE. Fig, 76F. 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. 76s 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 -4- 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 the 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 fluid vaiy, the pi'oblem 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 in a pump causes the slow motion of a large piston in a working cylinder. The pump draws water from a reservoir, and forces it into the working cylinder: during the outward stroke of the punip piston, the piston of the working cylinder stands still; during the inward stroke of the pump piston, the piston of the working cylinder moves outward with a velocity as much less than that of the pump piston as its area is greater. When the piston of the TRAINS OF 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 different machine. The principles of comparative motion in such converging and diverging trains are the same as in simple trains. 112 CHAPTER III. ON AGGREGATE COMBINATIONS. 190. The General Principles of aggregate combinations have already been given in Part I., Chapter II., Section 3. The pro- blems to which those principles are to be applied may be divided in to two classes. I. Where a secondary moving piece is connected at three, or at two points, as the case may be, with three or with two other pieces whose motions are given; so that the problem is, from the motions of three or of two points in the secondary piece, to find its motion as a whole, and the motion of any point in it. The solution of this problem is given in Articles 97 and 98. IT. Where a secondary piece, C, is carried by another piece, Bj and denoting the frame of the machine by A, there are given two out of the three motions of A, B, and C, relatively to each other, and the third is required. The motion of C relatively to A is the resultant of the motion of C relatively to B, and of B relatively to A; and the problem is solved by the methods already explained in Articles 99 to 107, inclusive. Professor Willis distinguishes the effects of aggregate combina- tions into aggregate velocities, whether linear or angular, pi'oduced in secondary pieces by the combined action of different drivers, and aggregate paths, being the curves, such as cycloids and trochoids, epicycloids and epitrochoids, described by given points in such secondary pieces. \, The following Articles give examples of two simple aggregate combinations. 191. Differential Windlass. In fig. 77, the axis A 1 carries two barrels of different radii, ^ being the greater, and r 2 the less. A running block containing a single J pulley is hung by a rope which passes \|U^ below the pulley, and has one end wound # Ixj 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 ar^ and the division which hangs from the smaller barrel moves in. the contrary direction with the velocity a r 2 (whose direction iis 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, 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 the two ends of the horizontal diameter of the pulley, and let A V a = ar lt and B V 4 = r 2 represent their velocities; join V ~V 6 cutting AB in O; this is the instantaneous axis. Now AO-t-OB:AO-OB::AY.+BV s :AV BT BY M AB: 200:: afa + rj: a^-r,); and hence the distance of the instantaneous axis from the centre or moving axis of the pulley is obviously .(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 1 2 . 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 Si S ]; and S 2 S^, of the respective pitches Fi 3 . 7 ft and p 2 , p being the greater, and let the screws in the first instance be both right-handed or both left-handed. Let ISf^ and N 2 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 114 THEORY OF MECHANISM. 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. Fig. 79 represents a compound screw in which the two screws are contrary-handed, and the relative velocity of the nuts N x N" 2 is that due to the sum of the two pitches ; or as they are iisually equal, to double the pitch of each screw. This combination is used in coupling railway-carriages. PART 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 actionv 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- logons 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 lias no real existence. Nevertheless, in investigations respect- ing the action of a distributed force upon the position and move- ments, as a whole, of a rigid body, or of a body which without error may be treated as rigid, like the solid parts of a machine, fixed or moving, that force may be treated as if it were concen- trated at a point or points, determined by suitable processes j 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 OF ACTION of that force. The MAGNITUDES of two forces are equal when, being applied to the same body in opposite directions along the same line of action, they balance each other. The magnitude of a force is expressed arithmetically by stating in numbers its 1'atio 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 as The grain = TT 1 TTTJ - of a pound avoirdupois. The ti'oy pound = 5,760 grains = 0-S2285714 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 0> 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 Ibs. avoirdupois. A pound avoirdupois is 0'4535926525 of a kilogramme. For scientific purposes, forces are sometimes expressed in Absolute Units. The "Absolute Unit of Force" is a term used to denote the force which, acting on an unit of mass for an unit of time, produces an unit of velocity. The unit of time employed is always a second. The 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 much matter as weighs one REPRESENTATION OF FORCES BY LINES. 117 unit of weight near the level of the sea, and in some definite latitude. In Britain the latitude chosen is that of London ; in France, that of Paris. In Britain the unit of weight chosen is sometimes a grain, sometimes a pound avoirdupois; and it is equal to 32'187 of the corresponding absolute units of force. In France the unit of weight chosen is either a gramme or a kilogramme, and it is equal to 9-8087 of the corresponding absolute units of force. Each of those coefficients is denoted by the letter g. 195. Measures of Force and Mass. If by the unit of force is understood the weight of a certain 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 mass, the mass of the standard itself, then the unit of force is the absolute unit; and the weight of the standard in such units is expressed by g; for g is the velocity which a body's own weight, acting unbalanced, impresses on it in a second. This will be specially treated of in Part Y. This is. the system employed in many scientific writings, and in particular,, in Thomson and Tait's Natural Philosophy. It has great advan- tages in a scientific point of view; but its use in calculations for practical purposes would be inconvenient, because of the prevailing custom of expressing forces in terms of the standard of weight. 196. Representation of Forces by Lines. A single force may be represented in a drawing by a straight line; an extremity of the line indicating the point of application of the force, the direction of the line, the direc- tion of the force, and the length of the line, the magnitude of the force, according to an arbitrary scale. For example, in fig. 80, the fact that the body B B B B is acted upon at the point O a by a given force, may be expressed by drawing from O x a straight line G! F! in the direction of the force, and of a length representing the magnitude of the force. If the force represented by O x Fj is balanced by a force applied either at the same point, or at another point 2 (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 O 2 F 2 , opposite in direc- 118 PRINCIPLES OF STATICS. tion, and equal in length to Oj .b\, 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 Oj O 2 , &c., and represented by lines O x -b\, 2 F 2 , &c. ; then either direction in the line L L (such as the direc- tion towards + L) is to be considered as positive, and the opposite direction (such as the direction towards L) as negative ; and if the sum of all the lines repre- senting forces which point -p- 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 BESULTAKT of any combination of forces applied to one body is a single force capable of balancing that single force which balances the combined forces; that is to say, the resultant of the combined forces is equal and directly opposed to the force which balances the combined forces, and is equivalent to the combined forces so far as the balance of the body is concerned. The com- bined forces are called components of their resultant. The resultant of a set of mutually balanced forces is nothing. The magnitudes and directions of a resultant force and of its components are related to each other exactly in the same manner with the velocities and directions of resultant and component motions. As to the position of the resultant, if the components act through one point, the resultant acts through that point also; but if the components do not act through one point, the position of the re- sultant is to be found by methods which will be stated further on. 198. Equilibrium or Balance is the condition of two or more forces which are so opposed that their combined action on a body produces no change in its rest or motion, and that each force merely tends to cause such change, without actually causing it. In treatises on statics, the word pressure is often used to denote any balanced force; although in the popular sense that word is used to denote a force, of the nature of a thrust or push, distributed over a surface. 199. Parallel Forces are forces whose directions of motion, are parallel, excepting couples and directly opposed forces. 200. Couples. Two forces of equal magnitude applied to the same body in parallel and opposite directions, but not in the same DISTRIBUTED FORCES IN GENERAL. 119 line of action (such as F, F, in fig. 82), constitute what is called a " couple." The arm or leverage of a couple (L, fig. 82) is the perpendicular distance between the Hues 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 s ' 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 footpounds, 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 (0 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 Centre of Gravity or the Centre 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 ~by the line of action of the resultant of every system of parallel forces having the given mutual ratios and applied to the given system . of points, whatsoever -may b& 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 at each point of their surfaces of application, or capable of being resolved into sets of parallel forces; hence, parallel distributed forces have alone to be considered ; and every such force is statically equivalent either to a single resultant, or to a resultant couple. The intensity of a 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 over a surface, such as pressure or friction. 203. Specific Gravity Heaviness Density Bulkiness. I. Specific Gravity is the ratio of the weight of a given bulk of a given substance to the weight of the same bulk of pure water at a standard temperature. In Britain the standard temperature is 62 Fahr. = 16 c -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 Ibs. avoirdupois to the cubic foot ; in French measures, in kilogrammes to the cubic decimetre (or to the litre). The values of the heaviness of water at 39 0< 1 Fahr., and at 62 Fahr., are respectively 62425 and 62-355 Ibs. to the cubic foot. III. The density of a substance is either the number of units of mass in an unit of volume, in which case it is equal to the heavi- ness, or the ratio of the mass of a given volume of the substance to the mass of an equal volume of water, in which case it is equal to the specific gravity. In its application to gases, the term THE INTENSITY OP 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, Ib. avoirdupois; in French measures, in cubic decimetres (or in litres) to the kilogramme. Rise of temper- ature produces (with certain exceptions) increase of bulkiness. The linear expansion of a solid body is one-third of its expansion in bulk. 204. The Centre of Gravity of a body or of a system of bodies, is the point always traversed by the resultant of the weight of the body or system of bodies, 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. Eesultant 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 "sum" is to be taken in the algebraical sense; that is to say, that forces acting in the same direction are to be added to, and forces acting in opposite directions subtracted from each other. When a system of forces acting along one straight 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 be repre- sented in direction and magnitude bu the sides of a parallelogram, their resultant is represented by 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 foi'ces 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. OA 2 Fig. 83. 210. Triangle of Forces. To balance the forces O A and OB, a^fovce is required equal and directly opposed to their resultant O C. This may be expressed by saying, that if the directions and RESOLUTION OF A FORCE INTO TWO COMPONENTS. 123 Fig. 84. magnitudes of three forces be represented by the three sides of a triangle, taken in the same order (such as O^A, A C, 0~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 angle between the other two. 211. Polygon of Forces. If a number offerees acting through the same point be represented by lines equal and parallel to the sides of a closed polygon, taken in the same order, those forces balance each other. To fix the ideas, let there be five forces acting through the point (fig. 84), and represented iu direction and magnitude by the lines Fj, F 2 , F 3 , F 4 , F 5 , which are equal and parallel to the sides of the closed poly- gon O A BCD O; viz.: F 1 = and ii A ; F 2 = and n AB; F 3 = and 11 B C; F 4 = and 11 C D; F 5 = and II D O. Then, by the principle of the parallelogram of forces, the resultant of F x and F 2 is O B; the resultant of F x , F 2 , and F 3 is O C ; the resultant of F x , F 2 , F 3 , and F 4 is O D, equal and opposite to F 5 , so that the final resultant is nothing. The closed polygon may be either plane or " gauche" that is, not in one plane. 212. Principles of the Parallelepiped of Forces. The simplest gauche polygon is one of four sides. Let AOBCEFGH (fig. 85), be a parallelepiped 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 F 15 F 2 , F 3 , acting through 0, be represented by the three edges A~O, OB, O C, of a parallelepiped, the dia- gonal O H represents their resultant, and a fourth force F 4 equal and opposite to O H balances them. 213. Resolution of a Force into two Components. In order that a given tingle force may be resolvable into two ^* Fig. 85. components acting iu 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 Hues of action should be in one plane. 121 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 intersect it in one point 0. Though draw C A || Y, cutting O X in A, and C B || O X, cutting O Y in B. Then will 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 O C. The magnitudes of the forces are in the following proportions: OC :0 A :OB AOB:sinBOC:sinAOC 214. Resolution of a Force into three Components. In order that a given single force may be resolvable into three components acting in given lines inclined to each other, it is necessary that the lines of action of the components should intersect the line of action of the given force in one point. Returning to fig. 85, let O H represent the given force which it is required to resolve into three component forces, acting in the lines O X, O Y, O Z, which intersect O H in one point O. Through H draw three planes parallel respectively to the planes Y O Z, Z O Y, X Y, and cutting respectively X in A, O Y in B, O Z in C. Then will O A, OB", (JO, represent the component forces required. Three forces respectively equal to, and directly opposed to O A, CTB, and OC", will balance O~lf.' 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, 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 witli the components by , /3, y, respectively; then the relations between the three rectangular components and their resultant are expressed by the following equations : X = Rcos; Y = Rcos,3; Z = Rcos*/; .......... (2.) (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 = R sin /3; Y = R cos /3 = R sin ; Z = 0;...(4.) R 2 = X 2 + Y 2 (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 (2 X, 2 Y, 2 Z); these will be the rectangular components of the resultant of all the forces; and its magnitude and direction will be given by the following equations : R2 = (2-X)2 + (2-Y) 2 + (S-Z) 2 ; (1.) 2-X , 2-Y 2-Z cos = - ; cos ,3 = -]-; cos y = j- ( 2 -) If the forces all act in one plane, two rectangular axes in that plane are sufficient, and the terms containing Z disappear from the equations. If the forces balance each other, the components parallel to each axis balance each other independently; that is to say, the three following conditions are fulfilled : 2-X = 0; Z-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 OP 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 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 011 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 being = ^-, and parallel to the Fig. 86. plane of the couple M. Then the forces at 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 effect of that combination is to shift the line of action of the force parallel to itself through a distance O A = -=; ; to the left if M is right-handedto the right if M is left-handed. . MAGNITUDE OF RESULTANT OP PARALLEL FORCES. 127 223. Moment of Force with respect to an Axis. In fig. 87, let the straight line F represent a force. Let O X be any straight line perpendicular in clii-ec- 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 X which is nearest to A, ' combined with the couple composed of F and - F', ' Oi ' 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 tlie 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 = ^ F A B ; represents one-half of the moment of F relatively to O X. SECTION 3. EESULTANT 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 Resultant of Parallel Forces Principle of the Lever. If three, parallel forces applied to one body balance each other, they must lie in one plane; the two extreme forces must act in the same direction; the middle force must act in the opposite direction; and the magnitude of each force must be proportional to the distance between the lines of action of the other two. Let a body (fig. 88) be maintained in equilibrio by two opposite coxiples acting in the same plane, and of equal moments, F A L A = F B L E , and let those couples be so applied to the body that the lines of action of two of those forces, - F A - F B , which act in the same direction, shall coincide. Then those two forces are equivalent to the single middle force F c = (F A + F E ), equal and opposite to the sum of the extreme forces + F A , + F B , 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 ACl=L A ; CB = L B ; A~B = L A + L B ; and consequently F A : F B : F : : ^B : A~C : A~B; (1.) This proposition holds also when the straight line AGE 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 plane \vith, 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 AE'and B D, cutting each other in F. In B D (produced in fig. 90) take B G = D F. 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 HELATIVE PROPORTIONS OF FOUR PARALLEL FORCES. 129 EG or D G = Fig, 91. OH'"' C H When the two given parallel forces are opposite and equal, they foini 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, iig. 91, and measure the distances between the points where it cuts the lines of action. Then eacli foi'ce 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 = ; 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 Eelative 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 Fig. 92. position; and in fig. 92 or fig. 93, let A, B, C, D repre- 130 PRINCIPLES OF STATICS. sent the points where the four lines of action cut the plane. Draw the six straight lines joining those four A 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 foi'ces 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:CDA:DAB:ABO B C D. Each of the four forces is equal and opposite to the resultant of the other three ; and each set of three forces are equal and 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 /X6S. 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 O, at right angles to each other and to the line of action of F. Let A B = y, 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 -XO + X, -YO + Y, respec- tively. According to the arrange- ment of the axes in the figure, x is to be considered as positive to the right, and negative to the left, of YO + 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. RESULTANT OF ANY SYSTEM OF PARALLEL FORCES IX ONE PLANE. 131 F' = F applied at B, combined with the couple constituted by F and F' with the arm y, whose moment is y F ; being positive in the case represented, because the couple is right-handed. Next, at the origin O, conceive a pair of equal and opposite forces, F" and F", to be applied, F" being equal and parallel to F 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 = #, whose moment is - x 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-oi-dinates 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 X, and - x F relatively to the axis OY right-handed couples being considered positive; and + Y lying to the left of + X, as viewed by a spectator looking from + X towards 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 OX,?/ also being considered as positive or negative, according to its direction; then, 2-F, 0; S-yF = 0. In summing moments, right-handed couples are usually con- sidered as positive, and left-handed couples as negative. 231. Let R denote the Resultant of any System of Parallel Forces in one Plane, and y r , the distance of the line of action of that resultant from the assumed axis O X to which the positions of forces are referred; then, 132 PRINCIPLES OF STATICS. In some cases the forces may have no single resultant, 2 F being =0; and then, unless the forces balance each other com- pletely, their resultant is a couple of the moment 2 y F. 232. Balance of any System of Parallel Forces. In order that any system of parallel forces, whether in one plane or not, may balance each other, it is necessary and sufficient that the three following conditions shall be fulfilled : First (As already stated) that the algebraical sum of the forces shall be nothing. Secondly 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 X, and x its perpendicular distance from O Y; then, 2-F = 0; 2-yF = 0; 2-aF = 0; 233. Let R denote the Eesultant of any System of Parallel Forces, and x r and y r the distances of its line of action from two rectangular axes; then, 2-aF 2-7/F - In some cases the forces may have no single resultant, 2 F being =0; 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 Ms = 2-yFj M, = -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. y ; that is to say, pro- portional to those moments in length, and pointing in the direction from which those couples must respectively be viewed in order that they may appear right-handed. Complete the rectangle whose sides are those lines; its diagonal will represent the axis, direction, and moment of the final resultant couple. Let M r be the moment of this couple; then and if e 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, 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 O X, and perpendicular to the plane Y Z, y parallel to O Y, and perpendicular to the plane Z X, and z parallel to O Z, and perpendicular to the plane X Y. x, y, and z are the rectangu- lar co-ordinates of A, 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 j>ositions 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. 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.) Thirdly, conceive all the parallel forces to act in lines parallel to the plane X Y. Then the distance of their resultant, and of tho centre of parallel forces from that plane is 2-F"' .(3.) 134 PRINCIPLES OF STATICS. If the forces have no single resultant, so that 2 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 giv r en plane is simply the mean of the distances of the points of the system from that plane. SECTION 4. OF ANY SYSTEM OF FORCES. 235. Resultant and Balance of any System of Forces in One Plane. Let the plane be that of the axes 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 2 Y ; its magnitude is given by the equation R 2 = (2-X) ? +(2'Y) 2 , ..................... (1.) .and the angle r which it makes with Xis found by the equations 2-X 2-Y cos a r = R ; sm* r =-^- ................. (2.) TThis 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 y cos , and hence the resultant moment of the system of forces about the axis O Z is (3.) and is right or left-handed according as M is positive or negative. The perpendicular distance of the resultant force II from is -I ...................................... w Let x r and y r be the co-ordinates of any point in the line of action of that resultant; then the equation of that line is* av2-Y-y P 2-X = M ............................ (5.) * The method of obtaining this result by Co-ordinate Geometry is the RESULTANT AND BALANCE OF ANY SYSTEM OP FORCES. 135 If M = the resultant acts through the origin O ; if M has magnitude, and R = (in which case S-X = 0, 2-Y = 0) the resultant is a couple. The conditions of equilibrium of the system of forces are = 0; M = 0. 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 of co- ordinates. As before, let in fig. 95, p. 133, denote the origin of co-or- dinates, and O X, O Y, O Z, the three rectangular axes: and let them be arranged so that in looking from X ) ( Y towards Z ) Y > towards O, rotation from Z J (X towards Y j shall appear right-handed. Let X, Y, Z, denote the rectangular components of any one of the forces; x, y, z, the co-ordinates of a point in its line of action. Taking the algebraical sums of all the forces which act along the same axes, and of all the couples which act round the same axes, following :-Let C=L, AB=E, ZXAB=*,.; and let E G=o> and O G = y r be the co-ordinates of the point E. Then by Trigonometry x sin a r sin AC = cos CO A = sin -cos r=cos OAC=sin CO A = cos DOG. L=DC+OD=FE+OD =EG-sinEGF + O G cos D G = x r sin r -y r ' cosu r nmltiplying by R L Pi = M = ov -II 'sin a r - y r R ' cos ec r G B^ 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; ( L ) Couples, about OX; ,outOX; M 1 =S(yZ-*Y)|) OY; M a =S(*X-*Z);V " OZ;M 8 = S(*Y-yX);J (2.) found as already explained in Article 235. The three forces are equivalent to a single force s ......... (3.) acting through O in a line which makes with the axes the angles given by the equations 2-X , 2-Y 2-Z ... cos = -g- ; cos fl=~^-; cos y = - R - ......... (4.) The three couples, M 1? M 2 , M 3 , are equivalent to one couple, whose magnitude is given by the equation M= ^(MJ + M' + MI), .................... (5.) and whose axis makes with the axes of co-ordinates the angles given by the equations M, M, M 3 ... cos X:= _i ; cos , = -; cos, = M 3 , .................. (6.) i-ii I denote respectively the angles m which 4 v \ made by tl | e axig M with OX) ol] The conditions of equilibrium of the system of forces may bo expressed in either of the two following forms : 2-X = 0; 2-Y = 0; 2-Z = 0; M 1 = 0;M 2 = 0; M, = 0; (7.) or R = 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 II acting through O. KESULTANT AND BALANCE OF ANY SYSTEM OF FORCES. 137 CASE II. When the axis of M is at right angles to tlie direction of R, a case expressed by the following equation : cos cos A + cos /3 cos p + cos y cos v = ; ...... (9.) (an equation of Co-ordinate Geometry) the resultant of M and R is a single force equal 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 (10.) CASE III. When R = 0, there is no single resultant; and the only resultant is the couple M. CASE IV. When the axis of M is parallel to the line of action of R, that is, when, either * = *; ft = ft- , = y, .................. (11.) or 7v= _ ft= _/3; ,= - v ; ............ (12.) there is no single resultant; and the system of forces is equivalent to the force R 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 + cos ft cos /3 + cos t, cos -/, ____ (13.) the couple M is to be resolved into two rectangular components, viz: M sin 6 round an axis perpendicular to R, and in "j the plane containing the direction of R and of I ,-. , v the ax is of M; M cos 6 round an axis parallel to R. The force R and the couple M sin 6 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 distance from O is (15.) The couple M cos d, 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); to a force, combined with a couple whose axis is parallel to the line of action of the force, as in Cases IV. and V. This can occur witk inclined forces only ; for the resultant of any number of parallel forces is either a single force or a couple. 237. Parallel Projections or Transformations in Statics. If two figures be so related, that for each point in one there is a corre- sponding point in the other, and that to each pair of equal and parallel lines in the one, there corresponds a pair of equal and parallel lines in the other, those figures are said to be PARALLEL PKOJECTIONS of each other. The relations between such a pair of figures is expressed alge- braically as follows : Let any figure be referred to axes of co- ordinates, whether rectangular or oblique ; let x, y, z, denote the co-ordinates of any point in it, which may be denoted by A : let a second figure be constructed from a second set of axes of co-ordinates, either agreeing with, or differing from, the first set as to rectangu- larity or obliquity; let x', y', z', be the co-ordinates in the second figure, of the point A' which corresponds to any point A in the first figure, and let those co-ordinates be so related to the co-ordi- nates of A, that for each pair of corresponding points, A, A', in the two figures, the three pairs of corresponding co-ordinates shall bear to each other three constant ratios, such as x' y' , z' - =a: =o: - = 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 parallelepiped. VI. A parallel projection of a pair of parallel plane surfaces, PARALLEL PROJECTIONS OR TRANSFORMATIONS IN STATICS. 139 \vliose areas are in a given ratio, is also a pair of parallel plane surfaces, whose areas are in the same ratio. VII. A parallel projection of a pair of volumes having a given ratio, is a pair of volumes having the same ratio. The following are the mechanical properties of parallel projec- tions in connection with the principles set forth in this section: VIII. If two systems of points be parallel projections of each other; and if to each of those systems there be applied a system of parallel forces bearing to each other the same system of ratios, then, the centres of parallel forces for those two systems of points will be parallel projections of each other, mutually related in the same 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. 140 CHAPTER TIT. DISTRIBUTED FORCES. SECTION 1. CENTRES OP GRAVITY. 238. Centre of Gravity of a Symmetrical Homogeneous Body. If a body is homogeneous, or of equal specific gravity throughout, and so far symmetrical as to have a centre of figure ; that is, a point within the body, which bisects every diameter of the body drawn through it, that point is also the centre of gravity of the body. Amongst the bodies which answer this description, are the sphere, the ellipsoid, the circular cylinder, the elliptic cylinder, prisms whose bases have centres of figure, and parallelepipeds, whether right or oblique. 239. The Common Centre of Gravity of a Set of Bodies whose several centres of gravity are known, is the centre of parallel forces for the weights of the several bodies, each considered as acting through its centre of gravity. (See Article 234, p. 133.) 240. Planes of Symmetry Axes of Symmetry. If a homogeneous body be of a figure which is 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 s d x is the volume of the layer, and v= the volume of the whole body, being the sum of the volumes of CENTRE OF GRAVITY FOUND BY SUDTKACTION". 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 given by the equation Find, by a similar process, the distances y , z , of the centre of gravity from the other two co-ordinate planes, and its position will be completely 'determined. If the centre of gravity is previously known to be in a particular plane, it is 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, Jet w be the mean heaviness of the layer sdx, so that W = fwsdx, is the weight of the body. Then 243. Centre of Gravity found by Addition. When the figure of a body consists of parts, whose respective 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 figui-e of a homogeneous body, whose centre of gravity is sought, can be made by taking away a figure whose centre of gravity is D , 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, W x its weight. Let A B E be the smaller figure, whose centre of gravity G 2 is known, W 2 its weight. Let E B C D be the figure whose centre of gravity G 3 is sought, made by taking away ABE from A C D, so that its weight is 142 PRINCIPLES OF STATICS. Join G! G 2 ; G 3 will be in the prolongation of that straight line beyond G r In the same straight line produced, take any point as origin of co-ordinates. Make OG i = x 1 ; O (jr 2 = x 2 , O G 3 (the unknown quantity) = x z . Then w x -w 2 245. Centre of Gravity Altered by Transposition. In fig. 98, let A B D be a body of the weight W , whose centre of gravity G is known. Let the figure of this body be altered, by trans- posing a part whose weight is W 1} from the position E F to the position V D H, so that the new figure of the body is A B H E. Let G! be the original, and G 2 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 G 3 , in a direction G G 3 parallel to G 2 G ls and through a distance given by the formula. ,..(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, , be the co-ordinates, rectangular or oblique, of any point in the figure of the first body; x f , y', z', those of the corresponding point in the second body; x , y , z , the co-ordinates of the centre of gravity of the first body; x' , y' 0) z' , those of the centre of gravity of the second body, then .(5.) This theorem facilitates much the finding of the centres of gravity of figures which are parallel projections of more simple or more symmetrical figures. STRESS ITS INTENSITY. 143 For example, let it be supposed that the centre of gravity of a sector of a circular disc has been found (Case IX. Article 44), and let it be required to find the centre of gravity of a sector of an elliptic disc. In fig. 99, let A B' A B' be the ellipse, ADA = 2 a, and B' O B' = 2 h, its axes, and C' O D' the sector whose centre of gravity is required. About the centre of the ellipse, 0, describe the circle, A B A B, whose radius is the serai- axis major a. Through C' and D' respectively draw E 0' C and FD'D, parallel to O B, and cutting the circle in C and D respectively; the circular sector C 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 Ficr. 99. Then the co-ordinates of the centre of gravity G' of the sector of the elliptic disc are (6.) HG'=y'.=^?j 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 \ip 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 PRINCIPLES OF STATICS. The following table 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 Y^-j- One inch of mercm-y (that is, weight of a column of mercury at 32 Fahr., one inch high), 7073 0-4912 One foot of water (at 39-l 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.),... 5-1D625 0-036085 One atmosphere, of 29-922 inches of mercury, or 760 millimetres, 2116-4 147 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, 27847 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 Tlmist 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 STRESS 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 251. In Stress of Varying Intensity, but of One Sign, there is all tension, or all pressure, or all shear in one direction. In fig. 100, let A A be the given plane surface at which the stress acts; O 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 piismatic surface generated by the motion of a straight line parallel to O Z round the outline of A A, and at the other 100 Cn( * ^ a sul 'f ace 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 \vhere_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 by dxdy, and so small that the stress on each is sensibly uniform; the entire area being f f The volume of the ideal solid will be z-dxdy ...................... (2.) So that if it be conceived to consist of a material whose heaviness is w =, the amount of the stress will be equal to the weight of the solid ; that is to say, 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 symbolically, take the straight line in question for the axis Y; conceive the substance to be divided into bands by lines parallel to Y; let y denote the length of one of these bands, and d x its breadth, so that ydx is its area, and S= \ydx the area of the whole sm-face. 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 a being a constant coefficient; then the amount or resultant of the stress is ? = Jpydx = aJxydx; (5.) and the perpendicular distance of the centre of stress from Y is / pxydx x z ydx ; x = J -j = - (6.) jpydx P 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 OP 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 of not tending to preserve a definite shape, and the possession of this property by a body in perfection throughout all its pai'ts, constitutes that body a perfect fluid. A necessary consequence of that property is the following prin- ciple, which is the foundation of the whole science of hydro- statics: I. In a perfect fluid, when still, the pressure exerted at a given point is normal to the surface on which it acts, and of equal intensity for all positions of that surface. The following are some of the most useful consequences of that principle : II. A surface of equal pressure in a still fluid mass is everywhere perpendicular to the direction of gravity ; that is, horizontal through- out. In other words, the pressure at all points at the same level is of equal intensity. III. The intensity of the pressure at the lower of two points in a still fluid mass is greater than the intensity at the higher point, by an amount equal to the iveight of a vertical column of the fluid whose heigM 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 l 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 (1.) In a gas, such as air, w varies, being nearly proportional to p ; but in a liquid, such as water, the variations of w are too small to be considered in practical cases. For example, let the upper of the two points be the surface of a mass of water where it is exposed to the air; then p is the atmos- pheric pressure; let the depth x of the second point below the surface be given in feet, and let the temperature be 39 0> 1; then Pi in Ibs. on the square foot =p$ + 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 143 PRINCIPLES OF STATICS. the excess of the pressure of the water above the atmospheric pressure, which may be thus expressed, p'= p 1 -p = Q2 -425 x nearly .................... (3.) IY. 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 volumo; 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 O 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, dx its breadth, x the distance of its centre from O Y; then y dx is its area, x sin 6 the depth at which it is immersed; and if w be the weight of unity of volume of the fluid, the intensity of the pressure on that band is p = wx sin 6 .............................. (4. ) The whole area of the pressed siirface, being the sum of the areas of all the bands, is S = \y d x; the whole pressure upon it is P = \pydx-w sin 6 I x y d x j ................ (5.) the mean intensity of the pressure is J y d x fxyd j- - Jydx COMPOUND INTERNAL STRESS OP SOLIDS. 149 and the distance of the centre of pressure from O Y is \xpydx Jxydx (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 v measured along the slope, so that its lower edge is immersed to the depth x l sin 9. Then its centre of gravity is immersed to the depth x 1 sin 6 -f- 2, and the mean intensity of the pressure in Ibs. ou the square foot, is P 62 4 a;, sin* . The breadth y is constant; so that the area of the surface is S = x y ; and the total pressure is ..... ................... . The distance of the centre of pressure from the upper edge is *o = f *r>- ...................... (10.) Next, let the upper edge, instead of being at the surface of the water, be at the distance x 2 from it, so as to be immersed to the depth x z sin 6. Then the centre of gravity of the pressed surface is immersed to the depth (x l + x z ] sin + 2, and the meau intensity of the pressure upon it, in Ibs. on the square foot, is P _ 624 (x l + a? 2 ) sin 6 --_- the area of the surface is (a^ - x 2 ] y, and the total pressure on it p = 62-4 (aj-^y sin The distance of the centre of pressure from the line O Y is 254. Compound Internal Stress of Solids. Tf a body be con- ceived to be divided into two parts by an ideal plane traversing it in any direction, the force exerted between those two parts at the plane of division is an internal stress. 150 PRINCIPLES OF STATICS. According to the principles stated in the preceding article, the internal stress at a given point in a fluid is normal and of equal intensity for all positions of the ideal plane of division. In a solid body, on the other hand, the stress may be either normal, oblique, or shearing; and it may vary in direction and intensity, as the position of the ideal plane of division varies. 255. Conjugate Stresses Principal Stresses. If two planes traverse a point in a body, and the direction of the stress on the first plane is parallel to the second plane, then the direction of the stress on the second plane is parallel to the first plane. Such a pair of stresses are said to be conjugate; and if they are both normal to their planes of application (and consequently perpendi- cular to each other) they are called principal stresses. Three con- jugate stresses, or three principal stresses, may act through one point; but in the present treatise it is sufficient to consider two. Fig. 101 represents a pair of conjugate oblique tensions acting in the direction X X and Y Y through a prismatic particle A B D. The rectangular directions in which principal stresses that is, direct pulls and thrusts act, through a given point in a solid, are called axes of stress. In a fluid, the stress at a given point being of equal intensity in all directions, every direction has the property of an axis of stress. A solid may be in the same condition with a fluid as to stress; but it may also have the principal stresses at a given point of different intensities. In a mass of loose grains, the ratio of those intensities has a limit depending on friction : in a firm continuous solid, the principal stresses at a point may bear any ratio to each other, and may be either of the same or of opposite kinds. 256. The Shearing Stress, on two planes traversing a point in a solid at right angles to each other, is of equal intensity. 257. A Pair of Equal and Opposite Principal Stresses; that is, a pull and a thrust of equal intensity acting through a particle of a solid in directions at right angles to each other, are equivalent to a pair of shearing stresses of the same intensity on a pair of planes at right angles to each other, and making angles of 45 with the first pair of planes. 258. Combination of any Two Principal Stresses. PROBLEM. A pair of principal stresses of any intensities, and of the same or opposite kinds, being given, it is required to find the direction and intensity of the stress on a plane in any position at right angles to the plane parallel to which the two principal stresses act. COMBINATION OF ANY TWO PRINCIPAL STRESSES. lol Let X and O Y (figs. 102 and 103) be the directions of the two principal stresses; OX being the direction of the greater stress. Let PI be the intensity of the greater stress; and p z that of the less. Fig. 102. Fig. 103. The kind of stress to which each of these belongs, pull or thrust, is to be distinguished by means of the algebraical signs. If a pull is considered as positive, a thrust is to be considered as negative, and vice versd. It is in general convenient to consider that kind of stress as positive to which the greater principal stress belongs. Pig. 102 represents the case in which p l and p z are of the same kind; fig. 103 the case in which they are of opposite kinds. In all the following equations, the sign of p z is held to be implied in that symbol; that is to say, when p 2 is of the contrary kind to p v the sign applied to its arithmetical value, in computing by means of the equations, is to be reversed. Let A B be the plane on which it is required to ascertain the direction and intensity of the stress, and O N a normal to that plane, making with the axis of greatest stress the angle On N take O M this will represent a normal stress on A B of the same kind with the greater principal stress, and of an intensity which is a mean between the intensities of the two principal stresses. Through M draw P M Q, making with the axes of stress the same angles which O N makes, but in the opposite direction ; that 152 PRINCIPLES OF STATICS. Is to say, take"M"P="M~Q = Ml). On the line thus found set off from M towards the axis of greatest stress, Mii = ^- 1 -~-^ - Join O K. Then will that line represent the direction, and intensity of the stress on A B. In fig. 102, p-L and p 2 are represented as being of the same kind ; and M K is consequently less than O M, so that O li falls on the A A same side of O X with O N; that is to say, n r < x n. In fig. 103, /?! and p 2 are of opposite kinds, M E, is greater than O M, and O R falls on the opposite side of O X to M; that is to say, n r > x n. The locus of the point M is a circle of the radius 9 ^ 2 , and that of the point R, an ellipse whose semi-axes are p^ and p 2 , 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 x and p y denote the original intensities of a pair of principal stresses acting at right angles to each other through one particle of a solid. Suppose that with these there is combined a shearing stress of the intensity q, acting in the same plane with the original pulls or thrusts; it is required to find the new inten- sities and new directions of the principal stresses. To assist the conception of this problem, the original stresses referred to are represented in fig. 104, as acting through a particle of the form of a square prism. The principal stresses, both original and new, are represented as tensions, although any or all of them might be pressings. 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 x ; the arrows B B, the less original tension p y ; 0, 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 FRICTION. 153 principal tension, p^, is represented by the arrows E, E; it deviates to the right of p x through an angle which will be denoted by t. Tl e less new principal tension p l is represented by the arrows F, F ; it deviates through the same angle to the right of p y . Then the intensities of the new principal stresses are given by the equations, Pl= and the double of the angle of deviation by either of the following-, tan 2* = ^_ ? -; or cotan 2 <; =^^ (4.) The greatest value of ( is 45, when p x = p y . The new principal stresses are to be conceived as acting normally on the faces of a new square prism. 260. Parallel Projection of Distributed Forces. In applying the principles of parallel projection to distributed forces, it is to be borne in mind that those principles, as stated in Article 237, are applicable to lines representing the amounts or resultants of distri- buted forces, and not their intensities. The relations amongst the intensities of a system of distributed forces, whose resultants have been obtained by the method of projection, are to be arrived at by a subsequent process of dividing each projected resultant by the projected space over which it is distributed. 261. Friction is that force which acts between two bodies at their surface of contact so as to resist their sliding on each other, and which depends on the force with which the bodies are pressed together. It is a kind of shearing stress. The following law respecting the friction of solid bodies has been ascertained by experiment : The friction which a given pair of solid bodies, with their surfaces in a given condition, are capable of exerting, is simply proportioned 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 aorae substances, especially those whose surfaces are sensibly indented by a moderate pressure, such as timber, the friction between a pair of surfaces which have remained for some time at rest relatively to each other, is somewhat greater than that between the same pair of surfaces when sliding on each other. That excess, how- ever, of the friction of rest over the friction of motion, is instantly destroyed by a slight vibration; so that the friction of motion is alone to be taken into account, as contributing to the stability of a structure. The friction between a pair of surfaces is calculated by multiply- ing the force with which they are directly pressed together, by a factor called the coefficient of friction, which has a special value depending on the nature of the materials and the state of the surfaces. Let F denote the friction between a pair of sur- faces; N, the force, in a direction perpendicular to the surfaces, with which they are pressed together; and / the coefficient of friction; then (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 , and its reciprocal 1 : /, for various materials condensed from the tables of General Morin, and other sources, and arranged in a few comprehensive classes. The values of those constants which are given in the table have reference to the friction of motion. 155 SURFACES. 9 / 1 / Dry masonry and brickwork, Masonry and brickwork with wet 31 to 35 25i 0-6 to 0-7 0-47 1^7 to 1-43 2-1 Masonry and brickwork, with slightly damp mortar, Wood on stone, 36 22 35 to 16 0-74 about 0'4 0-7 to 0'3 1-35 2-5 1-43 to 3-33 27 0-51 1-96 18 0-33 3 Earth on earth, ,, ,, dry sand, clay, 14 to 45 21 to 37 0-25 to 1-0 0-38 to 0-75 4 to 1 2-63 to 1-33 Earth on earth, damp clay, ,, ,, wet clay ,, ,, shingle and gravel, 45 17 35 to 48 14 to 26' 1-0 0-31 0-7 to 1-11 25 to '5 1 3-23 1-43 to 0-9 4 to 2 1 1*0 to 2 2 to -04 5 to 25 26|o to 31 5 to -6 2 to 1-67 t wet, 13i<> to 144 24 to -26 4-17 to 3-85 ll'o 2 5 Metals on elm, dry, 114 to 14 .2 to -25 5 to 4 Bronze on lignum vitae, constantly 3? 05? 20? 28 53 1'89 ypet ... . 184 33 3 15 to 194 27 to '38 3-7 t 2 -86 Leather on metals, dry, 294 56 1-79 , ,, wet, 20 36 278 , ,, greasy, 13 8i 23 15 435 6 '67 8ito lli 15 to '2 6 '67 to 5 . ,, wet and clean,.. I ,, damp and slimy, Smooth surfaces, occasionally ) " 16 8 r 4 to 4 3 14 07 to '08 3-33 7-14 14-3 to 12 -5 Smooth surfaces, continually ) 3 05 20 Smoothest and best greased surfaces, If to 2 03 to -036 33 -3 to 27 '6 PART IV, THEORY OF STRUCTURES. CHAPTER I. SUMMARY OF PRINCIPLES OF STABILITY AND STRENGTH. SECTION 1. OF STRUCTURES IN GENERAL. 262. A Structure consists of portions of solid materials, put together so as to preserve a definite form and arrangement of parts, and to withstand external forces tending to disturb such form and arrangement. As the parts of a structure are intended to remain at rest relatively to each other, the forces which act on the whole structure, and on each of its parts, should be balanced, so that the mechanical principles on which the permanence and efficiency of structures depend for the most part belong to STATICS, or the science of balanced forces. The mate-rials of a structure may be more or less stiff, like stone, timber, and metals, or loose, like earth. In the present chapter are given a summary of mechanical principles applicable to structures. 263. Pieces Joints Supports Foundations. A structure consists of two or more solid bodies, called its pieces, which touch each other and are connected at portions of their surfaces, called joints. This statement may appear to be applicable to structures of stiff materials only; but, nevertheless, it comprehends masses of earth also, if they are considered as consisting of a very great number of vei-y small pieces, touching each other at innumerable joints. Although the pieces of a structure are fixed relatively to each other, the structure as a whole may be either fixed or movable relatively to the earth. A fixed structure is supported on a part of the solid material of the earth, called the foundation of the structure; the pressures by STABILITY, STRENGTH, AND STIFFNESS. 157 vbich the structure is supported, being the resistances of the vuiious parts of the foundation, may be more or less oblique. A movable structure may be suppoi'ted, as a ship, by floating in water, or as a carriage, by resting on the solid ground thi-ough wheels. When such a structure is actually in motion, it partakes to a certain extent of the properties of a machine; and the deter- mination of the forces by which it is supported requires the con- sideration of kinetic as well as of statical principles; but when it is not in actual motion, though capable of being moved, the pressures which support it are determined by the principles of statics; and it is obvious that they have their resultant equal and directly opposed to the weight of the structure. 264. The Conditions of Equilibrium of a Structure are the three following: I. That the forces exerted on the whole structure by external bodies shall balance each other. The forces to be considered under th's head are (1.) the Attraction of the Earth that is, the weight of the structure; (2.) the External Load, arising from the pressures exerted against the structure by bodies not forming part of it nor of its foundation; (these two kinds of forces constitute the gross or total load); (3.) the Supporting Pressures, or resistance of the foundation. Those three classes of forces will be spoken of together as the External Forces. II. That the forces exerted on each piece of the structure shall balance each other. These consist of (1.) the Weight of the piece, and (2.) the External Load on it, making together the Gross Load; and (3.) the Resistances, or forces exerted at the joints, between the piece under consideration and the pieces in contact with it. III. That the forces exerted on each of the parts into which each- piece of the structure can be conceived- to be divided shall balance each other. Suppose an ideal surface to divide any part of any one of the pieces of the structure from the remainder of the piece; the forces which act on the part so considei'ed are (1.) its weight, and (2.) (if it is at the external surface of the piece) the external force applied to it, if any, making together its gross load; (3.) the stress, or force, exerted at the ideal surface of division, between the part in question and the other parts of the piece. 265. Stability, Strength, and Stiffness. It is necessary to the permanence of a structure, that the three foregoing conditions of equilibrium should be fulfilled, not only under one amount and one mode of distribution of load, but under all the variations of the load as to amount and mode of distribution which can occur in the use of the structure. Stability consists in the fulfilment of foe first and second condi- tions of equilibrium of a structure under all variations of the load within given limits. A structure which is deficient in stability 158 THEORY OP STRUCTURES. gives way by the displacement of its pieces from their proper posi- iions. When a structure, or one of its parts, is flexible, like the chain of a suspension bridge, or in any other way free to move, its stability consists in a tendency to recover its original figure and position after having been disturbed. Strength consists in the fulfilment of the third condition of equi- librium of a structure for all loads not exceeding prescribed limits; that is to say, the greatest internal stress produced in any part of any piece of the structure, by the prescribed greatest load, must be such as the material can bear, not merely without immediate breaking, but without such injury to its texture as might endanger its breaking in the course of time. A piece of a structure may be rendered unfit for its purpose, not mei'ely by being broken, but by being stretched, compressed, bent, twisted, or otherwise strained out of its proper shape. It is neces- sary, therefore, that each piece of a structure should be of such dimensions that its alteration of figure under the greatest load applied to it shall not exceed given limits. This property is called stijftiess, and is so connected with strength that it is necessary to consider them together. SECTION 2. BALANCE AND STABILITY OF FRAMES, CHAINS, AND BLOCKS. 266. A Frame is a structure composed of bars, rods, links, or cords, attached together or supported by joints, such as occur in carpentry, in frames of metal bars, and in structures of ropes and chains, fixing the ends of two or more pieces together, but offering little or no resistance to change in the relative angular positions of those pieces. In a joint of this class, the centre of resistance, or point through which the resultant of the resistance to displacement of the pieces connected at the joint acts, is at or near the middle of the joint, and does not admit of any variation of position consis- tently with security. The line of resistance of a frame is a line traversing the centres of resistance of the joints, and is in general a polygon, having its angles at these centres. 267. A Single Bar in a frame may act as a TIE, a STRUT, or a BEAM. I. A tie has equal and directly opposite forces applied to its two ends, acting outwards, or from each other. The bar is in a state of tension, and the stress exerted between any two divisions of it is a pull, equal and opposite to the applied forces. A rope or chain will answer the purpose of a tie. The equilibrium of a movable tie is stable; for if its angular posi- A SINGLE BAR. 159 tion be deviated, the forces applied to its ends, which originally were directly opposed, now constitute a couple tending to restore the tie to its original position. II. A strut has equal and directly opposite forces applied to its two ends, acting inwards, or towards each other. The bar is in a state of compression, and the stress exerted between any two divi- sions of it is a thrust equal and opposite to the applied forces. It is obvious that a flexible body will not answer the purpose of a strut. The equilibrium of a movable strut is unstable; for if its angular position be deviated, the forces applied to its ends, which originally were directly opposed, now constitute a couple tending to make it deviate still farther from its original position. In order that a strut may have stability, its ends must be pre- vented from deviating laterally. Pieces connected with the ends of a strut for this purpose are called stai/s. III. A beam is a bar supported at two points, and loaded in a direction perpendicular or oblique to its length. CASE I. Let the supporting pressures be parallel to each other and to the direction of the load; and let the load act between the points of support, as in fig. 105; where P s. t represents the resultant of the gross load, in- ^ 1 * eluding the weight of the beam itself; L, the ~W r }\*- point where the line of action of that resultant ||$ j intersects the axis of the beam ; R^ R 2 , the p r W two supporting pressures or resistances of the Fl S- 105 - props parallel to, and in the same plane with P, and acting through the points S 1? S 2 , in the axis of the beam. Then, according to the principle of the lever, Article 2'25, page 128, each of those three forces is proportional to the distance between the lines of action of the other two; and the load is equal to the sum of the two supporting pressures; that is to say, P : R, : R 2 : : S^ 2 : L^, : ITS^; ............ (1.) andP = R 1 + R 2 .......................... (2.) CASE II. Let the load act beyond the points of support, as in fig. 106, which represents a cantilever or project- ing beam, held up by a wall or other prop at S x , held down by a notch in a mass of masonry or s otherwise at S 2 , and loaded so that P is the re- sultant of the load, including the weight of the beam. Then the proportional Equation 1. re- mains exactly as before; but the load is equal to the difference of the supporting pressures; that is to say, P = R 1 -R 2 . ICO THEORY OF STRUCTURES. In these examples the beam is represented as horizontal; but the same principles would hold if it were inclined. CASE III. Let the directions of the supporting forces R 1} R 2 , be now inclined to that of the resultant of the load, P, as in fig. 107. This case is that of the equilibrium of three forces treated of in Article 209, page 122, and consequently the following principles apply to it : The lines of action of the supporting forces and of the resultant of the load must be in one plane. 107 They must intersect in one point (C, fig. 107). Those three forces must be proportional to the three sides of a triangle A, respectively parallel to their directions. PROBLEM. Given, the resultant of the load in magnitude and position, P, the line of action of one of the supporting forces, R x , and the centre of resistance of the other, S 2 ; required, the line of action of the second supporting force, and the magnitudes of both. Produce the line of action of R I( till it cuts the line of action of P at the point C; join C S 2 ; this will be the line of action of R 2 ; construct a triangle A with its sides respectively parallel to those three lines of action; the ratios of the sides of that triangle will give the ratios of the forces. To express this algebraically, let i lt {, be the angles made by the lines of action of the supporting forces with that of the resultant of the load ; then P : R! : R 2 : : sin (r x + ?' 2 ) : sin i 2 : sin t\ (4.) The same piece in a frame may act at once as a beam and tie, or as a beam and strut; or it may act alternately as a strut and as a tie, as the action of the load varies. The load tends to break a tie by tearing it asunder, a strut by crushing it, and a beam by breaking it across. The power of materials to resist those tendencies will be considered in a later section. 268. Distributed Loads. Before applying the principles of the present section to frames in which the load, whether external or arising from the weight of the bars, is distributed over their length, it is necessaiy to reduce that distributed load to an equiva- lent load, or sei'ies of loads, applied at the centres of resistance. The steps in this process are as follows : I. Find the resultant load on each single bar. II. Resolve that load, as in Article 267, Equation 1, page 159, FRAMES OF TWO BARS. 161 into two parallel components acting through the centres of resist- ance at the two ends of the bar. III. At each centre of resistance where two bars meet, combine the component loads due to the loads on the two bars into one resultant, which is to be considered as the total load acting through that centre of resistance. IV. When a centre of resistance is also a point of support, the component load acting through it, as found by step II. of the pro- cess, is to be left out of consideration until the supporting force required by the system, of loads at the other joints has been deter- mined; with this supporting force is to be compounded a force equal and opposite to the component load acting directly through the point of support, and the resultant will be the total supporting force. In the following Articles of this section, all the frames will be supposed to be loaded only at those centres of resistance which are not points of support; and, therefore, in those cases in which components of the load act directly through the points of support also, forces equal and opposite to such components must be com- bined with the supporting forces as determined in the following Ai'ticles, in order to complete the solution. 269. Frames of Two Bars. Figures 108, 109, and 110, repre- sent cases in which a frame of two bars, jointed to each at the point L, is loaded at that point with a given force, P, and is sup- Fig. 109. Fig. 110. ported by the connection of the bars at their farther extremities, Si> S 2 , with fixed bodies. It is required to find the stress on each bar, and the supporting forces at S x and S 2 . Resolve the load P (as in Article 213, page 123) into two com- ponents, R 1( Pv 2 , acting along the respective lines of resistance of the two bars. Those components are the loads borne by the two bars respectively; to which loads the supporting forces at S 15 S 2 , are equal and directly opposed. The symbolical expression of this solution is as follows: Let i 1} ?', be the respective angles made by the lines of resistance of the bars with the line of action of the load; then P : R! : Rj : : sin (^ + ^) : sin i, : sin ^. 1G2 THEORY OF STRUCTURES. The inward or outward direction of the forces acting along each bar indicates that the stress is a thrust or a pull, and the bar a strut or a tie, as the case may be. Fig. 108 represents the case of two ties; fig. 109 that of two struts (such as a pair of rafters abutting against two walls); fig. 110 of a strut, L S 1} and a tie, L S 2 (such as the jib and the tie-rod of a crane). A frame of two bars is stable as regards deviations in the plane of its lines of resistance. With respect to lateral deviations of angular position, in a direction perpendicular to that plane, a frame of two ties is stable; so also is a frame consisting of a strut and a tie, when the direction of the load inclines from the line S x S 2 , joining the points of sup- port. A frame consisting of a strut and a tie, when the direction of the load inclines towards the line Si S 2 , and a frame of two struts in all cases, are unstable laterally, unless provided with lateral stays. These principles are true of any pair of adjacent bars ivhose farther centres of resistance are fixed; whether forming a frame by them- selves, or a part of a more complex frame. 270. Triangular Frames. Let fig. Ill represent a frame, con- sisting of three bars, A, B, 0, connected at the three joints 1, 2, 3, viz., C and A at 1, A arid B at 2, B and C at 3. Let a load P x be applied at the joint 1 in any given direction; let supporting forces, P 2 , P 3 , be applied at the joints 2, 3; the lines of action of those two forces must be in the same plane with that of P I} and must either be parallel to it or intersect it in one point. The latter case is taken first, because its solution comprehends that of the former. The three external forces balance each other, and are therefore proportional to the three sides of a triangle respectively parallel to their clirec- V* tions. In fig. 112, let A B C be such a triangle, in which C A represents P I} A"B P 2 , Fig. 112. B (J P 3 , Draw C parallel to the bar C, and A parallel to the bar A, meeting in the point 0, and join B 0, which will be parallel to B. The lengths of the three lines radiating from O will represent the stresses on the bars to which they are respectively parallel. When the three external forces are parallel to each other, the triangle of forces A B C of fig. 112, becomes a straight line C A, as POLYGONAL FRAME. 1G3 Fig. 113. in fig. 113, divided into two segments by the point B. Lot straight lines radiate from O to A, B, C, respectively pai-allel to the bars of the frame; then if the load A be applied at 1 (fig. Ill), A B applied at 2, and B applied at 3, are the supporting forces required to balance it ; and the radiating lines O A, O B, O 0, represent the stresses on the bars A, B, C, respec- tively, as before. From O let fall O H perpendicular to C A, the common direction of the external forces. Then that line will represent a component of the stress, which is of equal amount in each bar. When C A, as is usually the case, is vertical, O H is horizontal; and the force represented by it is called the "horizontal thrust" of the frame. Horizontal Stress or Resist- ance would be a more precise term ; because the force in question is a pull in some parts of the frame, and a thrust in others. In fig. Ill, A and C are struts, and B a tie. If the frame were exactly inverted, all the forces would bear the same proportions to each other; but A and C would be ties, and B a strut. The trigonometrical expression of the relations amongst the forces acting in a triangular frame, under parallel forces, is as follows : Let a, b, c, denote the respective angles of inclination of the bars A, B, C, to the line H (that is, in general, to a horizontal line); viz., the angles A O H, B O H, C O H of fig. 113, then Horizontal Stress OH- load A tan c tan a, (1.) Supporting / A B = O H (tan a + tan 6) ; Forces \ BC = OH-(tan& tanc); Th \e 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, sucli as is formed by A, B, 0, and D, when inverted, is called by mathematicians, a funicular polygon, because it may be made of ropes. It is to be observed, that the stability of an unstayed polygon of ties is of the kind which admits of oscillation to and fro about the position of equilibrium. That oscillation may be injurious in practice, and stays may be required to prevent it. 274. Bracing of Frames. A 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 I/raced frames contains an improvement sug- gested by Prof. Clerk Maxwell in 1S67. BRACING OF FRAMES. 167 Fig. 118. occurs in iron roofs, consisting of two struts or rafters, A and E, and three tie-bars, B, C, and D, form- ing a polygon of five sides, jointed at ~ 1, 2, 3, 4, 5, loaded vertically at 1, and a 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 b, 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 0. To each joint in the frame, fig. 117, there corre- sponds, in fig. 118, a triangle, or other closed polygon, having its sides respectivelv parallel, and therefore proportional, to the forces that act at that joint. For example, Joints, 1, 2, 3, 4, 5, Polygons, EAaceE; AB6A; Bc&B; DdcD; D E eD. The order of the letters indicates the directions in which the forces act relatively to the joints. Another method of treating simple cases of bracing is illustrated by fig. 119. A and B are two struts, forming the two halves of Fig. 119. Fig. 120. 163 THEORY OP STRUCTURES. one straight bar; C and D are two equal tie-rods ; E, a strut brace. A vertical rod P is applied at the joint 1, between A and B; two vertical supporting pressures, each denoted by R = P -i- 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 B A parallel to A and B, O C parallel to C, and D parallel to D. Make D = O C; join CD; this line will be parallel to the brace E, and perpendicular to O A. Through D and C draw vertical lines D B, C A ; these, being equal to each other, are to be taken to represent the two sup- porting pi-essures B, ; and their sum D B + A C will represent the load P. The equal tensions on C and D will be represented by O C and D, 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 Bare 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 **\ IT' the joints C, C, by the rods C B, C B, lig. 121. These rods also serve to make the re- sistance of the tic-beam A A to being bent act so as to prevent the SECONDARY AND COMPOUND TRUSSING. 1G9 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 tniss 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. 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 ihe 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 143; it consists of a strut, 1 3 (the rafter itself), 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 123. Let W denote the weight of the roof; then ^ "VV 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, eacli equal to ^ W, acting through the ends of the rafter; then ^ \V is to be considered as directly supported at 3, W at 2, and W + W = ^ W at 1 ; therefore the load at the joint 1 is Let * be the inclination of the rafters to the horizon ; then by the equations of Article 270. 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 , n . (2.) Secondary Truss 1435. 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 r. gainst 4, a thrust equal to the component perpendicular to the rafter of the load Wj which thrust is given by the equation E M = i Wcost ........................ (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. 1 = 8 W C0tan W cosec . = 1 w (cogec . _ _ (4.) The difference between the thrusts on the two divisions of the rafter, R 35 -R 51 = | W sin i, id the component along the rafter of the load at the point 5. (3.) Smaller Secondary Trusses, 1 7 5, 5 6 3. These trusses are similar in every respect to the larger secondary trusses, except that the load on each point is one-half, and consequently each of the stresses is reduced to one-half of the corresponding stress in the Equations 3 and 4. (4.) Itesultant Stresses. The pull on the middle division of the great tie-rod 2 3 is simply that due to the primary truss, 123. The pull on the tie 4 7 is simply that due to the secondary truss 143. The pulls on the ties 5 7, 5 G, are simply those due to the smaller secondary trusses, 1 5 7, 5 6 3. But agreeably to the Theorem stated at the commencement of this article, the pull on the tie 1 7 is the sum of those due to the larger secondaiy truss 143, and the smaller secondary truss 175. The pull on 6 4 is the sum of those due to the primary truss 123, and to the larger secondary truss 143. The pull on 6 3 is the sum of those due to the primary truss 1 2 3, to the larger secondary truss 143, and to the smaller secondary truss 5 G 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 completely divided into two parts by an ideal surface, the stresses along the bars lokich are intersected by that 172 TIIEORY OF STRUCTURES. surface, lalance the external forces which act on each of the two parts 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 plane of the frame. Take the vertical line of intersection of these two planes for an axis of co- ordinates, say for the axis of y, and any convenient point in it for the origin 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 # = 2 ' X; the total force along y 2 * Y; and the moment of the couple acting round 3 = 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 problem is or may be indeterminate. The formulae to which this reasoning leads are as follows : Let x be positive in a direction from the plane of section towards the part of the structure which is considered in determining 2 X, 2 Y, and M; let 4- y be 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 plane of section, make the angles i v i z , i s , with x. Let n v n z , n 3 , be the perpendicular dis- tances of those three lines of resistance from 0, distances lying upwards } ~ ^ -, . ., , f positive , r -, > from (J x being considered as < ' , . downwards J { negative. Let R 1; R 2 , R s , be the 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-^ + R 9 cos ? 9 + R 3 cos ?' 3 ; ^ 2 Y = Rj sin i^ + R 9 sin ?' 2 +.E 3 siu ? 3 ; > (1.) - M = R! TO! + R 2 n 2 + R 3 n s ; ) from which the three quantities sought, R p R 9 , R 3 can be found. Speaking with reference to the given plane of section, 2 X may be called the normal stress, 2 Y, the shearing stress, and M, the BESISTANCE OF A FRAME AT A SECTION*. 173 moment of flexure, or bending stress; for it tends to benrl the frame at the section under consideration. M is to be considered as I negative f accor< ^ n o as ^ tends to make the frame become con- upwards ) downwards. J The following is one of the simplest examples of the solution of a problem by the method of polygons, and the met/tod of sections. Fig. 121 represents a truss of a form very common in carpentry (already referred to in Article 275), and consisting of three struts, A C, C 0, C A, a tie-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 A C, 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 : ) R = - P cosec i. f ) (2.) (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 x v y v 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 F x = 0; F,= -P M = - P. 174 THEORY OF STRUCTURES. Then, observing that for the strut AC, n = 0, and that for the tie A A, n t/ v we have, by the equations 1 of this Article K cos i + H = F, = ; R sin i = - P Hf i .-M-+P lJ whence we obtain, from the last equation, from the first, or from the second } ............... (3.) R = - . - - P cosec i cos \ ) 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 = F a = 0, and T= -H= -Pcotani ..................... (4.) In the example just given, the method of sections is tedious and complex as compared with the method of polygons, and is intro- duced for the sake of illustration only. 278. Balance of a Chain or Cord. A loaded chain may be looked Tipon as a polygonal frame whose pieces and joints are so numerous that its figure may without sensible error be treated as a continuous curve. The following are the principles respecting the equilibrium of loaded chains and cords which are of most importance in practice. I. Balance of a Chain in general. Let D A C, in fig. 123, repre- sent a flexible cord or chain supported at the points C and D, and 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 ^~~ l 5? 3T~ two points in this chain ; * 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 chain 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, C D, fig. 124, to represent the total load, and from its ends draw C O and D 0, parallel to two tangents at the points of support of the chain, and meeting in 0; 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 A; this will represent the horizontal component of the tension of the chain at every point, and if O B be parallel to a tangent to the chain at B (fig. 123), A B will repi-esent the portion of the load sup- ported between A and B, and O B the tension at B. To express this algebraically, let H = O A = horizontal tension along the chain at A; E, B = pull along the chain at B; P = A B = load on the chain between A and B; = ZX PB (fig. 123) = Z A O B(fig. 124) = inclination of chain at B; then, P = Htani;R= J (P 2 + H 2 ) = H sec i (1.) To deduce from these formula? an equation by which the form of the curve assumed by the chain can be determined when the dis- tribution of the load is known, let that curve be referred to rect- angular, horizontal, and vertical co-ordinates, measured from the lowest point A, fig. 123, the co-ordinates of B being, AX = a?, X B = y, then tan i- ~- =75. , a differential equation, which enables 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, TO, its components perpendicular and parallel to the joints respectively. Then the conditions of stability are the following: &' 125< I. In orde)' that the block may not slide, the obliquity of the 176 THEORY OF STRUCTURES. pressure must not exceed the angle of repose (Article 251, page 154), that is to say, ^? ................................ (1.) II. In order that the block may be in no danger of overturning, the ratio which the deviation of the centre of pressure from the centre of figure of th? joint bears 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 of stability of friction, the second, that of stability of position. In a structure composed of a series of blocks, or of a series of courses so bonded that each may be considered as one block, which blocks or courses press against each other 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 C x of the joint 1, 1, to be known, and also the amount and direction of the pressure, as Fig. 120. indicated by the arrow traversing C T . 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 C 2 . By continu- ing this process there are found the centres of pressure C 3 , C 4 , &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 i*esultant 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 pressures" 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 joint by more 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 of that joint. Conceive a line to pass through all the limiting positions of the centre of resistance of the joint, so as to enclose a space beyond which that centre must not be found. The product of the iveight 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 the centre of resistance; for that, according to Article 222, is the moment of the couple, which, being combined with a single force equal to the weight of the structure, transfers the line of action of that force parallel to itself through a distance equal to the given horizontal distance of the centre of resistance from the centre of gravity of the structure. The applied couple usually consists of the thrust of a frame, or an arch, or the pressure of a fluid, or of a mass of earth, against the structure, together with the equal, opposite, and parallel, but not directly opposed, resistance of the joint to that lateral force. To express this symbolically, let t be the length of the diameter of the joint where it is cut by the vertical plane traversing the centre of gravity of the structure and parallel to the applied thrust; let./ be the inclination of that diameter to the horizon; let q t be 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 ')tco S j; (1.) the sign < * I being used according as the centre of resistance, 178 THEORY OF STRUCTURES. and the vertical line through the centre of gravity, lie towards { opposite sides ) of the middle of ^ 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-whbt (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'} cosj w hb fi (3.) This quantity is divided by points into three factors, viz. : (1.) n (q q) cosj, a numerical factor, depending 011 ilia figure of the structure, the obliquities of its co-ordinates, and the direction in which the applied force tends to overturn it. (2.) w, the specific gravity of the material. (3.) hbt 2 , a geometrical i'actor, depending on the dimensions of the structure. Now the first factor is the same in all structures having figures of the same class, with co-ordinates of equal obliquity, and exposed to similarly applied external forces; that is say, to all structures whose figures, together with the lines of action of the applied forces, are parallel projections of each other, with co-ordinates of equal obli- quity; hence for any set of structures which fulfil that condition, the moments of stability are pi-oportional 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 stilftiess, 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 : Fracture. ,-,, (Extension Tearing. Elementary j Compression Crushing. ( 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. Ill 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 different kinds as there are different 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 pix>duces, or is liable to produce, tivice, or very nearly twice, the effect, in the shape of stress and strain, which an equal dead load would produce. The mean intensity of the stress pro- duced by a suddenly applied load is no greater than that produced l;y 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 OP SAFETY, IgJ factor of safety in those cases in which the calculation of strength is based, not upon the greatest straining action of the load, but upon a mean straining action, which is exceeded by the greatest straining action in a certain proportion. In such cases the apparent factor of safety is the product obtained by multiplying the real factor of safety by the ratio in which the greatest straining action exceeds the mean. Another class of cases in which the 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 OF SAFETY. 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 : Real Factor of Safety, 6 exceeds Mean s $,.' Effort, nearly. o^tf Steam engines acting against a constant resistance Single engine, ............................... 1'6 9'6 Pair of engines driving cranks at right ) j.^ g.g angles, .................................... j Three engines driving equiangular) J.Q~ g.^ cranks, ................................... j Ordinary cases of varying effort and ) g.Q J2'0 resistance, ................................ } Linesof shafting in inillwork; apparent ] factor of safety for twisting stress | due to motive power, to cover allow- ^ from 18 to 36 ances for bending actions, accidental [ extra load, &c., ........................ \ 182 TIIEORY OP STRUCTURES. Almost all the experiments hitherto made on the strength of materials give coefficients or moduli of ultimata strength; that is, coefficients expressing the intensity of the stress exerted by the most severely strained particles of the material just before it gives way. In calculations for the purpose of designing framework or machinery to bear a given working load, there are two ways of using the factor of safety, one is, to multiply the working load by the factor of safety, so as to determine the breaking load, and use this load in the calculation, along with the modulus of ultimate strength : the other is, to divide the modulus of ultimate strength by the factor of safety, and thus to find a modulus or coefficient of working stress, which is to be used in the calculation, along with the 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 vibi'ations 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 until the piece breaks, care being taken, especially when the breaking point is approached, to increase the load by small qtiantities at a time, so as to get a sufficiently precise result. The proof strength requires much more time and trouble for its determination than the ultimate strength. One mode of approxi- mating to the proof strength of a piece is to apply a moderate load and remove it, apply the same load again and remove it, two or three times in succession, observing at each time of application of the load the strain or alteration of figure of the piece when loaded, by stretching, compression, bending, distortion, or twisting, as the case may be. If that alteration does not sensibly increase by re- peated applications of the same load, the load is within the limit of proof strength. The effects of a greater and a greater load being successively tested in the same way, a load will at length be reached whose successive applications produce increasing disfigurements of the piece; and this load will be greater than the proof strength, which will lie between the last load and the last load but one in the series of experiments.- ELASTICITY OP A SOLID. 183 It was formerly supposed that the production of a set that is, a disfigurement which continues after the removal of the load was a test of the proof strength being exceeded ; but Mr. Hodgkinson shewed that supposition to be erroneous, by proving that in most materials a set is produced by almost any load, how small soever. The strength of bars and beams to resist breaking across, and of axles to resist twisting, can be tested by the application of known weights either directly or through a lever. To test the tenacity of rods, chains, and ropes, and the resist- ance of pillars to crushing, more powerful and complex mechanism is required. The apparatus most commonly employed is the hydraulic press. In computing the stress which it produces, no reliance ought to be placed on the load on the safety valve, or on a weight hung to the pump handle, as indicating the intensity of the pressure, which should be ascertained by means of a pressure gauge. This remark applies also to the proving of boilers by water pressure. From experiments by Messrs. Hick and Luthy 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 culled a " modulus ff pliability" that is to say, Intensity of Stress Modulus of btmness -^ : ; Strain Modulus of Pliability = = : Intensity or otress 287. The Elasticity of a Solid consists of stiffness, or resistance to change of figure, combined with the t>o\ver of recovering the 184 THEORY OP STRUCTURES. original figure when the straining force is withdrawn. If that recovery is complete and immediate, the body is perfectly elastic; if there is a set, or permanent change of figure, after the removal of the straining force, the body is imperfectly elastic. The elasticity of no solid substance is absolutely perfect, but that of many sub- stances is nearly perfect when the stress does not exceed the proof strength, and may be made sensibly perfect by restricting the stress within small enough limits. Moduli or Coefficients of Elasticity are the values of moduli of stiffness when the stress is so limited that the value of each of those moduli is sensibly constant, and the elasticity of the body sensibly perfect. 288. Resilience or Spring is the quantity of mechanical ivork* 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 stiffness, means the length of an imaginary vertical column of the material to which the modulus belongs, whose weight would cause a pressure on its base equal in intensity to the stress expressed by the given modulus. Hence Height of a modulus in feet Modulus in Ibs. on the square foot "Heaviness of material in Ibs. to the cubic foot' Modulus in Ibs. on the square inch Weight of 12 cubic inches of the material' Height of a modulus in inches Modulus in Ibs. on the square inch Heaviness of material in Ibs. to the cubic inch' Height of a modulus in metres _ Modulus in kilogrammes on the square metre Heaviness of material in kilogrammes to the cubic metre* SECTION 2. OP RESISTANCE TO DIRECT TENSION. 290. Strength, Stiffness, and Resilience of a Tie. The word tie is here used to denote any piece iu framing or in mechanism, such * Mecliamcal Work, which will be fully treated of in Part VI., maybe defined as the product of & 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 ?=pS (1.) If the sectional area varies at different points, the least area is to be taken into account in calculations of strength. The elongation of a tie produced by any load, P, not exceeding the proof load, is found as follows, provided the sectional area 13 uniform : Let x denote the original length of the tie, A# the elongation, and = the extension; that is, the proportion .which that elongation bears to the original length of the bar. being the numerical measure of the strain. Let E denote the modulus of direct elasticity, or resistance to stretching. Then = |i 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/'-=-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/ o3-=-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 -f- 2 ; so that the work done in stretching the tie to the proof strain, that is, its resilience or spriny, is /S /_/* Sx ~2~'~E -^'"2~" The coefficient f 2 E, by which one-half of the volume of the tie is multiplied in the above formula, is called the MODULUS OP RESILIENCE. A sudden pull of / S-=-2, or one-half of the proof load, being applied to the bar, will produce the entire proof strain of /'-^-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. y"'S-=-2, through a given spaco, is the same with the work per- formed by the action, through the same space, of a force increasing at an uniform rate from up to/' S. Hence a tie, to resist with safety the sudden application of a given pull, requires to have twice the strength that is necessary to resist the gradual application and steady action of the same pull. This is an illustration of the principle, that the factor of safety for a live load is twice that for a dead load. 291. Thin Cylindrical and Spherical Shells. Let r denote the radius of a thin hollow cylinder, such as the shell of a high-pressure boiler ; t, the thickness of the shell; /, the ultimate tenacity of the material, in pounds per square inch ; p, the intensity of the pressure, in pounds per square inch, re- quired to burst the shell. This ought to be taken at six TIMES the effective working pressure effective pressure meaning the excess of the pressure from within above the pressure from without, which last is usually the atmospheric pressure, of 14*7 Ibs. on the square inch or thereabouts. Then and the proper proportion of thickness to radius is given by tki formula, t > ...(2.) Thin spherical shells are twice us strong as cylindrical shells of the same radius and thickness. The tenacity of good wrought-iron loiler-plates is about 50,000 Ibs. SECTION 3. OF RESISTANCE TO DISTORTION AND SHEARING. 292. Distortion and Shearing Stress in General. In framework and mechanism many cases occur in vhich the principal pieces, such as plates, links, bars, or beams, being themselves subjected to ten- fciou, 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, <>xerted STlll'NGTIl OP A CYLINDRICAL SHAFT. 187 in directions making angles of 45 with the shearing stress. Hence it follows that a body may give way to a shearing stress either by actual shearing, at a plane parallel to the direction of the shearing force, or by tearing, in a direction making an angle of 45 with that force. The mai.nei 1 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 v be the distortion, expressed by the tangent of the difference between each of the distorted angles of the prism and a right OMgle; then f-ct... ....a.) is the modulus of transverse elasticity, or resistance to distortion. One mode of expressing the distortion of an originally square prism is as follows : Let denote the proportionate elongation of one of the diagonals of its end, and the proportionate shorten- ing of the other; then the distortion is C The ratio , of the modulus of transverse elasticity to the modulus Ail of direct elasticity defined in Article 287, page 184, has different values for different materials, ranging from to ^. For wrought- 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 shafts which transmit motive power. The moment is called the twisting moment, and at each cross- section of the bar it is resisted by an equal and opposite moment of stress. Each particle of the shaft is in a state of distortion, and exerts shearing stress. In British measures, twisting moments are expressed in inch-lbs. 294. Strength of a Cylindrical Shaft.- A cylindrical shaft, AB, 188 THEORY OF STRUCTURES. fig. 127, being subjected to the twisting moment of a pair of equal and opposite couples applied to the cross-sections, A and B, it is required to find the condition of stress and strain at any intermediate cross-sec- tion, such as S, and also the angular displacement of any cross-section rela* Fig. 127. tively to any other. From the uniformity of the figure of the bar, and the uniformity of the twisting moment, it is evident that the condition of stress and strain of all cross-sections is the same ; also, because of the circular figure of each cross-section, the condition of stress and strain of all particles at the same distance from the axis of the cylinder must be alike. Suppose a circular layer to be included between the cross-section S, and another cross-section at the longitudinal distance dx 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 S. Then if there be two points at the same distance, r, from the axis of the cylinder, one in the one cross-section and the other in the other, which points were originally in one straight line parallel to the axis of the cylinder, the twisting moment shifts one of those points laterally, relatively to the other, through the distance r d 6. Consequently, the part of the layer which lies between those points is in a con- dition of distortion, in a plane perpendicular to the radius r; aud the distortion is expressed by the ratio d 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 The STRENGTH of the shaft is determined in the following man- ner: Let J (1 A .) 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 MOMENT OF STRESS. 199 and the sum of all such moments, being the MOMENT OP STRESS, or MOMENT OF RESISTANCE of the given cross-section of the beam to breaking across, is given by the formula, <; (2.) or making/ 1 When the breaking load is in question, the coefficient / is what is called the MODULUS OF RUPTURE of the material. When the pi-oof 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 Ibs. on the square inch for wrought iron, 4,500 for cast iron, and from 1,000 to 1,200 for wood. The factor denoted by I in the preceding equation is what is called the " geometrical moment of inertia " of the cross-section of the beam. For sections whose figures are similar, or are parallel projections of each other, the moments of inertia are to each other as the breadths, and as the cubes of the depths of the sections, and the values of y^ are as the depths. If, therefore, b be the breadth and h the depth of the rectangle circumscribing the cross-section of a given beam at the point where the moment of stress is greatest, we may put I = n'bh*, (3.) yi = m'h, (4.) n' and m' being numerical factors depending on the form of section, and making n' + 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 stiffness, and then the unknown quantity is the breadth, b. Sometimes, as when THEORY OP STRUCTURES. the cross-section is circular or square, we have b = h; and then we have A 3 , instead of b A 2 in Equation 5, which is solved so as to give A by extraction of the cube root. The following are the formulae for these calculations: ( " h M tr\ b= ^fv> : < 6 -) and when A = b, h=( .. w ,.(6 A.) EXAMPLES OF THE NUMERICAL FACTORS is EQUATIONS 3, 4, 5 AND 6. Form of Cross-Sections. "-& '=7T -jftr 1 12 1 2 i 6 (including square) II. Ellipse- Vertical axis A, ) Horizontal axis b, > (including circle) ) IIL Hollow rectangle, b h b' h'; -. -* also 'I -formed section, Kt,; where b' is the sum of the breadths of the lateral '" hollows, .I... JT 1 = 0-0491 1 2 jr_ 1 = 0-0982 1 (l 6/A ' 8 \ 1 2 6 V 1 "^! 3 "/ IV. Hollow square . A 2 A'' s ('-^ 1 2 K 1 -*) V. Hollow ellipse h( l -vf) 1 -7T / ft'/t'*\ VI Hollow circle 2 S2 v & A 3 ; 64 \ 1 ~jf) 1 2 '(l ^ VlL Isosceles triangle; base b, height A ; y t measured 32 V A 4 / 1 38 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 tear a proportion to the load which it has to carry, sufficiently great to requii-e 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 OP BEAM. 201 beam of uniform cross-section. Let W' be the external working load, #! its factor of safety, s 2 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 l W' Compute the weight of the beam from that provisional breadth, and let it be denoted by B/ Then ^-, - is the proportion in which the SJL W - s 2 B' gross breaking load exceeds the external part of that load. Conse- quently, if for the provisional breadth b' there be substituted the exact breadth, the beam will now be strong enough to bear both the proposed external load W, and its own weight, which will now be ' and the true gross breaking load will be As the factor of safety for a steady load is in general one-half of that for a moving load, s^ may be made = 2s 2 ; in which case the preceding formulae become 2B ' W ' - In all these formula?, 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: - (7.) the value of m for an uniformly distributed load and rectangular cross-section being ^; and nfhA. being = nfb h 2 , Equation 5, Article 299; I, h and A being the length, depth, and sectional area of the beam, /the modulus of rupture, and n a factor depend- ing on the form of cross-section. The weight of the beam will be expressed by B = w'JA; ........................... (8.) w' being the weight of an unit of volume of the material, and Jc 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 W~8nfh which increases in the simple ratio of the length, if the proportion I x h 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 s. 2 B = W ; that is, (10.) v ' 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 : 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 tenacity. resistance to direct crushing instead of the Blocks whose lengths are less than about once-and-a-half their diameters offer greater resistance to crushing than that given by the rules; but in what proportion is uncertain. The modulus of resistance to direct crushing 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 by 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. v ,.,////// /.,-! JX Fig. 135. Fig. 136. Fig. 137. Fig. 138. II. Crushing ly 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 OP 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. . III. Crushing by bulging, or lateral swelling and spreading of the block which is crushed, is characteristic of ductile and tough materials, such as wrought iron. Owing to the gradual manner in which materials of this nature give way to a crushing load, it is difficult to determine their resistance to that load exactly. That resistance is in general less, and sometimes considerably less, than the tenacity. In wrought iron, the resistance to the direct crush- ing of pillars or struts of moderate length, as nearly as it can be 2 4 ascertained, is from -^ to = of the tenacity. o IV. 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. PART Y, PRINCIPLES OF KINETICS. CHAPTER I. SUMMARY OF GENERAL PRINCIPLES. NATURE AND DIVISION OF THE SUBJECT. THE present Chapter contains a summary of the Principles of Kinetics. 302. Effort ; Resistance ; Lateral Force. Let F denote a force applied to a moving point, and 6 the angle made by the direction, of that force with the dii'ection of the motion of the point. Then, by the principles of Article 215, the force F may be resolved into two rectangular components, one along, and the other across, the direction of motion of the point, viz : The direct force, F cos 6. The lateral force, F sin 0. A direct force is 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 ( - 6) if ti 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 OF 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 weights at a given locality on the earths 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 d t 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 0, as before, be the angle which the unbalanced force F makes with the body's path during an indefinitely short interval, d t. P d t = F cos 6 d t is accelerating impulse if 8 is acute; ~Rd t = F cos (K - 6) d t is retarding impulse if 6 is obtuse ; Q d t = F sin & d t is deflecting impulse. 315. A Deviating Force is one which acts unbalanced in a direc- tion perpendicular to that of a body's motion, and changes that direction without changing the velocity of the body. 316. Centrifugal Force is the force with which a revolving body reacts on the body that guides it, and is equal and opposite to the deviating force with which the guiding body acts on the revolving body. In fact, as has been stated in Article 193, every force is an action between two bodies; and deviating force and 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 203 PRINCIPLES OF KINETICS. point is the product of the mass of the body into one-half of the square of its velocity, that is to say, it is represented by 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 OP PERCUSSION. 209 be the same as those of the actual body in every position of the body. 326. The subjects to which the principles of kinetics relate will be classed in the following manner: I. Uniform Motion. II. Varied Translation of Points and Rigid Bodies. III. Rotations of Rigid Bodies. IV. Motions of Fluids/. 21Q CHAPTER IT. 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 force and of balance (Articles 55, 56); at the same time it is to be observed, that the framing of those definitions has been guided by experimental knowledge. 211 CHAPTER III. ON THE VARIED TRANSLATION OF POINTS AND RIGID BODIES. SECTION 1. LAW OP 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 'momentum equal to the impulse producing it. (See Articles 330, 331.) 329. General Equations of Dynamics. To express the second law of motion algebraically, two methods may be followed : the first method being to resolve the change of momentum into direct variation and 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 d v d z s m d-t = m 'di*'> and from Articles 77 and 78, that the rate of deviation of its momentum is 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): P or - E ^F (cs 8 = m r = m ~Tfi>. ............... 0-) Q^F sin * = ............... ............. (2,) 212 PRINCIPLES OF KINETICS. The radius of curvature r is in the direction of the deviating force Q. /Second method. As in Article SO, let the velocity of the body be resolved into three rectangular components, -j, ~^, ^ ; so that the three component rates of variation of its momentum are d 2 x d* ?/ d 2 z Also let the unbalanced force F, making the angles *, /3, y, with the axes of co-ordinates, and its impulse per unit of time, be resolved into three components, F*, F y , F z . Then we obtain d 2 x F z = F cos * = m - F. = Fcosy = m-^; three equations, which are substantially identical with the Equa- tions 1 and 2. 330. Mass in Terms of Weight. A body's own weight, acting unbalanced on the body, produces velocity towards the earth, increasing at a rate per second denoted by the symbol g, whose numerical value is as follows: Let x denote the latitude of the place, h its elevation above the mean level of the sea, ^ = 32*1695 feet, or 9-8051 metres, per second; being the value of g for A 45 and h 0, and R = 20900000 feet, or 6370000 metres, nearly, being the earth's mean radius; then g = ffl (! -0-00284 cos 2 A) (l ~) ............ (1.) For latitudes exceeding 45, it is to be borne in mind that cos 2 A 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 = 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 forc3 F, the change of velocity in the direction of F produced in a second will be F F. m ~ W ' whence 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 weight is employed as the unit of force, instead of the absolute unit, the corresponding unit of mass becomes g times the unit just mentioned: that is to say, in British measures, the mass of 32-2 Ibs. ; 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 V Q be the velocity of the body at the beginning of an interval of time t, its velocity at the end of that time is the mean velocity during that time is VQ + V g t , g v arid the vertical height fallen through is h = v,t + g -f-.. ...M.\ 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 by v-v = gt, 214 PRINCIPLES OF KINETICS. the acceleration, aud compared with Equation 3; giving the follow- ing results : When the body falls from a state of rest, v is to be made = ; so that the following equations are obtained : v-gt;h = g -j- = ^- (5.) " " 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 V Q 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 being a rise equal to the fall due to the initial velocity v . 333. An Unresisted Projectile, or a projectile to whose motion there is no sensible resistance, has a motion compounded of the vertical motion of a falling body, and of the horizontal motion due to the horizontal component of its velocity of projection. In fig. 139, let O represent the point from which the projectile is originally projected in the direction A, making the angle X A = 6 with a, horizontal line X in the same vertical plane with O A. Let horizontal distances parallel to O X be denoted by x, and verti- cal ordinates parallel to Z by z, positive upwards, and negative downwards. In the equations of vertical motion, the symbol h of the equations of Article 332 is to be replaced by - z, because of h Fig. 139. an d z being measured in opposite directions. Let v be the velocity of projection. Then at the instant of pro- jection, the components of that velocity are, horizontal, --, = V Q cos 6; vertical, -j- = V Q sin 6 j AN UNIFORM EFFORT OR RESISTANCE. 215 and after the lapse of a given time t, those components have become ' -T- = v cos 6- constant ; -v sin d t ~ V m Hence the co-ordinates of the body at the end of the time t are horizontal, x = V Q cos 6 t; vertical, the Equations 2 being those of which the differential coefficients are Equations 1, and because t = '^>, those co-ordinates are thus related, x = v cos 6 t ; \ V Q sin 0- t- g ~; \ ............... an equation which shews the path O B C of the projectile to be a parabola with a vertical axis, touching O A in 0. 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 2 -t- cos 2 = 1), from the last form of which is obtained the equation which, being compared with Equation 4 of Article 332, shews that the relation between the 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, 216 PRINCIPLES OF KINETICS. where / 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 f g for g in the equations of Article 332, those equations are transformed into the equations of motion of the body in question, h being taken to represent the distance traversed by it in a positive direction. In the apparatus known by the name of its inventor, Attwood, for illustrating the effect of uniform moving forces, this principle is applied in order to produce motions following the same law with those of falling bodies. Two weights, P and R, of which P is the greater, are hung to the opposite ends of a cord passing over a finely constructed pulley. Considering the masses of the cord and pulley to be insensible, the weight of the mass to be moved is P + E, and the moving force P - E, being less than the weight in the ratio, P-R consequently the two weights move accoitfing 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. 5 - f* - - ' -*? ...... w 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 v 2 W v 2 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 OP ANGULAR VELOCITY. 217 A pair of free bodies attracting each other have both deviated motions, the attraction of each guiding the other; and their devia- tions of momentum are equal in equal times; that is, their devia- tions of motion are inversely as their masses. In a machine, each revolving body tends to press or draw the body which guides it away from its position, in a direction from the centre of curvature of the path of the revolving body; and that tendency is resisted by the strength and stiffness of the guiding body, and of the frame with which it is connected. 336. A Revolving Simple Pendulum consists of a small mass A, suspended from a point C by a rod or cord C A of insensibly small weight as compared with the mass A, and revolving in a circle about a vertical axis C B. The tension of the rod is the resultant of the weight of the mass A, acting vertically, and of its centrifugal force, acting horizontally; and therefore the rod will assume such an inclination that Fig. 140. height B G weight gr radius A~B ~ centrifugal force ~ v z " ' where r = A B. Let n be the number of turns per second of the pendulum; then v = 2 v nr; and therefore, making B C = h, . ,0-8154 foot 9-7848 inches . x = (in the latitude of London) - ^ - = ^T -- '(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, (I-) Let ar be substituted for v in the value of deviating force of Article 335, and that value becomes .... ....(2.) 218 PRINCIPLES OF KINETICS. In the case of a body revolving with uniform velocity in a circle, like the bob A of the revolving pendulum of Article 336, a = 2 if n, where n is the number of revolutions per second, so that (3.) from which equation the height of a revolving pendulum might be deduced with the same result as in the last Article. 338. A Simple Oscillating Pendulum consists of an 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. : wr-G, CA acting along C A, and balanced by the tension of the Fig. 141. rod or cord, and W sin Z_ D C A = W ^=5, O 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, - = 2 z-A /-; and and the following statement shews the connection between a simple oscillating arid revolving pendulum, viz., that the length 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 AXGULA'R 'MOMENTUM. 219 SECTION 2. YARIED 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 those bodies, arid 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 ai-e 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 ti'anslation, 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 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 pf that resultant angular 220 PRINCIPLES OF KINETICS. momentum is called the axis of angular momentum of the system. The term angular momentum, was introduced by Mr. Haywavd. 342. Angular Impulse is the product of the moment of a couple of forces (Article 200) into the time during which it acts. Let F be the force of a couple, I its leverage, and dt the time during which it acts, then Fldt 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 dt; and let I be the perpendicular distance from the fixed point to the line of action of the first force. Then the couple in question exerts the angular impulse 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 so that relatively to the fixed point, the variation of the body's angular momentum is 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 the 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 eftect the equilibrium or motion of the system. Also, the resultant of all the couples thus formed is nothing; therefore, the COLLISION. 221 resultant of their angular impulses is nothing; therefore, the 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 dii-ection 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 mi, m z , be the masses of the two bodies, and u^ u 2 , their velocities before the collision, whose directions should be indicated by their signs. Then the velocity of their common centre of gravity is and tiiis is not altered by the collision. 222 CHAPTER IT. ROTATIONS OF KIGID BODIES. 346. The Motion of a Rigid Body, or of a body which sensibly preserves the same figure, has already been shewn in Part I., Chapter II., to be always capable of being resolved at each instant into a translation and a rotation; and by the aid of the principles explained in Section 3 of that chapter, the component rotation can always be conceived to take place abont 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 carved 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 II., Section 3. SECTION 1. ON MOMENTS OF INERTIA, EADII OF GYRATION, AND CENTRES OF PERCUSSION. 347. The Moment of Inertia of an indefinitely small body, or " physical point," relatively to a given axis, is the product of the mass of the body, or of some quantity proportional to the mass, such as the weight, into the square of its perpendicular distance from the axis : thus in the following equation : ?=,*=^r,... ....(i.) 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 1 4- g; the former, when the unit is the moment of inertia of an unit of weight at the end of an arm, whose length is nnity ; 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 of feet. 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, (1.) 349. The Moment of Inertia of a Rigid Body is the sum of the moments of inertia of all its parts, and is found by integration; that is, by conceiving the body to be divided into small parts of regular figure, multiplying the mass of each of those parts into the square of the distance of its centre of gravity from the axis, adding the products together, and finding the value towards which their sum converges when the size of the small parts is indefinitely diminished. For example, let the body be conceived to be built up of rectangular molecules, whose dimensions are d x, d y, and d z, the volume of each d x d y d z, and the mass of unity of volume w. Then r'wdxdydz (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 z for 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, 224 PRINCIPLES OF KINETICS. When symbols of integration ai-e used, tins becomes / / \i s w'dxdydz I / \w-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 x is that is to say, the, moment of inertia, of a body round a given axis may be found by adding together the sum of the products of the masses of the particles, each multiplied by the square of each of its distances from a pair of planes cvtting 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 1^ = 2- Wz' + S-Wz 2 ; I 2 = 2- War' + S-W*, 2 ...... (2.) 352. Moments of Inertia Hound 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 I be the moment of inertia of a body about an axis traversing its centre of gravity in any given direction, and I the moment of inertia of the same body about an axis parallel to the former at the perpendicular distance r n : then 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 OP GYRATION. 225 given axis ( ?0 ), and to the perpendicular distance between these axes (r ). That is to say, (2.) COROLLARY II. The moment of inertia of a body about an axis traversing its centre of gravity in a given direction, is less than the moment of inertia of the same body about any other axis parallel to the first. COROLLARY III. The moments of inertia of a body about all axes parallel to each other, which lie at equal distances from its centre of gravity, are equal. 353. Combined Moments of Inertia. THEOREM. 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, I its moment of inertia about an axis traversing its own centre of gravity parallel to the given common axis, and r the distance of its centre of gravity from that common axis. Then the moment of inertia of that body about the common axis, according to Article 352, Equation 1, is Consequently, the combined moment of inertia of the system of bodies is (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 $o 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 I gives the moment of inertia ; that headed &, the square of the radius of gyration. The mass of an unit of volume is in each case denoted by w. Q 226 PRINCIPLES OP KINETICS. BODT. AXIS. W ID 5 I S here of radius r 4*iw 8T W r 5 2r 2 II. Spheroid of revolution polar semi-axis a, equa- 3 43-war 2 15 5 III. Ellipsoid semi-axes, a, I c Axis 2a 3 15 5 IV. Spherical shell external 3 42 r 2 a 2 2 , be drawn parallel to O X, so as to form a stepped or serrated outline, consisting of lines pai-allel to O X and 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 AC, 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 ; and the whole work performed, on the same supposition during the whole motion 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 I'ectangles is an approximation to the real mode of variation represented by the continuous line E G, and is a closer approximation the closer and the more numerous the parallel ordinates are, so the sum of the rectangles is an approximation to the exact representation of the work performed against the conti- nuously varying resistance, and is a closer approximation the closer and more numerous the ordinates are, and by making the ordinates numerous and close enough, can be made to differ from the exact representation by an amount less than any given difference. But the sum of those rectangles is also an approximation to the area O E G F, bounded above by the continuous line E G, and is a closer approximation the closer and the more numerous the ordi- nates are, and by making the ordinates numei-ous and close enough, can be made to differ from the area O E G F by an amount less than any given diffei-ence. Therefore the area E G F, bounded by the straight line O F, which THE WORK PERFORMED AGAINST FRICTION. 251 represents the motion, by the line E G, whose ordinates represent the values of the resistance, and by the two ordinates O E 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 OT. 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 woi'k 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 fiat pivot, the leverage is two-thirds of the radius of the pivot. For a collar, let r and r be the inner and outer radii; then the leverage of the friction is 2 r*-r* 252 THEORY OF MACHINES. In the cup and ball pivot, the end of the shaft, and the step on which it presses, present two recesses facing each other, into which ai'e fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere, or a lens having convex surfaces of a some- what less radius than the concave surfaces of the cups. The moment of friction, of this pivot is at first almost inappreciable, from the extreme smallness of the radius of the circles of contact of the ball and cups; but as they wear, that radius and the moment of friction increase. By the rolling of two surfaces over each other without sliding, a resistance is caused, which is called sometimes " rolling friction," but more correctly rolling resistance. It is of the nature of a couple resisting rotation; its moment is found by multiplying the normal pressure between the rolling surfaces by an arm whose length depends on the nature of the rolling surfaces; and the work lost in an unit of time in overcoming it is the product of its moment by the 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., with good mineral ungiients, 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 body. Then the force must act upon the driven body in the direction of its motion. Every force being a pair of equal and opposite actions between a pair of bodies, the same force which accelerates the driven body is a resistance as respects the driving body. For example, during the commencement of the stroke of the piston of a steam engine, the velocity of the piston and of its rod is accelerated ; 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 h, 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 1 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 If the acceleration be at any different rate per second, the force necessary to produce that acceleration, being the resistance on the driving body due to the acceleration of the driven body, bears 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, g = 9 - 81 nearly.) To express this by symbols, let the weight of the driven body be denoted by W. Let its velocity at a given instant be t\ feet per second; and let that velocity increase at an uniform rate, so that at an instant t seconds later, it is v 2 feet per second. Let /"denote the rate of acceleration ; then /=^; (2.) 254 THEORY OF MACHINES. and the force R necessary to produce it will be given by the pro- portion, ff :f::W:R; that is to say, R = g 9* w The factor , in the above expression, is called the MASS of the driven body; and being the same for the same body, in what place soever it may be, is held to represent the quantity of matter in the body. (See Article 195, page 117 ) The product - of the mass of a body into its velocity at any instant, is called its MOMENTUM ; so that the resistance due to a given acceleration is equal to the increase of momentum divided \>y 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 R varies along with it. In this case, the value, at a given instant of the rate of acceleration, is represented by / = , and the cr- responding value of the force is (4) 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 by the increase of the half-square of its velocity. To express this by symbols, in the case of an uniform rate of acceleration, let s denote the distance moved through by the driven body during the acceleration ; then ......................... (5.) WORK OF ACCELERATION. 255 which being multiplied by Equation 3, gives for the work of accele- ration, fa?UfcJS',JM?l ( ,) In the case of a variable rate of acceleration, let v denote the mean velocity, and ds the distance moved through, in an interval of time d t so short that the increase of velocity dv is 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, - g dt W = v dv; ............................ (8.) 9 and the integration of this expression (see Article 29) gives for the work of acceleration during a finite interval, 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 v 2 , and not on the intermediate changes of velocity. If a body falls freely under the action of gravity from a state of rest through a height h, so that its initial velocity is 0, and its final velocity v, the work of acceleration performed by the earth on the body is simply the product W h of the weight of the body into the height of fall. Comparing this with Equation 6, we find This quantity is called the height, or fall, due to the velocity v ; and from Equations 6 and 9 it appears that the work performed in producing a given acceleration is the same ivith that performed in Lifting the driven body through the difference of the heights due to Us 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. 256 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 fW vl-vl) \ 9 "T~ J 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 a lt to the value a 2 , the increase of the velocity of the particle in question is v a -v 1 = r(a a ^a 1 ) ........................ (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 inci'ease 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, 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 pai'ticle of the body; and the rate of acceleration produced by gravity, g 32 - 2 is a common divisor. It is therefore sufficient to add together the EEDUCED INERTIA. 257 products of the weight of each particle (w) into the square of its radius-vector (r 2 ), and to multiply the sum so obtained (2 w r 2 ) by the increase of the half-square of the angular velocity y(aj - a?)Y and divide by the rate of acceleration due to gravity (g). The result, viz.: is the work of acceleration sought. In fact, the sum 2 w r 2 is t/te weight of a body, which, if concentrated at the distance unity from (Jte axis of rotation, would require the same work to produce a given increase of angular velocity which the actual body requires. 405. Keduced Inertia. If in a certain machine, a moving piece whose weight is \V 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 reqxiired if a weight n- 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 (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 ^ is the initial, and r 2 the final velocity of the driving point, the work of accelera- tion of the whole machine is This operation may be called the reduction of the inertia to the driving point. Mr. Moseley, by whom it was first introduced into the theory of machines, calls the expression (1.) the "coefficient of steadiness." In finding the reduced inertia of a machine, the mass of each rotating piece is to be treated as if concentrated at a distance from its axis equal to its radius of gyration j; so that if v represents the velocity of the driving point at any instant, and a the corresponding angular velocity of the rotating piece in question, we are to make in rerforming the calculation expressed by the formula (1.) 258 THEORY OP MACHINES. 406. Summary of Various Kinds of Work. In order to present at one view the symbolical expression of the various modes of per- forming work described in the preceding articles, let it be supposed that in a certain interval of time d t the driving point of a machine moves thi'ough 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 R, 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 ~ t undergoes the acceleration d v. Then the whole work performed during the interval in ques- tion is d/i-SW + ds-SwIU 2 n' 2 W ............ (1.) The mean total resistance, reduced to the driving point, may be Computed by dividing the above expression by the motion of the driving point d s = v d t, giving the following result : g dt SECTION 2. OF ENERGY, POWER, AND EFFICIENCY. 407. Condition of Uniform Speed. According to the first law of motion, in order that a body may move unifoi'inly, the forces applied to it, if any, must balance each other; and the same principle holds for a machine consisting of any number of bodies. When the direction of a body's motion varies, but not the velocity, the lateral force required to produce the change of direction depends on the principles set forth in Article 335; but the condition of balance still holds for the forces which act along the direction of the body's motion, that is, for the efforts and resistances ; so that, whether for a single body or for a machine, the condition of uniform velocity is, that the efforts shall balance the resistances. In a machine, this condition must be fulfilled for each of the single moving pieces of which it consists. It also follows, from the principles of statics, that in any body, system, or machine, that condition is fulfilled when the sum of the products of the efforts into the velocities of their respective points of action is equal to the sum of the products of the resistances into the velocities of the points where they are overcome. ENERGY - POTENTIAL ENERGY. 259 Thus, let v be the velocity of a driving point, or point where an effort P is applied; v' the velocity of a working point, or point where a resistance E is overcome; the condition of uniform velocity for any body, system, or machine is 2-Pv = 2-R w '. ............... . .......... (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 v ; reducing the above equation to Pv = 2-Rv' .......................... (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 ai-e 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 then the condition of uniform speed may be thus expressed : P-2-wR; ............................ (3.) that is, the total effort is equal to the sum of the resistances reduced to the driving point. 408 Energy Potential Energy. Energy means capacity for performing work, and is expressed, like work, by the product of a force into a space. The energy of an effort, sometimes called "potential energy" (to distinguish it from another form of 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 case admit of its descending, that weight is said to possess potential energy to the amount of lOOx 20 = 2,000 foot-pounds; which means, that in descending from its actual elevation to the lowest point of its course, the weight is capable of performing work to that amount. To take another example, let there be a reservoir containing 10,000,000 gallons of water, in such a position that the centre of gravity of the mass of water in the reservoir is 100 feet above the lowest point to which it can be made to descend while overcoming resistance. Then as a gallon of water weighs 10 Ibs., the weight of the store of water is 100,000,000 Ibs., which being multiplied by the height through which that weight is capable of acting, 100 feet, gives 10,000,000,000 foot-pounds for the potential energy of the weight of the store of water. 2GO THEORY OF MACHINES. 409. Equality of Energy Exerted and Work Performed, or the Conservation of Energy. When an effort actually does drive its point of application through a certain distance, energy to the amount of the product of the effort into that distance is said to be exerted ; and the potential energy, or energy which remains capable of being exerted, is to that amount diminished. When the energy is exerted in driving a machine at an uniform speed, it is equal to the work performed. To express this algebraically, let t denote the time during which the energy is exerted, v the velocity of a driving point at which an effort P is applied, s the distance through which it is driven, v' the velocity of any working point at which a resistance R is overcome, s' the distance through which it is driven ; then s = v t ; s = v' t ; and multiplying Equation 1 of Article 407 by the time t, we obtain the following equation : -Ru'* = 2-Ps = :2 Rs'; ............ (1.) which expresses the equality of energy exerted, and work per- formed, for constant efforts and resistances. When the efforts and resistances vary, it is sufficient to refer to Articles 400 and 29, to shew that the same principle is expressed as follows : 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 ACCELERATING 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 effort required to produce a given acceleration d v in a body whose weight is W, when the time d t 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 : W vdv W d(v*} '' 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 2 ) _ I a d a Let d t be the time, and diadt the angular motion in which that acceleration is to be produced ; let P be the accelerating effort, and I its leverage, or the perpendicular distance of its line of action from the axis; then, according as the time dt, or the angle di, is given, we have the two following expressions for the accelerating couple: _I ada_ I d(a-) ^g' di ~ g2di"' 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 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 f, in which the driving point moves through the distance ds = vdt, 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 262 THEORY OF MACHINES. both of which are derived from the equation P ds = P v dt = v dv ( 2 9 413. Work During Retardation 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 R, 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, The total actual energy of a system of bodies, each moving with its own velocity, is denoted by 9 and when those bodies are the pieces of a machine, whose velocities A RECIPROCATING FORCE. 263 bear definite ratios (any one of which is denoted by n) to the velo- city of the driving point v, their total actual energy is ~'^W, ........................... (3.) being the product of the reduced inertia (or coefficient of steadiness, as Mr. Moseley calls it) into tlie height due to the velocity of tfa driving point. The actual energy of a rotating body whose angular velocity is a, and moment of inertia 2 "W r 2 = I, is that is, the product of the moment of inertia into the height due to th? velocity, a, of a point, 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 i* to be taken into account. For example, if it be wished to determine how many turns a wheel of a locomotive engine, i-otating 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 engiae, and it is inquired how high it will ascend up a perfect'^ 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 effort and as an equal and opposite resistance, according to the direction of motion of the body. Such a force is the weight of u 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 2G4 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 /* a height through which the common centre of gravity of them all is alternately raised and lowered. Then the quantity of energy A2W, is stored while the centre of gravity is rising, and restored while ife 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 perfoi-ming 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 tlte resistances, exactly as if the speed were uniform and the reciprocating forces null; and all the equa- tions of Articles 407 and 409 are applicable to periodic motion, pro- vided that in the equations of Article 407, and Equation 1 of Article 409, P, 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. 2G5 These principles are illustrated by the steam engine. The velo- cities of its moving parts are continually varying, and those of some of them, such as the piston, are periodically reversed in direc- tion. But at the end of each period, called a revolution, or double- stroke, every part returns to its original position and velocity; so that the equality of energy and work, and the equality of the mean effort to the mean resistance reduced to the driving point, that is, the equality of the mean effective pressure of the steam on the 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 iu 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 it* 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- locating 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 Efficiency 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 236 THEORY OF MACHINES. machines is to bring their efficiency and counter-efficiency as near 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 ; R x , the mean xiseful resist- ance; s v the space through which it is overcome in the same interval; R 2 , any one of the wasteful resistances; s. 2 , the space through which it is overcome; then (1.) and the efficiency of the machine is expressed by /o \ " In many cases the lost work of a machine, 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 (3.) 1 + A + 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 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 principle applies to a train of successive machines, each driving that which follows it; and to counter-efficiency as well as to efficiency. 419. Power and Effect Horse Power. The power of a machine THE PRINCIPLE OF VIRTUAL VELOCITIES. 2G7 is the energy exerted, and the effect, the useful work performed, in some interval of time of definite length, such as a second, a minute, an hour, or a day. The unit of power called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour. The effect is equal to the power multiplied by the efficiency ; and the power is equal to the effect multiplied by the counter-efficiency. The loss of power is the dif- ference between the effect and the power. As to the French " Force de Cheval," see Article 392, page 244. It is equal to 0'9863 of a British horse-power ; and a British horse-power is 1-0139 of a French force de cheval. 420. General Equation. The following general equation pi-e- 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; ^ the velocity at the beginning, and v 2 the velocity at the end, of the interval in question, with which a given particle of the machine of the weight W is moving; g, the acceleration which gravity causes in a second, or 32-2 feet per second, or 9*81 metres per second. folds', the work performed in overcoming any resistance during the interval in question ; / P d s, the energy exerted during the interval in question. The second and third terms of the right-hand side, when positive, are energy stored; when negative, energy restored. The principle represented by the equation is expressed in words as follows : The energy exerted, added to the energy restored, is equal to the energy stored added to the work performed. 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 woi'ds as follows: T/te effort is equal to the sum of the resistances reduced to t/te 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 the general equation of Article 420, obtained in the following manner: Let n denote either the ratio borne at a given instant by the velocity of a given working point, where the resistance 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 n denote the corresponding ratio for the mean velocity of a mass whose weight is W, undergoing acceleration or retardation, and either the rate of acceleration 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 n, which the velocity of the mass W bears to that of , , , . . ... .dv'n'dv. dv the driving point, is constant, we may put r- = , where -r (It d t at denotes the rate of acceleration of the driving point; and then the third term of the foregoing expression becomes =- 2 n' 2 W, as gdt in formula 2 of Article 406, page 258. 422. Forces in the Mechanical Powers, Neglecting Friction Purchase. The mechanical powers, considered as means of modi- fying motion only, have been considered in Section 6, Part II., pages 107 to 110. When friction is neglected, any one of the mechanical powers may be regarded as an uniformly-moving simple machine, in which one, effort balances one resistance; and in which, consequently, according to the principle of virtual velocities, or of the equality of energy exerted and work done, tJie effort and resistance are to each other inversely as the velocities along their lines of action of the points where they are applied. In the older writings on mechanics, the effort is called the power, and the resistance the weight; but it is desirable to avoid the use of the word "power" in this sense, because of its being very commonly used in a different sense viz., the rate at which energy is exerted by a prime mover; and the substitution of " resistance" for " weight" is made in order to express the fact, that the principle just stated applies to the overcoming of all sorts of resistance, and not to the lifting of weights only. The weight of the moving piece itself in a mechanical power may either be wholly supported at the bearing, if the piece is FORCES IN THE MECHANICAL POWERS. 2G9 balanced; or if not, it is to be regarded as divided into two parallel components, one supported directly at the bearing, and the other being included in the effort or in the resistance, as the case, may be. The relation between the effort and the resistance in any mechanical power may be deduced from the principles of statics; viz.: In the case of the LEVER (including the wheel and axle], from the balance of couples of equal and opposite moments ; in the case of the INCLINED PLANE (including the wedge and the screw), from the parallelogram of forces; and in the case of the pulley, from the composition of parallel forces. The principle of virtual velocities, however, is more convenient in calculation. The Mai load in a mechanical power is the resultant of the effort, the resistance, the lateral components of the forces acting at the driving and working points, and the weight directly carried at the bearings; and it is equal and directly opposed to the i-e-action of the bearings or supports of the machine. By the purchase of a mechanical power is to be understood the ratio borne by the resistance to the effort, which is equal to the ratio borne by the velocity of the driving point to that of the working point. This term has already been employed in connec- tion with the pulley. The following are the results of the principle of virtual velocities, as applied to determine the purchase in the several mechanical powers : I. LEVER. The effort and resistance are to each other in the inverse ratio of the perpendicular distances of their lines of action from the axis of rotation or fulcrum ; so that the purchase is the ratio which the perpendicular distance of the effort from the axis bears to the perpendicular distance of the resistance from the axis. Under the head of the lever may be comprehended all turning or rocking primary pieces in mechanism which are connected with their drivers and followers by linkwork. II. WHEEL AND AXLE. The purchase is the same as in the case of the lever; and the perpendicular distances of the lines of action of the effort and of the resistance from the axis are the radii of the pitch-circles of the wheel and of the axle respectively. Under the head of the wheel and axle may be comprehended all turning or rocking primary pieces in mechanism which are connected with their drivers and followers by means of rolling contact, of teeth, or of bands. By the "wheel" is to be understood the pitch-cylinder of that part of the piece which is driven ; and by the " axle," the pitch-cylinder of that part of the piece which drives. III. INCLINED PLANE, and IV. WEDGE. Here the purchase, or ratio of the resistance to the effort, is the ratio borne by the whole velocity of the sliding body (represented by B C in fig. 7GE, 270 THEORY OF MACHINES. and C c in fig. 76r, page 109) to that component of the velocity (represented by B D in fig. 76E, and C e in fig. 76r, page 109) which is directly opposed to the resistance : it being understood that the effort is exerted in the direction of motion of the sliding body. The term inclined plane may be used when the resistance to the motion of a body that slides along a guiding surface consists of its own weight, or of a force applied to a point in it by means of a link; and the term wedge, when that resistance consists of a pressure applied to a plane surface of the moving body, oblique to its direction of motion. Y. SCEEW. Let the resistance (R) to the motion of a screw be a force acting along its axis, and directly opposed to its advance; and let the effort (P) which drives the screw be applied to a point rigidly attached to the screw, and at the distance r from the axis, and be exerted in the direction of motion of that point. Then, while the screw makes one revolution, the working point advances against the resistance through a distance equal to the pitch (p); and at the same time the driving point moves in its helical path through the distance J (4 K 2 i 2 +p 2 ); therefore the purchase of the screw, neglecting friction, is expressed as follows : E, length of one coil of path of driving point pitch VI. PULLEY. In the pulley without friction, the purchase is the ratio borne by the resistance which opposes the advance of the running block to the effoi-t exerted on the hauling part of the rope; and it is expressed by the number of plies of rope by which the running block is connected with the fixed block. VII. The HYDRAULIC PRESS, when friction is neglected, may be included amongst the mechanical powers, agreeably to the definition of them given at the beginning of this Article. By the resistance is to be understood the force which opposes the outward motion of the press-plunger; and by the effort, the force which drives inward the pump-plunger. The intensity of the pressure exerted between each of the two plungers and the fluid is the same; therefore the amount of the pressure exerted between each plunger and the fluid is proportional to the area of that plunger; so that the purchase of the hydraulic press is expressed as follows: Pv transverse area of press-plunger t P . transverse area of pump-plunger* STEAM ENGIN2 INDICATOR. 271 and this is the reciprocal of the ratio of the velocities of those plungers, as already shewn in Article 185, page 110. The purchase of a train of mechanical powers is the product of the purchases of the several elementary parts of that train. The object of producing a purchase expressed by a number greater than unity is, to enable a resistance to be overcome by means of an effort smaller than itself, but acting through a greater distance; and the use of such a purchase is found chiefly in machines driven by muscular power, because of the effort being limited SECTION 3. OF DYNAMOMETERS. 423. Dynamometers are instruments for measuring and record- ing the energy exerted and work performed by machines. They may be classed as follows : I. Instruments which merely indicate the force exerted between a driving body and a driven body, leaving the distance through which that force is exerted to be observed independently. II. Instruments which record at once the force, motion, and work of a machine, by drawing a line, straight or curved, as the case may be, whose abscissae represent the distances moved through, its ordinates the resistances overcome, and its area the work per- formed (as in fig. 149, page 249). A dynamometer of this class consists essentially of two principle parts : a spring whose deflection indicates the force exerted between a driving body and a driven body ; and a band of paper, or a card, moving at right angles to the direction of deflection of the spring with a velocity bearing a known constant proportion to the velo- city with which the resistance is overcome. The spring carries a pen or pencil, which marks on the paper or card the required line. The Steam Engine Indicator is an example of this class of instruments. III. Instruments called Integrating Dynamometers, which re- cord the work performed, but not the resistance and motion separately. 424. Steam Engine Indicator. This instrument was invented by Watt, and has been improved by other inventors, especially M'Naught and Richards. Its object is to record, by means of a diagram, the intensity of the pressure exerted by steam against one of the faces of a piston at each point of the piston's motion, and so to afford the means of computing, according to the principles of Articles 395 and 400, first, the energy exerted by the steam in driving the piston during the forward stroke; secondly, the work lost by the piston in expelling the steam from the cylinder during the return stroke; and thirdly, the difference of those quantities, 272 THEORY OF MACHINES. which is the available or effective energy exerted by the steam on the piston, and which, being multiplied by the number of strokes per minute and divided by 33,000 foot-pounds, gives the INDICATED HORSE-POWER. The indicator in a common form is represented by fig. 150. A B is a cylindrical case. Its lower end, A, contains a smaller cylinder, fitted with a piston, which cylinder, by means of the screwed nozzle at its lower end, can be fixed in any convenient position on a tube communicating with that end of the engine-cylinder where the work of the steam is determined. The communication between the engine-cylinder and the indicator-cylinder can be opened and shut at will by means of the cock K. When it is open, the intensity of the pressure of the steam on the engine-piston and on the indi- cator-piston is the same, or nearly the same. The upper end, B, of the cylindrical case con- tains a spiral spring, one end of which is at- tached to the piston, or to its rod, and the other to the top of the casing. The indicator-piston is pressed from below by the steam, and from above by the atmosphere. When the pressure of the steam is equal to that of the atmosphere, the spring retains its unstrained length, and the piston its original position. When the pressure of the steam exceeds that of the atmosphere, the piston is driven outwards, and the spring compressed ; when the pressure of the steam is less than that of the atmosphere, the piston is driven inwards, and the spring extended. The compression or extension of the spring indicates the difference, upward or downward, between the pressure of the steam and that of the atmosphere. A short arm, C, projecting from the indicator piston-rod carries at one side a pointer, D, which shews the pressure on a scale whoso zero denotes the pressure of the atmosphere, and which is graduated into pounds on the square inch both upwards and downwards from that zero. At the other side the short arm has a longer arm jointed to it, carrying a pencil, E. F is a brass drum, which rotates backward and forward about a vertical axis, and which, when about to be used, is covered with a piece of paper called a " card." It is alternately pulled round in one direction by the cord H, which wraps on the pulley G, and pulled back to its oi-iginal position by a spring contained within itself, The cord H is to be connected with the mechanism of the steam engine in any convenient manner which shall ensure that the velocity of rotation of the drum shall at every instant bear a Fig. 150. STEAM EXGIKE INDICATOR. 273 constant ratio to that of the steam engine piston : the back and forward motion of the surface of the drum representing that of the steam engine piston on a reduced scale. This having been done, and before opening the cock K, the pencil is to be placed in con- tact with the drum during a few strokes, when it will mark on the card a line which, when the card is afterwards spread out flat, becomes a straight line. This line, whose position indicates the pressure of the atmosphere, is called the atmospheric line. In fig. 151 it is represented by A A. The cock K is opened, and the pencil, moving up and down with the variations of the pressure of the steam, traces on the card during each complete or double stroke a curve such as B C D E B. The ordinates drawn to that curve from any point in the atmospheric line, such as H Iv and H G, indi- cate the differences between the pressure of the steam and the at- mospheric pressure at the corre- F -^ 151 spending point of the motion of the iston. The piston. The ordinates of the part BODE represent the pres- sures of the steam during the forward stroke, when it is driving the piston, those of the part EB represent the pressures of the steam when the piston is expelling it from the cylinder. To found exact investigations on the indicator-diagrams of steam engines, the atmospheric pressure at the time of the experiment ought to be ascertained by means of a barometer; but this is generally omitted; in which case the atmospheric pressure may be assumed at its mean value, being 14-7 Ibs. on the square inch, or 21 16'3 Ibs. on the square foot, at and near the level of the sea. Let A O = H F be ordinates representing the pressure of the atmosphere. Then () F V parallel to A A is the absolute or true zero line of the diagram, corresponding to no pressure; and ordi- nates drawn to the curve from that line represent the absolute intensities of the pressure of steam. Let OB and L E be ordi- nates touching the ends of the diagram ; then O~X i-epresents the volume traversed by the piston at each single stroke ( = s A, -where s is the length of the stroke and A the area of the piston) ; The area O B C D E L O represents the energy exerted by the steam on the piston during the forward stroke; The area O B E L represents the work lost in expelling the steam during the return stroke; The area B C D E B, being the difference of the aoove areas, T 274: THEORY OF MACHINES. represents the effective work of the steam on the piston during the complete stroke. Those areas can be found by the rules of Article 34, page 17; and the common trapezoidal rule, D, page 21, is in general sufficiently accurate. The number of intervals is usually ten, and of orclinates eleven. The mean forward pressure, the mean back pressure, and the mean effective pressure, are found by dividing those three areas respec- tively by the volume s A, which is represented by O L. Those mean pressings, however, can be found by a direct process, without first measuring the areas, viz. : having multiplied each ordinate, or breadth, of the area under consideration by the proper multiplier, divide the sum of the products by the sum of the multipliers, which process, when the common trapezoidal rule is used, takes the following form : add together the halves of the endmost ordinates, and the whole of the other ordinates, and divide by the number of intervals. That is, let 6 be the first, b n the last, and b : , 5 , &c., the intermediate breadths; then let n be the number of intervals, and b m the mean breadth; then 1(1 ~{ n\ and this represents the mean forward pressure, mean back pressure, or mean effective pressure, as the case may be. Let p e be the mean effective pressure; then the effective energy exerted by the steam on the piston during each double stroke is the product of the mean effective pressure, the area of the piston, and the length of stroke, or p e ^s; ............................... (2.) and if N be the number of double strokes in a minute, the indicated power in foot-pounds per minute, in a single-acting engine, is (3.) from which the indicated horse-power is found by dividing by 33,000. In a double-acting engine the steam acts alternately on either side of the piston; and to measure the power accurately, two indi- cators should be used at the same time, communicating respectively with the two ends of the cylinder. Thus a pair of diagrams will be obtained, one representing the action of the steam on each face of the piston. The mean effective pi-assure is to be found as above for each diagram separately, and then, if the areas of the two faces of the piston are sensibly equal, the mean of those tiuo results is to be taken as the general mean effective pressure ; which being multi- plied by the area of the piston, the length of stroke, and twice the STEAM ENGINE INDICATOR. 275 number of double strokes or revolutions in a minute, gives the indicated power per minute ; that is to say, if $>" denotes the general mean effective pressure, the indicated power per minute is p" A-2Ns; .............................. (4.) If the two faces of the piston are sensibly of unequal areas (as in "trunk engines"), the indicated power is to be computed separately for each face, and the results added together. If there are two or more cylinders, the quantities of power indicated by their respective diagrams are to be added together. The reactions of the moving parts of the indicator, combined with the elasticity of the spring, cause oscillations of its piston. In order that the errors thus produced in the indicated pressures at particular instants may be as small as possible, and may neutralize each other's effects on the whole indicated power, the moving masses ought to be as small as practicable, and the spring as stiff as is consistent with shewing the pressures on a visible scale. In Richard's indicator this is effected by the help of a train of very light linkwork, which causes the pencil to shew the move- ments of the spring on a magnified scale. The friction of the moving parts of the indicator tends on the whole to make the indicated power and indicated mean effective pressure less than the truth, but to what extent is un- certain. Every indicator should have the accuracy of the graduation of its scale of pressures frequently tested by comparison with a standard pressure gauge. The indicator may obviously be used for measuring the energy exerted by any fluid, whether liquid or gaseous, in driving a piston ; or the work performed by a pump, in lifting, propelling, or compressing any fluid. 27G CHAPTER III. OF REGULATING APPARATUS. 425. Regulating Apparatus Classed Brake Fly Governor. The effect of all regulating apparatus is to control the speed of machinery. A regulating instrument may act simply by con- suming energy, so as to prevent acceleration, or produce re- tardation, or stop the machine if required; it is then called a brake; or it may act by storing surplus energy at one time, and giving it out at another time when energy is deficient: in this case it is called a fly; or it may act by adjusting the power of the prime mover to the work to be done, when it is called a governor. The use of a brake involves waste of power. A fly and a governor, on the other hand, promote economy of power and economy of strength. SECTION 1. OF BRAKES. 426. Brakes Defined and Classed. The contrivances here com- prehended under the general title of Brakes are those by means of which friction, whether exerted amongst solid or fluid particles, is purposely opposed to the motion of a machine, in order either to stop it, to retard it, or to employ superfluous energy during uniform motion. The use of a brake involves waste of energy, which is in itself an evil, and is not to be incm-red unless it is necessary to convenience or safety. Brakes may be classed as follows : I. Block-brakes, in which one solid body is simply pressed against another, on which it rubs. II. Flexible brakes, which embrace the periphery of a drum or pulley. III. Pump-brakes, in which the resistance employed is the friction amongst the particles of a fluid forced through a narrow IV. Fan-brakes, in which the resistance employed is that of a fluid to a fan rotating in it. 427. Action of Brakes in General. The work disposed of by a brake in a given time is the product of the resistance which it pro- duces into the distance through which that resistance is overcome iu a given time. BLOCK -BRAKES. 277 To stop a machine, the brake must employ work to the amount of the whole actual energy of the machine, as already stated in Article 417. To retard a machine, the brake must employ work to an amount equal to the difference between the actual energies of the machine at the greater and less velocities respectively. To dispose of surplus energy, the brake must employ work equal to that energy; that is, the resistance caused by the brake must balance the surplus effort to which the surplus energy is due; so that if n is the ratio which the velocity of rubbing of the brake bears to the velocity of the driving point, P, the surplus effort at the driving point, and R the resistance of the brake, we ought to have ' It is obviously better, when practicable, to store surplus energy, or to prevent its exertion, than to dispose of it by means of a brake. When the action of a brake composed of solid material is long- continued, a stream of water must be supplied to the rubbing surfaces, to abstract the heat that is produced by the friction, according to the law stated in Article 402, page 252. 428. Block-Brakes. When the motion of a machine is to be controlled by pressing a block of solid material against the rim of a rotating drum, it is advisable, inasmuch as it is easier to renew the rubbing surface of the block than that of the drum, that the drum should be of the harder, and the block of the softer material the drum, for example, being of iron, and the block of wood. The best kinds of wood for this purpose are those which have con- siderable strength to resist crushing, such as elm, oak, and beech. The wood forms a facing to a frame of iron, and can be renewed when worn. When the brake is pressed against the rotating drum, the direc- tion of the pressure between them is obliquely opposed to the motion of the drum, so as to make an angle with the radius of the drum equal to the angle of repose of the rubbing surfaces (denoted" by DQ = R -> whence it follows that the efficiency and counter-efficiency are given by the following equations: P D H cos * cos (0 + ) l-/tan/3 ,, , Efficiency = ^ = ^ = CQS ^ . CQS (;p9= -j- ; (4 A.) c l+^tan* tan * When the lines of action of the forces are parallel, we have sin ft and sin = +1 or 1, as the case may be; and the formulae take the following shape : When I and m lie at contrary sides of 0, the piece is a " lever of the first kind; " and When I and m lie at the same side of 0; EFFICIENCY OF MODES OF CONNECTION IN GENERAL. 233 If m > I, the piece is a lever of the second kind; " and 1 fr P .(5A.) ', the piece is a "lever of the third kind;" and V=K ( 5 *) (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 ; iu the former case it is approximate, in the latter exact. Through O, perpendicular to O C, draw U V, cutting the lines of action of the given force and of the effort in U and V respectively. The point where this transverse line cuts the small circle B B coincides exactly with T when the forces are parallel, and is very near T when they are inclined; and in either case the letter T will be used to denote that point. Then PoOU TV 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 ; 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 b N ; in which N is the normal pressure exerted by the pitch-surfaces on each other; b, a constant arm, of a length depending on the nature of the surfaces (for example 0-002 of a foot = 0-6 millimetre for cast iron on cast iron, see page 252); and a the relative angular velocity of the surfaces. The useful work per second is expressed by ufN, in which/ is the coefficient of friction of the surfaces, and u the common velocity of the pitch lines. Hence the counter-efficiency is Let Pi and p. 2 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 Pi Pz therefore /I 1\ = u ( + ); Vft pJ which value being substituted in Equation 1, gives for the counter- efficiency the following value : KM) It is assumed that the normal pressure is not greater than is EFFICIENCY OF SLIDING CONTACT IX GEXEUAL. 203 necessary in order to give sufficient friction to communicate the motion. It is evident, from the smallness of b, that the lost work in this case must be almost always a very small fraction of the whole. 442. Efficiency of Sliding Contact in General. In tig. 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 V W, the common component of the velocities of the two particles along their line of connection RTF. C T 0, parallel to V W, and perpendicular to R T P is a common tangent to the two acting surfaces at the point T ; the arrow A represents the direction in which the driver slides relatively to the follower; and the arrow B, the direction in which the follower slides relatively to the driver. Along the line of connection, that is, normal to the acting sur- faces at T, lay off T P to represent the effort exerted by the driver on the follower, and TR(=-TP)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 R 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 V W, in each second ; while the useful resistance, T If, is overcome through the distance T U. Hence the useful work per second is T U T R; the lost work is V W R S; and the counter-efficiency is Fig. 157. .(1.) 296 THEORY OF MACHINES. Let the angle U T V = , the angle U T W = fl, and let/ be the coefficient of friction. Then we have VW . , US f =tan * + tan ft; r^/i and consequently c = l+/(tan + tan /8) ...................... (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 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 formula? 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 Let u be the common velocity of the two pitch-circles ; 6 , 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. so that the counter-efficiency is u cos Let T! and r 2 be the radii of the two pitch circles; then we have u u i=%-r; 2 =-; and consequently (2.) EFFICIENCY OF BANDS. 297 If two pairs of teeth at least are to be in action at each instant fas in the case of involute teeth, and of some epicycloidal teeth), and if the pitch be denoted by p, we have t see e =%', and therefore where ^ and r? 2 are the number of teeth in the two wheels. In many examples of epicycloidal teeth, especially where small 'o Q pinions are used, the duration of the contact is only = or .- of that o 4 assumed in Equation 3; and the work lost in friction is less in the same proportion. 444. Efficiency of Bands. A band, such as a leather belt or a hempen rope, which is not perfectly elastic, requires the expenditure of a certain quantity of work first to bend it to the curvature of a pulley, and then to straighten it again; and the quantity of work so lost has been found by experiment to be nearly the same as would be required in order to overcome an additional resistance, varying directly as the sectional area of the band, directly as its tension, and inversely as the radius of the pulley. In the follow- ing formula? for leather belts, the stiffness is given as estimated by Reuleaux (Constructionslehrefur Maschinenbau, 307). Let T be the mean tension of the belt; S, its sectional area; r, the radius of the pulley; b, a constant divisor determined by experiment; R.', the resistance clue to stiffness; then ' ST 6 (for leather) = 34 inch = 87 millimetres. To apply this to an endless belt connecting a pair of pulleys of the respective radii r, and r 2 , let T\ and T 2 be the tensions of the two sides of the belt. Then the useful resistance is T^ - T 2 the T 4- T mean tension is -i-^ -j an d the additional resistance due to stiffness is 2 ftn consequently the counter-efficiency is Sfl I]. (N-l)6(r, r,)' 298 THEORY OF MACHINES. T N denoting ~. The sectional area, S, of a leather belt is given by the formula where p denotes the safe working tension of leather belts, in units of weight per unit of area; its value being, according to Morin, 0-2 kilogramme on the square millimetre, or 285 Ibs. on the square inch. The ordinary thickness of the leather of which belts are made is about 0-16 of ail 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 x being the tension at the driving-side of the belt, exclusive of centrifugal ' 'fusion : (5.) and the following formula for the counter-efficiency : 12 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., EFFICIENCY OF LINKWORK. 299 where D is the diameter of the rope, and b' = 54: millimetres = 2-125 inches. In all the forrnulse, ~ is to be substituted for ~. Ihe proper value of D 2 is given by the formula T =^;. p .(8.) where p' = 1,000 for measures in inches and Ibs. ; and p' = 0'7 for measures in millimetres and killogrammes. 445. Efficiency of Linkwork. In fig. 158, let C 1 T v C 2 T 2 be two levers, turning about parallel axes at C 1 and C 2 , and connected with each other by the link T x T 2 ; T x and T 2 being the connected points. Fig. 158. The pins, which are connected with each other by means of the link, are exaggerated in diameter, for the sake of distinctness. Let C x T! be the driver, and C 2 T 2 the follower, the motion being as shewn by the arrows. From the axes let fall the perpendiculars Ci Pj, C 2 P 2 , upon the line of connection. Then the angular velocities of the driver and follower are inversely as those perpen- diculars; and, in the absence of friction, the driving moment of the first lever and the working moment of the second are directly as those perpendiculars; the driving pressure being exerted along the line of connection T z T 2 . Let M 2 be the working moment ; and let M be the driving moment in the absence of friction ; then we have 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, T! AI and T 2 Ao, draw small circles about the connected points. Then draw a straight line, Q! Aj B x Q 2 A 2 B 2 , touching both the small circles, and in such a position as to represent the line of action of a force that resists the motion of both pins in the eyes of the link. This will be the line of action of the resultant force exerted through the link. Let fall upon it the perpendiculars Ci Qi, C 2 Q 2 ; these will be proportional to the actual driving moment and working moment respectively; that is to say, let ]\I X be the driving moment, including friction; then M M.-C.Q, M i = p~o ^2^2 Comparing this with the value of the driving moment without friction, we find for the counter-efficiency and for the efficiency 1_M _C.Q.-Q I P 1 -- 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 purcJiase, when friction is neglected. Let c denote the counter-efficiency of a single sheave, as depend- ing on its friction on the pin, according to the principles of Article 373, page 290. Let c' denote the counter-efficiency of the rope, when passing over a single sheave, determined by the principles p of Article 444, the tension being taken as nearly equal to ; where R is the useful load, or resistance opposed to the motion of the running block. R -j- n is also the effort to be exerted on the hauling part of the rope, in the absence of friction. Then the counter-efficiency of the tackle will be expressed approximately by (cc')"; .............................. (1.) so that the actual or effective purchase, instead of being expressed by n, will be expressed by n(cc')- n .............................. (2.) EFFICIENCY OF CONNECTION BY MEANS OF A FLUID. 301 447. Efficiency of Connection by means of a Fluid. When 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 maybe taken at 0-9 nearly; while with a carefully made cupped leather collar the counter-efficiency of a plunder may be taken at the following value : '- w in which d is the diameter of the plunger; and b a constant, whoso value is from O'Ol to 0-015 of an inch, or from 0-25 to 33 of a millimetre. For if c be the circumference of the plunger, and p the effective pressure of the liquid, the whole amount of the pres- sure on the plunger is *-j- ; and the pressure required to overcome the friction is p c b, The efficiency and counter-efficiency of the intervening mass of fluid remain to be considered; and if that fluid is a liquid, and may therefore be regarded as sensibly incompressible, these quan- tities depend on the work which is lost in overcoming the 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, - is the velocity of the stream of fluid. Let b denote the wetted border or circumference of the passage ; then, is what is called the hydraulic mean depth of the passage. In a cylindrical pipe, m = diameter. Let I be the length of the 302 THEORY OP MACHINES. passage, and w the heaviness of the liquid. Then the loss of pres- sure in overcoming the friction of the passage is , fl wv z in which g denotes gravity, and / a coefficient of friction whose value, for water in cylindrical cast-iron pipes, according to the experiments of Darcy, is /= 0-005 d being the diameter of the pipe in feet. Let p be the pressure on the driven or following piston ; then the pressure on the driving piston is p + p' ; and the counter- ejficiency of 'the fluid is which, being multiplied by the product of the counter-efficiencies of the two pistons, gives the counter-efficiency of the intervening liquid. When the intervening fluid is air, there is a loss of work through friction of the passage, depending on principles similar to those of the friction of liquids; and there is a further loss through the escape by conduction of the heat produced by the compression of the air. The friction which has to be overcome by the air, and which causes a certain loss of pressure between the compressing pumps and the working machinery, consists of two parts, one occasioned by the resistance of the valves, and the other by the friction along the internal surface of pipes. To overcome the resistance of valves, about five per cent, of the effective pressure may be allowed. The friction in the pipes depends on their length and diameter, and on the velocity of the current of air through them. It is nearly proportional to the square of the velocity of the air. A velocity of about forty feet per second for the air in its com- pressed state has been found to answer in practice. The diameter of pipe required in order to give that velocity can easily be com- puted, when the dimensions of the cylinders of the machinery to be driven, and the number of strokes per minute, are given. When the diameter of a pipe is so adjusted that the velocity of the air is 40 feet per second, the pressure expended in overcoming its friction may be estimated at one per cent, of the total or absolute * When the diameter is expressed in millimetres, fov~ substitute 5-* EFFICIENCY OF CONNECTION BY MEANS OF A FLUID. 303 pressure of the air, for every five hundred diameters of the pipe that its length contains. Although the abstraction from the air of the heat produced by the compression involves a certain sacrifice of motive power (say from 30 to 35 per cent.) still the effects of the heated air are so inconvenient in practice, that it is desirable to cool it to a certain extent during or immediately after the compression. This may be effected by injecting water in the form of spray into the com- pressing pumps; and for that purpose a small forcing pump of about Y^th 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 Ibs. on the square inch, or 10,333 kilogrammes on the square metre); let^j be the absolute working pressure of the air, so that p -p is the effective pressure; then the counter-efficiency due to the escape of heat is, From examples of the practical working of compressed air, when used to transmit motive power to long distances, it appears that in order to provide for leakage and various other imperfec- tions in working, the capacity of the compressing pumps should be very nearly double of the net volume of uncompressed air required; and it has also been found necessary, in working the compressing pumps, to provide from three to four times the power of the machinery driven by the compressed air. 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 plane, 134. of chain or cord, 174. of couples, 126. of forces in oue line, 118. of inclined forces, 122. of parallel forces, 131, 132. of structures, 157. Balanced forces, motion under, 210. Bauds, classed, 97. connection by, 72, 97, 98. efficiency of, 297. length of, 99. motion of, 97. principle of conrection by, 97. Bar. 153. Beam, 158. allowance for weight of, 200. limiting length of, 200. in linkwork, 101. Bearings, 71. friction of, 251. Belt, with speed cones, 100. Bending moment, at a series of sec- tions, 193. Bending moment, greatest, 194. Bending moments, calculation of, 190. Bending, resistance to, 189. moment of, 190. Bevel-wheels (see Wheels). Blocks and tackle, 105. efficiency of, 300. Blocks, stability of a series of, 158, 175. Bodies, 30. rigid, 47. Bracing of frames, 163, 167, 168. Brake, 241. Brakes, 276. block, 277. Bulkiness, 121. Buoyancy, centre of, 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. I Cinematics, 31. x 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, pitch (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). Coupled parallel shafts, 101. Couples, 118, 119. equivalent, 125. parallelogram of, 126. polygon of, 126. resultant of, 125. with parallel axes, 126. Coupling, double, Hooke's, 105. Hooke's, 104. Oldham's, 96. Coupling-rod, 101 (see Linkwork). Crank-rod, 101 (see Linkwork). r. resistance to JS9. 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, 289. of a sliding piece, 288. of modes of connection in mechan- ism, 293. of primary pieces, 287. of bands, 297. of linkwork, 299. of blocks and tackle, 300. of fluid connection, 301. of a screw, 293. of rolling contact, 294. of sliding contact, 295. of teeth, 296. Effort, 205. accelerating, 260. when speed is uniform, balances resistances, 215. Elasticity, 183. coefficients of, 184. modulus of, 184. Elementary combinations, 72. classed generally, 72. 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. Sfime, of a number, 1. ing 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. EX. 307 Forces, parallelogram of, 122. parallelepiped of, 123. polygon of, 123. representation of by line, 117. resolution of, 122, 123, 124. resultast and component of, 118. triangle of, 122. Fractions, continued, 2. Frames, 71. bracing of, 166. equilibrium and stability of, 158. of two bars, 161. polygonal, 163, 164, 165. resistance of, at a section, 171. triangular, 162, 163. Friction, 153, 154. coefficient of, 154. moment of, 251. of liquid, 235. of solid bodies, law of, 153. tables of, 155. work done against, 251. Frictional 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. efficiency 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, with friction, 233. surface of equal pressure in, 231. without friction, motion of, 230. Live load, 180. Load, 179. dead, 180. live, 180. working, 179. Logarithms, common, 4, 5, 6. MACHINE, efficiency of (see Effi- ciency), action of, 243. general equation of the action of, 267. moving pieces in, primary and secondary, 72. Machines, 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. Merrih'eld's trapezoidal rule, 19, 20. Modulus of elasticity, 184. height or length of, 184. of pliability, 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, 223. 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. 309 Motion, second law of, 211. 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, 2G6. horse, 241, 266. Powers, mechanical (see Mechanical powers). Press, hydraulic (see Hydraulic press). Pressure, 144. centre of, 121. intensity of, 121. Primary moving pieces, efficiency of. 287. motions of, 72. Prime factors, 1 . Prime movers, 240. Projection, parallel, 138, 153, 178. Projectile, unresisted, 214. Proof strength, 182, 183. Pull (see Tension). Pulley-blocks (see Tackle). Pulley (mechanical power), 107. RACKS, toothless, 74. smooth, 74. straight and circular wheels, 75. Radius, geometrical, 81. of gyration, 208. real, 81. Ratio, 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, 238. of mouthpieces, 238. of rolling, 252. useful and prejudicial, 241. Resolution offerees, 122. Rest, 31. Resultant, 118. momentum, 207. of any system offerees, 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 LtfDEX. fligid body, motion of, 47, 222 (see Rotation). Rigidity or stiffness, 183. coefficients of, 183. Rod (see Crank-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, 9a 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 of, 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. 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 liukwork, 104. Struts, 158. Supports, 156. Surface, 30. System of parallel forces, 131. TACKLE, 105. connection by, 73, 105. efficiency of, 300. Tearing, resistance to, 184. Teeth, arc of contact of, 88. dimensions of, 91. efficiency of, 296. epicycloidal, 89. involute, for circular wheels. 88. 89. of mitre or bevel-wheels, 91, 92. of non-circular wheels, 92. of spur wheels and racks, 86. of wheels, 81. of wheel and trundle, 90. pitch and number of, 81. sliding of, 87. traced by rolling curves, 86. Tension, 144, 184. Testing, 182. Thrust, 144. Tie, 158. strength of, 184. Time, measure of, 35. Tooth, face of, 81. flank of, 81. Torsion (see Wrenching). Trains of mechanism, 73, 111. of wheelwork, 83, 84, 85. Ti-ansformation (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 Rotation). 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 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. 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The labour expended on ' Suggestive Thoughts ' must have been immense, and the result is as nearly perfect as human fallibility can make it. ... Apart from the selections it contains, the book is of value as an index to theological writings. As a model of judicious, logical, and suggestive treatment of a subject, we may refer our readers to the manner in which the subject ' JESUS CHRIST ' is arranged and illustrated in ' Suggestive Thoughts.' "Gtasffw News. A BOOK NO FAMILY SHOULD BE WITHOUT. New issue of this important Work-Enlarged, in part Re-written, and thoroughly Revised to date. TWENTY-SIXTH EDITION. Royal Svo, Handsome Cloth, los. 6d. A DICTIONARY OF DOMESTIC MEDICINE AND HOUSEHOLD SURGERY, BY SPENCER THOMSON, M.D., EDIN, L.R.C.S., REVISED, AND IN PART RE-WRITTEN, BY THE AUTHOR, AND BY JOHN CHARLES STEELE, M.D., OF GUY'S HOSPITAL. With Appendix on the Management of the Sick-room, and many Hints for the Diet and Comfort of Invalids. In its New Form, DR. SPENCER THOMSON'S "DICTIONARY OF DOMESTIC MEDICINE" fully sustains its reputation as the " Representative Book of the Medical Knowledge and Practice of the Day applied to Domestic Requirements. The most recent IMPROVEMENTS in the TREATMENT OF THE SICK in APPLIANCES for the RELIEF OF PAIN and in all matters connected with SANITATION, HYGIENE, and the MAINTENANCE of the GENERAL HEALTH will be found in the New Issue in clear and full detail ; the experience of the Editors in the Spheres of Private Practice and of Hospital Treatment respectively, combining to render the Dictionary perhaps the most thoroughly practical work of the kind in the English Language. Many new Engravings have been introduced improved Diagrams of different parts of the Human Body, and Illustrations oi the newest Medical, Surgical, and Sanitary Apparatus. ** All Directions given in such a form as to be readily and safely followed. FROM THE AUTHOR'S PREFATORY ADDRESS. 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" There can be little doubt that it is destined to take a high place among book* of this class." Kotet and Queries " A treasure to every reader who may be fortu- nate enough to possess it. Its perusal is like in- haling essences ; we have the cream only of the great authors quoted. Here all are seeds or gems. " Engliih Journal of Education. " Mr. Sonthgate's reading will be found to ex- tend over nearly the whole known field of litera- ture, ancient and modern." Gentleman' t Maga- zine. ' We have no hesitation in pronouncing it one of the most important books of the season. Credit is due to the publishers for the elegance with which the work is got up, and for the extreme beauty and correctness of the typography." Morning Chronicle. u Of the numerous volumes of the kind, we do not remember having met with one in which the selection was more judicious, or the accumulation of treasures so truly wonderful. "-.Jfomin^ Herald. " The selection of the extracts has been made with taste, judgment, and critical nicety." Homing Pott. " This is a wondrous book, and contains a great many gems of thought." Daily New$. " As a work of reference, it will be an acquisi- tion to any man's library." Publisher/ Circular. " This volume contains more gems of thought, refined sentiments, noble axioms, and extractable sentences, than have ever before been brought to- gether in our language." The Field. u All that the poet has described of the beautiful in nature and art, all the axioms of experience, the collected wisdom of philosopher and sage, are garnered into one heap of useful and well-arranged instruction and amusement." The Era. " The collection will prove a mine rich and in- exhaustible, to those in search of a quotation." "Will be found to be worth ifr weight in gold by literary men," J%e Buildtr. " Every page is laden with the wealth of pro- fonndert thought, and all aglow with the loftiest inspirations of genius." Star. "The work of Mr. Sonthgate far outstrips all others of ite kind. To the clergyman, the author, the artist, and the essayist, .' Many Thoughts of Many Minds ' cannot fall to render almost incal- culable service.* Edinburgh Mercury. " We have no hesitation whatever in describing Mr. Southgate'i as the very best book of the class. There is positively nothing of the kind in the lan- guage that will bear a moment's comparison with it" Maneheiter Weekly Advertiter. " There is no mood in which we can take It up without deriving from it instruction, consolation, and amusement. We heartily thank Mr. Southgate for a book which we shall regard as one of our best friends and companions." Cambridge Chronicle. " This work possesses the merit of being a MAGNIFICENT GIFT-BOOK, appropriate to all times and seasons ; a book calculated to be of use to the scholar, the divine, and the public man." Freemason' i Magazine. " It is not so much a book as a library of quo- tations, "/tariot. " The quotations abound in that thought which is the mainspring of mental exercise." Liver- pool Courier. " For purposes of apposite quotation, it cannot be surpassed." Bristol Time*. " It is impossible to pick out a single passage in the work which does not, upon the face of it, jus- tify its selection by its intrinsic merit." Dortet Chronicle. " We are not surprised that a SECOND SERIES of thia work should have been called for. Mr. Sonthgate has the catholic tastes desirable in a good Editor. Preachers and public speakers will find that it has special uses for them." Edinburgh Daily Review. " The SECOND SERIES fully sustains the de- erved reputation of the FIRST." John Bull. LONDON ? CTTABLES Q-RTFFTN * COMPANY. UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. STACK M.?7 ;4UL07198; "STACK JULQ7198; | #* REGElVEt > JUN10 193 i STACK ANis EX Form L9-100n*-9,'52(A3105) 444