F?. S. HuTCHEON LIBRARY THE UNIVERSITY OF CALIFORNIA SANTA BARBARA PRESENTED BY ELEANOR HUTCHEON WORKS OF THE LATE JOSEPH LIPKA PUBLISHED BY JOHN WILEY & SONS, INC. Graphical and Mechanical Computation An aid in the solution of a large number of problems which the engineer, as well as the student of engineering, meets in his work, ix + 264 pages. 6 by 9. 207 fig- ures, 2 charts. Cloth. Also published in two parts Part I. Alignment Charts. xiv + 119 pages. 6 by 9. 130 figures, 2 charts. Cloth. Part II. Experimental Data. Pages 120 to 259. 6 by 9. 77 figures. Cloth. BY S. R. CUMMINGS AND JOSEPH LIPKA Alignment Charts for the Engineer By S. R. Cummings.S.M., Research Engineer, The Hoo- ver Co., and the late Joseph Lipka, Ph. D. Part I. Air and Steam. Twenty charts for various en- gineering equations and formulas, designed for practical use by the engineer and student of engineering. 9$ by 12. Loose leaf, in heavy paper envelope. BY HUDSON, LIPKA, LUTHER AND PEABODY The Engineers' Manual By Ralph G. Hudson, S. B., Professor of Electrical Engineering, Massachusetts Institute of Technology, assisted by the late Joseph Lipka, Ph.D., Howard B. Luther, S. B., Dipl. Ing., Professor of Civil Engineer- ing, University of Cincinnati, and Dean Peabody, Jr., S. B., Associate Professor of Applied Mechanics, Massachu- setts Institute of Technology. A consolidation of the more commonly used formulas of engineering, each arranged with a statement of its appli- cation. Second edition, iv + 340 pages. 5 by 7|. 238 figures. Flexible binding. BY R. G. HUDSON AND JOSEPH LIPKA A Manual of Mathematics By Ralph G. Hudson, S.B., and the late Joseph Lipka, Ph. D. A collection of mathematical tables and formulas cover- ing the subjects most generally used by engineers and by students of mathematics, and arranged for quick ref- erence, iii -(- 132 pages. 5 by 7J. 95 figures. Flexible binding. A Table of Integrals By Ralph G. Hudson, S.B., and the late Joseph Lipka, Ph.D. Contains a Table of Derivatives, Table of Integrals, Nat- ural Logarithms, Trigonometric and Hyperbolic Func- tions. 24 pages. 5 by 7J. Paper. [GRAPHICAL AND MECHANICAL COMPUTATION/ PART n. EXPERIMENTAL DATA BY JOSEPH LIPKA, PH.D. LATE ASSISTANT PROFESSOR OF MATHEMATICS IN THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED ** IN THE REPRINTING OF THIS BOOK, THE RECOM- MENDATIONS OF THE WAR PRODUCTION BOARD HAVE BEEN OBSERVED FOR THE CONSERVATION OF PAPER AND OTHER IMPORTANT WAR MA- TERIALS. THE CONTENT REMAINS COMPLETE AND UNABRIDGED. COPYRIGHT, 1918, BY JOSEPH LIPKA Printed in U. S. A. PREFACE This book embodies a course given by the writer for a number of years in the Mathematical Laboratory of the Massachusetts Institute of Technology. It is designed as an aid in the solution of a large num- ber of problems which the engineer, as well as the student of engineering, meets in his work. In the opening chapter, the construction of scales naturally leads to a discussion of the principles upon which the construction of various slide rules is based. The second chapter develops the principles of a network of scales, showing their application to the use of various kinds of coordinate paper and to the charting of equations in three variables. Engineers have recognized for a long time the value of graphical charts in lessening the labor of computation. Among the charts devised none are so rapidly constructed nor so easily read as the charts of the alignment or nomographic type a type which has been most fully developed by Professor M. d'Ocagne of Paris. Chapters III, IV, and V aim to give a systematic development of the construction of alignment charts; the methods are fully illustrated by charts for a large number of well-known engineering formulas. It is the writer's hope that the simple mathematical treatment employed in these chapters will serve to make the engineering profession more widely acquainted with this time and labor saving device. Many formulas in the engineering sciences are empirical, and the value of many scientific and technical investigations is enhanced by the discovery of the laws connecting the results. Chapter VI is concerned with the fitting of equations to empirical data. Chapter VII considers the case where the data are periodic, as in alternating currents and volt- ages, sound waves, etc., and gives numerical, graphical, and mechanical methods for determining the constants in the equation. When empirical formulas cannot be fitted to the experimental data, these data may still be efficiently handled for purposes of further computation, interpolation, differentiation, and integration, by the numerical, graphical, and mechanical methods developed in the last two chapters. Numerous illustrative examples are worked throughout the text, and a large number of exercises for the student is given at the end of each chapter. The additional charts at the back of the book will serve iii iv PREFACE as 'an aid in the construction of alignment charts. Bibliographical references will be found in the footnotes. The writer wishes to express his indebtedness for valuable data to the members of the engineering departments of the Massachusetts Institute of Technology, and to various mathematical and engineering publications. He owes the idea of a Mathematical Laboratory to Professor E. T. Whittaker of the University of Edinburgh. He is especially indebted to Capt. H. M. Brayton, U. S. A., a former student, for his valuable suggestions and for his untiring efforts in designing a large number of the alignment charts. Above all he is most grateful to his wife for her assistance in the revision of the manuscript and the reading of the proof, and for her constant encouragement which has greatly lightened the labor of writing the book. JOSEPH LIPKA. CAMBRIDGE, MASS., Oct. 13, 1918. CONTENTS. CHAPTER I. SCALES AND THE SLIDE RULE. ART. PAGE 1. Definition of a scale I 2. Representation of a function by a scale I 3. Variation of the scale modulus 2 4. Stationary scales 5 5. Sliding scales ' 7 6. The logarithmic slide rule 9 7. The solution of algebraic equations on the logarithmic slide rule 1 1 8. The log-log slide rule . 13 9. Various other straight slide rules 15 10. Curved slide rules 16 Exercises 18 CHAPTER II. NETWORK OF SCALES. CHARTS FOR EQUATIONS IN TWO AND THREE VARIABLES. 11. Representation of a relation between two variables by means of perpen- dicular scales 20 12. Some illustrations of perpendicular scales 21 13. Logarithmic coordinate paper 22 14. Semilogarithmic coordinate paper 24 15. Rectangular coordinate paper the solution of algebraic equations of the 2nd, 3rd, and 4th degrees 26 1 6. Representation of a relation between three variables by means of perpen- dicular scales 28 17. Charts for multiplication and division 30 18. Three-variable charts. Representing curves are straight lines 32 19. Rectangular chart for the solution of cubic equations 35 20. Three- variable charts. Representing curves are not straight lines 37 21. Use of three indices. Hexagonal charts 40 Exercises 42 CHAPTER III. NOMOGRAPHIC OR ALIGNMENT CHARTS. 22. Fundamental principle 44 (I) Equation of form f^u) + f 2 (v) =/ 3 (w) or /,() }M = MW) Three parallel scales 45~54 23. Chart for equation (I) 45 24. Chart for multiplication and division 47 vi CONTENTS ART. PAGE 25. Combination chart for various formulas 48 26. Grasshoff 's formula for the weight of dry saturated steam 50 27. Tension in belts and horsepower of belting 52 (II) Equation of form /i() +/() +/() + = /(*) or /,() ./, (t>) ./.(w) . . . = /<(*) Four or more parallel scales 55~63 28. Chart for equation (II) 55 29. Chezy formula for the velocity of flow of water in open channels 56 30. Hazen-Williams formula for the velocity of flow of water in pipes 57 31. Indicated horsepower of a steam engine 6l Exercises , 64 CHAPTER IV. NOMOGRAPHIC OR ALIGNMENT CHARTS (Continued). (HI) Equation of form/i() = / 2 () MW) or/i() = f t (v) f * w Z chart. . 65-67 32. Chart for equation (III) 65 33. Tension on bolts with U. S. standard threads 66 (IV) Equation of form ' = * Two intersecting index lines 68-75 34. Chart for equation (IV) 68 35. Prony brake or electric dynamometer formula 69 36. Deflection of beam fixed at ends and loaded at center 70 37. Deflection of beams under various methods of loading and supporting 71 38. Specific speed of turbine and water wheel 73 (V) Equation of form fi(u) = fa(v) ' fz(w) /(/) . . . Two or more inter- * secting index lines '. 76-87 39. Chart for equation (V) 76 40. Twisting moment in a cylindrical shaft 77 41. D'Arcy's formula for the flow of steam in pipes 79 42. Distributed load on a wooden beam 80 43. Combination chart for six beam deflection formulas 84 44. General considerations 87 (VI) Equation of form -JT\ = fT\ Parallel or perpendicular index lines 87-91 45. Chart for equation (VI) 87 46. Weight of gas flowing through an orifice 89 47. Armature or field winding from tests 90 48. Lame formula for thick hollow cylinders subjected to internal pressure 91 (VII) Equation of form /,() - /,() = / 3 (w) - f t (q) or /i() :/(*) = /a(ttO : /(?) Parallel or perpendicular index lines 9i~95 49. Chart for equation (VII) 91 50. Friction loss in flow of water 94 Exercises 95 CONTENTS YD CHAPTER V. HOMOGRAPHIC OR ALIGHMEHT CHARTS (Continued). (VIE) Equation of form /,() +/,(*) = Parallel or perpendicular index lines ................................................... 97-104 51. Chart for equation (Vlli) ........................................... 97 52. Moment of inertia of cylinder ........................................ 99 53. Bazin formula for velocity of flow in open channels ..................... 101 54. Resistance of riveted steel plate ..................................... ". IOT (IX) Equation of form Three or more concurrent scales 104106 55. Chart for equation (IX) 104 56. Focal length of a lens 106 (X) Equation of form /.()+/,(*)-/,() =/() Straight and curved scales 106-113 57. Chart for equation (X) 106 58. Storm water run-off formula 107 59. Francis formula for a contracted weir no 60. The solution of cubic and quadratic equations no (XI) Additional forms of equations. Combined methods 114-117 62. Chart for equation of form /,()+/,(*) /,(*>) =/(f) ................... 114 63. Chart for equation of form /,() ./ 4 (a) +/^) /,() = 1 ................. 114 64. Cfctforeo^ionoffann+=... ..... 05 65. Chartforequadonofform^ )+ ^ = i ...................... ..... . 66. Chart for equation of form/if.) -/.(j) +/,(*) -fifa) = / 5 (.) .............. 116 67. Chart for equation of form /!()/,()+/,() / 4 (w)=/ i ( J )+/(w) ....... 117 ............................................ "7 for Chapters III, IV, V ......................... 118 CHAPTER YL EMPIRICAL FORMULAS If OH-PKRIODIC CURVES. 68. Experimental data lao (C The straight line 122-127 69. The straight fine, j = Joe 122 70. The straight Foe, jr = + fcr 125 V1H CONTENTS ART. PAGE (II) Formulas involving two constants 128-139 71. Simple parabolic and hyperbolic curves, y = a* 6 128 72. Simple exponential curves, y = a^ z 131 73. Parabolic or hyperbolic curve, y = a + bx n (wnere n is Known) 135 74. Hyperbolic curve, y = , , or - = a + bx 137 (III) Formulas involving three constants 140-152 75. The parabolic or hyperbolic curve, y = ax b + c 140 76. The exponential curve, y = ae bx + c 142 77. The parabola, y = a + bx + ex 2 145 78. The hyperbola, y = \- c 149 79- The logarithmic or exponential curve, log y = a + bx -\- ex 2 or y = aeP I ~*~ cx . . 151 (IV) Equations involving four or more constants 152-164 80. The additional terms ce dx and cx d 152 81. The equation y = a + bx + ce dx 153 82. The equation y = ae bx + ce dx 156 83. The polynomial y = a + bx + ex 2 + dx 3 + 159 84. Two or more equations 161 Exercises 164 CHAPTER VII. EMPIRICAL FORMULAS PERIODIC CURVES. 85. Representation of periodic phenomena 170 86. The fundamental and the harmonics of a trigonometric series 170 87. Determination of the constants when the function is known 173 88. Determination of the constants when the function is unknown 174 89. Numerical evaluation of the coefficients. Even and odd harmonics 179 90. Numerical evaluation of the coefficients. Odd harmonics only 186 91. Numerical evaluation of the coefficients. Averaging selected ordinates. ... .192 92. Numerical evaluation of the coefficients. Averaging selected ordinates. Odd harmonics only 198 93. Graphical evaluation of the coefficients 200 94. Mechanical evaluation of the coefficients. Harmonic analyzers 203 Exercises 207 CHAPTER VIII. INTERPOLATION. 95. Graphical interpolation 209 96. Successive differences and the construction of tables 210 97. Newton's interpolation formula 214 98. Lagrange's formula of interpolation 218 99. Inverse interpolation 2IO Exercises 22 , CONTENTS CHAPTER IX. APPROXIMATE INTEGRATION AND DIFFERENTIATION. ART. PAGE 100. The necessity for approximate methods 224 101. Rectangular, trapezoidal, Simpson's, and Durand's rules 224 102. Applications of approximate rules 227 103. General formula for approximate integration 231 104. Numerical differentiation 234 105. Graphical integration 237 106. Graphical differentiation 244 107. Mechanical integration. The planimeter 246 108. Integrators 250 109. The integraph 252 no. Mechanical differentiation. The differentiator 255 Exercises 256 CHAPTER VI. EMPIRICAL FORMULAS NON-PERIODIC CURVES. 68. Experimental data. In scientific or technical investigations we are often concerned with the observation or measurement of two quanti- ties, such as the distance and the time for a freely falling body, the volume of carbon dioxide dissolving in water and the temperature of the water, the load and the elongation of a certain wire, the voltage and the current of a magnetite arc, etc. The results of a series of measurements of the same two quantities under similar conditions are usually presented in the form of a table. Thus the following table gives the results of observa- tions on the pressure p of saturated steam in pounds per sq. in. and the volume v in cu. ft. per pound : p = 10 20 30 40 50 60 v = 37-80 19.72 13.48 10.29 8.34 6.62 40 30 20 10 I \ \ \ k \ ^ ' . --^^ **-^ ) 10 20 30 40 50 60 (P) FIG. 68. We represent these results graphically by plotting on coordinate paper the points whose coordinates are the corresponding values of the measured quantities and by drawing a smooth curve through or very near these points. Fig. 68 gives a graphical representation of the above table, where the values of p are laid off as abscissas and the values of v as ordi- nates and a smooth curve is drawn so as to pass through or very near the plotted points. ART. 68 EXPERIMENTAL DATA 121 The fact that a smooth curve can be drawn so as to pass very near the plotted points leads us to suspect that some relation may exist between the measured quantities, which may be represented mathematically by the equation of the curve. Since the original measurements, the plotting of the points, and the drawing of the curve all involve approximations, the equation will represent the true relation between the quantities only approximately. Such an equation or formula is known as an empirical formula, to distinguish it from the equation or formula which expresses a physical, chemical, or biological law. A large number of the formulas in the engineering sciences are empirical formulas. Such empirical for- mulas may then be used for the purpose of interpolation, i.e., for comput- ing the value of one of the quantities when the value of the other is given within the range of values used in determining the formula. It is at once evident that any number of curves can be drawn so as to pass very near the plotted points, and therefore that any number of equations might approximate the data equally well. The nature of the experiment may give us a hint as to the form of the equation which will best represent the data. Otherwise the problem is more indeterminate. If the points appear to lie on or near a straight line, we may assume an equation of the first degree, y = a + bx, in the variables. But if the points deviate systematically from a straight line, the choice of an equa- tion is more difficult. Often the form of the curve will suggest the type of equation, parabolic, exponential, trigonometric, etc., but in all cases, we should choose an equation of as simple a form as possible. Before proceeding any further with this choice we may test the correctness of the form of the equation by "rectifying" the curve, i.e., by writing the assumed equation in the form (i) f(y} = a + bF(x) or (2) / = a + bx', where / = f(y) and x' = F(x), and plotting the points with x' and y' as coordinates ; if the points of this plot appear to lie on or very near a straight line, then this line can be represented by equation (2) and hence the original curve by equation (i). We shall use the method of rectifica- tion quite freely in the work which follows. Having chosen a simple form for the approximate equation we now proceed to determine the approximate values of the constants or co- efficients appearing in the equation. The method of approximation employed in determining these constants depends upon the desired degree of accuracy. We may employ one of three methods: the method of selected points, the method of averages, or the method of Least Squares. Of these, the first is the simplest and the approximation is close enough for a large number of problems arising in technical work; the second re- quires a little more computation but usually gives closer approximations; 122 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI while the third gives the best approximate values of the constants but the work of determining these values is quite laborious. All three methods will be illustrated in some of the problems which follow. After the constants have been determined the formula should be tested by performing several additional experiments where the variables lie within the range of the^ previous data, and comparing these results with those given by the empirical formula. We shall now work two illustrative examples to indicate the general method of procedure. (I) THE STRAIGHT LINE. 69. The straight line, y bx. The following table gives the results of a series of experiments on the determination of the elongation E in inches of annealed high carbon steel wire of diameter 0.0693 in. and gage length 30 in. due to the load W in pounds. w E EW W* E c ' " Ef A' A" A'" o O O O O O 5 0.0130 0.650 2,500 0.0130 0.0131 0.0131 O I I 100 0.0251 2.510 10,000 0.0260 0.0261 0.0262 - 9 10 II 15 0.0387 5 805 22,500 0.0390 0.0392 0.0393 - 3 5 - 6 200 0.0520 10.400 40,000 0.0520 0.0522 0.0524 o + 2 4 225. 0.0589 13-253 50,625 0.0585 0.0587 0.0589 + 4 + 2 250 0.0659 !6-475 62,500 0.0650 0.0653 0.0655 + 9 + 6 + 4 260 0.0689 !7-9!4 67,600 0.0676 0.0679 0.0681 +13 + 10 + 8 2 1235 0.3^25 67.007 255-725 38 36 34 1-7-8= 4.8 4-5 4-3 SA* = 356 270 254 The plot. The data are plotted on a sheet of coordinate paper about 10 inches square and ruled in twentieths of an inch or in millimeters. If we wish to express the elongation as a function of the load, we plot the load on the horizontal axis or as abscissas, if the load as a function of the elongation we plot the latter as abscissas. In Fig. 69 we have plotted the values of W as abscissas and the values of E as ordinates. The scales with which these values are plotted are generally chosen so that the length of the axis represents the total range of the corresponding vari- able, and so that the line or curve is about equally inclined to the two axes. There is no advantage in choosing the scale units on the two axes equal. Care should be taken not to choose the units either too small or too large; for in the former case the precision of the data will not be utilized, and in the latter case the deviations from a representative line ART. 69 THE STRAIGHT LINE, y = bx 123 or curve are likely to be magnified. The drawing of a good plot is evi- dently a matter of judgment. It is best to mark the plotted points as the intersection of two short straight lines, one horizontal and one vertical. 0.07 0.06 0.05 0.04 (E) 0,03 0.02 0.01 50 100 (W) FIG. 69. 150 200 250 The representative curve and its equation. We now draw a smooth curve passing very near to the points of the plot, so that the deviations of the points from the curve are very small, some positive and some negative. In Fig. 69, the points seem to fall approximately t on a straight line. This should be tested by moving a stretched thread or by sliding a sheet of celluloid with a fine line scratched on its under side among the points and noting that the points do not deviate systematically from this thread or line. Having decided that a straight line will approximate the plot, we assume that an equation of the first degree, E = a + bW, will approximately represent the relation between the measured quantities. In this example we may evidently assume that E = bW since a zero load gives a zero elongation. 124 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI The determination of the cdnstant. We shall now determine the con- stant b in the equation E = bW. This may be done in several ways. The three methods which are generally employed are as follows : I. Method of selected points. Place the sheet of celluloid on the coordinate paper so that the scratched line passes through the point W = o, E = o, and then rotate the sheet until a good average position among the plotted points is obtained, i.e., until the largest possible num- ber of points lie either on the line or alternately on opposite sides of the line, in such a manner that the points below the line deviate from it by approximately the same amount as the points above it. Then note the values of W and E corresponding to one other point on this line, prefer- ably near the farther end of the line. Thus we read W = 250, E = 0.0650. Substituting these values in the equation E = bW, we have 0.0650 = 250 b, and hence b = 0.000260, and finally E = 0.000260 W. Since the choice of the "best" line is a matter of judgment, its position, and hence the value of the constant, will vary with different workers and often with the same worker at different times. II. Method of averages. The vertical distances of the plotted points from the representative line are called the residuals; these are the differ- ences between the observed values of E and the values of E calculated from the formula, or E E c , where E c = bW', some of these residuals are positive and others are negative. If we assume that the "best" line is that which makes the algebraic sum of the residuals equal to zero, we have 2(E - bW] =o or ZE - bZW = o, 2E 0.3225 hence b = ^TFT> = = 0.000261, 2W 1235 and we may call this an average value of b. By this method it is no longer necessary to shift the line among the points so as to get an average position. III. Method of Least Squares. In the theory of Least Squares * it is shown that the best line or the best value of the constant is that which makes the sum of the squares of the differences of the observed and cal- culated values a minimum, i.e., 2 (E - bW} 2 = minimum. Hence the derivative of this expression with respect to b must equal zero, or ^ 2 (E - bW) z = o, or S W(E - bW) = o, or 2WE - bZW* = o, and b = j * See Bartlett's " The Method of Least Squares," or any other book on this theory. ART. 70 THE STRAIGHT LINE, y = a + bx 125 We form two columns, one giving the values of RW and the other the values of W 2 , and adding these columns, we find b = 67.007/255,725 = 0.000262. We may now compare the results obtained by each of the three methods. For this purpose we complete the table by computing the values of E from the formulas I. E = 0.000260 W; II. E = 0.000261 W; III. E = 0.000262 W. These are marked E c *, E c u , E c m , in the table. To discover how closely the computed values agree with the observed values we form the residuals A 1 = E - E c \ A 11 = E - E c n , A 111 = E - E. Disregarding the signs of these residuals, we add them and divide by their number, 8, and find the average residual to be 0.00048, 0.00045, 0.00043, respectively. We also find the sum of the squares of the resid- uals to be 356, 270, 254, respectively. We may therefore draw the fol- lowing conclusions: all three methods give good results; the method of Least Squares gives the best value of the constant but requires the most calculation; the method of averages gives, in general, the next best value of the constant and requires but little calculation; the graphical method of selected points requires the least calculation but depends upon the accuracy of the plot and the fitting of the representative line. 70. The straight line, y = a + bx. For measuring the temperature coefficient of a copper rod of diameter 0.3667 in. and length 30.55 in., the following measurements were made. Here, C is the temperature Centi- grade and r is the resistance of the rod in microhms. c ' C rC V r= r c m A 1 A" A 1 " 19.1 76.30 364-81 1.457-33 76.19 76.19 76.26 -t-o.ii +0.1 1 +0.04 25.0 77.80 625.00 1,945.00 77.91 77.92 77.96 O.II 0.12 0.16 30.1 36.0 79-75 80.80 906 .01 1296 .00 2,400.48 2,908.80 79-39 8i.ii 79-41 81.14 79-43 81.13 +o. 3 b -0.31 +0-34 -0-34 +0.32 -0.33 40.0 82.35 1600 .00 3,294.00 82.27 82.31 82.28 +0.08 +0.04 +0.07 45-1 83.90 2034.01 3.783-89 83.75 83-80 83.76 +0.15 +0.10 +0.14 85.10 2500.00 4,255-00 85.18 85.24 85.16 0.08 0.14 0.06 2245.3 566.00 9325.83 20,044 . 50 1.20 I.I9 I. 12 S -j- 7 = 0.171 0.170 0.160 SA Z = 2852 2869 2646 The plot (Fig. 70) appears to approximate a straight line, so that we shall assume the relation r = a + bC. We shall determine the con- stants, a and b, by the three methods. L Method of selected points. Use a sheet of celluloid to determine the approximate position of the best straight line, and note two points 126 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI on this line; thus, C = 20, r = 76.45, and C = 48, r = 84.60. Substi- tuting these values in the equation r = a + bC, we get 76.45 = a + 20 b and 84.60 = a + 48 b, from which we determine a = 70.63 and b = 0.291, so that our relation becomes r = 70.63 + 0.291 C. 85 84 83 82 81 80 (r) 79 78 77 76 75 / / / f / / / + / 7 / / 7 / 10 15 20 25 30 85 40 46 50 FIG. 70. II. Method of averages. Since we have to determine two constants, we divide the data into two equal or nearly equal groups, and place the sum of the residuals in each group equal to zero, i.e., I, (r - a - bC} = o or Sr = na + bZC, where n is the number of observations in the group. Thus, dividing the above data into two groups, the first containing four and the second three sets of data, and adding, we get 314-65 = 4 a + 1 10.2 b and 251.35 = 3 a -f- 135.1 b, AST. 70 THE STRAIGHT LINE, y = a + bx 12? from which we determine a = 70-59 and b = 0.293, so that our relation becomes r = 70-59 + 0.293 C. III. Method of Least Squares. The best values of the constants are those for which the sum of the squares of the residuals is a minimum, i.e., 2 (r a bC) z = minimum; hence the partial derivatives of this expression with respect to a and b must be zero ; thus, Z(r-a- bC? = o, 1 S (r - a - 6C) 2 = o, or 2[2(r-a- bC] (-i)j = o, 2 [2 (r - a - bQ (-Q] = o, or Zr = an + where n is the number of observations. We solve these last two equations for a and b. (Note that these equations may be formed as follows: substitute the observed values of r and C in the assumed relation r = a -f- bC; add the n equations thus formed to get the first of the above equations; multiply each of the n equations by the corresponding value of C and add the resulting n equations to get the second of the above equations.) We now compute the values of rC, C 2 , ZC, ZrC, and ZC 2 , and substi- tute these in the equations for determining a and b. We thus get 566.00 = 7 a + 245.3 b, 20,044.50 = 245.3 a + 9325-83 b, from which we determine a = 70.76 and b = 0.288, so that our relation becomes r = 70.76 + 0.288 C. Comparison of results. We note that the various results agree very well with the original data and with each other. We compute the resid- uals and find that the average residual is smallest by the third method and is approximately the same by the first two methods. The computa- tion necessary in applying the method of Least Squares is very tedious. The method of selected points requires the fitting of the best straight line, and this becomes quite difficult when the number of plotted points is large. We shall therefore use the method of averages in most of the illustrative examples which follow. 128 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI (II) FORMULAS INVOLVING TWO CONSTANTS. 71. Simple parabolic and hyperbolic curves, y = ax 6 . As stated in Art. 68, when the plotted points deviate systematically from a straight line, a smooth curve is drawn so as to pass very near the points; the shape of the curve or a knowledge of the nature of the experiment may give us a hint as to the form of the equation which will best represent the data. Simple curves which approximate a large number of empirical data are the parabolic and hyperbolic curves. The equation of such a curve is y = ax 6 , parabolic for b positive and hyperbolic for b negative. In Fig. 710, we have drawn some of these curves for a = 2 and b = 2, I, 0.5, 0.25, 0.5, 1.5,2. Note that the parabolic curves all pas through the points (o, o) and (i, a) and that as one of the variables increases the other increases also. The hyperbolic curves all pass through the point (i, a) and have the coordinate axes as asymptotes, and as one of the variables increases the other decreases. There is a very simple method of verifying whether a set of data can be approximated by an equation of the form y = ax*. Taking loga- rithms of both members of this equation, we get log y = log a + b log x, and if x' = log x, y' = log y, this becomes y' = log a + bx' t an equation of the first degree in x' and y' ; therefore the plot of (x r , y') or of (log x, log y) must approximate a straight line. Hence, ART. 71 PARABOLIC AND HYPERBOLIC CURVES, y I2 9 // a set of data can be approximately represented by an equation of the form y = arc 6 , then the plot of (log x, log y) approximates a straight line. Instead of plotting (log*, logy) on ordinary coordinate paper, we may plot (x, y) directly on logarithmic coordinate paper (see Art. 13). We determine the constants a and b from the equation of the straight line by one of the methods described in Art. 70. Example. The following table gives the number of grams S of anhy- drous ammonium chloride which dissolved in 100 grams of water makes a saturated solution of 6 absolute temperature. 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2 5$ 2.51 2.52 2.53 2.54 2.55 2.56 2.57 75 70 65 60 55 (S) 50 45 40 35 30 25 ^ ^ /.(? /.75 (log 1.60 1.55 1 50 / X 4 ./ X x /x X . Other simple curves that approximate a large number of experimental results are the exponential or logarithmic curves. The equation of such a curve may be written in the form y = a^ x , where e is the base of natural logarithms; the form y = ab* is sometimes used. In Fig. 720, we have drawn some of these curves for a = i and b =2, i, 0.5, 0.5, i, 2. Note that these curves all pass through the point (o, a) and have the x-axis for asymptote. EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI 3.0 2.5 2.0 There is a very simple method of verifying whether a set of data can be approximated by an equation of the form y = a&*. Taking logarithms of both members of this equa- tion we get log y = log a + (b log e) x, and if y' = log y, this equation becomes y' = log a + (b log e) x, an equation of the first degree in x and y' ; therefore the plot of (x, y'} or of (x, log y) must approximate a straight line. Hence, // a set of data can be ap- proximately represented by an equation of the form y = ae* 1 , then the plot of (x, log y) ap- proximates a straight line. Instead of plotting (x, logj) on ordinary coordinate paper, we m?.y plot (x, y} directly 1.5 1.0 0.5 (x) y = ae l FIG. 72 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 on semilogarithmic coordinate paper (see Art. 14). The con- stants a and b are determined from the equation of the straight line by one of the methods described in Art. 70. Example. Chemical experiments by Harcourt and Esson gave the results of the following table, where A is the amount of a substance re- maining in a reacting system after an interval of time /. t A log/ log A AC A 2 94.8 0.3010 .9768 94-9 O.I 5 87.9 0.6990 .9440 87.7 +0.2 8 81.3 0.9031 .9101 81.0 +0.3 ii 74-9 I .0414 .8745 74.8 + 0.1 14 68.7 I . 1461 .8370 69.1 -0.4 i? 64 .0 I . 2304 .8062 63.8 +0.2 27 49-3 i -43*4 .6928 49-0 +0.3 3i 44.0 i .4914 6435 44-i O.I 35 39-i i -5441 5922 39-6 -0-5 44 31-6 I-643S 4997 31.2 +0.4 ZA -i- 10 = 0.26 The points (/, A) are plotted in Fig. 726. This curve appears to be exponential, so that we plot (/, log A) and (log/, A); it is seen that the plot of (/, log A) approximates a straight line. We may therefore assume an equation of the form A = ae*' or log A = log a + (b log e) t. ART. 72 SIMPLE EXPONENTIAL CURVES, y - o** (I* 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 133 95 90 ?5 Of) 2.00 1.95 1.90 1.85 1.80 1.75 (log A) 1.70 1.65 1.60 1.55 1.50 1.45 \ *>! \ V,, 'x, \ \ \ \ s^ \ \ \ \\ \ ^ \ \ 75 70 65 <^) 60 55 50 45 4JS 35 30 c >; Si \ V \ . \ X \ \ \ z \ \ r A" \ \1 ^ ^ ^ ^ 5 \ S Nj \ \ \ \ ^L ', \ f s \ \ \ \ \ \ 4 \ V \ \ \ \ \ \ \ \ s \. s \ \ S \ S N\ Vx \ X ) 5/0/5 20 25 30 35 40 45 FIG. 726. We shall use the method of averages to determine the constants. Divid- ing the data into 2 groups and adding, we get 9.5424 = 5 log a + 40 (b log e}, 8.2344 = 5 log a + 154 (b log e): :. b\oge= 0.0115, log a = 2.0005. b = 0.0265, a = loo.i, since log e = 0.4343. .-. log ,4 = 2.0005 0.0115 /, or ^4 = 100. i g--<65. We now compute the values of A and the residuals, and note the close agreement between the observed and the calculated values of A . 134 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI Example. The following table gives the results of measuring the electrical conductivity C of glass at temperature 6 Fahrenheit. 0.09 0.08 0.07 0.06 0.05 (CO 0.04 0.03 0.02 0.01 f y.u n / y 8.8 o r ^ / -t / ft / / j 8.5 * i / V / 1 f / * I / 4 7 / ^ i y 8.1 o n / Q Z / / / r 7.9 7.8 7.7 V 1 / I A 1 x / -^ "*"*" / \ / > 60 80' 100 120 140 160 180 20Q~' 22Q 24 FIG. 72 c. C log logC C, A 58 O I 7634 00 0.0019 86 O.OO4 i -9345 7.6021 10 0.0039 -fo.oooi 148 O.OlS 2.1703 8.2553-10 0.0185 0.0005 1 66 O.O29 2.22OI 8.462410 0.0292 O.OOO2 188 0.051 2.2742 8.7076 10 0.0510 O 202 0.073 2.3054 8.8633 10 0.0728 +0.0002 210 0.090 2.3222 8.9542-10 0.0891 O.OOIO ART. 73 PARABOLIC OR HYPERBOLIC CURVE, y = a + bx n 135 In Fig. 72c, the pofnts (9, C) and (0, log C) are plotted; the latter plot approximates a straight line. We may therefore assume the equa- tion C = ae, or log C = log a -f- (b log e) 0. We use the method of averages to determine the constants. Omitting the first set and dividing the remaining data into two groups of three sets, we get 24.3198 - 30 = 3 log a + 400 (b log e), 26.5251 - 30 = 3 log a + 600 (b log e). .'. bloge = o.ono, logo = 6.6399 10. .'. b = 0.0253, a = 0.000436. .'. log C = 6.6399 10 + o.ono 6, or C = 0.00436 e - 0258 ', We now compute the values of C and the residuals and note the re- markably close agreement between the observed and computed values of C. 73. Parabolic or hyperbolic curve, y = a + bx n (where n is known). In using this equation, it is assumed that from theoretical considerations we suspect the value of n. It is evident that If a set of data can be approximately represented by an equation of the form y = a + bx n , where n is known, then the plot of (x n , y} approximates a straight line. Example. A small condensing triple expansion steam engine tested under seven steady loads, each lasting three hours, gave the following results; I is the indicated horse-power, w is the number of pounds of steam used per hour per indicated horse-power. (From Perry's Ele- mentary Practical Mathematics.) / w wl w a A 36.8 12. S 460.0 12.6 o. 31.5 I2. 9 406.4 12.8 +o. 26.3 I3-I 344.5 13.0 +0. 21 .O 13-3 279-3 13-4 o. IS.8 I4.I 222.8 14.0 + 0. 12.6 14-5 182.7 14.6 o. 8-4 I6. 3 136.9 16.1 +0. "SA -i- 7 = o.n Fig. 73 gives the plot of (I, w). This is not a straight line. But if we plot (/, wl), i.e., the total weight of steam used per hour instead of the weight per indicated horse-power, we find that this plot approximates a straight line. Hence, we may assume the linear relation wl = a + bl. This relation may also be written w = b + a/ 1, so that the plot of (l/J, w) also approximates a straight line. We use the method of averages to 136 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI determine the constants. Dividing the data into two groups, the first three and last four sets, and adding, we have 1210.9 = 3 a + 94.66, 821.7 = 4 a + 57-86. .*. b = 1 1. 6, a = 37.8. .'. wl = 37.8 -f- II.67, or w=ii.6 + ^y We now compute the values of w and the residuals. 16 15 14 13 / 450 400 350 300 250 ( I) 200,. 150 100 50 n [ / I / \\ V / \\ t / \\ / \\ ^ \ /\ 9 \ / \. / \ / > ft- + / V > / / "v / 4 V / x 3.1. (We may similarly introduce the permeability, B/H, and note that the plot of (B/H, B) approximates a straight line.) Hence, we assume a TT relation of the form = a + bH. Using the method of averages, 100 FIG. 74&. omitting the first three values of H, and dividing the remaining data into two groups containing five and four sets respectively, we get the equations . 3.621 = 5 a + 49.1 b, 13.88 = 4 a + 235.5 b. /. b = 0.0560, a = 0.174. /. -= = 0.174 + 0.0560 H or B = r-^ z := B 0.174 + 0.0560 H We now compute B and the residuals and note the close agreement between the observed and computed values. 140 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI (HI) FORMULAS INVOLVING THREE CONSTANTS. 75. The parabolic or hyperbolic curve, y = ax b + c. It is often im- possible to fit a simple equation involving only two constants to a set of data. In such cases we may modify our simple equations by the addition of a term involving a third constant. Thus the equation y = ajc* may be modified into y = ax 6 -\- c. If b is positive, the latter equation repre- sents a parabolic curve with intercept c on OF; if b is negative, the equation rep- resents a hyperbolic curve with asymptote y = c. In Fig. 750, we have sketched the curves y = 2 ^- 5 , y = 2 x - 5 + 2, y = 2 x~- 5 , y = 2 x~- 5 + 2 to illustrate the relation of the simple types to the modified types. In Art. 71 it was shown that if we suspect a relation of the form y = ax 11 , we can To verify this by observing whether the plot of (log x, log y) approximates a straight line. Now the form y = ax 11 + c may be written log (y c} = log a + b log x, so that the plot of (log x, log (y c)) would approximate a straight line. To make this test we shall evidently first have to determine a value of c. We might attempt to read the value of c from the original plot of (x, y}. In the parabolic case we should have to read the intercept of the curve on OY, but this may necessitate the extension of the curve beyond the points plotted from the given data, a procedure which is not safe in most cases. In the hyperbolic case, we should have to estimate the position of the asymptote, but this is generally a difficult matter. The following procedure will lead to the determination of an approxi- mate value of c for the equation y = ax* + c. Choose two points (xi, yi) and fa, yd on the curve sketched to represent the data. Choose a third point (x 3 , yd on this curve such that x 3 = ^XiX 2 , and measure the value of y 3 . Then, since the three points are on the curve, their coordinates must satisfy the equation of the curve, so that yi axi b + c, y z = ax z b + c, y 3 = ax 3 b + c. 75 PARABOLIC OR HYPERBOLIC CURVE, y = a* 6 + c 141 Now, since #3 = Vjc^. therefore x 3 b = Vxi b x z b , or y, - c and therefore and = Vaxf - c), = yiy* - - 2 y 3 It is evident that the determination of c is partly graphical, for it depends upon the reading of the coordinates of three points on the curve sketched to represent the data. The curve should be drawn as a smooth line lying evenly among the points, i.e., so that the largest number of the plotted points lie on the curve or are distributed alternately on opposite sides and very near it, Having determined a value for c, we plot (log x, log (y c)). If this plot approximates a straight line, the constants a and b in the equation log (y ~ c ) = log a -\- b log x may then be determined in the ordinary way. Example. In a magnetite arc, at constant arc length, the voltage V consumed by the arc is observed for values of the current i. (From Steinmetz, Engineering Mathematics.) , i V V - 30.4 log (V- 30.4) logi v e A o.S 1 60 129.6 .1126 9.6990 10 158.8 + 1-2 i 1 2O 89.6 95 2 3 o.oooo 10 120.8 -0.8 2 94 63.6 8035 0.3010 10 94.0 4 75 44-6 .6493 0.602 1 10 75-i O.I 8 62 31-6 4997 0.9031 10 61.9 +0.1 12 56 25.6 .4082 i .0792 10 56.0 We plot (i, V) and note that the curve appears hyperbolic with an asymptote V c, and hence we assume an equation of the form V= a* 6 + c. To verify this we must first determine a value for c. Choose two points on the experimental curve; in Fig. 756, we read ii = 0.5, FI = 160 and t2 = 12, F2 = 56. Choose a third point such that is = Vi^ = V6 = 2.45, and measure Vs = 88. Then F^z - F 2 = (160) (56) - (88) 2 = 1216 F! + Vt - 2 F 3 160 + 56-2 (88) 40 30.4. Now compute the values of V 30.4 and log ( V 30.4) and plot (log i, log (V 30.4)). This last plot approximates a straight line so that the choice of the equation V = ai b + c is verified. To determine the constants in the equation log (V - 30.4) = log a + b log i, 142 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI we use the method of averages, dividing the data into two groups of three sets each, and find 5.8684 = 3 log a, 4-5572 = 3 log a + 2.58446. b = 0.507, log a = 1.9561, a = 90.4. .*. log (V - 30.4) = 1.9561 - 0.507 log*, or V = 30.4 + 90.4 i- 0807 . Finally, we compute the values of V and the residuals. (log irl 6 9. 7 9. 8 9. 9 0. 0. 1 0. 2 0. 3 4 5 0. 6 0. 7 8 0. d i. 1. 1 160 p n \ r> J \ ^ 2 \ ^ ^ (V) 100 \ ^ v* V,- i I / S"' on \ c \ tfl '** on \ S ^ r ff X V fr 7 "^ ^ f T 50 .1 ^ ^-^ * ~~, - ~ ^ .. - ^ /.4 ) . > ; t < \ i ( '' I ? y / y / / / 2 FIG. 75&. 76. The exponential curve, y = ae 6 "" + c. The simple exponential equation y = ae? z may have to be modified into y = ae?> x + c in order to fit a given set of data. In the latter curve, the asymptote is y c. In Fig. 760, we have sketched the curves y = 2 e - 11 , y = 2 eP- lx + i y = 2 e- lx , ;y = 2 e-^- 1 * + I. In Art. 72 it was shown that if we suspect a relation of the form y = ae* x , we can verify this by observing whether the plot of (x, logy) approximates a straight line. Now y = aeP x + c may be written log (y c) = log a + (b log 0) x, so that the plot of (x, log (y c)) would approximate a straight line. Evidently we shall first have to determine a value for c. We proceed to do this in a manner similar to that employed in Art. 75. Choose two points (xi, y\) and (x%, y%) on ART. 76 THE EXPONENTIAL CURVE y = ae 6 * + c 143 the curve sketched to represent the data, and then a third point (x 3 , ys) on this curve such that x 3 = \ (x\ + #2) and measure the value of y 3 . Since the three points are on the curve. yi = ae 6 * 1 + c, y 2 = ae 6 * + c, ;y 3 = ae 6 * 3 + c, or log = (6 log c) *!, log (If) 3 2 ~ = (6 log e) * 2( log *_ = (6 log e} x 3 . FIG. 760. y = ae 6 * + c 9 10 Now, since therefore and log - (b log e) x 3 = \ (xi + x z ), [(b log e) Xi + (b log e) x*], - c y* -c a ' a Hence y 3 c = V(y x c) (y z c), and c = - 2 y 3 If the data are given so that the values of x are equidistant, i.e., so that they form an arithmetic progression, we may verify the choice of the equation y = ae bx + c and determine the ^constants a, b, and c in the following manner. Let the constant difference in the values of x equal In. If we replace x by x + h, we get y' = ae?( x+K ) + c, and therefore, for the difference in the values of y, &y = y' - y = ae?>( I+ V - a(* x = ad 31 (ef> h - l), and log Ay = log a (& h i) + (b log e) x. 144 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI This last equation is of the first degree in x and log Ay so that the plot of (x, log Ay) is a straight line. To apply this to our data, we form a column of successive differences, Ay, of the values of y, and a column of the logarithms of these differences, log Ay, and plot (x, log Ay) ; if the equation y = aefi 1 -f- c approximates the data, then this last plot will approximate a straight line. We may then determine b log e and log a (^ h i) and hence a and b in the ordinary way, and finally find an average value of c from 2y = aSe 6 * + nc, where n is the number of data. Example. In studying the skin effect in a No. oooo solid copper conductor of diameter 1.168 cm., Kennelly, Laws, and Pierce found the following experimental results; F is the frequency in cycles per second, L is the total abhenrys observed. F L L- si, 860 log (L 51,860) L. A 60 53.912 2052 3.3122 53.952 40 306 53.767 1907 3 2804 53.668 +99 888 53,143 1283 3.1082 53,140 + 3 1600 52,669 809 2.9079 52,699 -30 2040 52,499 639 2.8055 52,506 - 7 3065 52,215 355 2.5502 52,212 + 3 3950 52,082 222 2-3464 52,068 + 14 5000 5L965 I5 2.0212 5i,972 . - 7 In Fig. 76^, the points (F, L) are plotted; the curve appears to be exponential with an asymptote L = c. We shall try to fit the equation L = ae^ F + c. First determine an approximate value for c by choosing two points on the experimental curve, FI = 875, LI = 53,140, and F 2 = 5000, L 2 = 51,980, and a third point F s = | (Fi + F 2 ) = 2938, L 3 = 52,250. Then c = * ' ^- = 51,860. Now compute (L 51,860) Li\ -+- L,z 2 1>3 and log (L 51,860), and plot (F, log (L 51,860)); this plot approxi- mates a straight line, thus verifying the choice of equation. We deter- mine the constants in the equation log (L 51,860) = log a + (b log e) F by the method of averages. Dividing the data into two groups of four sets each and adding, we have 12.6087 = 4 log a + 2854 b log e, 9.7233 = 4 log a -f 14,055 b log e. and or bloge = -0.0002576, log a = 3.3360, b = 0.0005931, a = 2168. iog (L - 51,860) = 3.3360 - 0.0002576 F, L = 51, 860 + 2168 ^-0005931 F ART. 77 THE PARABOLA, y = a+bx+cx> 145 We now compute L and the residuals, and note the close agreement between the observed and computed values except for the first two values of F. If we omit these two values in computing a and b, these constants have slightly different values, but the agreement between the observed and computed values of L is about the same. 54,000 53,800 53,600 53,400 53,200 53.000 52.800 52,600 52,400 52.200 S 2,000 51.800 L 4 X \ 3.2 3.1 o n \ \ \ \ V \ + \ \ N ^ \ \ 2.9 2.8 J 2.6^ 2.5 2 4 \ ^ \ \ f- ^ \ 2> 1 ^ V \ y >t H<" \ \ \ \ \ \ \ \ \ s s^ s 2.9 2.2 2.1 Ik' ^ X ^ \ \ ^* . V NS \ \ 1,000 2,000 3,000 4.000 5.01 (F) FIG. 766. 77. The parabola, y = a + bx + ex 2 . The equation of the straight line y = a + bx may be modified by the addition of a term of the second degree to the form y = a + bx + cx z . This is the equation of the ordi- nary parabola. We may verify whether this equation fits a set of experi- mental data by one of the following methods. 146 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI (l) Choose any point (**, y k ) on the experimental curve; then = o, + bxk + cXk 2 , and y-y k = b(x-x k )+c(x 2 -x k 2 ), or ?~ x x k This last equation is of the first degree in x and ^ ^- so that the plot of (x, 1 will approximate a straight line. \ x x k / (2) If the values of x are equidistant, i.e., if they form an arithmetic progression, with common difference h, then if we replace x by x + h in the equation, we get y' = a + b (x + 7z) + c (x + h) 2 and Ay = y' y = (bh + ch 2 ) + 2 chx. This last equation is of the first degree in x and Ay, so that the plot of (x, Ay) will approximate a straight line. Hence, if a set of data may be approximately represented by the equation y = a + bx + ex 2 , then (i) the plot of (x, y ~ yk ), where (x k , y k ) are the \ x x k / coordinates of any point on the experimental curve, will approximate a straight line, or (2) the plot of (x, Ay), where the Ay's are the differences in y formed for equidistant values of x, will approximate a straight line. The following examples will illustrate the method of determining the constants. Example. In the following table, 6 is the melting point in degrees Centigrade of an alloy of lead and zinc containing x per cent of lead. (From Saxelby's Practical Mathematics.) i e * - 36.9 0-l8l 0-i8i *-j6.9 6 C A 87.5 292 50.6 III 2.20 295 -3 84.0 283 47-i 102 2.17 285 77-8 270 40.9 89 2.18 268 + 2 63-7 235 26.8 54 2.01 234 +1 46.7 i97 9-8 16 1.63 199 2 36.9 181 182 I In Fig. 770, we have plotted (x, 6). We shall try to fit an equation of the form e = a + bx + ex 2 to the data. To verify this choice, observe that the curve passes through the point x k = 36.9, 6 k = 181, and plot the (/\ _ o T \ x, J ; this last plot approximates a straight line. (In plotting the ordinates for the straight line a scale unit ten times as large as that used for the ordinates of the experimental curve has been used: any further increase in the scale unit would simply magnify the devia- ART. 77 THE PARABOLA, y = a + bx + c* e - 181 14? f\ _ _ Q _ tfons.) We may now assume the relation =a' + b'x, and use the method of averages to determine the constants. Dividing the data into two groups of three and two sets respectively and adding, we get 6-55 = 3 a' + 249.3 b f , 3.64 = 2 a' + 110.46'. .'. b' = 0.0130, a' = 1. 10. _ = 1. 10 + 0.0130 x, or = 141.4 + 0.620* + 0.0130 x*. We now compute 6 and the residuals. 300 290 280 270 260 250 (e) 240 230 220 210 200 190 180 40 50 60 (x) FIG. 770. 70 Example. The following table gives the results of the measure- ments of train resistances; V is the velocity in miles per hour, R is the resistance in pounds per ton. (From Armstrong's Electric Traction.) 148 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI V R AK V> R, A 20 5-5 3-6 400 5-70 O.2O 40 9-i 5-8 i, 600 9.08 +O.O2 60 14-9 7-9 3,600 14.82 +0.08 80 22.8 i-5 6,400 22.86 O.O6 IOO 33-3 12.7 10,000 33-22 +0.08 120 46.0 14,400 45-90 +0.10 2 420 131-6 36,400 In Fig. 776, the plot of (V, R) appears to be a parabola, R = a + bV + cF 2 . Since the values of V are equidistant, we shall verify our choice of equation by a plot of (V, A.R) ; this last plot approximates a straight line. We may therefore assume AR = (bh + eft 2 ) -\-2chV, where h = 20. 50 45 40 35 30 (R)25 20 15 10 20 40 80 100 120 (AP FIG. We determine the constants in this last equation by the method of averages, using the five sets of values of V and A.R. Dividing these data into two groups of three and two sets respectively and adding, we get 17.3 = 3(6& + c#) + 120 (2 eft), 23.2 = 2 (bh -f eft 2 ) + 1 80 (2 eft). /. 2 eft = 0.117, bh + eft 2 = 1.08. .'. c = 0.0029, b 0.004. /. R = a - 0.004 V + 0.0029 F 2 . AXT. 78 THE HYPERBOLA, y a + bx 149 We determine a by substituting the six sets of values of V and R, and summing, thus 2R = 6 a - 0.004 S V + 0.0029 S F 8 , or 131.6 = 60 0.004 (4 2 ) + 0.0029 (36,400), and therefore a = 4.62. Hence, finally, R = 4.62 0.004 V + 0.0029 F 2 . We now compute the values of R and the residuals; the agreement between the observed and calculated values of R is very close. 78. The hyperbola, y = j + c. a *f- o;c tion of the equation y This equation is a modifica- discussed in Art. 74. In the latter a + bx equation, x = o gives y = o, while in the former, x = o gives y = c. We JC may verify whether the equation y = jr f- c fits a set of experi- a-\-bx mental data as follows. Choose any point (x k , curve; then y k = ,*\ + c, and *(*- on the experimental (a + foe) (a + (a + to) + (a + This last equation is of the first degree in x and -, so that the plot of (x, -j will approximate a straight line. Hence, if a set of data may be approximately represented by the equation y = * , + c, the plot of (x, ^ V where (x k , yk) are the coordinates of a point on the experimental curve, will approximate a straight line. Example. The following table gives the results of experiments on the friction between a straw-fiber driver and an iron driven wheel under a pressure of 400 pounds; y is the coefficient of friction and jc is the slip, per cent. (From Goss, Trans. Am. Soc. Mech. Eng., for 1907, p. 1099.) X y x 0.65 y-o.i29 y - 0.129 * y 0.65 0.129 0.129 0.129 0.87 0.217 O.22 0.088 50 0-253 0.228 0.88 0.228 0.23 0.099 32 0.256 0.232 0.90 0.234 0.25 0.105 .38 0.264 0.238 o-93 0.275 0.28 0.146 .92 0.274 0.248 1.16 0.318 0.51 0.189 70 0.326 0.304 i. 80 0.400 0.271 25 0-394 0.388 2.12 0.410 1-47 0.281 5-23 0.410 0.411 3-00 0-435 2-35 0.306 7.68 0-435 0.451 150 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI In Fig. 78 we have plotted the points (x, y); the experimental curve X appears to be an hyperbola with an equation of the form y = |- c. To verify this we note the point x F= 0.65, y = 0.129 on the curve, and plot the points Ix, ^ f~^-\ This last plot approximates a straight line. We may therefore assume the relation = a + bx, and y 0.129 0.5 0.4 0.3 (y) 0.2 0.1 0.5 2.5 FIG. 78. we shall determine the constants by the method of averages. As the first three points do not lie very near this straight line, we shall use only the last five sets of data, and dividing these into two groups of three and two sets respectively and adding, we get 8.87 = 3 a + 3.89 b, 12.91 = 2 a + 5.12 b. .: b = 2.77, a = 0.64. x 0.65 x 0.65 = 0.64 + 2.77 x or y = ^ h 0.129. y - o 129 2.77 x - 0.64 If we had used all eight points in determining the constants, we should have obtained 9.12 = 4 a + 3.586, 19.86 = 4 a + 8.08 b. :. b = 2.39, a = 0.14. x ~ 0.65 = 0.14 + 2.39 x or x - 0.65 2.39 x + 0.14 0.129. ART. 79 THE LOGARITHMIC CURVE, log y =a + bx + cy? We have computed both y and y' and note that the agreement with the observed values is probably as close as could be expected. 79. The logarithmic or exponential curve, log y = a + bx + c* 2 or y = ae bx + cae \ These equations are modifications of the logarithmic form log y = a + bx and the exponential form y = a&*. The equation y = ae bl+c ^ may be written log y = log a + (b log e) x -\- (c log e) x 2 , and so is equivalent to the form log y = a + bx + cx z . This last equation is similar in form to the equation y = a + bx + ex 2 discussed in, Art. 77, and the equation may be verified and the constants determined in a similar way. Hence, if a set of data may be approximately represented by the equation hg y ~ where log y = a + bx + ex 2 , then (i) the plot of (x, \ (xk, yk) are the coordinates of a point on the experimental curve, will approxi- mate a straight line, or (2) the plot of (x, A log y) , where the A log y are the differences in log y formed for equidistant values of x, will approximate a straight line. Example. The following table gives the results of Winkelmann's experiments on the rate of cooling of a body in air ; is the excess of tem- perature of the body over the temperature of its surroundings, t seconds from the beginning of the experiment. t logo log 8 log 118.97 log 8 -log 1 18.97 e e A I 118.97 2.07544 118.97 O 12. 1 116.97 2.06808 0.00736 O.OOo6o8 116.99 O.O2 25-8 114.97 2.06059 0.01485 0.000576 114.97 41-7 112.97 2.05296 0.02248 0.000539 112.90 +0.07 59-7 82.0 110.97 108.97 2.04520 2.03731 0.03024 0.03813 0.000507 0.000465 110.90 108.90 +0.07 +0.07 109.0 106.97 2.02926 0.04618 0.000424 107.15 -0.18 In Fig. 79 we have plotted the points (t, 6}. According to Newton's law of cooling, 6 = ae* 1 or log = a + bt, and so we have also plotted the points (/, log 6} ; this last plot has a slight curvature. We shall therefore assume the law in the form log e = a + bt + ct 2 . To verify this, we note the point tt = o, dk = 118.97 on the experimental curve, and plot the points (/, }', this plot approximates a straight line, so that we may assume ~ = b + ct. We use the method of averages to determine the constants. Dividing the data into two groups of three sets each and adding, we get -0.001723 = 36 + 79-6 c, -0.001396 = 36 + 250.7 c. 152 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI .'. c = 0.000001911, b = 0.000625. log 6 log 118.97 7^- = 0.000625 + 0.000001911 / or log e = 2.07544 - 0.000625 t + 0.600001911 P. We now compute 6 and the residuals and note the close agreement between the observed and calculated values. 11R 0? s Q. 00 06 Q 000% 116.97 114.97 (e) 112.97 110.97 108.97 106.97 C X, \ N 4 X ^ ^ \ w '""*** \: s^ K ^ c f ^c ^3 ~^^ ^ ff >! ^x ^ ^ ^ ""v *>t 7^ ^^ ^ -^ 8 / / + bx} or y = a + bx + ce dx . The equation y = a + bx + ce? x may fit an experimental curve although no part of the curve is approximately a straight line; this means that the values of the term ce dx are not negligible for any values of x. If the values of x are equidistant, we may verify that this equa- tion is the correct one to assume by the following method. Let the constant difference in the values of x be h. If we replace x by x + h, we get (y)s 4 3 2 1 8 9 10 y = a * and, therefore, for the difference in the values of y, Ay = y' - y = bh + ce dx (e dh - i). If Ay and Ay' are two successive values of Ay, then Ay' = bh + *(*+ (e* - i), and the difference in the values of Ay is A'y = Ay' - Ay = ct** (e dh - i) 2 . log A 2 y = log c (*> - i) 2 + (d log e) x. Hence, The last equation is of the first degree in x and log A 2 y so that the plot of (x, logA 2 y) will approximate a straight line. From this straight 154 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI line we may determine the constants log c (e dh i) 2 and d log e and therefore c and d in the usual way. We now write the equation in the form y ce dx = a -\- bx, and from the straight line plot of (x, y ce dx ), we determine the constants a and b. In Fig. 8ib we have plotted the equations y = 0.5 + x, y = 0.5 + x o.oi e 1 , y = 0.5 +- x o.ooi e x , y = 0.5 + x + o.oi e x , y = 0.5 +- x + o.ooi e x . Example. The following data are the results of experi- ments made with a gasometer by means of which the amount of air which passes into a re- ceiving tank can be measured; x is the vacuum in the tank in inches of mercury, y is the number of cu. ft. of air per minute passing into the tank. (Experiments made by W. D. Canan at the Mass. Inst. of Tech.) FIG. 8 1 b. X V ' r = y 1 y logr r. Vc A 8 17 49 0-32 9.5051 10 0.322 17 10 37 55 0.18 9.2553 - 10 0.179 37 12 So .61 O.II 9.0414 10 0.099 .51 o.oi H .62 .67 0.05 8.6990 10 0-055 .61 + 0.01 16 7i 73 0.02 0.031 70 +O.OI 18 .80 79 O.OI 0.017 77 +0.03 ' 20 85 85 0.009 .84 + 0.01 22 9i .91 0.005 90 +0.01 24 .96 97 O.OI 0.003 97 O.OI 26 .02 3 O.OI 0.002 3 o.oi 28 . 10 .09 o.oi O.OOI 09 +O.OI In Fig. Sic we note that the plot of (x, y) approximates a straight line for values of x > 14, and we shall fit an equation of the form ART. 8 1 THE EQUATION + bx + ce** 155 y' = a + bx to this part of the data. Using the method of averages and dividing the data into two groups of four and three sets, we have 7.27 = 40 + 76 b, 6.08 = 3 a + 78 b, .: b = 0.03, a = 1.25 and y' = 1.25 + 0.03*. 2.1 2.0 / p.ff 0.4 5.2 -r S 3.0 ~ .ff /* r s /_ f 1.9 1.8 1.7 1.5 1.4 1.3 1.2 1.1 1 n \ d S \ / \ ^ / ''^ S * y& ^ f \ , '', / f s* \ 7 A / \ A j \ / <*" 7 i n \ y ^ > 2 1 * \ 1 x YCt ^ Jrl [ X <^ ** ^^^ , n S /0 12 1.4 16 18 20 (x) FIG. 8 ic. 22 24 26 2S Now compute the values of y' and the residuals r = y' y (by taking r = y' y instead of r = y y', the residuals are positive and easier to handle in the subsequent calculations). Plot (x, r} for values of x < 14 and study the nature of this plot; this seems to be a simple exponential, r ce dx ] verify this by plotting (x, log r) and note that this plot approximates a straight line. Using the method of averages determine the constants in the equation log r ='log c + (d log e) x; thus 8.7604 10 = 2 log c + 18 d log e, 7.7404 10 = 2 log c + 26 d log e. 156 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI /. d log e = 9.8725 - 10 = -0.1275, logc = 0.5277. /. d = - 0.294, c = 3-37- /. log r = 0.5277 - 0.1275 x, and r = 3.37 e- - 894 * The final equation is y = 1.25 + 0.03 x - 3.37 e" - 294 *. Now compute y and the residuals, and note the close agreement be- tween the observed and calculated values. 82. The equation y = ae ba> + ce dx . A part of the experimental curve may be represented by a simple exponential y = ad 3 *, i.e., a part of the plot of (x, log y) approximates a straight line. We then study the deviations, r = y y c = y ae**, of this exponential curve from the rest of the experimental curve. The plot of (x, r} may be repre- sentable by another exponential, r = ce dx , where the values of r are negligible for that part of the experimental curve to which y = aeP* has been fitted. The entire curve can then be represented by the equa- tion y = a& x + ce dx . The equation y = aeP x -f- ce dx may fit an experimental curve although no part of the curve can be approximated by the simple exponential y = a^ x . If the values of x are equidistant, we may verify that this equation is the correct one to assume by the following method. Let the constant difference in the values of x be h. Consider three succes- sive values x, x + h, x + 2 h and their corresponding values y, y', y". We evidently have y = ae*> x + ce dx , y ' = ag* (*+>>) -f *(*+*> = ae* x (* h + ce dx e dh , y" = ae 6 (*+ 2 *) + <#*<*+*> = a#*& + ce dl e zdh . Now eliminate & x and e dx from these three equations by multiplying the first equation by g( b+d )' 1 , the second by (e 6 * + e dh ), and adding the results to the third equation. We get y" _ (gW + gdh-) y' or y = (ef' h + e dh ) y - This is an equation of the first degree in y'/y and y" /y so that the plot of (y'/y, y" /y) will approximate a straight line. From this straight line determine the constants e* h + c* and e^^, and hence b and d as usual. We now write the original equation ye~ dx = ae( b ~ d)x + c. This is a linear equation in gC^* and ye~ dx so that the plot of (e^ 6 -"")', ye"* 1 } would approximate a straight line. From this straight line determine the values of the constants a and c. ART. 82 THE EQUATION V = ae * + "* 157 In Fig. 820, we have plotted the equations y = e~ x , y = e~ x + 0.5 " t.O . x 1.5 (x) y = a<* x + ce dx FIG. 820,. Example. The following are the measurements made on a curve recorded by an oscillograph representing a change of current i due to a change in the conditions of an electric circuit t. (From Steinmetz, Engineering Mathematics.) 1 I logi ' r = t' i logr r e c A 2.10 O.3222 4.94 2.84 0-4533 2.85 2.09 -fo.oi O.I 2. 4 8 0-3945 4-44 1.96 0.2923 1 .96 2.48 0.2 0.4 2.66 2.58 0.4249 0.4Il6 3-99 3-22 1.33 0.64 0.1239 9.8062 10 1-34 0.63 2.65 2-59 +0.01 O.OI 0.8 2.00 O.30IO 2.10 O.IO 9.0000 10 0.14 1.96 +0.04 I .2 1.36 0-I33S i-37 O.OI 0.03 1-34 +O.O2 1.6 0.90 9.9542 - 10 0.89 0.01 O.OI 0.88 +O.O2 2.0 P.58 9.7634 - 10 0.58 o 0.58 O 2-5 0-34 9.5315 - 10 0-34 o o 0-34 3-0 O.20 9.3010 10 0.20 o 0.20 O In Fig. 826 we note that the right-hand part of the plot of (t, i) appears to be exponential. We verify the choice of i' = ad 1 by plotting (/, log i) and noting that this plot approximates a straight line for values of '58 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI i > 0.8. We therefore assume log i' = log a + (b log e) t, and using the method of averages for the values of t > 0.8, we have 9.8511 - 10 = 3 log a + 4.8 b log e, 8.8325 - 10 = 2 log a + 5.5 b log e. 0.6934, and bloge = 9.5356 - 10 = -0.4644, log a b = 1.07, a = 4.94, \ogi' = 0.6934 - 0.4644 /, or i' = 4.94 er l - mt . 5n n s* \ N \ ^ , ^ \ 4 f 7^ -^ s s \ 5 ' \ X s \ 3 7 \ 'N XN / \ N n 9 / \ \ \ 9 n \ \ \ n | ] \ \ ? \ % f, /) /) 1 \ \ 1 5 g ^ 1 , \ \ <> 9 9' \ \ s \ r? L V "N s >> \ ^"*" q o \ \ \ 5 1 A- i \ \ \ 97 1 \ \ \ \ \ \ g /j \ \ \ s. 1 \ n e y s T v \ ^ X v Q C /.t ^) V \ "s \, s ' \ ~. --, ^\ 9f \ \ ^ ^-^ Q V ^ \ ,9,9 C > 0. 5 /. 1. (t 5 ) 2. 2. 5 3.( ? Now find the values of i' and the residuals r = i' i; these residuals are practically negligible for values of t > 0.8. We plot (/, r) and try to fit an equation to this curve. This again appears to be exponen- tial and we verify this by plotting (/, log r) ; the plot approximates a ART. 83 THE POLYNOMIAL y = a + bx + c* 2 + dx? + 159 straight line, except for t = 0.8. We therefore assume r = ce di or log r = log c + (d log e) t. Using the method of averages for t < 0.8, we have 0.7456 = 2 log c + o.i d log e, 9.9301 10 = 2 log c + 0.6 d log e. .'. dloge = -1.6310, logc = 0.4544. .'. d = -3-76, c = 2.85, and log r = 0.4544 - 1.6310 /, or r = 2.85 g- 376 '. The final equation is i = 4.94 e~ l - mt 2.85 e- 3 - 1 . We now compute i and the residuals and note the very close agree- ment between the observed and computed values of i. 83. The polynomial y = a + bx + ex* -f- dx 3 + . The equa- tion y = a + bx + ex 2 may be modified by the addition of another term into y = a + bx + cx z + (fo 3 . If the values of x are equidistant, we may verify the correctness of the assumption of the last equation by the following method. Let the constant difference in the values of x be h. Then the successive differences in the values of y are Ay = (bh + cW + dh 3 } + (2 ch + 3 dtf) x + 3 dfcc, A 2 ? = (2 cW + 6 d& 3 ) + 6 d/* 2 x, A 3 ;y = 6 dh*. Hence the plot of (x, A 2 ;y) will approximate a straight line, and the values of A 3 3> are approximately constant. From the equation of the straight line we may determine the constants c and d, and writing the original equation in the form (y ex 2 dx 3 ) = a + bx, the plot of (x, ycx^ dx 3 ) will approximate a straight line, from which the constants a and b may be determined. Another method of determining the constants a, b, c, d in the equation y = a + bx + ex* + dx? consists in selecting four points on the experimental curve, substituting their coordinates in the equation, and solving the four linear equations thus obtained for the values of the four quantities a, b, c, and d. In a similar manner the polynomial y = a + bx + cx z -\- + kx n may be determined so that the corresponding curve passes through n + i points of the experimental curve; it is simply necessary to sub- stitute the coordinates of these n + i points in the equation and to solve the n + i linear equations for the values of the n + i quantities, a, b, c, . . . , k. If the values of x are equidistant, we can show that the plot of (x, A n-1 ;y) is a straight line and that A n ;y is constant, where A n-1 y and A";y are the (n i)st and nth order of differences in the values of y. Thus, if a sufficient number of terms are taken in the equation of the polynomial, this polynomial may be made to represent any set of data exactly; but it is not wise to force a fit in this way, since the deter- mination of a large number of constants is very laborious, and in many i6o EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI cases a much simpler equation involving fewer constants may give much more accurate results in subsequent calculations. We shall work a single example to illustrate the method of determin- ing the constants. Example. We wish to fit a polynomial equation to the following data: &.0 5.5 5.0 4.5 4.0 3.5 3.0 y) 2.5 2.0 1.5 n 20 / / /I i 0.19 0.19 0.14 n 12 j / 1 ^ / \ i >/ / / / ? / n iff / : / 0.08 06 / ^ / / 7 n 04 0.5 n X X Q Q9 ^ ^ n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (X) FIG. 83. z V Av AV A'y Vc A o o O.2I2 0.039 O.OI9 O.I O.2I2 o .251 0.058 O.OI4 0.2IO + O.002 0.2 .463 0.309 0.072 0.09 0.463 O o-3 772 0.381 0.091 o .0 9 0.770 + 0.002 0.4 153 0.472 O.IIO 0.08 I.I52 -j-o.ooi -5 -625 0.582 0.128 O.O I 1.625 0.6 .207 0.710 0.149 o.o 4 2.209 O.OO2 0.7 .917 0.859 0.163 o.o 8 2.92O 0.003 0.8 3-776 I .022 0.181 3-776 O 0.9 4.798 I .203 4-797 +O.OOI I.O 6 .001 S-998 +0.003 ART. 84 TWO OR MORE EQUATIONS 161 In Fig. 83 we have plotted (x, y). We form the successive differences and note that the third differences are approximately constant, and that the plot of (x, A 2 y) approximates a straight line (Fig. 83). We may therefore assume an equation of the form y = a + bx + cx z + dx 3 , or y = bx + ex 2 + dx 3 , since the curve evidently passes through the origin of coordinates. To determine the constants b, c, and d, select three points on the experimental curve; three such points are (0.2, 0.463), (0.5, 1.625), an< 3 (0.8, 3.776). Substituting these coordinates in the equation, we get 0.463 = 0.2 b + 0.04 c + 0.008 d, i .625 = 0.5 b + o. 25 c + 0.125 d, 3.776 = 0.8 b + 0.64 c + 0.512 d. Solving these equations for b, c, and d, we have b = 1.989, c = 1.037, d 2.972 and hence the equation is y = 1 .989 x + 1 .037 x* + 2.972 x 3 . We now compute the values of y and the residuals. 84. Two or more equations. It is sometimes impossible to repre- sent a set of data by a simple equation involving few constants or even by a complex equation involving many constants. In such cases it is often convenient to represent a part of the data by one equation and another part of the data by another equation. The entire set of data will then be represented by two equations, each equation being valid for a restricted range of the variables. Thus, Regnault represented the relation between the vapor pressure and the temperature of water by three equations, one for the range from 32 F. to o F., another for the range from o F. to 100 F., and a third for the range from 100 F. to 230 F. Later, Rankine, Marks, and others represented the rela- tion by a single equation. The following example will illustrate the representation of a set of data by two simple equations. Example. The following data are the results of experiments on the collapsing pressure, p in pounds per sq. in. of Bessemer steel lap-welded tubes, where d is the outside diameter of the tube in inches and / is the thickness of the wall in inches. (Experiments reported by R. T. Stew- art in the Trans. Am. Soc. of Mech. Eng., Vol. XXVII, p. 730.) t d p *j logP p. A 0.0165 225 8.2175 1 2.3522 230 I 0.0194 383 8.2878 - 10 2.5832 381 + 2 0.0216 524 8-3345 - 10 2.7193 533 -9 0.0214 536 8.3304 10 2.7292 Si? +19 162 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI 6000 tfcnn 8. (log */&) 2 8.3 S.4 2.8 2.7 ,st 2.4 2.3 i / i / I y * ^ / KOQO / / A > / 4500 4000 / j / I / j / / X 3500 (P) 3000 2560 2000 icnn f T / / /- + / ^ / V - r ^ \ J' i J 1000 ? / 500 n A" fi / , ^ T 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 O.JO (Vd) FIG. 84. t d P P e A * d P P e A 0.0228 S92 570 + 22 0.0370 1779 1821 42 0.0250 670 764 - 8 4 0.0374 1860 1856 + 4 0.0253 870 790 + 80 0-0375 1879 1865 + 14 0.0277 928 1002 - 74 o . 0400 2147 2085 + 62 0.0298 964 Il87 223 0.0403 2224 2112 + 112 0.0299 1184 II 9 6 12 o .0436 2280 2403 -123 0.0309 1251 1284 - 33 o .0442 2441 2455 14 0.0316 1319 1346 27 0.0477 2962 2764 + 198 0.0309 1419 1284 + 135 0.0527 3170 3204 - 34 0.0343 1680 1583 + 97 0.0628 4095 4194 - 99 0.0349 1762 1636 + 126 0.0815 SS6o 5741 -181 ABT. 84 TWO OR MORE EQUATIONS 163 It should be noted that a set of corresponding values of t/d and P are not the results of a single experiment but the averages of groups containing from two to twenty experiments. Following the work of Prof. Stewart, we have plotted (t/d, P), Fig. 84, and note that the experimental curve approximates a straight line for all values of t/d except the first four, i.e., for values of t/d > 0.023. We may therefore assume P = a + bl-j). If we use the method of W selected points to determine the constants a and b we may choose the points t/d = 0.065, P = 4250, and t/d = 0.030, P = 1215 as lying on the straight line; we then have 4250 = a + 0.065 6, 1215 = a + 0.0306. .'. b = 86,714, a = -1386 and P = 86,714^)- 1386. This result agrees with that given by Prof. Stewart. If we use the method of averages to determine the constants a and b we divide the last 22 sets of data into two groups of n each, and get 12,639 = 110 + 0.32316, 30.397 = 1 1 a + 0.5247 b. .: b = 88,085, a = -1438, and P = 88,055 Q - H38. In our table we have given the values of P computed from this last formula. The values of P computed from the first formula agree very closely with these. It is seen that the percentage deviations are in general quite small though large in a few cases, varying from 0.2 per cent to 10 per cent, which is to be expected from the nature of the experiments. We now attempt to fit an equation to the first four sets of data. The /A* k- addition of a modifying term of the formic (-3) or ce d to the above formula is not successful here. We shall therefore follow Prof. Stew- art's work and attempt to fit an equation of the parabolic form, P = /A 6 ft \ af-J . We verify this choice by plotting Mog-^, logP) and observing that this plot approximates a straight line. (The fewness of the ex- periments for values of t/d < 0.023 is a handicap here.) Assuming 164 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. V! and using the method of averages, we find 4.9354 = 2 log a + (6.5053 - 10) b, 5.4485 = 2 log a + (6.6649 - 10) b. .: b = 3.11, a = 80,580,000 /A3.11 and P = 80,580,000 (-, We compute the values of P from this formula. The entire set of data have thus been represented by means of two simple equations, each valid for a restricted range of the variables.* EXERCISES. [Note. The exercises which follow are divided into two sets. The type of equation that will approximately represent the empirical data is suggested for each example in the first set. For the examples in the second set, the choice of a suitable equation is left to the student.] I . Temperature coefficient ; r is the resistance of a coil of wire in ohms, 6 is the tem- perature of the coil in degrees Centigrade, [y = a + bx] r\ 10.421 I 10.939 11.321 I 11.799 I 12.242 12.668 10.50 | 29.49 42.70 | 60.01 | 75.51 91.05 2. Galvanometer deflection; D is the deflection in mm., I is the current in micro- amperes, [y = a + bx] 29.1 D T~| 0.0493 48.2 72.7 J2.O 118 140 165 J 54 o-i97 j. 0.234 0.274 199 0.328 0.0821 I 0.123 3. Volt-ampere characteristic of 118 volt tungsten lamp; e is the terminal voltage, is the current. \y = ax b ] e I 2 4 8 16 25 32 50 64 I loo 125 0.0245 0.0370 0.0570 e 0.0855 1 ISO 0.1125 1 80 0.1295 200 0.1715 218 O.2OOO 1 O.26O5 0.2965 t 1 0.3295 0-3635 0.3865 0.4070 4. Pressure- volume of saturated steam; v is the volume in cu. ft. of I pound of steam, p is the pressure in pounds per sq. in. [y = ax b ] 26.43 | 22.40 [ 19.08 14.70 I 17.53 | 20.80 16.2 I4-04 28.83 33-71 9-147 45-49 7-995 52-52 5. Chemical concentration experiment; x is the concentration of hydrogen ions, y is the concentration of undissociated hydrochloric acid. \y = ax b ] 1.68 1.22 0.784 1.32 0.676 0.216 0.426 "00747 0.092 0.047 0.0096 0.0049 0.00098 0.0085 0.00315 0.00036 0.00014 0.000018 6. Vibration of a long pendulum; A is the amplitude in inches, / is the time since it was set swinging. \y I 2 3 4 5 6 10 4-97 2-47 1.22 0.61 0.30 0.14 * Prof. Peddle in " The Construction of Graphical Charts " has fitted the equation t/d = 0.00274 ^P + o.ooooooooii P 2 to Prof. Stewart's data. EXERCISES 165 7. Newton's law of cooling; is the excess of the temperature of the body over the temperature of its surroundings, t is the time in seconds since the beginning of the experi- ment. \y - e bx ] 19.9 345 18.9 19-30 14.9 28.80 12.9 53-75 70-95 8.9 6.9 8. Barometric pressure; p is the pressure in inches of mercury, h is the height in ft. above sea level, [y = ae*>*\ 886 29 2753 4763 27 6942 10,593 9. Electric arc of length 4 mm. ; V is the potential difference in volts, * is the current in amperes. \y = * + - J 2-97 345 3-96 4-97 5-97 6.97 7-97 V 67.7 65-0 | 63.0 61.0 | 58.25 | 56.25 | 55.10 54.30 10. Speed of a vessel; H.P. is the horse power developed, is the speed in knots. H.P. 290 560 1144 ii 12 1810 2300 1 1 . Hydraulic transmission ; H.P. is the horsepower supplied at one end of a line of pipes, u is the useful power delivered at the other end. I = a. + bx 2 H.P. 150 96.5 138 172 .250 196 300 206 12. Magnetic characteristic of iron; H is the number of gilberts per cm., a measure of the field intensity, B is the number of kilolines per sq. cm., a measure of the flux density, [y = 13.0 14.0 154 16.3 17.2 40 17.8 60 80 18.5 18.8 13. Focal distance of a lens; p is the distance of the object, p' is the distance of its image. 140 22.50 1 23.20 1 23.80 1 24.60 1 26.20 29.00 14. Pressure-volume in a gas engine; p is the pressure in pounds per sq. in., v is the volume in cu. ft. per pound. \y = ax b + c] 53-8 85.8 113.2 7-03 5-85 1.90 3-50 | 2.50 15. Law of cooling; is the temperature of a vessel of cooling water, / is the time In minutes since the beginning of observation, [y = ae bas + c] '-, , i-^ ,- 74-5 I 92.0 85.3 79-5 67.0 60.5 53-5 45-0 39-5 i66 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI 1 6. Straw-fibre friction at 150 pounds pressure according to Goss's experiments; y is the coefficient of friction for a straw-fibre driver and an iron driven wheel, x is the slip, 0-153 0.179 0.213 "J 0.271 0.313 0-359 0.368 0.381 0.386 0.405 0.56 o. 5 8 f 0.61 0.411 0.78 0-432 0-99 0.458 1. 10 0.463 1.04 0.465 1.22 0-47 1.40 L_ i-75 1.94 2.OO 2.25 2-33 3-15 2-79 17. Expansion of mercury according to Regnault's experiments; j is the coefficient of expansion between o C. and t C. \y = a + bx + c* 2 ] 7 |0.< 150 250 300 360 .00018179 |o.oooi82i6 10.00018261 0.0001832310.00018403 0.00018500 0.00018641 18. Velocity of water in Mississippi River; v is the velocity, D is the depth. [y = a + bx + ex 2 ] O.I 0.2 0.3 0.4 0-5 0.6 0.7 0.8 0.9 3-1950 3.2299 3-2532 3.26II | 3.2516 3.2282 3.1807 3.1266 3-0594 2-9759 19. Solution of potassium chromate; s is the weight of potassium chromate which will dissolve in 100 parts by weight of water at a temperature of t C. [log y = a+bx+cx z ] o IO 27.4 42.1 61.5 62.1 66.3 70.3 - 20. Load-elongation of annealed high carbon steel wire of diameter 0.0693 an d gage length 30 in.; Wis the load in pounds, E is the elongation in inches, [y 50 100 1 150 1 200 1 225 ] 250 260 280 290 300 310 0.0130 0.0251 0.0387 (0.0520 0.0589 (0.0659 0.0689 0.0746 0.0778 0.0807 0.0842 33Q 340 350 0.0916 I 0.0980 | o.i i ii 21. Load-elongation of wire of Ex. 20 in hard-drawn condition; W is the load in pounds, E is the elongation in inches, [y = a -f- bx + cx d ] W 0.0280 0.0562 0.0849 400 1 500 600 700 800 850 900 0.115010.1471 0.1820 0.2191 0.2628 0.2879 0.3166 0.6 o-9 1.2 i-5 1.8 2.1 2-4 3-o 1.27 0.88 0.63 0.46 o-33 0-25 0.18 O.IO 22. Empirical curve. \y = ae*"" + ce das ] _JLl__J_M_ y I 3.00 | 1.89 23. Magnetic characteristic of iron; H is the number of gilberts per cm., a measure of the field intensity, B is the number of kilolines per sq. cm., a measure of the flux density (cf. Ex. 12). 3-o 8.4 6 8 10 1 1. 2 13.0 14.0 15 15-4 20 30 40 60 80 16.3 17.2 I 7 .8 18.5 18.8 24. Speed of a vessel; / is the indicated horsepower, v is the speed in knots. [y = a + bx + ex 1 + dx-] 8 9 10 II 12 13 14 15 16 17 18 1000 1400 1900 2500 3250 4200 5400 6950 8950 11,450 15,400 EXERCISES I6 7 25. Test on square steel wire for winding guns; 5 is the stress in pounds per sq. in., is the elongation in inches per inch. S 5000 I 10,000 20,000 30,000 40,000 50,000 60,000 70.000 80,000 E -I- |o.c OOOI9 O.OOO57 O.OOO94 O.OOI34 O.OOI73 O.OO2I6 O.OO256 O.OO297j 0.00343 100,000 110,000 0.00390 (0.00444 26. Flow of water over a Thomson gauge notch; Q is the number of cu. ft. of water, H is the head in feet. 4.2 6.1 1.6 8-5 n-5 2.4 14.9 23-5 27. Friction between belt and pulley; 9 is the arc of contact in radians between belt and pulley, P is the pull in pounds applied to one end of pulley to raise a weight W at the other end. ".- HIT 6 5.62 6.93 8.52 10.50 12.90 15.96 19.67 24.24 29.94 28. Electric arc of length 2 mm.; V is the potential difference in volts, i is the cur- rent in amperes. I 1.96 I 2.46 I 2.97 I 3.45 I 3.96 I 4.97 5.97 I 6.97 7.97 9.00 V\ 50.25 I 48.70] 47.90 I 47.50 I 46.80 I 45.70 45.00 I 44.00 43.60 43.50 29. Normal induction curve for transformer steel; H is the number of gilberts per cm., B is the number of lines per sq. cm. H\ _ I.O 1.3 2.1' B | 425 I 800 I 1750 2.9 2850 34 4.1 4300 I 6100 4-5 6725 5-2 5-9 7800 | 8600 Jio, 7-5 I 9-Q I 0,20o|lI,I50| II.O 12,200 30. Pressure- volume in a gas engine; p is the pressure in pounds per sq. in., v is the volume in cu. ft. per pound. i 39-6 44-7 53-8 V 10.61 9-73 8-55 85-8 113-2 135-8 178.2 6.23 5-18 4-59 3-87 31. Melting point of alloy of lead and zinc; 6 is the temperature in degrees Centi- grade, x is % of lead. 40 50 I 60 I 70 I 80 I 90 1 86 32. Empirical curve. 205 226 250 276, 304 5 7 9 II 13 11.03 14.03 17-53 21-55 26.12 6.42 I 8 33. Candle-power of an incandescent lamp; H is the age of the lamp in hours, C.P. is the candle-power. H C.P. 24.0 17-6 50Q 16.5 750 1000 1250 15.8 15.3 14.9 34. Insulation resistance-current passes through insulator and galvanometer; D is the deflection of the galvanometer, / is the time in minutes. 8.0 6.2 6 I 7 ... 5.0 | 4.4 4.0 9 3-5 3-3 II 12 13 3.0 | "27" | 2.5 JiJ-JS- 2.5 | 2.4 i68 EMPIRICAL FORMULAS NON-PERIODIC CURVES CHAP. VI 35. Experiments with a crane; / is the force in pounds which will just overcome a weight w. 8-5 12.8 300 400 500 25-6 600 I 70Q I 34-2 800 36. Copper-nickel thermocouple; t is the temperature in degrees, p is the thermo- electric power in microvolts. 24 37. Law of falling body; 5 is the distance in cm. fallen by body in / sec. 38. Loads, which cause the failure of long wrought-iron columns with rounded ends; P/a is the load in pounds per sq. in., l/r is the ratio of length of column to the least radius of gyration of its cross-section. l/r 140 1 80 220 260 300 340 380 420 P/a \ 12,800 7500 5000 3800 2800 | 2100 I 1700 1300 39. Heat conduction of asbestos; is the temperature in degrees Fahrenheit, C is the coefficient of conductivity. 32 212 | 392 572 752 1 1 12 1.048 1.346 | 1.451 1.499 1.548 1.644 40. Rubber-covered wires exposed to high external temperatures; C is the maximum current in amperes, A is the area of cross-section in sq. in. 3.2 5.9 9.0 22.0 42.0 68.0 84.0 I 102.0 0.001810 I 0.004072 I 0.007052 0.02227 0.05000 0.09442 0.1250 I 0.1595 41. Pressure-volume relation for an air compressor; p is the pressure, v is the volume. p | 18 21 | 26.5 | 33.5 | 44 62 0-635 0.556 | 0.475 I 0.397 I 0.321 0.243 42. Power delivered by an electric station; w is the average weight of coal consumed per hour per kilowatt delivered, / is the load factor. f 0.25 0.20 0.15 0.10 w 2.843 3-012 3-293 3-856 43. Temperature at different depths in an artesian well; is the temperature in degrees C., d is the depth. d 28 | 66 173 I 248 I 298 400 I 505 548 11.71 12.90 16.40 23.75 I 26.45 27.70 44. Resistance of copper wire; R is the resistance in ohms per 1000 ft., D is the diameter of wire in mils. D R 289 182 102 57 iS 10 3.234 10.26 | 32.80 | 105.1 45. Hysteresis losses in soft sheet iron subjected to an alternating magnetic flux; B is the flux density in kilolines per sq. in., P is the number of watts lost per cu. in. for I cycle per sec. 40 0.0022 I 0.0067 60 So 100 0.0128 j 0.0202 I 0.0289 0.0387 120 EXERCISES I6 9 46. Volt-ampere characteristic of a 60 watt tungsten lamp: V is the number of volts, / is the number of milli-amperes. V 2 5 10 20 3 40 50 60 70 1 80 90 100 I V 49 no 80 120 117 130 1 80 140 227 1150 272 160 3ii 170 348 1 80 383 1 4H 190 1 200 443 210 473 220 I 501 526 553 577 597 618 639 663 682 1 702 722 743 47. Calibration of base metal pyrometer (40% Ni and 60% Cu); V is the number of millivolts, / is the temperature in degrees F. 10 I 12 I 14 I 16 t | o 146 255 320 | 396 475 | 553 | 634 | 714 48. Tests on drying of twine; / is the drying time in minutes (time of contact of twine with hot drum), W is the percentage of total water on bone dry twine at any time, E is the percentage of total water on bone dry twine at equilibrium, d is the di- ameter of the twine in ins. (a) d = 0.102 ins., E = 18.7%. t o 0-44 0.88 i-3i i-75 W-E 29-5 15-4 9-4 5-1 3-i 6.2%. 1 i. ii 2.23 3-34 4-45 1 5-56 W-E 30.3 17.4 12.4 8.2 4-9 1 3-3 CHAPTER VII. EMPIRICAL FORMULAS PERIODIC CURVES. 85. Representation of periodic phenomena. Periodic phenomena, such as alternating electric currents and alternating voltages, valve-gear motions, propagation of sound waves, heat waves, tidal observations, etc., may be represented graphically by curves composed of a repetition of congruent parts at certain intervals. Such a periodic curve may in turn be represented analytically by a periodic function of a variable, i.e., by a function such ihatf(x -f- k) = f(x), where k is the period. Thus the functions sin x and cos x have a period 2 TT, since sin (x + 2 TT) = sin x and cos (x -\- 2 TT) = cos x. Again, the function sin 5 x has a period 2 7T/5, since sin 5 (x + 2 7r/5) = sin (5 x + 2 TT) = sin 5 x, but the func- tion sin x -\- sin 5 x has a period 2 TT, since sin (x -f 2 TT) + sin 5 (x + 2 TT) = sin x + sin 5 x. Now, any single-valued periodic function can, in general, be expressed by an infinite trigonometric series or Fourier's series of the form y = f(x) = a + a\ cos x + 0,1 cos 2 x + + a n cos nx -\- + 61 sin x + bz sin 2 x -f- -}- b n smnx -\- - - , where the coefficients a^ and bk may be determined if the function is known. This series has a period 2 TT. But usually the function is un- known. Thus, in the problems mentioned above, the curve may either be drawn by an oscillograph or by other instruments, or the values of the ordinates may be given by means of which the curve may be drawn. Our problem then is to represent this curve approximately by a series of the above form, containing a finite number of terms, and to find the approximate values of the coefficients a k and b k . The following sections will give some of the methods employed to determine these coefficients. 86. The fundamental and the harmonics of a trigonometric series. In Fig. 86a we have drawn the curves y = a v cos x, y = bi sin x, and y a\ cos x -f b\ sin x. The maximum height or amplitude oi y = a\ cos x is fli and the period is 2 TT. The amplitude of y = bi sin x is bi and the period is 2 TT. Now we may write 170 ART. 86 TRIGONOMETRIC SERIES 171 and letting Vc we may write y = Ci sin (x + = cos *), where c* = Va k 2 + ft* 2 and 0* = tan" 1 a k /b k . The wave represented by y c k sin (kx -f- <#>*) is called the kth har- monic, its amplitude is c k , its phase is fa, its period is 2 T/, since in Iktx + ^ + 0J = sin [k x + 2 T + **] = sin (feac + fc ), and its frequency, or the number of complete waves in the interval 2 T, 13 k. The trigonometric series is often written in the form y = c +ci sin Ot+'s of the form 3V = 0o + a>\ cos x r + + bi sin x r + + fffc COS kx r + + bk sin kx r -\- where r takes in succession the values o, I, 2, . . . , n I. We may now solve these n equations for the a's and &'s. We shall first state two theorems in Trigonometry concerning the sum of the cosines or sines of n angles which are in arithmetic progression, viz.: Tcos (a + f/3) = cos a + cos (a + )3) + cos (a + 2 0) + sin - + cos (a + [n - i] |8) = cos sin AKT. 88 DETERMINATION OF THE CONSTANTS sin (a + rj3) = sin a + sin (a + /3) + sin (a + 2 /3) + 175 sm - 2 . an If we let a = o and 8 = 1 , these become n 2 IT sin lir I (n i) TT 2, 2 TT sin iir L \n i TT . . , cos rl = ;- cos = O, since sin ITT = o, n . lir n sm n 2, 2?r sm lir . I (n l) IT . . . sin rl = r- sin = o, since sin lir = o. n . lir n * We may prove these theorems as follows: By means of the well-known trigonometric identities 2 cos u sin v = sin (u -\- v) sin (u v), 2 sin u sin v = cos ( ) cos (u + v) we may write the identities 8 I 8\ I p\ 2 cos a sin - = sm I a + - 1 sin I a -- 1 2 cos (a +5) sin | = sin f a+^?)-sinf a + | V 2cos(a+[-i]/3)sin = sin Adding, we get sin { a + ~~^ / sin^ -W.fS.!, Adding, we get 2 sin | ^sin (o+r/3) =cos fa - | J = 2 sin fa + - - /3j sii . w/3 sm 2 . -r sm 176 EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII for all values of / except / = o, when ^cos rl = 5} cos o = n, ** n ** I = n, when 5} cos ^ = 5} *** cos zrir = n. Since x r = r , we may finally state that o, except when / = o or / = n = n, when / = o or / = n. ^sin lx r = o for all values of /. To determine a we merely add the n equations, and get 5j;y r = na + + a k ])cos kx r + + a k J)sin kx r + = na , since all the other terms vanish. To determine a k we multiply each of the n equations by the coefficient of ak in that equation, i.e., by cos kx r , and add the n resulting equations; we get 2)y r cos kx r = a 2J COS kx r + - + a* 2} cos 2 kx r + ~\~ a p 2) COS pX r COS kx r + ' ' ' + 6p^sln pX r COS kx r + . Now, ]vcos kx r = o; cos kx r * = % J) cos (P + *) Xr + ^ 2} cos ^ ~ ^^ r = 0; cos fe^ r * = ^ ^sin (^? + k) x r + | ^sin (p k) x r = o; ., , n = n, if k = 2 * We use the trigonometric identities 2 cos a cos ti = cos ( + u) + cos (M t>). 2 sin cos v = sin ( + v ) + sin ( i). 2 sin u sin = cos ( ^) cos ( + ). ART. 88 Hence, DETERMINATION OF THE CONSTANTS r cos kx r = - a k , except when k - i t , when k = - 177 To determine bk we multiply each of the n equations by the coefficient of b k in that equation, i.e., by sin kx r , and add the n resulting equations; we get r sin kx r in kx r Now, + b k sin kx r r sin kx r r sin kx r * = r sin kx r * = p) x r - (i cos 2 fcc r ) = - -- 2 2, 2. Hence, ^ r sm ^ !Cr " ~~ ^ Collecting our results, we have finally o = - 2yr = - (yo + y\ l (y - "2 - (y O r sin ^ P = - (y6 sin ^ (k p} x r =* o; s (^ f ^) *r - o; = - . if 2 = o, if k = . + y n -i cos kx n -J , + ^-i sin kx n . L ). * We use the trigonometric identities 2 cos cos = cos (M + f ) + cos (w ). 2 sin M cos v = sin (M + ) + sin (M v). 2 sin sin t - cos (u v) cos ( + p). 178 EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII If n is an even integer, -our periodic curve is now represented by the equation y = a + ai cos x + + a k cos kx + + a n cos - x x + + bk sin fcc -f- + & sin I i ) x. 2~ l \ 2 / The n coefficients are determined as above. Thus a is the average value of the n ordinates. a n is the average value of the n ordinates taken alternately plus and ak or bk is twice the average value of the products formed by multiply- ing each ordinate by the cosine or sine of k times the corresponding value of*.* We note that each coefficient is determined independently of all the others. If we wished to represent the periodic curve by a Fourier's series con- taining n terms, but had measured m ordinates, where m > n, we should have to determine the coefficients by the method of least squares. The values of the ordinates as computed from this series will agree as closely as possible with the values of the measured ordinates. It may be shown that the expressions for the coefficients obtained by the method of least squares have the same form as those derived above. f * We may also derive the values of the coefficients as follows: In Art. 87, we have shown that J * y cos kx dx = ak J * cos 2 kx dx, since all the other terms vanish. If we replace the integrals by sums, and take for dx the interval 2 v/n, this becomes >.yr cos kx r = ak ^cos 2 kx r = ak, if k ^ o or k ^ = nak, if k = o or k = Hence, Vy r = noo, Vy r cos - x r = na n , ^y r cos kx r = Similarly we may show that ^y r sin kx r = bk. t See A Course in Fourier's Analysis and Periodogram Analysis by G. A. Carse and G. Shearer. ART. 89 NUMERICAL EVALUATION OF THE COEFFICIENTS 179 We shall illustrate the use of the above formulas for the coefficients by finding the fifth harmonic in the equation of the periodic curve passing through the 12 points given by the following data (Fig. 89). I I COS 5 Z sin 5 z y cos 5 z V sin 5 z 9.3 1.000 0.000 9.30 0.00 30 15.0 -0.866 0.500 -12.99 7.50 60 17.4 0.500 -0.866 8.70 -15.07 90 23.0 0.000 1.000 0.00 23.00 120 37.0 -0.500 -0.866 -18.50 -32.04 150 31.0 0.866 0.500 26.85 15.50 180 15.3 -1.000 0.000 '-15.30 0.00 210 4.0 0.866 -0.500 3.46 - 2.00 240 - 8.0 -0.500 0.866 4.00 - 6.93 270 -13.2 - 0.000 -1.000 0.00 13.20 300 -14.2 0.500 0.866 - 7.10 -12.30 330 - 6.0 -0.866 -0.500 5.20 3.00 2 = 3.62 - 6.14 cos 5 x r = 0.60; r sin 5 * = I - 02 - Hence the fifth harmonic is 0.60 cos 5 x 1 .02 sin 5 x. It is evident that the labor involved in the direct determination of the coefficients by the above formulas is very great. This labor may be reduced to a minimum by arranging the work in tabular form. These forms follow the methods devised by Runge * for periodic curves in- volving both even and odd harmonics (Art. 89), and by S. P. Thompson f for periodic curves involving only odd harmonics (Art. 90). 89. Numerical evaluation of the coefficients. Even and odd har- monics. (I) Six-ordinate scheme. Given the curve and wishing to determine the first three harmonics, i.e., the 6 coefficients in the equation y = OQ + a\ cos x + 02 cos 2 x + a 3 cos 3 x + bi sin x + bz sin 2 x, we divide the period from x = o to x = 360 t into 6 equal parts and * Zeit. f. Math. u. Phys., xlviii. 443 (1903), Hi. 117 (1905); Erlauterung des Rech- nungsformulars, u.s.w., Braunschweig, 1913. t Proc. Phys. Soc., xix. 443, 1905; The Electrician, 5th May, 1905. t If the period is taken equal to 2 ir/m instead of 2 IT, the representing trigonometric series has the form y = Oo + Ci cos md + a? cos 2 md + + bi sin md + > 2 sin 2 md + , where represents abscissas. By the substitution mO = x or = x/m, the series be- comes . N V = Oo + a,\ cos x + 02 cos 2 x + A + 61 sin x + 6j sin 2 x + , and this has a period 2 TT. The abscissas from 6 = o to 6 = 2 r/m now become the abscissas from x = o to x = 2 IT, and we proceed to determine the coefficients in the second series as outlined. Having determined the coefficients, we finally replace x by me. l8o EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII measure the ordinates at the beginning of each interval; let these be rep- resented by the following table: y 6o c :8o c y* 240 y* Here n = 6, and using the formulas on p. 177, we have = ^0 +yi +^2 +ya +^4 = y yi -\-y-i y 3 +y* 3 bz = ^osino -\-y\ sin 120 +;y 2 sin24O +y3sin36o +^ 4 sin48o 4-y 5 sin6oo We arrange the y's in two rows, 3-0 yi y* y* Sum VQ Vi 1)% v 3 Diff. w, w z where the z/s are the sums and the w's are the differences of the quantities standing in the same vertical column ; thus, v = y , v\ = y\ + 3*5, Wi = yi 3/5, etc. Since cos 240 = cos 120, cos 300 = cos 60, etc. We may now write 6 a = z>o + Vi + % + v 3 3 ai = v + vi cos 60 + v 2 cos 120 + v 3 cos 180 3 az = v + Vi cos 120 + z/2 cos 240 + v 3 cos 360 3 61 = Wi sin 60 + w-i sin 120 3 b z = Wi sin 120 + ^2 sin 240 We arrange the z>'s and w's in rows, Sum po Pi r\ Diff. go 2i 5i and we now write 6ao = po + pi, 6a 3 = qo q\, 3 ffi = So + i 2i, 3 2 = po - i pi, 3*i= ^r lf 3 *. = ^5i. Example. Determine the first three harmonics for the following data taken from Fig. 86&. ART. 89 NUMERICAL EVALUATION OF THE COEFFICIENTS 181 x o 6o 120 I 80 240 300 ,y_ 0.47 1.77 0.47 2. 2O 2. 2O 1.77 2.20 0.49 1.64 -1.64 2.20 -0.49 v 0.47 1.28 0.56 2.20 w 2.26 3.84 0.47 1.28 2.26 2. 2O 0.56 3.84 p -1-73 1-84 r 6.10 q 2.67 0.72 s -1.58 600 = o.n, 6a 3 = 1.95, 301 = 3.03, 302 = -2.65, 3 bi = 5-28, 362 = -1.37- Hence, OQ = 0.02, a\ = = i.oi, 02 = 0.88, as = 0.33, bi = 1.76, & 2 = 0.46, and y = 0.02 + i.oi cos * 0.88 cos 2 x + 0.33 cos 3 x + 1.76 sin x 0.46 sin 2 jc. The equation from which the curve in Fig. 866 was plotted was = cos x 0.87 cos 2 +0.35 cos 3 x + 1. 73 sin +0.50 sin 2 #0.35 sin 3 x. We observe the close agreement between the two sets of coefficients, the small discrepancies being due to the approximate measurements of the ordinates for our example. (II) Twelve-ordinate scheme. Given the curve and wishing to deter- mine the first six harmonics, i.e., the 12 coefficients in the equation y = a + a\ cos x + 3 W v& v$ Vi w Sum po pi pz PS T\ Diff. qo qi qz Si the equations may be written 12 a = qo 4~ i cos 60 4- pz cos 120 + p 3 cos 180 6 bi = r\ sin 30 6 & 2 = ^i sin 6o c + r-i sin 60 + ra sin 90 + 5 2 sin 120 Finally, if we arrange the p's, g's, and r's as follows : 2 Pi P3 Sum /, the equations become 12 Oo Diff. I2fl 6 = IQ 6ai 6 dz 6a 3 sin 60 + q 2 sin 30. 6 a 5 = g 2i s ' m 60 + 22 sin 30. -/>2 sn 30 6 a 4 = 6 6 3 = sin 30 6 61 = ri sin 30 + r 2 sin 60 -f- r 3 . 6 6 5 = r\ sin 30 r 2 sin 60 + r s . 6 6 2 = (si + 5 2 ) sin 60. = (si s 2 ) sin 60. ART. 89 NUMERICAL EVALUATION OF THE COEFFICIENTS 183 We may now arrange the above scheme in a computing form as fol- lows: Ordinates y yi yz yz y* y& yt yn yio y y yi Sum v { ) Vl Vz V$ V V 5 V, Diff. w\ Wz Wz W\ W& Sum po Diff. go Pi p2 2i 2z Pi p3 fl Ti 7*3 S\ Sz *i 9.0 Pz p3 ?3 ff2 Sum lo k Diff. /i tz Multipliers of the quan- tities in the same horizontal rows before Cosine terms Sine terms these are entered sin 30 = 0.5 ?2 -Pi Pi n sin 60 = 866 9i r 2 Si Sl sin 90 = 1.0 tfo Po -ps <2 la h r ti Sum of 1st column Sum of 2d column Sum 6d 6a 2 6 a s 12 a 661 66, 66, Difference 6a s 6a 4 12 a. 6h 6&4 Checks : y = ^o + fli + CL-L + 03 + ^4 + a& + a$. yi - yu = (&i + h) + \/3 (6, + 64) + 2 ft,. Result: y = OQ + i cos x + ag cos 2 x + + a 6 cos 6 x + fti sin x + bz sin 2 oc -J- + fts sin 5 . /r FIG. 89. i8 4 EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII Example. In the periodic curve of Fig. 89, the interval from x = o to x = 360 is divided into 12 equal parts and the ordinates y to yn are measured. o" 9-3 15.0 60 17.4 90 120 150 1 80 210 240 270 300 330 23.0 37-o 31.0 15-3 4.0 -8.0 -13.2 -14.2 -6.0 We shall determine the first six harmonics by the above scheme. Ordinates 9.3 15.0 17.4 23.0 37.0 31.0 15.3 6.0 14.2 13.2 8.0 4.0 Sum (i) 9.3 9.0 3.2 9.8 29.0 35.0 15.3 Diff. (w) 21.0 31.6 36.2 45.0 27.0 9-3 9.0 3.2 9.8 21.0 31-6 36.2 15-3 35-0 29.0 27.0 45-0 Sum (p) 24.6 44-o 32.2 9.8 W 48.0 76.6 36.2 \ Diff. (g) -6.0 26.0 -25-8 (*) -6.0 -13-4 24.6 44.0 48.1 -6.0 32.2 9.8 36.2 -25-8 Sum (/) 56.8 53-8 Diff. (/) 11.9 19.8 Multipliers Cosine terms Sine terms 0.5 0.866 t.O -12.9 -22.5 -6.0 -16.1 22.0 24.6 -9.8 19.8 56.8 53.8 24.0 66.3 36.2 -5.2 -11.6 11.9 Sum of 1st col. Sum of 2d col. -18.9 -22.5 8.5 12.2 56.8 53.8 60.2 66.3 - 5.2 -11.6 Sum Diff. -41.4 = 60! 3.6 = 6o 6 20.7 = 6a 2 -3.7=604 19.8 = 6a 3 110. 6=12 a c 3.0=12o 6 126. 5 = 6 6j -6.1 = 665 -16.8=66j 6.4=664 11.9 = 66 3 d= 6.90, 02 = 3.45,03 = 3.30, a = 9.22, bi =21.08, bz= 2.80, 63 = 1.98, 05 = 0.60, #4= 0.62, 05 = 0.25, 6 5 = i. 02, 6 4 = 1.07. Check: 9.3 = 9.22 - 6.90 + 3.45 + 3.30 - 0.62 + 0.60 + 0.25 = 9.30. 21.0 = (21.08 1.02) + 1.732 ( 2.80 + 1.07) + 2^(1.98) =21.02. Result: * y = 9.22 6.90 cos x + 3.45 cos 2 x + 3.30 cos 3 x 0.62 cos 4 x + 0.60 cos 5 x -f 0.25 cos 6 x + 21.08 sin x 2.80 sin 2 x + 1.98 sin 3 x + 1.07 sin 4 x 1.02 sin 5 x, or y = 9.22 + 22.18 sin (x 18.12) 4.44 sin (2 x 50.93) + 3.85 sin (3 x + 59-04) + 1-24 sin (4 x - 30.09) 1. 1 8 sin (5 x 30.47) 0.25 sin (6 x 90). * The coefficients of the fifth harmonic agree with those found by the direct process in Art. 88. The time and labor spent in the computation of all six harmonics by means of the above computing form is much less than that spent in the determination of the fifth harmonic alone by the direct process in Art. 85. ART. 89 NUMERICAL EVALUATION OF THE COEFFICIENTS The last result was obtained by using the relations dk cos kx + bk sin kx = Ck sin (kx + y& Sum Si $z s 3 54 Ss s$ Difi. di dz d 3 di d$ d$ replace the trigonometric functions by their values in terms of the sines of 15, 30, 45, 60, 75, 90, and collect terms, we may write 6ai = 4sini5+d4sin3O -f-i sin 30 = 0.5 d* d 4 St Sl sin 45 = 0.707 d 3 e\ -d* 83 Tl s t sin 60 = 0.866 dt -d 2 4 Si sin 75 = 0.966 di d 6 Si Si sin 90 = 1.0 -d< S 6 r* it Sum of 1st col. Sum of 2d col. Sum 6d 6 as 6a 5 661 66, 66 6 Diff. 6 a a 6 a 9 6 a 7 6611 66 9 6b 7 Checks: + an = O, Result : y = ai cos x + a 3 cos 3 x + + an cos 1 1 x + b\ sin x + b 3 sin 3 x + + b\\ sin 1 1 x. 50- I 15 30 45 60 75 90 105 120 136 150 165 FIG. 906. Example. Fig. 906 represents a half-period of an e.m.f. wave whose frequency is 60 cycles. We wish to find the odd harmonics up to the nth order. Choose the x-axis midway between the highest and lowest points of the complete wave and the origin at the point where the wave crosses the #-axis in the positive direction. Divide the half-period into 12 equal ART. 90 NUMERICAL EVALUATION OF THE COEFFICIENTS parts and measure the ordinates yi, y% . . . , yn. These are given in the following .table: x I 15 I 30 45 60 I 75 90 105 120 135 I 150 I 165 30 y I 4 I 21 19 t 27 | 29 33 46 38 50 We arrange the work in the above computing form. 4 21 19 27 29 33 37 + 69 - 75 = 31 33 30 50 38 46 51-33 Sums (s) 33 37 51 Diff. (d) 29 9 69 65 75 31 11 17 33 = 18 = -29 + 31 + 17 = 19 = Multipliers Cosine terms Sine terms 0.259 0.5 0.707 0.866 0.966 1.0 -4.4 -5.5 -21.9 -7.8 -28.0 13.4 11.0 -7.5 -5.5 21.9 7.8 -16.4 9.6 25.5 48.8 56.3 72.5 33.0 21.9 18.0 19.4 25.5 -48.8 -56.3 35.7 33.0 Sum 1st col. Sum 2d col. -13.3 -54.3 11.0 13.4 2.3 -2.0 130.9 114.8 21.9 18.0 6.3 2.2 Sum Diff. -67.6 41.0 24.4 -2.4 0.3 4.3 245.7 16.1 39.9 3.9 8.5 4.1 Divide by 6 d =-11.27 a u = 6.83 a 3 = 4.07 a 9 = -0.40 05 = 0.05 07 = 0.72 61 =40.95 6,1= 2.68 63 = 6. 65 69 = 0. 65 65 = 1.42 67=0.68 +an= -11.27+4.07+0.05+0.72-0.40 + 6.83 = -b n = 40.95 -6.65 + 1.42 -0.68 +0.65 -2.68 = 33.01= Check: ai+a 3 + bi-b 3 + - Result: y = 1 1 .27 cos x + 4.07 cos 3 x + 0.05 cos 5 x + 0.72 cos 7 x 0.40 cos 9 x + 6.83 cos 1 1 x + 40.95 sin x + 6.65 sin 3 # + 1 .42 sin 5 x + 0.68 sin 7 x + 0.65 sin 9 x + 2.68 sin 1 1 x. (Ill) Odd harmonics up to the seventeenth. Given a symmetric curve and wishing to determine the coefficients in the equation y = a\ cos x + 0,3 cos 3 x + + an cos 1 7 x + bi sin x + b s sin 3 x + + bn sin 17 x, we choose the origin at the point where the wave crosses the axis, so that yo = o, divide the half -period into 18 equal parts, and measure the 17 ordinates yi, y t , . . . , y n . Thus we have X 10 20 30 170 y y\ y* y 3 . . . yn If we use the same method as that employed in deriving the n-ordi- nate scheme, we shall arrive at the following 17-ordinate computing form. This form is self-explanatory. ' ' , 192 EMPIRICAL FORMULAS PERIODIC CURVES yn y\6 yi& yu. y\s yu yn CHAP. VII Sum $i Sz S3 S^ Si Sfi S 7 s s* Diff. d l d* dz d, d 5 d, d 7 d, Si s* S3 r\ dz d, de 1 -si S4 ~S 9 -r s -d, -d, C3 Sum Multipliers Cosine terms Sine terms sin 10 = 0.1737 sin 20 -0.3420 sin 30 0.5000 sin 40 0.6428 sin 50 0.7660 sin 60 0.8660 sin 70 0.9397 sin 80 0.9848 sin 90 = 1.0000 d, d< d 6 d s d* d a d 2 d, Ci e-i e 3 -dt -d* d 6 di di -d> -d t d 7 d< di d, -d 7 -d z -d 3 -di d. e\ S 2 83 S 4 S 5 Se s^ Si sg Ti r-t r.-i s; 84 S3 Si s t S 2 Si s, 83 S2 s? Se 4 r* Sum of 1st col. Sum of 2d col. Sum Diff. 9 0l 9 an 9o 3 9a 15 9a 6 - 9a 13 9a 7 9 a n 9fl9 9 61 9& 17 90s 96 16 9 6 5 96,3 9 6 7 96n 96, Check: a\ + a 3 + a 5 + = o, Result : y = a\ cos x + a 3 cos 3 x + + an cos 1 7 x + bi sin x + 63 sin 3 x + + bn sin 1 1 x. Similar computing forms may be constructed for symmetrical waves containing odd harmonics up to the seventh, ninth, etc., orders. 91. Numerical evaluation of the coefficients. Averaging selected ordinates.* We are to determine the coefficients in the trigonometric series y = a + ai cos x + az cos 2 x + -f- a* cos kx -\- + bi sin x + b% sin 2 x + + bk sin kx + Let a n and b n represent the coefficients of any harmonic. We divide the period 2 IT into n equal intervals of width 2 ir/n and measure the ordi- nates at the beginning of these intervals. We have the table x I x * These methods have been developed by J. Fischer-Hinnen, Elekrotechm'sche Zeitschrift, May 9, 1901, and S. P. Thompson, Proc. of the Phys. Soc . of London, Vol. XXIII, 1911, p. 334. See, also, a description of the Fischer-Hinnen method by P. M. Lincoln, The Electric Journal, Vol. 5, 1908, p. 386. ART. 91 NUMERICAL EVALUATKJN OF THE COEFFICIENTS 193 Substituting these pairs of values in our series, we have n equations of the form y r = 00 + 0,i COS X T + 02 COS 2 X r + + d k COS kx r + - + bi sin x r + b z sin 2 x r + + b r sin kx r -K , where r takes in succession the values o, 1,2,3, ...,n i; adding these n equations, we get where the summation is carried from r = otor = n I. If we let /3 = jfe in the expressions for V cos (a + r|8) and w 2 sin(a + r|8) derived in the note on p. 175, these become v^ / , 2ir\ sin kir / . k(n I)TT\ . . . >,cos a+r = - ,, . .cos ' +3?1 yi+ ' ' ' +^-1 -^'n-l)- The first set of n ordinates start at x = ir/2 n and are at intervals of 2 ir/n, and the second set of n ordinates start at x -- (- - and are at 2 n n intervals of 2 tr/n ; thus the period from x = ir/2 n to x = 2 IT + ir/2 n is divided into 2 n equal parts each of width - Hence, n If, starting at x ir/2 n, we measure 2 n ordinates at intervals of ir/n, the average of these ordinates taken alternately plus and minus is equal to the 196 EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII sum of the amplitudes, taken alternately plus and minus, of the nth, 3 nth, 5 nth, . . . sine components. Thus to determine the sum of the amplitudes, taken alternately plus and minus, of the 5th, I5th, 25th, . . . sine components, merely average the 10 ordinates taken alternately plus and minus, at intervals of 180 -r- 5 = 36, starting at x = 180 -5- 10 = 18, i.e., at x = 18, 54, 90, . . ., 342 (Fig. 9ic); therefore b& - 6i6 + 625 - = T V CXis - 3*54 + 3*90 - 3*126 + 3*162 - ym + 3*234 ~ 3*270 + 3*306 - ^42). If the I5th, 25th, . . . harmonics are not present, then &5 = lv (3*18 - 3*&4 + 3*90 - ym + ym - ym + 3*234 - 3*270 + 3*306 - 3*342). We may also note that the set of 2 n ordinates measured for deter- mining the 6's lie midway between the set of 2 n ordinates measured for determining the a's, so that to determine any desired harmonic we actually measure 4 n ordinates, starting at x = o and at intervals of 7T/2 . We use the 1st, 3d, 5th, ... of these ordinates for determining a, and the 2d, 4th, 6th, ... of these ordinates for determining b. If the higher harmonics are present, these must be evaluated first. The absolute term a is obtained from the relation yo = a + a\ + a* + a s + . We shall now illustrate the methods developed by an example. Example. Given the periodic wave of Fig. 89 and assuming that no higher harmonics than the 6th are present, we are to determine the co- efficients in the equation y = a + fli cos x + o 2 cos 2 x + + a 6 cos 6 x + b\ sin x + bz sin 2 x + -f b 6 sin 6 x. To determine a 6 and b 6 measure 12 ordinates at intervals of 30 be- ginning at x = o and x = 15 respectively (Fig. 916); then FIG. ART. 91 NUMERICAL EVALUATION OF THE COEFFICIENTS 197 = T* (jo - yso + iyeo - ysx) + ' + ym - ym) = A (9-3 - 15-0 + 174 - 23.0 + 37.0 - 31.0 + 15.3 - 4.0 - 8.0 + 13.2 - 14.2 + 6.0) = 0.25. = iV (yi5 ^45 + yib ym + + yzu, y^b) = rV (13-0 - 16.0 + 19.5 - 31.0 + 35.3 - 23.8 + 10.5 + 5.7 - 10.0 + 14.5 - 11.0-0.5) = 0.52. 30- 70, 234 270 -10- FIG. 9 ir. To determine a 5 and 65 measure 10 ordinates at intervals of 36, beginning at x = o and x = 18 respectively (Fig. 91^) then 0-5 = iV (yo - ^36 + yn - yios + + ym - ym) = iV (9-3-I5-3-I-I8.8-32.8+33.0-I5.3-I.O+9.5-I5-0+84) = 0.04. b& = TV (yis yiA + yw ym + + ym ywt) = T V (13-8-16.8+23.0-36.8+25.5-9.0-7.7+13.4-13.2+1.5) = -0.63. FIG. gid. 198 EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII To determine a 4 and bi measure 8 ordinates at intervals of 45, be- ginning at x = o and x = 22\ respectively (Fig. 91^); then 04 = I (y* - y^ + ^90 - ym + + y m - y> K ) = I (9-3 - 16.0 + 23.0 - 35.3 + 15.3 + 5.7 - 13.2 4- ii.o) = -0.03. &4 = | (^22.5 ^67.5 + ^112. 5 + ^292.5 ~ ^SST.s) = I (H-S - 18.0 + 35.0 - 27.7 + 7.7 + 8.8 - 14.7 + 3.0) = i .08. To determine a 3 and b$ measure 6 ordinates at intervals of 60, be- ginning at x = o and x = 30 respectively (Fig. 916); then as = i (yo - yw + yuo - ym + ym - ym) = I (9-3 - 174 + 37-0 - 15-3 - 8.0 + 14.2) = 3.30. bz = \ (^30 ^90 + ym yzio + ^270 ~ ^SSo) = i (i5-0 - 23.0 + 31.0 - 4.0 - 13.2 + 6.0) = 1.97. To determine a^ and bz measure 4 ordinates at intervals of 90, beginning at x = o and x = 45 respectively (Fig. 916); then 02 + a 6 = I (yo - :V9o + ym -yvo) = I (9-3 - 23.0 + 15.3 + 13-2) = 3-7O, .-. 02 = 345- b* - b 6 = \ (y& - ym + 3^25 - y 3 is) = I (16.0 - 35.3 - 5.7 + 11.0) = -3.50, .-. 6 2 = -2.98. To determine a\ and bi measure 2 ordinates at intervals of 180, beginning at x = o and x = 90 respectively (Fig. 916); then an + a 3 + a & = \ (y Q - y m ) = \ (9.3 - 15.3) = -3.00, /. ai = -6.26. bi - b 3 + b 5 = (y 90 - ym) =\ (23.0 + 13.2) = 18.10, /. 61 = 20.60. To determine a we have a + ai + a 2 + as + a 4 + a & + fl e = yo = 9-3. * <*<> = 8.63. Result: y = 8.63 6.26 cos x + 3.45 cos 2 x -\- 3.30 cos 3 x 0.03 cos 4 x 0.04 cos 5 x + 0.25 cos 6 x + 20.60 sin re 2.98 sin 2 x + i .97 sin 3 x + i .08 sin 4 x 0.63 sin 5 x + 0.52 sin 6 re. This result agrees quite closely with that of Art. 89, p. 184; the differ- ences in the values of the coefficients are due to the fact that by the method of Art. 89 only the ordinates at o, 30, 60, . . . , 330 are used, whereas by the method of this Art. a large number of intermediate ordi- nates are used. If the curve is drawn by some mechanical instrument, the present method will evidently give better approximations to the values of the coefficients; but the labor involved in using the computing form on p. 183 is much less than that used in measuring the selected ordinates above. 92. Numerical evaluation of the coefficients. Averaging selected ordinates. Odd harmonics only. If the axis is chosen midway between the highest and lowest points of the wave and the second half-period is ART. 92 NUMERICAL EVALUATION OF THE COEFFICIENTS merely a repetition below the axis of the first half-period, then only the odd harmonics are present. If the ordinates at x = x r and x = x r + TT are designated by y r and y r +v respectively, then y r+K = y r . In the method of averaging selected ordinates, the 2 n ordinates are spaced at intervals of ir/n and are taken alternately plus and minus; then y r+T is at a distance TT = n (ir/ri), or n intervals, from y r , and since n is odd, y r+r will occur in the summation with sign opposite to that with which y r occurs, so that, e.g. + - v + 3V I 2W - y 0+w + ;y 1+ ;' - '+ . . . 2 y r = -(3^-yi'+ 3v ). n Hence we need merely divide the half-period into n equal intervals and average n ordinates. We may therefore restate our rules for determining the coefficients if the wave contains odd harmonics only. If, starting at x = o, we measure n ordinates at intervals of ir/n, the average of these ordinates taken alternately plus and minus is equal to the sum of the amplitudes of the nth, 3 nth, 5 nth, . . . cosine components. If, starting at x = ir/2 n, we measure n ordinates at intervals of TT/W, the average of these ordinates taken alternately plus and minus is equal to the sum of the amplitudes, taken alternately plus and minus, of the nth, 3 nth, 5 nth, . . . sine components. Furthermore, a = o since the sum of the ordinates over the entire period is zero. FIG. 92. Example. Assuming that the symmetric wave of Fig. 92 contains no higher harmonics than the 5th, we are to determine the 1st, 3d, and 5th harmonics. Applying the above rules we have EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII 05 = i (yo - y6 + yn - y = J (o 8.6 + 6.3 27.7 + 19.0) = 2.20. b 6 = I (yis-yM+ygo-ym+yiwH i (11.3-2.7+20.5-25.5+10.7) = 2.86. &s = % (yo - yeo + yw) = l(o - 2.8 + 26.5) = 7.90. b 3 = % (yso - yw + yiso) = ? (10.7 - 20.5 + 16.6) = 2.27. a\ + a 3 + a 5 = i (yo) = o, .*. ai = 5.70. bi - b 3 + 6 5 = 1 (y) = 20.5, .'. &! = +19.91. Result: y = 5.70 cos x + 7.90 cos 3 x 2.20 cos 5 x + 19.91 sin x + 2.27 sin 3 x + 2.86 sin 5 x. We may compare this result with that obtained for the same curve by the use of the computing form on p. 187. If only the 1st and 3d harmonics had been present in the above wave, we should have as = $ (yo y< + yiaO ; b 3 = I (y 3 o yw + yiw) ; 01 + 3 = yo = o; 61 - 6 3 = ysx). If all the odd harmonics up to the ninth had been present in the above wave, we should have 09 = i (yo y2o + yio yeo + yso yioo + y\ ym + yi0 ; 69 = i (yio - yso + yso - y?o + y 9 o - yno + yiao - yi 5 o + ym) ; 7 = j (yo y25.71 + ysi.43 ~ >'77.14 = ^ (yi2.se yss.57 + = i (yo - yse + y?2 - + ^9 = 1 (yo y yns.71 yi4i.4s + yieT.iO ; 65 = J (yw - ys4 + y^ - yiae + y 9 = | (j 3 o yoo + yiw) ; + yi4 4 ) yo = o; i Similar schedules may be formed for determining the odd harmonics up to any order. 93. Graphical evaluation of the coefficients. Various graphical methods have been devised for finding the values of the coefficients in the Fourier's series, but these are less accurate and much more laborious than the arithmetic ones. The graphical methods, while interesting, are of little practical value in rapidly analyzing a periodic curve, so that we shall describe here only one of these methods the Ashworth-Harrison method.* If, for example, we divide the complete period into 12 equal intervals and measure the 12 ordinates, we shall have the table 30 60 90 120 150 1 80 210 240 270 300 330 yo yi y2 ys y4 ys ye y7 ys yg yio yn * Electrician, Ixvii, p. 288, 1911; Engineering, Ixxxi, p. 201, 1906. Other methods are briefly mentioned and further references are given in Modern Instruments and Methods of Calculation, a handbook of the Napier Tercentenary Celebration. ART. 93 GRAPHICAL EVALUATION OF THE COEFFICIENTS We have already shown (p. 181) that 6 ai = ^.y r cos x r = y cos o + yi cos 30 + + y n cos 330, 201 6bi r sin x r = y sin o + y\ sin 30 + yu sin 330. It is evident that if we consider the y's as a set of co-planar forces radiating from a common center at angles o, 30, 60, . . . , the sum of the horizontal components is equal to 6 ai and the sum of the vertical components is 6 b\. To facilitate the finding of these sums we may draw the polygon of forces, starting at a ' point and laying off in succession the ordinates, each making an angle of 30 with the preceding, as in Fig. 930 (proper regard must be had for the signs of the ordinates). The polygon of forces may be constructed rapidly by means of a protractor carrying an ordinary measuring scale along the diameter. Then, OA, the projection of the resultant OP on the horizontal, is equal to 6 a\, and OB, the projection of the resultant OP on the vertical, is equal to 661. Further- more, if we write a\ cos x + b\ sin x = c\ sin (x + 00, then the length FIG. 930. of OP is 6 ci and the angle FOB is 0i. In Fig. 930 we have made the construction for the determination of ai, bi, c\ t and 0i for the periodic curve drawn in Fig. 89 using the table of ordinates on p. 184. We find OA 134, di = -41.4, OB = 6bi = 126.0, OP = 6 Z POB = 0i = -18.1; hence ai = 6.9, bi = 21.0, Ci = 22.3, 0i = 18.1. These results agree very closely with those obtained on p. 184. We may find a 2 and b z by laying off in succession the ordinates, each making an angle of 60 with the preceding; we proceed similarly in finding the other coefficients. A separate diagram must be drawn for each pair of coefficients. More generally, if we divide the complete period into n equal intervals of width 2 ir/n and measure the n ordinates, then (p. 177) 202 EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII cos kx r = y Q cos 0+3-1 cos k I j + : 3/0 sin o +3>i sin k I y n -i cos k i * - flt= "5%, COS^ACr = 'VnCOSO + Vi COS k\ I 4- -\-M i COS k n 2 Hence, if we construct the polygon of co-planar forces by starting at a point and laying off in succession the ordinates, each making an angle 2 kir/n with the preceding, then OA, the projection of the resultant OP on the horizontal, is equal to a*/2, and OB, the projection of the result- ant OP on the vertical, is equal to nbk/2, except when k = o or k = n/2, when we get the values na , nbo, na n /2, nb n /2, respectively. Further- more, the length of OP is n/2 (or n) -3 A -2 FIG. 936. FIG. 93c. times the amplitude Ck and the angle between OP and OB gives the phase k of the complete harmonic Ck sin (kx + 0*). Example. Analyze graphically the periodic curve in Fig. 866. As in the example on p. 181, we shall find the first three harmonics from the data o 60 120 | 1 80 240 300 0.47 1.77 Here 6 a 3 = y y\ + 3 fll = OA (Fig. 936) 3 bi = OB (Fig. 936) 3 d = OP (Fig. 936) 3 02 = OA (Fig. 93c) 3 6 2 = OB (Fig. 93^) 3 c 2 = OP (Fig. 93 c) 2.20 3-5 2.20 o.n; 1-95; 3-09; 5-35; 6.25; -2.67 -1-35 3.00 a, ; a 2 = 1.64 0.49 0.02. 0-33- 1.03. 1.78. 2.08, 0.89. -0.45. 1. 00, 30. 2 = -6o c ART. 94 MECHANICAL EVALUATION OF THE COEFFICIENTS 203 Result: y = 0.02 + 1.03 cos x 0.89 cos 2 x + 0.33 cos 3 x + 1.78 sin x 0.45 sin 2 x = 0.02 + 2.08 sin (x + 30) + sin (2 x - 60) - 0.33 sin (3 x - 90). Note the close agreement of this result with that obtained by the arithmetic method on p. 181. 94. Mechanical evaluation of the coefficients. Harmonic analyzers. A very large number of machines have been constructed for finding the coefficients in Fourier's series by mechanical means. These instru- ments are called harmonic analyzers. The machines have done useful work where a large number of curves are to be analyzed. Among these analyzers we may mention that of Lord Kelvin,* Henrici,f Sharp, J Yule, Michelson and Stratton,|j Boucherot,^[ Mader,** and Westinghouse.ff We shall briefly describe the principles upon which the construction of two of these instruments depend. JJ The harmonic analyzer of Henrici. This is one of a number of ma- chines which use an integrating wheel like that attached to a planimeter or integrator to evaluate the integrals occurring in the general expres- sions for the coefficients i c 2r i r 2ir i r 2 * a = I y dx, ak = - \ y cos kx dx, b/, = - I y sin kx dx 2 7T JQ IT JQ IT Jo given on p. 174. If the curve in Fig. 94 / a represents a complete period of the curve to be analyzed, then evidently r ydx = area OABCDBO; so that, if the tracing point of a planimeter is allowed to follow the curve OABCDBO, the integrating wheel will give the reading 2 TTOQ, from which ao may be computed. * Proc. Roy. Soc., xxvii, 1878, p. 371; Kelvin and Tait's Natural Philosophy. t Phil. Mag., xxxviii, 1894, p. no. I Phil. Mag., xxxviii, 1894, p. 121. Phil. Mag., xxxix, 1895, p. 367; The Electrician, March 22, 1895. H Phil. Mag., xlv, 1898, p. 85. 1[ Morin, Les Appareils d' Integration, 1913, p. 179. ** Elektrotech. Zeit., xxxvi, 1909; Phys. Zeit., xi, 1910, p. 354. ft The Electric Journal, xi, 1914, p. 91. ft Brief descriptions of all but the last of these may be found in Modern Instruments and Methods of Calculation, a handbook of the Napier Tercentenary Celebration, 1914. For the principle of the planimeter and integrator, see pp. 246, 250. 204 EMPIRICAL FORMULAS PERIODIC CURVES CHAP. VII Integrating by parts, we may write -U": ~rS ycoskxdx = Ij-ysinkx vsmkxdx = -;- \ kir Now if the planimeter carries two integrating wheels whose axes make at each instant angles kx and ir/2 kx with the y-axis, and the point of inter- section of these axes is capable of moving parallel to the y-axis, then as the tracer point passes around the boundary OABCDBO, these wheels give readings proportional to I sin kx dy and / sin ( kx\ dy = I cos kx dy, from which the values of a& and bk can be found. In one form of the instrument the curve is drawn on a horizontal cylinder with the ;y-axis as one of the elements. A mechanism is attached to a carriage which moves along a rail parallel to the axis, by means of which a tracer point follows the curve while the cylinder rotates; the mechanism allows the axes of the integrating wheels to be turned through an angle kx while the cylinder ro- tates through an angle x. Coradi, the Swiss manufacturer, has per- fected the instrument so that sever- al pairs of coefficients may be read with a single tracing of the curve. 360 X FIG. 946. FIG. 94C. The Westinghouse harmonic analyzer. -This machine, constructed by the Westinghouse Electric and Mfg. Co., is particularly useful in ART. 94 MECHANICAL EVALUATION OF THE COEFFICIENTS 205 analyzing the alternating voltage and current curves represented by a polar or circular oscillogram. Fig. 946 gives one period of a periodic curve drawn on rectangular coordinate paper. In Fig. 94*;, the same curve is represented on polar coordinate paper. This is done by constructing a circle of any convenient radius, called the zero line or reference circle and locating any point P by the angle 6 = x and the radial distance r = y from the zero line. Thus the points marked P, A, and B in Figs. 946 and 94C are corresponding points. If only the odd harmonics are present, the second half-period of the curve in Fig. 946 will be a repetition below the re-axis of the first half-period; in this case, the diameters at all angles of the curve in Fig. 946 will be equal, and equal to the diameter of the reference circle. The re- lation between r and -6, r = /(0) = ai cos 6 + a 2 cos 2 6 + + a k cos kd + -f bi sin 6 -f b z sin 2 d + + b k sin kO + , is the function to be analyzed. This is done as follows. The circular record of the periodic curve, drawn by hand from the rectangular record or directly by the circular oscillograph,* is transferred FIG. 94I 13 4 2 I 15 2 7 16 3 o 3 17 29 33 42 44 45 30 31 29 15. Determine the first three harmonics for the periodic curve from which the fol- lowing measurements were taken; use the method of selected ordinates in Art. 91; assume that all higher harmonics are absent. o" 10.0 5-o _45!_ 5-3 60 90 120 135 150 1 80 7-2 6.0 y -6.8 300 10.9 I 315 " 8.9 3. IO.O JO -17.3 1 -4.7 210 225 240 10.7 1-3.4 -25-9 16 Determine the first three harmonics for the periodic curve drawn, in Fig. 86b; use the method of selected ordinates in Art. 91. 17. Determine the first six harmonics for the periodic curve drawn in Fig. 89; use the method of selected ordinates in Art. 91; assume that all higher harmonics are absent. 1 8. Determine the first and third harmonics for the symmetric periodic curve given by the following data; use the method of selected ordinates in Art. 92; assume that all higher harmonics are absent 60 66.5 22.4 14.9 19. Assuming that the harmonics higher than the fifth are negligible, determine the odd harmonics of the symmetric periodic curve from which the following measurements were taken; use the method of selected ordinates in Art. 92. * X 30 60 90 120 150 3 1 I ( 8 3 ^6^ 719 676 54 660 702 940 90 IOO 1 08 4 12 5 6 54 J44l 639 y o 470 678 940 1086 920 I62 C 375 20. Use the method of selected ordinates in Art. 92 to determine the ninth harmonic of the curve given by the table in Ex. 14. 21. Analyze graphically the curve in Ex. 7, CHAPTER VIII. INTERPOLATION. 95. Graphical Interpolation. Having found the empirical formula connecting two measured quantities we may use this in the process of interpolation, i.e., in computing the value of one of the quantities when the other is given within the range of values used in the determination of the formula. It is the purpose of this chapter to give some methods whereby interpolation may be performed when the empirical formula is inconvenient for computation or when such a formula cannot be found. Let the following table represent a set of corresponding values of two quantities X where y is a known or an unknown function of x. Our problem is to find the value of y = y k for a value of x = Xk between x and x n . A simple graphical method consists in plotting the values of x and y as coordinates, drawing a smooth curve through or very near the plotted points, and measuring the ordinate y* of the curve for the abscissa x*. The value of y k thus obtained may be sufficiently accurate for the purpose in hand. Thus from the curve in Fig. 726, we read t = 10, A = 77.0, and / = 30, A = 45.0. If we use the empirical formula derived on p. 133, A = 100.1 e-o-0265', or log^4 = 2.0005 0.0115 /, we compute t = 10, A = 76.8 and / = 30, A = 45.2. By comparison with the table on p. 132 we note that the measured values of A for / = 10 and / = 30 agree about as closely with the computed values as the neigh- boring observed values agree with their corresponding computed values. Here, the last significant figures in the values of A were used in construct- ing the plot. On the other hand, in Fig. 7ic, we read v = 40, p = 10.00, whereas the empirical formula on p. 131 gives v = 40, p = 9.4.2. The residual is 0.58, much larger than the residuals in the table on p. 130 for neighbor- ing values of v. Here, the plot was constructed without using the last significant figures in the values of the quantities. It is of no advantage to construct a larger plot since the curve between plotted points is all the more Indefinite. 209 2IO INTERPOLATION CHAP. VIII For most problems the arithmetic or algebraic methods to be explained in the following sections give much bettei results. 96. Successive differences and the construction of tables. Given a series of equidistant values of x and their corresponding values of y, Xi x X n x + nh we define the various orders of differences of y as follows: 1st difference = A 1 : a = y\ y , a\ = y% y it . . . , a n -\= y n 2d difference = A 2 : b = di a , bi = a<> a lf . . . , & n _ 2 = a n 3d difference = A 3 : c = b\ bo, c\ = b 2 bi, . . . , c n -z= b kth difference = A* : ko = ji j 0f k\ = j ji, . . . . These may be tabulated as follows: x y A 1 A J A 3 A ... A* ... XQ = X ^ Xi == XQ ~i~ ft >i b a\ Co Xt = X + 2h y% bi do CL% Cl x 3 = x + 3 h y 3 bz O3 ko X^ = x^ + 4 h * kl Xvr-i = x + (n - i) h a n -i Xn = X + nh where a quantity in any column of differences is written between two quantities in the preceding column and is equal to the lower one of these minus the upper one. We may apply the above definitions in the formation of the differences of y when y = /(#) ; thus, Ay = f(x + A) - /(*) = A/(*) ; tfy = A/(* + A) - A/(*) = A'-/(*) ; etc. E.g., if y = x 2 - 2 x + 2, Ay = [(x + h)* - 2 (x + A) + 2] - [x 2 - 2 x + 2] = 2hx+ (h* -2 A); = 2 A 2 . We note that A 2 y = 2 & 2 , so that the second differences are constant for all values of x. A.RT. g6 SUCCESSIVE DIFFERENCES Similarly, if y = x n where n is a positive integer, Ay = (x 211 n (n - i) x n - 2 /* 2 + > , n (n - i) (n - 2} x n ~ 3 h 3 + &y=n(n - i) (n - 2) . . . 3 - 2 i A" = |n h n ; hence the nth differences of x n , where n is a positive integer, are constant, and hence the nth differences of any polynomial of the nth degree Ax n + Kx + L, where n is a positive integer, are constant. If in forming the differences of a function some order of differences, say the nth, becomes approxi- mately constant, then we may say that the function can be represented approximately by a polynomial of the nth degree, where n is a positive integer. The formation of the differences for various functions is illustrated in the following tables: (2) X (i) y y = ** A' A* I i 7 2 8 12 19 6 3 27 18 37 6 4 64 24 61 6 5 125 30 91 6 6 216 36 127 6 7 343 42 169 6 8 512 48 217 9 729 X y A A* 5-16 137-39 4-03 5-21 141.42 O.08 4.II 5.26 145-53 O.O8 4.19 5.31 149.72 0.08 4-27 5-36 153-99 0.08 4-35 5-41 158.34 0.08 4-43 5.46 162.77 0.08 5.51 167.28 4-51 0.09 4.60 5-56 171.88 212 INTERPOLATION LHAP. VIII (3) y = (4) y X y A' A , * y A 1 20 2.7144 611 8.4856 445 46 21 2.7589 -14 612 8.4902 43i 46 22 2.8020 12 613 8.4948 419 46 23 2.8439 -13 614 8.4994 406 46 24 2.8845 -II 615 8.5040 395 46 25 2.9240 616 8.5086 (5) Train-resistance (6) Speed of a vessel V R V speed in resist, in Ibs. A 1 A 2 speed in mi. per hr. per ton knots per hr. horse- power A 1 A J A' 20 5.5 8 I,OOO 3-6 400 4 9.1 2.2 9 1,400 IOO 5-8 500 60 14.9 2.1 10 1,900 IOO 7-9 600 50 80 22.8 2.4 II 2,500 150 10-5 750 50 IOO 33-3 2.2 12 3,250 2OO 12.7 950 50 120 46.0 13 4,200 250 ] 2OO IOO 14 5,400 350 ] 550 IOO 15 6,950 450 ; 000 50 16 8,950 500 2500 17 11,450 (7) y = log x (8) y = log sin x x y A x y A A' A 500 2.6990 i o' 8.2419-10 8 e )6 9 501 2.6998 i 10' 8.3088-10 -8 9 9 i 80 20 502 2.7007 I20' 8.3668-10 -69 9 i >ii fi6 503 2.7016 i3o' 8.4179-10 -53 8 i ^58 8 504 2.7024 i4o' 8.4637-10 -45 9 ( ^3 10 505 2.7033 i 50' 8.5050-10 -35 9 \78 506 2.7042 2 0' 8.5428-10 In the above tables we note the following: In (i), y = x 3 and A 3 is constant. In (2), y = x 3 and A 2 is constant since we have carried the work to two decimal places and A 3 does not sensibly affect the second decimal place. AXT. 96 SUCCESSIVE DIFFERENCES 213 If the computation had been carried to six decimal places, A 2 would not be constant but A 3 would be. In (3), A 2 is approximately constant, so that if we desire to work to four decimal places, \/x could be represented by a polynomial of the second degree within the given range of values of x. In (4), A 1 is approximately constant so that \^x could be represented by an equivalent polynomial of the first degree. In (5) and (6), A 2 and A 3 are approximately constant, so that R may be approximately represented by a polynomial of the second degree in V, and / by a polynomial of the third degree in V. In (7), log x may be approximately represented by a polynomial of the first degree, and in (8), log sin x by a polynomial of the third degree within the given range of values of x. In general, it is evident that we may stop the process of finding suc- cessive differences much sooner the smaller the number of digits required and the smaller the constant interval h. We should stop immediately if the differences become irregular. The formation of differences is often valuable where a function is to be tabulated for a set of values of the variable. Thus, suppose we wish to form a table for y = irx 2 /4, expressing the area of a circle in terms of the diameter, for equidistant values of x. Since we have a polynomial of the second degree, A 2 ;y is constant, and if h = I and the work is to be carried to 4 decimal places, we need merely compute the values of y for x = i, 2, 3 and form the corresponding differences; proceeding back- wards, we repeat the value of A 2 y = 1.5708, add this to Ay = 3.9270 and get 5.4978, add this to 7.0686 and get 12.5664, which is the value of y for x = 4. We proceed in the same manner to get the values of y for suc- cessive values of x. y = T*/4 A 1 A 1 * 5 0.7854 69 2.3562 3.I4I6 1.5708 70 3.9270 7.0086 1.5708 71 54978 12.5664 1.5708 72 7.0686 3739-28 109.17 3848.45 1-57 110.74 3959-19 1.57 112.31 4071.50 1.57 4185.38 113.88 19-6350 73 For larger values of x where we wish to work to two decimal places only, we take A 2 ;y = 1.57 and proceed as above. Suppose we wish to tabulate the function y = x 3 . Here A 3 is con- stant so that we merely compute the part of the accompanying table in heavy type. Then we extend the column for A 3 by inserting 6's, extend the columns for A 2 and A 1 by simple additions and subtractions, and thus determine the values of x 3 for all integral values of x. 214 INTERPOLATION CHAP. VIII X y-s* A A A I I I I 6 I I 6 7 6 2 8 12 19 6 3 27 18 37 6 4 64 24 61 6 5 125 30 91 6 216 The same procedure may be followed in the construction of a table for a function where a certain order of differences is only approximately con- stant. Thus, in forming table (4) of cube roots, we note that for that portion of the table Ay is approximately 0.0046 so that we can find the values of \/x by simple additions; we must check the work by direct computation every few values in order to find when A 2 y changes its value. 97. Newton's interpolation formula. We shall now express the value of y for any value of x. From the definitions of successive differ- ences we have iVi = yo + a ; y-i = yi + ai = (yo + do) + (do + b ) = y + 2 a -f b ', y 3 = yz + 02 = (yo + 2 a + b ) + (a + 2 & + CD) = y + 3 o + 3 &o + c ; y* = yz + as = (> + 3 a + 3 6 + c ) + (ao + 3 b + 3 c + do) = > + 4 ao + 6 6 + 4 c o + do] We note that the coefficients are those of the binomial expansion, and this suggests that n(n - i) t , n(n - i) (n - 2) , m ' > v 1 / where n is a positive integer. If this equation is true, then, replacing y by a, the first difference, we may also write , , n (n - i) n(n - i}(n- 2) d n = a + woo H r- c H 1- L i^ r.. /. ( t rn.(n - i) (n - 2) | n (n - i) j_ /- T ^ , = yo + (n + i) do H ART. 97 NEWTON'S INTERPOLATION FORMULA 215 where the coefficients are again those of the binomial expansion with n replaced by n + i. Thus we have shown that if equation (I) is true for any positive integral value of n, it is true for the next larger integral value. But we have shown (I) to be true when n = 4, therefore it is true when n = 5 ; since it is true for n 5, therefore it is true for n = 6 ; etc. Hence (I) is true for all positive integral values of n. Now if some order of differences, say the kth order, is constant, i.e., & k y = ko, then y is a polynomial of the kth degree in n, and equation (I) may be written A + Bn + C 2 + + Ln* = y + no* + n ( " ~ T) b + . . . , n (n - i) . . . (n - k + i) , ~\j~ ~ ko ' The right member of this equation is also a polynomial of the kth degree in n, and since these polynomials are equal for all positive integral values of n (i.e., for more than k values of ), they must be equal for all values of n, integral, fractional, positive, and negative. Hence if the kth order of differences is constant, we have L5 for all values of n. This fundamental formula of interpolation is known as Newton's interpolation formula. In this formula, y is any one of the tabulated values of y and the differences are those which occur in a line through y and parallel to the upper side of the triangle in the tabular scheme on p. 210. Newton's formula is approximately true for the more frequent case where the differences of some order are approximately constant; all the more so if n < i. We can always arrange to have n < i ; for if we wish to find the value of y = Y for x = X, where X lies between the tabular values Xi and Xi+i, we use Newton's formula with y,- and the correspond- ing differences a<, bi, d, . . . , so that X = x f + nhandn = r^ h=i, w= (3.4 3)71=0.4; .-. (3.4)' = 27 + (o.4) (37)+ (-4)(--6) ( 24 ) + ($4) (-Q-6) (-1-6) (fi) 2 6 = 39.304. (3) To compute \/2^5; y = 2.8439, h = i, n =(23.5 - 23)71 = 0.5; /. ^23.5 = 2.8439 + \ (0.0406) + | (o.ooii) = 2.8643. If we use the ordinary interpolation formula of proportional parts, -^23.5 = 2.8439 + \ (0.0406) = 2.8642, which would be correct to three decimals only. (4) To compute ^612.25; 3-0 = 8.4902, h=i, = (612.25-612)71 =; /. v"6i2.25 = 8.4902 + \ (0.0046) = 8.4914. (5) To computed when 7=65; ^0 = 14-9, A = 20, n = (65 6o)/2o = i : ; /. R = 14.9 + J (7.9) - TJ\ (2.4) = 16.7. (7) To compute log 501. 3; 3-0 = 2.6998, h= i, n= (501. 3-501)71 =0.3; /. log 501.3 = 2.6998 + 0.3 (0.0009) =2.7001. (8) To compute log sin i 16'; 3-0 = 8.3088 10, h = 10', n = (i 1 6' - i io')/io' = 0.6; .'. log sin i 16' = (8.3088 10) + 0.6 (0.0580) 0.12 ( 0.0069) + 0.056 (0.0016) = 8.3445 I0 correct to 4 decimals. If we use the ordinary formula of proportional parts, we have log sin i 16' = 8.3088 10 + 0.6 (0.0580) = 8.3436 10, correct to 2 decimals only. If the value of x for which we wish to determine the value of y is near the end of the table we may not have all the required differences. To take care of this case Newton's formula is slightly modified. If we invert the series of values of x in the tabular scheme on p. 210, and form the differences, we have yn y n -i -On-l Ci Co ART. 97 NEWTON'S INTERPOLATION FORMULA 2 17 t y\ and , . n (-a 3 ) Starting at y\ and applying Newton's formula, we get n (n i) , . n (n i) (n 2\ , , n(n-i)i n (n - i) (n - 2) _ = y 4 - wa 3 H --- ^ - 6 2 - ^ - ci + Comparing the result with the scheme on p. 210, we note that the differ- ences are those which occur along a line parallel to the lower side of the triangle in that scheme. Here y\ is any value of y, and if X lies between # 4 and x 3 , then X = # 4 nh, and n = (x* X)/h. Example. To compute ^24.8. In table (3), y 4 = 2.9240, h = i, n = (25 - 24.8)71 = 0.2; .*. "V/24.8 = 2.9240 0.2 (0.0395) H ' - (0.0011) = 2.9162. If a series of corresponding numerical values of two quantities are given, we may use Newton's formula for finding the polynomial which will represent this series of values exactly or approximately. For this purpose we replace n by (x Xo)/h. Thus, in table (i), h = i, X Q = I, n = x I ; i / \ 7 V 20 V In table (5), h = 20, F = 20, w = = - 1 ; = 4.1 + 0.015 v + 0.00275 y 2 . The values of .R computed by this formula agree quite closely with those in the table. In table (6), h = i, F = 10, n = V - 10; /. / = 1900 + (V - io) 600 + (V ~ IO) 2 (F ~ JI) 150 (F-io)(F-n)(F-i2) 6 = -6850 + 2042 V - 200 F 2 + 8| F 3 . The values of 7 computed by this formula agree quite closely with those in the table; thus, F = 12 gives / = 3254. Various formulas of interpolation similar to Newton's have been de- rived which are very convenient in certain problems. Among these may be mentioned the formulas of Stirling, Gauss, and Bessel.* * For an account of these formulas, see H. L. Rice, Theory and Practice of Interpola- tion, and D. Gibb, Interpolation and Numerical Integration. 2l8 INTERPOLATION CHAP. VIII 98. Lagrange's formula of interpolation. Newton's' formula is applicable only when the values of x are equidistant. When this is not the case, we may use a formula known as Lagrange's formula. Given the following table of values of x and y, X \ a 2 C.I 3':. we are to find an expression for y corresponding to a value of x lying be- tween a\ and a n . We take for y an expression of the (n i)st degree in x containing n constants, and determine these n constants by requiring the n sets of values of x and y to satisfy the equation. But instead of assuming the form y = A + Bx + Cx 2 + the equivalent form y = A (x - a 2 ) (x - a 3 ) (x - a 4 ) + B (x - ai) (x - as) (x - a 4 ) + C (x - ai) (x - 02) (* - a*) + Nx"^, we may assume . . (x - a n } . . (x - a n } ..(*- On) + TV (x - ai) (x - 02) (x - a 8 ) . . . (x - a n -i), where the w terms in the right member of the equation lack the factors (x Ci), (x o s ), (x a n } respectively. Since (a\, yi) is to satisfy this equation, y\ = A (ai - 02) (ai - c 3 ) (ai - a 4 ) . . . (ai - a), since all the other terms contain the factor (ai ai) and therefore vanish. Similarly, j 2 = B (0-2 - aO (02 - a s ) (02 - a 4 ) . . . (a 2 - a n ), y 3 = C (a 8 - ai) (a 3 - a 2 ) (a s - a 4 ) . . . (a s - a n ), Hence, [ N (a n - ai) (a w - - a 3 ) . . . (a n - a n _i (ai a%) (ai ^2 . . (ai a n ) -, etc., . (x-aj (a 2 ai) (flz and, finally, (*- -i a n ) Z (a 2 ai i (x a?) . . . (x ) (a 2 a 3 ) . . - fln-i) ' ((h ~ an} (a n - - a 2 ) (a - a_0 We note that in the term containing yk, the numerator of the fraction lacks the factor (x a*) and the denominator lacks the corresponding factor (a* a*). Lagrange's formula is in convenient form for logarith- mic computation. ART. 99 INVERSE INTERPOLATION 2I 9 Example. In the table on p. 132 we have ' 7 / J 4 17 3i A 68.7 64.0 44.0 and we are to find the value of A when t (27-17) (27-31) (27-35) (14-17) (14-31) (14-35) 14) (27- 17) (27-35) , 6 +4 I .. ' . ' 39-1 27. Using Lagrange's formula, (27-14) (27-31) (27-35) (17-14) (17-31) (17-35) (27-14) (27-17) (27-31) (3i -14) (31 -17) (31 -35) = -20.5 + 35.2 + 48.0 - 13.4 = 49.3, which agrees exactly with the observed value. Example. In the table on p. 157 we have O.I 0.2 I O.4 (35 -14) (35 -17) (35 -3i) 2.48 0.8 and we are to find the value of i when / / = 0.2 and t = 0.4, 2.66 I 2.58 | 2.00 0.3. Using only the values * = 2.66 H^ + ^!f^i='-33 + '.*9-.6, Using all four values of /, i = 2.68. Using the empirical equation i = 4.94 er ljat 2.85e~ 3 - 76 ' (on p. 159), we get i = 2.66. Gauss's interpolation formula for periodic functions. When the data are periodic we may find the empirical equation as a trigonometric series by the method of Chapter VII and use this equation for purposes of in- terpolation, or we may use an equivalent equation given by Gauss: sin % (x 02} sin \ (x a 3 ) . . . sin \ (x a n ) y = sin ^ sin a t a 2 ) sin \ (ai a 3 ) (x aO sin \ (x o 3 ) sn a sin \ (x a n ) a n ) ai) sin \ (a 2 - a 3 ) . . . sin - a n ) It is evident that y = y\ when x = a\, y = Ji when x = a%, etc., so that the equation is satisfied by the corresponding values of x and y. 99. Inverse interpolation. Given the table X #0 Xi Xt x& . . . *n y y yi yz y 3 . . . yn we may wish to find the value of x corresponding to a given value of y. If the values of x are equidistant we may use Newton's interpolation formula. Here we know y n , y , a , &o, CQ, . . . , and substituting these values in the formula we have an equation which is to be solved for n. If only the first order of differences are taken into account, then y n = yo + noo, and n = , the ordinary formula for inverse inter- a polation by proportional parts. 220 INTERPOLATION CHAP. VIII Example. In table (7), given log x = 2.7003, to find x. ' and x = x + nh = 501+0.56(1) =501.56. If only the first and second differences are taken into account, then y n = yo + na + -- b , a quadratic equation which can easily be solved for . Example. In table (5), given R = 27.3, to find V. Here 27.3 = 22.8 + n (10.5) + * (n ~ ^ (2.2), or i.i w 2 -f 9.4 n 4.5 = o; hence n = T 5 T = 0.455 and x = V + nh = 80 + (0.455) 20 = 89.1. The empirical formula R = 4.62 0.004 ^ ~H 0.0029 V 2 on p. 149 gives V = 89.1, R = 27.3. But if the third and higher orders of differences have to be taken into account, the method would require the solution of equations of the third and higher degrees. In such cases as well as in the case where the values of x are not equidistant, we may use Lagrange's formula and merely in- terchange x and y; i.e., (a] a 2 ) (a! a 3 ) . . . (a 2 fli) (a 2 fit) . . . Example. In table (8), given log sin # = 8.3850 10, to find x. Using only the following values, log sin x I 8.3088 - 10 1 8.3668 - 10 x 70' 80' 5.4179 10 90' we have x = 70 ' (0-0182) (-0.0329) go/ (0.0762) (-0.0329) (-0.0580) (-0.1091) (0.0580) (-0.0511) , (0.0762) (0.0182) (0.1091) (0.0511) = 70' (-0.0946) + 80' (0.846) + 90' (0.249) = 83.47' = i 23.47'. We may also use a method of successive approximations as follows: From Newton's formula we write (n - i) b + \(n - i) ( - 2) c + Applying this to the above example, and taking only the first differences into account, we get as a first approximation, n = y~y = (8.3850 ~ 10) ~ (8.3668 - 10) = 182 = Q Co 0.0511 511 ART. 99 INVERSE INTERPOLATION 221 Taking also second differences into account and introducing the value of n\ for w in the denominator, we get as a second approximation, _ y y 0.0182 _ 182 _ *** ~ OQ + \ (wi i) b ~ 0.0511 + 0.0017 ~ 5 2 8 We may continue in this way approximating more and more closely to the value of n. In this example it will be unnecessary to carry the work to third differences since A 3 is negligible. Hence n = 0.345, and x = x + nh = i 20' + (0.345) (10') = i 23.45'. We may check this by direct interpolation. Here yo = 8.3668 10, h = 10', and n = 0.345; hence, y = 8.3668 - 10 + 0.345 (0.0511) - 0.113 (-0.0053) = 8.3850-10. Example. Find the real root of the equation X s + 5 x 1=0. We form a table of differences of the function y x* -J- 5 x I. -19 -7 -6 o 6 12 The root lies between x = o and x = I, and we are to find the value of x when y = o. Using the method of successive approximations we have o 4- i i y y - yo 6 + |( l_ l)6 = f = o. 2 8 57 , fl<> + 1 (2 - I) bo + I (2 ~ I) (** ~ 2) C 6^+1* 249 % = 0.1968, 0.1985. 4- i (s - i) (3 - 2) c 6 - 2.4096 + 1.4483 nh = 0.1985. 5-0387 Hence, From the table we note that x = 0.1984 is the root correct to 4 decimals. X 0.1985 0.19845 0.1984 y 0.00032 0.00006 0.00019 222 INTERPOLATION CHAP. VIII EXERCISES 1. Tabulate the values and differences of the following functions; h is the common interval. (a) x 2 , from x = 5 to x =' 12 when h = i ; and from * = 3 to x =3.1 when h - o.oi. (b) V#, from x = i to x = 10, when h = i, and from x = 563 to * = 570 when h = i. (c) -, from x = 60 to x = 70 when h = i, and from x = 260 to x = 262 when h = 0.2. (d) r- (volume of a sphere), from D = I to D = 1.8 when A = o.l. (e) log x, to 4 decimals, from * = 356 to x = 362 when h = i . (f) tan x, to 4 decimals, from x = 32 to x = 33 when h = 10'. (g) log cos x, to 4 decimals, from x = 88 10' to * = 89 20' when h = 10'. (h) and we may easily verify that A = i h (y + 4^! + 3k) If we have an even number of intervals and apply this formula to the successive areas under the parabolic arcs, we get (yo + 4 yi + 2 ^2 + 4 Js + 2 y 4 + +2 ;y n _ 2 + 4 y^j + y B ) +y-0 To apply Simpson's rule we must divide the interval into an even number of parts, and the required area is approximately equal to the sum of the extreme ordinates, plus four times the sum of the ordinates with odd subscripts, plus twice the sum of the ordinates with even sub- "scripts, all multiplied by one-third the common distance between the ordinates. (4) Durand's rule.* If we have an even number of parts and apply Simpson's rule to the interval from Xi to x n -\ and the Trapezoidal rule to the end intervals, + ! yn-a + J y- 2 + I yn-i) + (I y n -i + * y.)]. Applying Simpson's rule to the entire interval from x to x n , 4=A[$yo+ Jyi + f y, + $y, + - + $ y*-s + ! y- + $ y-i + f y*]. Adding, 2 A =h[ty +Wyi+2y2 + 2y 3 + . . . +2y n _ 3 + 2y n _ 2 +^y n * Given by Prof. Durand in Engineering News, Jan., 1894. ART. 102 APPLICATIONS OF APPROXIMATE RULES 227 Hence, A D = h [A (yo + y) + H (yi + y-i) -f = A [0.4 (jo + y w ) + i.i (yi + y*-i) + ^ Collecting our rules, we have (1) A R = h(y + yi + y z + - - + y n -i), or A R ' = h(yi + y 2 + y 3 + + yn). (2) 4 r = A ft On, + y n ) + yi + y 2 + (3) -4s = I ^ [(yo + yn) + 4 (yi + y + ys + + 2 (ys + (4) A D = h [0.4 Cvo + y) + i.i (yi + y n - 102. Applications of approximate rules. We shall give some ex- amples illustrating the application of these rules. C l dx I. Area. Evaluate I . This is equivalent to finding the area *J 2 X between the curve y = i/x, the x-axis, and the ordinates x = 2 and x = 10. If we divide the interval into 8 parts, then h = I ; we have the table + y*- + ye + + ?-)] X 2 3 4 5 o 7 8 9 10 y i 4 i i I 1 i iV A R = i (i + 1 + i + +*} = 1.8290; AR = i (i + 1 + i+ + iV) = 14290; A T = i B (H- A) + *+ + *]- 1.6290; i.i 4^ = i [0.4 (i + T V) r i0 dx By actual integration, / = t/2 X In x *)+2(t + i + ]= 1.6109; i + i + + 1] = 1.6134- = In io In 2 = In 5 = 1.6094. We note that Simpson's rule gives the best approximation (within o.i % of the true value), with Durand's next. If we take h = \, Thus the Trapezoidal rule with 16 ordinates does not give the accuracy given by Simpson's rule with 8 ordinates. 2. Area. The half-ordinates in feet of the mid-ship section of a vessel are 12.5, 12.8, 12.9, 13.0, 13.0, 12.8, 12.4, ii. 8, 10.4, 6.8, 0.5, and the ordinates are 2 feet apart; find the area of the whole section. 228 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX \A T = 2 ft (12.5 + 0.5) + 12.8 + . . . + 6.8] = 224.8; A 8 = l [(12.5 + 0.5) + 4 (12.8 + 13.0 + 12.8 + 1 1.8 + 6.8) + 2 (12.9 + 13.0 + 12.4 + 10.4)] Hence, A T = 449.6 sq. ft., A s = 452.2 sq. ft. 3. Work. Given the following data for steam 226.3 V 2 4 6 8 10 p 68.7 3i-3 19-8 14-3 n-3 where v is the volume in cu. ft. per pound and p is the pressure in pounds per sq. in.; find the work done by the piston. Work = / p dv ; this is equivalent to finding the area under the curve obtained by plotting (v, p). W T = 2 ft (68.7 + 11.3) + 31.3 + 19.8 + 14.3] = 210.80; W s = .f [(68.7 + 11.3) + 4 (31.3 + 14.3) + 2 (19.8)] = 201.33. By the methods of Chapter VI we find the empirical formula con- necting v and p to be pv 1 - 12 = 148, and hence, W rw p =l pdv = i4SI t/2 Jz = 148 .,-0.12 - O.I2 110 \2 199.31 This last value differs from the value given by Simpson's rule by about I %. 4. Mean effective pressure. Indicator diagram. Fig. 1020 is a re- production of an indicator diagram; to find the mean effective pressure. FIG. 1020. The mean effective pressure P is the area of the diagram divided by the length of the diagram, since the area represents the effective area of the piston in sq. in. and the length represents the length of the stroke in ft. Since the total area enclosed by the curve is the difference between the area bounded by a horizontal axis, the end ordinates, and the upper part of the curve, and the area bounded by the same straight lines and the lower part of the curve, we need merely measure the lengths of the ordi- nates within the curve. The diagram is 3.5 ins. long. We divide the interval into 8 parts; then h = T 7 ff , and we measure the ordinates o, 0.40, 0.63, 0.91, 0.98, i. oo, 0.92, 0.74, o. ART 102 APPLICATIONS OF APPROXIMATE RULES 229 yV [040 + 0.63 + + 0.74] = 2.44; ? 7 s [4 (0.40 + 0.91 + 1. 00 + 0.74) + 2 (0.63 + 0.98 + 0.92)] 2.52. p - s 3-5 3-5 J, and we measure the We divide the interval into 14 parts; then h ordinates o, 0.30, 0.42, 0.54, 0.68, 0.88, 0.96, 0.98, i.oo, i. 02, 0.97, 0.89, 0.78, 0.64, o. A T = I [0.30 + 0.42 + + 0.64] = 2.52. A s = & [4(0-30+0.54 H +o.64)+2 (0.42+0.68+ +0.78)1 = 2.55. Hence, / / P = ff ->-& - 0.73. We note that A s with 9 ordinates has the same value as A T with 15 ordinates. 5. Velocity. Given a weight of 1000 tons sliding down a i% grade (Fig. 1026) with a frictional resistance of 10 Ibs. per ton at all speeds. The total resistance is 30,000 Ibs. (a frictional resistance of 10,000 Ibs. and a grade resistance of 20,000 Ibs.). Let the following table express the accelerated force F as a function of the time / in seconds : joo> FIG. IO2&. 100 200 300 1 400 1 500 I 600 I 700 800 900 F |2o,ooo| 19,000 16,000 ii,ooo|5oooj loooj 500018500 Find the velocity acquired by the body in 1000 seconds. 2,000,000 1,000,000 Since F = m X a, and m therefore, a = = m 16.1 F g . dv and dt 1,000,000 We form a table for the acceleration a. a, (0.322 0.306 200 300 0.258 | 0.177 400 I 500 O.oSl O.OI6 600 700 16.1 hence, v 800 ! -0.081 1 0.137 - 900 IOOO -I5~ooo adt. 0.177 1 -0.209 1-0.242 Here, h = 100, so that V T = ioo [(0.322 0.242) + (0.306+0.258+ 0.209)] =24.2 ft. per sec. *>s = A - [(0.322 0.242) + 4 (0.306 + 0.177 0.016 0.137 0.209) + 2 (0.258 + 0.081 0.081 0.177)] = 24.2 ft. per sec. 6. Volume. If S x is the area of a cross-section of a solid made by a plane perpendicular to the .T-axis, then the volume of the solid included between the planes # and x n is V = I " S x dx. In order to integrate, Jx, we must know the analytical expression for S z as a function of x. Otherwise we employ the approximate formulas; the values of S x are the ordinates and h is the common distance between the cutting planes. 230 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX A buoy is in the form of a solid of revolution with its axis vertical, and D is the diameter in ft. at a depth p ft. below the surface of the water. P o 0-3 0.6 0.9 1.2 i-5 1.8 D 6.00 5-90 5.80 5-55 5-25 4.70 4.20 D 2 36.00 34.81 33-64 30.80 27.56 22.09 17.64 Find the weight of water displaced by the buoy (i cu. ft. of sea water weighs 64.11 Ibs.). Here, X1.8- -D*dp, and h = 0.3, 4 0.3 7T hence, V s = J -- [(36.00 + 17.64) + 4 (34.81 + 30.80 + 22.09) O T" + 2 (33.64 + 27.56)] = 41.38 CU. ft., and the weight of water displaced = 2652.87 Ibs. The areas in sq. ft. of the sections of a ship below the load-water plane and 3 ft. apart are 7500, 7150, 6640, 5680, 4225, 2430, 260, where the load-water plane has an area of 7500 sq. ft. Find the dis- placement in tons (35 cu. ft. of sea water weigh I ton). V T '-=3 [K7500+260) + (7150+6640+5680+4225+2430)] = 90,015 cu. ft. Vs = f [(75<>+26o) +4(7150+5680+2430) +2(6640+4225)] = 90,53001. ft. Hence, the displacement is 2572 tons by the Trapezoidal rule and 2587 tons by Simpson's rule. 7. Moment of inertia. The moments of inertia of an area about the axes are /*= **l?dx, /, The evaluation of these integrals is equivalent to finding the areas under the curves with y 3 or x 2 y as ordinates and x as abscissas. The half-ordinates in ft. of the mid-ship section of a vessel are 12.5, 12.8, 12.9, 13.0, 13.0, 12.8, 12.4, ii. 8, 10.4, 6.8, 0.5, and the ordinates are 2 ft. apart. Find the moment of inertia of the entire section about the axis. Here, J x = 2 / ^ y 3 dx, h = 2, and the values of r 3 are Jo 1953.1, 20972, 2146.7, 2197.0, 2197.0, 2097.2, 1906.6, 1643.0, 1124.9, 314.4, o.i, and applying Simpson's rule, J* = ! (I) [(I953-I + o.i) + 4 (2097.2 + + 314.4) + 2 (2146.7 + . + 1124.9)] = 22,266.1. ART. 103 GENERAL FORMULA FOR APPROXIMATE INTEGRATION 231 8. Pressure and center of pressure. The pressure on a plane area perpendicular to the surface of the liquid, between depths x and x n , is p = w I n xydx, where w is the weight of the liquid per unit volume, y is t/*c the width of the area at a depth x beneath the surface. The depth of the center of pressure of such an area is given by x /xn I xyd /*o ! All these integrals can be evaluated approximately. 9. Center of gravity. The coordinates of the center of gravity of an area are /* Moment about OY Area f = J - T fydx Moment about OX Area The half-ordinates in ft. of the mid-ship section of a vessel are '< 12.5, 12.8, 12.9, 13.0, 13.0, 12.8, 12.4, ii. 8, 10.4, 6.8, 0.5, and the ordinates are 2 ft. apart. Find the center of gravity of the section. xydx Moment about OY /' / Jo ydx Area and applying Simpson's rule to the table, x 2 4 y 12.5 12.8 12.9 xy 25.6 51.6 I3-Q 78.0 M s = [(o +10.0) +4 (25.6 + = 2018.9. A s = I [(12.5 + 0.5) +4 (12.8+ = 226.1. _ 2018.9 Hence, x = 8 10 12 14 16 18 20 13.0 12.8 12-4 1 1. 8 10.4 6.8 o-5 104.0 128.0 148.8 165.2 166.4 122.4 IO.O 122.4) +2 (51.6+ + 6.8) +2 (12.9 + 8-93 ft. + 166.4)] +10.4)] 103. General formula for approximate integration. We may derive a general formula for approximate integration by integrating any of the formulas of interpolation. Thus, Newton's formula (p. 215), na Q n (n i) ' n (n - i) . . . (n - k + i) ~~ ' 232 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX where x = XQ + nh, is true for all values of n if some order of differences is constant or approximately constant. Multiplying by dn and inte- grating term by term between the limits o and n, we have r* "*' C n C n b c n I y n dn = y I dn + a I n dn + - / n (n - i) dr /0 t/O vO [? *SO Since x = XQ + nh, therefore, n = ; and dn = -,- dx. Hence, n n Thus, if the differences after some order, as the &th, are negligible, we may use this formula to get the approximate area between the curve, the x-axis, and the ordinates x = x and x = x n . The process is equivalent to approximating the equation of the curve by a polynomial of the &th degree. The differences a , b , c , . . . are those which occur in a line through y parallel to the upper side of the triangle in the scheme on p. 210. Similar integration formulas can be derived from the other interpolation formulas. If the interval from x to x n is large, it is well to divide this into smaller intervals, apply the formula to each of the smaller intervals, and add the results. In this way we may derive the formulas of Art. 101 and similar formulas as special cases of the above general formula. Let us first note that by means of the rule for the formation of the successive differences of a function (p. 210) we may express the differences do, b , CQ, . . . in terms of y , yi, y z , - - Thus, ao = y\ yo, bo = ai a = (j2 yi) (yi yo) = yz 2 yi + yo, c = bi-b = [(y 3 - 2y z + y 1 ) - (y 2 '-2yi do = y 4 y 3 + 6 y t 4 yi + y , e* = Js - 5 ^4 + 10 ;y 3 - 10 y 2 + 5 yi - y , , - ko = y k - ky k -i H -- T- - y k -z - where the coefficients in the right members of these equations are the binomial coefficients, taken alternately plus and minus. ART. 103 GENERAL FORMULA FOR APPROXIMATE INTEGRATION 233 (i) Let n I and b , c , . . . all zero, i.e., approximate the curve (Fig. ioia) from x to Xi by a straight line, y = A + Bx. Then f y* Xl ydx = A [yo + iflo] = * bo J Applying this result to each interval and adding, we get the Trapezoidal rule: A T (2) Let n = 2 and c , J , all zero, *.e., approximate the curve (Fig. ioia) from x to x z by a parabola, y = A + -Brc + Cx 2 . Then /\T I y o)] = t ^ bo + 3 yi + Applying this result to n intervals, where n is a multiple of 3, and adding, we get Simpson's three-eighths rule: A s '= (4) Let n = 6 and the differences beyond the 6th order negligible, i.e., approximate the curve (Fig. ioia) from x to x 6 by a parabola of the 6th degree, y = A + Bx + Cx 2 + + Hx 6 . Then r ydx = h[6y + i8a + 27 6 Substituting the values of Oo, &o > /o in terms of the y's and re- placing T*iV /o by T^J / , thus neglecting fiofo which will be fairly small, we get Weddle's rule: A w = I 'ydx = fV h [yo + 5 yi + ^2 + 6y 3 + y4 + 5 J?> + ye]. Jxo We may apply this rule to n intervals where n is a multiple of 6. 234 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX r 6 dx . x We divide the interval into 6 equal parts, so that h = o.i. From the table 2 2.1 2.2 2-3 2.4 2.5 2.6 I 2 I 2.1 I 2.2 I 2^3 I 2.4 I 2^5 I 2^6 = o.i \- (- + -^ + + + + + 1 = 0.2624493; |_2 \2 2.6/ 2.1 2.2 2-3 ^ 2.4 2.5] + = 0.2623645; A, - (o . l)+5 . + . +6 . ++5+ .. = . 2623643 . By integration, r*= J 2 x \nx\ =ln2.6 In2 = ln 1.3=0.2623637. AT agrees with A to 4 decimals, while AS, AS', and ^4^ agree about equally well with A to 6 decimals. 104. Numerical differentiation. We are to find the slope of the curve y = f(x) at any point when the curve is drawn or a table of values of equidistant ordinates are given, i.e., we are to find ~ when the analyti- cal form of the function is unknown. Graphically, we must construct the tangent line to the curve at the given point. The exact or even approximate construction of the tangent line to a curve (except for the parabola) is difficult and inaccurate.* dy We may derive an expression for -j- by differentiating Newton's in- terpolation formula. Newton's formula . y n = yo -f- na H n (n i) n (n - i) . . . (n - k + i) \k k , is true for all values of n if some order of differences, as the &th, is con- stant or approximately constant. Since x = XQ + nh, therefore, dx hdn, and - = T ~r- . -=^ = ^. 2 * * - T ~r- . dx h dn dx 2 * See Art. 106 on graphical differentiation. dn* ART. 104 NUMERICAL DIFFERENTIATION 235 Hence, The values of these coefficients are tabulated for values of n between O and I at intervals of o.oi.* For the tabulated values x , Xi, . . . , x n , we have n = o, so that for these values of x we have the simpler formulas dy If the value of x for which -7- is required is near the end of the table, we may use similar formulas derived from the modified Newton's formula for end-interpolation (p. 217). d'V d?'V Example. Find -j- and -7-^ for x = 3 and x = 3.3 from table (l) on p. 211 and check the results by differentiating y = x 3 . Since x = 3 is a tabulated value we apply the second set of formulas : | = [37- 2 '(24)+f(6)]=2 7 ; g- [24- 6] = 18. From y = x 3 , ^ = 3^ = 27, | = 6x = 18. For x = 3.3 we apply the first set of formulas, where #0 = 37. &o = 24, c = 6, n = 0.3. Then ^ = [37 + (- 04)^+ (0-47) f]= 32.67; g = [ 24 + (-0.7) 6]= 19.8. From y = x*, ^ = 3 x* = 32.67, g = 6 x = 19.8. Example. Rate of change. The following table gives the results of ob- servation ; 6 is the observed temperature in degrees Centigrade of a vessel of cooling water, t is the time in minutes from the beginning of observation. 92.0 85-3 79-5 I 74-5 I 70-2 To find the approximate rate of cooling when t = I and / = 2.5. * See Rice, Theory and Practice of Interpolation. 236 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX From the table of differences / e Ai A 1 A> o 92.0 I 85-3 -6. 7 0.9 -5-8 O.I 2 79-5 0.8 -5-0 O.I 3 74-5 0.7 -4-3 0.4 4 70.2 I.I -3-2 5 67.0 when/=i, w = o and = -5.8 -^ (0.8) +^ (-o.i) = -6.23; when / = 2.5, n = 0.5 and^ = [-5.0 + o + (- 0.25) f^ 4 )] S-oa. Example. Maximum and minimum. The following table gives the results of measurements made on a magnetization curve of iron; B is the number of kilolines per sq. cm., n is the permeability (Fig. 104). o I 2 3 4 5 6 7 8 I 9 IO 1 II 1 12 1120 14 i 370 too WO 100 wo /*) >00 too wo ( 57C > 73< ) 865 985 1090 H75 1245 1295! 1330 I34o|i32o|i25o 930 72 ^ v ^ X x^ X \ S / \ / / \ \ / ) 1 2 3 4 5 6 7 FIG. 8 9 (B) 104. 10 11 12 13 If 15 To find the maximum permeability. In Fig. 104 the maximum perme- ability appears to be in the neighborhood of B = 10. We therefore tabu- late the differences of /* in the neighborhood of B = 10. 1330 1340 1320 1250 1 1 2O - 20 " - 70 _ -130 ART. 105 - GRAPHICAL INTEGRATION For values of B between B = 9 and B = 10, we have 237 For a maximum, TB = > hence 6n? + 6n n = o, and n = 0.94. Therefore, B = B + nh = 9.94. We find the corresponding value of p by the interpolation formula, M = 1330 + (0.94) (10) + (0.0282) (-30) + (o.oioo) (-20) = 1340. If we take account of A 1 and A 2 only, we get J =o, or n = | = 0.83, and B = 9.83. Then /* = 1336 + (0.83) (10) + (0.0275) (-30) = 1337-5- \ 105. Graphical integration. Let us find the value of the definite integral / f(x) dx or the area under the curve y = f(x) by graphical methods. We draw the curve y = f(x) (Fig. 1050) and along the ordinate at P (x, y} erect the ordinate y' whose value is a measure of the area under FIG. 1050. Fio. 1056. the curve y = f(x) from the initial point A (x = a) to the point P, i.e., y' = I }(x) dx. Thus for every point P (x, y) we have a corresponding point P' (x, y'). The curve traced by the point P' (marked / in the figure) is called the integral curve and the curve traced by the point P (marked A in the figure) is called the derivative curve. Evidently, if P and Q are two points on the A-curve and P' and Q' are their correspond- ing points on the /-curve, the difference of the ordinates of P' and Q', y" y', is a measure of the area under the arc PQ. The practical construction of the integral curve consists of the follow- ing steps (Fig. 1056). (i) Divide the interval from XQ to x n into n equal or unequal intervals and erect the ordinates y 0l y\, . . . , y n . 238 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX (2) Measure the areas xoA AiXi = yi, x A oA z x 2 = yz', . , XoA A n x n = y n '. These areas may be found by means of a planimeter or by the construction of the mean ordinates. Thus, the area x A AiXi is equal to the area of a rectangle whose base is x Xi and whose altitude is the mean ordinate mi within that area. Similarly, the area XiAiA&i is equal to the area of a rectangle whose base is x\x 2 and whose altitude is the mean ordinate m 2 within that area. Estimate the mean ordinates mi, m z , m 3 , . . . , m n within the successive sections. Then y\ = mi (*o*i), yz = y\ + m? (xix 2 ), y* = y 2 f + m$ (xyxa), . . . , y n f = yn'-i + m n (* n -i*n). If the intervals are all equal, i.e., x Xi = XiX 2 = . . . = x n -\x n = AJC, then y' = 2m AJC. (We shall later give a more exact construction for the mean ordinate.) (3) At xi, x 2> x 3 , . . . , x n erect or- dinates xiBi, XzB z , . . . , XnB n equal respectively to y\ , y 2 f , . . . , y n ', and draw a smooth curve through the points Bo, Bi, B 2 , . . . , B n . This last curve will approximate the required integral curve. Example. Construct the integral curve of the straight line y = I x be- tween x = o and x = 2. (Fig. 105^.) Divide the interval from x = o to x = 2 into 10 equal parts and erect the ordinates given in the table; here, AJC = 0.2. FIG. * y mAx / = 2mA* 0.9 0.18 o o 18 .6 .8 .0 .6 .4 .2 .2 -4 - .6 - .8 .0 0.7 0.5 0-3 O.I O.I -0.3 -0.5 -0.7 -0.9 o. 14 O . IO 0.06 0.02 o .02 o .06 O. IO 0.14 -0.18 0.32 0.42 0.48 0.50 0.48 0.42 0.32 0.18 It is evident that the mean ordinate in each section is merely one-half the sum of the end ordinates, so that the values of m are easily found. Erect the ordinates y' and draw a smooth curve through the ends of the /'z ordinates. The curve will approximate the parabola y' = I (i x) dx */o _ ART. 105 GRAPHICAL INTEGRATION 239 Example. The following table gives the accelerations a of a body sliding down an inclined plane at various times /, in seconds. To find the velocity and distance traversed at any time, if the initial velocity and initial distance are zero. o I IPO 0.320! 0.304 200 I 300 I 400 0.256) 0.176 | 0.080 500 0.016 600 0.080 700 I 800 0.136! 0.176 900 -0.208 1000 0.240 Since v = I a di and s = / v dt, the time- velocity curve is the integral curve of the time-acceleration curve, and the time-distance curve is in turn the integral curve of the time-velocity curve. In Fig. 105^, we have plotted t as abscissas and. a as ordinates. Th.- units chosen are I in. = 100 sec., and I in. = 0.16 ft. per sec. per sec. 0.32 0.16 (a) -0.16 -0.32 JU.UUV 60 000 ^ ^ ' y X 50,000 40,000 30,000 20,000 10,000 n V t ^ 00 --, \ ^ ^ ^, -^ / ^^ X S "^ x 80 60 / N X / ^ s / X v^ / ^^ \ / / / xl ^ ^ \ A ? . / ^ ^^ \ \ 40 * / ^ " -^ 1 1 -. \ / ^ ? 20 / X / - " ' JOO 200 300 400 500 600 700 800 900 1000 FIG. I05 is the angle between P"Q' and P'Q', then dS= Idn + U 2 . Therefore, dS = I ds - al d + Hence the total area swept out by PQ is ids - alfd + %l 2 C FIG. io7c. Now, if PQ comes back to its original position without turning com- pletely around, then the total angle of rotation / d = o, so that s-it, where 5 is the total displacement of any point on the circumference of the integrating wheel. But if PQ comes back to its original position after turning completely around, then S = Is - 2iral + irP. 248 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX The most common type of planimeter is the Amsler polar planimeter * (Fig. loyc). Here, Fig. 107^, by means of a guiding arm OQ, called the polar arm, one end Q of the tracer arm PQ^s constrained to move in a circle while the other end P is guided around a closed curve c-c-c- . . . which bounds the area to be measured. Then the area Q'P'PP"Q"QQ' is swept out twice but in opposite directions and the corresponding dis- placements of the integrating wheel cancel, so that the final displacement gives only the required area c-c-c- .... The circumference of the wheel is graduated so that one revolution corresponds to]a certain definite number of square units of area. P' FIG. 107^. The ordinary planimeter used for measuring indicator diagrams has / = 4 in. and the circumference of the wheel is 2.5 in.; hence one revolu- tion corresponds to 4 X 2.5 = 10 sq. in. The wheel is graduated into 10 parts, each of these parts again into 10 parts, and a vernier scale allows us to divide each of the smaller divisions into 10 parts, so that the area can be read to the nearest hundredth of a sq. in. The indicator diagram on p. 228 gives a planimeter reading of 2.55 sq. in., which agrees with the result found by Simpson's rule with 15 ordinates. The polar planimeters used in the work in Naval Architecture usually have a tracer arm of length 8 in., and a wheel of circumference 2.5 in., so that one revolution corresponds to 20 sq. in., thus giving a larger range for the tracing point. If the area to be measured is quite large, it may be split up into parts and the area of each part measured; or the area may be re-drawn on a smaller scale and the reading of the wheel multiplied by the area-scale of the drawing. f * This instrument was first put on the market by Amsler in 1854. f If PQ (Fig. 107 d) turns completely around, the required area is 5 + T (OQY. ART. 107 MECHANICAL INTEGRATION. THE PLANIMETER 249 If very accurate results are required, account must be taken of several errors, (i) The axis of the integrating wheel may not be parallel to the tracer arm PQ. This error can be partly eliminated by taking the mean of two readings, one with the pole to the left of the tracer arm, the other with the pole to the right* (Fig. loye). This cannot be done with the ordinary Amsler planimeter because the tracer arm is mounted above the polar arm, but can be done with any of the Coradi or Ott compensation planimeters; one of these instruments is illustrated in Fig. loj/. (2) The integrating wheel may slip; some of this slipping may be due to the irregularities of the paper and has been obviated by the use of disc planimeters, in which the recording wheel works on a revolving disc instead FlG of on the surface of the paper. Various types of linear planimeters have been constructed. These differ from the polar planimeters in that one end of the tracer arm is FIG. constrained to move in a straight line instead of in a circle. Planimeters of the linear type form part of the integrators described in Art. 108. Various other types of planimeters have been constructed, which do not have an integrating wheel. One of the best known of these is that of Prytz, also known as the hatchet planim- eter. f In this form of the instrument (Fig. ioyg) the end Q forms a knife- edge so that Q can only move freely along the line PQ. When P traces the FIG. iojg. given curve, Q will describe a curve such that PQ is always tangent to it. * For a proof of this statement, see Instruments and Methods of Calculation, p. 196. t For the theory of this instrument, see F. W. Hill, Phil. Mag., xxxviii, 1894, p. 265. 250 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX Prytz starts the instrument with the point P approximately at the center of gravity G of the area to be measured, moves P along the radius vector to the curve, completely around the curve, and back along the same radius vector to G. The required area is then given approximately by J 2 0, where / is the length PQ and is the angle between the initial and final positions of the line PQ. 1 08. Integrators. The Amsler integrator is practically an extension of the linear planimeter. In the latter instrument, the end Q of the tracer arm PQ of constant length /, is constrained to move in a straight line X'X, while the tracing point P describes a circuit of the curve. If the axis of the integrating wheel attached to PQ makes a variable angle ma with X'X (Fig. io8a) at each instant, the point P will have for ordinate y m = I sin ma, and the area described by P will be I / sin ma dx. On the other hand, the area described by P is equal to / times the displace- ment of any point on the circumference of the integrating wheel; hence / sin ma dx is equal to the displacement of a point on the circumference of an integrating wheel whose axis makes an angle ma with X'X. FIG. io8a. FIG. 1086. Now, given a curve c-c-c- . . . (Fig. 1086), Area = I y dx = / / sin a dx = I I sin a dx. Moment of area i f , , if,,.. PC, \ j v/ v = - I -y 2 dx = - I I 2 sin 2 a dx = - I (l cos 2 a) dx about X'X 2j^ 2J 4 J I 2 C I 2 C = - I dx -- I sin (90 2a)dx 4J 4 J - I sin (90 2 a) dx, since I dx = o, the arm PQ returning to its original position when P makes a complete circuit of the curve. Moment of inertia _ i_ of area about X'X ~ -\ - 3 1' \4 i 3 r . . i 3 r . , = - I sm a dx I sin 3 a dx. ART. 108 INTEGRATORS 251 Now , I sin a dx, I sin (90 2 a) dx, and I sin 3 a dx, and hence the area, moment, and moment of inertia can be measured by three in- tegrating wheels whose axes at any instant make angles a, 90 2 a, and 3 a, respectively, with X'X. The Amsler $-wheel integrator (Fig. io8c) consists of an arm PQ and 3 integrating wheels A , M, and /. The instrument is guided by a carriage which rolls in a straight groove in a steel bar; this bar may be set at a proper distance from the hinge of the tracer arm by the aid of trams. The FIG. io8c. line X'X, which passes through the points of the trams and under the hinge, is the axis about which the moment and moment of inertia are measured. The radius of the disk containing the Af-wheel is one-half the radius, and the radius of the disk containing the /-wheel is one-third the radius of the circular disk D to which they are geared. Therefore, the axis of the M-wheel turns through twice, and the axis of the /-wheel turns through three times the angle through which the tracer arm PQ or the axis of the A -wheel swings from the axis X'X. The integrating wheels are set so that in the initial position, i.e., when P lies on X'X, the axes of the A- and /-wheels are parallel to X'X while the axis of the M-wheel is perpendicular to X'X. Then, when the tracer arm PQ makes an angle a with X'X, the axes of the A-, M-, and /-wheels make angles a, 90 2 a, and 3 a, respectively, with X'X. Further- more, the graduations of the M-wheel are marked so that these gradua- tions move backward while the graduations on the other wheels move 252 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX forward. Hence, when P has completed the circuit, and if a, m, and i are the displacements of points on the circumferences of the A-, M-, and I- wheels, respectively, we have I 2 P /* Area la; Moment = -m; Moment of Inertia = -a i. The wheels are graduated from i to 10 so that a reading of 5, for example, means 5/10 of a revolution. The constants by which these readings are multiplied depend upon the length of the tracing arm and the circumferences of the integrating wheels. In the ordinary instru- ment, / = 8 in. and the circumferences of the A-, M-, and /-wheels are C A = 2.5 in., C M = 2.5 in., d = 2.34375 in. Thus, to find the area, a must be multiplied by' 8 X 2.5 = 20; moment, m " " " " - X 2.5 = 40; 4 8 3 moment of inertia, a " " " X 2.5 = 320, 4 8 3 and * " X 2.34375 = 100. Finally, if a\, az, m\, mz, and i\, i z are the initial and final readings of the A-, M-, and /-wheels, we have Area = 20 (az a\); Moment = 40 (w 2 m\) ; Moment of Inertia 320 (az a\) 100 (iz ii). 109. The integraph. This is a machine which draws the integral curve, y' = I f(x) dx, of the curve y = f(x} . The most familiar type of such machines is the one invented by Abdank-Abakanowicz in 1878. The theory of its construction is very simple. A diagram of the machine is given in Fig. 1090. The machine is set to travel along the base line of the curve to be integrated, and two non-slipping wheels, W, ensure that the motion continues along this axis. The scale-bar slides along the main frame as the tracing point P, at the end of the bar, describes the curve y = f(x) to be integrated. The radial-bar turns about the point Q which is at a constant distance a from the main frame. The motion of the re- cording pen at P\ is always parallel to the plane of a small, sharp-edged, non-slipping wheel w, and by means of the parallel frame- work ABCD, the plane of the wheel w is maintained parallel to the radial bar [since w is set perpendicular to AB which is parallel and equal to CD throughout the motion, and the radial bar is set perpendicular to CD]. As the point P describes the curve y = /(*), the angle 6 between the radial-bar and the ART. 109 THE INTEGRAPH 2 53 axis, and consequently the angle 6 between the plane of the wheel and the axis, are constantly changing, and the recording pen at PI draws a curve with ordinate y' such that its slope dx and therefore, y ' = 1 Cf ( x ) dx=^X area ORP, so that the curve drawn by PI is the integral curve of the curve traced by P. W FIG. 1090. If we now set the machine so that the point P traces the integral curve, then the recording pen PI will draw its integral curve We may thus draw the successive integral curves y', y", y'" ..... Fig. 1096 gives the integral curves connected with the curve of loads of the shaft of a Westinghouse-Rateau Turbine. The curve of loads is repre- sented by the broken line in the figure. By successive integration we get the shear curve, the bending moment curve, the slope curve, and the deflection curve. The distance marked "offset" is the distance OOi in Fig. (254) ART. no MECHANICAL DIFFERENTIATION 255 no. Mechanical differentiation. The Differentiator. This is a dy machine which draws the derivative curve y' = -/- of the curve y = f(x). CLx Since the ordinate of the derivative curve is equal to the slope of the in- tegral curve, it is necessary to construct the tangent lines at a series of points of the integral curve. We have already mentioned (Art. 106) the use of a strip of celluloid with two black dots on its under side to deter- FIG. no. mine the direction of the tangent. This scheme is used in a differentiating machine constructed by J. E. Murray.* In a differentiating machine recently constructed by A. Elmendorf,f a silver mirror is used for de- termining the tangent. The mirror is placed across the curve so that the curve and its image form a continuous unbroken line, for then the surface of the mirror will be exactly normal to the curve, and a perpen- dicular to this at the point of intersection of the mirror and the curve will give the direction of the tangent line. If the surface of the mirror de- * Proc. Roy. Soc. of Edinburgh, May, 1904. f Scientific American Supplement, Feb. 12, 1916.] 256 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX viates even slightly from the normal, a break will occur at the point where the image and curve join. It is claimed that with a little practice a re- markable degree of accuracy can be obtained in setting the mirror. Fig. no gives a diagram illustrating the working of this machine. The tracing point P follows the curve y = f(x) so that the curve and its image in the mirror MP form a continuous unbroken line; then the arm P T, which is set perpendicular to the mirror, will take the direction of the tangent line to the curve. The link PR, of fixed length a, is free to move horizontally in the slot X'X' of the carriage C. The vertical bar SU passes through R and is constrained to move horizontally by heavy rollers. The point Q slides out along the tangent bar PT and also vertically in the bar SU, carrying with it the bar QP r . If we choose for the #-axis a line XX whose distance from X'X' is equal to QP' t then the point P' will draw a curve whose ordinate is equal to y' = RQ. But RQ/a is the slope of the tangent PT, hence, y' = a X -f^-, and the curve drawn by P' is the derivative curve of the curve traced by P. The machine is especially useful for differentiating deflection-time curves to obtain velocity-time curves, and by a second differentiation, acceleration-time curves. It is also helpful in solving many other problems. EXERCISES. Apply the approximate rules of integration (Art. 101) to the following examples: /* 1 - dx 1. Evaluate J , when h = o.i, and when h = 0.05, and compare the results with the values obtained by integration. 2. Evaluate \ sin x dx, when h = ^ , and when h= , and compare the results /o 6 12 with the values obtained by integration. 3. The arc of a quadrant of an ellipse whose eccentricity is 0.5 is given by * pi . _ J vi 0.25 sin 2 < d. Evaluate the integral when h = 9. rdx ,. =. , when h = 0.5. - v x 3 - x + i 5. The semi-ordinates in ft. of the deck plan of a ship are 3, 16.6, 25.5, 28.6, 29.8, 30, 29.8, 29.5, 28.5, 24.2, 6.8; these measurements are 28 ft. apart. Find the area of the deck. 6. Given the following data for superheated steam io ^ 13 Find the work done. V 2 4 6 1 8 P 105 42.7 25-3 I 16.7 EXERCISES 257 7. The length of an indicator diagram is 3.6 in. The widths of the diagram, 0.3 in. apart, are o, 0.40, 0.52, 0.63, 0.72, 0.93, 0.99, i. oo, i. oo, i. oo, i. oo, 0.97, o. Find the mean effective pressure. 8. The length of an indicator diagram is 3.2 in. The widths of the diagram, 0.2 in. apart, are i. oo, 1.68, 1.62, i. oo, 0.64, 0.48, 0.36, 0.26, o. Find the mean effective pressure. 9. The speed of a car is v miles per hour at a time t seconds from rest; 1 o 5 1 10 15 20 25 30 V 3-7 7-5 10.9 13-0 13-7 H Find the distance traversed in 30 seconds. 10. 5 is the specific heat of a body at temperature 6 C. o 2 | 4 6 1.00664 1-00543 I 1-00435 1-00331 | 1.00233 | 1.00149 1.00078 Find the total heat required to raise the temperature of a gram of water from o C. to 12 C. (total heat = f^sde). /BI 11. The areas in sq. ft. of the sections of a ship above the keel and two feet apart are 2690, 3635, 4320, 4900, 5400. Find the total displacement in tons. 12. A reservoir is in the form of a volume of revolution and D is the diameter in ft. at a depth of p feet beneath the surface of the water. p o 16 | 32 48 | 64 80 I 96 D no 105 I 100 86 I 66 48 J 27 Find the number of gallons of water the reservoir holds when full. 13. A plane board is immersed vertically in water. The widths of the board in ft. parallel to the surface of the water and at depths ft- apart are 4.0, 3.6, 3.0, 1.7, 1.3, i.o, 0.8, 0.6, o.i. Find the pressure on the board and the depth of the center of pressure when the surface of the water is level with the top of the board. 14. The half-ordinates in ft. of the mid-ship section of a vessel at intervals 2 ft. apart are 12.2, 12.5, 12.6, 12.7, 12.7, 12.5, 12. 1, II.5, IO.I, 6.5, 0.2. Find the position of the center of gravity of the section. 15. The shape of a quarter-section of a hollow pillar is given by the following table. The axes of x and y are the shortest and longest diameters. x'm. o 0.25 0.50 075 5.83 1. 00 5-64 ISO 5-48 ITS 5-22 2.00 2.25 2.50_ 4-35 ^75 3-88 3-00 3-25 3-25 out- side y\ in. in- side y* in. 6 5-95 5-90 5T6 4-99 4.68 2-34 5 4.90 4.78 4-65 4-45 4.22 3-8o 3-40 2-77 2.08 o Find the moments of inertia of the section about the x- and y- axes. 258 APPROXIMATE INTEGRATION AND DIFFERENTIATION CHAP. IX 16. Apply the formulas for numerical differentiation (p. 235) to table (2) y = x 3 on p. 211, and find ~ and -3-^ when x = 5.31 and * = 5.33. Check the results by actual differentiation. 17. Apply the formulas for numerical differentiation (p. 235) to table (8) y = log sin x on p. 212, and find -3- and -j - when x = i 20' and x = i 24'. Check the results by actual differentiation. 18. In the following table, 5 is the distance in ft. which the projectile of a gun travels along the bore in / sec. 0.0360 | 0.0490! 0.0598 0.0695 0.0785 0.0871 0-0953 I 0.1032 | 0.1109 I 0.1184 ds fdt d?s dH //dt\ 3 , Find the velocity v = -r = i 1 -7- , and the acceleration a = -j^ = ~ j~i / I ;r ) when 5 = 5ft. 19. Use the data given in Ex. 6 to find the rate of change of the pressure with re- spect to the volume, dp/dv, when v = 4 and v = 5. 20. Use the data given in Ex. 9 to find the acceleration, a = -j- , when / = 10 and t = 12. 21. Find the minimum value of the polynomial which has the values 4 I 6 ii 27 22. The following table gives the results of measurements made on a normal in- duction curve for transformer steel; B is the number of kilolines per sq. cm.; ju is the per- meability. B 625 870 1035 1350 1465 1520 I 1480 | 1430 TO 1370 II 1280 1130 Find the maximum permeability. 23. Construct the integral curve of the parabola y = x | x- as x varies from o to 2. 24. Construct the integral curve of the sine wave y = 2 sin 2 x as x varies from o to IT. 25. The following table gives the accelerations a of a body sliding down an inclined plane for various distances 5 in ft. s \ o | loo 200 300 [ 400 500 | 600 700 800 | 900 I 1000 -I I- a [o.32o|( a [0.32010.304 0.256 0.176 | 0.080 -0.016 | -0.080 -0.136-0.1761-0.2081-0.240 Use the method employed in the illustrative example on p. 239 for drawing the integral curves and determining the velocity, v - y 2 fa ds, and the time, t = J - ds, for any distance, if v = o and / = o when s = o. 26. The following table gives the accelerations a of a body at various velocities v in ft. per sec. 0.405 0.360 | 0.283 Draw the integral curves to determine the time, t for any velocity, if / = o and 5 = when v = o. 0.179 0.069 I 0.013 /- dv, and the distance, s = \ -dv, a J a ' EXERCISES 259 27. In the following table o I 4 6 8 "5 15 38,000 38,500 38,500 35,500 27,500 19,000 15,700 xx) I 3850 P is the resultant pressure in pounds on the piston of a steam engine at distances s inches from the beginning of the stroke. Draw the integral curve to find the work done as the piston moves forward. ( Work = J P ds. J 28. A car weighs 10 tons. It is drawn by a pull of P Ibs.; / is the time in seconds since starting. t o 2 5 | 8 | 10 13 | 16 I 19 P 1020 980 882 I 720 I 702 650 I 713 I 722 805 If the retarding friction is constant and equal to 410 Ibs., draw the integral curve to find the speed of the car at any time. ( Momentum = J (P 410) dt. J 29. In the following table t I 0.00490 0.00598 I 0.00695 0.0078510.00871 0.00953 1 0.01032 0.01109 0.01184 869 987 | 1074 H42 I H95 1242 | 1277 1309 1335 v is the velocity of projection in ft. per sec. in the bore of a gun at time t sec. from the beginning of the explosion. U s = 2 ft. when / = 0.00490 sec., draw the integral curve to show the relation between the distance and the time. 30. A beam 10 ft. long is loaded as in the following table, where w is the weight per unit length at distances x ft. from the free end. 2.5 3-7 I 5-5 4 5 6 7 8 9 IO 7-7 9-7 11.2 12.2 n.8 10.2 7.2 Draw integral curves to show (i) the shearing force, s = \w dx and (2) the bending moment, M = ( s dx. 31. The following table gives the measurements for every 15 of the intensity of illumination of a lamp. Angle 6 c-p _2_J5_ 60 5! 88.0 99-5 45 I 60 86.5 ; 50.0 _75__90_ 25.0 28.0 135 20.0 150 165 180 15.0 13-0 12.5 Apply the method of the illustrative example on p. 242 to find the m.s.c.p. for various sections of the lamp. 32. In the following table IO 20 30 156 608 1308 60 70 80 90 100 4076 4942 5676 6236 6588 5 is the distance in ft. traversed by a body weighing 2000 Ibs. in t sec. Draw the deriva- tive curves to show the velocity and acceleration at any time. Also draw the curve showing the relation between the kinetic energy and the force. 33. The observed temperature 6 in degrees Centigrade of a vessel of cooling water at time t in min. from the beginning of observation are given in the following table: t I I < Draw the derivative curve to show the rate of cooling at any time. I 2 3 5 7 10 15 20 85-3 79-5 74-5 67.0 60.5 53-5 45-o 39-5 INDEX. Adiabatic expansion formula, 48 chart for, 33, 49 Alignment or nomographic charts (see also Charts, alignment or nomo- graphic) fundamental principle of, 44 with curved scales, 106 with four or more parallel scales, 55 with parallel or perpendicular index lines, 87, 91, 97 with three or more concurrent scales, 104 with three parallel scales, 45 with two intersecting index lines, 68 with two or more intersecting index lines, 76 with two paraHel scales and one inter- secting scale, 65 Approximate differentiation, 224 Approximate integration, 224 Area, by approximate integration rules, 227 by planimeter, 246 Armature or field winding formula, 90 chart for, 90 Bazin formula, 101 chart for, 102, 116 Center of gravity, by approximate inte- gration rules, 231 Chart, alignment or nomographic, for adiabatic expansion, 49 armature or field winding, 90 Bazin formula, 102, 116 Chezy formula for flow of water, 58 D'Arcy's formula for flow of steam, 81 deflection of beams, 72, 73, 86 discharge of gas through an orifice, 89 distributed load on a wooden beam, 83 focal length of a lens, 106 Francis formula for a contracted weir, 109 friction loss in pipes, 94 Chart, Grasshoff's formula, 51 Hazen- Williams formula, 60 horsepower of belting, 54 indicated horsepower of a steam en- gine, 63 Lame formula for thick hollow cylin- ders, 92 McMath "run-off," formula, 49 moment of inertia of cylinder, 100 multiplication and division, 47 prony brake, 70 resistance of riveted steel plate, 103 solution of quadratic and cubic equa- tions, 112 specific speed of turbine and water wheel, 75 storm water run-off formula, 108 tension in belts, 54 tension on bolts, 67 twisting moment in a cylindrical shaft, 78 volume of circular cylinder, 49 volume of sphere, 49 Charts, hexagonal, 40 Chart with network of scales, for adiabatic expansion, 33 chimney draft, 38 elastic limit of rivet steel, 34 equations in three variables, 28 equations in two variables, 20 multiplication and division, 30, 31 solution of cubic equation, 36 temperature difference, 39 Chezy formula for flow of water, 56 chart for, 58 Chimney draft formula, 37 chart for, 38 Coefficients in trigonometric series evalu- ated, by six-ordinate scheme, 179 by twelve-ordinate scheme, 181 by twenty-four-ordinate scheme, 185 for even and odd harmonics, 179 INDEX Coefficients in trigonometric series evalu- ated, for odd harmonics only, 1 86 for odd harmonics up to the fifth, 187 for odd harmonics up to the eleventh, 189 for odd harmonics up to the seventeenth, 191 graphically, 2OO mechanically, 203 numerically, 179, 186, 192, 198 Constants in empirical formulas deter- mined by method of averages, 124, 126 method of least squares, 124, 127 method of selected points, 124, 125 Coordinate paper, logarithmic, 22 rectangular, 21 semilogarithmic, 24 D'Arcy's formula for flow of steam, 79 chart for, 8 1 Deflection of beams, 70, 71, 84 chart for, 72, 73, 86 Differences, 210 Differentiation, approximate, 224 graphical, 244 mechanical, 255 numerical, 234 Differentiator, 255 Discharge of gas through an orifice, 89 chart for, 89 Distributed load on a wooden beam, 80 chart for, 83 Durand's rule, 226 Elastic limit of rivet steel, 32 chart for, 34 Empirical formulas, determination of constants in, 124, 125, 173, 174 for non-periodic curves, 120 for periodic curves, 170 involving 2 constants, 128 involving 3 constants, 140 involving 4 or more constants, 152 Equations, solutions of (see Solutions of algebraic equations) Experimental data, 120, 170 Exponential curves, 131, 142, 151, 153, 156 Focal length of a lens, chart for, 35, 40, 106 slide rule for, 15 Fourier's series, 170 Francis formula for a contracted weir, no chart for, 109 Friction loss in pipes, 94 chart for, 94 Fundamental of trigonometric series, 170 Gauss's interpolation formula, 219 Graphical differentiation, 244 Graphical evaluation of coefficients, 200 Graphical integration, 237 Graphical interpolation, 209 Grasshoff's formula, 50 chart for, 51 Harmonic analyzers, 203 Harmonics of trigonometric series, 170 Hazen-Williams formula, 57 chart for, 60 Hexagonal charts, 40 Horsepower of belting, 52 chart for, 54 Hyperbola, 149 Hyperbolic curves, 128, 135, 137, 140 Index line, 44 Indicated horsepower of steam engine, 6l chart for, 63 Integraph, 252 Integration, approximate, 224 applications of, 227 by Durand's rule, 226 by rectangular rule, 223 by Simpson's rule, 226, 233 by trapezoidal rule, 225 by Weddle's rule, 233 general formula for, 231 graphical, 237 mechanical, 246 Integrators, 250 Interpolation, 209 Gauss's formula for, 219 graphical, 209 inverse, 219 Lagrange's formula for, 218 Newton's formula for, 214, 217 Isopleth, 44 Lagrange's interpolation formula, 218 INDEX xiii Lame formula for thick hollow cylinders, 91 chart for, 92 Least Squares, method of, 124, 127 Logarithmic coordinate paper, 22 Logarithmic curve, 151 Logarithmic scale, Maxima and minima by approximate differentiation formulas, 236 McMath "run-off" formula, 48 chart for, 49 Mean effective pressure by approximate integration rules, 228 Mechanical differentiation, ,255 Mechanical integration, 246 Moment, by integrator, 250 Moment of inertia, by approximate integration rules, 230 by integrator, 250 Moment of inertia of cylinder, 99 chart for, 100 Multiplication and division, charts for, 30, 31, 41, 47 Newton's interpolation formula, 214, 217 Nomographic or alignment charts (see Alignment or nomographic charts) Numerical evaluation of coefficients, 179, 186, 192, 198 Numerical differentiation, 234 Numerical integration, 224 Numerical interpolation, 215 Parabola, 145 Parabolic curves, 128, 135, 140 Periodic phenomena, representation of, 170 Planimeter, Amsler polar, 248 compensation, 249 linear, 249 principle of, 246 Polynomial, 159 Pressure and center of pressure, by approximate integration rules, 231 Prony brake, 69 chart for, 70 Rates of change, by approximate differ- entiation formulas, 235 Rectangular coordinate paper, 21 Rectangular rule, 225 Resistance of riveted steel plate, 101 chart for, 103 Scale, definition of, I equation of, 2 logarithmic, 2 representation of, I Scale modulus, 2 Scales, network of, 20 perpendicular, 20 sliding, 7 stationary, 5 Semilogarithmic coordinate paper, 24 Simpson's rule, 226, 233 Slide rule, circular, 16 for electrical resistances, 15 for focal length of lens, 15 Lilly's spiral, 1 8 logarithmic, 9 log-log, 13 Sexton's omnimetre, 17 Thacher's cylindrical, 18 Solutions of algebraic equations, by means of parabola and circle, 26 by means of rectangular chart, 35 by means of alignment chart, no by method of inverse interpolation, 221 on the logarithmic slide rule, 1 1 Specific speed of turbine and water wheel, 73 chart for, 75 Storm water run-off formula, 107 chart for, 108 Straight line, 122, 125 Tables, construction of, 213 Temperature difference, 37 chart for, 39 Tension in belts, 52 chart for, 54 Tension on bolts, 66 chart for, 67 Trapezoidal rule, 225 Trigonometric series, 170 determination of constants in, 173, 174 Twisting moment in a cylindrical shaft, 77 chart for, 78 J1V INDEX Velocity, by approximate integration rules, Volume of sphere, 50 229 chart for, 49 Volume, by approximate integration rules, 229 Weddle's rule, 233 Volume of circular cylinder, 48 Work, by approximate integration rules, chart for, 49 228 THE LIBRARY UNIVERSITY OF CALIFORNIA Santa Barbara THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW. 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