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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I I . -,,, j^. ^-Tf^ -\ t f I MATHEMATICAL MONOGRAPHS EDITED BY Mansfield Merriman and Robert S. Woodward. Octavo, Cloth. No. 1. No. 2. No. 3. No. 4. No. S. No. 6. No. 7. No. 8. No. 9. No. 10. No. 11. No. 12. No. 13. No. 14. No. 15. No. 16. No. 17. No. 18. No. 19. History of Modern Mathematics. By David Bugenb Smith. |i.oo net. Synthetic Projective Geometry. By George Bruce Ualsted. Ii.oo net. Determinants. By Labnas Gifford Weld. 1 1. 00 net, Hyper'jolic Functions. By James Mc- Mabon. %i.oo net. Harmonic Functions. Byerly. Ii.oo net. By WnxiAM B. Qrassmann's Space Analysis. By Edward W. Hyde. Ii.oo net. Probability and Theory of Errors. By Robert S. Woodward. Ii.oo net. Vector Analysis and Quaternions. By Alexander Macfarlame. |i.oo net. Differential Equations. By William WooLSEY Johnson. |i.oo net. The Solution of Equations* By Mansfield Merriman. |i.oo net. Functions of a Complex Variable. By Thomas S. Piske. $i.oo net. The Theory of Relativity. By Robert D. Carmichael. 1 1.00 net. The Theory of Numbers. By Robbrt D. Carmichael. |i.oo net. Algebraic Invariants. By Lbonasd B. Dickson. I1.25 net. Mortality Laws and Statistics. By Robert Henderson. I1.25 net. Diophantine Analysis. By Robert D. Carmichael. I1.25 net. Ten British Mathematicians. By Alex- ander Macfarlanb. $1.35 net. Elliptic Integrals. By Harris Hancock. I1.25 net. Empirical Formulas. By Thkodors R. Running. Ix.40 net, > PUBLISHED BY JOHN WILEY & SONS, Inc., NEW YORK. CHAPMAN &. HALL, Limited, LONDON. I MATHEMATICAL MONOGRAPHS EDITED BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD No. 19 / EMPIRICAL FORMULAS • • •• "* BT J^^ - THEODORE Ri*' RUNNING Associate Professor of Mathematics. University of Michigan FIRST EDITION NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited 1917 \ Copyright, 1917, BY THEODORE R. RUNNING pnna or . BRAUNWORTH A,' CO. BOOK MANUrACTURCRB ■OOOKLVN. N. V. 7 ^ > o #• ^ PREFACE This book is the result of an attempt to answer a number of questions which frequently confront engineers. So far as the author is aware no other book in English covers the same ground in an elementary manner. It is thought that the method of determining the constants in formulas by the use of the straight line alone leaves little to be desired from the point of view of simplicity. The approxi- mation by this method is close enough for most problems arising in engineering work. Even when the Method of Least Squares must be employed the process gives a convenient way of obtain- t ing approximate values. For valuable suggestions and criticisms the author here expresses his thanks to Professors Alexander Ziwet and Horace . W. King. p^ X. Jx. Jx. o^ UNiVERsmr of Michigan, 191 7. CONTENTS PAGE Introduction 9 CHAPTER I I. y=a'^bx'^cx*-\-dx*'^ . . . -\-qo^ 13 Values of x form an arithmetical series and A^y constant. 11. y=a+-^+-^+4+ • • • +Zh 22 X X^ X^ 7^ Values of - form an arithmetical series and A*y constant. X III. - ^a-\-hX'\-cx'^-\-dx^-\- . . . •\-qo^ 25 y Values of x form an arithmetical series and A**- constant. y rV. y'^-a-\-hx-^cx'^'\-dx^-^ . . . •\-qo^ 25 Values of x form an arithmetical series and A**;y* constant. CHAPTER II V. y-alf • 27 Values of x form an arithmetical series and the values of y form a geometrical series. VI. y=-a-\-hc' 28 Values of x form an arithmetical series and the values of Ay form a geometrical series. ^"-^ VII. logy =aH-&c* 32 Values of x form an arithmetical series and the values of A log y form a geometrical series. VIII. y—a'\-hx-\-cd' ^^ Values of x form an arithmetical series and the values of A*y form a geometrical series. ^X IX. y = ioa+»*+«*' 37 Values of x form an arithmetical series and the values of A* log y constant. 5 6 CONTENTS PAGE X. y^ks'f 37 Values of x form an arithmetical series and the values ^"^ A* log y form a geometrical series. XI. y= -^ 38 Values of x form an arithmetical series and A*- constant. y CHAPTER III ^ XII. y=ax^ 42 Values of x form a geometrical series and the values of y form a geometrical series. XIII. y =a+6 log x-\-c log*jc .44 Values of x form a geometrical series and A^y constant. XIV. y=a-\-bxf 45 Values of x form a geometrical series and the values of Ay form a geometrical series.^ XV. y^aic?^ 49 Values of x form a geometrical series and the values of A log y form a geometrical series. CHAPTER IV XVI. {x^a)(y^h) =c 53 Points represented by (x—xky ^ | lie on a straight Une. -^ XVIa. y=aid'+' / 56 Points represented by ( log — , log — ) lie on a \x—xk y ytj straight Hne. XVII. y=ae^-\-be^ 58 Values of x form an arithmetical series and the points repre- sented by I'-^j ^^^^] He on a straight line whose slope, \yk yi J My is positive and whose intercept, 5, is negative, and also M^+^B is positive. XVIII. y =e^(c cos bx-\-d smbx) 61 Values of x form an arithmetical series and the points repre- sented by l^^^j 5!i±-M lie on a straight line whose slope, \ yk yk J M, and intercept, B, have such values that M^-\-/^ is negative. CONTENTS 7 PAGE XIX. y=ax^-\-hofi 65 Values of x form a geometrical series and the points repre- sented by (^^S y^±l\ lie on a straight line whose slope, \ yt yk I My is positive and intercept, 5, negative, and also JIf 24.45 positive. XlXa. y=aofc' 72 Values of x form a geometrical series and the points repre- sented by ixny log 5!^ii J lie on a straight line. CHAPTER V XX. y =Oo+fli cos X'{-a2 cos 2x+ai cos3ic+ . . . -h^ cos rx 74 -\-bismx-\-b2sm2x-\-biSm$x-\' . . . -\-br sin rx Values of y periodic. CHAPTER VI Method of Least Squares 90 Application to Linear Observation Equations. Application to Non-linear Observation Equations. CHAPTER VII Interpolation 100 Differentiation of Tabulated Fimctions. CHAPTER Vni Numerical Integration 114 Areas. Volumes. Centroids. Moments of Inertia. APPENDIX Figures I to XX 132-143 Index 144 .»* EMPIRICAL FORMULAS INTRODUCTION In the results of most experiments of a quantitative nature, two variables occur, such as the relation between the pressure and the volume of a certain quantity of gas, or the relation between the elongation of a wire and the force producing it. On plotting the sets of corresponding values it is found, if they really depend on each other, that the points so located lie approximately on a smooth curve. In obtaining a mathematical expression which shall represent the relation between the variables so plotted there may be two distinct objects in view, one being to determine the physical law underlying the observed quantities, the other to obtain a simple formula, which may or may not have a physical basis, and by which an approximate value of one variable may be computed from a given value of the other variable. In the first case correctness of form is a necessary considera- tion. In the second correctness of form is generally considered subordinate to simplicity and convenience. It is with the latter of these (Empirical Formulas) that this volume is mostly concerned. The problto of determining the equation to be used is really an indeterminate one; for it is clear that having given a set of corresponding values of two variables a number of equations can be foimd which will represent their relation approximately. Let the coordinates of the points in Fig. i represent different sets of corresponding values of two observed quantities, x and y. If the points be joined by segments of straight lines the broken 9 10 EMPIRICAL FORMULAS line thus formed will represent to the eye, roughly, the relation between the quantities. It is reasonable to suppose, however, that the irregular dis- tribution of the points is due to errors in the observations, and that a smooth curve drawn to conform approximately to the distribution of the points will more nearly represent the true relation between the variables. But here we are immediately confronted with a difficulty. Which curve shall we select? a I/l 1 U .^ / ^ ^ > .•^ / r , ^ ^ / r i a / C^ "- A \ # Fig. I. or J? or one of a number of other curves which might be drawn to conform quite closely to the distribution of the points? In determining the form of curve to be used reliance must be largely placed upon intuition and upon knowledge of the experiments performed. The problem of determinmg a simple equation which will represent as nearly as possible the curve selected is by far the more difficult one. Ordinarily the equation to be used will be derived from a consideration of the data without the intermediate step of drawing the curve. Unfortunately, there is no general method which will give the best form of equation to be used. There are, however, a number of quite simple tests which may be applied to a set of INTRODUCTION 11 data, and which will enable us to make a fairly good choice of equation. The first five chapters deal with the application of these tests and the evaluation of the constants entering into the equations. Chapter VI is devoted to the evaluation of the constants in empirical formulas by the Method of Least Squares. In Chapter VII formulas for interpolation are developed and their applications briefly treated. Chapter VIII is devoted to approximate formulas for areas, volumes, centroids, moments of inertia, and a number of examples are given to illustrate their application. Figs. I to XX at the end of the book show a few of the forms of curves represented by the different formulas. A few definitions may be added. Arithmetical Series. A series of numbers each of which, after the first, is derived from the preceding by the addition of a constant number is. called an arithmetical series. The con- stant number is called the common difference 6, 6.3, 6.6, 6.9, 7.2, 7.5 .. . and 18.0, 15.8, 13.6, 11.4, 9.2 .. . are arithmetical series. In the first the common difference is .3, and in the second the common difference is —2.2. Geometrical Series, A series of niunbers each term of which, after the first, is derived by multiplying the preceding by some constant multiplier is called a geometrical series. The constant, multiplier is called the ratio. 1.3, 2.6, 5.2, 10.4, 20.8, 41.6 . . . and 100, 20, 4, .8, .16, .032 . . . are geometrical series. In the first the ratio is 2, and in the second it is .2. Differences are frequently employed and their meaning can best be brought out by an example. 12 EMPIRICAL FORMULAS X y Ay A*y A»y A*y I I0.2 2 II. I 0.9 I.I 0.2 0.0 3 12.2 1.3 0.2 1.2 1.2 4 5 6 13.5 16.2 18.0 2.7 1.8 1.0 1.4 -0.9 -0.8 -2.3 0.1 2.8 -3.5 2.4 2.7 7 19.0 2.0 8 22.0 3.0 In the table corresponding values of x and y are given in the first two columns. In the third column are given the values of the first differences. These are designated by Ay. The first value in the third column is obtained by subtracting the first value of y from the second value. The column of second differ- ences, designated by A^y, is obtained from the values of Ay in the same way that the column of first differences were obtained from the values of y. The method of obtaining the higher differ- ences is evident. CHAPTER I I. y—a+bx+co(^+dx'^+ . . . +qaf^ Values of x form an arithmetical series and A'^y constant. In a tensile test of a mild steel bar, the foUowing observa- tions were made (Low's Applied Mechanics, p. i88): Diameter of bar, unloaded, 0.748' inch, IT = load in tons, x= elongation in inches, on a length of 8 inches. w I 2 3 4 5 6 X Ax 0.0014 0.0013 S 0.0027 0.0013 0.0040 o.oois 0.0055 0.0013 0.0068 0.0014 0.0082 Plotting W and x, Fig. 2, it is observed that the points lie very nearly on a straight line.* Indeed, the fit is so good that it may be almost concluded that there exists a linear relation, between W and x. From the figure it is found that the slope of the line is 0.00137 and that it passes through the origin. The relation between W and x is therefore expressed by the equation x=o,ooi^'jW, The observed values of x and the values computed by the above formula are given in the table below. w Observed x Computed x I 2 3 4 5 6 0.0014 0.0027 0.0040 O.OOSS 0.0068 0.0082 0.00137 0.00274 0.00411 O.00S48 0.00685 0.00822 * By the use of a fine thread the position of the line can be deter- mined quite readily. 13 14 EMPIRICAL FORMULAS The agreement between the observed and the computed values is seen to be quite good. It is to be noted, however, that the formula can not be used for computing values of x outside ¥0 .0080 y ► 4»T& .0072 .0068 y / / r- / 0061 / .0060 / /■ .0056 y / iOOR? J\ r 0048 / r 0014 / .0040 A y .0036 .0032 y / y / .0028 / r .0024 / T .0020 jflOlB / V / .0012 .0008 ■ / / A .0004 / r ^ 1 J L 4 1 1 4 1 i \ \ 1 S 6 MO Fig. 2. the elastic limit. In the experiment 6 tons was the load at the elastic limit. It is not necessary to plot the points to determine whether they lie approximately on a straight line or not. Consider the general equation of the straight line Starting from any value of x^ give to x an increment, Aa:, and y will have a corresponding increment, Ly, y'\-Ly=m{^-\-^oi)'\-'k\ y=mx+k; Ay=mAx, DETERMINATION OF CONSTANTS 15 From this it is seen that, in the case of a straight line, if the increment of one of the variables is constant, the increment of the other will also be constant. From the table it is observed that the successive values of W differ by unity, and that the difference between the successive values of x is very nearly constant. Hence the relation between the variables is expressed approximately by x=inW+k, where tn and k have the values determined graphically from the figure. By the nature of the work it is readily seen that the graphical determination of the constants will be only approximate imder the most favorable conditions, and should be employed only when the degree of approximation required will warrant it. Satisfactory results can be obtained only by exercising great care. Carelessness in a few details will often render the results useless. Understanding how a graphical process is to be carried out is essential to good work; but not less important is the practice in applying that knowledge. In experimental results involving two variables the values of the independent variable are generally given in an arithmetical series. Indeed, it is seldom that results in any other form occur. It will be seen, however, that in many cases where the values of the independent variable are given in an arithmeti- cal series it will be convenient to select these values in a geomet- rical series. As a special case consider the equation y = 2-'^x+x!^. If an increment be assigned to Xy y will have a corresponding increment. The values of x and y are represented in the table below. Ay stands for the number obtained by subtracting any value of y from the succeeding value. A^y stands for the num- 16 EMPIRICAL FORMULAS ber obtained by subtracting any value of Ay from its succeeding value. The values of x have the common difference 0.5. X o.S I.O i-S 2.0 2-5 30 35 4.0 y 0.7s 0.00 -0.25 O.CX) 0.7S 2.00 3-75 6.00 Ay -0.7s —0.25 0.25 0.7S I-2S 1.75 2.25 A^y 0.50 0.50 0.50 0.50 0.50 0.50 The values A^y, which we call the second differences, are constant. These differences could equally well have been computed as follows: y+Ay = 2 —^(x+Ax) + (x+Axy, Ay= — 3 (Ax) + 2x(A:r) + (Ax)^, Ay+A^y= -siAx) + 2ix+Ax){Ax) + {Axy, A^y = 2 (Axy = 0.5 since Ax = 0.5. From this it is seen that whatever the value of Ax (in y = 2-'^x+x^) the second differences of the values of y are constant. Consider now the general case where the nth differences are constant. For convenience the values of y and the successive differences will be arranged in columns. The notation used is self-explanatory. Vi Ay\ y2 A2yi Ay2 A3yi etc.. yz A2y2 A^yi Ayz A3y2 etc.. y^ A^ya A^y2 Ay4 A^ys etc., y-o Ays t^y\ • • • • • • ye • • • ^ DETERMINATION OF CONSTANTS 17 From the above it is clear that y2=yi+Ayi, y3=y2+Ay2, =yi+Ayi+A(yi+Ayi). ^yi+2Ayi+A^yi. 3^4=^3+ Ays, ==yi+2Ayi+A^yi+A(yi+2Ayi+A^yi), =yi+3Ayi+3A^yi+A3yi. y6=y4+Ay4, = yi +sAyi +3A^yi +A^yi +A(yi +sAyi +sA^yi +A^yi) = yi +4Ayi +6A^yi +4A^yi + A^yi. In the above equations the coefficients follow the law of the binomial theorem. Assuming that the law holds for yt it will be proved that it holds for y^+i. By hypothesis 3^=yi+(ife"i)Ayi+ ^^"'y""'W ffcife^^^^A3yx+etc. . . . . (i) 13 If this equation is true, then yt+i=yi+(^-i)Ayi+ ^ ""V "" A ^yi |2 + ^^ ^-^ — — ^A^y 1 + etc. 3 +A[yi + ()fe-i)Ayi+^^-^^^-^A2y = y, + My,+^(^A2y+M*:^^ 4 J ^ -J 18 EMPIRICAL FORMULAS This is the same law as expressed in the former equation, and therefore, if the law holds for yky it must also hold for yt+i. But we have shown that it holds for y^, and therefore, it must hold for ys. Since it holds for ys it will hold for y^. By this process it is proved that the law holds in general. If now the first differences are constant the second and higher differences will be zero, and from (i) yk=yi+ik-i)Ayi. If the second differences are constant the third and higher differences will be zero, and it follows from (i) that In general, then, if the nth differences are constant yt=yi+{k-i)Ayi+- p -A^yi+- ^-^ — - — —A^yi \l 13 . (^-x)(^~2)(^-3)(^~4) . . . (k-n) .. , . + . . . + n A yi. (2) The law requires that the values of x form an arithmetical series, and hence Xt=xi+{k—i)Ax; from which follows *-^+' « Substituting this value of * in equation (2) it is found that the right-hand member becomes a rational integral function of Xk of the «th degree. Equation (2) takes the form yt=a+bxk+cx.^^+dx,.^+ . . . +qX)c''. \/ Since Xt, and y* are any two corresponding values of x and y the subscripts may be dropped and there results the following law: taB^l^^i^H ^mt^^ DETERMINATION OF CONSTANTS 19 If two variables, x and y, are so related thai when values ofjc are taken in an arithmetical series the nth differences of the cor- responding valines of y are constant y the law connecting the variables is expressed by the equation y=a+bx+cx^+dx?+ • • • +?«*. The nth differences of the values of y obtained from observa- tions are seldom if ever constant. If, however, the nth differ- ences approximate to a constant it may be concluded that the relation between the variables is fairly well represented by I. As an illustration consider the data given on page 131 of Merriman's Method of Least Squares. The table gives the velocities of water in the Mississippi River at different depths for the point of observation chosen, the total depth being taken as imity. At surface. 0.1 depth. 0.2 0.3 0.4 o.S 0.6 0.7 0.8 0.9 3 . 1950 3 2299 3.2532 3.2611 3.2516 3.2282 3.1807 3.1266 3.0594 2.9759 Av +349 + 233 + 79 - 95 -234 -475 -541 —672 -835 A*if -116 -154 -174 -139 — 241 — 66 -131 — 163 A»» - 38 - 20 + 35 - 102 + 175 - 6s - 32 A*v + 18 + 55 -137 + 277 — 240 + 33 Ah + 37 — 192 +414 -517 +273 From the above table it is seen that the second differences are more nearly constant than any of the other series of differ- ences. Of equations of form I, y==a+bx+cx^y where x stands for depth and y for velocity, will best represent the law connecting the two variables. It should be emphasized, however, that the fact that the second differences are nearly constant does not show that I is the correct form of equation 20 EMPIRICAL FORMULAS to be used. It only shows that the equation selected will represent fairly well the relation between the two variables. It might be suggested that if an equation of form I with ten constants were selected these constants could be so deter- mined that the ten sets of values given in the table would satisfy the equation. To determine these constants we would sub- stitute in turn each set of values in the selected equation and from the ten equations thus formed compute the values of the constants. But we would have no assurance that the equation so formed would better express the law than the equation of the second degree. For the purpose of determining the approximate values of the constants in the equation y=a+bx+cx^ (i) from the data given proceed in the following way: Letx=X+oco, ./:' V /- y=Y+yo, ^ r where xq and yo are any corresponding values of x and y taken from the data. The equation becomes Y+yo^a+b{X+xo)+c{X'{'Xoy = a+hocQ+cX(?+{b+2Coco)X'\-cX^. Y = {h+2cxQ)X+cX^) (2) since yo^a+hoco+coci?. Dividing (2) by X it becomes ^4l+2cc^-\-cX (3) F This represents a straight Kne when X and — are taken as Ji. coordinates. The slope of the line is the value of c and the intercept the value of b+2CXo. The numerical work is shown in the table and the points represented by [X, — :] are seen DETERMINATION OF CX)NSTANTS 21 in Fig. 3. The value of c is found to be -\o,^6^ When oco==o, the intercept, 0.44 is the value of 6. For x^^X, the value of .5 Y X .4 .3 ^ ^^^ ' ^ ^^ 1 .2 .1 "--- <. ■^^^-^ M -.2 -.3 - '^ ■^ ""^ . * .1 .2 .3 .4 \ .6 .7 .8 .9 PlG. 3. yo is taken from the table to be 3.1950, therefore each value of Y will be the corresponding value of y diminished by 3.1950. / -z. "? j^ Cr -? X y X Y Y X .44X— .76X* a=y-.44* -.76X* Computed y .0 3.1950 0.0000 0.0000 3.1950 3.1948 I 2 .3 3.2299 3-2532 3. 261 I I 2 3 0.0349 0.0582 0.0661 0.3490 0.2910 0.2203 0.0364 0.0576 0.0636 3 3 3 1935'^ 1956 1975 3.2312 3.2524 3.2584 .4 s .6 7 8 3.2516 3.2282 3.1807 3.1266 3.0594 4 5 .6 •7 .8 0.0566 0.0332 —0.0143 — 0.0684 -0.1356 O.1415 0.0664 —0.0238 -0.0977 — 0.1695 0.0544 0.0300 —0.0096 —0.0644 -0.1344 3 3 3 3 3 1972 1982 1903 . 1910' .1938 3 • 2492 3.2248 3.1852 3 . 1304 3.0604 9 2.9759 •9 -0.2191 -0.2434 —0.2196 3.195s 2.9752 10 )31.9476 fl= 3.1948 The numbers in column 6 were found after 6 and c were deter- mined in Fig. 3. The sum of the numbers in the seventh column divided by ten gives the value of a. In the last column are written the values of y computed from the formula y=3.i948-f-.44a?— ^6x2. (4) 22 EMPIRICAL FORMULAS II. :y = a+-+-+_+ . . . ^. Values of - form an arithmetical series and A y are constant. Another method of determining the constants is illustrated in the following example: Let it be required to find an equation which shall express approximately the relation between x and y having given the corresponding values in the first two columns of the table below. I 2 3 4 5 6 7 8 9 X y I X X y Ay A*y 2 Com- puted y I.O 4.000 1.0 1. 000 4.00 -0.68 0.04 2.00 4.000 1.2 2.889 0.9 I. Ill 3.32 —0.64 0.03 1.50 2.889 1.4 2.163 0.8 1.250 2.68 —0.61 0.05 I. 14 2.163 1.6 1.656 0.7 1.429 2.07 -0.56 0.05 0.87 1.656 1.8 1.284 0.6 1.667 i-Si -0.51 0.03 0.67 1.284 2.0 1. 000 0.5 2.000 1. 00 -0.48 0.04 0.50 1. 000 2.2 0.777 0.4 2.500 0.52 -0.44 0.36 0.777 2.4 0.597 0.3 3-333 0.08 0.25 0.597 In column 3 are given values of - in arithmetical series X and the corresponding values of x and y are written in columns 4 and 5 of the table. The values of y were read from Fig. 4. It is seen that the second differences of the values of y given in column 7 are nearly constant, and therefore the relation between the variables is represented approximately by the equation y-'H^+ii) ■ ^^ This becomes evident if x be replaced by - in I. The law X may then be stated: // two variables, x and y, are so related that when values of - X are taken in arithmetical series the wth differences of the corre- DETERMINATION OF CONSTANTS 23 sponding values of y are constant, the law connecting the variables is expressed by the eqimtion II . b . c . d . q A • 8 . 1 • 5 Values of -^ .6 .7 • 8 • 9 1 V ^ L 3 K V / k « .A /^ ;^ \ S^ A ^— o id2 • \ kv > y s N y /^ ? ^ v,^ / ^ J ^ . > [y / >< ^^ • / ^ / ^^^^ _^ / — ^' 1 1. 2 1. i h 6 1.1 3 Val 2. ues of a 2, 2 % 1 2. 6 2. 8 3.0 «l^ 1® OS Fig. 4. If in equation (s) - be replaced by X, then and y=a+ftX+cJt2, y+A3;=a+6(X+AZ)+c(Z+AX)2. By subtracting (6) from this equation and from (7) Ay+A2y = 6AZ+2(;(AZ)(Z+AZ)+(;(AZ)2. . Subtracting (7) from (8) A2y = 2c(AZ)2; A^y "^ 2(AZ)2' (6) (7) (8) 24 EMPIRICAL FORMULAS From column 7 it is seen that the average value of A^y is 0.04, and as AX was taken —.1, 0.04 Writing the equation in the form -5-+KJ- z y- 2 «2 n / / yi ' « * / 14> t / / / t X 1 y / c / •0 ' / 1 /^ / 1 •4 L . 2 ^ r. i . ij .1 3 . 7 . B . 9 1. X Fig. 5. it is seen that it represents a straight line when - and y— X 2 3 are the coordinates. From Fig. 5 6 is foimd to be 3 and a to be — i. The for- mula is y=-i+3 ©-&)• The last colmnn gives the values of y computed from this equation. The following, taken from Saxelby's Practical Mathe- matics, page 134, gives the relation between the poten- tial difference V and the cur- rent ^-4 in the electric arc. Length of arc =2 mm., A is given in amperes, V in volts. A Observed V. z A Computed V , 1.96 50.25 .5102 50.52 2.46 48.70 .4065 48.79 2.97 4790 ■ 3367 47.62 3.45 47.50 .2899 46.84 3.96 46.80 .2525 46.22 4.97 45.70 .2012 45.36 5.97 45.00 .1675 44.80 6.97 44.00 .1435 44.40 7.97 43.60 .1255 44.10 9.00 43.50 .iiii 43.85 DETERMINATION OF CONSTANTS 25 Fig. 6 shows V plotted to — as abscissa. The slope of this line is 12.5 divided by .75 or 16.7. The intercept on the V—ax is is 42. This gives for the relation between V and A F=42-f 16.7 Although the points in Fig. 6 do not follow the straight line very closely the agreement between the observed and the computed values of V is fairly good. 55 60 45 40 35 r — I — I — I — I — I — I — I — I — i — I— —J — > .1 .2 A Fig. 6. III. - = a+bx+cx^+d(x^+ . . . +qx^, y Values of x form an arithmetical series and A**— constant. .6 1 3^ ■Z- If two variables, x and y, are so related that when values of X are taken in an arithmetical series the »th differences of the cor- responding valines of - are constant, the law connecting the variables is expressed by the eqimtion III - = a+bx+cx^+dx^+ . . . +qx'^. y This becomes evident by replacing y in I by -. The con- stants in III may be determined in the same way as they were in I. IV. y2 = a+bx+cx^+dcfi+ . . . +qx''. Values of x form an arithmetical series and A" y* constant. // two variables, x and y, are so related that when values of x are taken in an arithmetical series the wth differences of the cor- 26 EMPIRICAL FORMULAS responding values of y^ are constant^ the law connecting the variables is expressed by the equation IV :^=^a-{-bx-{-co(^+d:fi+ . . . +gx". This also becomes evident from I by replacing y by y^. The method of obtaining the values of the constants in formulas III and IV is similar to that employed in formulas I and II and needs no particular discussion. CHAPTER II V. y = ah\ Values of x form an arithmetical series and the values of y a geometrical series. // two variables, x and y, are so related that when values of X are taken in an arithmetical series the corresponding valtces of y form a geometrical series, the relation between the variables is expressed by the equation V y = ab^. If the equation be written in the form logy = loga+(log6):x;, it is seen at once that if the values of x form an arithmetical series the corresponding values of log y will also form an arith- metical series, and, hence, the values of y form a geometrical series. The law expressed by equation V has been called the com- poimd interest law. If a represents the principal invested, b the amount of one dollar for one year, y will represent the amount at the end of x years. The following example is an illustration under formula V. .In an experiment to determine the coefficient of friction, /i, for a belt passing round a pulley, a load of W lb. was hung from one end of the belt, and a pull of P lb. applied to the other end in order to raise the weight W, The table below gives cor- responding values of a and /x, when a is the angle of contact between the belt and pulley measured in radians. a IT 2 2ir 3 5^ 6 IT 7^ 6 4ir 3 3^ 2 5![ 3 6 P 5.62 6.93 8.52 10.50 12.90 15.96 19.67 24.24 29.94 >- 27 28 EMPIRICAL FORMULAS The values of a form an arithmetical series and the values of P form very nearly a geometrical series, the ratio being 1.23. The law connecting the variables is The constants are determined gi-aphically by first writing the equation in the form log P = log a+a log b and plotting the values of a and P on semi-logarithmic paper; or, using ordinary cross-section paper and plotting the values of a as abscissas and the values of log P as ordinates. Fig. 7 gives the points so located. The straight line which most nearly passes through all of the points has the slope .1733 and the intercept .4750. The slope is the value of log b and the intercept the value of log a. log a =0.4750, log J =0.1733; 6 = 1.49. Vi' K' %»• »• %'' Yi' %^ ^^ ^ Values of oc Fig. 7. or The formula expressing the relation between the variables is P= 3(1.49)", VI. y-=a+b<f. Values of x form an arithmetical series and the values of Ly form a geometrical series. // two variables, x and y, are so related that when values of x are taken in an arithmetical series the first differences of the values DETERMINATION OF CONSTANTS 29 of y form a geometrical series, the relation between the variables is expressed by the equation VI y = a+bc'. By the conditions stated the »th value of x will be Xn=xi+{n-i) Air, and the series of first differences of the values of y will be Ayi, Ayir, Ayir^, Ayir^, A^^ir* . . . Ayir""^. The values of y will form the series yu yi+Ayi, yi+Ayi+rAyi, yi+rAyi+r^Ayi . . . yi+Ayi+fA3;i+r2Ayi+r3Ayi+ . . . +r""^Ayi. The »th value of y will be represented by yn=yi+Ayi i-r ,n-l I— r From the wth value of x »— 1 = X n — Xl Lx Substituting this value in the above equation there is ob- tained yn=yi+^yi =a+b(^, i—r Ax I—r Avi" Avi "^ where a stands for yi-\ — =^, b for ^— r ^ , and c for r^. i—r I— r Let it be required to find the law connecting x and y having given the corresponding values in the first two lines of the table. X o .1 .2 -3 .4 -5 .6 -7 .8 -9 I.O y Ay y 1.300 0.140 1.300 1.440 0.157 1-439 1.597 0.177 1-597 1.774 0.200 1.774 1-974 0.224 1-973 2.198 0.254 2.198 2.452 0.285 2.452 2.737 0.323 2.738 3.060 0.363 3-059 3.423 0.407 3.421 3.830 3.830 30 EMPIRICAL FORMULAS Since the values of Ay form very nearly a geometrical series the relation between the variables is expressed approximately by y^a+bc". The constants in this formula can be determined graphically in either of two ways. First determine a and then subtract this value from each of the values of y giving a new relation y—a = b(f; which may be written in the logarithmic form log {y-a)=^logb+x log Cy and b and c determined as in Fig. 7; or, determine c first and plot (f as abscissas to y as ordinate giving the straight line y^a+b{c'\ whose slope is b and whose intercept is a. First Method, The determination of a is very simple. Select three points P, Q^ and R on the curve drawn through the points represented by the data such that their abscissas form an arithmetical series. Fig. 8 shows the construction. P^{xo,a+b(f'); Q=(xo+Ax, a+b(f'c^); R= (:r6+2A:x:, a+b(f'(?^). Select also two more points S and T such that S={xo-{-Ax,a+b(f')\ T= (:xk)+2Aa;, a+b(f'c^). The equation of the line passing through Q and R is y= ^ Lx ^^ -{ocQ+Ax)+a+b(f^c^. (i; DETERMINATION OF CONSTANTS 31 The equation of the line through the pomts S and T is y= \^ ^ -^ '-(x,^+Ax)+a+b(f\ . (2) These lines intersect in a point whose ordinate is a. For, multiplying equation (2) by c^ and subtracting the resulting equation from (i) gives y = a. Fig. 8 gives the value of a equal to 0.2. The formula now becomes log (y — .2) =log b+x\og c. In Fig. 9 \og(y — .2) is plotted to x as abscissa. The slope of the line is 0.5185 which is the value of log c, hence c is. equal to $.;i. The intercept is the ordinate of the first point or 0.0414, which is the logarithm of 6, hence b is equal to i.i. The formula is >'-0.2-f 1.1(3.3)^ 32 EMPIRICAL FORMULAS The last line in the table gives the values of y computed from this formula. Second Method, For any point (a;,y) the relation between X and y is expressed by and for any other point (a;+A:r, y+Ay) by y+A3; = a+&(f{;^. From these two equations is obtained Ay = 6(f(c^-i) log Ay = log &(c^* — i) +:x; log c. .6 If now log Ay be plotted to X as p abscissa a straight 3 line is obtained .4 or • / J k J Y i / 1 / ■ / / / / / / A .J L .: 2 .. Va lues of a • \ A J .1 J i. £ whose slope is log c. 8 The value of c hav- 'I mgbeendetennined, the relation .1 Fig. 9. y^a+h{(f) will represent a straight line pro- vided y is plotted to (? as abscissa. The slope of this line is b and its intercept a. VII. \ogy=a+h(f. Values of x form an arithmetical series and the values of A log y form a geometrical series. If two variables, x and y, are so related that when values of x are taken in an arithmetical series the first differences of the cor- DETERMINATION OF CONSTANTS 33 responding values of log y form a geometrical series, the relation between the variables is expressed by the eqimtion VII logy = a+Jc'. This is at once evident from VI when y is replaced by log y. The only difference in the proof is that instead of the series of differences of y the series of differences of log y is taken. VIII. y^a+bx+cd". Values of x form an arithmetical series and the values of A*y form a geometrical series. // hvo variables, x and y, are so related that when values of x are taken in an arithmetical series the values of the second differ- ences of the corresponding values of y form a geometrical series, the relation between the variables is expressed by the equation VIII y^a+bx^-cd^ The »th value of x is represented by Xn—Xi'\'{n — l)^X. The values of y and the first and second differences may be arranged in columns yi Ayi y2 Ay2 A2yi ya Ays A2y2 y4 Ay4 A2y3 ys Ays A2y4 ye etc. etc. etc. 34 EMPIRICAL FORMULAS Since the second differences of y are to form a geometrical series they may be written A^^fi, rA^yi, r^^^yi, r^^^yi . . . r^"^A^yi. The series of first differences will then be A>^i , Ayi -h A^y 1 , Ay 1 -|- A^y i -|-r A^y i , Ay i -|- A^y i -|-r A^y i -f-r^A^y i Ayi+A2yi+rA2yi+r2A2yi+ . . . +r^"^A2yi. The »th value of y will be equal to the first value plus all the first differences. For convenience the wth value of y is written in the table below. yn=yi +Ayi +Ayi+A2yi + Ayi + A^yi +rA2yi + Ayi + A^yi +rA2yi +r^^yi + Ayi + A^yi +rA2yi +r^^^yi +r^A2yi +Ayi+A2yi+rA2yi+r2A2yi+r3A2yi+ . . . +r'*"^A2yi. Adding gives r yn=yi + (w-i)Ayi+A2yi ? — ^-+- — - + ^ }-- — ? Li— r I— r i—r i — +-— ^+ I— r , i-f"-' 1 I—r J The first two terms on the right-hand side represent the sum of all the terms in the first column of the value of yn* The remaining terms contain the common factor A^yi. The terms inside the bracket are easily obtained when it is remembered that each line, omitting the first term, in the value of y form a geometrical series. It is easily seen that the value of y» may be written DETERMINATION OF CONSTANTS 35 A2y] A2y] >»♦. JW=yi+(«-i)Ayi+^^-^(«-2)-^^^(r+f2+r3+. . .+r-=') I— r i—r I— f I—r I— f =4+5(«-i)+Cr-^ where ^=^,_^, B=Ay,+^, and C=-^3. (i— r)2 ^ I — r (i— ^) From the value of Xn is obtained w--i = Ax Substituting this in the value of y« it is found '=a+bxn+cd^'*. Since :![:» and yn stand for any set of corresponding values of X and y the resulting formula is Vni y^a+bx+cd". In the first two columns of the following table are given corresponding values of x and y from which it is required to find a formula representing the law connecting them. X y Ay A^y log A^y (2.00)* y— 1.01(2.00)* Computed y .0 1.500 .048 .023 -1.6383 1. 000 .490 1.492 .2 1.548 .071 .026 — I. 5850 1. 149 .388 1.550 .4 1. 619 .097 .028 — I 5528 1.320 .286 1.620 .6 1. 716 .125 .034 ^I 4685 1. 517 .184 I.71S .8 1. 841 .159 .039 — I 4089 1.742 .082 1. 841 I.O 2.000 .198 .043 ^I ■3665 2.000 — .020 1.999 1.2 2.198 .241 •051 — I .2924 2.300 -.125 2.196 1.4 2.439 .292 .059 — I .2291 2.640 — .227 2.440 1.6 2.731 .351 .067 — I 1739 3.032 -.331 2.735 X.8 3". 082 3.500 .418 3.482 4.000 -.435 -•540 3085 3 506 3.0 • • • • • • • • 36 EMPIRICAL FORMULAS Since the values of x form an arithmetical series and the second differences of the values of y form approximately a geometrical series, it is .evident that the relation between the variables is fairly well represented by y = a+bx+c(P. Taking the second difference or log A2y =log c(J^- i)2+(log d)x. Plotting the logarithms of the second differences of y from the table to the values of x, Fig. lo, it is foimd that log J = .3000 -1.0 -1.1 I" -1.6 -1.7 \ ^ _> ^ ^ \ y x^ X \ y y '^ ^, J H Y /^ M ^, > [y H \, ■ ^ / ^ M ^, 1 /" ^> \ ^ ^ ( ) J i J I .( 5 .1 raluei JD 1. BOtO 2 L > i 1. « JL B 2. i) .8 .4 1 ;^ CO 0-3 n2 -i4 -.6 Fig. 10. or J = 1.995, approximately 2. The intercept of this line, — 1:6500, is equal to log c(J^ — i)^. Since .02239 =c(2** — 1)2, C = I.OII. DETERMINATION OF CONSTANTS 37 Plotting y— (1.01)2* to Xy Figi lo, the values of a and b are found to be a= 0.5, 6= -0.5x5. The formula derived from the data is y=o.5-o.5i5a;+(i.oi)2*. In the last colmnn of the table the values of y computed from the formula are written down. Comparing these values with the given values of y it is seen that the formula reproduces the values of y to a fair approximation. IX. y = io«+^+«^*. Values of x form an arithmetical series and A* log y constant. If two variables, x and y, are so related that when values of x are taken in an arithmetical series the second differences of the values of log y are constant, the relation between the variables is expressed by the equation This becomes evident from I when y is replaced by log y. log y = a+bx+c:x^y which represents a parabola when logy is plotted to x. The constants are determined in the same way as they were in formula I. X. y=ksr/. Values of x form an arithmetical series and values of A' log y form a geometrical series. If two variables., x and y, are so related that when values of x are taken in an arithmetical series the second differences of the corresponding values of log y form a geometrical series, the relation between the variables is expressed by the equation X y = kff. 38 EMPIRICAL FORMULAS This becomes evident by taking the logarithms of both sides and comparing the equations thus obtained with VIII. X becomes log y = log * + (log s)x + (log g)d\ This is the same as VIII when y is replaced by log y, a by log ky b by log Sy and c by log g* XL y= ^ a+bx+cx^ Values of x form an arithmetical series and A*- are constant. If two variableSy x and y, are so related that when values of x are taken in an arithmetical series the second differences of the corresponding valines of - are constant, the relation between the variables is expressed by the equation X XI y = a+bx+cx^ Clearing equation XI of fractions and dividing by y - = a+bx+cx^. y X • This is of the same form as I, and when - is replaced by y the law stated above becomes evident. If a is zero XI becomes _ I ^'b+ac' which, by clearing of fractions and dividing by y, reduces to - = b+cXy y a special case of III. * For an extended discussion of X see Chapter VI of the Institute of Actuaries' Text Book by George King. \ DETERMINATION OF CONSTANTS 39 If c is zero XI becomes a special case of XVI, or X y X . which is a straight line when - is plotted to x, y Corresponding values of x and y are given in the table below, find a formula which will express approximately the relation between them. 2 3 4 5 6 7 8 9 o I 2 3 4 5 y X a1 y A2^ X y Y Y X X 2.S«a y o.ooo I • • • • • ■ • • ■ 4 ■ 1.333 0.075 .100 .050 — ■9 -2.703 3.003 .050 1. 143 0.175 .150 .050 — .8 — 2 . 603 3.254 •075 0.923 0.325 .200 .050 — .7 -2.453 3- 504 .100 0.762 0.525 .250 .050 — .6 -2.253 3.755 .125 0.645 0.775 .300 .051 -. •5 — 2.003 4.006 .150 0.558 I 075 .351 .049 - 4 -1.703 4.257 .175 0.491 1.426 .400 .047 - 3 -1.352 4.507 .201 0.438 1.826 .447 .058 - 2 -0.952 4.760 .226 0.396 2.273 .505 .040 — I -0.503 5. 030 .248 0.360 2.778 ■545 .054 0.000 0.331 3.323 .599 .056 I 0.545 5. 450 .298 0.306 3.922 .655 .051 2 1. 144 5.720 .332 0.284 4.577 .706 .03s 3 1.799 5-997 .352 0.265 5.283 .741 • • • • 4 2.505 6.262 .383 0.249 6.024 • • • • • • • 5 3.246 6.492 .399 Com- puted y 0.000 1.329 I/- 140 0.929 0.760 0.644 0.558 0.491 0.438 0.395 0.360 0.331 0.305 0.284 0.265 0.249 The values of x form an arithmetical series and since the • X • second differences of - are nearly constant the values of y will y be fairly well represented by y= or X X a+bx+cx^' - = a+bx+cx^, y This represents a parabola when - is plotted to x. Let X=x—ij F = --2.778. y 40 EMPIRICAL FOIOiULAS From these equations are obtained X The formula becomes - = 7+2.778. y Y+2,y7S--a+b{X+i)+c(X+iy =a+b+c+{b+2c)X+cX^. Since the new origin lies on the curve a+6+c = 2.778, the equation reduces to Y=(b+2c)X+cX^y or -— = b+2c+cX. Y . This represents a straight line when — is plotted to X. The value obtained for c from Fig. 11 is 2.5. The value of b could be obtained from the intercept of this line but the approximation DETERMINATION OF CONSTANTS 41 will be better by plotting — 2.501:2 to a;. In this way is obtained y the line y From the lower part of Fig. 11 the values of a and b are found to be a = .025, ^ = .2525. Substituting the values of the constants in XI the formula becomes X y= r. .025+. 25250^+2.5^^2 In the last column of the table the values of y computed from this equation are given and are seen to agree very well with the given values. CHAPTER m XII. y^a^. Values of x form a geometrical series aijd the values of y form a geometrical series. Ij Pwo variables, x and y, are so related thai when the values of X are taken in a geometrical series the corresponding values of y also form a geometrical series, the relation between the variables is expressed by the equation XII y = ax\ From the conditions stated equations (a) and (6) are obtained. Xn^xif-^, (a) yn=yiFP-\ {b) where r is the ratio of any value of x to the preceding one and i? is the ratio of any value of y to the preceding one. Taking the logarithm of each member of (a) log Xn = \ogxi + {n-i) log r, loga;n-loga;i n — I = -. . logr Also by substituting this value of n — i in the value of y» in equation (6), log Xi — log X\ yn=yiR ""' log XI / 1 \ log Xn = yiR l«8r |^2?*ogrj DETERMINATION OF CONSTANTS 43 where and log xn a^yiR *°«'- The following data (Bach, Elastizitat und Festigkeit) refer to a hollow cast-iron tube subject to a tensile stress; x represents the stress in kilogrammes per square centimeter of cross-section and y the elongation in terms of -^ cm. as unit. X 9-79 20.02 40.47 60.92 81.37 lOI . 82 204.00 408.57 y 0.33 0.695 1.530 2.410 3 295 4.185 8.960 19.490 log X. . . oipQoS I. 3014 1.6072 I . 7847 I . 9104 2.0078 2.3096 2.6II3 logy... —0.4815 -0.1580 0.1847 0.3820 0.5178 0.6217 0.9523 1.2898 Comp. y.... 0.324 0.714 1. 541 2.416 3.323 4.252 9.132 19.600 Selecting the values of x which form a geometrical series, or nearly so, it is seen that the corresponding values of y form approximately a geometrical series, and, therefore, the relation between the variables is expressed by the equation or y = ax , log 3^ = log a +6 log X, If now logy be plotted to logo; the value of b will be the slope of the line and the intercept will be the value log a. Fig. 12 gives 6 = i.iT In computing the slope it must be remembeted that the horizontal unit is twice as long as the vertical unit. The intercept is —1.5800 or 8.4200—10, which is equal to log 0.0263. The formula is .V y = .02630:' The values of y computed from this equation are written in the last line of the table. They agree quite well with the observed values. 44 EMPIRICAL FORMULAS XIII. y=a+b log x+c log% Values of log x form an arithmetical series and A^y constant. If two variables, x and y, are so related that when values of log X are taken in an arithmetical series the second differences of the corresponding values of y are constant the relation between the variables is expressed by the equation xin y = a+b log x+c log^jc. This becomes evident from I by replacing x by log x. The law can also be stated as follows : If the values of x form a geo- 1.4 1.2 1.0 .8 ^\ o s.2 !• -.2 -.4 -.6 -.8 ^ ,^ ,^ ^*^ ^ > ^ x^ ^ ^ / ^ ,^ X / r ^rf* ^ y^ x' y r^' X B ,\ ) 1 L 1. .1 1 .2 L 3 1. .4 1. 5 1. 6 1. VaJ 7 L lues 8 1. oil 9 5 Og i S 2, C .1 2. 2 2. .3 2. 4 2. S 2. 6 2. 7 2.8 Fig. 12. metrical series and the second differences of the corresponding values of y are constant the relation between the variables is expressed by the equation y =a+6 log x+c \o^x. If c is zero the formula becomes y = a+b log Xy which is V with x and y interchanged. t/ DETERMINATION OF CONSTANTS 45 Formula XIII represents a parabola when y is plotted to logo;. The constants are determined in the same way as the constants in I. XIV. y=^a+hx\ Values of x form a geometrical series and values of ^y form a geometrical series. If two variables, x and y, are so related that when the values of X are taken in a geometrical series the first diferences of the cor- responding values of y form a geometrical series, the relation between the variables is expressed by the equation XIV y = a+bx\ As in XII the «th value of x is x„=a;ir"-' {c) The series of first differences of y may be written Ayi, Ayii?, ^ylB?, ^yiK? . . . Ayii?*-2, and the values of y are yi, yi+Ayi, yi+Ayi+AyiiJ, yi+Ayi+AyiiJ+Ayii?^ . . . yi+Ayi+Ayii2+Ayii22+Ayiie3+ . . . +Ayii?*-2. That is the «th value of y will be y«=yi+Ayi+Ayii?+Ayii22+Ayiie3+ . . . +Ayii?*-2 =yi+Ayi(i+i2+i22+J23+ . . . +i?-2) =yi+Ayi ^_^ {d) Taking the logarithm of each member of (c), log Xn=\og xi+{n-i) log r log^^-log^ n — I ; . logr \y 46 EMPIRICAL FORMULAS Substituting this value of »— i in the nth value of y given in (J), yn=yi+^yi log xn —log x\ i-R log xi / 1 \\ogxn . . log XI / 1 \ ^ i-R i-R \ I = a+6(io0^°^^» =a+6io*°«^»' Let it be required to find the law connecting x and y having given the values in the first two lines of the table. X 2 3 4 5 6 7 8 y 4.21 5.25 6.40 7.65 8.96 10.36 II. 81 log JC .3010 .4771 .6021 .6990 .7782 .8451 .9031 X 2 2.5 3.125 3.906 4.883 6.104 7.630 y 4.210 4.720 5.388 6.290 7.515 9. no 11.275 logjc .3010 .3979 .4948 .5918 .6887 .7856 • • • • Ay .510 .668 .902 1.225 1 . 595 2.165 • • • • log Ay - .2924 -.1752 — .0448 .0881 .2028 .3358 • • • • y— 2.72 1.49 2.53 3.68 4.93 6.24 7.64 9.09 log(y-2.72) .1732 .4031 .5658 .6928 •7952 .8831 .9586 Computed y 4.21 5.25 6.41 7.65 8.98 10.36 II. 81 In the fourth line values of x are given in a geometrical series with the ratio 1.25. In the fifth line are given the cor- responding values of y read from Fig. 13. The first differences of the values of y are written in the seventh line. These differ- ences form very nearly a geometrical series with the ratio 1.336. Since the ratio is nearly constant the law connecting x and y is fairly well represented by the equation y = a+bxf^. There are two methods which may be employed for deter- mining the values of the constants, either one of which may serve as a check on the other. ■■ i^ DETERMINATION OF CONSTANTS 47 First Method, Select three points, A, P, and Q on the cxirve, Fig. 13, such that their abscissas form a geometrical series and two other points, R and 5, such that R has the same ordinate as A and the same abscissa as P, S the same ordinate as P and the same abscissa as Q, The points may be represented as fol- 5^ lows: 12 11 10 8 o m o 3 «7 A = (xo, a+bx(y); .. P={xor,a+bxoY); . Q^ixor^a+bxo'r^y, R={xor, a+bxa"); S^(xor^,a+bxoY). 3 The equation of the 2 line passing through P and Q is A A ! / / f A V • X V 1 ^^ /, / ^^ / n^ A, V ^ p y' / / y^ / • 2 ! 3 \ s 1 c 1 8 Values of x Fig. 13. XQr(r—i) r—i The equation of the line passing through the points R and X(f(r—i) r — i These two lines intersect in a point whose ordinate is a. In Fig. 13 xo is taken equal to 2 and r equal to 2. The value of a is foimd to be 2.72. The formula then becomes or y—2,T2 = bx^j log C}'— 2.72)=log6+^logJC. 1 L^ 48 EMPIRICAL FORMULAS In Fig. 14 log (y— 2.72) is plotted to log x and b and c determined as in XII. It is seen that the points lie very nearly on a straight line. The values of c and h are read from Fig. 14. log 6 = 9.7840—10; b= .61. The law, connecting x and y then is y = 2.72+.6ijc^' .4 .2 o r -»6 I / / . / / J / c / / / / / / f J / J / J V > / / / / A / /^ / f ^ / t^ / / / / / / / / f / ^ / .1 .2 .3 .4 ^ .6 .7 Values of log x Fig. 14. .8 .9 1.0 1.0 .9 .8 I At .6 .4 .2 .1 09 The values of y computed from- this formula are written in the last line of the table. Second Method. From the equation y^a-^-iof ^ DETERMINATION OF CONSTANTS 49 we have y+Ay=a+bxl^r^; Ay = 6a:*(r*'-i); log Ay = log 6(r*-i) +c log op. This is the equation of a straight line when log Ay is plotted to logic. Fig. 14 shows the points so plotted and from the line drawn through them the values of 6 and c are obtained. ^ = 1.3, 6 = .6i. a is foimd by taking the average of all the values obtained from the equation a is equal to 2.72. XV. y=aio^\ Values of X form a geometrical series and A log y form a geometrical series. If two variables J x afui y, are so related that when values of X are taken in a geometrical series the first differences of the cor- responding valtces of log y form a geometrical series, the relation between the variables is expressed by the eqtiation J>x' XV y^aid This equation written in the logarithmic form is logy=loga+bxf^. Comparing this with XIV it is evident that if the values of x form a geometrical series the first differences of the corre- sponding values of log y also form a geometrical series. In an experiment to determine the upward pressure of water seeping through sand a tank in the form shown in Fig. 15 was filled with sand of a given porosity and a constant head of EUPIRICAL FORMULAS water of four feet maintained.* The water was allowed to flow freely from the tank at A . The height of the column of water in each glass tube, f six inches apart. was measured. In 1 the table below x t represents the dis- tance of the tube I from the water head in feet, and y the height of the column of water in the tube, also in feet. It is required to find the law connecting x Fig. 15. and>. Tube , , J 4 s 6 , , 9 I** :%6 ' 3I1J '■M^A isoio \ L "i'.. ih Jiu -'S.. Jog (y-b^) Computed y '■'^ z J617 "mo 195 - isiie - .oiiS .36J. 1.195 :;l - i - .J6I7 ~s - .644J .3769 - .5J.8 In the fifth line values of x are selected in a geometrical series and the corresponding values of y written in the next line. In Fig. 16 log {—A log y) is plotted to log*. The points lie on a straight line. On account of the small number of points used in the test we select formula XV on trial. From the formula y = aio'^ it follows that logy = loga+Aa;' * Coleman's Thesis, University of Michigan. DETERMINATION OF CONSTANTS 51 logyk = loga+bxt' log yt+i =log a+bxt'r^ log (A log y) = log b(r^-i) +c log x. If A log y is negative b is negative, in which case it is only necessary to divide the equation by — i before taking the logarithms of the two members of the equation. « y A A Y y f* 0.5^ y y ■ y ^ 5 y r ' -1 ?? / /^ o ^ y i / r- 4.5^ ^ y y ^— . —2 -.i I I ^•« 3 ^1 . 9i I L aluc Fig log .1 X ). 1 2 »\ ) A L The last equation above represents a straight line when log (a log y) is plotted to log x. The slope gives the value of c and the intercept gives log6(r*'-i). From Fig. i6 values of b and c are readily obtained. 6= —.02282. In the next to the last line the value of a is computed for each value of x from the equation log a = log y+ .022801:^*. 52 EMPIRICAL FORMULAS The average of these values of a gives a =2.314. The formula obtained is y= (2.3 14) 10 -.0228x^' The values of y computed from this equation are written in the last line of the table. The agreement is not a bad one. CHAPTER W XVI. ix+a)(y+b)=c. I Points represented by Ix—xtj ) lie on a straight line. If two variables, x and y, are so related that the points repre- sented by Ix—Xky -* ) lie on a straight line, the relation between \ y-yk/ the variables is expressed by the equation XVI {x+a)(y+b)=c. Let x—Xic=X, where Xt and y* are any two corresponding values of x and y. From the above equations y=Y+yic. Substituting these values of x and y in equation XVI we have (X+x,+a){Y+yt+b)=c, or XY+(y,+b)X+{x,-\-a)Y+(x,+a)(yt+b)^c. Since (xt, yt) is a point on the curve {xt+a)(yjc+b)=Cy and XY+(yk+b)X+(x,+a)7==o. 53 54 EMPIRICAL FORMULAS Dividing the last equation by Y X X+{yk+b)—+Xk+a=-o, or I Y__^ +^ ( X This represents a straight line when X is plotted to — . The theorem is proved directly as follows: If the points x—x , X — Xk lie on a straight line its equation will be Clearing of fractions X — Xx y-y^ =p{x—X))-\-q. x-x:,=p{x-x^{y'-yi)-\'q{y-y^. This is plainly of the form {x-{-a)(y-\-h)^c. The following tables of values is taken from Ex. i8, page 138 of Saxelby's Practical Mathematics. It represents the results of experiments to find the relation between the potential differ- ence V and the current A in the electric arc. The length of the arc was 3 mm. A (am- peres) y (volts X Y X Y 1.96 67.00 2.46 62.75 0.50 -4.25 2.97 59.75 I.OI -7.2s 3.45 58.50 1-49 -8.50 3.96 56.00 2.00 —11.00 4.97 53.50 3.0I -13-50 5.97 52.00 4.01 —15.00 6.97 51.40 5.01 — 15.60 7.97 50.60 6.01 —16.40 - ,1176 - .1393 - .1752 - .1817 — .2228 — . 2670 — .3210 - .3665 Com- puted F 66.99 62.74 59 80 57.80 56.19 53.94 52.44 51.36 50.55 Let A be taken as abscissa and F as ordinate and transfer the origin to the point (1.96, 67.00) by the substitution X=^ — 1.96, 7 = 7-67.00. DETERMINATION OF CONSTANTS 55 The values of X and Y are given in the third and fourth X lines of the table. The values of — are plotted to X in Fig. 17 OD 0) ' ^^ — ' ' ^r--- > ' -^ .. ^" .^^ • J 5 i 3 4 ( Values of X ^ 6 ? Fig. 17. and are seen to lie nearly on a straight line. It is therefore concluded that the formula is {V+b){A+a)=c. By the equations of substitution this becomes (X+i.96+a)(F+67.oo+6) =c, or A XF+(67.oo+6)X+(i.96+»0F=o. Dividing by 7(67.00+6) X _ I y 1.96+g Y 67.00+6 67.00+6* The slope of this line is — L2_JI^. From Fig. 17 67.00+6 67.00+6 and the intercept is 67.00+6 = •045; Solving these equations From formula i.o6+a (^=0.151, &= -44-78, {;= 46.89. 56 EMPIRICAL FORMULAS These values give (.4+o.i5i)(F-44.78)=46.89. In the last line of the table are written the values of V com- puted from the above formula XVIa. y=aio*+^ (I y y \ log -^^j log — ) lie on a straight line. x-xu yt yt/ If two variables, x and y, are so related that the points repre- sented by I log — , log 2- ) lie on a straight line, the relation \x-xic y* yt/ between the variables is expressed by the equation b XVIa y=aIO*+^ By the condition stated y I 'v log^=w log ^+6, yt x-xt yt where oc* and y^ represent any two corresponding values of X and y. m is the slope of the line and b its intercept. Clear- ing the equation of fractions (log y-log yi){x''Xi) = w(log y-log y*)+&(ap-a;*), or log y{x-xt-m) = (6+log yi)x-\og yk{xu+m) -bxu. iQg y _ (^+log yQa: -log yu{xu+m) -bxt x—xt—nt Ax+B x+C B-AC iH log a-\ x+C b x+C DETERMINATION OF CONSTANTS 67 Therefore 6 ft For the purpose of determining the constants the equation is written in the form logy=loga+^^, • (logy-loga)(a;+c)=6, . Let logy=logF+logy*, and Then follows (log F+log y,-log a){X+Xi,+c) =6, X log F -f-log Y{xk -\-c) -f-ZOog yu -log a) + (log y* -log a) (a;* H-c) = 6. But (log yic - log a) (ap*+c) = 6, since the point (jc*, )^t) lies on the curve. X log F+log F(:r*+(;)+Z(log y^-log a) =o. Dividing this equation by X ^ log F = - (oct+c) -^|:- +log a-log y*. Replacing log F and X by their values log ^ = - {xi,+c)—^ log — +log a-log y*. From this it is seen that if log — be plotted to log ~ y* oc-ocfc ^'y* a straight line is obtained whose slope is —{xn+c) and whose intercept is log a — log y^. If the slope of the line is represented by M and the intercept by B c^—M—Xtf loga=B+logyt. 68 EMPIRICAL FORMULAS By writing XVIa in the logarithmic form a line is obtained whose slope is b. XVII. y=a€"+6A > 1 yt yt I lie on a straight line whose slope, M^ is positive and intercept, B^ is negative, and M^-\-4B positive. If two variables J x and y, are so related that when values of x are taken in an arithmetical series the points represented by y}±2.yk+2\ li^ 0fi a straight line whose slope, M, is positive , yt yt / and whose intercept, B, is negative and also M^+4B is positive tJie relation between the variables is expressed by the equation XVII y^ae'^+b^, . Let {xk,yk), {x+^x, yk+i), {xic+2^x, yk+2) be three sets of corresponding values of x and y where the values of x are taken in an arithmetical series. We can then write the three equations, provided these values satisfy XVII. )^, = ae'^*+6/^*, ....... (i) )'*+i=a€''^V^''+&/V^, (2) y*+2=a€^'^*e2^^^+6e'^V^^^ (3) Multipl3dng (i) by e^^ and subtracting the resulting equation from (2) yu^i-e'^y.^be^^'Ke'^^-'e'^^) (4) Multipl3dng (2) by e^^ and subtracting from (3) Multiplying (4) by e^^^ and subtracting from (5) there results DETERMINATION OF CONSTANTS 59 or yk+2 _ r^Axi^dAx\ Vk+l ^(c -\-d)Ax^ yt yt ' ' The values of c and d are fixed for any tabulated function which can be represented by XVII, and therefore, the last equation represents a straight line when ^^^ is plotted to 2!^:^. yt yk The slope of the line is and the intercept is It is seen that M is positive, B negative, and M^+4B posi- tive, for and In the first two lines of the table are given corresponding values of x and y. It is desired to find a formula which will •e:q)ress tjie relations between them. X I.O 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 y + .3762 + .0906 — .1826 - .4463 - .7039 - .9582 — 1.2119 -1.4677 —1.7280 yk+i y* + .241 — 2.0IS +2.444 +1.577 +1.361 +1.265 +1.211 y*+2 yt -.485 -4.926 +3.855 +2.147 +1.722 +1.532 +1.426 g-A12x y^-.mx + .662 + .319 + .539 + .071 + .439 -.131 + .359 - .295 + .290 - .429 + .236 - .538 + .192 - .626 + .157 - .698 + .127 - .757 Computed y + .371 + .087 -.18s - .447 - .704 - -957 —1. 210 -1.464 -1.723 Plotting the points represented by 1^^^, ^!^\ Fig. 18, \ yt yt / a straight line is obtained whose equation is — = 1.97 .90, yk yk 60 EMPIRICAL FORMULAS if =1.97, B= — .96. Since M is positive, B negative, and AP+4B positive, it follows that the relation between the variables is expressed approximately by XVTE. It has been shown that the slope .4 / 4 • / 8 2 / « / •• + I t 1 1. / / -3 -4 / i / / 1 V -5 / .2 .1 -.1 «4 o d 13 >--.4 -%5 -.6 -7 -^ -2 -i „ 1 Values of ^^ Fig. 18. / / - / / / • / / / / / • / / / I .5 I .; i 1 u s 5 .7 Values oir**"* Fig. 19. of the line is equal to e^^-\-ef^, and the intercept is equal to _^(c+d)Aa; Since Ax is .5 eV^ = i.97, = .96. From these equations are obtained the values of c and d, ^=-.247, ^ d = .i65.^ N DETERMINATION OF CONSTANTS 61 The formula is now Dividing both sides of this equation by e*^^^* gives the equation which represents a straight line when ye"*^^* is plotted to ^-.412* 'pjjg values of these quantities taken from the table are plotted in Fig. 19 and are seen to lie very nearly on a straight line. This line has the slope 2.00 and intercept — i.oi. Sub- stituting these values of a and b in the formula it becomes ;y = 2e-2*^*-i.oie-^«*'. It is seen that the errors in the values of y computed from this formula are in the third decimal place. The values are as good as could be expected from a formula in which the con- stants are determined graphically. For a better determination of the constants the method of Chapter VI must be employed. XVIII. y=€^{ccosbx+dsmbx). Values of x form an arithmetical series, and the points (^ — , 5!L_ ? ] \ yt yu / lie on a straight line. Also M^+4B is negative. ( // two variables, x and y, are so related that when values of X are taken in an arithmetical series the points represented by y*+l 2!!±? ] lie on a straight line whose slope M and intercept B have such values that M^+4B is negative, the relation between the variables is expressed by the equation XVIII y = 6^(c cos bx+d sm bx). Let X and yt be any two corresponding values of the variables. We have the three equations yt=ef^(c cos bx+d sin bx), (i) yt+i =^€f^[c cos (J)X+bAx)+d sm (bx+bAx)] 62 EMPIRICAL FORMULAS = d^'e^^^[c{cos bx cos bAx — sin bx sin bAx) +d(sin bx cos iAic+cos bx sin bAx)] =€^ef^[{c cos bAx+d sin JAit:)cos bx + {d cos 6Ax — c sin JAx)sin Jrrj. . (2) The value yit+2 can be written directly from the value of y + 1 by replacing A^ by 2Ajc. yk->f2=ef^e^'^^[{c cos 2bAx+d sin 26A:c)cos 6jc + {d cos 26A:c— c sin 26Aji[:)sin bx] (3) Subtracting (i) multiplied by e^(c cos bAx+d sin bAx) froiri (2) multiplied by c we have cyk+i — ^"^(^ cos bAx+d sin bAx)yk =:C€^ef^{d cos bAx—c sin 6Aa:)sin foe —def^ef^^c cos 6A^+d sin 6Arc)sin to = -(c2+(P)6^6^sm6Aa:sin6x (4) Similarly cyk+2-'e^*^(c cos 2bAx+d sin 2bAx)yt = -((;2+(PV*e2^sin2JA:rsinto (5) Multipl)dng equation (4) by e*^ sin 26Aji[: and subtracting it from (s) multiplied by sin bAx c sin bAxyk-{-2—e'^'^{c cos 26Aa: sin bAx+d sin 26Aa: sin 6Ajc)yfc — ce"^ sin 2bAxyt+i +e^'^{c cos 6Ax sin 26Arc +d sin bAx sin 26Aa:)y* = o. Simplifying c sin bAxyk+2'-c€f^ sin 2bAxyk+i +ce^'^ sin ftAjcy* =0. Dividing by c sin bAxyk, ^-^ = 2 cosbAx^^^-e"^. yk yk DETERMINATION OF CONSTANTS 63 The values of a and b will be fixed for any tabulated function which can be represented by XVIII, and therefore, the last equation represents a straight line when ^^^^ is plotted to 2-^. The slope of the line is y* M = 2€f^ COS h^x, and the intercept B=-(? aAx It is evident that AP+4B is negative. It is possible that in a special case AP+^B might be zero, but then b would be zero and hence y=cef^y which is formula V. Corresponding values of x and y are given in the first two columns of the table below. It is required to find a formula which will represent approximately the relation between them. y y*+i yt yk+2 yt .OSz co^hx tSinhx .O&r cosbx y Com- X «-^cos bX puted y + .300 • • • • I. 0000 I . 0000 .0000 1 . 0000 + .300 + .308 z + .011 • • • • I . 0833 + .8646 + .5812 + .9366 + .012 + .018 2 - .332 + .04 — 1. 11 I. 1735 + .4950 + 1.7556 + .5809 - .571 - .327 3 - .636 —30.2 -57.8 I. 2712 — .0087 -114.59 — .0111 +57.3 - .634 4 — .803 +1.92 + 2.42 I. 3771 — .5100 — I . 6864 - .7023 +1.143 — .804 5 — .761 +1.26 + 1.20 I. 4918 - .8732 - .5581 —1.3026 + .584 — .761 6 - .48s + .95 + .60 I.6I6I - .9998 + .0175 -1.6159 + .300 - .48s 7 — .017 + .64 + .02 1.7507 - .8557 + .6048 —1. 4981 + .011 — .012 8 + .537 + .04 — i.ii I . 8965 - .4797 + I. 8291 — .9098 - .590 + .545 9 +1.027 —31.6 —60.4 2.0544 + .0262 — 38.1880 + .0538 +19.08 + 1.035 10 +1.298 +1.91 + 2.42 2.2255 + .5250 — I. 6212 + I. 1684 + 1 .III +1 . 299 In Fig. 20 the points represented by ( ^^, ^^ ) are plotted. \ yt yt I They lie very nearly on the straight line whose equation ist 64 EMPIRICAL FORMULAS Since (1.878)2 —4(1.18) is negative the relation between the variables is expressed approximately by the equation y = ef^{ccosbx+dsinbx). It was shown that the slope of the line is equal to 2 (cos bAx)ef^^ and the intercept equal to-e^*^. Since Ax is equal to unity we have 2^ cos 6 = 1.875, ^ = 1.175, log cos b = 9.9370— ID, 6=30° 10' ap- proximately, a = .08. The formula is now / / 2JD / IS 4 ^ / f 1.4 L3 > / / / « 9k ^ .8 / / / *4 8 •« 9 y / > / - .2 -A -s2 "A A /- / i 'A / / "A *L0 / / -L2 ^ I .- V .( S .i i 1 .0 I. 2 1. 4 1. 6 1. 8 2. Values Of ^^17-^ Fig. 20. y = e-^^{c COS 30-J:r+(i sin 30^0;), where 30^ is expressed in degrees. Dividing the equation by e-®"* cos 30^0; 6'^^ COS 30^0?' =c+d tan 30^0:®, which is a straight line when -775 =^^^ — r—=, is plotted to tan e-^^ cos 30^0: 30^:*;°. In Fig. 21 these points are plotted and are seen to lie nearly on a straight line whose slope is — .496 and intercept DETERMINATION OF CONSTANTS 65 .308. Two of the points are omitted in the figure on account of the magnitude of the coordinates. Substituting the values of constants just foimd in the formula the equation expressing the relation between x and y is y«e*®8*(.3o8 cos 30^01;°— .496 sin 30^°). u S.1.0 8 s 0.5 The last column in the table gives the values of y computed from the equation. The agreement with the original values is fairly good. In case c is zero XVIII becomes the equation for damped vibrations, y=def^ sin bx. ^ f -0J5 •^_ ^ ^ V, "^ ^ w. V ^ ^ ^ ^ ^ ■ "^ [^ -1. & -L -0. 5 ( 1 0.5 1.0 1J5 2JQ Values of tan Z0}4x° Fig. 21. XrX. y = ax'^+b(xf^. fyt+i yk+2 Values of x form a geometrical series, and the points i - \ yt yk ) lie on a straight line, whose slope, Jf , is positive, and whose intercept, By is negative, and M*+4B positive. // two variables, x and y, are so related that when values of X are taken in a geometrical series the points represented by y]^^]^±l\ iIq Qfi d straight line whose slope, M, is positive, , yt yt / and intercept, B, negative, and also M^+4B positive, the relation between the variables is expressed by the equation XIX y = axf+bx^. Let X and yt be any two corresponding values of the variables. The following equations are evident: yu^axf'+bxf^, (i) yt-^i^-axfr'+bxfr^, ..... (2) 66 EMPIRICAL FORMULAS yt.H2=a^r2^+6^V^ (3) yt+i-f'y* =bx^(r^-r'), (4) yt+2-r'yj:+i =bo(f'r^{r^-r') (5) Multiplying equation (4) by 7" and subtracting it from equation (5) there results or . y* y* It is seen that the slope of this line is positive and the inter- cept negative, and M^+4B positive. In the table* below, the values of x and y from x^.o$ to « = .55 are taken from Peddle's Construction of Graphical Charts. yt+i yk+i .55 .85 y Com- X y X y yk yk X X .55 X puted y .05 .283 .05 .283 .192 .078 1.470 .283 .10 .402 .10 .402 .282 .141 1.426 .402 .15 .488 • • • • • • • •352 .199 1.385 .488 .20 .556 .20 .556 1.420 1.965 .413 .255 1.347 .556 .25 .613 • • • • .466 .308 1. 315 .612 .30 .658 • • • • .516 .359 1.276 .658 .35 .695 • • • • .561 .410 1.238 .697 .40 .730 .40 .730 1.383 1. 816 .609 .459 1.208 .730 .45 .757 • • • • .645 .507 1. 174 .757 .50 .78c • • . • .683 .555 1. 142 .780 .55 .800 • • • • .720 .602 1. 114 .799 .60 .814 • • • • .755 .648 1.078 .814 .65 .826 • • • • .789 .693 1.047 .826 .70 .835 • • • • .822 .738 1. 016 .835 .75 .840 • • • • .854 .783 0.984 .840 .8c .845 .80 .845 1. 313 1.520 .885 .829 0.955 .846 In column 3 the values of x are selected in geometrical ratio and the corresponding values of y are given in column 4. The points (y}LtLyh±l\ are plotted in Fig. 22, and although the * See Rateau*s ** Flow of Steam Through Nozzles." DETERMINATION OF CONSTANTS 67 three points do not lie exactly on a straight line the approxima- tion is good. The slope of the line is 4.10 and the intercept —3.86 which give the equations 2^+2'' =4.10, |C+d_ =3-86. Yalaes of x-^ M SO JSn .40 .60 .60 .70 JSO .90 Valuesol^ Fig. 22 AND Fig. 23. Solving these equations the values of c and d are found to be The formula now is e = i.40, ^=.55. 68 EMPIRICAL FORMULAS Dividing both members of this equation by r^* ^66 which represents a straight line when -^ is plotted to r^^. The slope of the line is equal to a and the intercept equal to h. From Fig. 23 ♦ a=— .685, 6 = 1.522. The formula after the constants have been replaced by their numerical values is y = i.522r^*-.68sx^'*^ The last column of the table shows that the fit is quite good. If the errors of observation are so small that the values of the dependent variable can be relied on to the last figure derivatives may be made use of to advantage in evaluating the constants in empirical formulas. But when the values can not be so relied on, or when the data must first be leveled graphically or otherwise, the employment of derivatives may lead to very erroneous results. This will be illustrated by two examples worked out in detail. The first step in the process is to write the differential equa- tion of the formula used and then from this equation find the values of the constants. Consider the formula y = ef^{c cos bx+d sin hx). Looking upon a and b as known constants and c and d as constants of integration, the corresponding differential equation is y' — 2ay+(a2+62)y=o. Dividing this equation by y y y DETERMINATION OF CONSTANTS 69 which, if the data can be represented by XVIII, represents a straight line whose slope is 2a and whose intercept is — (a^+fr^). Corresponding values of x and y are given in the table. y y' y t tL X «.06x cos .o8x tan .o&r y X 062 y y De- grees Min- utes g-VM* cos .o8x + .3000 I .0000 I. 0000 .0000 + .3000 I -}- .2750 4 35.02 10.04 I. 0618 .9968 .0802 + .2598 3 + .2441 -.03t2 -.0068 — .1401 -.0279 9 I. 1275 .9872 .1614 + .2193 3 + .2065 -.0459 -.0066 — .1976 -.0319 13 45.06 I. 1972 .9713 .2447 + .1776 4 + .1622 — .0481 —.0078 — .2965 — .0481 18 20.08 I. 2712 .9492 .3314 .4228 + i^st 5 + .1102 -.OSS7 -.007s — .5054 — .0681 22 55 10 1.3499 .9211 + 6 + .0506 -.0635 -.0086 —I .2549 — .1700 27 30.12 1-4333 .8870 .5206 .+ .0398 7 — .017s — .0721 —.0080 +4 .1200 +.4571 32 05.14 1.5220 .8472 .6270 — .0136 8 — .0937 — .0805 —.0087 + .8591 +.0928 36 40.16 1.6161 .8021 .7446 .0723 9 — .1786 -.0894 -.0985 —.0091 + • SOU +.0510 41 15.18 I. 7160 .7519 .8771 ^ .1384 10 — .2726 —.0091 + :ilg +.0334 45 50.20 I. 8221 .6967 1.0296 — .2147 II — • 3757 - . 1078 -.0085 + +.0226 SO 25.22 1.9348 .6372 1.2097 1.4284 — .3047 12 — .4881 -.1168 —.0087 + .2393 +.0178 55 00.24 2.0544 .5735 — .4143 .5518 13 — .6093 -.1257 —.0091 + .2063 +.0149 59 35.26 2.1815 .5062 1.7036 — 14 — .7396 -.1348 —.0089 + .1823 +.0120 64 10.28 2.3164 .4357 2.0659 — .7328 IS — .8788 - . 143s —.0084 + .1633 +.0096 68 45.30 2.4596 .3624 2.5722 — .9859 16 — 1 .0264. TK 20.32 2.6117 .2867 3.3414 — 1 .3707 17 — 1 :.i8i4 77 55. 34 2.7732 L .2098 4.6735 —a 1.0306 The values of y' and y^ are obtained by the formulas yn = i2h {yn-2—iyn-i+ 8y«+i-3/»+2), y"n = p iyn-2 - i(>yn-i +2>oyn - i6y„+ 1 +yn+2), where A = Ajc = i. These formulas are derived in Chapter VI. Plotting the points represented by (—,—), Fig. 24, it is seen that they lie nearly on a straight line whose slope is .12 and intercept —.01. Therefore 2a = .i2, a2+ft2 = .oi, a = .06, b = .08. We have then y = e'^^{c COS .Q&x+d sin .o8«). 70 EMPIRICAL FORMULAS Dividing this equation by e^' cos .o8x e^^ cos .68x =(;+(/ tan ,o8x. -1.2 -1.0 -. s -. B n4 -. 2 ■- / . 2 . 4 . 6 . B 1.0 i \ / / / X ■ > / > / / / A ^ ^ L' / / V A Y y / / / < / r / / y / / •:2 Fig. 24. This represents a straight line when e"®* cos .08:*: is plotted to tan .o8x. The slope is d and intercept c. From Fig. 25 c= .3. 'the law connecting the variables is represented by 3; = eO^(.3 cos .oSic— .5 sin .oSoc). DETERMINATION OF CONSTANTS 71 The values of y computed from this formula agree with those given in the second column of the table. Consider formula XIX The corresponding differential equation is ■ v~ £Q>ft2 006 .4 .2 1 N \ \. 0^ 4 -.2 \ k \ -.4 -A \ \ \ \ -s8 * \ \ \ \ \ \ \ ^D \ ■1^ \ \ ■ILA \ \ 1 t I 3 1 8 s 09 Valued of taa .08x Fig. 25. where c and d are known constants and a and b constants~of integration. The differential equation represents a straight line when — ^ is plotted to — . The slope is c+d—i and the inter- ne y cept is —cd. The values of x and y in the table below are the same as those given in the discussion of XIX. 72 EMPIRICAL FORMXTLAS # // xy' «y' X y y y y y .05 .283 10 .402 15 .488 1.503 -6.933 .462 — .320 20 • 556 1.240 -4.133 .446 -.297 25 .613 1. 015 -4.967 .414 -.506 30 .658 0.803 -3.267 .366 -.447 35 •695 0.720 —0.400 .363 — .071 40 .730 0.623 -3.533 .341 -.774 45 • 757 0.492 —1.500 .292 — .401 50 .780 0.433 —1.067 .277 -.342 55 .800 0.338 -2.633 .232 -.996 60 .814 0.2SS -0.633 .188 — .280 65 .826 0.213 —1.200 .167 — .614 70 • 835 0.135 -1.767 • 113 -1.037 75 .840 80 .845 The values of y' and y" were computed by means of the formulas used in the preceding table. In Fig. 26 the points -^, --^ 1 are plotted and, as is seen, the points do not deter- <y y ^ ^ mine a line. It is clear that the constants can not be determined by this method. XlXa. y-daf(f. Points represented by (x„, log -^ — ) lie on a straight line. \ yn / If two variables^ x and y, are so related that when volumes of x are taken in a geometrical series the corresponding values of y are such that the points represented by i Xny log ^^^ j lie on a straight line, the relation between the variables is expressed by the equation XlXa y = aa^c'. Using logarithms: log yn = log a+b log Xn+Xn log Cy log y„+i =log (^b log Xn+rxn log c+b log r. DETERMINATION OF CONSTANTS 73 Subtracting the first equation from the second log Mi = (r''i)xn log c+b log r. By plotting log Mi to acn a line is obtained whose slope is (r — i)log c, and since r is known c can be determined. • o -.1 * ".2 -.4 o o Q o O 1-6 -?8 o o • -.9 -1.0 o — n .1 .2 .3 Values of -y- Fig. 26. .4 J5 From the first equation log yn-Xnlogc=b log Xn+log a. If then log yn—Xn log c be plotted to log Xn a line is obtained whose slope is b and whose intercept is log a. CHAPTER V XX. y=ao+(iiCOSx-\-(hcos2x-\-aiCOSsx+ . . . -f-OrCosfX 4-6isinx+ftisin 2x4-6jsin3x+ . . . 4-ftr_iSin (r— i)x. Values of y periodic. The right-hand member of XX is called a Fourier Series when the nxmiber of terms is infinite. In the application of the formula the practical problem is to obtain a Fourier Series, of a limited nimiber of terms, which will represent to a sufficiently close approximation a given set of data. The values of y are given as the ordinates on a curve or the ordinates of isolated points. In what follows it is assumed that the values of y are periodic and that the period is known. We will determine the constants in the equation / I . ' . y =00+^1 cos x+a2 cos 2x+a3 cos 3X,+bi sin x+b2 sin 2x, so that the curve represented by it passes through the points given by the values in the table. X o° 60° 120° 180° 240° 300° 360° y I.O 1-7 2.0 1.8 i-S 0.9 1.0 Substituting these values in the equation we have the fol- lowing six linear relations from which the values of the six constants can be determined: i.o = ao+ ^1+ (i2-\-ci3j i'7 = ao+^ai''la2-as-{ — ^bi-{ — ^62, , 2 2 2.0=00-^^1— ha2+as-\ — ^bi ^62, 2 2 74 DETERMINATION OF CONSTANTS 76 1.8=00— ^1+ ^2 — 03, 1 .5 = flo - 5^1 - ha2 +a3 Hi H — ^62, 2 2 0.9 = 00+1^1 — 2^2—^^3 -bl ^62. 2. 2 ' J Multipl)^g each of the above equations by the coefficient of ao, (in this case unity) in that equation and adding the result- ing equations we obtain (i) below. Multiplying each equation by the coefficient of ai in that equation and adding we obtain (2). Proceeding in this manner with each of the constants a new set of six equations is obtained. I f ,600 = 8.9. • (i) 301 = -1.25 (2) 3^2= -.25 (3) 603 = . 10 (4) 3h = .6s^3' ....... (s) 3*2 = . 15^3- • (6) oo=tI, oi=--A, <3t2=— a, az^-h, &i=MV3, 62=1^^3. The equation sought is y =fj— A cos x—^ cos 2x+^ cos zx-\-\^y/T, sin x+^y/j, sin 2x, , It reproduces exactly each one of the six given values. The solution of a large number of equations becomes tedious and the probability of error is great. It is, therefore, very desirable to have a short and convenient method for com- puting the numerical values of the coefficients.* *The scheme here used is based upon the 12-ordinate scheme of Rimge. For a fuller discussion see "A Course in Fourier *s Analysis and Periodogram Analysis " by Carse and Shearer. 76 EMPIRICAL FORMULAS t f • S. • Take the table of six sets of values X o° 60° 120° 180° 240** 300" y yo yi y2 ya y* ys where the period is 2v. For the determination of the coeflBcients the following six equations are obtained: 2 2 2 2 y3 = ^ — ^1- + ^2 — ^3, VT, . V^, y4 = ao-§fl^l- 1^2+^3 -hx-\ ^62, 3^6 = 00+1^1-1^2 — ^3 ^61 ^62. 2 2 Proceeding in the same way as was done with numerical equations the following relations are obtained: 6ao=yo+ yi+ ^2+^3+ 3^4+ ys, 3^1=^0+ \y\- hy^-yz- -?y4+^y6, 3^2 =yo- \y\- 1^2+^3- ^y4-^y6, 6a3=j'o- )'i+ )'2-)'3+ ^4- ys, r . ^^3 . ^3 ^^3 "^ 2 2 2 2 \/^ V^ V^ V3 3*2 = +— ^y 1 ^3^2 + — ^3'4 -yh W J 22 22 For convenience in computation the values of y are arranged according to the following scheme: yo yi y2 y3 y4 ys Sum z;o z;i z>2 Difference w^o 2e;i tt;2 ' DETERMINATION OF CONSTANTS 77 vo V2 Wo Wi W2 Sum po Difference Pi ?i 6ao 3^1 3^2 6a3 Sum Difference ro n Sl po+Ph Po-^piy To— Sly 3^2 = ^1. (6) J It is evident that the equations in set (b) are the same as those in set (a). For the numerical example the arrangement would be as follows: l.O 1.7 . 2.0 . 1.8 1-5 0.9 Vo 2.8 3-2 2.9 Wo -.8 .2 I.I 2.8 3-2 t -.8 .2 Po 2.8 2-9 6.1 ro I.I -.8 1-3 » 91 I f •3 6ao = 3^1 = 3^2 = 6^3 = 3*1 = 3*2 = = +8 Sl i.90, •25, .25, .10, * -•9 78 EMPIRICAL FORMULAS It IS seen that the computation is made comparatively simple. The values of the v's are indicated by vo, the first one. The values of the p% etc., are indicated in the same way. 8-ORDiNATE SCHEME. The formula for eight ordinates which lends itself to easy computation is y = ao+ai cos 6+a2 cos 26+a3 cos ^d+a^ cos 46 +bi sin d+b2 sin 2^+63 sin 3^. For determining the values of the constants eight equations are written from the table: 45° 90° 135° 180° 225° 270° 315° y yo yi yi ys y* ys y6 3-7 yo=(io+ (ii+a2+ ^3+04, v2 V^2 V^2 v2 yi =floH di a3—a4-\ 6i+&2H &3, 2 2 2 2 y2=flo —^2 +^4+ fti — fts, \/2 V2 V^2 V^ y3=flo fli H ^3— ^4H &1— &2H 2J3, 2 2 2 2 V^2 V^2 v2 V2 y6=flo di -\ as—CA bi+b2 63, 2222 y6 = flo —(i2 +a4—bi + bzy , V2 V2 V2, , V2, y7=floH ^1 ^3— ^4 01—02 0^. 2222 From which are obtained the following eight equations: 8ao=yo+ yi+y2+ y3+y4+ y5+y6+ y?, \^2 y/2 y/2 y/~2 4^1 = yo -\ yi ys - y4 ys ^ yr, 2222 4^2 =yo ~y2 +y4 -ye, V2 . V2 . V2 V2 4^3 =yo yi -\ y3-y4+: — ys y?, 2222 DETERMINATION OF CONSTANTS 79 8a4=yo- yi+y2- yz+y^- ys+ye- yr, V2 , , V2 -— yi+y2+-— ys 2 2 yi - ys 461 = 462= \/2 V^2 463 = — y 1 - y2 +-^-y3 V2 V2 — —ys-y^ — — yr, 2 2 + ys - yr, V2 , VI — —ys+yQ — — yr. 2 2 2 ' 2 For. the purpose of computation the values of y will be arranged as follows : yo yi y2 ya y4 3'5 yc yr Sum t;o Difference w/o V2 Wi V3 V2 W2 vz Wo Sum Difference po Sum Difference Pi &aQ=po+ • ph To 4^2 = qo, 4^3 =ro V2 -su Sa^^po—pu 4*1 = ^2 H ri, 2 4^2 =gi, /^bz=-'r2^ ri. 2 The process will be made clear by an example: Wl Wz n e y W2 r2 45° 90° 135° 180° 225° 270° 315° 360° 4 — 2 — I 2 3 3 —I 2 4 I 80 EMPIRICAL FORMULAS For computation the arrangement is as follows: ' 4 —2 —I 2 3 3-12 vo 7 I Wo I -s 7 I — 2 4 Po S S ro qo 9 -3 ^1 Soq = lO, 4ai = I -f V2, 4^2 = 9) 4^3 = I+fV2, 8^4 = 0, 4*1= -fv^, 4*2= -3, 4*3=-fV2. — 2 O 4 o -5 o -5 -5 The formula becomes y = 1. 25 — .634 cos ^+2.25 cos 2^+1.134 cos 3^ — .884 sin 6— .75 sin 2d— .884 sin 3^. io-Ordinate Scheme yo yi y2 yz ^4 y9 ys yi ye ys Sum Vo Vl V2 vz V4. V5 Difference Wi W2 Wz Wa ■ Vo Vl V2 Wi W2 V5 po va pi V3 p2 Simi W4 Ws Sum h 12 Difference qo qi 92 Difference nil W2 DETERMINATION OF CONSTANTS 81 1000 = 5^1 = 5^2 = 5^3 = 5^4 = 1006 = In the above equations Ci = cos36°, C2 = cos 72®, po+pi+p2, '■qo+Ciqi+C2q2, 'P0+C2P1—C1P2, '■qo—C2qi—Ciq2, '■pO''Clpl+C2p2, qo-qi+q2, '•Slll-\-S2l2, = 52^1 +51^2, ■S2h—Sil2y 'Siini—S2fft'2' Si = sin 36°, 52 = sin 72°. In the schemes that follow, as in the lo-ordinate one, only the results will be given. Sum Difference 12-Ordinate Scheme yo vo yi yn Wi y2 yio yz y^ y^ V2 W2 V4 ys 7i "Oh ye z'e Sum Difference Sum t'o ^1 ^2 'Oz 2^6 H ^4 pQ p\ p2 pz ?o ?1 ?2 po p2 pi pz lo u Wi W5 W2 W4. Sum n 72 Difference ^i ^2 Wz n rz Difference ?2 fe 82 EMPIRICAL FORMULAS i2ao=/o+/i, 6ai=goH — -qi+iq2y 6a4=^po+ps''^(Pi+p2), 6^6 =?o -qi+hq2, i2ae=lo—li, 66i = ri+— ^r2+r3, 2 662 =-^(^1+^2), 6^3= /i, 664=— ^(51-^2), 2 16-ORDINATE Scheme Sum Difference yo yi ^2 ^3 ^4 yi5 yi4 yi3 712 t'O Wi V2 W2 V3 yn V4 W5 ^6 yio 1'6 Wq yr y9 V7 ys Vs Sum Difference Vg po qo Vl V7 Pl qi V2 vz V5 p2 ?2 p3 ?3 V4: P4. DETERMINATION OF CONSTANTS 83 Wl W2 W3 W4 W7 We W5 Sum ri r2 rs U Difference si S2 ss ) pO pi p2 ^^ p4: pZ ^2 Sum /o h h Sum k h Difference wo wi Difference xq 8ai =goH 5'2+Cig'i+C2g3, 8^2= Wo -I Wl, 2 8^3 =?o ?2 — Cig'3+C2?l, 2 8tl4 = ^, 8^5 = ?o ?2 + Cig'3 — C2gi , 806 = Wo Wl, 2 807 = ^0 H ?2 — Cigi — 02^3 , 2 86i = r4 H r2 +Cir3 +C2f i, 2 862 = ^2 H (^1+^3), 2 863= -^4 H — ^r2+Clfl-C2r3, 864 = ^1— -^3, 84 SJs EMPIRICAL FORMULAS V2 =^4 r2+Ciri—C2r3y 2 866 = ■S2-\ (^1+53), 2 867= -r4 r2+Cir3+C2ri. 2 Ci=cos 22^® = sin 67!°, C2 = sin 22|® = cos67^°. 20-Ordinate Scheme yo yi y2 ya y^ ys ye yi y% y^ yic yi9 yi8 yi7 yi6 yi5 yi4 yi3 yi2 yw Sum vo 1^1 V2 V3 Va vs vq V7 Vs V9 Z'lC Difference W\ W2 Wz ; W4: W5 Wq Wi Ws Wg 2^0 Vl V2 V3 va V5 t'lO V3 vs V7 . ^6 Sum />0 Pl p2 p3 P^ ps Difference ?o ?1 92 ?3 ?4 Wi W2 W3 W4 W5 w^ Ws W7 Wq Sum n r2 ra r4 rs Difference S\ S2 ^3 S4 />o Pi p2 90 91 92 ^ P^ p3 94 9s Sum /o h I2 Smn ko ki k2 Difference Wo /o /l /2 mi m2 Wo 1fl2 Ifl] I fl f3 Sxmi /o Simi no ni Sum Ol O3 DETERMINATION OF CONSTANTS 85 Sl Si S2 S3 il hi g2 h2 Sum Difference 2000 = k) ioai=qo+qi sin 72°+g2 sin S4°+g3 sin 36°+g4 sm I8^ ioa2=wo+wi sin 54^+^2 sin 18°, ioa3=5'o— ?3 sin 72°— 54 sin $4^+qi sin 36°— 92 sin 18°, ioa4=/o— fc sin S4°+/i sin 18°, 10^5 = ^0"" ^2, 1006=^0— W2 sin 54°— wi sin 18°, ioa7=5'o+?3 sin 72°— 54 sin S4°— gi sin 36°— 52 sin 18°, ioas=lo—h sin S4°+/2 sin 18°, ioa9=qo—qi sin 72°+g2 sin S4°— ^3 sin 36°+g4 sin 18°, 2oaio=wo~"Wi, lofti =r6+r4 sin 72°+r3 sin S4°+r2 sin 36°+ri sin 18°, 10*2 =g2 sm 72°+gi sin 36"*, 1063= — r5+r2 sin 72°+ri sin 54°— r4 sin 36°+r3 sin 18°, 10^4=^1 sin 72°+A2 sin 36°, 10^5= ^1—^3, ioft6=gi sin 72°-g2 sin 36°, io&7= — rs— r2 sin 72°+ri sin S4°+r4 sin 36°+r3 sin i8^ 10^8= — fe sin 72°+Aisin 36°, ioft9=r6— r4 sin 72°+r3 sin 54°— r2 sin 36°+ri sin 18°. 24-ORDiNATE Scheme yo yi y2 yz y^ ys y^ y? ys yg yio yn yi2 ^23 ^22 ^^21 y20 yi9 ^18 ^17 ^16 yi5 ^14 ^13 Sima vo vi V2 vz V4: vs vq V7 vs vq z;io vn z;i2 Difference wi W2 w^3 ^^4 tcs ^e W w'8 ^9 w'lo w'n 86 EMPIRICAL FORMULAS Sum Difference Vo Vi V2 V3 V4 Vs Ve ^'12 ^11 Z^IO V:) Vg V7 Po pi p2 p3 p4: p5 p6 yo qi q2 qs q^ qb Wi W2 Ws W^ Ws Wq Wll Wio Wo Ws W7 Sum Difference ^1 r2 S2 ^3 ^4 ^4 Tb S5 re Sum Difference po pi p2 p3 p6 pb p4: h h h h mo mi 1712 si Sb S2 S4. Sum Difference ki ni h h mo m2 k2 «2 fill Sum Difference 2400 = i2ai = i2a2 = I2a3 = 12^4 = I2a5 = I2a6 = I2a7 = I2a8 = I2ag = i2aio = i2aii= 24ai2 = go ho gi hi Sum Difference Co fo ei =go+gu =qo+k^+W2q3+h^q2+Ciqi+C2q6y =fno+ifn2+^V^mi, ^qo-q^+W^iqi-qz-qb), = ho+^hiy -- qo+C2qi - hVJq2 - W^qz +^^4 +Ci js, =/o, =?o-C2gi-|V3g2+^V2?3+^g4-Cig6, ■■qo-q^+W^i-qi+qz+qb), ■-nio+^fn2-^Vsmi, =go-Cigi+iV3g2-^Vjg3+ig4-C2g6, S3 k3 DETERMINATION OF CONSTANTS 87 i2fti = C2ri+§r2+^V2r3+^V3r4+Cir6+r6, 12b2 = iki+^V^k2+k3, 1 2*3 = r2 - re H-^v^ (ri +r3 — rs) , i2J5 = Ciri+Jr2-^V2r3-|V3r4+C2r5+r6, I2J6 = ^l~"^3, i2&7 = Ciri-^r2-|V2r3+^V3r4+C2r5-r6, I2&8 = iV3(wi-W2), i2ft9=r6-r2+jV2(ri+r3-r5), I2ftl0=53+i(^l+^5)-|V3(52+54), i2Jii = C2ri-ir2+§V2r3-^V3r4+Cir5-r6, Ci^ ^ = .96593, 2V2 C2 = ^~^ =.25882. 2V2 As an illustration let it be required to find a Fourier series of 24 terms to fit the data given in the table below. x"" y «« y x"" y :r° y 00 149 90 159 180 178 270 179 15 137 105 178 195 170 285 185 30 128 120 189 210 177 300 182 45 126 135 191 225 183 315 176 60 128 150 189 240 181 330 166 75 135 165 187 255 179 345 160 149 137 160 128 126 128 13s 159 178 189 191 189 187 178 166 176 182 185 179 179 181 183 177 170 vo 149 297 wi -23 294 302 310 320 338 357370374366357178 -38 -50 -54 -50 -20 -I 8 8 12 17 88 EMPIRICAL FORMULAS 149 297 294 302 310 320 33» 178 357 366 374 370 357 Po 327 654 660 676 680 677 338 qo -29 -60 -72 -72 -60 -37 -23 -38 -50 -54 -50 — 20 17 12 8 8 — I n - 6 -26 -42 -46 -51 — 20 Sl -40 -50 -58 -62 -49 327 654 660 676 -40 - 50 -58 338 677 680 665 133 I 1340 676 ki -49 - 62 /o -89 - 112 -58 flto — II -23 - 20 »i 9 12 66s 133 1 — II -23 ■ 676 1341 1340 2671 Co • — 20 go -31 -23 h — II -9 /o 9 The formula becomes y = 167.167 — 19.983 cos X--3.410 cos 2X+5.470 cos 3X — 1.292 COS4X+.249 cos Sx+,T$ cos 6X+.310 cos 7^ +.458 cos 8x— .304 cos 9X — .090 cos lox— .243 cos iix — .083 cos I2X— 12.779 sin X— 16.624 sin 2x— .323 sin3x + i.5i6sin4a:+i.46i sin sx— 2.583 sin6a!:+.32i sin ^x — .216 sin 8X+.676 sin 90;- .459 sin loic- .639 sin iix. In what precedes the period was taken as 27r. This is not necessary; it may be any multiple of 2x. The process of finding a Fourier series of a limited number of terms which represent data whose period is not 2x will be best set forth by an example. In the table below the period is x/3 and the values of y are given at intervals of -k/i^^. The 12-ordinate scheme can be used by first making the substitution x=\0 or ^=3rr. DETERMINATION OF CONSTANTS 89 a;° 0° y a:° 0** y x"" e° y GO GO +27.2 40 120 +9.8 80 240 -II-5 ID 30 +34.5 50 150 +8.5 90 270 -17. s 20 60 +21.5 60 180 +0.2 100 300 —17.2 30 90 + io.i 70 210 -7.1 no 330 + i.S 27.2 34-5 21-5 10 .1 5 (.8 8-5 0. 2 I.S - -17.2 — 17 •5 — I] t-5 ■ -7.1 ^0 27.2 36.0 4.3 ■ -7-4 — ] [-7 1-4 0. 2 W\ po 33.0 27.2 0.2 38.7 36.0 1-4 27 .6 21 4-3 1-7 [-3 15.6 7-4 27.4 37.4 2.6 7-4 33 15.6 27.0 38.7 21.3 34.6 27.6 6.0 27.4 2.6 37-4 -7-4 fl 48.6 60.0 27.6 k 30.0 30.0 Si 17.4 17.4 48.6 27.6 27.0 6.0 • 21.0 21.0 The formula is y = 5+9.994 cos ^+8.7 cos 2^+3.5 cos 3^+.oo6 cos 5^ + 17.31 sin ^+5.023 sin 2^+3.5 sin 3^— .01 sin 5^. Replacing by its value 3X, y ==5+9-994 cos 3:^+8.7 cos 6a!:+3.5 cos 9a!:+.oo6 cos i$x +17.31 sin 3:^+5.023 sin 6X+3.5 sin 90;— .01 sin 1501;. CHAPTER VI EMPIRICAL FORMULAS DEDUCED BY THE METHOD OF LEAST SQUARES In the preceding chapters we computed approximately the values of constants in empirical formulas. The methods em- ployed were almost wholly graphical, and although the results so obtained are satisfactory for most observational data, other methods must be employed when dealing with data of greater precision. It is not the purpose of this chapter to develop the method of least squares, but only to show how to apply the method to observation equations so as to obtain the best values of the constants. For a discussion of the subject recourse must be had to one of the nmnerous books dealing with the method of least squares.* It was found in Chapter I that the equation y=a+bx+ca^ (i) represents to a close approximation the relation between the values of x and y given by the data X y X y o 3-1950 ■5 3.2282 .1 3.2299 .6 3.1807 .2 3-2532 -7 3.1266 •3 3.261 1 .8 3-0594 •4 3-2516 -9 2-9759 * Three well-known books are: Merriman, Method of Least Squares; Johnson, Theory of Errors and Method of Least Squares: Comstock, Method of Least Squares. 90 DEDUCED BY THE METHOD OF LEAST SQUARES 91 where x represents distance below the surface and y represents velocity in feet per second. Substituting the above values of x and y in (i), the following ten linear observation equations are found: a+oft+ 0^=3.1950, a+.ib+.oic =3.2299, a+.2ft+.o4c=3.2S32, ^+-3ft+-09C=3.26ii, a+.4ft+. 16^=3.2516, ^+-Sft+-2Sc =3.2282, a+.6ft+.36(; = 3.1807, ^+.76+49^=3.1266, a+.8&+.64(; =3.0594, a+.9ft+.8ic = 2.9759. Here is presented the problem of the solution of a set of simultaneous equations in which the niunber of equations is greater than the ntmiber of imknown quantities. Any set of three equations selected from the ten will suffice for finding values of the unknown quantities. But the values so found will not satisfy any of the remaining seven equations. Since all of the equations are entitled to an equal amount of confidence it would manifestly be wrong to disregard or throw out any one of the equations. Any solution of the above set must include each one of the equations. The problem is to combine the ten equations so as to obtain three equations which will yield the most probable values of the three imknown quantities a, ft, and c. It is shown in works on the method of least squares that the first of such a set of equations is obtained by multiplying each one of the ten equa- tions by the coefficient of a in that equation and adding the result- ing equations. The second is obtained by multiplying each one of the ten equations by the coefficient of ft in that equation and adding the equations so obtained. The third is obtained by multiplying each of the ten equations by the coefficient of 92 EMPIRICAL FORMULAS c in that equation and adding the equation so obtained. The process of computing the coefficients in the three equations is shown in the table. The coefficients of a, ft, and c are represented hy A, B, and C respectively, and the right-hand members are designated by N. The number 5, which stands for the numeri- cal sum of -4 , 5, C and iV, is introduced as a check on the work. It must be remembered that this method of finding the values of the constants holds only for linear equations. The sum of the numbers in the column headed A A = 2-4-4 = io. The sum of the numbers in the column headed AB = 2^45=4.5. The sum of the numbers in the column headed -4C = 2-4C = 2.85. Also the sum of the numbers in the column headed -4iV = 2-4iV=3i. 7616. These sums give the coefficients in the first equation.* The second and third equations are obtained in the same way. The three equations from which we obtain the most probable values of the constants are: 10 a + 4.5ft +2.8sc =31.7616; 4.5a +2.8sft +2.02SC =14.08957; 2.8sa +2.o2sft+i.S333c= 8.828813. These are called normal equations. From them are obtained ^=+3-19513; . ft = + .44254; c=- .76531- The check for the first equation is 2i4^ + 2^5+2^C+2^iV' = 2^5=49.1116; for the second equation 2^5+255+25C+25iV = 255 = 23.46457; for the third equation 2^C+25C+2CC+2CiV = 2C5 = 15.2371x3. * Cf. Wright and Hayford, Adjustment of Observations. i^mui— i^^^^— Ml DEDUCED BY THE METHOD OF LEAST SQUARES 93 AA lO AB o I 2 3 4 5 6 7 8 9 45 AC o .OI .04 .09 .16 .25 .36 .49 .64 .81 2.85 AN 3 1950 3.2299 3 2532 3.2611 3- 2516 3.2282 3.1807 3.1266 3 0594 2.9759 31.7616 AS 4.1950 4.3399 4.4932 4.6511 4.8116 4.9782 5 . 1407 5.3166 5 • 4994 5.6859 49. II 16 AB BB BC BN BS .01 .001 .32299 •43399 .04 .008 .65064 .89864 .09 .027 .97833 I 39533 .16 .064 1.30064 1.92464 .25 •125 1.61410 2 . 48910 .36 .216 1.90842 3.08442 .49 .343 2.18862 3.72162 .64 .512 2.44752 4.39952 .81 .729 2.67831 5.11731 4.5 2.85 2.025 14.08957 23.46457 AC BC CC CN CS .0001 .032299 •043399 .0016 .130128 .179728 .0081 . 293499 .418599 .0256 .520256 . 769856 .0625 . 807050 I . 244550 :i296 I. 145052 1.850652 .2401 1.532034 2.605134 .4096 I. 958016 3.519616 .6561 2.410479 4.605579 2.85 2.025 I . 5333 8.828813 15.237113 94 EMPIRICAL TORMULAS The formula is 3^ =3.19513 +.442S4X-. 76531^2, For the purpose of comparison the observed values and the computed values are written in the table, v (called residual) stands for the observed value minus the value computed from the formula. Observed Compute 1 A X y y V V^ 3 1950 3 1951 — .0001 .00000001 .1 3 2299 32317 — .0018 .00000324 .2 3-2532 3 2530 + .0002 .00000004 .3 3.261I 3 2590 + .0031 .00000441 ■4 3- 2516 3 2497 + .0019 .00000361 .5 3.2282 3-2251 + .0031 .00000961 6 3.1807 3.1851 -.0044 .00001936 7 3.1266 3.1299 -.0033 .00001089 8 3 0594 3.0594 .OCXX5 .00000000 9 2-9759 2.9735 + .0024 .00000576 + 0001 .00005493 This method derives its name from the fact that the siun of the squares of the residuals is a minimum. A discussion of this will be found in the books referred to above. In case the formula selected to express the relation between the variables is not linear the method of least squares cannot be applied directly. In order to apply the method the formula must be expanded by means of Taylor's Theorem. Even when the formula is linear in the constants it may be advantageous to make use of Taylor's Theorem. In order to make this trans- formation clear we will apply it to the formula just considered. Suppose that there have been found approximate values of a, by and c, oo, bo and cq, say, then it is evident that corrections must be added in order to obtain the most probable values of the constants. Let the corrections be represented by Aa, AJ, and Ac, And let (i=ao+Aa, b = bo+Aby c=Co+Ac, DEDUCED BY THE METHOD OF LEAST SQUARES 95 The formula was y=a+bx+cx^. This may be written y=f{ay by c) =/(ao+Aa, bo+Ab, co+Ac). Expanding the right-hand member /(oo+Aa, J0+A6, co+Ac)=f{ao,bo,co)+^Aa+^Ab+^Ai 9^0 oOo qco +-^(AaAc)+-^(AjAc)l+ . . . 9ao9co 9^o9^o J where -^ stands for the value of the partial derivative of 900 /(a, b, c) with respect to a and ao substituted for a, — ^ stands for the value of the second partial derivative of f(a, b, c) with respect to a and oo substituted for a, etc. If oq, Jo, and co have been found to a sufficiently close approximation the second and higher powers of the corrections may be neglected. 9^0 dbo dco The formula becomes y-/(ao, bo, Co) =-^Aa+-^Ab+^Ac, 9^ doo dco or y— (ao+box+cox^) =Aa+xAb+x^Ac, 96 CMPIKICAL rOSMULAS Selecting for the values oi oo. bo. and co those found in Ch^ter I, the new set ot observation equations are Aa+ oAb+ oAc= .0002, Afl+.iAJ+x>iAc= —.0013, Afl+.2Aft+.04Air= joooS^ Aa+.3Ab+X)g^=^ 0027, Aa+.4Aft+.i6A£:= .0024, Aa+.s^+.25Ac= .0034, Aa+.6A6+.36Ac= —.0045, Aa+.7A6+49Ac= — .0038, Aa+.8Ab+,64Ac= —.0010, Aa + .9A6 + .8 1 Ac = .0007 . From these are obtained the three normal equations ioAa+4.5 A6+2.85 Ac=— .0004, 4.5Aa+2.85 AJ+2.025 Ac=— .00203, 2.85Aa+2.02sA6+i.5333Ac = - .002059. Solving Aa= +.00033, AJ = +.00254, Ac= -.00531, which added to the values of oo, Jo, and cq, give J= .44254, ^=-•76531- the same as just found. The above process may be applied to linear equations con- taining more than three constants. But as the method of pro- cedure is quite evident from the above the general statement of the process will be made with reference to equations con- taining only three constants. 2SZ£kia^daMl DEDUCED BY THE METHOD OF LEAST SQUARES 97 Let the observation equations be represented by aix+biy+ciz=ni pi, a2X+b2y+C2Z=n2 p2, azx+bzy+czz==m ps. amX+bmy+CtnZ = nm Pm- The normal equations will then be 2pa^ • X + l^pab • y + ^pac • z = S/^a«, S/>aft 'X+7:pP -y+Xpbc • z = Xpbn, l^pac ' x+2pbc ' y+Zpc^ • z = Xpcn, where a, b, c, and n are observed quantities, and x, y, and z are to be determined, />i, />2, />3 • • . /^m are the weights assigned to the observation equations. In the problem treated at the beginning of the chapter the weight of each equation was taken as xmity. It was stated on a preceding page that when a formula to be fitted to a set of observations is not linear in the constants it must be expanded by Taylor's Theorem. Take as an illustration a problem considered in Chapter IV. The formula considered was y=f(A, B, m, n) =Ax"'+Bx'', w _ymo dAo ^- dBo =x^. ^=Aoxrlogx, ■^'=Box'^ log*; 9«o 98 EMPIRICAL FORMULAS y=f{A,B,m,n) = /(^o, 5o, mo, no) +^A^ +^^ f ^A«,- 9-^ 9/wo 9wo Aw; y-/(i4o,5o, wo,«o) = 3/.A^ +4-A5+^Am+^A/^. 9^0 9^0 9^0 9wo The observation equations will be of the form a/ AA dAo dBo ^'^ ■AB+^Afn+-^An-=y-yo. dmo dno Assume the approximate values found in Chapter IV. A= 1.522, 5= -.685, w= .55, n= 1.4. .14 XT' Ao Bo logjc 5ox''« log X. .05 .10 15 .20 .25 .19 .28 .35 .41 .47 .02 .04 .07 .10 .14 1.522 - .685 — 2 . 996 . — 2.303 -1.897 — 1.609 -1.386 - .88 - -99 — 1.02 — 1. 01 - .98 .03 .06 •09 .12 .14 .30 .52 .19 1.204 •94 .15 .55 .14 X logic i4o«"*" logic.. .. fiox**" log X. . . The new observation equations become .i9Ai4 + .02A5— .88Aw+.03A» = .28Ai4 + .04A5— .99Aw+.o6Aw = .35Ai4 + .07A5 — 1 .02AW + .09AW = .0004, .0002, — .0001, DEDUCED BY THE METHOD OF LEAST SQUARES 99 .4iAi4+.ioAJ5— i.oiAw+.i2A«= .0000, .47Ai4+.i4AJ5— .gSAfn+,i4An^ .0013, .S^AA+.igAB— .94Aw+;isAw= — .0001, .S6A^+.23A5— .9oAw+.i6Aw= — .0019, .6oAi4+.28A5— .84Am+.i7A»= — .0016, .64A4+.33A5— .78Aw+.i8A»= — .0001, .68Ai4+.38A5— .72Aw+.i8A»= — .0001. .72Au4+.43A5— .66Afn+.iSAn= .0011. From these the four normal equations are obtained 2.96oAi4 + i.32iAjB--4.637Aw+ .8o6Aw= —.00071, i.32iAi4+ .642 A5 — 1. 802 Aw + .359A»= —.00031, — 4.637Ai4 — 1.802A5+8.737AW— i.253Aw = +.ooo85, .8o6Ai4+ .359AJB — 1.253AW+ .22iA»=— .00023. From which AA = — .0068, A5=+.0II2, Aw =—.0022, Af^=— .0070. These corrections being applied the final formula becomes /■ CHAPTER Vn INTERPOLATION.— DIFFERENTIATION OF TABULATED FUNCTIONS Interpolation In Chapter 11 we found that the formula XI. y ^ .02S+.2S2sa;+2.sx2 represents to a fair degree of approxunation the values of y given by the data. Any other value of y, within the range of values given, can be obtained in the same way. This rests on the assxmiption that the formula derived expresses the law con- necting X and y. For example, the value of y corresponding toa: = i.os will be When a formula is used for the purpose of obtaining values of ^ y, within the range of the data given it is called an inter- >o(^2^on formula. Interpolation denotes the process of calcu- \ l|Lting under some assumed law, any term of a series from values of any other terms supposed given.* It is evident that empirical formulas cannot safely be used for obtaining values outside of the range of the data from which they were derived. * For a more extended discussion of the subject the reader is referred to Text-book of the Institute of Actuaries, part II (ist ed. 1887, 2nd ed. 1902), p. 434; Encyklopadie der Mathematischen Wissenchaften, Vol. I, pp. 799-820; Encyclopedia Britannica; T. N. Thiele, Interpolationsrechnung. As to relative accuracy of different formulas, see Proceedings London Mathematical Society (2) Vol. IV., p. 320. 100 INTERPOLATION 101 There are two convenient formulas for interpolation which will be developed.* The first one of these requires the expression for yx+n in terms of yx and its successive differences, yx represents the value of a fimction of x for any ctosen value of x, and yx+n represents the value of that fimction when x+n has been sub- stituted for X. yx+i=yx+Ayx; yx+2=yx+Ayx+A(yx+Ayx) =yx+2Ayx+A^yx; yx+3 =yx+2Ayx+A^yx+A(yx+2Ayx+A^yx) =yx+3^yx+3^^yx+^^yx; yx+^=yx+3^yx+3^^yx+^^yx+A(yx+3Ayx+3A^yx+A^yx) = yx+4^yx+6A^yx +4A^yx+A^yx. These results suggest, by their resemblance to the binomial expression, the general formula , . .n(n—i).o , «(w— i)(w— 2) ., , . yx+n=yx+nAyx+ ^ ' A^yx-\ — ^ p -A^yx+etc. If we suppose this theorem true for a particular value of n, then for the next greater value we have yx+n+1 =yx+nAyx+^ — -A^yx+— f^ ^A^yx+etc, ^ 13 +Ayx+nA^yx+ , ^^ A^y^+etc, =yx+(n+i)Ayx+ I A^yx+- V^ ^A^y^+eta " 13 The form of the last result shows that the theorem remains true for the next greater value of n, and therefore for the next * See Chapter III, Boole's Finite Differences. 102 EMPIRICAL FORMULAS greater value. But it is true when «=4, therefore it is true when n = S' Since it is true ioi n = s it is true when « = 6, etc. If now o is substituted for x and x for w, it follows that yz=yo+xAyo+-^ — -A^yo+— p ^A^yo+etc. If A'*y.=o, the right-hand member of the above equation is a rational integral function of x of degree n—i. The formula becomes , . ,x{x—l).o , x(x — l)(x — 2) .^ , yx=yo+xAyo+^ — -A?yo+-^ p -A?yo+ . . . ^ x{x-l)(x-2) . . . ^^''^'^^\ n^l . . . (l) Formula (i) will now be applied to problems. It must not be forgotten that in applying this formula x is taken to represent the distance of the term required from the first term in the series, the common distance of the terms given being taken as unity. I. Required to find the value of y corresponding to x = ./^2$ having given the values under XIX. In the interpolation formula x = .$, yo yi y2 y^ Ayo A^yo. . . . A^yo. . . . y=yo+2^yo-{A^yo+TEA^yo = .730+.oi3S+.ooo5+.oooi = .744. This is the same as given by XIX. 2. Find the value of y corresponding to x- formula will have the value f if we take yo X = 2. • 730 .757 .780 .800 .027 .023 .020 — .004 -.003 .001 2.3. X in the • —.1826 when INTERPOLATION 103 yo yi y2 yz yi ys -.1826 -•4463 -•7039 - .9582 — 1.2119 -1.4677 -.2637 -.2576 -•2543 -•2537 - -2558 .0061 •0033 .0006 —.0021 —.0028 — .0027 —.0027 .0001 .0000 —.0001 Ayo yx=yo+xLy(i-\ --, '-A^yo + -^ p^ ^ A^yo 2 3 ■1 — ^^ ^-^ — — ^ A^yo + etc. (-f) l(-f)(~i), = -.i826+f(-.2637)+^^^-^(.op6i) + ^' \''^' (-.oo28) 2 6 j^iAzMzilizii) {,0001) 24 = -•3417- 3. The following example is taken from Boole's Finite Differ- ences. Given log 3.14 = .4969296, log 3.15 = .4983106, log 3.16 = .4996871, log 3.17 = .5010593; required an approximate value of log 3.14159- yo y\ y2 yz A>'o. A2yo A^yo .4969296 .4983106 .4996871 .5010593 .0013810 .0013765 .0013722 -.0000045 —.0000043 .0000002 Here the value of x in the formula is equal to 0.159. y:c = 4969296+(.i59)Cooi38io)+ ^^^^^^^^~^ (-. 0000045) .i59(.i59~i)(.i59-2) ^ orwv.^o>i 1 ^.0000002) o =.4971495. 104 EMPIRICAL FORMULAS This is correct to the last decimal place. If only two terms had been used in the right-hand member of the formula, which is equivalent to the rule of proportional parts, there would have been an error of 3 in the last decimal place. The rapid decrease in the value of the differences enables us to judge quite well of the acau'acy of the results. The above formula <:an be applied only when the values of x form an arithmetical series. In case the series of values given are not equidistant, that is, the values of the independent variable do not form an arithmetical series, it becomes necessary to apply another formula. Let ya, y*, yc, y^, . . . y* be the given values corresponding to a,b,Cyd,,,.k respectively as values of x. It is required to find an approximate expression for y^, an unknown term corresponding to a value of x between x^a and x=k. Since there are n conditions to be satisfied the expression which is to represent all of the values must contain n constants. Assume as the general expression y^=A+Bx+Cx^+Dofi+ . . . +Nxf' -1 Geometrically this is equivalent to drawing through the n points represented by the n sets of corresponding values a parabola of degree «— i. Substituting the sets of values given by the data in the equation above n equations are obtained from which to determine the values of -4, B, C, etc., ya=A+Ba+Ca?+Da?+ . . . Na""-^-, yi,=A+Bb+Cb^+Dl^+ . . . iVi'*"^; yu=A+Bk+Ck^-\rD]^+ . . . NV"-^. But the solution of these equations would require a great deal of work which can be avoided by using another but equiva- lent form of equation. INTERPOLATION "Let yx=A{x'-b){x—c){x'-d) . , . . {x—k) +B(x—a){x—c)(x—d) . . . {x—k) +C(x-a){x-b)(x-d) . . . {x-k) +D{x'-a){x—b){x—c) . , . {x-k) + etc. to n terms. 105 Each one of the n terms on the right-hand side of the equation lacks one of the factors x—a^ x—b^ x—c, x—d, . . . x—k, and each is affected with an arbitrary constant. The expression on the right-hand side of the equation is a rational integral function of x. Letting x = a gives and ya=A{a—b){a—c)(a—d) . . . a—k. A = Ja (a—b){a—c){a—d) . . . a—k' Letting x = b gives B = yb {b-a){b-c){b-d) . . . (b-k)' Proceeding in the same way we obtain values for all of the constants and, finally, {x--b)(x—c)(x—d) . . . (x — k) yx=ya +yi +y* (a--b){a—c){a—d) . {x—a){x—c){x--d) . {b-a){b-c){b-d) . {x—a)(x—b)(x--d) . +yd {c—a){c—b){c—d) . {x—a){x—b){x—c) . {d^a){d-b){d-c) : +y\ {x'-a){x—b){x'-c) (k-a){k-b){k-c) . (a — k) . (x — k) . (b-k) . (x—k) . (x — k) . (d-k) (2) 106 EMPIRICAL FORMULAS This is called Lagrange's theorem for interpolation. 1. Apply formula (2) to the data given imder formula XIX for finding the value of y corresponding to ii;= 0.425. Select two values on either side of the value required, (^'^''Z^^ ya = .69S, J = 40, y& = .730, ^ = 45, yc=.757» ^=.50, yd =.780. X in the formula must be taken as 0.5. ,_/.,,>, K-§)(-f) ,/ ..^x f(-^)(-l) y-(-695)(_,)(_,)(_3)+(.73o)(^)(_^)(_^) =.744. • 2. Required an approximate value of log 212 from the fol- lowing data: log 210 = 2.3222193, log 211 = 2.3242825, log 213 = 2.3283796, log 214 = 2.3304138. log 212 = {2 ^22210^) (^)(-^)(-^) +(2 .242820 (^)(~^)(-^) = 2.326359. This is correct to the last figure. In case the values given are periodic it is better to use a formula involving circular functions. In Chapter V the approxi- mate values of the constants in formula XX were derived. This formula could be used as an interpolation formula. But on account of the work involved in determining the constants it is INTERPOLATION 107 much more convenient to use an equivalent one which does not necessitate the determination of constants.* The equivalent formula given by Gauss is _ sm^(x — b) sin^ix—c) . . . sinJC^— ^) +yi +yc sin ^(a — 6) sin f (a — c) . sin ^(6— a) sin ^{b—c) . sin^ix—a) sm^(x—b) . sin ^{c—a) sin ^{c—b) . + etc. . . sinj(a— ^) . sin^(a::— ^) . sin ^(6—^) . sm^ix—k) sin^(c— ^) (3) It is evident that the value of ya is obtained from this formula by putting x=a. The value of y& is obtained by putting x = by and yc by putting x=c. The proof that (3) is equivalent to XX need not be given here. Let it be required to find an approximate value of y cor- responding to ic=42*^ from the values given. X y 30° 40° 50° 10. 1 9.8 8.5 From (3) • o • / o\ . >, sini sm (-4) _L sm y—5) sm(— 10 ; , . sin 6^ sin (-4^) sm 5 sm (—5 ) +(8.S) n • ^O • I sm 6 sm i sm 10° sin 5° - ('Tr^ ^\ (-°^75)(-o698) I /„ o>i (-1 045) (-0698) I /-O -X (-1045) (-0175) ■^^"•^^ (.i736)(.o872) = 9.618. * Trigometrische Interpolation, Encyklopadie der Mathematischen Wissenchaften, Vol. II, pt. I, pp. 642-693. 108 EMPIRICAL FORMULAS A better result would have been obtained by using four sets of values. Differentiation of Tabulated Functions It is frequently desirable to obtain the first and second derivatives of a tabulated function to a closer approximation than graphical methods will yield. For that piupose we will derive diflferentiation formulas from (i) and (2). From |2 ^ Is . . ,XVX— IKo . XyX—l)\X—2) .^ By differentiating it follows that 2 13 + 4^-"^+^^^-V o+ (4) Differentiating again yJ'=A^yo+{x-i)A?yo+{hx^'-x+H)A'yo+ (s) As an illustration let it be required to find the first and second derivatives of the fimction given in the table below and determine whether the series of observations is periodic* The consecutive daily observations of a function being 0.099833, 0.208460, 0.314566, 0.416871, 0.514136, 0.605186, 0.688921, 0.764329, show that the function is periodic and deter- mine its period. * Interpolation and Numerical Integration, by David Gibb. INTERPOLATION 109 From the given observations the following table may be wntten: X y =f{x) 1 0.099833 2 0.208460 3 0.314566 4 0.416871 5 0.514136 6 0.605186 7 0.688921 8 0.764329 (- (- (- (- 0.108627 0.106106 0.102305 )o.097265 )o.09io5o )o.o83735 )o.o754o8 From (4) y'l^ .108627 — .000427 .001260 —.000010 . 109887 -.000437 . 109450 /a = . 102305 .002520 -.000437 000392 000019 104825 —.000411 000411 . 104414 A* .002521 .003801 .005040 .006215 .007315 .008327 (+) (+) (+) (+) .001280 .001239 .001175 .001100 .001012 A« .000041 .000064 .000075 .000088 /2 /4 = . IO6IO6 .001900 .108006 — .000429 .107577 .097265 .003108 .100373 .000389 .099984 — .000413 — .000016 — .000429 — .000367 — .000022 — .000389 For the remaining first derivatives the order must be reversed and the resulting sign changed. y'5= — .097265 .002520 y'6= — .091050 .003108 — .000010 .000413 —.000016 .000392 097275 002933 .002933 .091066 003500 .003500 .094342 .087566 110 EMPIRICAL FORMULAS y7=- 08373s .000019 •083754 .004025 .079729 003658 /8= -.075408 000367 —.000022 004025 From (5) •075430 .004501 .070929 .004164 000337 004501 y"i = — .002521 .001280 y'2= — .003801 .001239 .001318 .000038 .001318 .001298 -.002503 .000059 — .001203 .001298 y"a=-. 005040 .001175 /'4 = — .006215 .001100 .001244 .000069 .001244 .001181 .000081 -.003796 -.005034 .001181 y"6=-. 005040 y\= — .006215 — .001239 -.001175 — .006279 -.007390 .000038 .000059 — .006241 -.007331 y'7=-. 007315 A= — .008327 — .001100 — .001012 — .008415 -.009339 .000069 .000081 — .008346 — .009258 X y y r y I •099833 . 109450 — .001203 — .0121 2 . 208460 .107577 -.002503 — .0120 3 .314566 . 104414 -.003796 — .0121 4 .416871 .099984 -.005034 — .0121 5 6 7 8 INTERPOLATION 111 y y' y" y" y .514136 .094342' - .006241 — .0121 .605186 .087566 - .007331 — .0121 .688921 .079729 .008346 — .0121 .764329 .070929 — .009258 — .0121 Jl Since — is very nearly constant and equal to —.0121, the corresponding differential equation is y+.oi2iy=o, whose solution is y^A cos o.iioc+B sin o.wx. This shows that y is a period function of oc, and its period is 27r J , or 57.12 days. O.II Convenient formulas for the first and second derivatives may also be obtained by differentiating Lagrange's formula for inter- polation. Using five points the formula is _ {x—})){x-'C){x—S){x—e) (x—a){x—c){x'--d)(x—e) y^-y^ (a-bXa-cXa-dXa-ey^' {b-a){b-c){b-d){b-e) , {x—a){x-'b)(x—d){x—e). (x-'a)(x'-b){x'-c)(x—e) '^^' {c-a){c-b){c-d){c-e) ^^"^ {d-aXd-bXd-cXd-e) {x-a)(x-b)(x-c){x-d) , v ■^^^ (e-a){e-b){e-c){e-d) ^ ^ Selecting the points at equal intervals and letting e—d=d-'C=c—b = b—a=hf and differentiation ya=-^[-25ya+48y6-36yc+i6yd-3yJ, /&=— t[- sya-ioyt+iSyc- 6yd+ yd 112 EMPIRICAL FORMULAS /d = — 7[- ya+ 6>-i8yc+ioyd+3yJ, y'e = -^-^[ 3> - le^ft+sfij^c - 48ytf + 2syJ. Differentiating again y'« = ^[3Sya-io43;6+ii4yc- S6yd+iiyj, y"6=m5[iiya- 203^6+ 6yc+ 4yd->'J, 12^2 y'c=^^[-ya+ i6y6- 3oyc+ i6yd-yJ, y'e = ^[i lya - s6>+ 1 H^c - io4ytf +3SyJ- The results of applying these formulas to the function given are expressed in the table below. X y y y I .099833 .109451 — .001203 2 . 208460 . 107583 — .002524 3 .314566 .104415 — .003804 4 .416871 .099986 -.005045 S .514136 •094347 — .006221 6 .605186 .087568 — .007322 7 .688921 .079733 -.008334 8 .764329 .070929 — .009258 These results agree fairly well with those previously obtained. It is probable that the formulas derived from the interpolation formula give the most satisfactory results. INTERPOLATION 113 As another application let us find the maximum or minimum value of a function having given three values near the critical point. Let ya, ybj and yc be three values of a function of x near its maximum or minimimi corresponding to the values of x, a, b, and c respectively. From (2) ^ (x-b)(x-c) (x-aXx-c) (x-a)(x-b) ^' ^"^ {a-bXa-cy^'ib-^aXb-cy^' (c-a)(c-6)' Equating to zero the first derivative with respect to x y. =ya 2X'-b'-C 4-^6 2X—a x = {a-bXa-c) ' '^ (b-aXb-c) ya(b^ - c") +y,{c^ - a^) +yc{a' - ^ ) 2\ya{b - c) +yj,{c - a) +yc{a - b)] f-yc 2x—a-'b {c—aXc—b) =0; (6) This is equivalent to drawing the parabola y=-A+Bx+Cx? through the three points and determining its maximiun or minimum. From the table of values 6.0 6.5 7.0 10.05 10.14 10.10 the abscissa of the maximum point is foimd from (6). ^^ (io.o5)(~6.75) + (io.i4)(i3) + (io.io)(~6.25) 2[(io.o5)(-.s)+(io.i4)(i) + (io.io)(-.s) = 6.596 y = 10.1424, CHAPTER VIII NUMERICAL INTEGRATION Areas An area bounded by the curve, y=f{x), the axis of x, and two given ordinates is represented by the definite integral =r ydXy where the ordinates are taken at x = a and x=n. It may be said that the definite integral represents the area imder the curve, or that the area imder the curve represents the value of the definite integral. If a function is given by its graph, it is possible, by means of the planimeter, to find roughly the area boimded by the curve, two given ordinates and the jc— axis, or, what amoimts to the same thing, the area enclosed by a curve. This method is used in finding the are^ of the indicator diagrams of steam, gas or oil engines, and various other diagrams. The approximations in these cases are close enough to satisfy the requirements. If, however, considerable accuracy is sought, or whenever the function is defined by a table of numerical values another method must be employed. Mechanical Quadrature or Numerical Integration is the method of evaluating the definite integral of a fimction when the fimc- tion is given by a series of numerical values. Even when the function is defined by an analytical expression but which can- not be integrated in terms of known functions by the method of the integral calculus, numerical integration must be resorted to for its evaluation. The formulas employed in numerical integration are derived from those established for interpolation. 114 NUMERICAL INTEGRATION 115 In interpolation it was found that the order of differences which must be taken into accoxint depends upon the rapidity with which the differences decrease as the order increases. This is also true of numerical integration. It is the same as saying that if the series employed does not converge the process will give unsatisfactory results. An illustration will be given later. Formulas for numerical integration will be derived from (i) of Chapter VII. In this formula it was assumed that the ordinates are given at equal intervals. yz=yo+xAyQ+^ — ^A^yoH- — | -A^yo ^ x(x-i)(x-2){x'-s) ^. ^ x(x-i){x-2)(x-s)(x-4) ^.^ k ° Is ^ x(x-i){x-2)ix-s)(x-4)(x-5) ^f, ^ ^^>^ Integrating the right-hand member, J\ yxdx=yQ I dx+Ayo I ocdx-\ — p^ I x{x — i)dx Jo Jo \2 J - I x{x — i){x-'2)dx Jo + 4 Asyo ■ I x(x-'i)(x-'2)(x — ^)dx Jo J I x{x—i){x'-2)(x—:y)(x—4)dx \-^jJx(x^i)ix--2)(x-3)ix-4){x^5)dx+ . . . =wyo 2 \3 2/ |2 \4 / I3 \S 2 3 / I4 116 EMPIRICAL FORMULAS \6 4 3 /Is + \7 2 4 3 / P The data given in any particular problem will enable us to compute the successive differences of yo up to A"yo. On the assumption that all succeeding differences are so small as to be negligible the above formula gives an approximate value of the integral. It is only necessary to assign particular values Let w=2, then I yzd:i: = 2yo+2Ayo+iA2yo, A23;o=Ayi-Ayo=y2-yi-yi+yo, =3'2-2yi+yo. Substituting these values in the above integral it becomes I yxdx = 2yo+2yi-2yo+iy2'-iyi+iyoy _yo+4yi+y2 3 This is equivalent to assuming that the curve coincides with a parabola of the second degree. If the common distance between the ordinates is h, the value becomes ydx=ih(yQ+4yi+y2) (7) X If ;^=3 '3 ydx=syo+^^yQ+l^^yQ+i^^yo, Ayo=yi— yo, A^yo=Ayi-Ayo=y2-'2yi+yo, i NUMERICAL INTEGRATION 117 A^yo = A^yi — A^yo = Ay 2 — Ay 1 — Ay 1 +Ayo =y3-3>'2+3)'i-yo. Substituting these values in the equations, 1 ydx=syo+h^i-h^o+iy2-hi+ho+hz-h^2+h^i-ho, =lyo+lyi+ly2+ly3, =1(^0+3^1+3^2+^3). If the common distance between the ordinates is h the formula becomes XZh =U(yo+3yi+3y2+ys) (8) This is equivalent to asstmiing that the curve coincides with a parabola of the third degree. If there are five equidistant ordinates, h representing the distance between successive ordinates r y^^=^4(y»+>^4)+64(yi+y3) + 24y2^^ ... (9) b 45 If the area is divided into six parts boxmded by seven equi- distant ordinates the integral becomes I y(/:i: = 6yo+i8Ayo+27A2yo+24A3yo+ WA^yo +HA5yo+ittrA«yo. Sine 3 the last coefficient, t^, differs but slightly from -nr and by the assumption that A^yo is small the error will be slight if the last coefficient is replaced by tu* Doing this and replacing Ayobyyi-yo, A^yo by y2 - 2yi +yo, A3yo by y^-^yz+Syi-yoy 118 EMPIRICAL FORMULAS A^jb by y4-4)'3+6y2 -4)'i+yo, ^^yo by ys - 5^4 + loya - ioy2 + syi - yo, A^yo by /-6y5+iSy4-2oy3+iS)'2-6yi+yo, gives the formula Jrth I- y(/:i: = ^%o+y2+y4+y6+s(yi+)'5)+6y3]. . (lo) The application of these formulas is illustrated by finding the area in Fig. 27. Fig. 27. By (7) 4=1^(^0+4^1 + 2^2+4^3 + 2^4+4^6 + 2^6+4^7+2^8+4^9 + 2yi0+4)'ll+yi2). By (8) 4=fA(yo+3)'i+3y2+2y3+3y4+3)'6+2y6+3>^+3y8+2y9 +3yio+3>'ii+yi2). By (9) -4=AA[i4(yo+2y4+2y8+yi2)+64(yi+y3+y6+y7+y9+yii) + 24(3/2 +y6+y 10)]. By (10) -4=iir%o+y2+y4+2y6+y8+yio+yi2+s(yi+)'6+y7+yii) +6(3/3+^9)]. I. A rough comparison of the approximations by the use of these formulas will be obtained by finding the value of — . The value of this definite integral is log 13 = 2.565. It 1 X is also equal to the area imder the curve y= X NUMERICAL INTEGRATION 119 from x = i to a;«=i3. Dividing the area up into 12 strips of unit width by 13 ordinates the corresponding values of x and y are X I 2 3 4 s 6 7 8 9 10 II 12 13 y I i i i i i i i 1 ■h 1 11 1^ 1^ By (7) i4=Mi+2+f+i+f+f+f+Ht+f+A+i+A] = 2.578, error .5%; By (8), ^ = 2.585, error .8%; By (9), -4 = 2.573, error .3%; By (10), A = 2.572, error .3%. 2. The accuracy of the approximation is much increased by taking the ordinates nearer together, as is shown by the following evaluation of ^^ dx r i+x The value of this integral is equal to the area under the curve _ I from a;=o ta x = i. Dividing the area into twelve parts by thirteen equidistant ordinates the value of I — -— is foimd to be Jo i+x By (7), 0.69314866, error 0.00000148; By (8), 0.69315046, error 0.00000328; By (9), 0.69314725, error 0.00000007; By (10), 0.69314722, error 0.00000004. The correct value is, of course, log^ 2, which is 0.693 14718. Formulas (7) and (8) are Simpson's Rules, (10) is Weddle's Rule. 120 EMPIRICAL FORMULAS 3. Apply the above formulas to the area of that part of the semi-ellipse included between the two perpendiculars erected at the middle points of the semi-major axes. Let this area be divided into twelve parts by equidistant ordinates. Since the equation of the ellipse is ^ y'f — these ordinates are By (7), -4=0.9566099^6; By (8), i4 =0.9566080^6; By (9), -4=0.956611406; By (10), A =0.956611406. The correct value to seven places is 0.95661 1506. In the application of these formulas it is highly desirable to avoid large differences among the ordinates. For that reason the formulas do not give so good results when applied to the quadrant of the ellipse. 4. The area under the curve, Fig. 28, determined by the following sets of values: .6 .8 3.0 2.5 2.0 1.6 1.0 0.5 . / V J Y N ^ 1 / / i 1 i 1 S A 3 1. 1. 2 3) X Fig. 28. .2 .4 i.o 1.2 1.0 1.5 2.2 2.7 2.6 2.3 2.1 is by (7) A =i-Ki-o+6.o+4.4+io.8+s.2+9.2+2.i) = 2.58, and by (8), 4 =1-1(1.0+4.5+6.6+5.4+7.8+6.9+2.1) = 2.5725. NUMERICAL INTEGRATION 121 This area is represented by the definite integral | ydx. : The area found is therefore the approximate value of this integral 5. Find the area xinder the curve determined by the points x\i 1.5 1.9 2.3 2.8 3.2 3.6 4.0 4.6 4.8 5.0 — I' _^^_— — — ^-— ^-^— ^— ^^— ^— ^^-^— ^— — ^-^— ^— ^-^— ^^— ^— ^— — .^— — — ^^^— __ y\o .40 1.08 1.82 2.06 2.20 2.30 2.25 2.00 1.80 1.5 The points located by the above sets of values are plotted in Fig. 29 and a smooth curve drawn through them. The area Fig. 29. is divided into strips each having a width of .4. Rectangles are formed with the same area as the corresponding strips. The eye is a very good judge of the position of the upper boimd- ary of each rectangle. Adding the lengths of these rectangles and multiplying the sum by .4 the area is foimd to be 6.644. By Simpson's Rule, formula (7), are found for h = ,2, A = .4, i4 =6.639, A =6.645. The graphical determination of areas can be made to yield a close approximation by taking narrow strips, and where the points are given at irregular intervals the area can be found more rapidly than by the application of Simpson's Rules. 122 EMPIRICAL FORMULAS 6. A gas expands from volume 2 to volume 10, so that its pressure p and volume v satisfy the equation pv = ioo. Find the average pressure between v = 2 and z; = 10. The average pressure is equal to the work done divided by 8. The work is equal to the area under the curve />=— from V z; = 2 to z; = 10, which is £ That this area represents the work done in expanding the volume from 2 to 10 becomes evident in the following way. Let s represent the surface inclosing the gas, ps will then be the total pressure on that surface. The element of work will then be dW=psdnj when dn lepresents the element along the normal. W=fpsdn. But sdn=dVj and W=j^pdv. This is the equation above. The average pressure over the change of volume from 2 to 10 is 160.944-^8 = 20.118. 7. Find the mean value of sin^ x from x=o to x = 2w. Plot the curve y = sm? x by the following values of x and y: X X 12 TT 6 4 ■K 3 Sir 12 TT 2 .0670 .2500 .5000 .7500 •9330 I. 0000 ^l^ 2t J^ S^ inr 12 3 4 6 12 •9330 -7500 .5000 .2500 .0670 NUMERICAL INTEGRATION 123 X T I37r 12 77r 6 4 47r 3 I77r 12 3^ 2 O .0670 .2500 .5000 .7500 •9330 I. 0000 a; 19F 12 3 Ttt 4 IITT 135 12 27r •9330 -7500 .5000 .2500 .0670 O Applying Simpson's Rule, formula (7), the area is found to be TT. The mean value is the area divided by 27r or .5. 8. A body weighing 100 lb. moves along a straight line without rotating, so that its velocity v at time / is given by the following table: /sec V ft./sec 1.47 1.58 ^.67 1.76 1.86 Find the. mean value of its kinetic energy from / = i to / = 9. / I 3 5 7 9 v^ .' 2.1609 2 . 4964 2 . 7889 3.0976 3 4596 Kinetic energy . 3. 355 3.876 4-331 4.810 5-372 Plotting kinetic energy to /, the area under the curve is 34.755. This divided by 8 gives the mean kinetic energy as 4.357- Volumes Fig. 30 explains the ap- plication of the formulas , to the problem of finding the approximate volume of an irregular figure. The area of the sections at right angles to the axis of x are: Ai = lk(yi+4y5+y4^y A2=lk(y6+4yg+y8)y ^3 = ^0^2+4^7+^3). Fig. 30. 124 EMPIRICAL FORMULAS If the areas of these sections be looked upon as ordinates, h being the distance between two adjacent ones, it is evident that the volume may be represented by the area under the curve drawn through the extremities of these ordmates. V^\h{Ai+^2+Az) Substituting the'values of A\, A2, and Az in this equation, the volume becomes F = JA[P(yi+43'5+)'4) +^^Cv6+4y9+y8) +P(y2+4y7+y3)] =iA%i+y2+y3+y4+40'5+y6+y7+y8)+i6>'9] In order to apply formulas (8), (9) and (10), the solid would have to be divided differently, but the method of application is at once evident from the above and needs no further discussion. 1. The following arp values of the area in square feet of the cross-section of a railway cutting taken at intervals of 6 ft. How many cubic feet of earth must be removed in making the cutting between the two end sections given? 91, 95, 100, 102, 98, 90, 79. These cross-section areas were obtained by the apphcation of Simpson^s Rules. By (7), 7=^-6(91+380+200+408+196+360+79) =3428; By (8), 7=1-6(91+285+300+204+294+270+79) =3426.8. 2. -4 is the area of the surface of the water in a reservoir when full to a depth A. hit, . . A sq.ft. 30 25 20 15 10 26,700 22,400 19,000 16,500 14,000 10,000 5,000 NUMERICAL INTEGRATION 125 Find (a) the volume of water in the reservoir, (b) the work done in pumping water out of the reservoir to a height of loo ft. above the bottom until the remaining water has a depth of lO ft. F =1(26,700+89,600+38,000+66,000+28,000+40,000+5,000) =488,833 cu. ft. rzo Work=w I A{ioo—h)dhy where w= weight of i cu.ft. of water = 62.3 lb. The value of this integral will be approximately the area under the curve determined by the points h 30 25 20 15 10 — — • — t i4(ioo— A), j 1,869,000 1,680,000 1,520,000 1,402,500 1,260,000 multiplied by 62.3. This area is equal to 1^(1,869,000+6,920,000+3,040,000+5,610,000+1,260,000) = 31,165,000. Multiplying this by 62.3 gives the work equal to 1,941,579,500 ft.-lb. 3. When the curve in Fig. 29 revolves about the ic-axis, find the volume generated. The areas of the cross-sections corresponding to the given values of x are given in the following table: X .2 .4 .6 .8 I.O 1.2 ^ .... IT 2.257r 4.847r 7.297r 6.767r 5.297r 4.4i7r By (7) F = 5.8627r = 18.416. By (8) F = 5.8o37r= 18.231. 4. When the curve in Fig. 30 revolves about the nc-axis, find the volume generated from x = i toa;=4.2. From the curve the following sets of values are obtained: 126 EMPIRICAL FORMULAS :t;|i.oi.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 3^0 .11 .29 .53 .87 1.37 1. 71 1.90 2.01 'fX O .012 .084 .281 .757 1.877 2.924 3.610 4.040 X y_ f 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 2.06 2.12 2.2 2.27 2.30 2.28 2.25 2.20 4.2444.4944.84 5.153 5.290 5.198 5.062 4.84 The volume is by (7) ^•ii(i49-oo4)=3i.2 cu. units. Centroids Let the coordinates of the centroid of an area be represented by X and y. Then from the calculus rxydx ^ 1 ydx - I y^dx y=P — " I ydx The integral in the numerator of the value of x may be represented by the area bounded by the curve Y=xyy the x-auds and the two ordinates a; = a and x = b. The original area is bounded by the curve whose ordinates are represented by y, the ic-axis and the two ordinates x = a and x = b. The integral in the numerator of the value of y may be represented by the area bounded by the curve Y=y^, the ic-axis and the two ordinates x = a and x = b. For a volume generated by revolving a given area about the x-2ods • TT I y^xdx TT I y^dx NUMERICAL INTEGRATION 127 / If 10 . 1 ir idu repKSSi- tie le When the volume is irregular Axdx f. Adx The process of finding the coordinates of the centroid of the area in Fig. 28 is shown in the table: X .2 .4 .6 .8 1.0 1.2 y I.O IS 2.2 2.7 2.6 2.3 2.1 xy 0.00 0.30 0.88 1.62 2.08 2.30 2.52 y' 1. 00 2.25 4.84 7.29 6.76 S.29 4.41 yH 0.000 0.450 1.936 4.374 S.408 5.290 5.292 The area under the curve F=a;y is i^[o.oo+i.2o+i.76+6.48+4.i6+9.2o+2.52] = 1.688; - 1.688 . ^ = —-^ = •654. 2.55 The area under the curve F=J3^ is ^[1.00+9.00+9.68+29.16+13.52+21.16+4.41] = 2.931 - 2.931 , As was pointed out before, large changes in the ordinates must be avoided. For the volume generated by revolving the area about the ic-axis -_7rA^[o.ooo+ 1.800+3.872 + 17.496+ 10.816+21. 160+5. 292] X — 7rr5[i.oo+9.oo+9.68+29.i6+ 13.52 +21. 16+4.41] =^=.687. 8793 128 EMPIRICAL FORMULAS Moments of Inerila The expression for the moment of inertia of an area about the y-axis is Iy= I x^ydx. About the x-axis Ix= I xy^dy. When the equation of the curve is known these integrals can be calculated at once, but when this is not the case approxi- mate methods must be resorted to. I. The process of finding the approximate values of these integrals is shown in the table below. The values of x and y are taken from Fig. 28. X .2 .4 .6 .8 1.0 1.2 y I.O 15 2.2 2.7 2.6 2.3 2.1 x^ O.CXX) 0.060 0.335 0.972 1.664 2.300 3.024 Jy' 0-333 1. 125 3. 549 6.561 5.859 4.056 3.087 If the values of x^y be plotted to x we will have a curve imder which the area represents the moment of inertia of the area in Fig. 28 about the y-axis. Dividing this by the area foimd before, there results for the radius of gyration about the y-axis if/ = .526. Plotting ^ to X and finding the area under the curve so determmed /x= 4.6136, and 22x2 = 1.788. NUMERICAL INTEGRATION 129 2. The form of a quarter section of a hollow pillar, Fig. 31, is given by the following table. Find the moment of inertia of the section about the axes of x and y. y .6 I •i: ^ r-^ ^. A '^^ ^ ^v. [N N w. \ Ji N 'v \ \ \ .2 \ \ k, \ \ a \ \ \ \ \ .2 .3 Fig. 31. .4 X X F x^Y y X y^X .00 .050 .00000 .00 .100 .00000 •05 .055 .00014 •05 .108 .00027 .10 .068 .00068 .10 .116 .00116 ■ 15 .078 .00175 IS .120 .00270 .20 .096 .00384 .20 .125 .00500 • 25 .116 .00725 .25 .130 .00812 .30 .148 .01332 .30 • 133 .01197 .35 .2CX5 .02450 •35 .140 .01715 .40 .300 .04800 .40 .150 .02400 •45 •215 •043S4 •45 .215 .04354 •50 .000 .00000 .50 .000 .00000 In the above table X stands for the width of the area parallel to the x-aixis and Y for the width parallel to the y-axis. The area is 0.066. The moment of inertia about the y-axis is r- .066 130 EMPIRICAL FORMULAS The moment of inertia about the ^-axis is where if stands for the radius of gyration. The values of the above integrals were computed by for- mula (7). APPENDIX If a chart could be constructed with all the dlfiferent forms of curves together with their equations which may arise in representing different sets of data it would be a comparatively simple matter to select from the curves so constructed the one best suited for any particular set. Useful as such a chart would be its construction is clearly out of the question. The most that can be done of such a nature is to draw a number of curves represented by each one of the simpler equations. A word of caution is, however, necessary here. A particular curve may seem to the eye to be the one best suited for a given set of data, and yet, when the test is applied, it may be found to be a very poor fit. It is of some aid, nevertheless, to have before the eye a few of the curves represented by a given formula. The purpose of the following figures is to illustrate the changes in the form of curves produced by slight changes in the constants. Figs. I, II, III, and IV show changes produced by the addition of terms, Figs. V to XIX changes in form produced by changes in the values of the constants, and Fig. XX the changes in form brought about by varying both the values of the constants and the nimaber of terms. A discussion of all the figures is imnecessary. A few words in regard to one will suffice. Formula XIV, for example, y = a+bx'', an equation which can be made to express fairly well the quantity of water flowing in many streams if x stands for mean depth and y for the discharge per second, represents a family of triply infinite number of curves. Fixing the values of b and c and varying the value of a does not change the form of the curve, but only moves it up or down 131 132 EMPIRICAL FORMULAS along the y-axis. Keeping the values of a and h constant and varying the value of c, the formula will represent an infinite number of curves all cutting the y-axis in the same point. In the same way, keeping the value of a and c constant and vary- ing the value of J, an infinite number of curves is obtained, all of which cut the y-axis in a fixed point. In Fig. XIV the quantity a is constant and equal to unity, while h and c vary. To one trained in the theory of curves the illustrations are, of course, of no essential value, but to one not so trained they may be of considerable help. The text should be consulted in connection with the curves in any figure. The figures are designated to correspond to the formulas discussed in the first five chapters. y 1 ft / 1.0 lA L4 L2 1 0.8 / // h / X «^ ?^^ (22^ f r ^ — ^ (6) ^ 0.6 0.4 0L2 -OJ! •0.4 ?^ "^ W^ ^ tD ^ V, ^ \ ^ \ ^ • \ \ -08 V \ (1) i/=l-.lx (2) i/=l-.la;+.01a;2 (3) i/=l-.la;+.01x*-.001x5 (4) y=l-.lx4-.01a;2-.001a;'-L .OOOlx* (5) y=l-.lx+.01x2-.001x'+ .0001x*-.00001z* (6) y=-l-.lx4-.01x2_.ooix«-f- .OOOlx*- .OOOOlx'+.OOOOOlx*' See formula I, page 13 8 9 10- u u m Fig. I. APPENDIX 133 y 1.4 1 \\ \ i =«« — 0.8 0.6 0.4 OJt -0.2 -0.4 -ae -03 w^ fftH? ^ ^ 15W / * ' 1 » I i' I } i \ ( I ( i 3 r J i f ) 1 1 1 L e » (1) i/=l-l/a; (2) y=l-l/x+l/x^ (3) i/=l-l/a;+l/x*-l/x» (4) y=l-l/x+l/x^-l/x^'\-l/x* (5) i/=l-l/x+l/x2-l/x»+ l/x*-l/x* (6) i/=l-l/x+l/x*-l/x«4- l/x*-l/x*+l/x« See formula II, page 22 Fig. n. 8, 2.8 V / \ / \ &6 1 / / 1 2.4 J J 2.2 1 (8)/ 2 / // f 1.8 ,/ / /; / 1.0 - J / 4 /^ N^ 1.4 / ^ ^ ^0^ -^ ^N ^ \ 1.2 ^ ^ 4 ^ V^ • M \ s> 0.6 \ 0.4 0.2 I I \ i ' ^ ( J K 1 I I i I i h (1) -«l_.la; y (2) -=l-.lx+.01x* {S) -=l-.lx+.01x*-.001x' y (4) -=l-.lx+.01x2-.001a8i- y .OOOlx* (6) -=l-.lx+.01x«-.001x3+ ' y .0001x*-.00001x* — 1 (6) -=l-.lx+.01x2-.001x5+ y .OOOlx*- .OOOOlx^+.OOOOOlx* See formula III, page 25 Fig. m. 134 EMPIRICAL FORMULAS y, t L8 •u y. ^ 1 -= ^ ^ (2L ^ ^ 0i8 08 *^ ■^ ■^ — i«v. ' > ^ 3 \ 02 • \ «-04 •06 -^ ' 1 1 i 1 1 I i d > C 1 1 f 1 i 1 II I I L L r'Sj (1) i/«=l-.lx (2) i/*-l-.lx+.01x* (3) |/»-l-.la:+.01x«-.001x» (4) y«=l-.lx+.01x'-.001x»+ .OOOlx* (5) y«=l-.lx+.01x2_.ooix»+ .0001x*-.00001x* (6) y«=l-.lx+.01x2-.001x'+ .0001x*-.00001x^+.000001x« See formula IV, page 25 Fig. IV. s M tA U ti S / ./ 1 r / / / / / / 4' / / / r J / f\ / ^ / / r ^ -^ / r / ^ lA / / y / ^ ^ "^ 1.4 6 L/ ^ -" ti^ r (»). — — 1 ^ — ■ -(7)- -__ 08 06 ^ s^ ^ ^~ -^ ^ $^ ^~ ^ - ■ — — M \ :v ^ v-._ .1% ■^ " " ■— . ^ ^ ^ 2*2.^ ■*" r: — __ ___ 1 I 1 i 1 i < \ 4 \ ] ( r 1 > 11 > 1 1 L B 11 S V \ u (1) i/-(.5)^ (2) l/=(.6)* (3) l/=(.7)^ (4) i/=(.8)^ (5) y=(.9)^ (6) y=(.95)^ (7) y=.99)* (8) y=(1.01)* (9) y=(1.05)* (10) y=(l.l)* (11) y=(1.2)« See formula, V, page 27 Fig. V. APPENDIX 135 V 16 3> ?y< ^ :::::: == , . _^ 1.6 y^ $; "" (a (4)^ ^ -.^-' ^-' -^ 1.4 ^ ^ ^ m^ -^ ^_ 12 >^ ^ :^ ■^ (7) 0) — ' ' — _ ^ 1 08 1 ,^^ === I^ — :;; — — ^ ^ == ^ mr ^1) — — 06 ■ ^ ^ vS --^ (12) " " ^^ -^ 0.4 v ^ i^, "^ ^-^ ^ "^ 0..2 N N. N^ v^ •Oi2 s [\ V \ \ -04 \ s -(L8 \ -U.B^ > I J 1 I ( 4 I ( ( \ r \ \ 11 9 1] L L I OB (1) y=:2-(.5)* (2) y=2-(.6)* (3) i/=2-(.7)* (4) i/=2-(.8)* (5) i/=2-(.85)* (6) i/=2-(.9)*' (7) i/=2-(.95)* (8) y=2-(.97)* (9) y=2-(.99)* 10) j/=2-(1.01)* 11) i/=2-(1.03)* 12) i/=2-(1.05)* 13)j/=2-(1.07)* 14) j/=2-(1.08)* See formula VI, page 28 Fig. VI. y 10 9 J y^ "-^ _^ — — 8 V ^ x^ "^ // '% y ^^ / / / ,^ 7 h V / ^ r^ -^ « TT /— -p^ "^ ■ 6 1 7/ y / (5J^ ''^ Fa / . r «i^ — // y ^^ --^ (6) ^^ 4 8 i — ^ J7), =i2i ' 2 1 N % ^ ^ ^ — .42> \^ ^S^2 f->a )^ ■^ , — _ ~^ ■ • D ] L 1 { 1 ' < 5 ( 1 ' r 1 ) » 1 1 I 1 2 m (1 (2 (3 (4 (5 (6: (7 (8 (9 (lo: (11 (12 (13 logi/= log|/= log J/= log|/= log J/= logy= logi/= log J/= logy= logy= log J/= logy= logy= base= -.6(.6)^ -.5(.6)* -.5(.7)* -.5(.8)* -.5(.9)* -.5(.95)* -.5(.98)* -.5(1.02)* -.5(1.1)* -.5(1.2)* -.5(1.3)* -.5(1.5)* -.5(2)* See formula VII, page 32 Fig. Vn. 136 EMPIRICAL FORMULAS y t .^ (1) 1.A (2) \4i -^ ^ ■ J3)^ ^ 1.4 ^ ■^^ ^, U N^ S.I71 ^ ^ 1 f N \ ^ \ V M^ S(8> \ \ \ N •^ 0.6 \ \ s. X s a4 \ X s ^^ a2 \ ^-^ \ \ 02 \ a4 \ i a 4 5 fl > fl fl IC > u L U 1 « (1) y=2-.01a>-(.5)* (2) i/=2-.03a!-(.5)* (3) y=2-.05x-(.5)* (4) y»2-.08a;-(.5)* (5) y*2-.lx-(.5)* (6) y=2-.l2z-(.5)* (7) i/=.2-.15i-(.5)* (8) i/=2-.2x -(.5)* See formula VIII, page 33 Fig. VIII V ^■■■■B e ' i 6 4 1 / \ - / 8 \\ // V) \ \ / 2 \ \ / / \" X! 1/ / 1 ::^ ^ r—- ^ I6i^ 1 :^ ^ 1 % a 4 6 « 1 8 8 U 1 1 L U 1 _• (1) j/s=lO-81--3to + .03a;* (2) y =10.54-. 24a; +.02z* (3) j/=10-27-.12a;+.01a;« (4) i/= 10-135 -.Oftr+. 005a;* (5) i/=10--135 + .06x- OOSx* (6) y=10--54+.24a;-.02z« See formula IX, page 37 Fig. IX. APPENDIX 137 V ' i \ L6 \ ' L6 \ 3^ (3) 1.4 ^ =^ 1.8 \ \r — ■** 1 \ / / a8 > V J 7 0.0 \ ^ / / 0.4 02 ^ ^ y^ (5/ — " J2- I i 1 i ^ 1 1 5 6 7 i 9 1 1 1 i 2 w (1) i/=(1.01)* (1.05)(l-2)^ (2) y=(1.01)* (1.05)<1-1®^^ (3) y=(1.01)* (1.05)<1-1S^* (4) y=(.6)* (2) (1-24)^ (5) i/=(.5)* (2)a-23)^ (6) y=(.5)^ (2)Cl-2)^ See formula X, page 37 Fig. X. 1 i 10 9 f \ " 8 7 6 f \ X V \ \ 5 f^ \ \ ^ \ \ ^. \ s. 4 \ X v^ 8 -^^ s^ N N, ^ \ <v s,,^ 2 1 \ Kt \ ^ ■^ - --^ N .^^ "^ ;;^ ^— -^ ■■^••. ^ - — ^^"^"" 1 — — ' c i ! I 4 6 fl 1 8 1 S ) 1 } I L U 8 '« ^^^ ^ .2-.lx+.05x* ^^^ ^=.2-.lx+.07? <'> «'=.2-.J+.lx« <*> ''-.2-.1X+.2X' <^> ''=.2-.J+.4x' See formula XI, page 38 Fig. XI. 138 EMPIRICAL FORMULAS y 10 y y y • r\ / — / r . / ^ -I8V- — — — ^ -^ "^«lLl J£L — __ \s "^ /iti — *2L. -- ■ _ — » 1 m t i J i (1 i 1 « 1 S u J 1 I a I » (i) y=5x-* (2) i/=5z-2 (3) y«5x-l (4) y=5x""*^ (5) i/=5x--2 (6) y-Sx--* See formula XII, page 42 Fig. Xn. (1) y=l+log x+.l log* x; 1/=— 1.5 (min.) when log x=— 5 (2) i/=H-log x+.011og« x; l/=— 24 (min.) when log x=— 50 (3) l/=H-.21ogx+.31og*x (4) j/=l— logx+log*x (5) y=l— logx+.51og*x See formula XIII, page 44 10 11 U 13 u u Fig. Xm. APPENDIX 139 »i i 5^ ^ ^ ;;^ "^ ^ ■ — 1 '^ . ft) "^ ^ X N, ^^ ^^ > S(to \ 1 i i i > < i i i ( ) r 9 9 1 1 1 1 i « (1) i/«H-.008xl-'' (2) y^l+.007x^-^ (3) i/«l+.006a;l-'» (4) i/«l-.002a^ (5) y=l-.003a:2.1 (6) j/=l-.004a:2.2 See formula XIV, page 45 Fig. XIV, y, 1 s ^ s \ ^: \ \ \ s^ \ \ \ I ^ ^ \ V \ \ \, ^ \ \ \ \ V ^ '^ ^s, ::^: s^ s. "^ ^ .^ V sX ^ ^ ^- 1 i 1 I i ' i < i 1 r c \ ( ) I D L 1 . I e » (1) i/=(2.0) 10- -01^ (2) |/=(1.6) 10--02ajl-^ (3) |/=(1.2) l0--03x^-* (4) y=(1.0) 10- O*^^-^* (5) i/=(0.8) lO--0635^'2* (6) y=(0.6) l0--0««^"^^ See formula XV, page 49 Fig. XV, 140 EMPIRICAL FORMULAS «! 1 "^■^ t V i\\- ' -^ = ^ \ ^ "Vp: ^ \ x^ ^ ^ y^ 1 // K N s. // \ <: \ //» / V \ V < ^^ f ^ ^ ^ "^ ...^ f -- ->^ — J' 1 2 a 4 6 6 7 8 1 » U » u 1 u i » (1) (1/-2) (x-.5) = -l (2) (i/-2) (x+.75) = -1.5 (3) (y-2) (x+1) 2 (4) (1/+.1) (a:+4)=8.2 (5) (y+.l) (x+3)=6.3 (6) (y+.l) (x+2)=4.2 See formula XVI, page 53 Fig. XVI. 9 i 6 6 7 8 9 Fig. XVIfl. W U IS 24 (1) y=J 10a?+24 -12 (2) y=J 10a;-24 2 (3) v=i 10a:+2 2^ (4) y=To 10^ .5 (6) j/=-10af+l See formula XVIa, page 56 APPENDIX 141 (1) j/=.5e.01^+e-05a; (2) j/=:2e.05«_.5e-l* (3) i/=2.25e-05a;_.75e.lit (4) y=1.8e-01a;_.3e-l« (5) v=1.92e--^^-A2e- (6) j/=2e--0^-e--01aJ ^=4.2e--2a5-3.5e--25af j-.2z_4.ie-.25ar ,-.01«_j3e-.15x .-.OU (7) l/= (8) y=4.5e- (9) i/=.25e- (10) i/=e- l«_l.le-.2x (11) i/=.27e--01a;-.77e--25a; (12) y=e-'^-2e--^^ See formula XVII, page 68 10 u ja « Fig. XVII. V k 2 1^ 1.0 P- 1.4 \ '^ ^ 1.2 \ V ^ 1 ") XiSi \ N. 08 J2L V \ Oifl (5) \ ai 2 \ J-"— •^_"^ N \— 0.2 f R' ^'-^ (.«y / N < :;^ \ \ \/ / / y S ^ o.i v /^ > / 06 \ / *"->> ^ . - i i J S 6 7 9 t • 1 1 1 1 « ^ (1) y=e«01a;(1.5 cos .Ix— .5 sin .Ix) (2) y=e"~'23J(1.5cos.5z— .5sin.5x) (3) j/=e~'^(.6 cos .lx+.8 sin .Iz) (4) y=e'^{.2 cos .3x— .1 sin .3x) (5) y=e-02x(.4cos.l6x+.17sin.l6x) (6) 2/=.5e~'^ sin X See formula XVIII, page 61 Fig. XVni 142 EMPIRICAL FORMULAS t u 14 1^ k y r — — ^ -^ (fl)/ 7^ / 14 1 lU- ^ -^ / / ^ ^ ■^ as / ?i r^ ^ ^ -- ^ ^ ^ ^ .^'^ (S) --. -^ a4 02 ^ r < 1 < I 1 ] XT X. > { 1 1 9 L 1 1 T& (1) y=-:2x-l-x-2 (2) y«3x-5-2.2x-6 (3) y=2.3a;-8-2x-85 (4) y=.lx-l+.5x-2 (5) 1/=. 33a;- .0012x3 (6) y=.25x-5+.05x-8 See formula XIX, page 65 h K 3 9 1 -1 ^ r k 1 A V \, ^ A A \ s. ^ /.^ /^ sW \ <: > ^^ -^ ^ . <:l ^ ^- . 1 \ 4^ . '^^^" 1 u s t 1 \ i , 1 > « 1 8 fl K ) U I U {"^ (1) y=15xl.5(.4)* (2) y=3x2(.5)* (3) j/=3x- 2(1.5)* (4) y=-.5xl-5(.75)* See formula XlXa, page 72 Fig. XlXa. APPENDIX 143 y 200 190 180 170 160 150 140 130 120 110 100 A i2l ^ . * ^ W ^ \ -rf* ^^ >a^^ .^,0^ C3) ?^^ Iff > ^ =- -^ / ^ < s (2r ^■ N vl*i y X N N ^, > ^ \ ^ Vl / \ ■^ f "^ 3^ y \ i T_ 1 r ¥ »r X- FiG. XX. (1) i/= 166.25-14.5 cos a;-2.75 cos 2a;-10 sin « (2) y= 167.83-20 cos x-4.33 cos 2x+o.5 cos 3x- 13.28 sin «- 17.32 sin 2x (3) j/=167.62-17.5 cos x-2.75 cos 2x+3 cos 3a;-1.38 cos 4x-12.42 sin x-18 sin 2x- 2.42 sin 3x (4) i/=167.08-17.22cosa;— 3.5 cos 2x+5.5 cos 3x— 0.83 cos 4x— 2.78 cos 5x+ 0.76 cos 6a;— 12.14 sin a;— 19.05 sin 2x— sin 3x— 1.73 sin 4x+1.14 sin 5x See formula XX, page 74