' 1 HE MEASURE STEVEN IRLF THEORY OF MEASUREMENTS A MANUAL FOR PHYSICS STUDENTS BY JAMES S. STEVENS Professor of Physics in tlie University of Maine ILLUSTRATED NEW YORK D. VAN NOSTRAND COMPANY 25 PARK PLACE 1915 7 COPYRIGHT, 1915 BY D. VAN NOSTRAND COMPANY THE SCIENTIFIC PRESS ROBERT DRUMMOND AND COMPANY BROOKLYN. N. Y. PREFACE THIS book is designed to be used in either of two ways: 1. As a Text-book. The work outlined would require two semester hours for its completion. By extending the discussions and problems, it may be made to cover three semester hours; or by omitting portions of the theory, the student may gain a working knowledge of the subject in a shorter time. A " rule of thumb " knowledge of adjusting observations, however, is not to be recommended. 2. As a Laboratory Guide. The work would cover a three years' course in the physical laboratory. During the first year, the student would make use of those portions which are devoted to methods of estimating precision, and the propagation of errors; in the second year the methods of adjustment of observations would be used; and in the third year the student should be prepared to discuss his results by the use of empirical formulae and curves. The work of the second year is well adapted to students in junior courses in engineering, the adjustment of data obtained from surveys being especially appropriate. iii 345123 iv PREFACE The use of the graphic method would be illustrated through- out the entire course. Considerable space is devoted to the theory of prob- ability. This subject is a fascinating one to students, presumably on account of its human interest. J. S. S. UNIVERSITY OF MAINE, ORONO, ME., January, 1915. CONTENTS PREFACE iii NOTATION . . vii CHAPTER I INTRODUCTION 1 MEASUREMENTS 2 ERRORS 3 CHAPTER II PROBABILITY 4 THE CURVE 9 THE INTEGRAL 14 THE ARITHMETICAL MEAN 17 A CONSTANT INTERVAL 17 WEIGHTS 18 CHAPTER III THE ADJUSTMENT OF OBSERVATIONS 22 SHORT METHODS 32 ILLUSTRATIONS 32 CHAPTER IV THE PRECISION OF MEASUREMENTS 36 GRAPHIC REPRESENTATION 36 v vi CONTENTS PAGE PRECISION MEASURES AND THEIR DERIVATION 37 AVERAGE DEVIATION 38 PROBABLE ERROR 38 CHAPTER V THE PROPAGATION OP ERRORS 44 THE DIRECT PROBLEM 45 THE CONVERSE PROBLEM 49 FRACTIONAL METHOD 50 BEST MAGNITUDES AND RATIOS 51 CHAPTER VI PLOTTING 54 METHOD ILLUSTRATED 55 CONSTRUCTION AND INTERPRETATION OF CURVES 57 APPLICATION TO LABORATORY PROBLEMS 57 CHAPTER VII NEGLIGIBILITY 62 APPLICATIONS 63 CRITERIA 66 SIGNIFICANT FIGURES . 69 , CHAPTER VIII EMPIRICAL FORMULAE AND CONSTANTS 72 METHOD OF PROCEDURE 72 TYPES OF CURVES 75 ILLUSTRATIONS 76 NOTATION The following notation will be used in this book: y Simple probability; P= Compound probability; ' Q = Negative probability; m= Single observation; M = Mean of observations; 2= True value of a component observation; =The resultant true value; x= Error; 0= Residual; ft Measure of precision; &= Probability of error zero; r= Probable error of a single observation; fo= Probable error of the mean; 6= Error in a component measurement; A = Error in a result; o.d.= Average deviation of a single observation; A.D.- Average deviation of the mean; p= Weight (Latin pondus)', E= Huge error. vii THEORY OF MEASUREMENTS CHAPTER I INTRODUCTION LOBD KELVIN has told us that one's knowledge of science begins when he can measure what he is speaking about and express it in numbers. Every year a vast num- ber of measurements are made in physical, chemical, and engineering laboratories, as well as in laboratories for advanced research. We are unable, however, to state concerning any one of these measurements that the result is absolutely correct. One of the most precise measure- ments in physical science is that of the wave-length of light. The wave-length of cadmium light, measured by a Michelson interferometer and a Rowland grating, was found to be Ac = 0.000064384722 cm. (Michelson) ^=0.00006438680 cm. (Rowland). Or we may put it in another way, 1 meter contains 1553163.6 wave-lengths (Michelson), 1 meter contains 1553164.1 wave-lengths (Fabry and Perot). 2 THEORY OF MEASUREMENTS These measurements were made by different observers using different methods. They are remarkable for their agreement and they give us the wave-length of light with sufficient accuracy for all purposes. But they are not correct, and it is not at all likely that we shall ever know the true length of a wave of cadmium light. It is of great importance, however, that we should be able to pass judgment upon the accuracy of measurements like these. To borrow another illustration from the domain of optics, we may assume the value for the velocity of light to be 29986030 km. per second. Here the number 30 is a measure of precision, which, if omitted, would make the statement ridiculous, since the velocity of light has not been measured with sufficient accuracy to be regarded as correct to a single kilometer. A student who brings in a value for g, resulting from measurements with a simple pendulum, of 982.436 centi- meters per second per second, is likely to look upon the result with complacency until he is made to see that all figures after the 8 are useless because erroneous. MEASUREMENTS Measurements are usually classified as follows: 1. Direct when, for example, a distance is measured with a tape line. 2. Indirect when the density of a cylinder is deter- mined by measurements of its length, diameter, and mass. 3. Conditioned when the third angle of a triangle is restricted by the values of the other two angles. Measurements not so conditioned are called independent. INTRODUCTION 3 ERRORS Errors in measurement may be divided into two general 1. Those which may be eliminated, in part at least, by improving adjustments and taking greater care in the method employed. These include errors in instruments, such, for example, as are caused by faulty measuring sticks, imperfectly graduated circles, and poorly adjusted balance beams. The obvious method of correcting these errors is to sub- stitute good instruments for poor ones; or, when possible, to eliminate the errors by compensation. Personal Errors. These are errors characteristic of the individual observer. If we swing a pendulum before a class of students and ask them to indicate the times of greatest displacement by tapping with their pencils, the result will well illustrate the personal equation. In the experiment in wireless telegraphy by which time messages were sent from Paris to Arlington, each observer was carefully rated in order that this error should be guarded against. Mistakes are, unfortunately, too common in students' laboratory work to call for an extended explanation. Errors in reading scales, in computation, and in tabulation may be classed as mistakes, and these should become less and less frequent as the student gains experience. 2. The second class consists of errors which are inde- terminate in their nature, and may not be entirely elimi- nated, however much care we may take in our measure- ments, or by the use of the highest grade of apparatus. It is these errors with which we are concerned in the dis- cussions in this book. CHAPTER II PROBABILITY PROFESSOR JEVONS in his Principles of Science states a truism which has an important bearing upon the theory of probability. " Perfect knowledge alone can give cer- tainty, and in Nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowl- edgeknowledge mingled with ignorance producing doubt." We may interpret this to mean that from the point of view of Omniscience everything exists as certainty. The path of a leaf falling from a tree, as well as that of a mote dancing in a sunbeam is known with as great a cer- tainty as that of a heavy body dropped to the earth. For finite minds, however, where certainty is impossible, the ability to pass upon the probability that an event will happen in a certain way is the best substitute attainable. What is highly probable in minds of a certain order of intelligence, may be improbable to others. To a trained meteorological observer, it may appear extremely prob- able that it will rain to-morrow; to one who bases his weather predictions upon popular superstition, it may be improbable. Again it may seem probable to certain people that spirit hands produced music at a stance, while to others the probability becomes negative. 4 PROBABILITY 5 We may look at the subject in another way. When a championship game of baseball has been finished, the result becomes a certainty in the minds of the spectators; while in various parts of the country, where the result has not as yet been reported, bets continue to be made expressing the probability of what to other minds is a certainty. If a coin is tossed up under normal conditions, the probability that it will fall " heads " is one out of two, or one-half. This is also the probability that it will fall " tails." We have, then, from our notation, y = %, or, since we shall deal with more than one event, P = \ and Q J. Since the coin must fall in one way or the other, we have P+Q= 1 (the symbol for certainty). If n coins are thrown, we have, by the binomial theorem The first term expresses the probability that all will come down heads, the second that all but one will come down heads, etc. If we take n = 6, the chances that all will be heads may be expressed by the fraction ^j, and that five will be heads by / c . PROBLEMS 1. By expanding the binomial to the proper number of terms, show that the chances for four, three, two, one, and no heads may be expressed by f , If > if > A> 6 THEORY OF MEASUREMENTS 2. By using an additional term prove that it is impossible to throw seven heads with six coins. 3. Plot a curve with the data obtained, and preserve it for future use. 4. If the class is sufficiently large (say 20) it will prove an interesting exercise to take the mean results of 64 trials by each student and compare with the results obtained by theory. A better acquaintance with the laws of probability may be obtained by putting aside the formula and solving the following problems by an appeal to reason. PROBLEMS 1. If two dice are thrown what is the probability that the sum of the numbers will be five? We first determine the possible results with two dice, which may be obtained by considering that each number on one die may appear with each number on the other. This gives us 6X6 = 36. Five may be obtained either by a four and a one, or a two and a three. Now the four may be on the first die and the one on the second, or the one may be on the first and the four on the second. The same is true for the two and three. Thus we have four possibilities, giving a prob- ability of ^. 2. What is the probability of throwing one ace with a single die in one throw? 3. What is the probability of throwing no ace with a single die in one throw? 4. What is the probability of throwing one ace in two trials? PROBABILITY 7 5. What is the probability of throwing two aces in two trials? 6. What is the probability of throwing no ace in two trials? 7. What is the probability of throwing only one ace in two trials? 8. A bag contains eight red, six black, and five green balls. What is the probability of drawing first a red and then a black ball in two trials? 9. The class record shows that each student in a class of twenty-five usually solves one problem out of three assigned. What is the probability that an assigned problem will be solved? 10. In Merriman's Least Squares we have this problem: Let a hundred coins be thrown up each second by each of the inhabitants of the earth. How often will a hundred heads be thrown in a million years? (It will prove inter- esting for the student to guess the answer to this problem before solving it. The number of inhabitants may be taken as one and one-half billion.) Before leaving the subject of probability a few general considerations may be suggested. It is a rather prevalent notion that antecedent happenings have an effect upon present probability. If a coin has come down heads five times in succession, it is pretty difficult to convince the average man that the chances for tails on the sixth throw are not greater than the chances for heads. Of course, if the conditions are normal, the probability of throwing heads is just one-half. (The derivation of the word chance is an interesting one.) Such expressions as " the turning 8 THEORY OF MEASUREMENTS of luck " indicate the strong hold this feeling has upon the majority of people. Again, we should not become over-confident from the results of the laws of probability. Interesting applica- tions may be found in connection with the periodic law of Mendelee*ff; the law of Prout, which states that the atomic weights of the other elements are exact multiples of that of hydrogen; and the kinetic theory of gases as developed by Clausius and Mayer. A case in which the probability almost becomes a certainty is illustrated by the coincidence of the seventy spectral lines in iron vapor with those in solar light. The possible arrangement of the seventy lines would be 70X69X68X . . . 4X3X2X1. But as all possible arrangements would not apply, the probability of a chance coincidence has been estimated by Kirchhoff as one in one trillion. Do we know that iron exists in the sun? The following illustration may serve to weaken our feeling of certainty: Imagine a fly watching two coin- cidence pendulums which come together on the eighty- first swing. While we are about it, let us imagine that in the mind of a fly, one second represents a year. The fly will watch the vibrations for ten seconds (years) and report to another fly which has just come up, that quite an extended series of observations fails to discover the two pendulums in coincidence. The pendulums are watched through fifty, sixty, and seventy swings and it becomes a PROBABILITY 9 law in flyland that pendulums do not get together. When the law has become established by observations extending over eighty fly-years, the bell rings and the assumption based upon a very reasonable law of probability breaks down. The Probability Curve. Referring to problem 3, page 6, it will be seen that the curve takes the form indicated in Fig. 1. Any similar data from experiments which follow the laws of probability would yield such a curve as this. FIG. 1. A familiar illustration is afforded by target practice. If the target is marked off into divisions, the distances of these divisions from the center may be taken to represent the magnitude of the errors. If a series of shots be fired by expert marksmen under normal conditions, they will form the basis for a curve which will resemble Fig. 1. Since the errors are plotted along the x-axis and their corresponding probabilities along the t/-axis; and since the curve is seen to be symmetrical with respect to the 10 THEORY OF MEASUREMENTS 2/-axis, and the z-axis is an asymtote, it will be seen that its equation must take some such form as A rigid deduction of this probability equation has been given by Gauss. Let x represent an error and y its probability. Then (1) (2) Let us suppose that n observations are taken on the quan- tities si, and 22. The most probable values of these quan- tities will make P maximum, For compound probability P = 2/12/2 . . . 2/ dP df(x 2 ) dP _ df(xi) + _df(x 2 ) df(xj ' /(*.)&! df(x a ) (4) We have . . . . (5) PROBABILITY 11 Since if we differentiate log/(zi) with respect to xi, we have Substituting (5) in (3) and (4), we have ^7/* .. x7/"h -. y-7/V r=0 (6) "2 , / \^^n_rt /ys The equations may be simplified as follows: Consider that z has been measured n times, giving mi, m^ . . . m n as results. Then z\ mi =2/1 z\m n x n Since mi, W2, and w n are constants, we have dxi 1 dx2 1 dxn 1 /0 x ^ = 1; -j =1; T~ = l> .... (8) d^i d^i dzi and similar results follow for dz2. This greatly simplifies Eqs. (6) and (7) and produces =0. ... (9) 12 THEORY OF MEASUKEMENTS Errors and Residuals We must now make a brief digression in order to illus- trate the difference between errors and residuals. Take the following measurements, of equal weight: mi =430.6 w 2 = 429.9 w 4 = 430.8 By universal custom the mean of these results, 430.6, is taken as the best attainable value. (Later a mathe- matical proof of this will be given.) The differences between the measurements and the mean are called residuals, and are as follows: vi= 0.0 2,2= -0.7 2,3 =+0.5 The algebraic sum of the residuals always equals zero, and this may serve as a check upon the work. Now if we had some way of knowing that 430.6 was the correct result, we would transform those residuals into errors and desig- nate them by xi, X2, 3, and x. A residual, then, is the difference between a measure- ment and the best attainable result; an error is the dif- ference between a measurement and the true result. PROBABILITY 13 It is obvious, first, that we cannot determine the values of the errors; and secondly, that the sum of the errors will approach the sum of the residuals as we increase the number of measurements. Going back to Eq. (9), we may assume that n is suf- ficiently large so that we can apply the above law. Then, since the sum of the residuals is always zero, one may take the sum of the errors to be zero, and write From this it follows that < is a constant, which we may call c. If we substitute the values of (xi), $(#2), and (j>(x n ) for (5) in (9), we obtain - | CT , | Since this holds for any number of observations, the corresponding terms are equal. Omitting subscripts we have in general ex. f(x)dx Substitute values from (1) and integrate. We have 14 THEORY OF MEASUREMENTS c is negative (why?) and may be replaced for the sake of convenience by 2h 2 . We may also replace e k ' by k. Then y = ke~ h ***, which is the equation of the probability curve. It is similar in form to the equation suggested to express the curve in Fig. 1. This is the most important equation in the theory of precision of measurements and its meaning should be clearly understood. PROBLEMS 1. Show from the curve that positive and negative errors are equally likely to occur. 2. Show that k represents the probability of the error zero. 3. Explain why h is called the measure of precision. 4. Show that the curve is horizontal over the origin. 5. Show that a point of inflection occurs when THE PROBABILITY INTEGRAL In order to express the probability that a certain group of errors will be made we integrate between the limits concerned. As this involves compound probability, we have p = ^_| e-tofifa. For certainty P = 1 and the limits become + oo and oo . PROBABILITY 15 This is a well-known integral and is discussed in various \A h treatises on the calculus. It equals -7, and therefore _ 1 _hdx L 7 7 A/ , . dxh ' VTT and the probability equation becomes For compound probability The probability that an error lies between +x and x is double the probability that it lies between x and 0. We have This may take the form This is the usual form of the probability integral, h may be determined in a manner to be explained later; then by use of a table the probability of any desired magnitude may be obtained. The student is referred to any of the larger text-books on Least Squares for illustrations. 16 THEORY OF MEASUREMENTS EXERCISES In order to become familiar with the terms h and k, the following curves should be plotted: 1. Consider h constant. Give k the values 1, 2, 3, 4. Plot a curve for each value. 2. Consider k constant. Give h the values }, \, 1, 2. In each case select a series of rather small numbers for x and use the formula 3. If time permits, a set of curves should be con- structed in which both h and k are variables, x and y having constant values. The Term " Least Squares " If we take the product of a number of single probabili- ties, we obtain An inspection of this equation shows us that the probabil- ity is greatest when the expression in the parenthesis is least. But these numbers are the squares of the errors (residuals, see page 12). It follows that the most probable values of observed quantities are those which make the sum of the squares of the residuals the least. From this law is derived the term Least Squares. PEOBABILITY 17 THE ARITHMETICAL MEAN We are now prepared to prove that in a set of observa- tions, the arithmetical mean has the greatest probability. Let M be the mean of observations mi, m% . . . m n . Then Mmi, M m<2, . . . M m n are residuals. We may make the sum of their squares a minimum. . . . 2(M-mn)=Q. M = " vi ' "** ' "* n , the arithmetical mean. n A CONSTANT INTERVAL. FORMULA n(n 2 1) For a constant interval the arithmetical mean should not be employed. This formula comes from formulae used in developing normal equations. See Mellor's Higher Math- ematics, page 327. If n is odd, the middle term does not appear. Use an even number of measurements. 18 THEORY OF MEASUREMENTS ILLUSTRATIVE PROBLEMS 1. Kundt's experiment: 30.7, 43.1, 55.6, 67.9, 80.1, 92.3, 104.6, 116.9, 129.2, 141.7, 154.0, 166.1 cm. Ans. 12.3 cm. 2. Time of vibration of magnet bar: 3.25, 9.90, 16.65, 23.35, 30.00, 36.65, 43.30, 50.00, 56.70, 63.30, 69.80, 76.55, 83.30, 89.90, 96.65, 103.15, 109.80, 116.65, 123.25, 129.95, 136.70, 143.35. Ans. 6.67. 3. A problem illustrating this method may easily be suggested from measurements with a planimeter; also from the acceleration curve with a dropped fork, WEIGHTS Heretofore we have considered all our measurements to be of equal value. Suppose the length of a small object was measured with a micrometer gauge, a vernier reading to hundredths of a millimeter, a vernier reading to tenths of a millimeter, and a meter stick. The following results might be recorded: 1. Micrometer 3.542 cm. 2. Vernier A 3.544 " 3. Vernier B 3.54 " 4. Meter Stick 3.55 " The mean of these measurements, 3.544, is obviously not the best value, since some represent greater precision than others. If one attaches to these results a number indicating their relative values, such a number is called the PROBABILITY 19 weight and is usually designated by p. (Latin, pondus, weight. Compare pound.) Further on, rules will be given for finding the weight of observations, but for the present we may use our judgment. No. 4 is evidently the least accurate and we may give it the weight 1. No. 3 comes next and we may call it 3. No. 1 and No. 2 are about alike in weight, with perhaps a little advantage in favor of No. 1. We may, therefore, assign 8 to No. 2 and 10 to No. 1. Putting our results in tabular form, we have: No. Obs. (m) Weights (p) Weighted Obs. (pm) 1 3.542 10 35.420 2 3.544 8 28.352 3 3.54 3 10.62 4 3.55 1 3.55 = 77.942 Spra -T- Sp = weighted mean =3.543 The weighted mean may be proved to have the greatest probability in a manner similar to that employed for the unweighted mean. PROBLEMS 1. In establishing a north and south line, the following readings were taken: N. 6' E., N. 4' W., N. 50" W., N. 00'. All readings were taken with equal care, but the first was taken twice and the third three times. Find the best value. 20 THEORY OF MEASUREMENTS 2. Joule's values of the mechanical equivalent of heat have been weighted by Rowland as follows: 442.8 (0); 428.7 (2); 425.8 (2); 426.0 (5); 427.5 (2); 429.1 (1); 428.0 (3); 422.7 (1); 426.8 (10); 428.0 (1); 427.1 (3); 426.3 (1). He concludes that 426.9 best represents the result of Joule's work. 3. The sum of the angles of an equilateral triangle is found to measure 180 9'. If the first angle has been meas- ured six times, the second three times, and the third once, how should the error be distributed among the angles? Additional problems in weighting will be given after the subject of precision of measurement has been dis- cussed. Subjects for Discussion The following topics are suggested for those who may wish to pursue the subject further: 1. The part played by the theory of probability in the work of the U. S. Weather Bureau. 2. The bearing of the laws of probability upon popular superstitions. Read the Vice-Presidential address of Pro- fessor A. G. Webster, at Atlanta, Ga., Dec. 30, 1913. (Science, Jan. 9, 1914.) 3. Exceptional phenomena. Read Chapter 29 in Jevons's Principles of Science. 4. Relation to gambling. PROBABILITY 21 Dr. H. G. Burnham, of Chicago, thinks that the best way to wipe out gambling in America is to teach the children in the schools the laws of chance. He feels sure that the result of this would be that in childhood they would steer clear of the slot machine, and that when they grow up, they would shun the book maker and every other gambling magnate. To quote further: " The ordinary gambler does not stop to count his chances when he sees that by betting a dollar he may win one hundred dollars. If he had been taught in school to see that in reality the chances were 200 to one against him, and that he was betting a dollar against fifty cents, he would keep his money in his pocket." 5. The use of the theory of probability in mortality tables. CHAPTER III THE ADJUSTMENT OF OBSERVATIONS IT is a frequent experience in making measurements that our results do not check. The measurements may be conditioned (see page 2). The sum of the angles of a triangle, for example, may not come out just 180. Or they may be measurements which have a less degree of interdependence. If we take as the starting-point and measure a dis- tance, mi = 6.2 feet above 0; then W2=4.3 above mi; then m2 = 10.6 above 0; we observe that there is a discrep- ancy somewhere. In this case, an obvious method of adjustment would be to add 0.03 to the first and second observations and subtract it from the third. Since the discrepancy is 0.1, this adjustment would correct it to 0.01. Theory. While this method would answer very well for the simple case in question, it would break down when applied to more extended measurements. The theory of the method of adjustment is as follows: Suppose we have unknown quantities mi, m^ . . . m n , and suppose n measurements are made upon these quan- 22 THE ADJUSTMENT OF OBSERVATIONS titles. Then if a, b, c, ... Z are known constants and M the resulting measured quantity, we have Since there are more equations than there are unknowns it follows that in general no system of values will exactly satisfy them. Each equation has a most probable value for its unknown terms, but in each case there will be left a small residual v. We may write the equation: For simplicity we may designate all the terms inde- pendent of mi by K. = vi, and by symmetry Square both sides of these equations and add: 24 THEORY OF MEASUREMENTS The greatest probability occurs when the right-hand member is a minimum. Placing the first derivative equal to zero, we have =0. Similar equations may be written for W2 . . . m. These are called " normal equations " and their solution gives the most probable values of the observations under consideration. Rules. We may now summarize the methods of adjust- ing observations as follows: 1. Write an observation equation for each observation. 2. Form a normal equation for each unknown by multi- plying each observation equation by the coefficient of the unknown in that equation and adding the results. 3. Solve the normal equations by any method. These results are the most probable values. i After these " best values " have been found, the residuals should be computed, and the sum of their squares (2y 2 ) found. This value is less than that for any other possible values of the residuals. ILLUSTRATIVE PROBLEM 1. Given the following observation equations, = 3 mi -f 2ra2 + 2ms = 25 m\-\- 7772+^3 = 16 THE ADJUSTMENT OF OBSERVATIONS 25 Find the best values and the sum of the squares of the residuals. We first find the normal equations by the rule. For mi, = 9 25 ra 2 + m 3 = 16 = 32 For W2, 12wi -f- 16w2 4ms = 12 2wi-f- For = 3 = 50 = 10 = 16 6mi+2m 2 +7m 3 = 73 Grouping the normal equations, 26 THEORY OF MEASUREMENTS Solving, mi = 6.00 m 2 = 6.13 m 3 = 3.26 To obtain the residuals, we substitute these values in the observation equations, 18-24.52+3.26=- 3 t>i = 0.26 6+12.26+6.52= 25 t; 2 =-0.22 6.13+ 3.26 =10 t> 3 = 0.61 6+ 6.13+3.26= 16 v 4 = 0.61 2v 2 = 0.860 It would be an interesting exercise to try other values for mi, W2, and mz and compare the value of Zy 2 resulting, with the above values. If, for example, we choose mi = 6, W2 = 6, ma = 3, it is obvious that the second residual is unity, which alone is greater than the value of 2v 2 . The following observations were made on a triangle: Angle A =45; angle = 80; angle C = 54. A+B = 126; A+<7 = 100; A++C = 180. Find best results. The observation equations are mi = 45 The student should construct the normal equations and find the best values for mi, m2, and m^. THE ADJUSTMENT OF OBSERVATIONS 27 MEASUREMENTS OF A LINE B C ~ i~~ F ~ FIG. 2. =4.Q units FG= 7.8 units BC = 5.l AC= 9.0 CD = 6.9 BD = 11.9 DE = 2.0 DF= 8.2 #^ = 6.1 A=32.0 Find the best values of the above measurements made along a line. Here we have 10 observation equations which will reduce to six normal equations containing six unknown quantities, which may be solved in the usual way. The problem is inserted for the purpose of illustration; its solution is hardly worth while. FORMULAE When the number of equations is large and the con- stants are not whole numbers, it saves time and affords a check upon the work to use the following formulae: . . a n 2 28 THEORY OF MEASUREMENTS The normal equations become bami+bbm 2 + . . . almi-\-blm2-\- . . . Urn* = IM The following example will illustrate the use of these formulae : Observation Equations 3w2 = 15.1 2wi+2ra 2 = 13.9 mi 2ra2 = 2.1 aa= +36+4+1 = +41 a&=-18+4-2=-16 66 = +9+4+4= +17 aM= +90.6+27.8+2.1 = +120.5 bM= -45.3+27.8 -4.2 =-21.7 41mi+16w 2 = 120.5 16mi+17m 2 =-21.7 Let the student compare these results with those found without the use of the formula. THE ADJUSTMENT OF OBSERVATIONS 29 PROBLEMS 1. Adjust the following values and find mi = +3.06 m 2 = -1.30 2mi+ra 2 =+4.81 2. The following observations are taken from a student's field book: M = 147 14' 04.5" N= 88 58 06.1 0= 93 51 26.7 P= 29 56 16.6 59 55.0 3 39.2 45 53.0 1 57.8 8 30.0 FIG. 3. Find the best values of the angles. 30 THEORY OF MEASUREMENTS A = 22 13' 59" 5 = 100 18 40 C= 57 27 33 = 122 32 41 +(7 = 157 46 10 C+A= 79 41 29 180 00 3 FIG. 4. Find the best values of the angles. 4. Adjust the following angles: BOC = 2 COD = 3 DOE = 4 FIG. 5. THE ADJUSTMENT OF OBSERVATIONS 31 No. Observed Angles. 1 70 20' 51" 2 52 35 10 3 '. 50 41 25 4 95 10 41 5 v 91 11 57 6 .,....; 198 27 14 Observe that angle 6 = angle 2-f-angle 3+angle 4. 5. Clairaut's empirical formula for the relation between the length of a seconds pendulum and the latitude is l=L +A sin 2 L. Suppose the following observations to be made: L i 0' 0.990564 18 27 0.991150 48 24 0.993867 58 15 0.994589 67 4 0.995325 Substitute the values in the given equation for observa- tion equations; then form normal equations for LQ and A. Mellor gives these normal equations: 0.993099 =L +0.44765A, 0.994548 = L -|-0.70306A. 32 THEORY OF MEASUREMENTS SHORTER METHODS The following abbreviated methods are sometimes employed : MAYER'S METHOD Make all the coefficients of mi positive and add the results to form a normal equation for mi. Similarly for m,2 . . . m n . Solve the normal equations as before. The results are not so accurate as those obtained by the longer method, but are satisfactory for most purposes. PROBLEMS Observation equations. x-y+2z= 3 -5z= 5 Solve by the regular method and by Mayer's method. Some idea of their relative accuracies may be obtained by computing Zv 2 in each case. METHOD BY DIMINISHING THE CONSTANT TERM Suppose we have given A+B=W 4A-B = 19 2A +35 = 25 An inspection of these equations shows that A = 6 and B = 4 approximately. THE ADJUSTMENT OF OBSERVATIONS 33 Let a and b equal the difference between the true and approximate values of A and B. Then (6+a) + (4+6) -10 = 4(6+a)- (4+6) -19 = 2(6+a) +3(4+6) -25 =0 a+6 = 4a-6+l = 2a+36-l=0 Forming normals and solving, we have o=-0.153 b= +0.406 A =6-0. 153 = 5.847 5 = 4+0.406=4.406 The student should compare these results with those obtained by the regular method. This method is especially well adapted to the adjust- ment of angles observed at a station. Considerable mathematical computation is avoided. The following problem from the United States Lake Survey is solved in Merriman's Least Squares. No. Between Stations. Observation. 1. Bunday and Wheatland 44 25' 40 // .613 2. Bunday and Pittsford 80 47 32 .819 3. Wheatland and Pittsford '. . 36 21 51 .996 4. Pittsford and Reading 91 34 24 .758 5. Pittsford and Bunday 279 12 27 .619 6. Reading and Quincy 62 37 43 .405 7. Quincy and Bunday 125 00 18 .808 34 THEORY OF MEASUREMENTS We may set up the observation equations: mi = 44 25' 40".613 mi+m 3 = 80 47 32 .819 w 3 = 36 21 51 .996 7?i4= 91 34 24 .758 360-(wi+m 3 )=279 12 27 .619 w 6 = 62 37 43 .405 125 00 18 .808 FIG. 6. On account of the nature of the constant terms the solution would be extremely tedious. We may let vi, vs, v, and VQ be the most probable corrections to be applied. Then mi=4425'40".613-{-t;i w 3 = 36 21 51 .996 +v 3 m 4 = 91 34 24 .758+^4 m 6 = 62 37 43 .405+^6 THE ADJUSTMENT OF OBSERVATIONS 35 This gives rise to simpler observation equations: vi=Q ^4-^3=4-0.210 =- 0.228 +0.420 The right-hand members denote seconds. The normal equations are = +0.402 = +0.402 = +0.420 = +0.420 From which v 3 =+0 .022 t; 4 =+0 .126 v 6 = +0 426 The adjusted values now become, No.l .................... 44 25' 40 r/ .635 2 ........... ____ ... 80 47 32 .653 3 .................... 36 21 52 .018 4, .........7. ........ 91 34 24 .884 5 .......... .. . ........ 279 12 27 .347 6,. ............ ....... 62 37 43 .531 7.. , 125 00 18 .932 CHAPTER IV THE PRECISION OF OBSERVATIONS As stated in the introductory chapter, there are no measurements made with such a degree of accuracy that we may regard them as absolutely correct. The wave- length of cadmium light has been measured with an accu- racy which is marvelous, but its exact value will probably never be known. Since measurements differ among them- selves in accuracy, it is desirable to have some method of indicating this fact, and the term " Precision of Obser- vations " or " Precision of Measurements " is applied to the method of procedure employed in determining the relative accuracy of measurements. Strictly speaking, the term " deviation 'Ms a better one than " accuracy " because the latter term seems to presuppose a knowledge of exact results. Classes. There are three classes of precision measures: 1. Mean error; 2. Average deviation; 3. Probable error. Graphic Method. These may be illustrated graph- ically by use of the probability curve (Fig. 7). Since YOX may be regarded as a probability area, and since, as will be shown later, the probability that the true result lies within the limits indicated by the 36 THE PRECISION OF OBSERVATIONS 37 precision measure is one-half, it follows that we may express our precision in one of three ways: 1. By taking the distance from along the z-axis to the point of inflection of the curve. This is represented by OM and expresses the mean error or mean square error. 2. By taking the abscissa of the ordinate passing through the center of gravity. This is OD and expresses the aver- age deviation. 3. By taking the abscissa of the ordinate which divides YOX into equal parts. This is OP and expresses the probable error. By reference to Problem 5, page 14, it will be seen that the mean error, /z= - , the distance OM in the figure. The average deviation, OD, may be found by use of the formula in mechanics l/2h 2 38 THEORY OF MEASUREMENTS 4769 The probable error, OP ^r . This will be deduced later. Of these precision measures, it may be said that the mean error, " the square root of the arithmetical mean of the squares of the errors," is seldom used; the probable error is the most accurate; and the average deviation is the easiest to determine. THE AVERAGE DEVIATION If m is the mean of n measurements and v\, V2, #3, . . . t> the residuals (neglecting signs), the average deviation would be a.d. = . n Taking the numbers 10, 8, 9, 10, the mean is 9.25, and vi = 0.75, v 2 = 1.25, ^ = 0.25, and v 4 = 0.75. Sv = 3.00 and a.d. = 0.75. We may neglect the signs of the residuals. This is the average deviation of a single observation. The average deviation of the mean, A.D. = 7=. In Vn the problem just solved, A.D. = ^ = 0.38. a PROBLEMS It will prove of interest if the instructor will assign problems taken from the student's laboratory note-book in illustration of this topic. THE PROBABLE ERROR The term probable error is rather misleading, but its meaning may be made clear by the following definition: THE PRECISION OF OBSERVATIONS 39 The probable error is a number placed after a result with a plus and minus sign between them; and it indicates that it is an even wager that the true result lies between the indicated limits, and that it does not so lie. At first thought, this definition seems vague and not indicative of a very high order of precision. Let us take the value of the velocity of light as 299860zb30 km. per second. This means that it is just as likely that the true value lies between 299830 and 299890 as that it lies between zero and 299830, plus 299890 to infinity. This narrows the field and indicates an order of precision that is suf- ficiently high. FORMULAE FOR PROBABLE ERROR Going back to the probability integral, we must take the value of x when P = |. This gives J = 2/ vV \ X e~ h2xS dhx. From integration tables we find hx = 0.4769. The particular value of x which fulfills this condition will be denoted by r. Then hr = 0.4769. Take the equation for compound probability (page 15). Since h is the measure of precision, we wish to give it such a value as will render P a maximum. ah 40 THEORY OF MEASUREMENTS (Divide by ftn-i 6 - But h=<^. Therefore r = 0.4769 J = 0.6745* / . \ n \ n This is the probable error for a single observation when errors are considered. In order to change errors to residuals (a; to v), we make use of the following procedure: Let the sum of the squares of the errors differ from the sum of the squares of the residuals by some constant, say u 2 . Then Now u 2 = : n n n1' Substituting in the formula for r, = 0.6745- n THE PRECISION OF OBSERVATIONS 41 This is the formula for single observations for residuals. Since probable errors are inversely proportional to the squares of the number of observations taken (see any text- book on Least Squares), we have n: l::l/ro 2 : 1/r 2 , ro = r/Vn. ro thus represents the probable error of the mean and / Zy 2 may be written ro = 0.6745* /^ r-r. \n(n 1) We may now summarize the relation between the probable error, the measure of precision, and the weight as follows: 1. The measure of precision varies inversely as the probable error (since hr = constant) . 2. Weights are proportional to the squares of the pre- cision measures. 3. Weights are inversely proportional to the squares of the probable errors. PROBLEMS 1. Given the following measurements of the length of a line: 70.6 cm. 70.5 cm. 70.4 cm. 70.5 " 70.6 " 70.5 " 70.7 " 70.8 " 70.6 " Find the mean square error, the average deviation, and the probable error. 42 THEORY OF MEASUREMENTS 2. Assume that the lines OP, OD, and OM are drawn to scale in Fig. 7, and compare their relative lengths with the results of No. 1. 3. Given the measurements and probable errors: 427.320.04 427.300.16. Find the relative weight and the relative precision. 4. Twenty measurements of a line give a probable error of the mean of 0.06. How many additional measure- ments are required to reduce the probable error to 0.03? 5. An angle is measured 9 times with each of two tran- sits. The first gives a value of 41 32' 14"8".2. The second gives 41 32' 12"7".l. Find the best value for the angle. 6. When the best value is found from No. 5, its prob- able error may be found by use of the formula r/ 2 : r a 2 :: p a : p f where r/= probable error in the best value; r a = probable error in the first value; p a = weight of first value; p f = weight of first value + weight of second value. The precision measure is sometimes expressed as a fraction or a decimal with reference to the magnitude of THE PRECISION OF OBSERVATIONS 43 the measurement concerned. Thus in Problem 3, we may say that 0.04 is the precision measure, or we may say that the measurement is reliable to about 0.01%, or to one part in 10,000. PROBLEMS 1. Show from the data on page 37 that r=0.85 a.d. = 0.67 M . It follows that for a constant value of n, ro = 0.85 A .D. 2. The following results have been obtained from nine measurements in each case: 94.31cm. A. D.= 0.031 9436 " a.d.= 0.090 94.35 " r =0.025 94.33 " r = 0.007 94.34 " reliable to 0.03% 94.35 " precise to 2 parts in 10,000 Reduce these different precision measures to A.D.'s and write them in order of the reliability of the results. CHAPTER V THE PROPAGATION OF ERRORS IN the study of the propagation of errors we have two classes of problems the direct and the converse. In the first class we determine how the errors in component measurements affect the reliability of the results; and in the converse problem we determine with what accuracy we should make our component measurements in order to secure a required accuracy in the result. In solving problems under the first case, we proceed as follows: If Z represents the true result, and X its error; and z\, 22, z are true values of component measurements, and xi t X2, x n are their corresponding errors, then, , (22+0:2) . . By Taylor's theorem v _dZ dZ dZ * " Consider that we have a series of such terms, designated by SX, Sa?i, 2^2, etc., and square both sides: 44 THE PROPAGATION OF ERRORS 45 We have neglected terms containing the products of the errors, such as 2x 1X2, since positive and negative errors are equally likely to occur. If we divide both sides by n, we may substitute r 2 (the probable error in the final result) for - and ri 2 , f2 2 , r n 2 , the probable errors of 'the Ti components for , etc. The reason for this is evident from an inspection of the probable error formula. Thus we have Hereafter we shall replace r by A and r\, r% . . . r n by Si, 62, ... S n . These refer to any precision measure as well as the probable error. A means the deviation in the result due to deviations in the components, Si, fe, s. ILLUSTRATIVE PROBLEMS 1. The two sides of a rectangle measure 450.6 and 500.4. Find the deviation in the area. The formula s = (50) 2 (0.6) 2 +(45) 2 (0.4) 2 = 900+324 = 1224 A = 35 approximately. We may say, then, that the area (expressed in square feet), is 2250 with a deviation measure of 35. 46 THEORY OF MEASUREMENTS 2. It may be readily seen from the formula that if the equation has the form A =a+6, the value of A 2 becomes 5 fl 2 -N 6 2 . This may be applied to the measurement of a line which has to be made in sections. 3. If the equation contains a constant multiplier, A=ca, we have A = c5. If the radius of a circle is 10 cm. with a deviation of 0.1, the deviation of the resulting length of the circumfer- ence is 27rXO. 1 = 0.63. PROBLEMS 1. The formula for the radius of a sphere as given by a spherometer is If Z = 7.230.04cm., and a = 0.530.008 cm. find the length of the radius and its deviation measure. 2. If the length of a pendulum is lOOitO.l cm., and the period of one vibration 1.01 0.003 seconds, what is the deviation in 0? 3. The mass of a body in air is 300.1 g; in water 200.2 g. Find the deviation in the value of its specific gravity. 4. What is the deviation in the value of the velocity of liquid flow from a height of 50dbl cm.? V 2 = 2gh. ,5. A mass of 100 g. is revolved with a speed of 80 cm. per second on the end of a corti 50 cm. long. The mass measurements are 101, 100, 99; the speed measurements THE PROPAGATION OF ERRORS 47 82, 80, 78; the length measurements 51, 49, 50. Find the centrifugal force and its deviation. 6. Find the deviation in the mean of two quantities which differ by C. 7. If d is the deviation in log a, what is the deviation in a? 8. In No. 2, assume that the deviation found is an average deviation, A.D. Express the result fractionally, decimally, and as a probable error. 9. In determining a refractive index, the value of i is 408 / , and of r 326 / . Find the value of the index and its deviation. In solving these problems, the student should observe the relative effects of the deviations in the components upon the result. In No. 2, for example, it will be seen that a deviation in t is a much more serious matter than an equal deviation in I. A large number of problems illustrating this process may be found in Goodwin's Pre- cision of Measurements, PROBLEMS The following problems are of especial interest to engineering students : 1. In the triangle ABC, AB = 500'd=0.04' Z.A =3210', ZC = 604'. Find BC and its deviation. The formula for the area of a triangle with the given conditions should be written down and the usual method employed. We have the deviations given for the angles and need them for the sines of the angles. For a deviation of 10 minutes in 32 degrees, we find the deviation in the sine of 32 degrees 48 THEORY OF MEASUREMENTS by subtracting the sine of 32 from that of 32 10'. Or we may differentiate with respect to the angle. 2. In a triangle ABC, we have A B = 240.4 0.04, AC = 290.6 0.003, Z.4=44 20'1'. Find the area and its deviation. Use both methods referred to above in deal- ing with the angle A. 3. The formula for the elastic modulus of a rectangular PI 3 bar supported at its ends is E= 3> where P is the mass applied at the center, I the length, d the deflection produced, b and h the breadth and thickness. We have given the following measurements : b h d 0".331 0".490 0".206 0".333 0".492 0".205 0".329 0".491 0".206 0".330 0".490 0".207 Take the length two feet and P = 40 Ibs. Find E and its deviation. 4. Find the number of calories and its deviation, from a current of 6.20.06 amperes, through a resistance of 200.1 ohms, for 30 minutes (correct to a single second). 5. Observations on a resistance of about 10 ohms give values as follows: Correct to 0.09%, correct to one part in one hundred; a probable error of 0.001. Write these down in the order of their relative reliabilities. THE PROPAGATION OF ERRORS 49 THE CONVERSE PROBLEM In the cases already considered, we have discussed the effects of deviations in component measurements upon the deviation in the result. We are commonly called upon to determine in advance to what degree of precision we shall make our measurements in order to reach a given precision in our result. For example, we are required to measure the volume of a cylinder whose length is about 10 cm. and whose radius is about 3 cm. with such a degree of accuracy that the deviation in the volume (about 283 cm. 3 ) shall not exceed 3 cm. 3 It is customary to so adjust the devia- tions pertaining to each variable that each will have an equal effect upon the final result. This is called the method of equal effects. Representing the effects of the various deviations upon the results by A a , A&,_. . . A n , we have A 2 =A a 2 +A & 2 + . . . A n 2 = nA 2 . A = A a Vr&. In our problem we may write r ~A/^~V2~ The deviations in the length and radius should not separately produce a deviation in the area of more than 2.1 cm. 3 From our general equation, page 45, we have Therefore dl = 0.074 cm. Likewise A, = ^r : 5 r = 27rrZSr= 188.63, = 2.1. d r = 0.011 cm. dr 50 THEORY OF MEASUREMENTS We conclude that we must measure the length of the cylinder with an accuracy of 0.074 cm., and the radius with an accuracy of 0.011 cm. THE FRACTIONAL METHOD These problems may be solved in another way. It follows (not quite directly) from the method of equal effects that the deviation in a final result due to a devia- tion in a component bears the same relation to the final result as the deviation in the component bears to the com- ponent, if the exponent is unity. In other words ~r~r- If ^ ne component has any other exponent than unity, the fraction must be multiplied by the exponent, following the method of differentiation. We have in the last problem -j- = -, since the radius A. T appears in the second power. A Since Aj = A r = ;= we may write v2 Ai A r 1 A 1 3 Z = 3 = 0.0074; dl= 0.074 . This is seen to check with the former method. THE PROPAGATION OF ERRORS 51 PROBLEMS 1. With what accuracy must we measure the radius of a circle to obtain an accuracy of 0.1% in the area? (First method.) 2. With what accuracy should I and t be measured in a simple pendulum to obtain a value of g correct to one unit? (Second method.) 3. With what accuracy should h and a be measured with a spherometer to give a value in the radius correct to 0.03%? This problem may be solved in a literal form or values may be assigned to a and 6 and the result computed. (First or second method.) These are illustrative problems. They should be extended, so far as time permits, to include various problems from the student's laboratory note-book. The impor- tance of this part of the subject is obvious. BEST MAGNITUDES AND BEST RATIOS No one can work very long in a physical laboratory without observing that the results come out much better by using certain magnitudes and ratios than they do with others. In measuring current with a tangent galvanom- eter very poor results would be obtained for angles between 0-20 and 70-90. Around 45, however, they are satisfactory. In using a slide-wire Wheatstone bridge, the two segments of the wire must be about equal for satisfactory results. 52 THEORY OF MEASUREMENTS In the first case, our formula is I = c tan . _ d(c tan ) cd dj * cos 2 <' Dividing / cos 2 c tan sin 2<* This is a minimum when < = 45 and shows that the devia- tion in the result due to a deviation in $ is least at 45. For the Wheatstone bridge we may let x = the unknown resistance, R the known, a and b the segments of the wire, and c its total length. Then aR x = ca It should be easy for the student to prove that the best result occurs when a = b. These illustrations might be indefinitely extended. The student is referred to Holman's Precision of Measure- ments for a number of interesting problems. EXERCISES 1. Show that the relative error in measuring the area of a circle decreases with increasing radius. 2. In drawing a simple harmonic motion curve, discuss the effect of an error in the phase upon the displacement for various points on the curve. THE PROPAGATION OF ERRORS 53 3. A body rolls down an inclined plane for a definite period of time. Discuss the effect of an error in measur- ing the angle of the plane upon the error in velocity. Neglect friction. 4. How would an error of one minute in measuring the angle of incidence compare with a similar error in measur- ing the angle of refraction in their effect upon the refrac- tive index? Take z=45 and r = 30. 5. What is the " best value " of the limiting angle in measuring the coefficient of friction? ABC 6. In Lami's theorem sin a sin sin 7' take A =59.8, = 69.8, C= 102.8, a = 146, 0=139, 7 = 75. Discuss the effect upon each term of an error of 0.5 in measuring a, /3, and 7. CHAPTER VI PLOTTING Definitions. It may be assumed that students for whom this book was prepared have a knowledge of the fundamental principles of curve tracing. A few familiar definitions and illustrations will, however, be given. A plot is a graphic representation of the relation of two quantities of which one is a function of the other. The origin is the point of departure from which all distances are reckoned. The coordinates are horizontal and vertical distances from the origin. The axis of abscissas is the horizontal line through the origin. The axis of ordinates is the vertical line through the origin. The slope of the curve is the angle it makes with the z-axis. When the points for a curve have been established, there are two methods of procedure: If the curve appears to be regular or to correspond with well-known types, we should draw an average line through the points in such a manner that about as many will lie on one side as on the other. If the curve does not fulfill these conditions, the adjacent points should be connected by straight lines. 54 PLOTTING 55 EXERCISES In these exercises, the student should use his judgment as to the method to be followed. 1. Plot a simple interest curve for $100 for ten years at 5 per cent. 2. Plot a compound interest curve starting from the same origin. 3. The maximum temperature for each day of a certain month was as follows: 1 31 11 29 21 11 2 22 12 24 22 13 3 21 13 18 23 18 4 27 14 12 24 41 5 29 15 14 25 43 6 26 16 26 26 38 7 22 17 28 27 30 8 28 18 28 28 36 9 29 19 19 29 36 10 28 20 16 30 47 Plot the curve. 4. Compute the mean temperature for the month in No. 3, and plot a curve showing the departure from the mean for each day. ^. ; 5. The following plot will afford amusement as well as instruction: 56 THEORY OF MEASUREMENTS X V X V -0.0 1.5 1.3 -1.0 -1.0 1.2 1.6 -0.3 -2.0 0.7 2.0 -0.4 -3.0 0.0 3.0 -0.5 -3.5 -0.5 3.3 -0.5 -3.7 -1.0 3.5 -0.3 -3.5 -1.7 3.3 0.0 -3.0 -2.2 3.0 0.6 -2.5 -2.3 3.0 1.0 -1.0 -2.1 2.7 1.3 -0.5 -1.9 2.8 1.6 0.0 -2.3 2.5 1.5 0.5 -2.4 2.0 1.4 1.0 -2.5 1.0 1.6 0.8 -1.6 0.0 1.5 1.0 -1.3 6. A thermometer is found to have the following errors when calibrated by means of a standard: 10+0.03 12 -0.60 18 -0.45 22 -0.38 30 +0.71 Draw a curve of errors for the thermometer. Determination of Constants. of a straight line, Let us take the equation By definition - is the tangent of the angle which the PLOTTING 57 line makes with the z-axis. This is equal to a and deter- mines this constant. When x = Q, y = b. This determines the constant 6. Rearrangement of Data. It sometimes happens that if a curve is plotted in -one way it is meaningless or dif- ficult to interpret; while a rearrangement of data makes it perfectly intelligible. If, for example, a series of readings have been taken with a Boyle's law apparatus, the result- ing curve should be an equilateral hyperbola (pv = a). It will be found, however, that the readings one is ordinarily able to obtain are not sufficient to identify the curve. If we change the data and plot v and l/p we get a straight line. In a tangent galvanometer the formula is I = k tan 0. In order to get a series of variations in the current we introduce varying resistances. Since I = E/R, it follows that the reciprocals of the resistances will plot a straight line with tan EXERCISES 1. The following data come from an experiment with the Boyle's law apparatus: Volume. Pressure. Volume. Pressure. 26.5 134.3 34.9 102.9 27.4 130.5 36.2 99.3 27.9 126.4 37.6 95.7 28.2 122.9 39.2 92.3 29.2 118.4 40.7 88.3 31.3 114.5 42.5 85.3 32.4 110.5 44.0 81.9 33.6 106.7 45.8 78.9 68 THEORY OF MEASUREMENTS Discuss the accuracy of the observations from the plot. 2. The following table is made up from resistances and corresponding deflections with a tangent galvanometer: Resistances (Ohms). Deflections (Degrees). 90 5.3 50 8.7 30 12.8 20 16.7 10 24.0 5 30.6 3 34.5 1 39.3 42.5 a. Assume that we have a constant electromotive force and make a plot which will show the relation between the current and the deflection. 7 = fctan0, I = E/R. b. Plot the resistances with the cotangents of the angles and project until it cuts the s-axis. Interpret the curve. c. Find the value of E by Ohm's law. 3. Known volumes of a liquid were placed in a flask and weighed, giving data as follows: Volume of Liquid, cc. Total Mass (Liquid +Flask.) g. 20.8 204 37.6 221 61.3 250 86.5 287 108.0 307 136.5 336 PLOTTING 59 Plot the volume on the z-axis and the masses on the a. Find the mass of the flask from the plot. 6. Find the specific gravity of the liquid. 4. Two rulers were placed together at random and the reading on the centimeter scale was taken opposite each inch division on the English scale, giving data: Inches. Centimeters. 1 2 27.8 3 25.3 4 22.7 5 20.2 6 17.7 7 15.1 8 12.6 9 10.0 10 7.5 11 5.0 12 2.4 Plot the inch readings on the z-axis and the centimeter readings on the y-axis. a. Determine from the plot how the rulers were placed with reference to each other. 6. Find the ratio of the centimeter and inch by using the intercepts of the curve on the axes; also by using the ratio of the difference of two points 2/22/1 and x% x\. 5. The candle-power of a 16-c.p., 110-v. lamp was measured for different voltages across the terminals as follows : 60 THEORY OF MEASUREMENTS Volts. Candle-power. 60 0.4 70 1.1 80 2.3 90 4.2 100 8.3 105 10.3 118 16.6 128 22.3 138 25.8 150 44.7 160 54.5 170 72.2 180 79.7 190 93.9 200 115.7 210 120.3 Plot volts on x and candle-powers on t/-axis. a. Find candle-power at 110 v. 6. Find candle-power at 220 v. c. Is it possible to estimate the voltage at which the filament first begins to glow? 6. The elongation of a spring was measured for differ- ent loads, and the energy stored in the spring computed, giving data: Loads. Elongation. Energy. g. cm. g. cm. 0.0 50 2.3 57.5 100 5.4 270.0 150 8.8 660.0 200 11.9 1190.0 250 15.2 1900.0 PLOTTING 61 Plot loads (x) with elongations (y) and energy (y). a. Compute the constant of the spring (g/cm). b. We have = kL; Interpret this equation and compare with the curve. c. Compute the area between the load-elongation curve and the z-axis. What does this represent? The electrical resistance of lead at various temperatures indicated by means of a plot that the resistance would be zero at 273 C. The recent experiments of Professor Kamerlingh Onnes have justified this conclusion. CHAPTER VII NEGLIGIBILITY Importance. It is of the utmost importance that stu- dents in laboratory courses should come to know under just what conditions they are at liberty to neglect small quantities which appear in their work. A student who comes into physics from courses in mathematics is con- scious of a decided shock when he is told to throw away certain terms in an equation. It is hoped that a few illus- trations will convince him that such a process is not at all unscientific or inaccurate. We learn in trigonometry that the sine, tangent, and radian value of an angle may be used interchangeably if the angle is small. The following table will make this clear: Degrees. Radians. Sine. Tangent. 30' 0.00873 0.00873 0.00873 1 0.01745 0.01745 0.01746 2 0.03491 0.03490 0.03492 3 0.05236 0.05234 0.05241 4 0.06981 0.06976 0.06993 5 0.08727 0.08716 0.08749 62 NEGLIGIBILITY 63 It will be seen that for most purposes very little error would result from substituting the radian measure or the tangent for the sine. EXERCISES 1. Construct a circle with radius unity. Take an angle of 60 at the center and draw an arc and a chord. Call the arc dx and the chord dy (equal to the radius). Find the value of the arc by radians -^-, and the chord by the law of sines. In this case dy=l and dx = 1.0472. Find relation between dx and dy for 30, 10, 5, and 1. 2. What is the largest angle for which dx = dy to four places of decimals? The Pendulum Formula. In deducing the formula for the simple pendulum, we make use of an approximation, so that the equation T=2irVl/g is not quite correct. The equation tends to diminish the error as more and more terms are a used. Here K = -. EXERCISE Give e values of 60, 10, and 1 and assume that the longer formula gives the correct value of g. Find the error in each case due to using the shorter formula. Re- member that g appears as a square root. 64 THEORY OF MEASUREMENTS The Mirror Formula. In developing the formula for the circular mirror a similar approximation is noted. By using a parabolic mirror no approximation appears. EXERCISES 1. Show that light starting from the focus of a para- boloid of revolution will go in parallel lines upon reflection from any part of the mirror. 2. Construct a parabola whose equation is y 2 = 4px. Construct a circle internally tangent to this whose equa- tion is x 2 -\-y 2 2rx=0. Let r = 2p. Select points along the axis distant r/4, r/2, 3r/4, and r from the origin. Find the value of y on the circle and the corresponding value of y on the parabola in each case. The relation of these values may be used to measure the error due to the approximation. 3. Find the angular aperture at the center of the circle for each point taken. Approximate Squares and Square Roots. There is a short method of squaring numbers and extracting their square roots which involves an approximation. Let us take the number 1.01 and consider it made up of 1.0+0.01. If we square this by use of the binomial theorem, we have 1+0.02+0.0001 = 1.0201. This is nearly equal to 1.02. If, therefore, we square our first term and add twice the second term, we have a rule for squaring such numbers as these. The square of 1.0019 is 1.0038036, to which 1.0038 is a close approxi- NEGLIGIBILITY 65 mation. If the whole number is any other than unity, its value must be multiplied into the smaller term. By a similar process we may find the square root of 1.04 to be 1.02, and that of 4.04 to be 2.01. The Slide Rule The slide rule is an instrument that gives approximate values and well illustrates the principle of negligibility. The following table shows its accuracy for certain processes: Problem. Solution by Slide Rule. Solution by Five Place Logs. Deviation, % 152X391X31 1842000 1842400 0.022 86.4X0.028X4.95 11.97 11.975 0.042 496 -r- 381 1.303 1.3018 0.21 292-7-468 0.624 0.62394 0.063 25X0.07X1.151 0.00543 0.0054327 0.05 378X0.98 1 The Value of TT. Much energy has been expended in obtaining values of TT correct to a large number of decimal places. It has been carried out to 707 places. (See Seven Follies of Science by Phin.) If TT is carried out to six or seven places, it is sufficient for all purposes. The value 3^ will answer for the majority of cases. It will prove convenient to remember that 7r 2 = 9.87 to a very close approximation. 66 THEORY OF MEASUREMENTS PROBLEMS 1. Find the error due to using the value 3^ carried out to four decimal places, for TT. 2. Find the error due to using the value 9.87 for 7r 2 , when TT is carried out to four places. 3. We are accustomed to assume in physics that the coefficient of cubical expansion of a solid is three times that of the linear expansion. What error does this involve when a bar of brass, 100 cm. long at C., is heated to 100 C.? The error involved in using a mirror and scale (Poggen- dorff's method) is discussed in Stewart and Gee's Practical Physics, Vol. I, p. 55. CRITERIA FOR NEGLIBILITY It often happens that in a series of measurements of the same quantity, there are one or more values which do not compare well with the others. The question arises, what shall be done with these observations? They seem to have been taken with equal care with the others, and there seems to be no particular reason for rejecting them. Several criteria have been proposed for testing such obser- vations. The Huge Error. We find the mean and average deviation (a.d.) omitting the doubtful observation; then find the difference between the doubtful observation and the mean, and if this equals or is greater than 4 a.d., it should be rejected. Suppose we have a set of measurements: 10, 9, 10, 9, 12. The last measurement is questionable. The mean NEGLIGIBILITY 67 is 9.5. a.d.=0.5. 12-9.5 = 2.5. 4X0.5 = 2.0. The meas- urement should be rejected. PROBLEM The following length measurements (centimeters) have been taken: 60.1, 60.2, 60.3, 59.9, 58.3, 63.1. Apply the method to the last two results. Chauvenet's Criterion. This is a more elaborate method. We may let t stand for the ratio between the limiting error (x) and the probable error of a single observa- tion (r). If there are n errors, we may assume that nP will be the number less than x, and nnP the number greater than x. Then, by definition 2n-l 2n ' We have 2 f* =T= I e~*dt. (See page 15.) VTT^ If we equate these two values of P and solve, we may find a value of t corresponding to any value of n. A few common values of n are given: n t n t n l 3 2.05 7 2.67 11 2.96 4 2.27 8 2.76 12 3.02 5 2.44 9 2.84 13 3.07 6 2.57 10 2.91 14 3.12 68 THEORY OF MEASUREMENTS Since the limiting error, x = tr, we have only to find these two values and compare. Let us take the following data: 12, 13, 12, 13, 13, 15. The mean is 13 and S^ = 6. r = 0.6745 \,= 0.6745 x /f = 0.74. \ 71 1 \ O For n = 6, =2.57, x = tr = l&. The deviation of the term in question is 2.0 and it should be rejected. PROBLEM The following observations were taken with a transit, 43 52' 26".4 28 .5 27 .8 28 .0 31 .3 30 .2 27 .9 Test such observations as may be necessary by Chau- venet's Criterion. Criterion for Negligibility for Deviation in Components When the deviation in any component contributes but little to the deviation in the result, it is sometimes proper to neglect it altogether. We know that the error in the final result, . . . A B 2 . NEGLIGIBILITY 69 Suppose A 2 is the quantity under investigation. Let . . . A n 2 , where A2 2 has been omitted. A Ao = the reduction in the deviation in the result due to omitting A2. We may assume this value to be any- thing we please, depending upon the accuracy we require. If we insist, as some authors suggest, that it be equal to or less than T VA, we have A f0.9A. But A 2 2 = A 2 -A 2 = A 2 (1-0.9 2 )=0.19A 2 . A 2 = 0.43A. We may thus neglect a deviation if it contributes less than 0.43 of the total deviation of the result. A more rigid criterion could, of course, be determined by substituting a smaller value for the T V Significant Figures. This is a subject that may be properly treated under the head of negligibility. At each stage in a series of measurements the student should look over his work and eliminate such figures as lend nothing to its precision. A rather full treatment of this subject is given in Holman's Precision of Measurements, to which the student is referred. The following rules of precision are given: 1. When a rejected figure is five or over, increase the previous figure by one. If we decide to drop the 7 in 14.637, we should write it 14.64. 70 THEORY OF MEASUREMENTS 2. In the deviation measure, we should retain two significant figures. Thus a.d. = 0.062 ; r = 1 .6. 3. In our measured quantity, we should retain places corresponding to the second significant figure in the devia- tion measure. m = 368.7310.21 becomes 368.73, ra = 406.672.6 becomes 406.7. 4. In adding quantities, retain such places as corre- spond to the number having the largest deviation measure. Quantity. a.d. Result. 374.20 0.36 374.20 4768.121 0.012 4768.12 984.1698 0.0021 984.17 6126.49 5. In multiplication and division, find the quantity whose - - is the greatest. From this find the percentage precision and if this is 1 per cent or more, use four significant figures; if between 1 and 0.1 per cent use five significant figures; if between 0.1 and 0.01 per cent, use six significant figures. If we have to multiply 41.60.4 by 590.250.06, we note that is greater for the first than for the second. m As this is less than 1 per cent, we should retain five sig- nificant figures, giving 24556 as a result. NEGLIGIBILITY 71 PROBLEMS 1. We have given the number 504.628 with the fol- lowing deviation measures: a.d. = 0.21; r = 0.031; cor- rect to one part in one hundred; correct to two per cent. What is the proper expression for the number in each case? 2. Add 21.42 dhO.61 338. 161 0.042 543.1 1.5 3. Multiply 630.450.62 by 25.635 0.024. CHAPTER VIII EMPIRICAL FORMULAE AND CONSTANTS Definitions. A mathematical formula is one that is de- duced by a process of reasoning along mathematical lines. The formula for the distance passed over ;by a falling body is an illustration. S = v t+%at 2 is derived from certain definitions of velocity and acceleration, to which mathematical processes have been applied. An empirical formula cannot be derived in this man- ner, but depends upon the results of experiments which are treated in a manner to be described. The following is a typical empirical formula: = 980.6056-2.5028 cos 2Z-O.OOOOOS*. This has been made up from a large number of measure- ments of g under various conditions of latitude (I) and altitude (h). In order to explain the method employed in constructing empirical formulae, let us take an experiment illustrating the relation of the space passed over by a falling body, to the time of fall. 72 EMPIRICAL FORMULAE AND CONSTANTS 73 The following may be assumed to be the results of the experiment: t 9 1 1 2 4 3 9 4 16 The obvious relation existing between t and s will be disregarded, and we will endeavor to find an empirical FIG. 8. formula which fits the conditions. Our first step is to plot a curve. This is seen to have the characteristics of a parabola and we write the general equation, y = S+Tx+Ux 2 +, etc. x has been substituted for t, and y for s. From this are found observation equations: 74 THEOKY OF MEASUREMENTS 1=S+T+U The normal equations are: 30=4S+10!T+30t7 100 = 10/S+307M- 100*7 When these equations are solved, we have and our equation becomes y=x 2 . If the experimenter had used only three seconds and had recorded an 8 instead of a 9, the values would have been 17 = 0.5 Giving y= - 1+1.5* -f0.5z 2 . Upon repeating the experiment as the space measurement approaches closer to 9 this formula approaches the value !/=**. EMPIRICAL FORMULAE AND CONSTANTS 75 While no one would think of actually applying this method to so simple a case, it affords a good illustration of the manner in which all empirical formulae are con- structed. Classes of Curves. The following are some of the common curves from which these formulae may be con- structed, together with their formulae and illustrations: 1. Straight line. x = y. 2. Parabolic. y = S+Tx+Ux +, etc. 3. Cyclic. y = S+T sin x+T r cos x+, etc. 4. Logarithmic. y = be ax . 5. Hyperbolic. xy = a. The formula for velocity due to gravity, v = gt, illus- trates the first class. The space passed over by a falling body, the change in velocity of a river below its surface, and the growth of the United States in population, illustrate the second class. Cyclic curves may be used to represent the rise and fall of temperature, pressure, and humidity through a given interval. The logarithmic curve represents plots made from Newton's law of cooling, the absorption of light for varying thicknesses, and the gain due to com- pound interest. A familiar example of a hyperbolic curve is afforded by Boyle's law, pv = ct. This will plot a rec- tangular hyperbola. Rules. The method of procedure may be summarized in the following rules: 1. Write the observation equation. 2. Plot the curve. 76 THEORY OF MEASUREMENTS 3. Identify the curve and write its equation. 4. Form the normal equations. 5. Solve for the unknowns, and these give the values of the constants sought. It will be observed that every additional observation taken alters the values of the constants. EXERCISES AND PROBLEMS For the straight line and rectangular hyperbola an inspection of the data is generally sufficient to determine the value of the constant. For the parabola the following examples should be studied: 1. Vertical velocity curve on the Mississippi river. Depth. Velocity. . foot 3 . 1950 feet per second 0.1 3.2299 0.2 3.2532 0.3 3.2611 0.4 3.2516 0.5 3.2282 0.6 3.1807 0.7 3.1266 0.8 3.0594 0.9 2.9759 The curve is obviously a parabola. EMPIRICAL FORMULAE AND CONSTANTS 77 The first two observation equations are Write down the remaining eight observation equations, form the normals, and solve. The results are S = 3.195 13 7=0.44253 U =-0.7653, giving the formula F = 3.19513+0.44253X-0.7653X 2 . FIG. 9. It will prove of interest to determine the velocity for each depth and compare with that obtained by experiment. (Why should they fail to check?) 2. The growth in the population of the United States is a good illustration of this curve, although from the nature of the case the results are not very reliable. In Popular Science Monthly for April, 1910, page 382, a set of curves are drawn for various countries. It will be seen that Sweden and Norway, Turkey, Spain, and Italy may be fairly rep- resented by straight lines. The United States shows an 78 THEORY OF MEASUREMENTS easily recognized parabola. In the article referred to the complete solution of the census problem is given. 3. The following illustrates a cyclic curve: e Y e Y 20 200 14 20 38 220 -13 40 51 240 -24 60 64 260 -30 80 68 280 -28 100 70 300 -24 120 64 320 -13 140 53 340 - 2 160 16 360 20 180 20 Substitute 6 for ~# in the equation and plot the curve. m Use the first three terms and form an observation equation for each value of 9. We have for the first four observa- tion equations: 20=/S-r-T r sin sin 20+!T' cos 20 sin 40+r cos 40 sin 60+r cos 60 Normal equations for S, T, and T' are formed from these and their solution supplies the values of the constants in the formula. 4. We may use Winkehnann's data showing the rela- EMPIRICAL FORMULAE AND CONSTANTS 79 tion between the temperature of a cooling body at differ- ent times, as an illustration of a logarithmic curve. o ti -h 18.9 3.45 X6.9 10.85 14.9 19.30 12.9 28.8(0 10.9 40.10 8.9 53.75 6.9 >0.95 The logarithmic relations of these quantities is not obvious by direct inspection. It is reached by the fol- lowing process: We assume that the ratio at which a body loses heat ( dQ) is proportional to the difference between its temperature and that of its surroundings. This is ex- pressed by By definition of specific heat Substitute this in the equation above dQ k -=a0, where a=- and 00=0. From this we get by integrating log b log 0=at. -THEORY OF MEASUREMENTS Now if 0i represents temperature at time ti, and 62 tem- perature at time fe, we have log 6 log 2 = afe; 1 A From which a = - log =-> Take 02 = 19.9 and substitute the values of for 0i and show that a is a constant, approximately equal to 0.0065. Such an equation as ~"Jt =a ^t where the change in a quantity is proportional to the quantity itself, is called a " compound interest equation/' INDEX Adjustment of Observations 22 Approximate Squares and Square Roots 64 Arithmetical Mean 17 Average Deviation 38 Best Magnitudes and Best Ratios 51 Chauvenet's Criterion 67 Constant Interval 17 Constants in an Equation 56 Empirical Formulae and Constants 72 Errors Classes of 3 Fractional Method 50 Huge Error 66 Least Squares 16 Mean Square Error 37 Measurements Classes of 2 Negligibility 62 Normal Equations 24 Plotting 54 Precision of Observations 36 Probable Error 38 Probability 4 Probability Curve 9 Probability Integral 14 Propagation of Errors 44 Propagation of Errors Converse Problem 49 Residuals 12 Short Methods of Adjustment 32 Significant Figures 69 Slide Rule 65 Value of T 65 Weighting 18 81 D. VAN NOSTRAND COMPANY 25 PARK PLACE NEW YORK SHORT-TITLE CATALOG OF OP SCIENTIFIC AND ENGINEERING BOOKS This list includes the technical publications of the following English publishers : SCOTT, GREENWOOD & CO. JAMES MUNRO & CO.. Ltd. CONSTABLE & COMPANY, Ltd. TECHNICAL PUBLISHING CO. ELECTRICIAN PRINTING & PUBLISHING CO., for whom D. Van Nostrand Company are American agents. Descriptive Circulars sent on request. JUNE 1916 SHORT=TITLE CATALOG OF THE Publications and Importations OP D. VAN NOSTRAND COMPANY 25 PARK PLACE Prices marked with an asterisk (*) are NET All bindings are in cloth unless otherwise noted Abbott, A. V. The Electrical Transmission of Energy 8vo, *$5 oo A Treatise on Fuel. (Science Series No. 9.) i6mo, o 50 Testing Machines. (Science Series No. 74.) i6mo, o 50 Adam, P. Practical Bookbinding. Trans, by T. E. Maw.iamo, *2 50 Adams, H. Theory and Practice in Designing 8vo, *2 50 Adams, H. C. Sewage of Seacoast Towns 8vo, *2 oo Adams, J. W. 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