UC-NRLF 
 
- 
 
ELEMENTS 
 
 OF THE 
 
 PRECISION OF MEASUREMENTS 
 
 AND 
 
 GRAPHICAL METHODS 
 
 BY 
 
 H. M. GOODWIN, PH.D. 
 
 PROFESSOR OF PHYSICS 
 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
 
 McGRAW-HILL BOOK COMPANY 
 
 239 WEST 39-TH STREET, NEW YORK 
 
 6 Bouverie Street, London, E.C. 
 
 1913 
 
Engineering 
 Library 
 
 COPYRIGHT, 1908 
 
 REVISED, EXTENDED, AND COPYRIGHTED, 1913 
 BY H. M. GOODWIN 
 
 All rights reserved 
 
 PRESS OF GEO. H. ELLIS CO., BOSTOM 
 
PREFACE. 
 
 In its present form the "Elements of the Precision of Measure- 
 ments and Graphical Methods" represents the ground covered in a 
 brief course which has been given for a number of years at the Massa- 
 chusetts Institute of Technology to all students in connection with 
 their work in the Physical Laboratory. The author has been induced 
 to amplify the printed "Notes" on this subject and give them a wider 
 circulation in response to repeated requests to use them elsewhere. 
 Although prepared primarily to meet the needs of his own classes, it 
 is hoped, in the present form, they may prove useful in other tech- 
 nical schools and colleges where quantitative work forms a part of 
 the curriculum, and also to engineers whose work involves experi- 
 mental testing. In many laboratories far too little weight is attached 
 to the discussion of the magnitude and effect of sources of error 
 on a result. This has been forced upon the writer's attention as the 
 result of personal interviews with hundreds of graduate students 
 entering the Institute, who apply for excuse from laboratory work. 
 It is the exceptional student who has any conception how to figure 
 out the precision of a final computed result from the precision 
 of his individual measurements, and this is true even though his 
 laboratory note-book shows his work to have been carefully and credi- 
 tably performed. It is the author's firm conviction that one of the 
 most valuable and enduring benefits of physical laboratory training 
 to a student of Science or Engineering is the acquisition of the proper 
 view-point with which to approach an investigation, be it either 
 purely scientific or technical; that is, the ability to recognize the 
 essentials of a problem at the outset, so as to economize both time 
 and labor in its solution. Although the exercise of judgment, based 
 upon the personal experience of the investigator, is essential to the 
 "best solution" of any experimental problem, still it is desirable to 
 direct the student's attention to precision methods at an early stage 
 of his laboratory work. Experience has shown that this may be satis- 
 factorily done as soon as he has had a little practice in exact measure- 
 ments and can handle the elements of Differential Calculus. At the 
 Institute the course is given at the middle of the sophomore year, 
 after the student has performed some six or eight experiments on 
 fundamental measurements in Mechanics. Continued application of 
 the principles is then made in subsequent laboratory work throughout 
 the junior and senior years, and a precision discussion is regarded as 
 
 267862 
 
PREFACE 
 
 an important feature of the final thesis. It has been the writer's 
 experience that students have little trouble in understanding the gen- 
 eral principles involved, but meet with considerable difficulty in 
 applying these principles to concrete problems. For this reason the 
 subject is most satisfactorily taught to small sections by recitations 
 based on the text and the solution otnumerous problems selected from 
 the book and from the current laboratory work. A close correlation 
 of class-room and laboratory work is indeed highly desirable, and in 
 the Rogers Laboratory of Physics it is the practice to require with 
 each laboratory report a precision discussion of the data or a solution 
 of some precision problem related to the experiment. The laboratory 
 manuals have been written with this in view. 
 
 The method of treatment has been kept as brief as possible. A 
 full discussion of the subject, with proofs based on the Theory of 
 Probability and the Method of Least Squares, would so enlarge the 
 work as to defeat its end. Proofs of the few theorems and formula 
 which the student is asked to assume may be found in any good treatise 
 on Least Squares. An excellent treatment is that given in Bartlett's 
 "Method of Least Squares." A more exhaustive treatment of Pre- 
 cision Methods- may be found in Holman's "Precision of Measure- 
 ments." 
 
 A chapter on the solution of illustrative problems has been added 
 to assist students who find it necessary to work up the subject by 
 themselves. The collection of problems has been compiled from 
 recent examination papers. The chapter on Graphical Methods con- 
 tains specific directions for constructing graphs, and general directions 
 for obtaining therefrom the functional relationship between two 
 variables. For engineering students, as well as physicists, the method 
 of logarithmic plotting will be found of wide application. In the 
 Appendix several tables, of assistance in precision computations, 
 have been added. 
 
 In conclusion the author desires to express his indebtedness for 
 many suggestions to his colleagues who have so ably assisted him in 
 the instruction of this subject in recent years, and in particular to 
 Professor William J. Drisko, whose experience in teaching this and 
 related subjects has been most helpful. 
 
 H. M. GOODWIN. 
 
CONTENTS. 
 
 PART I. 
 PRECISION OF MEASUREMENTS. 
 
 Classification of Measurements 7 
 
 Precision Discussion of Direct Measurements 7 
 
 Determinate Errors 8 
 
 Indeterminate Errors 11 
 
 Law of Error 13 
 
 Method of Least Squares 14 
 
 Deviation and Precision Measures 15 
 
 Average Deviation of a Single Observation 16 
 
 Deviation of the Mean 16 
 
 Fractional and Percentage Deviation 17 
 
 Deviation vs. Precision Measures 17 
 
 Probable Error v . 18 
 
 Mean Error 19 
 
 Weights and Weighted Mean 20 
 
 Criterion for Rejecting Observations 20 
 
 Computation Rules and Significant Figures 21 
 
 Precision Discussion of Indirect Measurements 25 
 
 Separate Effects 27 
 
 Resultant Effects 29 
 
 Criterion for Negligibility of an Error 30 
 
 Equal Effects 32 
 
 Fractional Method of Solution . 33 
 
 PART II. 
 
 GRAPHICAL METHODS. 
 
 Nature of Problems 41 
 
 Procedure in Plotting 41 
 
 Determination of Constants of a Straight Line 46 
 
 Rectification of Curved Lines 47 
 
 Trigonometric Functions 48 
 
 Reciprocal Functions 50 
 
6 CONTENTS 
 
 PAGE 
 
 The Logarithmic Method. Exponential Functions; y = mxn . ... 52 
 Special Cases: 
 
 y = m(x + p)"> 58 
 
 y = m IQn*; y = menx 59 
 
 Precision of Plotting 59 
 
 Residual Plots , 60 
 
 Interpolation Formulae ' 63 
 
 Graphical Solution 64 
 
 Least Square Solution 64 
 
 PART III. 
 PROBLEMS. 
 
 Solution of Illustrative Problems 69 
 
 Problems 81 
 
 APPENDIX. 
 
 Table I. Mathematical Constants 99 
 
 Table II. Approximation Formulae 100 
 
 Table III. Squares, Cubes, and Reciprocals 101 
 
 Table IV. Four Place Logarithms 102 
 
 Table V. Sines, Cosines, and Tangents 104 
 
PRECISION OF MEASUREMENTS. 
 
 Classification of Physical Measurements. All physical 
 measurements may be classed as direct or indirect ac- 
 cording as the measurement gives the desired result di- 
 rectly, or as the result is obtained by combining the re- 
 sults of several measurements by means of some formula. 
 Examples of the first class are the measurement of a length 
 by means of a scale, the mass of a body by means of an equal 
 arm balance, and the electrical resistance of a wire by the 
 direct method of substitution. Examples of indirect meas- 
 urements are the determination of g, the acceleration due to 
 gravity, by means of a pendulum, involving the measure- 
 ment of the length and time of vibration of the pendulum, 
 the determination of the index of refraction of a substance 
 from measurements of the angle and the minimum deviation 
 of a prism by means of a spectrometer, and the determina- 
 tion of the specific heat of a substance by the method of 
 mixtures hi which the results of the measurement of a num- 
 ber of temperatures and weights are combined. The great 
 majority of problems arising hi practice come under the 
 second class. 
 
 Reliability of a Result. In order that the result of any 
 measurement, whether direct or indirect, may be of any 
 scientific or technical value, it is necessary to have some 
 numerical estimate or measure of its reliability. The im- 
 portance of such a measure cannot be overestimated. The 
 result of a test, of a study of an instrument or method, or 
 of the determination of a constant, may be rendered almost 
 worthless, unless the investigator is able to state the degree 
 of reliance which can be placed upon it. This phase of 
 an investigation should be kept constantly hi mind in all 
 laboratory work. The student's ability to intelligently dis- 
 
8 PRECISION OF MEASUREMENTS 
 
 cuss the reliability of his data is regarded as of no less im- 
 portance than his ability to perform accurate work. 
 
 By the precision or precision measure of a result, denoted 
 for brevity by p.m., will be always understood the best nu- 
 merical measure of its reliability which can be obtained after 
 all known sources of error have t>een eliminated or corrected 
 for. How this may be computed will be explained below. 
 By the accuracy of a result should, strictly speaking, be un- 
 derstood the degree of concordance between it and the true 
 value of the quantity measured. Since, however, the latter 
 is usually unknown, it is seldom that we can obtain a numeri- 
 cal measure of the absolute accuracy of a measurement. We 
 must in most cases be content with an estimated or computed 
 precision measure. The terms "accuracy" and "precision" 
 are often carelessly used indiscriminately. 
 
 The precision measure of a direct measurement is of no 
 less importance than of an indirect measurement. As the 
 precision of the latter depends primarily on the precision of 
 the separate components from which it is computed, the 
 method of determining a numerical estimate of the reliability 
 of a series of direct observations w r ill first be considered. 
 
 Classification of Errors. When any quantity is measured 
 to the full precision of which the instrument or method em- 
 ployed is capable, it will in general be found that the results 
 of repeated measurements do not exactly agree. This is 
 true not only of results obtained by different observers using 
 different instruments and methods, but also when the meas- 
 urements are made by the same observer under similar con- 
 ditions. The cause of these discrepancies lies in various 
 sources of error to which all experimental data are subject. 
 These may be grouped conveniently in two classes, de- 
 terminate and indeterminate errors. 
 
 Determinate Errors. Determinate errors are, as their 
 name indicates, of such a nature that their value can be de- 
 termined and their effect on the result thereby eliminated. 
 They may be classified as follows: 
 
 a. Instrumental Errors. These may arise from poor con- 
 
DETERMINATE ERRORS 9 
 
 struct ion or faulty adjustment of an instrument, as, for ex- 
 ample, a defect in a micrometer screw, faulty graduations 
 of scales and circles, eccentricity of circles, unequal balance 
 arms, etc. 
 
 b. Personal Errors. These may arise from characteristic 
 peculiarities of individual observers, as, for example, the 
 tendency to always record the occurrence of an event too 
 soon or too late. This frequently happens in recording 
 transit observations in which the "personal equation" of the 
 observer becomes an important factor. 
 
 c. Errors of Method or Theoretical Errors. These may 
 arise from using an instrument under conditions for which 
 its graduations are not standard. 
 
 The following illustrations will make clearer the nature of 
 the above sources of error. Suppose that the arms of a 
 chemical balance are slightly unequal in length. All weigh- 
 ings made with such a balance will be in error due to this 
 cause (if the balance be used in the ordinary way), by an 
 amount depending on the inequality in the length of the 
 arms. Repeated weighings of the same substance on the 
 same balance by the same method will, however, give no 
 indication of the presence of this source of error. The re- 
 peated independent weighings may indeed check among 
 themselves to the full sensitiveness of the balance, and yet 
 the result may be in error, due to the constant instrumental 
 error, by a very large amount. The presence of such an error 
 would only be detected by comparing the results of the weight 
 of the same body obtained on different balances or by differ- 
 ent methods of weighing, for the probability of the same in- 
 strumental error occurring to the same extent in different 
 instruments' is very small. 
 
 Again, suppose a length is measured by means of a grad- 
 uated scale at 20 C., while the scale is standard at some 
 other temperature, say C. Repeated measurements with 
 such a scale by the same method and under the same condi- 
 tions would probably show a very close agreement among 
 themselves, and give no clew to the presence of any constant 
 
10 PRECISION OF MEASUREMENTS 
 
 error. The result would, however, be too small, since the 
 value of the units of the scale would all be too large, due to 
 the expansion of the scale from to 20. The error thus 
 introduced by using the scale under conditions other than 
 those for which it is standard is, however, determinate in its 
 nature, since a knowledge of the coefficient of expansion of 
 the scale and of the temperature at which it is standard, 
 and also at which it is used, furnishes all necessary data, 
 for reducing the observed result to the value it would 
 have had, had the scale been standard at the time of the 
 measurement. The concordance of a series of observations 
 taken under similar conditions is, therefore, no criterion of 
 the absence of constant errors even of very large amount. 
 
 To detect and eliminate such errors, it is necessary to com- 
 pare the results of measurements of the quantity by different 
 methods, different apparatus, and, if possible, different ob- 
 servers, and to average such independent results by a special 
 method described later; for the probability of the same source 
 of error being present under such variable conditions is very 
 small. An interesting illustration of the presence of a con- 
 stant error escaping detection is to be found in the original 
 determination of the ohm by the British Association Com- 
 mittee. The excellent agreement of the observations among 
 themselves lead to the conclusion that the result possessed a 
 high degree of reliability. Later determinations by inde- 
 pendent methods and observers gave values which differed 
 from the B. A. value by over one per cent., an amount far 
 hi excess of the precision with which the B. A. determination 
 had been carried out. Attention was thus called to the 
 probable presence of some constant error which further in- 
 vestigation verified. 
 
 Residual Errors. After a result has been corrected as well 
 as may be for all known sources of determinate errors, there 
 may still remain in it small errors, the value of which cannot 
 be determined, and which, therefore, fall into the second 
 general class of errors, indeterminate errors. Thus, if the 
 instrumental error arising from inequality in the length of 
 
INDETERMINATE ERRORS 11 
 
 balance arms be corrected by a determination of the ratio 
 of the arms, this ratio will be known to only a certain degree 
 of precision, and hence the corrected result of a weighing 
 may still be in error by an amount depending on the preci- 
 sion with which the correction itself has been determined. 
 Or, again, correcting for the expansion of a scale involves an 
 experimental investigation of the coefficient of expansion of 
 the material of which the scale is constructed, and this con- 
 stant can be determined with only a certain degree of pre- 
 cision. A result corrected by means of this coefficient will, 
 therefore, still be uncertain beyond a certain point due to 
 the uncertainty in the value of the coefficient used. These 
 small errors remaining, because of the impossibility of com- 
 pletely correcting for constant errors, are called residual 
 errors: then- numerical value and algebraic sign cannot be 
 determined, but usually limiting values may be estimated 
 and assigned to them. For this reason they are properly 
 grouped and treated under the second general class of errors 
 mentioned, indeterminate errors. 
 
 Indeterminate Errors. Accidental; Residual. Experience 
 shows that, when a measurement is repeated a number 
 of times with the same instrument and by the same ob- 
 server under apparently the same conditions, the results 
 usually differ in the last place or sometimes last two 
 places of figures. Thus in so simple a measurement as the 
 determination of the distance between two lines with a scale 
 graduated in millimeters, successive measurements will not 
 agree to one-tenth millimeter if fractions of a millimeter are 
 estimated by the eye. Errors which give rise to such varia- 
 tions which at one time cause a result to be too high and 
 at another too low are due to causes over which the observer 
 has no control, such as sudden temperature fluctuations which 
 may give rise to unequal expansion of different parts of an 
 apparatus, or to changes in refraction, barometric changes, 
 shaking of the instrument due to mechanical jar or to the 
 wind, etc.; and, more important still, to physiological causes 
 arising from imperfections or fatigue of the eye or ear 
 
12 PRECISION OF MEASUREMENTS 
 
 of the observer. The magnitude and sign of errors arising 
 from such causes have been shown, however, to follow 
 a perfectly definite law, namely, the law of chance. The 
 nature of this law may be illustrated as follows. Suppose a 
 thousand shots be fired at a target by a skilled marksman 
 under conditions as nearly alike as possible. Experience 
 shows that the shots will be distributed in a manner which 
 at first sight seems entirely irregular, but which on more 
 careful examination will be found to be approximately in 
 conformity with a perfectly definite law. In an actual case 
 obtained with a target ruled in horizontal sections by lines 
 one foot apart, the centre line (corresponding to the bull's 
 eye) being in the middle of one of these spaces, the follow- 
 ing results were obtained: 
 
 In space No. of shots 
 
 1 
 4 
 10 
 89 
 190 
 212 
 204 
 193 
 79 
 16 
 2 
 
 If a plot be made with the number of shots falling hi the 
 several sections as ordinates and the distance of the corre- 
 sponding spaces from the central line as abscissae, we obtain 
 Figure 1. From this it appears that plus and minus devia- 
 tions of the shots from the central line are about equally 
 frequent, and that small deviations occur with much greater 
 frequency than large ones. If the number of shots (corre- 
 sponding to observations) be increased, the irregularities pres- 
 ent in the curve will tend to smooth out, and it can be 
 shown mathematically that in the limit the curve represent- 
 
 + 54 
 
 to +4* 
 
 + 44 
 
 " + 34 
 
 + 3* 
 
 " + 24 
 
 + 24 
 
 " +14 
 
 + 14 
 
 " + 4 
 
 + 4 
 
 " - 4 
 
 2 
 
 "-14 
 
 -14 
 
 " 24 
 
 -24 
 
 " 34 
 
 -34 
 
 "-44 
 
 -*4 
 
 "-54 
 
CURVE OF ERROR 
 
 13 
 
 ing the law of chance takes the general form shown in Fig- 
 ure 2, the equation of which is 
 
 Here y is the frequency of the occurrence of an error of the 
 magnitude z, and h is a constant, the value of which depends 
 on the character of the observations and which affords a 
 measure of their precision. The curve represented by this 
 
14 PRECISION OF MEASUREMENTS 
 
 equation is called the Curve of Error. By inspection it is 
 seen that: 
 
 First. Small errors occur more frequently than large ones 
 (curve of error has a maximum for x = o) ; 
 
 Second. Very large -errors are, unlikely to occur (curve is 
 asymptotic to the axis of X) ; 
 
 Third. Positive and negative errors of the same numerical 
 magnitude are equally likely to occur (curve of error is sym- 
 metrical with respect to axis of Y). 
 
 Since accidental and residual errors of a series of observa- 
 tions follow the law of chance, they may be properly subjected 
 to mathematical treatment based on this law. It must be 
 remembered, however, that, since the law itself represents 
 a limiting case, corresponding to an infinite number of ob- 
 servations, deductions from it apply to a finite number of 
 observations only with a certain probability which becomes 
 less the smaller the number of observations. 
 
 The Method of Least Squares. As already pointed out, 
 in the great majority of measurements the true value of the 
 quantity is unknown and cannot be determined. Were it 
 known, a measurement would be superfluous. All that we 
 can hope to obtain from our experimental data is the most 
 probable value of the quantity or quantities in question. 
 In many cases this is a simple matter; but in others, where 
 the number of observations is larger than the number of 
 unknown quantities to be determined, the problem may 
 become one of some difficulty. The branch of mathematics 
 which treats of the general problem of the adjustment of 
 errors of observation so that their effect upon the result is 
 reduced to a minimum, and the best representative values 
 of the desired quantities thus obtained, is called Least Squares, 
 the name being derived from the criterion upon which the 
 adjustment of the observations is based. This states that 
 the most probable values of a series of related observations 
 arc those for which the sum of the squares of the errors is 
 a minimum. Certain deductions from the theory of Least 
 Squares will be assumed as demonstrated in the course of 
 
ARITHMETICAL MEAN 15 
 
 this work. For the proofs the student is referred to Bart- 
 let t's Method of Least Squares or other treatises on the 
 subject. An illustration of the method as applied to the 
 computation of the constants of an empirical equation is 
 given under Graphical Methods. 
 
 The Arithmetical Mean. Deviation Measures. We will 
 now consider the precision discussion of a series of direct 
 measurements. Let a 1} a 2 , . . . a n be a series of observations 
 on a quantity, all of which possess an equal degree of prob- 
 ability. Under these conditions the most probable value 
 of the quantity is given by the arithmetical mean, m, of the 
 series, i.e. 
 
 
 (1) 
 
 Since the .true value of the quantity is unknown, the error 
 of each of the observations and of the mean, m, cannot be 
 determined. We can, however, obtain a numerical measure 
 of the amount by which each observation differs from the 
 mean value, and from this the probable deviation of the mean 
 can be computed. The difference between the value of any 
 observation of a series and the mean value of the series is 
 called the deviation of that observation from the mean. It 
 is to be distinguished from the absolute error of the obser- 
 vation, i.e., the difference between the observed value and 
 its true value, from which it may differ widely. Deviations 
 as thus computed follow the same law as indeterminate errors, 
 i.e., the law of chance, and are subject, therefore, to the 
 same mathematical treatment. They give a measure of 
 the magnitude of the accidental error of a measurement, but 
 evidently afford no indication of the presence or magnitude 
 of any constant errors which may be present. 
 
 If the numerical deviations d^ d%, d B , ... d n , be computed 
 for any series of observations as above, their algebraic sum 
 will be zero, since the sum of the positive deviations is equal 
 to the sum of the negative deviations. If, however, their 
 arithmetical mean be computed, disregarding their sign, the 
 
16 PRECISION OF MEASUREMENTS 
 
 result will be a number which expresses how much on the 
 average any single observation of the series taken at random 
 is likely to differ (plus or minus) from the mean, m. This 
 average value 
 
 n 
 
 (2) 
 
 is called the average deviation of a single observation, and will 
 be denoted by a.d. In recording data as o|, 0%, . . . a n , space 
 should always be left for computing the deviations d i} d%, 
 . . . d n , respectively, and their a.d. as follows: 
 
 a-i m d 1 
 m d 
 
 w== a.d.=~ 
 
 Looked at from another point of view, an a.d. is a numerical 
 measure of the amount by which a new observation taken 
 under the same conditions as before is likely to differ from 
 the mean value, m. It gives a numerical measure of the 
 reliability of any single observation of the series so far 
 as accidental errors affecting the measurement are con- 
 cerned. 
 
 Deviation of the Mean, A.D. In general, however, it is 
 the reliability of the mean that we desire to know rather 
 than that of a single observation. As the mean has a higher 
 degree of probability than any single observation from which 
 it is computed, it must evidently have a smaller deviation 
 than a single observation in proportion to its greater reli- 
 ability. It can be shown that an arithmetical mean com- 
 puted from n equally probable observations is Vn times as 
 reliable as any one observation. Hence, if the deviation 
 
DEVIATION MEASURES 17 
 
 measure of a single observation of a series is a.d., the devia- 
 
 tion measure of the mean of n such observation is only 
 
 Vn 
 
 as great; i.e., the deviation of the mean, denoted by A.D., is, 
 
 Thus, if the mean value of nine measurements of the dis- 
 tance between two lines is 1.3215 mm. and the average devia- 
 tion of any one of the measurements is found to be a.d. = 
 0.0033 mm., the mean will have a probable deviation not 
 
 0.0033 
 greater than 7= = 0.0011 mm. From this it will be seen 
 
 that in general it does not pay to increase the number of 
 observations beyond a certain limit, say nine or sixteen, as the 
 time and labor involved soon become excessive, without a 
 corresponding increase in the precision attained. 
 
 Fractional and Percentage Deviation Measures. It is fre- 
 quently convenient to express the reliability of a quan- 
 tity as a fractional or as a percentage part of the quantity 
 itself. Thus we have in very common use the two following 
 deviation measures derived from the preceding: 
 
 the fractional deviation of a single observation = ^-i 
 
 the percentage deviation of a single observation = 100 '- 
 
 a ' 
 
 A D 
 
 the fractional deviation of the mean = - - 
 
 m ' 
 
 A D 
 
 the percentage deviation of the mean = 100 - '-. 
 
 in 
 
 Since these measures are never computed to more than two 
 significant figures, see page 23, a and m, being approxi- 
 mately the same, may be used indiscriminately in the com- 
 putation. 
 
 Deviation Measure vs. Precision Measure. A little con- 
 sideration will make clear that all of the above deviation 
 measures give a measure of the magnitude of errors of that 
 
18 PRECISION OF MEASUREMENTS 
 
 type which has been classed as accidental. A result may 
 be seriously in error due to residual errors, and yet the ob- 
 servations show a good agreement among themselves, and 
 their deviation measure be correspondingly small. If the 
 magnitude of the residual errors can be estimated, we may 
 compute the true precision measure, abbreviated p.m., of 
 the result as follows: 
 
 Let the estimated magnitude of the residual errors be 
 r \j r 2) - r n> Let d.m. represent the value of the deviation 
 measure of the accidental errors. This may be expressed as 
 an average, fractional or percentage deviation, but, which- 
 ever is chosen, the residuals must be expressed in the same 
 way. It can then be shown that the most probable measure 
 of the reliability of the result will be given by the expression : 
 
 p.m. 2 = d.m. 2 + rf + r* + . . . +f**, (4) 
 
 or p.m. = -yd.ra. 2 + r\ + r 2 2 + . . . + r n 2 . (4a) 
 
 Thus the precision measure of a result differs from its devia- 
 tion measure in that it includes the effect of residual as well 
 as of accidental errors. In a great many cases the value of 
 the residuals is negligible compared with the magnitude of 
 the accidental errors. In this case p.m. = d.m. The synir 
 bol 8 will be used to represent the value of p.m. or d.m. in- 
 discriminately, as the latter is only a special case of the 
 former when Sr 2 is negligible. 
 
 It will be shown on page 31 that any single residual r k 
 may be regarded as negligible in computing p.m. if 
 
 r t = J p.m. (5) 
 
 Also that any number p of residuals are simultaneously neg- 
 ligible if 
 
 Vri' + r, + . . . + V = J p.m. (6) 
 
 The Probable Error and the Mean Error. In the discus- 
 sion of observations by the method of Least Squares and 
 in many foreign treatises certain other measures are in 
 common use; namely, the so-called probable error and the 
 
PROBABLE ERROR AND MEAN ERROR 19 
 
 mean error. The "probable error" of an observation is of 
 such a magnitude that the probability of making an error 
 greater than it is just equal to the probability of making 
 one less than it, both probabilities being one-half. The 
 probable error of a single observation and of the mean 
 of n observations are given by the expressions 
 
 p.e. = 0.6745 J ^ and P.E. = 0.6745 \J __ ^ - 
 
 ' n 1 r n (n 1) 
 
 respectively, where 3d 2 is the sum of the squares of the 
 deviations of the single observations from the mean. 
 The following approximate formulae are more convenient 
 forms to use for purposes of computation: 
 
 p.e. = 0.8453 , J^= and P.E. = 0.8453 JJ . 
 
 \n (n 1) n\n 1 
 
 The "mean error" ^ is defined as the square root of the 
 arithmetical mean of the squares of the errors. It is 
 seldom used except in treatises on Least Squares. 
 
 It can be shown from the equation of the curve of error 
 (p. 13) that, interpreted geometrically, p.e. = OP, the ab- 
 scissa of the ordinate which divides the area OXY into 
 equal parts; a.d. =OD, the abscissa of the ordinate pass- 
 ing through the center of gravity of the half area; and 
 p=OM, the abscissa of the point of inflection of the curve. 
 From this it follows that 
 
 0.4769 1 1 
 
 P .e.= --; .<*. 
 
 or p.e. = 0.85 a.d. = 0.67/*. 
 
 Although the probable error is frequently used by physi- 
 cists as a/ precision measure, the average deviation is 
 simpler, and will be adopted throughout the present work. 
 Weights. It frequently happens that it is necessary to 
 average a series of results which have not been taken under 
 like conditions, and which are not all equally probable ; i.e., 
 
20 PRECISION OF MEASUREMENTS 
 
 which do not have the same precision measures. In this case 
 it is first necessary to assign relative weights to the obser- 
 vations, so that, in taking the average, the more precise 
 measurements may be given a proportionally greater 
 " weight " than the less precise measurements. 
 
 It can be shown that the relative weights of a series of 
 observations are inversely proportional to the squares of 
 their precision measures; i.e., if p 1; p z , p 3 , . . . etc., are the 
 weights of a series of observations whose respective precision 
 measures are B l9 S 2 , $3, etc., respectively, 
 
 1 . i . i . , 
 
 PI P2 PS =:== ^ 7T2 jr| l*J 
 
 In determining the values of p } the nearest round numbers 
 satisfying the above proportion should be chosen. 
 
 Since the various precision and deviation measures differ 
 from each other only by a constant factor, any one of them 
 may be used in computing " weights." It is, of course, neces- 
 sary, however, that the same measure be used throughout 
 any given discussion; i.e., it is not permissible to express 
 the precision of one quantity as an average deviation, another 
 as a probable error, and a third as a percentage error. 
 
 The Weighted Mean. Having obtained the weights pi, p 2 , 
 p 8) etc., to be assigned respectively to a series of quantities 
 Wj, w 2 , m s , etc., the best representative value or weighted 
 mean will evidently be given by the expression 
 
 ViXm-i + VvX m + p X m* ... /ox 
 
 (8) 
 
 Pi + Pz + Ps + - 
 
 Rejection of Observations. In a series of measurements 
 taken under similar conditions, it not unfrequently happens 
 that an observation will differ quite widely from others in 
 the series, and the tendency to regard such an observation 
 as erroneous and to reject it is great, particularly among 
 beginners. If such an observation obviously contains a 
 mistake, as, for example, the recording of a wrong number, 
 the recording the wrong scale division, the incorrect adding 
 
REJECTION OF OBSERVATIONS 21 
 
 up of weights, etc., it may, of course, be legitimately re- 
 jected. If, however, no mistake is apparent, the observa- 
 tion should never be rejected without the most scrupu- 
 lously unbiassed judgment on the part of the observer or 
 the application of some mathematical criterion for the 
 rejection of doubtful observations. For the experienced 
 observer the former procedure is preferable, even though 
 several mathematical criteria, Peirce's, Chauvenet's, etc., 
 have been deduced, which are very satisfactory when 
 the number of observations considered is large. In most 
 physical work the number of observations is not very 
 great, however, and one widely discordant from the others 
 has an undue weight on the value of the mean. It is 
 frequently better to reject such an observation, even 
 though it contains no apparent mistake. A good cri- 
 terion to follow in such cases is the following: 
 
 Compute the mean and the average deviation a.d., 
 omitting the doubtful observation. Compute also the 
 deviation, d, of the doubtful observation from the mean. 
 If d >T 4 a.d., reject the observation, since it can be shown 
 that the probability of the occurrence of an observation 
 whose deviation is equal to four times the average devia- 
 tion is only one in a thousand. An error of this unusual 
 magnitude is called a Huge Error. 
 
 Computation Rules and Significant Figures. It is prob- 
 ably true that at least half the time usually spent on com- 
 putations is wasted, owing to the retention of more figures 
 than the precision of the data warrants, and to the failure 
 to use either logarithms or a slide rule instead of the 
 lengthy arithmetical processes of multiplication and divi- 
 sion. An important feature of physical laboratory work is 
 the proper use of significant figures in recording data and 
 in subsequent computations. The habit should be ac- 
 quired at the outset of rejecting at each stage of the work all 
 figures which have no influence on the final result. 
 
 Rules for the correct use of significant figures are dis- 
 cussed in the introduction to Holman's "Computation 
 
22 PRECISION OF MEASUREMENTS 
 
 Rules and Logarithm Tables" which may very advantageously 
 be used in connection with the laboratory work. A fuller 
 discussion, including the demonstration of these rules, is 
 given in Holman's "Precision of Measurements." The 
 following brief statement of the rules is essentially that 
 given in these works: 
 
 A Digit is any one of the ten characters 1, 2, 3, 4, 5, 6, 7, 8, 
 9,0. 
 
 A Significant Figure is any digit to denote or signify the 
 amount of the quantity in the place in which it stands. Thus 
 zero may be a significant figure when it is written, not merely 
 to locate the decimal point, but to indicate that the quantity 
 in the place in which it stands is known to be nearer to zero 
 than to any other digit. 
 
 For example, if a distance has been measured to the nearest 
 fiftieth of an inch, and found to be 205.46 inches, all five of 
 the figures, including the zero, are significant. Similarly, if 
 the measurement had shown the distance to be nearer to 
 205.40 than to 205.41 or to 205.39, the final zero would be 
 also significant, and should invariably be retained, since its 
 presence serves the most useful purpose of showing that this 
 place of figures had been measured as well as the rest. If 
 in such a case the quantity had been written 205.4 instead 
 of 205.40, the inference would be drawn either that the 
 hundredths of an inch had not been measured or that 
 the person who wrote the number was ignorant or careless 
 of the proper numerical usage. Failure to follow this 
 simple rule is a common source of annoyance and un- 
 certainty. 
 
 A zero, when used merely to locate the decimal point, is 
 not a significant figure in the above sense; for the position 
 of the decimal point in any measurement is determined solely 
 by the unit in which the quantity in question is expressed. 
 The number of decimal places in a result has, therefore, in 
 itself no significance in indicating the precision of a measure- 
 ment. For example, suppose a certain distance is found to 
 be 122.48 cm. with a A.D. of 0.12 cm. The percentage pre- 
 
RULES FOR SIGNIFICANT FIGURES 23 
 
 12 
 cision of the measurement is -J - X 100 = 0.10%. The re- 
 
 suit contains five significant figures, and its precision remains 
 the same, namely, 0.10%, whether it be expressed as 1.2248 
 meters, A.D. = 0.0012 m., or 1224.8 mm., A.D. 1.2 mm. 
 The statement that the distance is measured to 0.12 cm. gives 
 no idea of the precision of the measurement unless the distance 
 itself is stated. A fractional or percentage precision measure, 
 on the other hand, gives a definite idea of the precision of the 
 measurement without any further statement, as it involves 
 both the value of the quantity and its average deviation. 
 
 The following rules are deduced subject to the condition 
 that the accumulated errors in a computation shall not affect 
 the second unreliable place of figures in the final result by 
 more than one unit, even though as many as sixteen rejec- 
 tions of figures are made in the course of the computation. 
 This is a safe limit to assume for most physical work, as it 
 is seldom that more than this number of quantities or oper- 
 ations enter into any single computation. 
 
 Rule I. In rejecting superfluous figures, increase by 1 the 
 last figure retained, if the following figure (that rejected) is 5 
 or over. 
 
 Rule II. In all deviation and precision measures retain 
 two, and only two, significant figures. 
 
 The reason for this rule is as follows: consider the above 
 example where the length measured is m = 122.48 cm. with an 
 A.D. 0.12 cm. The significance of the A.D. is that the 
 place of figures in m occupied by the 4 is uncertain by 1 
 unit, and that the next place of figures occupied by 8 is un- 
 certain by 12 units, while the third decimal place would be 
 uncertain by at least 120 units; i.e., by an amount which 
 would render it practically worthless. In general, the place 
 of figures corresponding to the first significant figure of the 
 deviation measure is somewhat uncertain (from 1 to 9 units), 
 while the place corresponding to the second significant figure 
 in the deviation measure is uncertain by ten times this amount 
 (10 to 90 units, or, more exactly, 10 to 99 units). Beyond 
 
24 PRECISION OF MEASUREMENTS 
 
 this place the significance of additional figures is so slight as 
 to be of no value : hence, as deviations and precision meas- 
 ures are at best only estimates of the reliability of a result, 
 it is useless to compute them to places of figures which have 
 no real significance in the result to which they refer. 
 
 If the first significant figure of the precision measure is as 
 great as 8 or 9, in which case the place of figures in the data 
 corresponding to the second place in the precision measure 
 is unreliable by 80 to 90 units, it is usually sufficient to 
 retain but one significant figure in the precision measure. 
 
 Rule III. Retain as many places of figures in a mean re- 
 sult and in data in general as correspond to the second place 
 of significant figures in the deviation or precision measure. 
 
 Two places of doubtful figures are thus retained in data 
 and computations rather than one, so that accumulated 
 errors due to rejections in the course of a computation may 
 not affect the first place of uncertain figures in the result. 
 
 Rule IV. The sum or difference of two or more quanti- 
 ties cannot be more precise numerically than the quantity 
 having the largest average deviation. Hence, in adding or 
 subtracting a number of quantities, find the average deviation 
 of each, and then retain in each quantity as many places of 
 figures as correspond to the second place of significant figures 
 in the largest deviation. 
 
 Rule V. In multiplication or division the percentage 
 precision of the product or quotien cannot be greater than 
 the percentage precision of the least precise factor entering 
 into the computation. Hence, in computations involving 
 these operations, the number of significant figures to be re- 
 tained in each factor is determined by the number properly 
 retained under Rule III. in the factor which has the largest 
 percentage deviation. Computations involving a precision 
 not greater than ^ P er cent, should be made with a slide 
 rule. For greater precision logarithm tables should be 
 used. If multiplication and division must be resorted to, 
 the "short method" of rejecting all superfluous figures 
 at each stage of the operation should be adopted. 
 
INDIRECT MEASUREMENTS 25 
 
 Rule VI. In carrying out the operations of multiplication 
 and division by logarithms, retain as many figures in the man- 
 tissa of the logarithm of each factor as are properly retained 
 in the factors themselves under Rule V. 
 
 Precision Discussion of Indirect Measurements. We will 
 now consider the precision discussion of indirect measure- 
 ments; i.e., those in which the final result is a more or less 
 complicated function of one or more directly measured 
 quantities. Two distinct classes of problems may arise: 
 
 First. The precision measures of the directly measured 
 components are known (determined as above described), and 
 it is desired to find the precision measure of the final result. 
 
 Second. The desired precision of the final result is stipu- 
 lated at the outset, and the problem is to ascertain what 
 precision is necessary in the components, in order that the 
 accumulated effect of the errors in these on the final result 
 shall not exceed the prescribed limit. 
 
 The importance of these problems cannot be overesti- 
 mated; for, in the first case, a final result, be it the result 
 of chemical analysis, the value of a physical constant, the 
 algebraic expression of a law, or an efficiency test of an en- 
 gine, is practically worthless unless a numerical estimate 
 of its reliability can be stated. In fact, it may be worse 
 than worthless if carried out to indicate a higher precision 
 than the data warrant. And the second case is of equal 
 importance; for, unless an investigator makes a preliminary 
 precision discussion of his method and apparatus before 
 beginning work, so that he may know, at least approx- 
 imately, how precisely each quantity entering into the final 
 result should be measured, the chances are that much 
 time and labor will be wasted in measuring some compo- 
 nents more precisely than necessary, while others will be 
 measured to a degree of precision which will render impos- 
 sible the attainment of the desired precision in the final 
 result. 
 
 Notation. In the precision discussion which follows, the 
 following notation will be adopted. 
 
26 PRECISION OF MEASUREMENTS 
 
 M = the final computed result of any indirectly measured 
 quantity. 
 
 A = numerical precision measure of M. 
 
 m ij m z, . . . = directly measured quantities, which may 
 be either mean results or single observations. 
 
 Si, 52,... = the numerical precision measures of m 1} m z , 
 . . . respectively. 
 
 The values of 5 might be expressed as average deviations, 
 probable errors, or mean errors, discussed on page 19. In the 
 discussion of any given problem, however, the same kind 
 of precision measure must be used throughout; i.e., in any 
 given problem it is not proper to express the precision meas- 
 ures of some quantities as probable errors, others as average 
 deviations, and still others as percentage or fractional de- 
 viations. In the following discussion we shall always assume 
 values of 8 to be expressed as deviations. 
 
 A 1; A 2 , . . . , will be used to denote the deviations pro- 
 duced in M by deviations Si, 8 2 > m the components 
 %, ra 2 , . . . respectively. 
 
 From the above notation it follows that, 
 
 the fractional precision of the final result ; 
 
 100 = percentage precision of the final result ; 
 
 a $ 
 
 , ,...= the fractional precision of the components 
 
 7?ll m^ 
 
 m i> m z> - - respectively; 
 
 OJ $ 
 
 100 , 100 . . = the percentage precision of the com- 
 ra x m% 
 
 ponents m l} w 2 , . . . respectively. 
 In general 
 
 M = f (m lt ?n 3 , . . . m n ), (9) 
 
 which for brevity may be written M = f ( ), where the 
 form of the function is determined by the formula by which 
 M is computed from raj, ra 2 , etc. The first class of problems 
 may then be stated mathematically as follows : 
 
INDIRECT MEASUREMENTS 27 
 
 Case I. The Direct Problem. Given the precision meas- 
 ures 8 1? S 2 , . . . 8 n , of the component measurements m l} m 2 , 
 . . . m n , to compute the precision measure A of the result M. 
 
 The solution of this problem is obtained by finding, first, 
 the effect of the deviation in each component on M, and 
 then combining these separate effects to get the resultant 
 effect. The method of computation to be followed in this 
 last procedure depends upon the law to which the deviations 
 concerned are subject, and will be considered below. 
 
 Separate Effects. The effect of a deviation 8* in any com- 
 ponent m k will be to produce a deviation A* in M of an 
 amount 
 
 i.e., an amount equal to the rate at which the value of the 
 function M = f( ) changes, as m k changes (the other com- 
 ponents m 2 , ra 3 , . . . etc., remaining constant), multiplied 
 by the actual change 8# in m k , or, in other words, the partial 
 differential coefficient of the function with respect to m* 
 multiplied by the actual deviation 8* in m t . 
 
 Example 1. Find the deviation in the volume of a 
 sphere whose diameter is 10.013 cm., if the average devia- 
 tion in the measurement of the diameter is A.D. 0.012 cm. 
 
 V = *D*. 
 
 Comparing with the notation on page 26, it is evident that 
 M=V = f( ) = %TrD*, 
 
 m = D = 10.013 cm., and 5 = A.D. = 0.012 cm. 
 The computed value of V is 
 
 V = i X 3.1416 X 10.013* 
 
 = 525.52 c.c. 
 
 By (10) the deviation A in this volume produced by the 
 deviation 5 in the diameter is 
 
 A=A a ^ 3) . 5 
 
 = i IT. 3D 2 . 5 
 
 = i X 3.1 X 3 X 10 2 0.012 
 
 = 1.9 c.c. ; 
 
28 PRECISION OF MEASUREMENTS 
 
 i.e., the volume 525.5 c.c. is uncertain by 1 .9 c.c., or by 19 parts 
 
 0.012 
 D ~ 10 
 
 in 5300. A deviation of 100 ^ = 100 ^^ = 0.12% in the 
 
 A 19 
 
 diameter introduces a deviation of 100 y 100 -^ 0.36% 
 
 in the volume, i.e., a percentage deviation three times 
 as great. In this case V is a function of only a single vari- 
 able, hence the resultant deviation in V is given at once by 
 the above result. 
 
 Example 2. What will be the numerical deviation in the 
 value of g, as determined by a second's pendulum, due to 
 a deviation A.D. = 0.0020 second in the determination of 
 the time of vibration, and a deviation A.D. = 0.10 cm. in 
 the determination of the length? 
 
 9 = 
 
 Hence in the general notation M = g = f ( ) = ^ 
 
 mi=l=WO cm.; mz = t 1 sec.; 5i = 5i = 0.10 cm.g 
 5 2 = Si = 0.0020 sec. 
 
 The deviation Aj in g produced by the deviation Si in I is 
 by (10) 
 
 = ^~ X 1 X 0.10 
 
 sec. 
 
 i.e., a deviation of 0.10 cm. in the measurement of I will 
 
 cm 
 produce a deviation of 0.96 - 2 in the value of g. 
 
 Similarly the deviation A t in g due to the deviation 8 t in t 
 is by (10) 
 
 HfcS) 
 
 2 X3 - 1 1 2 8 X1 X 0.0020 
 
 cm. 
 
 = 3.8 
 
 sec. 
 
RESULTANT EFFECT 29 
 
 i.e., a deviation of 0.0020 sec. in the measurement of the time 
 
 cm. 
 
 will introduce an uncertainty in the value of g of 3.8 == 
 
 sec. 
 
 The negative sign simply indicates that a positive deviation in 
 t produces a negative deviation in g, and vice versa. Since all 
 deviations are equally likely to be plus or minus, in pre- 
 cision discussions no attention is usually paid to the sign 
 resulting from differentiation of a function. By a direct 
 application of (10) the effect of a deviation in any single 
 component on a final result may always be computed. A 
 much shorter method than the above, applicable in certain 
 special cases, will be pointed out below. 
 
 Resultant Effect. To find the combined or resultant effect A, 
 on the final result of the separate deviations AJ, A 2 , . . . etc., 
 produced by the components. 
 
 If for any reason the values of A b A 2 , ... etc., are of 
 specified magnitude and sign (in which case they would not 
 follow the general law of deviations), they should be com- 
 bined according to the formula 
 
 A = A! + A 2 + . . . + A n . (11) 
 
 As this case rarely occurs in practice, it need not be further 
 discussed here. 
 
 The important case to consider is that in which the values 
 of AJ, A 2 , . . . etc., are equally likely to be plus or minus and 
 of a magnitude determined by the general law of deviations. 
 If each A* is computed by formula (10), page 27, i.e., 
 
 these conditions will always be fulfilled, since the values of 
 8 k which determine A* are of the nature of true deviations. 
 Under these circumstances the most probable resulting devia- 
 tion A, in M, can be shown by the method of Least Squares to 
 be that obtained by combining the values of A* by the formula 
 
 A 2 = A X 2 + A 2 2 + . . . + A 2 n , (12) 
 
 or A = V Al 2 + A 2 2 + . . +An 2 . (12a) 
 
30 PRECISION OF MEASUREMENTS 
 
 This does not give us an exact solution of the problem, but 
 rather the solution which in the long run is better than that 
 obtained by any other method of combining the values of 
 A*. It is to be noted that by this method of computation 
 the effect of the sign of individual deviations A* is elim- 
 inated. The resultant deviation A is, of course, to be re- 
 garded as equally likely plus or minus. 
 
 Example 2 (continued). Thus in Example 2 the com- 
 bined effect of the deviations in I and in t on the value of g 
 is to be found by taking the square root of the sum of the 
 squares of the deviations Aj and &t, which 61 and <$t sep- 
 arately produce in g respectively; i.e., 
 
 * 
 
 Vo.96 2 + 3.8 2 
 
 cm. 
 
 = 3.9=^ 
 sec. 
 
 Hence a deviation of 0.10 cm. in Z and 0.0020 sec. in t will 
 
 make the value of g = 980 ===% uncertain by nearly 4 ==3. 
 sec. sec. 
 
 iterion for Negligibility of Deviations in Components. It 
 
 is frequently important to determine whether the deviation 
 arising from one or more components may be neglected 
 in computing the A of the final result. For this purpose 
 the following criterion may be deduced. 
 
 As explained under rules for significant figures on page 
 23, two significant figures are all that should be retained 
 in any deviation measure. A quantity which affects a re- 
 sult by only yg- the amount of its deviation or precision 
 measure will therefore affect it only in that place of significant 
 figures corresponding to the second place in the deviation 
 measure. This place is so uncertain that such an amount 
 may in general be regarded as negligible. Although the 
 assumption that $ p.m. or -fa d.m. is negligible is some- 
 what arbitrary, it has been found to be a convenient and 
 practical criterion to adopt. 
 
 Suppose, therefore, that the value of the A of some quantity 
 
NEGLIGIBILITY OF DEVIATIONS 31 
 
 M is made up of deviations A x , A 2 , . . . A n , arising from 
 various deviations S 1; 8 2 > S n , in components m lf m z , . . . 
 m n . May any of these A's, as A*, be neglected in com- 
 puting A, or, in other words, may any of the components, 
 as m t , be regarded as being without sensible error on Mf 
 To answer this question, let 
 
 -fA 2 2 +. .+A, 2 +...A n 2 
 
 and A' = V A x 2 + A 2 2 -f . . . + A n 2 with A* omitted. 
 Then, if A A' < T V A 
 
 or A' > 0.9 A, 
 
 by the above criterion A* may be considered as negligible. 
 But A/ = A 2 A' 2 
 
 = A 2 (1 2 0.9 2 ) 
 = 0.19 A 2 
 /. A* = 0.43 A. 
 
 Hence the deviation in any component m may be neglected 
 in computing the A of M , if its effect on M is equal or less 
 than 0.43 A. A still safer and more convenient criterion to 
 adopt, since the number of components considered is usually 
 small and hence the assumed formula of squares is less 
 rigidly applicable, is 
 
 A* ^ 0.33 A ^1 A. (13) 
 
 In the same way it can be shown that deviations in any 
 number, p, components are simultaneously negligible if 
 
 V Al 2 + A 2 2 + ...A p 2 < J A. (14) 
 
 The above criterion also applies to the rejection of residuals 
 in computing the value of the precision measure by the 
 
 formula p.m. = d.m. 2 + r-f + r 2 2 + . . . r n 2 , as stated 
 on page 18. 
 
 Case II. The Converse Problem. Given a prescribed pre- 
 cision A to be attained in the final result M, to find the allow- 
 able deviations 1} 8 2 > &> i n the components m l} w 2 , etc., re- 
 
32 PRECISION OF MEASUREMENTS 
 
 spectively, such that their combined effect on M shall not exceed 
 the value of A. 
 
 We have seen that when the deviations follow the law 
 of errors, 
 
 If the value of A is given and no further conditions im- 
 posed, there are evidently an infinite number of solutions 
 to the problem; i.e., an indefinite number of values can be 
 found for A b A 2 , . . . etc. (and hence for the correspond- 
 ing values of 8^ S 2 > etc.), which will satisfy the above 
 equation. 
 
 The most advantageous distribution of errors among the 
 components will evidently be that one by which the desired 
 precision is obtained with the minimum expenditure of 
 time and labor on the part of the experimenter. As this 
 will vary greatly with each individual problem, no mathe- 
 matical criterion can be formulated which will embrace 
 all cases. It is best, therefore, at least for a preliminary 
 distribution of errors among the components, to so adjust 
 them that the errors inherent hi each variable or component 
 shall produce the same effect on the final result. This is 
 spoken of as the solution of the problem for "equal effects." 
 
 Solving the formula for resultant effects subject to this 
 condition, i.e., 
 
 = A," _ 
 
 we have 
 
 * 
 
 hence for any component, 
 
 (16) 
 
 Having thus determined A*, the corresponding value of 
 can be found at once by equation (15). 
 
EQUAL EFFECTS 33 
 
 Example 3. How precisely should the time of vibration 
 and length of a seconds pendulum be measured in order that 
 the computed value of g may be reliable to one-tenth of one 
 
 percent.? g= ^-- 
 
 As the pendulum is stated to be a "seconds" pendulum, 
 t = one second and Z = 100 cm. approximately. It is further 
 
 stipulated that 100 ^0.10; hence the allowable resultant 
 
 deviation A in g must not be greater than A = g x 0.0010 
 
 cm. 
 
 = 0.98 =4 We are to find the allowable values of 8 t and Si, 
 sec. 
 
 which will give this precision. Solving the problem subject 
 to the condition of equal effects, -i.e., that the resultant de- 
 viation in g is caused equally by the deviation in t and in I, 
 
 we have 
 
 A 0.98 cm. cm. 
 
 A, = A* = -j= = ^- =2 = 0.70=^ . 
 yn y2 sec. sec. 
 
 Hence the allowable deviations in the time and length meas- 
 urements must be reduced to such a magnitude that they do 
 
 not separately produce a deviation in g greater than 0.70 ^ 
 
 sec. 
 
 respectively. But by the general equation (10), page 27 
 
 7 _ cm. 2 x 3. 1 2 X 100 cm. 
 
 or 0.70 =3 = - . S t 
 
 sec. 1 sec. 
 
 hence 5 t = 0.00037 sec. 
 Shnilarly, -*.,- 
 
 cm. 3 
 
 or 0.70 =r = =2 Si 
 
 sec. I 2 sec. 
 
 hence 81 = 0.073 cm. 
 
 The time of vibration of the pendulum should there- 
 fore be measured to 0.00037 sec., and its length measured to 
 0.073 cm. 
 
 The Fractional or Percentage Method of Solution. The 
 preceding formulae for obtaining the precision of a final 
 result from the known or estimated precision of the com- 
 
34 PRECISION OF MEASUREMENTS 
 
 ponent measurements, and for calculating the necessary 
 precision of the component measurements when the de- 
 sired precision of the final result is stipulated, are entirely 
 general, and by them any type of problem can be solved. 
 It is to be particularly noted throughout the preceding 
 discussion that the values of the precision or deviation 
 measures 8 and A are numerical deviations expressed in 
 the same units as the quantities to which they refer. 
 Percentage and fractional deviations should not be used 
 when applying the general formulae (10) to (12a), (15) 
 and (16). If in the statement of a problem, as in example 
 3, the fractional or percentage precision is given, the cor- 
 responding deviations 8 or A should first be computed 
 before proceeding with the solution. 
 
 There are, however, a large number of formulae which 
 may be discussed with a great saving of time and labor 
 by the use of percentage or fractional deviations. This 
 is the case whenever the function M =/(rai, m 2 , . . . m n ) 
 can be put in the form of a product of the general type 
 
 M = k . mf .m ...m* (17) 
 
 where k, a, b, : . . p are constants (positive, negative, 
 fractional, or integral). For all such cases a very simple 
 relation holds between the fractional or percentage devia- 
 tion in any component and the fractional or percentage 
 deviation which it produces in the final result. This 
 may be shown as follows. Applying the general for- 
 mula (10) for separate effects to the above special case, 
 we have for the deviation A! in M produced by Si in mi 
 
 9M 
 
 Dividing through by equation (17) 
 
 --0 . ^ (18) 
 
 M mi 
 
 CN 
 
 i.e., a fractional deviation -- in mi produces a fractional 
 
EQUAL EFFECTS 35 
 
 deviation a times as great in the final result. Thus, if 
 
 8 
 
 a = 2, a deviation of one per cent. (100 = 1) in mi will 
 
 tn\ 
 
 introduce a deviation of two per cent, in M, no matter 
 what the value of the remaining factors in the expression 
 may be. The separate effect of a known fractional or 
 percentage deviation in any component on the final 
 result may therefore be stated at once by inspection, 
 whenever the formula under discussion can be put in 
 the above form. 
 
 Since the formula for Resultant Effects (12a) may be 
 put in the form 
 
 M 
 
 the complete solution for any product function of the 
 type given by equation (17) may be written down at 
 once by inspection as 
 
 i 
 
 if the fractional deviations , etc., of the components 
 
 ?n\ 
 
 are known. 
 
 Similarly, the solution of the converse problem for a 
 product function is equally simple, as the condition for 
 equal effects, page 32, may be written 
 
 _Al = ^2_ _Ak_ __A^ 
 
 M M~ ~M~~ ' ~M 
 
 and hence the allowable fractional deviation in the final 
 result which any component as m k may produce is 
 
 M < sin M 
 where TT is the prescribed fractional deviation of the 
 
36 PRECISION OF MEASUREMENTS 
 
 final result which must not be exceeded. Having thus 
 
 determined the value of -A, we obtain at once the allow- 
 
 g 
 able fractional deviation - in the corresponding com- 
 
 ponent m k by inspection from the simple relation ex- 
 pressed by equation (18). 
 
 Since the exponents a, 6, c, etc., of the factors may 
 have negative as well as positive values, the above solu- 
 tions apply to formulae involving division as well as mul- 
 tiplication of factors. 
 
 It is to be especially noted that, if the function M in- 
 volves the sum or difference of several components, or 
 is a trigonometric or logarithmic function, no simple 
 relation exists between the fractional deviation of a com- 
 ponent and the fractional deviation which it produces in 
 the final result. The above special method of procedure 
 is, therefore, inapplicable to such cases. This will be 
 readily seen from the following simple example. Sup- 
 
 pose 
 
 M= a rrii+b m 2 . 
 
 Then A ' 
 
 and A i 
 
 from which it appears that -^ stands in no simple relation 
 
 <\ 
 to unless the term b m 2 happens to be negligible in 
 
 mi 
 
 magnitude compared with a Wi, in which case we should 
 have assumed at the outset for our precision discussion 
 that M = ami approximately. 
 
 It frequently happens, however, that apparently com- 
 plicated functions can be transformed into a simple 
 product of factors by changing variables or noting that 
 certain components may be neglected in the precision 
 discussion. When this is possible, the fractional method 
 may be applied with advantage to each factor. 
 
EQUAL EFFECTS 37 
 
 Example 4. The solution of problem 2 may be obtained 
 much simpler by the fractional or percentage method than by 
 the general method as worked out on page 28. For the 
 
 formula g = 7r 2 - = irH t- 3 - is evidently a simple product func- 
 tion of the variables I and t. To find the deviation in g 
 due to a deviation of Si = 0.10 cm. in I and a deviation 
 8 t = 0.0020 second in t, we find first the fractional deviation 
 in I and in t respectively. 
 
 di 0.10 cm. 
 
 / 100 cm. 
 
 = 0.0010, 
 
 and ^ = 0.0020 sec. = 
 
 t 1.0 sec. 
 
 Then by inspection, since g is directly proportional to the 
 first power of Z, 
 
 *' = 5 J = 0.0010; 
 
 and, since g is proportional to the second power of t (neglect- 
 ing sign), 
 
 ^ = 2- = 2 X 0.0020 = 0.0040. 
 
 = V/(0.0010) 2 + (0.0040) 2 = 0.0041 
 
 or A = 0.0041 X 980 = 2 = 4.0 -^ 
 sec. sec. 
 
 which is practically the same result previously obtained, 
 the slight difference arising from the use of but two signifi- 
 cant figures in the computation. 
 
 Example 5. Again, the solution of example 3, page 33, 
 may be simplified by using the fractional method. Thus, if 
 it is stipulated that g is to be measured to 0.10 per cent., 
 
 i.e., 100 ~0.10, the prescribed fractional deviation is 
 ^ 0.0010. Distributing this deviation by the criterion of 
 
 equal effects between the component measurements I and t 
 respectively, we have 
 
 . o.ooon. 
 
 99^/29 y/2 
 
 TT-l 
 
 But by inspection of the formula g = /2 it is seen that g is 
 
38 PRECISION OF MEASUREMENTS 
 
 proportional to the first power of I and to the second power 
 of t, therefore 
 
 A? 81 , At 8 t 
 = v and = 2 
 9 I 9 t 
 
 Hence 
 
 ,- = 0.00071, 
 
 or the length must be measured to 
 
 5z = 100 cm. x 0.00071 = 0.071 cm. 
 Similarly, 
 
 -' =4 X 0.00071 = 0.00036, 
 
 t i 
 
 or the time must be measured to 
 
 d t = 1 sec. X 0.00036 = 0.00036 sec. 
 
 These, it is seen, are practically the same values previously 
 obtained by the differentiation method on page 33, the dif- 
 ference in the second place of figures arising from the use of 
 only two figures in the computation. 
 
 Discussion of "Equal Effect" Solution. It not infre- 
 quently happens upon solving a problem as described 
 above, that some component (or components) can with 
 little additional time and labor be determined with a 
 much higher precision than the solution demands. In 
 this case such a component or factor should be so meas- 
 ured and then regarded as a constant in the precision 
 discussion, since the error in it will have a negligible 
 effect on the final result. The problem should then be 
 re-solved on the basis of one less variable, in which 
 case the remaining components which are more difficult 
 to determine, may be measured with somewhat less pre- 
 cision than was demanded by the first solution. 
 
 The proper adjustment of precision among components 
 so as to give the desired precision in the final result with 
 the apparatus at one's disposal and with the least ex- 
 penditure of time and labor, requires some experience 
 and good judgment on the part of the investigator. 
 Beginners will not go far astray, however, if they follow 
 the above criterion for equal effects. 
 
PART II. 
 GRAPHICAL METHODS. 
 
GRAPHICAL METHODS. 
 
 Nature of Problems The graphical method of discussing 
 experimental data is of great convenience and importance 
 when the problem under investigation is to determine the 
 law or fundamental relationship between two quantities. 
 This type of problem arises very frequently in scientific 
 and technical investigations. The graphical method is 
 also of great value for purposes of interpolation, discus- 
 sion of corrections, etc. 
 
 Procedure. The general procedure to be followed in dis- 
 cussing observations by the graphical method will be ex- 
 plained and illustrated by following through, step by step, a 
 specific problem. Suppose it is desired to find the relation 
 which holds between the resistance of a certain coil of wire 
 and its temperature, between 10 and 100 C.; that is, to 
 determine the formula by which the resistance can be com- 
 puted at any given temperature between these limits. The 
 experimental procedure would consist in making a series 
 of measurements of the resistance r of the wire at various 
 temperatures t from approximately 10 to 100 C. Suppose 
 that the result of such experiments gives the two following 
 columns of data, the resistance measurements being reliable 
 to 0.003 ohm, and the temperature measurements to 0.02 
 C., as shown by their respective deviation or precision 
 measures. 
 
 EXPERIMENTAL DATA. 
 
 r = resistance of t = temperature of coil 
 
 coil in ohms. in degrees C> 
 
 10.421 10.50 
 
 10.939 29.49 
 
 11.321 42.70 
 
 11.799 60.01 
 
 12.242 75.51 
 
 12.668 91.05 
 
42 GRAPHICAL METHODS 
 
 The Direct Plot. To obtain some clue to the relation 
 between r and t (supposing it unknown), a Direct Plot 
 should first be made. Plotting-paper suitable for this 
 work should be ruled with carefully adjusted pens, other- 
 wise the errors arising from* irregularity of ruling may 
 easily exceed those of only moderately accurate data. A 
 convenient size is about eight by ten inches, ruled either 
 in millimeters, or preferably, in twentieths of an inch. 
 
 First. Choice of Ordinates and A bscissce. The first thing 
 to decide upon is which data are to be plotted as ordinates 
 and which as abscissae. The usual convention of analytic 
 geometry should always be followed. If, as in the problem 
 under consideration, it is desired to obtain a relation in 
 which r is expressed as a function of t, then values of r 
 should be plotted as ordinates and values of t as abscissae. 
 If, on the other hand, it were desired to obtain a formula 
 for computing the temperature t corresponding to any 
 resistance r, as in resistance pyrometry, the converse 
 would be the case. 
 
 Second. Choice of Scales. By the "scale" of a plot is 
 meant the ratio of the number of units (inches, centi- 
 meters, etc.) of the plot to one unit of the data. Scales 
 of both ordinates and abscissa? should be clearly indicated 
 on the plot. Thus, if 100 is plotted so as to extend 
 over 10 inches, the scale is 10" : 100 = 1 : 10, or one-tenth. 
 This is usually expressed as 1 inch to 10 degrees. In 
 general, it is not feasible to choose the same scale for 
 both ordinates and abscissae, nor should the attempt be 
 made to have the origin fall on the plot. If equal scales 
 are chosen for both abscissae and ordinates, the locus 
 of the data is likely to be a line either nearly horizontal, 
 in which case the precision of the data plotted as ordi- 
 nates is sacrificed, or nearly vertical, in which case the 
 same is true of the abscissae. Moreover, the intersection 
 of a nearly horizontal line with lines parallel to the axis of 
 X can be read off only with difficulty and liability to error, 
 while its point of intersection with the axis of Y is much 
 
DIRECT PLOT 43 
 
 more definitely defined. In order, therefore, to preserve 
 equal precision in the interpolation of both co-ordinates, the 
 line should be inclined as nearly as may be at an angle of 
 45 with both axes. Deviations of 10 or so to either side 
 of this position are not serious. 
 
 The scales chosen should, furthermore, be convenient; i.e., 
 in aiming to distribute the data approximately 45 across 
 the plotting-paper, scales of one inch equal to 1, 2, 4, 5, or 
 10 units (or these units multiplied by 10 n where n is an 
 integer), should be chosen, but never such scales as one inch 
 to 3, 7, 6, 11 units. The latter scales make plotting not only 
 laborious, but very liable to error, whereas the former scales 
 permit data to be plotted with facility. In choosing scales 
 for plotting, the student should guard as carefully against 
 adopting excessively large scales as excessively small ones. 
 In the latter case the plot will be cramped and the precision 
 of the data sacrificed. In the former case the deviations 
 of the data from the general law which they follow are likely 
 to be so magnified to the eye that it is difficult or impossible 
 to draw a representative line. Moreover, such plots give an 
 exaggerated idea of the precision of the data. As an upper 
 limit, a safe rule to follow is to adopt a scale which permits 
 of easy interpolation to not more than two uncertain places 
 of figures in the data; i.e., to that place corresponding to the 
 second significant figure in the deviation or precision meas- 
 ure. This rule applies particularly to data extending over 
 narrow numerical limits, to corrections, etc. 
 
 In the present problem it is seen that the extreme variation 
 of r is about 2.3 ohms, and of t, 90. The scales should there- 
 fore be so chosen as to distribute these quantities well over 
 the paper. Scales of 1" = 0.4 ohm and I" = 10 evidently 
 fulfill this condition and are at the same time convenient. 
 It is evident, however, that some of the precision of the 
 data will be sacrificed in plotting with these scales, since it 
 is impossible on a plot of the size chosen to locate the last 
 significant figure of the data with any great degree of pre- 
 cision. It should also be noticed that the origin will not fall 
 
44 GRAPHICAL METHODS 
 
 on the plot. This is not at all necessary, and only in those 
 cases when the data for both variables simultaneously 
 approach small values (zero) is this likely to be the case. 
 It is, however, desirable (although not imperative), that the 
 zero value of the abscissae should fall on the plot, in order 
 to determine the intercept of the curve with the ordinate 
 through this point for reasons explained below. 
 
 Third. To plot the data. Data should be plotted as fol- 
 lows. Locate the abscissa of the first point along the axis 
 of abscissae and with a straight edge placed vertically through 
 this point draw a fine line about one-eighth of an inch long 
 approximately at the place where the corresponding ordinate 
 is to be located. Then locate the ordinate along the axis 
 of ordinates, and draw a short horizontal line intersecting 
 the first line drawn. The intersection is the desired position 
 of the point. Never locate the data by dots, as the preci- 
 sion attainable with the paper is not only sacrificed, but there 
 is also much greater liability to error in the operation of 
 plotting itself when the attempt is made to locate both 
 ordinate and abscissa at the same time. 
 
 Fourth. To draw the "Best Representative Line." The 
 data being plotted, the next step is to draw a smooth curve, 
 the equation of which shall best represent the law connecting 
 the two variables in question. Inspection of the general 
 form of this curve will usually give valuable information as 
 to the form of the equation sought. 
 
 If the points appear to lie along a straight line, the best 
 representative line may be located by moving a stretched 
 fine black thread among the points until a position is found 
 such that the points lie as nearly as may be alternately on 
 either side of the thread and in such a manner that the points 
 above the line deviate from it by the same amount as those 
 below. The exact criterion for locating the best represen- 
 tative line would be to so adjust it among the points that the 
 sum of the squares of the deviations of the points above the 
 line is equal to the sum of the squares of the deviations of the 
 points below the line. (Criterion of Least Squares.) When 
 
DIRECT PLOT 
 
 45 
 
 the best position of the thread has been found, the location 
 of two points through which it passes is noted, and a fine 
 straight line is then ruled through these points with a hard 
 pencil, or, better, with a ruling pen. 
 
 If the points cannot be uniformly distributed about a 
 straight line, but deviate systematically from it, then the 
 best representative curved line is to be drawn with a French 
 or a flexible curve. The line should in this case be drawn 
 as before, so that the points are distributed as nearly as may 
 
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 Temperature. 
 PLOT 1. 
 
 be on alternate sides. From the form of the resulting curve 
 its equation may often be inferred. The next step is to 
 determine the equation of the curve by transforming it 
 graphically into a straight line by some one of the special 
 methods of transformation described below. The numeri- 
 cal constants in the equation of a straight line can always 
 be readily determined directly from the plot. 
 
 In the problem under consideration, the points are seen, 
 Plot I., to lie very closely along a straight line A'A", the 
 
46 GRAPHICAL METHODS 
 
 deviations from the line being of an irregular and not of a 
 systematic character. The relation between r and t is there- 
 fore a linear one; i.e., of the first degree. To determine 
 completely the function r=f (t), we have to find the numeri- 
 cal value of the constants in tjje equation of this line A!A". 
 Determination of the Constants of a Straight Line. The 
 general equation of a straight line is 
 
 y = ax + 6, (1) 
 
 where a and b are constants. 
 
 The constant a = -^- is the tangent of the angle which 
 ax 
 
 the line makes with the axis of X. The value of a cannot in 
 general be found by reading off the angle with a protractor 
 and looking out the value of its natural tangent, as the angle 
 is usually distorted owing to the unequal scales used in plot- 
 ting. To determine a, read off the value of the ordinate 
 and abscissa x', y' and x", y", respectively, of any two points 
 on the line, preferably near the extremities. These points 
 will not in general be observed points. Then 
 
 y" y f 
 a = tan == x _^' 
 
 Thus the co-ordinates of two such points A' and A" are 
 seen to be x' = 6.0, yf = 10.30 ohms, and x" = 94.5, 
 y" = 12.76 ohms. Hence 
 
 12.76 10.30 2.46 
 
 94.5 - 6.0 85 
 
 The constant b is the value of y when x = 0; that is, it is 
 the intercept of the line (prolonged, if necessary) on the 
 axis of Y, read off on the scale of ordinates chosen. Thus 
 from the plot it is seen that the line A' A" cuts the ordinate 
 through x = at b = 10.13. The desired equation con- 
 necting r and t is therefore 
 
 r = 0.0278 * + 10.13. (2) 
 
RECTIFICATION OF CURVES 47 
 
 Whenever the data are such that a long extrapolation of 
 the line is necessary in order to make it cut the ordinate 
 through x = 0, or when this ordinate falls off the plot, the 
 value of b is found as follows. Substitute the value of a 
 as determined above, together with values of x* and y' of some 
 point on the line, in equation (1) and solve for b directly. 
 
 It is to be noted here that the precision of the constants in 
 equation (2) is less than the precision of the original data. 
 Values of r computed by this formula cannot at best be more 
 precise than one or two parts in 1,000, while the observed 
 values were stated to be reliable to 3 parts in 10,000; in 
 other words, the full precision of the data has not been 
 utilized in the plot of the size here chosen. The procedure 
 by means of which the precision of the constants as above 
 determined may be increased, and another place of signi- 
 ficant figures obtained, will be explained later. See Resid- 
 ual Plot, p. 60. 
 
 Rectification of Curved Lines. When the plotted data do 
 not lie along a straight line, the form of the smooth curve 
 best representing the points will often suggest the relation- 
 ship sought. Thus curves resembling any of the conic sec- 
 tions or trigonometric functions are usually readily recog- 
 nized. In all such cases it is usually necessary to transform 
 the curve into a straight line in order to determine the con- 
 stants in its equation. Suppose from inspection of the curve 
 that the relation y = F (x) is suggested. If F (x) can be 
 factored or written in the form 
 
 F (x) = af(x) + b, 
 
 where a and b are numerical constants and f(x) contains no 
 constants, the function suggested can be very readily tested 
 graphically. This includes evidently the special cases when 
 a = 1 and when 6 = 0; i.e., the functions, 
 
 y = af(x); y = f(x); and y = f(x) + b. 
 
 In all of these cases let f(x) = z, and for each value of x of 
 the data compute the corresponding value of z. Construct 
 
48 GRAPHICAL METHODS 
 
 a new plot with values of z as abscissae and the correspond- 
 ing (observed) values of y as ordinates. The general equa- 
 tion of the new line will then be 
 
 y = az + b, 
 
 jt 
 
 or that corresponding to the above special cases, 
 
 y = az; y = z; and y = z + &, 
 
 all of which are equations of a straight line, the constants 
 of which may readily be determined as described above. 
 Whether the assumed equation y = af(x) + b represents 
 the experimental data or not can thus be judged by the mag- 
 nitude and sign of the deviations of the plotted data from the 
 straight line. If the correct function has been assumed, the 
 values of the constants a and b should be corrected by means 
 of a residual plot, provided the precision of the data war- 
 rants it. 
 
 Problem. Trigonometric Functions. Suppose that with a 
 certain galvanometer the following deflections are pro- 
 duced by the currents, 7, respectively, and it is desired to de- 
 termine the law of the galvanometer, i.e., the form of the 
 function / = F (6), so that the curre'nt corresponding to any 
 deflection may be computed. 
 
 OBSERVED DATA. 
 
 Deflection 6. Current I. z = tan 6. 
 
 10.17 0.0704 0.1794 
 
 19.27 0.1368 0.3496 
 
 29.16 0.2184 0.5580 
 
 40.47 0.3348 0.8532 
 
 48.45 0.4430 1.128 
 
 55.90 0.5780 1.477 
 
 The data plotted directly with values of I as ordinates and 
 as abscissas are found to lie along a curve A, Plot II., which 
 evidently suggests the relation I = a tan where a is con- 
 stant ; f or / = when 6 = and / approaches a very great 
 
TRIGONOMETRIC FUNCTIONS 
 
 49 
 
 value (infinity), for 6 = 90. Comparing the suggested 
 equation with y = af (x), we see / (x) = tan 0. Hence, to 
 test the suggested equation, we compute the value z = tan 
 for each observed value of 0, and construct a new plot with 
 the values of / as ordinates as before, and values of z as 
 
 Deflections 6 Curve A. 
 Tangent Curve B 
 
 PLOT II. 
 
 abscissae. The line best representing these data is shown 
 in B. This line must necessarily pass through the origin, 
 since the current and corresponding deflection of the gal- 
 vanometer approach the value zero simultaneously. The 
 galvanometer is seen to follow the law of tangents between 
 and 60. Since the line B passes through the origin, the 
 value of the constant a, i.e., the tangent which the line 
 
50 GRAPHICAL METHODS 
 
 makes with the axis of X, is readily found from the co-or- 
 dinates x" ?/" of a single point M to be 
 
 The value of a, in this particular case, can also be obtained 
 as follows: Since tan 45 = 1, it follows from y = atanfl 
 that y = a, for = 45 or z = 1; i.e., the ordinate of curve 
 A at 45 or the ordinate of curve B at z = 1 gives the value 
 of a directly. By this method we find a = 0.393 from M', 
 curve A, and a = 0.392 from M", curve B, both values being 
 in good agreement with that obtained in the usual way. 
 The desired formula for the galvanometer is, therefore, 
 
 I = 0.392 tan 0. 
 
 If the observations had been extended beyond = 60 
 and it were found that the points corresponding to these 
 data regularly deviated from the straight line, the conclusion 
 would be that the instrument followed the law of tangents 
 only within certain limits, which could be thus determined. 
 
 Problem. Reciprocal Functions. Suppose the volume v of 
 a definite mass of gas kept at constant temperature is de- 
 termined at various pressures p with the following results, 
 and it is desired to find the law connecting p and v; e.g., to 
 determine the form of the function p = / (v). 
 
 Pressure p Volume v 
 
 in cm. of Hg. in c. c. v 
 
 37.60 41.90 0.02380 
 
 39.35 40.13 0.02493 
 
 43.59 36.51 0.02739 
 
 47.50 33.67 0.02971 
 
 54.34 29.65 0.03373 
 
 56.26 28.63 0.03497 
 
 58.28 27.70 0.03610 
 
 Constructing a direct plot from these data, we obtain a 
 slightly curved line A, Plot III. The volume diminishes as 
 the pressure increases, but not proportionally, since the data 
 do not lie along a straight line. The curve suggests an equi- 
 
RECIPROCAL FUNCTIONS 
 
 51 
 
 lateral hyperbola referred to its asymptotes as axes, the 
 equation of which is xy = const. If this suggested relation 
 be the correct one, i.e., if vp = const., or, otherwise written, 
 
 p = const. f-Y by changing the variable from v to z = -, 
 the resulting equation becomes p = const, z, the equation of 
 
 60 
 
 SCALE 
 
 DIRECT PL|OT 
 RECIPROCAL 
 
 ORO. 
 ABS. 
 PLOT ORD. 
 
 38 
 
 I 4 CM. 
 I"- 2CC. 
 l"= -4 CM. 
 \"m 0.002 
 
 .020 
 
 .030 
 
 .09* 
 
 Curve A, Volumes. 
 
 Curve B, Reciprocal of Volumes 
 
 PLOT III. 
 
 a straight line. Constructing, therefore, a second plot, with 
 the same values of p as ordinates and the reciprocal values 
 of v as abscissae, we obtain curve B, which should be a straight 
 line if the gas in question follows Boyle's law within the 
 errors of the experiment. This is seen to be the case. If 
 it were not the case, a study of the deviations of the data 
 
52 GRAPHICAL METHODS 
 
 from the straight line would afford a proper means of dis- 
 cussion of the deviations of the gas from Boyle's law. 
 
 Another method of treating this problem would be to 
 compute the product pv for each pair of values of p and v 
 and then to discuss .the values of the product graphically 
 as follows. With values of p as abscissae construct a plot 
 with corresponding values of pv as ordinates. If the data 
 satisfy the relation pv = const, within the experimental 
 error, the best representative line will be parallel to the 
 axis of abscissae, with the values of pv distributed alternately 
 and about equally on either side. If, on the other hand, 
 the gas deviates from Boyle's law, as many gases do even 
 under ordinary conditions, and as all gases do at very great 
 values of p, the resulting curve will give information not 
 only as to the amount of the deviations, but also as to the 
 method of correcting the assumed simple relationship to 
 make it better conform with experimental facts. 
 
 The Logarithmic Method. Exponential Functions. If the 
 data of a direct plot are found to deviate continually from 
 a straight line, they may very often be represented by an 
 exponential equation of the form y = mx n where m and n 
 are constants which may have any value. Cases of this 
 kind are of very frequent occurrence, and it is, therefore, 
 of great importance to be able to test this relationship and 
 to determine the numerical values of the constants m and 
 n. This can always be done by means of a Logarithmic 
 Plot; i.e., a plot constructed with the values of the loga- 
 rithms of y as ordinates and the corresponding logarithms 
 of x as abscissae. For, if we take logarithms of both sides 
 of the equation y = mx", we have 
 
 log y = n log x -\- log m. 
 
 Changing the independent variables x and y in this equation 
 to x' and y' respectively, by putting x' = log x and y' = log y, 
 and writing b = log m, the equation becomes y' = nxf -|- &. 
 This is the equation of a straight line of which the in_ 
 tercept on the axis of Y is b = log m, and of which the 
 
LOGARITHMIC METHOD 53 
 
 natural tangent of the undistorted angle which it makes with 
 the axis of X is n. Hence the constants in the -original equa- 
 tion y = mx n may be obtained at once by looking out the 
 number ra whose logarithm, b, is the intercept of the straight 
 line on the axis of Y, and by determining the tangent which 
 the line makes with the axis of X. 
 
 93 
 
 ECT PLOT 
 6. PLOT 
 
 GALE 
 ORD. l" 
 ABS. I" 
 ORD. I" 
 ABS ! 
 
 VVO CM. 
 O.ISECONJD 
 O.2000 
 O.I 000 
 
 J20 
 
 . 
 
 T* 
 
 T6 
 
 PLOT IV. 
 
 T.7 
 
 0.6 
 10 
 
 O.7 
 Tft 
 
 0* 
 
 Here, again, the values of the constants m and n as thus 
 determined are usually reliable to not more than 0.5%, and 
 hence, if the orginal data warrant it, they should be further 
 corrected by means of a residual plot. 
 
 Problem. The logarithmic method will now be illustrated 
 by discussing data obtained for a body falling freely under 
 the influence of gravity. Suppose experiments gave the fol- 
 lowing values for the distance s, through which a ball fell 
 in the time t, and it is desired to deduce the law between s 
 
54 GRAPHICAL METHODS 
 
 and t; i.e., to find the equation by which the distance s can 
 be computed for any value of t. 
 
 OBSERVED DATA. 
 
 Distance s in Time tin s f = log s. t' = log t. 
 
 centimeters. seconds. ^ 
 
 30.13 0.2477 1.4790 1.3939 
 
 85.26 0.4175 1.9308 I- 6207 
 
 150.39 0.5533 2.1772 L7430 
 
 223.60 0.6760 2.3495 1.8300 
 
 274.20 0.7477 2.4381 1.8737 
 
 A direct plot A, Plot IV., of s and t, shows at once that s 
 and t are not proportional. The regular deviation from a 
 straight line suggests an exponential curve, i.e., s = ml". 
 To test this relation, we construct on ordinary co-ordinate 
 paper a "logarithmic plot" B with s' = log s as ordinates 
 and if = log t as abscissae. Convenient scales which dis- 
 tribute these values about 45 across the paper are V = 0.2 
 for the ordinates and 1" = 0.1 for the abscissae. It is to be 
 noticed that the values of t, being less than unity, lead to 
 values of log t with negative characteristics. The abscissae 
 are, therefore, laid off to the left of the origin as indicated, 
 the plot thereby lying in the second quadrant. The data 
 are seen to lie very closely along a straight line, the constants 
 of which are to be determined as described on page 46. Thus 
 the intercept of the line on the axis of Y is b = log m = 2.688, 
 whence m = 488. The tangent which the line makes with 
 the axis of X is found to be n = 1.995 or n = 2.00 within 
 the error of plotting. 
 
 The desired equation is, therefore, s = 488 t 2 - 00 . Since the 
 law of falling bodies is known to be s = Jgtf 2 , it follows that 
 %g = 488, or the mean value of g from the data, within the 
 
 cm 
 error of direct plotting, is g = 2 X 488 = 976 ^ . 
 
 sec. 
 
 If the data s and t are reliable to more than about 0.5 per 
 cent., the constants m and n should be corrected by means 
 of a residual plot. 
 
 Attention should be called to one important point in this 
 
LOGARITHMIC METHOD 55 
 
 connection. In constructing a plot like the above in which 
 the intercept on the axis Y is to be determined, it is con- 
 venient to choose the units in which the abscissae are ex- 
 pressed such that the resulting line cuts the axis of Y without 
 a long extrapolation. By a suitable choice of units this 
 condition can always be attained, for increasing or dimin- 
 ishing the unit expressing the abscissas by a multiple of ten 
 does not affect the slope of the line, but simply shifts it 
 parallel with itself to or from the origin. 
 
 Logarithmic Plotting-paper. When the constants of a num- 
 ber of exponential curves of the type y = mx n are to be 
 determined, a great saving of time and labor may be effected 
 by using so-called logarithmic co-ordinate paper. Four 
 quadrants of such paper are shown in Plot V. The length 
 OX is laid off equal to OF and put equal to 10 or some in- 
 tegral power of 10 units. This is then subdivided into spaces 
 such that the distances 1-2, 1-3, 1-4, etc., are proportional 
 to the logarithms of 2, 3, 4, etc. Thus the point numbered 
 2 is located not at two-tenths the distance from to Z, 
 as in ordinary plotting-paper, but at log 2 = 0.301 of the 
 distance OX. The rulings thus become more and more 
 crowded together as they proceed from to X and Y. Con- 
 tinuing the rulings beyond 10 in either direction, it is evi- 
 dent that the unit square XOY repeats itself indefinitely, 
 since the value of the logarithm of any quantity multiplied 
 by 10*, where & is a positive or negative integer, is equal 
 to the logarithm of the original quantity plus k. Thus 
 the point marked 0.2 is laid off at a distance equal to 
 log 0.2 = log (2 X 10- 1 ) = log 2 -- 1 = 1 + 0.301 to 
 the left of 0; i.e., just the distance OX to the left of the 
 point marked 2, etc. 
 
 It is evident that a series of values of x and y which satisfy 
 the equation 
 
 y = mx* 
 
 will, if plotted directly on logarithmic paper, lie along a 
 straight line; for the paper has the effect of locating the 
 data y and x at points proportional to their logarithms, and 
 
56 
 
 GRAPHICAL METHODS 
 
 it has been shown on page 52 that this leads to a straight 
 line. We are thus saved the labor of looking out logarithms 
 and locating them on rectangular co-ordinate paper as pre- 
 viously explained. Moreover, since the scales of ordinates 
 and abscissae are here necessarily equal, the slope of the 
 
 
 
 
 
 
 
 
 
 f 
 
 9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ft 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 4- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 J 
 
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 5 
 
 1 
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 s 
 
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 f 
 
 2 
 
 3 
 
 4 
 
 5 
 
 
 7 
 
 8 ,Q 
 
 V' 
 
 
 
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 ft- 
 
 
 
 
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 ET f 
 
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 R >. 
 
 
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 A5S: 
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 E JN 
 
 S 
 
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 Y' 
 
 | 
 
 
 
 
 
 
 
 
 PLOT V. 
 
 resulting line is undistorted, and, therefore, the tangent 
 which it makes with the axis of X is obtained by measur- 
 ing off with a scale the distance y" y' and x" x' of two 
 
 y" y' 
 
 points on the line and taking their ratio, ,, , = n. 
 
 gives the desired value of the exponent of x. 
 
 This 
 
LOGARITHMIC METHOD 57 
 
 To find the value of the constant m, we note that in the 
 equation 
 
 log y = n log x -f- log m 
 
 when x = 1, 
 
 log y = log m. 
 
 Hence, since values read off on logarithmic paper correspond 
 to the numbers of which the spacings are the logarithms, 
 the intersection of the logarithmic plot with the ordinate 
 through x = 1 gives at once the value of m, whereas with 
 rectangular plotting paper the intercept with the Y-axis 
 gives the value of log m. 
 
 The following difference between logarithmic plots, drawn 
 on rectangular and on logarithmic paper, should also be noted. 
 Suppose the line representing the data plotted on logarith- 
 mic paper intersects the ordinate not through x = 1, but 
 through x = 10*, where k is any positive or negative in- 
 teger. Let ra' be the value of the intercept. To obtain 
 the value of m in the equation y = mx n from m', we have 
 y =i m' for x = 10*, i.e., 
 
 m' = m 10*" 
 
 m f 
 
 = 
 
 In the case of rectangular paper, on the other hand, the 
 line intersects the ordinate through x = log 10* = k at m". 
 To obtain m from this intercept, we have 
 
 w" = log y = nk log 10 + log m 
 
 = nk -f- log m 
 Hence log m = m" nk. 
 
 The solution of the problem discussed on page 54, by the 
 use of logarithmic paper is shown on a reduced scale in Plot 
 V. Values of s and t are plotted directly and lie, as is seen, 
 along a straight line. It is convenient to express here values 
 of s in meters instead of centimeters. The tangent which 
 this line makes with the axis of X measured off directly 
 along OY and OX' is found to be 2.00. Extrapolating the 
 
58 GRAPHICAL METHODS 
 
 line to cut the ordinate OY through x = 1, we see the inter- 
 cept to be 
 
 m = 4.89. 
 
 The equation connecting s and t is therefore 
 
 s=,4.89Z 2 - 
 
 which agrees, within the error of plotting, with that pre- 
 viously obtained with rectangular co-ordinate paper, when we 
 remember that in the above equation s is expressed in meters 
 instead of centimeters. 
 
 Equations of the Form y = m (x -f- ft)". 
 
 A slightly more complicated relation than that represented 
 by the equation y = mx n , which may also be treated by the 
 logarithmic method, is that represented by the formula 
 
 y = m (x +ft Y 
 
 where ft is a constant. If a logarithmic plot be made with 
 data xfli, z 2 2/2> etc., which satisfy an equation of this form, 
 the points will not lie along a straight line for both large as 
 well as small values of the variables. Suppose it is found 
 that for large values of x and y the curve is practically 
 straight, but for small values it becomes curved. Under 
 these circumstances it is worth while to see if the best repre- 
 sentative logarithmic plot cannot be rectified into a straight 
 line by assuming an equation of the above form, for by 
 putting z = x + ft the equation reduces to 
 
 y = mz n 
 
 from which m and n are easily determined, if values of y 
 and z are plotted logarithmically. It is only necessary 
 therefore to find the value of the constant ft to be added to 
 all values of x. This may be found as follows. Select two 
 points a;^ and x%y 2 near the ends of the original logarithmic 
 plot. Compute the ordinate 2/3 of an intermediate point 
 such that 
 
 log 2/3 = i -Logj/1 + i log 2/2 
 or 2/3 = V 2/i2/2 
 and look out its corresponding abscissa X B on the line. 
 
PRECISION OF PLOTTING 59 
 
 Then it follows, if the data satisfy an equation of the form 
 = m ( x + )*, that 
 
 log fa + ft = * log fa + ) + * kg fa + ft 
 or * 3 + j8 = V 
 
 2 
 
 from which = 
 
 Having thus obtained , proceed in the usual manner to 
 determine m and n from a new logarithmic plot of the equa- 
 tion y = m 2 n where 2 = x + ft. 
 
 Equations of the Form y = m 10"*; y = m c** 
 Data satisfying equations of the form 
 
 -^ = m I0 nx and y = m e**, 
 
 where e is the base of Naperian logarithms, may also be 
 treated graphically by the following special logarithmic 
 method. Taking logarithms of these equations, we obtain 
 
 log y = nx -{ log m 
 and log y = Mnx -f- log m 
 
 respectively, where M = 0.4343 is the modulus for reducing 
 Naperian to common logarithms. If, on ordinary plotting 
 paper, values of y' log y are plotted as ordinates and the 
 unchanged values of x as abscissae, the resulting curves will 
 be straight lines; the intercept on the axis of Y f or x = 
 will give y f = log m and the tangents of the lines with the 
 axis of X will give the values of n and Mn respectively. 
 
 Precision of Plotting. The question now arises as to the 
 precision of the constants deduced from a direct or rectified 
 plot. In discussing this question, we will consider only errors 
 inherent in the process of plotting and interpolation, and in 
 the plotting-paper itself. 
 
 The error of estimating tenths of the smallest division, 
 together with the uncertainty introduced by the width of 
 the lines locating the data and the inaccuracies in the paper 
 due to errors in ruling and unequal shrinkage, make 0.02 
 inch a fair estimate of the extreme precision of reading or 
 
60 GRAPHICAL METHODS 
 
 plotting. If the plot be 10 inches on a side (about the maxi- 
 mum size ordinarily employed), the fractional precision 
 
 attainable cannot therefore be greater than about -~ = 
 
 0.002, or 0.2 per cent. A more probable estimate of the 
 precision ordinarily attained in direct plots is 0.4 to 0.5 per 
 cent. Constants deduced from a direct plot of the size 
 considered can therefore be relied upon only to this degree 
 of precision; that is, in general, to three significant figures, 
 with the fourth doubtful. 
 
 If the experimental data are reliable to four or more signifi- 
 cant figures (i.e., to 0.1 per cent, or better), some of the pre- 
 cision will evidently be sacrificed in the direct plot unless a 
 much larger plot be made. In order that the full precision 
 of such data may be utilized, the direct plot should be fol- 
 lowed by a so-called residual plot, by means of which the 
 constants first obtained can be corrected and rendered more 
 precise. By this procedure the precision of the graphical 
 method may be greatly extended. The procedure to be fol- 
 lowed in constructing a residual plot will now be considered. 
 
 Residual Plot. A residual plot is one in which the de- 
 viations of the observed data from the "best representa- 
 tive line" are plotted on an enlarged scale. It serves to 
 correct the position of this line among the points, to correct 
 the numerical value of the constants, and to test whether 
 the data follows the assumed law within the precision of the 
 measurements. It is constructed as follows. Substitute 
 in the equation y = ax + 6, deduced for the best represen- 
 tative line, which may be either a direct or rectified plot, 
 the observed values of. x, and compute the corresponding 
 values of y. The differences between these computed values 
 of y and the corresponding observed values are called the 
 residuals. A study of the sign and magnitude of these re- 
 siduals furnishes much valuable information regarding the 
 representative character of the "best line" chosen, and the 
 graphical discussion of these constitutes the residual plot. If 
 a plot be made (preferably on the same paper and with the 
 
RESIDUAL PLOT 61 
 
 same scale of abscissae as the straight line plot), with the 
 values of the residuals r = y (observed) y (computed) as 
 ordinates, and the corresponding values of x as abscissae, we 
 obtain a graphical representation of the deviations of the 
 observed data from the line assumed to best represent them. 
 To better study these deviations, they should be plotted on 
 a large scale. In effect, the process is to project the "best 
 representative line" horizontally and to magnify the devia- 
 tions of the plotted data from it. If it is found that the 
 plotted residuals lie alternately and about equal distances on 
 either side of the horizontal line passing through the zero of 
 the residuals, the conclusion is that the original line is the 
 best line which can be drawn to represent the data. In gen- 
 eral, however, it will be found that a new line can be drawn 
 among the residuals which will distribute them more nearly 
 alternately on either side. The values of the tangent a and 
 intercept 6, found for the original representative line, should 
 therefore be corrected by the values of the tangent and in- 
 tercept respectively of the new best representative line of the 
 residual plot, read off of course on the scales on which it is 
 plotted. In this way the original constants may be corrected 
 to the fourth significant figure. It is sometimes necessary 
 to follow the first by a second residual plot when extreme 
 precision is desired. 
 
 If the residuals are found to deviate systematically from 
 the straight line, the conclusion is that the data cannot be 
 represented by the line in question within the precision of 
 the measurements. In such a case a new formula should be 
 sought. 
 
 Illustration of a Residual Plot. The procedure to be fol- 
 lowed in making a residual curve or plot will be illustrated 
 by the data given in the Problem discussed in Plot I., p. 45. 
 The equation of the best representative straight line for these 
 data was found to be 
 
 r = 0.0278 t + 10.13. 
 
 To test whether this equation is the best which can be 
 obtained to represent the given data, we proceed to compute 
 
62 
 
 GRAPHICAL METHODS 
 
 the residuals, as described above, by substituting the observed 
 values of t and computing /. 
 
 t observed. 
 
 r observed. 
 
 T* computed. 
 
 r-r 1 first 
 residuals. 
 
 r* computed. 
 
 r-r" second 
 residuals. 
 
 10.50 
 
 10.421 
 
 10.422 
 
 m 
 
 0.001 
 
 10.413 
 
 + 0.008 
 
 29.49 
 
 10.939 
 
 10.949 
 
 0.010 
 
 10.946 
 
 0.007 
 
 42.70 
 
 11.321 
 
 11.317 
 
 +0.004 
 
 11.317 
 
 +0.004 
 
 60.01 
 
 11.799 
 
 11.798 
 
 +0.001 
 
 11.802 
 
 0.003 
 
 75.51 
 
 12.242 
 
 12.229 
 
 +0.013 
 
 12.237 
 
 +0.005 
 
 91.05 
 
 12.668 
 
 12.661 
 
 +0.007 
 
 12.673 
 
 0.005 
 
 2 (r r') 2 = 437 
 
 2 (r r") 2 = 188 
 
 Inspection of these residuals, column 4, affords valuable 
 information, but they can be better studied graphically, 
 especially if the number of observations is great. The scale 
 to be chosen for the ordinates should not be greater than 
 about 1 inch to 0.01 ohm, since this will permit the residuals 
 to be plotted directly without interpolation to the last place 
 of significant figures of the data, while by estimation the plot 
 can be read to the next place of figures; i.e., to 0.0001 ohm, 
 which is more than ten times the precision of the data. The 
 plot may conveniently be made on the same sheet as the direct 
 plot, using the same scale of abscissae as shown in Plot I., 
 p. 45. The heavy horizontal line through represents the 
 line A' A!' projected horizontally. The residuals, plotted on 
 a magnified scale, are connected by dotted lines. Inspection 
 shows that the positive residuals preponderate, and that a 
 new line B'B" can be drawn which will distribute the residuals 
 more nearly alternately on either side of it. The original 
 line A' A!' should evidently have been drawn with a slightly 
 greater inclination. The value of the intercept of the new 
 line B'B" on the axis of Y (on the scale of residuals) is 0.011. 
 The tangent of the angle which it makes with the axis of X 
 is obtained from the ordinates and abscissae of two points 
 B' and B" on the line respectively: thus x f 5.00, y' = 
 0.0095; and x" = 95.05, y" = 0.0135. Hence 
 y"-y' _. 0.0135 -(-0.0095) = _ 
 F^"? ~ 95.05-5.00 
 
INTERPOLATION FORMULA 63 
 
 Hence the constants of the original equation should be 
 
 corrected by these amounts, thus becoming 
 V = 10.13 0.011 = 10.119 
 a! = 0.0278 + 0.00025 = 0.02805. 
 
 The corrected equation connecting r and t is, therefore, 
 r = 0.02805 t + 10.119. 
 
 This represents the original data much better than the first 
 equation obtained, as may be seen from the sign and magni- 
 tude of the new set of residuals r r" computed from the 
 corrected equation and given in the last column of the table. 
 There is now seen to be no systematic deviation among the 
 residuals, and the sum of their squares is seen to be much less 
 than in the case of the residuals from the first equation. 
 
 Interpolation Formulae. It frequently happens that ex- 
 perimental data whose locus differs slightly but progres- 
 sively from a straight line cannot be represented by a two 
 constant formula of the general exponential form y mx n . 
 This is the case, for example, with data on the coefficient 
 of expansion of many substances over wide ranges of tem- 
 perature. To obtain an algebraic relation for such cases, 
 interpolation formulae of the general form 
 
 y = a + bx + ex 2 + dx 3 + . . . 
 
 are usually assumed. The number of terms to be taken in 
 this equation (i.e., the number of constants to be determined) 
 depends upon the precision of the data and on the extent of 
 the deviation of the curve representing them from a straight 
 line. 
 
 The values of the constants in such an equation are in gen- 
 eral best determined analytically. For this purpose it is 
 necessary to know at least as many pairs of values of x and 
 y as there are constants to be determined. Thus, if the 
 equation assumed to represent the data be y = a -f- bx + ex 2 , 
 it is necessary to know at least three pairs of values of x and 
 y which, substituted in the equation, will lead to three 
 simultaneous equations, from which the values of the three 
 unknown constants, a, &, c, can be at once determined by 
 
64 GRAPHICAL METHODS 
 
 elimination. It is seldom necessary to carry the series be- 
 yond the fourth term da?; in fact, three terms are sufficient 
 for most purposes. 
 
 In general, however, the experimental data furnish many 
 more pairs of values of x and y than there are constants to 
 be determined. In all such cases the most probable value of 
 the constants can be determined by the graphical procedure 
 described below or by the method of Least Squares. 
 
 Graphical Solution. Let x^ l} x^, . . . x n y n , be the 
 numerical values of pairs of observations on the variables x 
 and ?/, which are assumed to satisfy the equation 
 y a + bx + ex 2 . 
 
 Any three pairs of values substituted in this equation will 
 give three simultaneous equations from which a, b, and c 
 can be computed, but the values of these constants will vary 
 to a certain extent according to which sets of values of x and 
 y are chosen. The simplest procedure by which to obtain 
 the best or most probable values of a, 6, and c, is to plot all 
 values of x and y and draw the best representative line 
 among them. Then select three points on this line, one 
 near each end and one half-way between for convenience, 
 determine their ordinates and abscissae, and with these three 
 pairs of values form three simultaneous equations and com- 
 pute a, b, and c. Having obtained the constants in this 
 manner, they may be further corrected by computing re- 
 siduals and studying these by means of a residual plot, 
 although this requires both care and judgment. A more 
 exact although more laborious method of procedure is the 
 analytical solution of the equation by the method of Least 
 Squares. 
 
 Least Square Solution As before, let Xjy lt z 2 7/ 2 , 
 x n yn, be numerical values of the observations, and 
 y = a -f- bx + ex 2 
 
 the equation the constants of which are to be determined. 
 This may be written 
 
 y a bx ex 2 = 0. 
 
LEAST SQUARE SOLUTION 65 
 
 If the observed values x, y, were free from all experimental 
 errors and the equation represented the law connecting them, 
 each pair would exactly satisfy the equation, with proper 
 numerical values of the constants. This, however, is not the 
 case, since all observations are liable to indeterminate error. 
 Hence, if the observations be substituted in the equation, the 
 right member will not in general equal zero, but will differ 
 from zero by some small quantity v called the residual error, 
 which may be plus or minus. Thus, by substituting the 
 observations in the assumed equation, we get the following 
 so-called "observation equations": 
 
 y l a bx l cxf = v lt 
 a bx cx 2 = v 
 
 yn a bXn CXn* = V, 
 
 from which the most probable values of the constants a, 6, 
 c, are to be determined. By the principle of Least Squares 
 those values of a, 6, c, are the most probable which make 
 the sum of the squares of the residual errors v a mini- 
 mum; i.e., those which make the value of 2,v 2 = v-f + 
 i> 2 2 + . . . Vn 2 a minimum. The expression 2,v 2 is a function 
 of the quantities a, 6, c, and the condition that it shall be a 
 minimum is that its first differential coefficient with respect 
 to these variables shall be zero, and its second differential 
 coefficient positive. The latter test need not be applied, 
 however, as inspection will distinguish between maximum 
 and minimum values, the limit of the former being evidently 
 infinity. 
 
 Applying this condition to the above observation equa- 
 tions, we have 
 
 d 
 
66 GRAPHICAL METHODS 
 
 Substituting the values of Vj, i? 2 , etc., and differentiating, we 
 obtain 
 
 & C 2 ) = 0, 
 a &*i czi 2 ) 
 
 a - 6Xn - CXn 2 ) Xn = 0, 
 
 a 6a^ cx)x+. 
 , 
 
 which may be simplified to the equations 
 
 = 0. 
 = 0. 
 0. 
 
 These are called the "normal equations," from which the 
 values of the constants a, b, c, may be computed by the 
 ordinary methods of elimination, there being now the same 
 number of equations as unknowns. It will readily be seen 
 that the process of substituting the values of ^y, 2xy, 
 2xy 2 , etc., in the normal equations and the subsequent so- 
 lution of the equations for the constants is a tedious proc- 
 ess, the labor involved increasing rapidly with the number 
 of constants to be determined. 
 
 For further details regarding the method of Least 
 Squares consult Bartlett's The Method of Least Squares, 
 Wright's Treatise on the Adjustment of Observations, or 
 Merriman's Least Squares. For special Graphical Meth- 
 ods see Peddle's The Construction of Graphical Charts. 
 
PART III. 
 SOLUTION OF ILLUSTRATIVE PROBLEMS, 
 
 AND 
 
 PROBLEMS. 
 
SOLUTION OF ILLUSTRATIVE PROBLEMS. 
 
 Before proceeding to the numerical solution of a pre- 
 cision problem, the student should first decide the follow- 
 ing questions: 
 
 First. Is the formula to be discussed in the simplest 
 form for precision treatment? It frequently happens, by 
 the omission of certain terms the deviations in which 
 evidently produce a negligible effect on the final result, 
 that an apparently complex formula can be reduced to 
 a more convenient form. If it can be reduced to a pro- 
 duct function, this should always be done. 
 
 Second. From a consideration of the form of function 
 to be discussed, a decision should be made as to which 
 method it is better to employ in the solution; that is, 
 whether to use the general " deviation method " involving, 
 differentiation of the function or the fractional or "in- 
 spection" method. 
 
 Third. Having decided these questions, the statement 
 of the problem should be studied; that is, all given data 
 should be systematically written down and inspected to 
 see if they are in the proper form for applying the method 
 of solution decided upon. If this is not the case, numer- 
 ical deviations S or A should be changed over into their 
 
 corresponding fractional deviations ; or -v^, or vice 
 
 m j\fJL 
 
 versa, as the case may be. Only after the problem has 
 been consistently stated should the actual solution be 
 begun. 
 
 These general directions are illustrated below by the 
 solution of several typical problems. 
 
70 SOLUTION OF ILLUSTRATIVE PROBLEMS 
 
 Problem i. Given the following mean values of the 
 weight of four substances with their respective deviation 
 measures: 
 
 MI = 3147.226 gms. A.D. = 0.312 gm. 
 
 102 = 100.4211 gms. reliable to 0.015 per cent. 
 
 w 3 = 1 .3246 gms. Probable error P.E. = 0.001 1 gm. 
 
 w 4 = 604.279 gms. reliable to 1 part in 5000. 
 
 (a) Indicate any superfluous figures in the above meas- 
 urements, considering each independent of the others. 
 
 Each quantity should be carried out to two places of un- 
 certain figures as indicated by the two significant figures in 
 its average deviation (Rule III., p. 24). The average devia- 
 tion of each measurement should therefore be computed for 
 each measurement if it is not already given. Computing 
 the average deviations and applying Rules I., II., and III., it 
 will be seen that the correct number of figures to be retained 
 is as follows: 
 
 101 = 3147.23 gms. A.D. = di= 0.31 gm. 
 w 2 = 100.421 gms. A.D. = 5 2 = 0.015 gm. as 100 4^ = 0.015. 
 
 J-UU 
 
 w&= 1.3246 gms. A.D. = 5 3 = 0.0013 gm. as P.E. = 0.85 A.D. 
 W) =604.28gms. AJ>.- ,- 0.12 gm. a.^'- JL . 
 
 (b) Which is the most and which the least precise of 
 these measurements? 
 
 When the quantities whose precision is to be compared are 
 not of approximately equal magnitude, their relative pre- 
 cision is found by comparing their fractional or percentage 
 deviations, but not their average deviations or probable 
 errors. Hence with the above data we must compute the 
 fractional or percentage deviation of each of the quantities. 
 
 For wi, 100 = 100 = 0.010 per cent. ; 
 
 100 = 100 = 0.015 per cent. ; 
 
 Wz iOO 
 
 wa, 100 - = 100 p = 0.10 per cent. ; 
 
 W, 100 ^ = 100^ = 0.020 per cent. 
 
 Therefore, the order of precision is wi t wz, w*, ws. It is to be 
 noted that, although ws is weighed to a much smaller fraction 
 
SOLUTION OF ILLUSTRATIVE PROBLEMS 71 
 
 of a gram than any of the other quantities, it has by far the 
 largest percentage deviation, and is to be regarded, there- 
 fore, as the least precise measurement. 
 
 (c) Find the sum of the measurements and its devia- 
 tion measure, retaining the proper number of significant 
 figures in the computation. 
 
 M = Wi -{- w 2 + w 3 -\- w. 
 
 The quantity having the largest A.D. is wi, its average de- 
 viation being 0.31 gm. In the units chosen to express the 
 measurements, the first and second decimal places are un- 
 certain. Therefore, by Rule IV., page 24, two decimal places 
 only should be retained in each of the other quantities to be 
 
 added. 
 
 3147.23 gms. 
 
 100.42 gms. 
 
 1.32 gms. 
 
 604.28 gms. 
 
 M= 3853. 25 gms. 
 
 The resultant deviation A of th3 sum M is 
 
 A! = ~ . 5i = 61 = 0.31 gm. 
 
 OWl 
 
 Similarly, A2 = 52 = 0.015 gm., which is negligible. 
 A 3 = 63 = 0.0013 gm., which is negligible. 
 A 4 =6 4 =0.12 gm. 
 
 Therefore, A = vV + 5 4 2 = Y/olH 2 + 0.12* = 0.33 gm. 
 
 (d) Find the product of the four quantities and its 
 precision measure. 
 
 M = WI . w 2 . w 3 . w 4 . 
 
 By Rule V., page 24, the least precise factor is 103, which 
 is good to only 0.10 per cent., and in which, therefore, five 
 significant figures should properly be retained in a compu- 
 tation, the last two being uncertain. Five figures should 
 likewise be retained in each of the other factors, and five 
 place logarithms should be used in the computation. 
 wi =3147.2 log =3.49793 
 wz = 100.42 log = 2.00182 
 tos=l .3246 log = 0. 12209 
 W* = 604.28 log = 2.78124 
 log M= 8.40308 
 or M= 252980000. gms. 4 
 
72 SOLUTION OP ILLUSTRATIVE PROBLEMS 
 
 The precision of M should be computed by the fractional or 
 inspection method, as it is a product function. By referring 
 back to problem (6) , it will be seen that the percentage pre- 
 cision of w\, wz, and w* is between five and ten times as great 
 as that of ws. Hence practically all of the uncertainty in the 
 product will result from the deviation in this factor alone. 
 As M is directly proportioned to the first power of ws, 
 
 100^= 100 6 i 
 
 M ws 
 
 and therefore the percentage deviation of the product 
 
 100 ~ = 0.10 per cent. Hence A = M x ~ = 250000. glnZ 4 
 
 (e) Suppose the quantities are to be combined by the 
 formula 
 
 M = Wi X w 2 w$ X w. 
 
 Compute M and its deviation measure. 
 
 Before substituting the values of w in the actual calcula- 
 tion of M, it is always well to note the approximate value 
 of the terms involved. From inspection of the data it is 
 evident that wix wz = 3 10000. approximately, and, as the 
 least precise factor w% is good to 0.015 per cent., this product 
 should be computed to six significant figures, of which the 
 last two will be uncertain. The second place of uncertain 
 figures thus falls in the units' place; anything beyond this 
 is, therefore, negligible. The term ws X w\ = 600. approxi- 
 mately, and, as this is to be subtracted from wiXwz, it is 
 useless to compute it beyond the units' place, i.e., three sig- 
 nificant figures are sufficient. Six place logarithms should 
 therefore be used in computing w\ x wz, and three place 
 logarithms, short multiplication, or a slide rule in computing 
 
 108 X W4. 
 
 Thus wi = 3147.23 gms. log w\ = 3.4979284 
 
 wz= 100.421 gms. log w 2 = 2.0018245 
 
 log wix wz =5.4997529 
 
 .' .wiXw 2 = 316048. gms. 2 
 Ws = 1.32 gms. log ws = 0.121 
 
 tC4 = 604. gms. log W* = 2.781 
 
 log wsXW4= 2.902 
 
 =798. gms. 2 
 
 AT =316048 gms. 2 -798 gms?= 315250 gms. 2 
 
SOLUTION OF ILLUSTRATIVE PROBLEMS 73 
 
 To determine the precision of M, we note that, if the for- 
 mula be treated in the form M = w\ x w% - wz X w*, we must use 
 the general differential method and find the effect of each 5 
 on M, and take the square root of the sums of the squares. 
 It is evident, however, since ws is good to 0.10 per cent., 
 and W4 to 0.02 per cent., that the first three significant figures 
 of the product ws . w are known exactly, and, therefore, the 
 deviations in ws and w* introduce no uncertainty in the final 
 result M . The whole uncertainty comes from the measure- 
 ments MI and v>2. As WB x w\ is also numerically small com- 
 pared with wi x wz we may, in the precision discussion, neglect 
 
 it and write 
 
 M =wi x W2 approximately, 
 
 and obtain the precision of M by the fractional method. 
 The resultant fractional deviation in M is 
 
 M 
 
 But ~ = - - = 0.00010 
 
 M wi 
 
 and T?-- =0.00015 
 
 M 102 
 
 Therefore, =>= 0.0001 J I 2 + 1.5 2 = 0.00018 
 
 or 100 M = 0.018 per cent. 
 
 and A = 320000. gms! 2 x 0.00018 = 58 gms? ; 
 
 that is, the value of M is uncertain by 58 units. 
 
 The above problem illustrates the manner in which the 
 number of significant figures to be retained in a measure- 
 ment depends entirely upon the way in which it enters 
 into the computation. Thus in (d) w 3 was required to 
 its full precision, while in (e) it might have been measured 
 much less precisely. 
 
 Problem 2. It is desired to determine the amount of 
 heat H generated in one hour by a certain incandescent 
 lamp, together with its deviation expressed in calories 
 and in per cent. Suppose mean measurements obtained 
 by an ammeter and voltmeter give 7 = 2.501 0.012 
 amperes, and E 109.72 0.34 volts respectively, and the 
 time of opening and closing the circuit is uncertain by 
 
74 SOLUTION OP ILLUSTRATIVE PROBLEMS 
 
 0.5 second at each operation. The value of H expressed 
 in calories is 
 
 # = 0.2390 /.#.. 
 
 The expression for H is, as it stands, a simple product 
 function of the variables /, E, aiid t, and cannot be further 
 simplified. The problem should therefore be solved by the 
 fractional method. 
 The data given are 
 
 7 =2.501 amp. 8 f = 0.012 amp. 
 E = 109.72 volts. 8 = 0.34 volt. 
 t = t<2 h = l hour = 3600 sec. 
 5^ = 5*2 = 0.50 sec., but d t is unknown. 
 
 The desired results are the value of H, its deviation A in 
 
 A 
 
 calories, and its percentage deviation 100 . 
 
 H 
 
 The first step is to change the given deviations 5 in each 
 component into their respective fractional deviations. 
 
 The fractional deviation in 7 is = = 0-0048. 
 
 The fractional deviation inE ia^ = ~ = 0.0031. 
 
 E 110. 
 
 To find the fractional deviation in t, we must first compute 
 its numerical deviation. Since t = t% t\ and 5^ = 5< 2 = 0.5 sec., 
 the numerical deviation in t is 5 = y /r A 1 2 +A 2 2 ; 
 
 dt 
 but AI = .5^=5^ 
 
 dt 
 
 and A 2 = 7 . ot a == 8t a 
 
 dt 2 2 2 
 
 Therefore 8 = V^O.5 2 + 0.5 2 = 0.70 sec., 
 and the fractional deviation in t is 
 
 This is seen to be negligible compared with the fractional 
 deviation in the current 7 and voltage E (see page 31). 
 Hence the resultant fractional deviation in H is 
 
 H 
 
SOLUTION OF ILLUSTRATIVE PROBLEMS 75 
 
 By inspection of the formula for H, since / and E both enter 
 as first power factors, 
 
 ^ = *' = 0.0048 
 
 * = * = 0.0031. 
 H E 
 
 Therefore, ^ = V/O0048 2 + O^OSl 2 = 0.0057, 
 
 or the percentage deviation in H is 100 -^ = 0.57 per cent. 
 
 ti 
 
 In computing the value of H, we note that the least pre- 
 cise factor is the current which is uncertain by 0.48 per cent., 
 and hence should be carried in the computation to four sig- 
 nificant figures. H should therefore be computed by four 
 place logarithms, four figures being retained in each factor, 
 including the constant for transforming Joules to calories. 
 H = 0.2390 x 2.501 x 109.7 X 3600 
 
 = 236100 calories. 
 
 The numerical deviation A in H is obtained at once from its 
 fractional deviation, as 
 
 A = 240,000 x 0.0057 = 1400 calories. 
 
 Problem 3. The candle power of a gas flame is meas- 
 ured against a standard candle by means of a photo- 
 meter, the flame being placed at the end of a bar 100 
 inches from the candle. Suppose the mean of a series 
 of disk settings gave a = 20.17 0.27 inches, a being the 
 distance of the disk from the candle. Compute the 
 candle power of the flame and its deviation, assuming 
 that the candle is burning at its normal rate. 
 
 L (flame) _(100-q) 2 
 L (candle) a* 
 
 /100 a\ 2 /100 aV 
 This may be written L = L / - / = ( - / 
 
 as L = 1 candle power = 1 c.p. = constant. 
 100. 20.17 
 
 20.17 
 
 L is a function of a single variable a; it is to be noted in 
 the precision discussion that the function cannot be regarded as 
 a fraction in which a deviation in the numerator is inde- 
 
76 SOLUTION OF ILLUSTRATIVE PROBLEMS 
 
 pendent of the deviation in the denominator. The formula 
 may, however, be simplified by writing it in the form 
 
 100 -a 
 
 L'=\IL=\X^- 
 
 and solving first for the deviation in L', after which the 
 desired deviation in L can be Easily found. 
 
 The deviation A' in L', due to a deviation 5 = 0.27 inch in 
 
 A , d/ lx !00-vA Ix 100 5 
 
 da \ a / a 2 
 
 = l(c.p.)* =r= x 0.27 cm. = 0.068 (c.p.)*. 
 20 cm. 
 
 To find now the deviation in L, we may proceed in either of 
 two ways: 
 
 First, General Method : 
 
 L' = /Z = L 
 
 where A is the desired deviation in L. 
 Therefore, A = 2y/L . A' 
 
 = 2x y/16 c.pTx 0.068 (c.p.)i 
 = 0.54 c.p. 
 
 Second, Fractional Method: 
 
 The fractional deviation in L' corresponding to A' is 
 A' 0.068 0.068 
 
 and, since L'L^, the fractional deviation in L' is one-half 
 the fractional deviation in L; 
 
 1 A 
 
 2 ' L' 
 
 2x 0.017 = 0.034. 
 
 .. . 
 
 Li 
 
 Therefore, A = 0.034 x L = 0.034 x 16 c.p. = 0.54 c.p. 
 
 The same result would of course be obtained by applying 
 the general differentiation method to the original formula 
 for L, but the resulting value of the differential coefficient is 
 somewhat more complicated. 
 
SOLUTION OP ILLUSTRATIVE PROBLEMS 77 
 
 On the assumption that the average light emitted by 
 the candle during the measurements is equal to one candle 
 power, the candle power of the gas flame is 15.67 + 0.54 c.p.; 
 
 54 
 i.e., it is known to only 100 -^- = 3.4 per cent., although the 
 
 27 
 original photometer setting a is good to 100 - = 1.4 per cent. 
 
 Problem 4. Suppose the index of refraction n of a 
 substance is to be determined by measuring the angle of 
 incidence i and the angle of refraction r of a ray of light. 
 If approximate measurements give z = 45 and r = 30, 
 how precisely should these two angles be measured to 
 give n to 0.2 per cent.? 
 
 The formula for n is 
 
 _sin i 
 '""sin r* 
 
 As this is not a product function of the variables i and r, we 
 must use the general deviation method if we treat the for- 
 mula in the above form. If, however, we change variables 
 to x and y, letting x = sin i and y = sin r, the formula becomes 
 n=-, to which we may apply the fractional method of solu- 
 
 tion for finding the allowable deviations in x and y. Having 
 done this, however, we still have two new problems to solve 
 by the general method, namely, the determination of the 
 deviations in i and r from the equations x = sin i and y = 
 sin r. Both methods lead, of course, to the same result. 
 We will solve the problem both ways. 
 First Solution. General Method: 
 
 sin i 
 n=- . 
 sm r 
 
 Given 100 - = 0.2; i=45; r = 30; to find ,- and S r . 
 
 We must first find the value of the A in n from the pre- 
 scribed percentage precision before proceeding to the solu- 
 tion. This necessitates knowing the approximate value of n, 
 which is easily obtained from the data; 
 
 sin 45 1.1 
 " = S30 = ^ ^ 2 = 1A *P P rox,mately. 
 
 Hence A = 1.4 X 0.0020 = 0.0028. 
 
78 SOLUTION OF ILLUSTRATIVE PROBLEMS 
 
 Distributing thia deviation between i and r by equal effects, 
 
 A 0.0028 ^n 
 A/= A r = = 7- = 0.0020. 
 
 Y/2 
 
 dn , cosi 
 
 But *--" 
 
 . - . = A< JET _ 0.0020 - - 0.0014 
 cos i cos 45 
 
 To express this allowable deviation in the angle in degrees, 
 we note that 
 
 1 = - = 0.017 radians, 
 
 therefore, * = = 0.082, or 4.9'. 
 
 dn , sini cosr 
 Similarly, A, = - . 5 r = sin2? . . r 
 
 = 0.0020 . qn = 0.00082, 
 
 cosr sin 45 cos 30 
 
 00082 
 or, expressed in degrees, ^.~r2f =0.048, or 2.9'. 
 
 The solution shows, therefore, that, to obtain a precision 
 of 0.2 per cent, in n, an instrument should be used capable of 
 reading to at least 3'. In practice one graduated to read to 
 minutes would be chosen. 
 
 Second Solution. Fractional Method: If we put z=sin i 
 and y=sin r, then n=-, and we may use the inspection 
 method as follows. The prescribed fractional deviation is 
 stated to be not greater than - = 0.0020. Hence, distribut- 
 ing this deviation between x and y by equal effects, we have 
 
 n n y/2 n y2 
 But by inspection of n =- it is seen that 
 
 Hence - = 0.0014 and ^ = 0.0014. 
 
 We have now two new problems to solve, namely, to find 
 8 t and 5 r from the above values of the allowable precision in 
 
SOLUTION OF ILLUSTRATIVE PROBLEMS 79 
 
 x and y. As z=sin i is a trigonometric function, we must 
 go back to the general differentiation method to find the 
 deviation in i corresponding to a deviation & x in x. 
 
 As ^ = 0.0014, 
 
 we have 8* = 0.0014 x = 0.0014 sin 45 = 0.0010. 
 
 d sin i 
 
 di 
 
 cos i . 
 
 6,=- = 0.0010 -r - =0.0014, 
 
 cos i V2 
 
 or in degrees 5, = = 0.082 = 4.9'. 
 
 Similarly, to find 5 r , given y = sin r and = 0.0014, we 
 
 have 
 
 d y = 0.0014 y = 0.0014 sin 30 = 0.00070. 
 
 Also 5,= . 5r= cos r . 5 r 
 
 - a00070 * - 
 
 or in degrees 6, = = .048 = 2.9'. 
 
 In this problem there is evidently no saving of labor by 
 transforming the function to the product form and first 
 using the fractional method, as the ultimate solution necessi- 
 tates going back to the general method. 
 
PROBLEMS 
 
 Questions and Problems. 
 
 1. Explain the terms: precision measure; deviation measure; 
 constant error; residual error; probable error; mean error; huge 
 error; indeterminate error; weighted mean; weights. 
 
 2. What is the geometrical significance of the average 
 deviation, probable error, and mean error in relation to the 
 curve representing the law of chance? 
 
 3. Is it practicable to reduce the average deviation of a 
 mean result to any desired value by increasing the number 
 of observations? Why? 
 
 4. If the mean value of the length of a rod computed from 
 nine measurements is 24.213 cm. A.D. = 0.012 cm., how many 
 more similar observations should be made in order that the 
 A.D. =0.0060 cm.? 
 
 5. Under what circumstances may an observation properly 
 be rejected, and why? 
 
 6. What determines the number of places of significant 
 figures to be retained at any part of a computation? 
 
 Under what circumstances should four, five, or seven-place 
 logarithms be used in a computation? 
 
 7. Do the number of significant figures in a result depend 
 upon the position of the decimal point? Explain reasons for 
 your answer. Does the precision of a result depend upon the 
 position of the decimal point? Why? 
 
 8. Explain why when adding or subtracting observed 
 quantities we are governed by decimal places in rejecting 
 figures, but, when multiplying or dividing, places of signifi- 
 cant figures, regardless of the decimal point, must be con- 
 sidered. 
 
82 PROBLEMS 
 
 9. Given the following measurements and their deviation 
 measures: 
 
 241.631 gms. reliable to 0.25 per cent. 
 
 1620.124 gms. A.D. = 0.81 gms. 
 
 10.005 gms. reliable to, 1 part in 1,000 
 
 7141.110 gms. P.E. = ti.603 gms. 
 
 (a) Indicate all superfluous figures. 
 
 (6) Which is the most and which the least precise quantity, 
 and why? 
 
 (c) How many figures should be retained in each quantity 
 in computing their sum? 
 
 (d) How many figures should be retained in each quantity 
 in computing their product? 
 
 (e) How many figures should be retained in each quantity 
 in computing the difference between the product of first two 
 and the product of the last two quantities? 
 
 10. Given the following measurements of the length of a 
 rod and their precision measures: 
 
 24.316 cm. A.D. = 0.028 cm. 
 
 24.3922 cm. fractional precision 12 parts in 1,000 
 
 24.358 cm. P.E. = 0.0121 cm. 
 
 24.3091 cm. reliable to 0.11% 
 
 24.3100 cm. A.D. 0.0172 cm. 
 
 (a) Indicate any superfluous figures. 
 
 (6) Indicate the order of reliability of the results. 
 
 (c) Compute the relative "weights." 
 
 (d) Compute the weighted mean. 
 
 11. The precision measures of four independent determina- 
 tions of the modulus of elasticity of steel are expressed as 
 follows: 1st, 0.60 per cent.; 2d, 2 parts in 1,000; 3d, 
 
 probable error = 4.2 ^ m "- 4th, average deviation = 10. gm '- 
 mm. mm. 
 
 The modulus of elasticity is about 20,000 2 '- 
 
 171171. 
 
 Which measurement is the most reliable? Find the rela- 
 tive weights of the four determinations. 
 
PROBLEMS 83 
 
 12. Independent determinations of the rate of a pendulum 
 by different methods and observers gave 
 
 0.70061 sec. A.D. = 0.00023 sec. 
 0.70047 " A.D. =0.00069 " 
 0.70056 " correct to 0.092% 
 0.70051 " P.E. = 0.00039 sec. 
 
 (a) Which is the most reliable observation? 
 (6) Compute the relative "weights" of the observations. 
 (c) Indicate how to compute the best representative value 
 of all the observations. 
 
 13. The dimensions of a right cylinder are found to be as 
 follows: 
 
 length = 12.183 cm. A.D. = 0.024 cm. 
 diameter = 4.242 cm. A.D. = 0.021 cm. 
 
 Find the volume and its deviation measure, indicating 
 proper number of significant figures at each step of the com- 
 putation. 
 
 14. The diameter of a spherical globe is found to be approxi- 
 mately six inches. If the average diameter varies by 0.1 per 
 cent., what variation in cubic inches will this produce in the 
 volume? If the variation in the diameter is 0.0020 inch, what 
 uncertainty will result in the volume? 
 
 15. The length of a physical pendulum is given by the 
 expression 
 
 where r = \ diameter of ball = -, 
 
 and h = distance of knife edge from the top of the ball. 
 
 Suppose h = 100.031 cm. A.D. = 0.027 cm. 
 
 d= 6.256cm. A.D. = 0.022 cm. 
 
 (a) Find the resultant deviation in I. 
 
 (b) How many significant figures should be retained in 
 each term of the formula, in computing I, and why ? 
 
 (c) Is it necessary to consider the third term in this 
 formula in a precision discussion of If Why? 
 
84 PROBLEMS 
 
 16. The mean time of nine coincidences of two beating 
 pendulums is 50.43 seconds. A.D. = 0.11 second. The 
 standard pendulum beating true seconds gains on the other 
 pendulum. Compute the true time of vibration of the latter 
 and its precision measure. 
 
 17. The time of swing of a half-second pendulum is measured 
 to 0.20 per cent. The length is measured to 0.10 mm. Find 
 the precision of the computed value of g. 
 
 18. Measurements with a spherometer gave for a certain 
 
 lens 
 
 h= 1.22110mm. A.D. = 0.00088 mm. 
 r = 35.735 mm. A.D. = 0.061 mm. 
 
 Compute R, the radius of curvature of the lens retaining 
 the proper number of significant figures throughout compu- 
 tation. What deviation would be introduced in R by a devi- 
 ation of 0.00088 mm. in h ? By a deviation of 0.061 mm. in r f 
 What would be the combined effect of these deviations on 
 
 R? Is the term negligible in the above precision discus- 
 sion, and if so, why? 
 
 19. A 10 ohm coil is standard at 15 C. What will be its 
 resistance at 30 C. and how precise will this be known if the 
 temperature is determined to 0.1 degree? 
 
 -4- 0.00388 (t 15)1 . 
 
 How closely must t be known in order that the resistance 
 may be depended upon to 0.02 per cent.? 
 
 20. The electromotive force of a Clark cell is 
 E t = 1.4340 [~1 0.00078 (t 15)1 . 
 
 How closely must t be measured in order that E may be 
 known to 0.05 per cent.? 
 
PROBLEMS 00 
 
 21. Given a coil of wire the resistance of which at 15 C. may 
 be assumed to be exactly 100 ohms. The wire is an alloy the 
 resistance of which at t may be computed by the expression 
 
 Rt=Ris fl + 0.00051 (t 15)1. 
 
 Calculate the amount of heat H generated in the wire in an 
 hour's run by a current of 11.273 amperes, if the wire is main- 
 tained at a temperature of 45 C. H=I 2 Rt. 
 
 If the temperature is known to 1.0, the duration of the 
 run to 1.0 second, and the current to 0.011 ampere, calcu- 
 late the deviation in H in Joules and in per cent. Are the 
 deviations in any measurements negligible, and if so, why? 
 
 22. The sensitiveness of a spirit level is = y X 206265". 
 
 (a) If I = 380. mm. approx. and h = 0.0597 mm. 0.0018 
 mm., how many figures would you keep in the value 01 
 
 (b) How many would you use in the value of the radian, 
 206265? How many in h? 
 
 (c) If I and 206265 are to introduce no deviation in the re- 
 sult in comparison with h, what is the allowable percentage 
 deviation, and how few figures may be used in each ? 
 
 (d) If you wished to determine with greater precision 
 than above, in which measurement would you use greater 
 care? 
 
 23. In calibrating a burette by drawing off water for each 
 10 cc. and weighing it in a flask, how precise should the weigh- 
 ings be made if the calibration is to be reliable to 0.01 cc.? 
 Suppose the calibration were made with mercury, how close 
 should the weighings be made? Is it necessary to note the 
 temperature of the water in this calibration and why? 
 
 24. It is desired to calibrate a flask to hold 1,000 cc. at 
 20 C., the calibration to be reliable to 0.5 cc. What weight 
 (using brass weights sp. gr. = 8.5) would you add to the 
 weight of the flask empty, so that it would exactly balance 
 the water having the desired volume? How precise would 
 you make your weighings? Would it be necessary to take 
 
86 PROBLEMS 
 
 into account the barometer reading in figuring the correc- 
 tion for reduction to vacuo? Why? 
 
 25. The per cent, of silver in a certain alloy is determined 
 gravimetric ally by weighing the amount of silver present as 
 silver chloride. Suppose an -''analysis gave the following 
 results: 
 
 Wt. of alloy 1.43252 gms. 0.00014 gm. 
 " "AgCl 0.19513 gms. 0.00021 gms. 
 
 Compute the per cent, of silver in the alloy and the preci- 
 sion with which this would be known. 
 
 Atomic weight silver == 107.93 0.02 
 Atomic weight chlorine = 35.45 0.03 
 
 26. A specific gravity determination by Archimedes' prin- 
 ciple gave the following results: 
 
 weight of substance in air = 10.2431 gms. .0004 gm. 
 weight of substance in distilled water at 20 C. = 9.0422 
 
 gms. .0010 gm. 
 density of water at 20 C. = 0.99825. 
 
 Compute specific gravity and its precision measure, using 
 the correct number of places of significant figures through- 
 out the computation. Is the correction for reduction of 
 weighings to vacuo negligible in this case, and why? 
 
 27. Given 
 
 wt = ( Ww - 6) [l + k (t - 20)]- 
 
 where w 2 o = 27.6231 grams = weight of bottle plus water. 
 b = 12.6193 grams = weight of bottle. 
 k = 0.000026 per degree C. 
 
 Dt and D 2 o = density of water at t and 20 respectively. 
 
 The density D diminishes 0.02 per cent, per degree C. 
 What is the greatest allowable value which t may have and 
 the correction term due to expansion of glass be negligible; 
 i.e., affecting the value of (W ZQ b) by less than 0.0001 gm.? 
 Is the correction due to change of density negligible for t = 22 
 
PROBLEMS 87 
 
 and why? If D 20 = 0.99827 and t = 25, compute wt, using 
 as few figures as practicable. 
 
 28. With what precision would it be necessary to meas- 
 ure the length and diameter of a right cylindrical column 
 approximately 12 inches long and 6 inches in diameter in 
 order to determine the volume to 0.10 per cent.? What 
 deviation in cubic inches would this correspond to? 
 
 29. If it is desired to compute the area of a circle approxi- 
 mately 10 sq. cm. in area to 5 parts in 10,000, how precise 
 should the diameter be known? How many places should be 
 retained in IT in the computation? 
 
 30. If the volume of a sphere is computed from a measure- 
 ment of its diameter, how precise should the latter be measured 
 in order that the former may be reliable to 0.1 per cent.; to 1 
 part in 500? 
 
 If the sphere is approximately six inches in diameter, what 
 precision in the diameter does a precision of 0.1 per cent, in the 
 volume require? If the A.D. of the diameter is 0.022 inch, 
 what is the precision of the volume? 
 
 31. The ratio of the length of the arms of a balance is given 
 by the expression 
 
 _ length right arm */Wi 
 
 length left arm VW7' 
 
 where Wi and W r are the observed weights of a given mass 
 when weighed in the left and right hand pan, respectively. If 
 the mass weighs approximately 20 grams, with what precision 
 must Wi and W r be determined in order that the ratio r may 
 be reliable to 0.01 per cent.? 
 
 32. The specific gravity of a platinum alloy is desired to 
 0.1 per cent. The method based on Archimedes' principle is 
 to be used. The weight of substance in air is about 42 grams. 
 The specific gravity is approximately 21. To what fraction 
 of a gram should each weighing be made? What corrections 
 should be applied? 
 
88 PROBLEMS 
 
 33. A freely falling body passes through the distance 
 h= %gt 2 in t seconds. It is desired to determine g to one- 
 tenth per cent, from a measurement of h and t. 
 
 (a) How precisely should h and t be measured? 
 
 (b) If h = 4 meters, what will be the allowable numerical 
 deviation in h and t? 
 
 (c) If h 4 meters and it is found that Bh = 1.0 mm. and 
 8* = 0.0014 second, what will be the resultant deviation in gf 
 
 34. If g is to be computed from the mean of a series of nine 
 observations on the time of swing of a pendulum, the length of 
 which is I = 1 meter, what must be the percentage precision 
 of t and I in order that g be precise to 0.1 per cent.? What 
 will be the value of ad. and A.D. in the case of t? How many 
 figures should be retained in IT? 
 
 35. The gas constant R is given by the formula 
 
 R = 2? where T = t + 273. 
 
 (a) How precisely must p, v, and t be measured in order 
 that R may be reliable to 0.1 per cent.? 
 
 (6) How large a deviation will be introduced in R by a 
 variation of 1 C. in the temperature alone at 20 C. ? 
 
 36. The indicated horse power (I. H. P.) of an engine is 
 
 P X L XA X N 
 given by the expression 7. H. P. = . 
 
 Suppose for a given engine the approximate values of these 
 quantities are as follows: 
 
 P = 50 Ibs. per sq. in. = mean effective pressure. 
 
 L = 2 feet == length of stroke. 
 
 A = Area of piston, the diameter, D, of which is 16 inches. 
 
 N = 100 = number of strokes per minute. 
 
 How, precisely, should P, L, D, and N be determined in order 
 that the computed horse power of the engine may be reliable 
 to 1 per cent.? To one-quarter of a horse power? 
 
 37. A rectangular steel rod of breadth b and depth d is 
 supported at its ends and loaded at its centre by a weight W. 
 
PROBLEMS 89 
 
 If I is the length of the rod between its supports and a is 
 the deflection at the centre, 
 
 WP 
 4Ebd 3 ' 
 
 where E is the modulus of elasticity. 
 
 Suppose measurements of these quantities gave 
 
 b= 8.113mm. 8 b = 0.042 mm. 
 
 d = 10.50 mm. 8d = 0.025 mm. 
 
 I = 1.000 meter precise to one part in 5,000 
 
 a = 2.622 mm. precise to 0.25% 
 
 W = 2 kgms. precise to 0.02 gram 
 
 Compute 
 
 (a) E, the modulus of elasticity. 
 
 (6) The deviation in E due to each component. 
 
 (c) The resultant deviation in E. 
 
 38. What would be the allowable deviations in the meas- 
 urement of b, d, I, and a for the beam defined in problem 37 
 if the value of E is to be reliable to \ per cent.? Assume 
 the error in the value of W to be negligible. 
 
 Do you think this precision could be readily attained? 
 Why? 
 
 39. The modulus of elasticity E of a cylindrical wire, of 
 length I, cross section q, which when loaded with a weight 
 w is elongated by an amount a, is given by the expression: 
 
 = ^. 
 aq 
 
 a = mi m where mi and m are mean micrometer read- 
 ings when the wire is under a load of w kilograms and no 
 load, respectively, q = \trd? where d is the mean diameter 
 of the wire. 
 
 Given the following data: 
 
 I = 200.11 cm. a.d. = 0.05 cm. 
 
 w = 10 kilograms accurate to 1 gram. 
 
 mi = 9.4255 mm. A.D. = 0.0024 mm. 
 
 m = 8.2233 mm. A.D. = 0.0012 mm. 
 
 d = 1.002 mm. correct to 0.2% 
 
90 
 
 PROBLEMS 
 
 (a) Compute the average deviation and the percentage de- 
 viation of q. 
 
 (b) Compute the deviation of a. 
 
 (c) Compute E, using proper number of significant figures. 
 
 (d) Compute resultant percentage error of E. 
 
 (e) Are any of the above dat^ more precise than necessary 
 and why ? 
 
 40. (a) If it is desired to determine E for the above sample 
 of wire to 0.5 per cent., how precisely should the compo- 
 nents a and q be measured, assuming that the deviations in 
 I and w can be made negligible? 
 
 (b) How precisely must d be measured to fulfil this condi- 
 tion? 
 
 (c) How precisely must Wi and m be measured? 
 
 (d) Do you think this degree of precision could be readily 
 attained, and why? 
 
 41. It is desired to determine the focal length of a plano- 
 convex lens from spherometer measurements, n 1.5 ap- 
 proximately. 
 
 . 1 
 
 (a) If F is desired to 0.1 per cent., how precisely must n 
 and R be determined? 
 
 (b) If preliminary measurements give r = 40 mm. and 
 h = 4 mm., approximately, how precisely should r and h 
 be determined to fulfil condition a? 
 
 42 The formula for computing the wave length by means of 
 a diffraction grating is for second order spectra X = . sin 0. 
 
 ifl 
 
 If the grating is ruled 17,296 lines to the inch, and a prelimi- 
 nary measurement gives = 53 approximately for the sodium 
 band, with what precision must be measured, and how 
 should the optical circle be graduated to give X to one part 
 in 10,000? 
 
PROBLEMS 91 
 
 43. / = K tan $ for a tangent galvanometer, whose galva- 
 nometer constant is 
 
 K = 1.963 0.002. 
 
 If the deviation in reading any deflection is 8$ = 0.1, com- 
 pute the value of / and its deviation measure for = 45 and 
 = 60. 
 
 44. The heating effect H of an electric current is to be 
 calculated from measurements of 7, t, and E, or 7, t, and R. 
 
 #=0.23907^; 77 = 0.2390 PRt. 
 
 (a) If H is to be reliable to 0.2 per cent., what must be the 
 percentage precision of each of the component measurements 
 in both cases? The allowable deviation in each measurement? 
 
 (b) If E= 110 volts and is measured to 0.5 per cent., 
 R = 100 ohms and is reliable to 1 part in 1,000, 81 = 0.03 
 amp. and 8t = 0.2 sec.; compute the precision of a ten-minute 
 run by each method, and state which method you consider the 
 better. 
 
 45 Given the following approximate values for a specific 
 heat determination: 
 
 s = sp. ht. of substance 
 w = wt. of substance = 300 grams 
 w = " " water = 500 grams 
 
 Wi = " " calorimeter = 100 grams 
 i = sp. ht. of calorimeter = 0.095 
 (ts t*) = fall of temperature of substance 
 
 = 100 - 20 = 80 C. 
 (t z ti = rise of temperature of calorimeter 
 
 plus water = 20 - 15 = 5 C. 
 (W Q + k) (I* - 
 
 7 : 
 w (t, - 
 
 , where k = w l 
 
 If s is desired to 0.5 per cent., with what precision should 
 each of the quantities w, w , Wi, Si, t f , ti, and 2 2 , be measured 
 or known? 
 
 Can the deviations in any of the above quantities be readily 
 made negligible? Solve the problem under these conditions. 
 
92 PEOBLEMS 
 
 46. The resistance of a metal bar is determined by measur- 
 ing the drop in potential between its ends and the correspond- 
 ing current flowing through it. Suppose mean measurements 
 gave 
 
 I = 11.431 amperes 0.022 ampere, 
 E= 0.5073 volt J).010 volt. 
 
 Compute the resistance of the bar and its deviation in ohms. 
 
 47. Suppose the mean of nine settings of a photometer disk 
 gave 
 
 a = 240.1 cm. 1.1 cm., 
 
 a being the distance of the disk from a gas flame whose candle 
 power, L, is desired, and 300 a, the distance of the disk 
 from a standard candle. Assuming the candle to be burning 
 at its normal rate, compute the candle power of the flame and 
 its percentage deviation. 
 
 L (flame) a 2 
 
 V (candle) "(300 a) 2 ' 
 
 48. If E = K log e , compute the deviation in E due to a 
 
 Pa 
 
 deviation of 0.1 per cent, in the measurement of p\ and p z 
 respectively, K being a constant. If the deviations in pi and 
 Pa were known to be of the same sign, what would be their 
 resultant effect on E? 
 
 49. Suppose a stop watch loses regularly at the rate of 1.2 
 seconds in 15 minutes and the uncertainty of stopping and 
 starting the second hand is 0.1 second in each case. If a 
 runner makes a one-mile record in 4 minutes 35f seconds by 
 the watch, calculate the true time and its deviation measure 
 in seconds and per cent. 
 
 50. To what fraction of a gram should a piece of aluminum 
 be weighed in air and in water, respectively, in order that its 
 computed specific gravity may be reliable to one part in a 
 thousand? The approximate weight of the sample in air is 
 25 grams, and its specific gravity is approximately 2.7. 
 
 51. Four independent observers determine a resistance of 
 about one ohm to 0.10 per cent., 0.0030 ohm, one part in five 
 
PROBLEMS 93 
 
 hundred, and with a probable error of 0.00085 ohm, respectively. 
 What are the relative reliabilities of the determinations and 
 their respective weights? 
 
 52. A certain 32 c.p. lamp takes a current of approximately 
 1 ampere at 110 volts. If its resistance under these conditions 
 is desired to 0.5 ohm, how precise should the current and 
 voltage be measured? 
 
 53. The mean of sixteen comparisons of a yard scale and a 
 standard meter scale gave the result: 
 
 1 yard = 0.91449 m.; A .D.= 0.00011 m. 
 
 If there is a residual error of 0.007 cm. in the meter scale 
 after all sources of constant error have been corrected for, to 
 what fraction of an inch is the value of the yard reliable? If the 
 meter scale were correct at C. and used at 20 C., how large 
 a constant error (in inches) would result if its expansion were 
 neglected, assuming that it expanded 0.0025 per cent, per 
 degree? 
 
 54. Two tuning forks of slightly different pitch "beat" with 
 each other 45.31 0.45 times per minute when sounded simul- 
 taneously. If the standard fork makes exactly 256 vibrations 
 per second, what is the rate of the second fork and its deviation 
 measure, assuming it to be of lower pitch than the standard? 
 
 55. Given the following data on the specific gravity of a sub- 
 stance lighter than water. 
 
 Weight of substance in air = 10.1321 gms. 0.0002 gm. 
 
 Weight of substance and sinker 
 
 immersed in water at 20.0 C. = 8.4418 gms. 0.0020 gm. 
 Weight of sinker immersed in 
 
 water at 20.0 C. = 10.4522 gms. 0.0010 gm. 
 
 Compute the specific gravity of the substance referred to water 
 at 20.0 C. and its numerical and percentage deviation. If the 
 density of water decreases 0.18% between 4 C. and 20 C., 
 compute the specific gravity referred to water at 4 C. How 
 precisely should the temperature of the water at 20 be deter- 
 mined if this last reduction is to be reliable to 0.05%? 
 
94 PROBLEMS 
 
 56. Suppose a pendulum approximately 550 feet in length is 
 swung from the top of the Washington Monument. How pre- 
 cisely should the length and time of vibration be measured in 
 inches and seconds, respectively, if the value of g computed 
 
 from these data is to be reliable -to 0.50 7 -r^ ? 
 
 (second) 2 
 
 57. Suppose that fifty 16 c.p. incandescent lamps are used on 
 the average 2 hours a day for 4 weeks. Each lamp takes 0.5 
 ampere at 110 volts. Calculate the total amount of energy 
 consumed in Joules. If this energy is measured by determin- 
 ing the average current and voltage by an ammeter and volt- 
 meter each of which reads uniformly 2% too high, how much 
 overcharge would there be on the lighting bill if the cost of 
 electrical energy is 10 cents per kilowatt hour? 
 
 If the average voltage and current are uncertain by 1.1 
 volts and 0.0050 ampere, respectively, what uncertainty 
 would there be in the total electrical energy consumed, and ia 
 its value in dollars and cents? 
 
 58. If the weight of a substance in air is 
 
 w = 49.7631 0.0012 grams 
 
 and it is desired to calculate its weight W in vacuo, how closely 
 should the density 8 of the substance, the density A of the 
 balance weights, and the density <r of the air in the balance 
 case be known in order that the correction to be added to w may 
 be computed to the nearest 0.0005 gram? 
 
 Given 8 = 0.8, A = 8.4, and <r = 0.0012 approximately. 
 
 59. The formula for calculating the temperature t of a mer- 
 curial thermometer when corrected for stem exposure is 
 
 t = ti + 0.000156 (tit a )n, 
 where ti = observed temperature, 
 
 t a = temperature of exposed stem, 
 n = number of degrees exposed at t a . 
 Suppose ti = 330 0.5, t a = 30 0.5, and 
 
 n = 200 5 approximately. 
 Compute t and its deviation measure. 
 
PROBLEMS 95 
 
 60. The formula for the latent heat of vaporization is 
 
 w 
 
 If the rise in temperature t z / 1 = 25 15 = 10, the fall in 
 temperature of steam t s 2 = 100 25 = 75, the condensed 
 steam = 20 grams, and the water equivalent wo + k = 1,200 
 grams approximately, calculate how precisely you would de- 
 termine each of these four factors if r is to be reliable to 0.5%. 
 (Assume r= 540 cal. for steam.) 
 
 rr 
 
 61. If I = - -, what is the allowable deviation in 
 
 -tti + 
 
 ohms in Ri, Rt, and R s , respectively, if / is to be reliable to 
 0.1% and the deviation in E is negligible? The approximate 
 values of the resistances are Ri = 10 ohms, and R 2 = 100 ohms, 
 and #3 = 1 ohm. If the above formula were 
 
 what would be the solution under the same conditions? 
 
 62. Suppose Ii=Ki sin and h=K* tan 0. The value of 
 Ki and K* are each known to 0.1%; = 45; 8 d = Q.l. 
 Compare the precision of / as calculated by the two formulae. 
 
 63. Suppose you had three areas, each equal approximately 
 to 10 sq. in., one being a circle, one a square, and one an 
 equilateral triangle. How precisely would you have to know 
 the diameter of the first and the average value of one side of 
 the last two, in order that the computed area may be reliable 
 to 0.01 sq. in.? 
 
 64. The capacity of a spherical condenser is given by the 
 
 /i 01 
 
 expression C= . Suppose 
 
 n = 10.0010 cms. 0.0019 cm.; 
 
 r z = 15.0000 cms. 0.0044 cm. ; 
 
 K= 2.0130 0.0012. 
 
 Calculate C and its resultant deviation measure; the percentage 
 deviation in the numerator; the deviation measure of the de- 
 nominator expressed in centimeters. 
 
96 PROBLEMS 
 
 65. Given a triangular lot of land whose three sides are a, b, c, 
 respectively, a = 120 feet approximately; b = 180 feet approx- 
 imately. The angle between a and 6 is about 45. It is 
 desired to find the area of the triangle to 0.12%, How pre- 
 cisely should a, b, and be measured? If a, b, and B are meas- 
 ured with the above precision, what would be the precision of 
 the computed value of c ? 
 
 Area = \ab sin 6 
 
 66. A sample of sodium chloride, NaCl, is analyzed by pre- 
 cipitating with silver nitrate and weighing the silver chloride, 
 AgCl. 
 
 Wt. of sample of NaCl = 0.5017 gm. 0.0005 gm. 
 Wt. of AgCl = 1.1817 gms. 0.0012 gm. 
 
 Assuming the atomic weight of sodium and chlorine to be 
 known to 0.1 per cent, and that of silver to be known to 
 0.03 per cent., calculate the per cent, of chlorine present in 
 the sample and the precision with which you can depend upon 
 this result. 
 
 Atomic weights: 
 
 Na = 23.00 01 = 35.46 Ag= 107.88 
 
 67. The index of refraction of a substance is given by the 
 
 sim 
 
 expression n = - 
 smr 
 
 If i = 45 10' and r = 305', compute the value of n and 
 its average and percentage deviation. 
 
 68. The index of refraction of a substance as determined by 
 the Pulfrich refractometer is given by the relation 
 
 n = \/N 2 sin 2 0, 
 
 where N is a constant of the instrument and is the measured 
 angle. 
 
 If N = 1.62100 0.00005 and = 45, approximately, howpre- 
 cisely should be measured in order that n may be reliable to 
 0.1 per cent.? 
 
 69. Given two resistances A and B, each known to -fa per 
 cent; A = 2 ohms approximately, B = l ohm approximately. 
 
PROBLEMS 97 
 
 If settings on a slide wire bridge are IA = 660 mm., and Zs 
 mm. approximately and these are each known to 0.2 mm., 
 compute the deviation measure of n. 
 
 1 A B1 B A 
 
 n 
 
 A B 
 
 70. What considerations determine the precision with which 
 the constants in the equation of a straight line may be ob- 
 tained by the graphical method? With plotting-paper 10 
 inches on a side and ruled in twentieths of an inch, explain 
 what the extreme limits of precision attainable are. 
 
 71. What is a residual plot, and what is its use? Explain 
 fully. 
 
 72. How would you determine by a graphical method the 
 value of the constants a, b, and c in the equation 
 
 y = a + bx + ex 2 
 from a series of values, x\ t y\, x z , y 2 , etc.? 
 
 73. What is a logarithmic plot and to what* class of prob- 
 lems is it applicable? Explain fully its use by an illustration, 
 first, using rectangular co-ordinate paper, and second, using 
 logarithmic paper. 
 
 74. The heat H generated in a coil of wire in a given time 
 varies as the square of the current /. How would you test 
 this relation graphically by a series of determinations of H 
 and If 
 
 75. How would you test graphically the law that the de- 
 flection of a beam loaded at its centre and supported at its 
 ends is proportional to the cube of its length? 
 
 76. From a series of measurements on the time of vibration 
 and corresponding length of a pendulum explain how you 
 could obtain the mean value of g by treating the data graphi- 
 cally, i-i 
 
 77. From a series of measurements on the strength of cur- 
 rent 7 flowing in a circuit of resistance R to which a variable 
 
98 PROBLEMS 
 
 (measured) electrometer force E is applied, explain how you 
 would find the mean value of R by treating the data by a 
 graphical method. 
 
 78. Suppose a current / is measured by a tangent galva- 
 nometer for which I=K tan 0>* From a series of values of 
 7 and corresponding values of explain how you would find 
 K by a graphical method. 
 
 79. The formula for the focal length of a lens is 
 
 iJUi. 
 / P P' 
 
 From a series of measurements on p and p f explain how 
 you would find the mean value of / by means of a graphical 
 method. 
 
APPENDIX. 
 
 TABLE I. 
 MATHEMATICAL CONSTANTS. 
 
 Quantity. 
 
 Value. 
 
 Remark. 
 
 Common 
 Logarithm. 
 
 c 
 
 2.71828 
 
 Base of Napierian, natural 
 or hyperbolic logarithms. 
 
 0.434295 
 
 loglO 
 
 0.434295 
 
 Factor to multiply Napier- 
 ian logs to convert into 
 Common logs. 
 
 9.637784 
 
 1 
 
 loglO 
 
 2.30259 
 
 Factor to multiply Com- 
 mon logs to convert into 
 Napierian logs. 
 
 0.362216 
 
 7T 
 
 3.14159 
 
 Ratio of circumference to 
 diameter. 
 
 0.497150 
 
 1/7T 
 
 0.318310 
 
 Reciprocal cf ir. 
 
 9.502850 
 
 X 2 
 
 9.86960 
 
 Square of IT. 
 
 0.994300 
 
 V^ 
 
 1.77245 
 
 Square root of x. 
 
 0.248575 
 
 1 radian 
 
 5717'45" 
 
 57.2958 = 206265" = arc 
 equal to radius. 
 
 
 1 degree 
 
 ^ = 0.017,4533 radian 
 
 
 
100 
 
 APPENDIX 
 
 TABLE II. 
 APPROXIMATION FORMULA. 
 
 It frequently happens in a computation that a factor of the 
 general form (1 a) n enters wjaere n is a constant and a is 
 a quantity whose numerical value is small compared with 
 unity. In such cases the approximate value of the factor 
 given by the first two terms of its expansion -may usually be 
 substituted in place of the factor itself without introducing an 
 appreciable error in the result, and the computation becomes 
 thereby decidedly simplified. If the factor is of the form 
 (m a') n where a' is small compared with m, it may be written 
 
 - 1 =m n (1 a) n and so reduced to the first form. 
 
 Table II. contains the approximate forms of the factor for 
 several values of n, together with the error which would be 
 introduced by using the approximation. 
 
 Factor. 
 
 Approximate 
 Form. 
 
 Resulting 
 Error. 
 
 Computed Error 
 if d = 0.01. 
 
 (1 a)* 
 
 la 
 
 n(n-l) 
 
 
 2 
 
 (1 a) 2 
 
 1 2a 
 
 a 2 
 
 0.0001 
 
 (1 a) 3 
 
 1 3a 
 
 3a 2 
 
 0.0003 
 
 (1 0)4 
 
 1 4a 
 
 6a 2 
 
 0.0006 
 
 (1 a)J 
 (1 a)i 
 (1 a)-* 
 
 1 \a 
 1 ia 
 1 T a 
 
 |a 2 
 -i<* 2 
 a 2 
 
 0.00001 
 0.00001 
 0.0001 
 
 (1 a)- 
 
 1 =p2a 
 
 3a 2 
 
 0.0003 
 
 (1 a)- 
 
 Iqp^a 
 
 fa 2 
 
 0.00004 
 
 (1 a)-* 
 
 1 T ia 
 
 fa' 
 
 0.00002 
 
 For (1 a) (1 6) (1 c) use (1 a b c). 
 
 For 
 
 I mi ma use mi - when mi and m% are nearly equal. 
 
 m 
 
APPENDIX 
 
 101 
 
 TABLE III. 
 SQUARES, CUBES, RECIPROCALS. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Recip. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Recip. 
 
 1.0 
 
 1.00 
 
 1.00 
 
 1.00 
 
 5.5 
 
 30.3 
 
 166. 
 
 .182 
 
 1.1 
 
 1.21 
 
 1.33 
 
 0.909 
 
 5.6 
 
 31.4 
 
 176. 
 
 .179 
 
 1.2 
 
 1.44 
 
 1.73 
 
 .833 
 
 5.7 
 
 32.5 
 
 185. 
 
 .175 
 
 1.3 
 
 1.69 
 
 2.20 
 
 .769 
 
 5.8 
 
 33.6 
 
 195. 
 
 .172 
 
 1.4 
 
 1.96 
 
 2.74 
 
 .714 
 
 5.9 
 
 34.8 
 
 205. 
 
 .169 
 
 1.5 
 
 2.25 
 
 3.38 
 
 .667 
 
 6.0 
 
 36.0 
 
 216. 
 
 .167 
 
 1.6 
 
 2.56 
 
 4.10 
 
 .625 
 
 6.1 
 
 37.2 
 
 227. 
 
 .164 
 
 1.7 
 
 2.89 
 
 4.91 
 
 .588 
 
 6.2 
 
 38.4 
 
 238. 
 
 .161 
 
 1.8 
 
 3.24 
 
 5.83 
 
 .556 
 
 6.3 
 
 39.7 
 
 250. 
 
 .159 
 
 1.9 
 
 3.61 
 
 6.86 
 
 .526 
 
 6:4 
 
 41.0 
 
 262. 
 
 .156 
 
 2.0 
 
 4.00 
 
 8.00 
 
 .500 
 
 6.5 
 
 42.3 
 
 275. 
 
 .154 
 
 2.1 
 
 4.41 
 
 9.26 
 
 .476 
 
 6.6 
 
 43.6 
 
 287. 
 
 .152 
 
 2.2 
 
 4.84 
 
 10.6 
 
 .455 
 
 6.7 
 
 44.9 
 
 301. 
 
 .149 
 
 2.3 
 
 5.29 
 
 12.2 
 
 .435 
 
 6.8 
 
 46.2 
 
 314. 
 
 .147 
 
 2.4 
 
 5.76 
 
 13.8 
 
 .417 
 
 6.9 
 
 47.6 
 
 329. 
 
 .145 
 
 2.5 
 
 6.25 
 
 15.6 
 
 .400 
 
 7.0 
 
 49.0 
 
 343. 
 
 .143 
 
 2.6 
 
 6.76 
 
 17.6 
 
 .385 
 
 7.1 
 
 50.4 
 
 358. 
 
 .141 
 
 2.7 
 
 7.29 
 
 19.7 
 
 .370 
 
 7.2 
 
 51.8 
 
 373. 
 
 .139 
 
 2.8 
 
 7.84 
 
 22.0 
 
 .357 
 
 7.3 
 
 53.3 
 
 389. 
 
 .137 
 
 2.9 
 
 8.41 
 
 24.4 
 
 .345 
 
 7.4 
 
 54.8 
 
 405. 
 
 .135 
 
 3.0 
 
 9.00 
 
 27.0 
 
 .333 
 
 7.5 
 
 56.3 
 
 422. 
 
 .133 
 
 3.1 
 
 9.61 
 
 29.8 
 
 .323 
 
 7.6 
 
 57.8 
 
 439. 
 
 .132 
 
 3.2 
 
 10.2 
 
 32.8 
 
 .313 
 
 7.7 
 
 59.3 
 
 457. 
 
 .130 
 
 3.3 
 
 10.9 
 
 35.9 
 
 .303 
 
 7.8 
 
 60.8 
 
 475. 
 
 .128 
 
 3.4 
 
 11.6 
 
 39.3 
 
 .294 
 
 7.9 
 
 62.4 
 
 493. 
 
 .127 
 
 3.5 
 
 12.3 
 
 42.9 
 
 .286 
 
 8.0 
 
 64.0 
 
 512. 
 
 .125 
 
 3.6 
 
 13.0 
 
 46.7 
 
 .278 
 
 8.1 
 
 65.6 
 
 531. 
 
 .123 
 
 3.7 
 
 13.7 
 
 50.7 
 
 .270 
 
 8.2 
 
 67.2 
 
 551. 
 
 .122 
 
 3.8 
 
 14.4 
 
 54.9 
 
 .263 
 
 8.3 
 
 68.9 
 
 572. 
 
 .120 
 
 3.9 
 
 15.2 
 
 59.3 
 
 .256 
 
 8.4 
 
 70.6 
 
 593. 
 
 .119 
 
 4.0 
 
 16.0 
 
 64.0 
 
 .250 
 
 8.5 
 
 72.3 
 
 614. 
 
 .118 
 
 4.1 
 
 16.8 
 
 68.9 
 
 .244 
 
 8.6 
 
 74.0 
 
 636. 
 
 .116 
 
 4.2 
 
 17.6 
 
 74.1 
 
 .238 
 
 8.7 
 
 75.7 
 
 659. 
 
 .115 
 
 4.3 
 
 18.5 
 
 79.5 
 
 .233 
 
 8.8 
 
 77.4 
 
 681. 
 
 .114 
 
 4.4 
 
 19.4 
 
 85.2 
 
 .227 
 
 8.9 
 
 79.2 
 
 705. 
 
 .112 
 
 4.5 
 
 20.3 
 
 91.1 
 
 .222 
 
 9.0 
 
 81.0 
 
 729. 
 
 .111 
 
 4.6 
 
 21.2 
 
 97.3 
 
 .217 
 
 9.1 
 
 82.8 
 
 754. 
 
 .110 
 
 4.7 
 
 22.1 
 
 104. 
 
 .213 
 
 9.2 
 
 84.6 
 
 779. 
 
 .109 
 
 4.8 
 
 23.0 
 
 111. 
 
 .208 
 
 9.3 
 
 86.5 
 
 804. 
 
 .108 
 
 4.9 
 
 24.0 
 
 118. 
 
 .204 
 
 9.4 
 
 88.4 
 
 831. 
 
 .106 
 
 5.0 
 
 25.0 
 
 125. 
 
 .200 
 
 9.5 
 
 90.3 
 
 857. 
 
 .105 
 
 5.1 
 
 26.0 
 
 133. 
 
 .196 
 
 9.6 
 
 92.2 
 
 885. 
 
 .104 
 
 5.2 
 
 27.0 
 
 141. 
 
 .192 
 
 9.7 
 
 94.1 
 
 913. 
 
 .103 
 
 5.3 
 
 28.1 
 
 149. 
 
 .189 
 
 9.8 
 
 96.0 
 
 941. 
 
 .102 
 
 5.4 
 
 29.2 
 
 157. 
 
 .185 
 
 9.9 
 
 98.0 
 
 970. 
 
 .101 
 
102 
 
 APPENDIX 
 
 TABLE IV. 
 FOUR PLACE LOGARITHMS. 
 
 3 2 
 
 
 
 
 
 
 
 
 
 
 
 PROPORTIONAL PARTS. 
 
 i 
 
 O 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 j 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 0000 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 4 
 
 8 
 
 12 
 
 17 
 
 21 
 
 25 
 
 29 
 
 33 
 
 37 
 
 11 
 
 0414 
 
 0453 
 
 0492 
 
 0531 
 
 0569 
 
 0607 
 
 0645 
 
 0682 
 
 0719 
 
 075.5 
 
 4 
 
 8 
 
 11 
 
 15 
 
 19 
 
 23 
 
 2G 
 
 30 
 
 34 
 
 12 
 
 0792 
 
 0828 
 
 0864 
 
 0899 
 
 0934 
 
 0969 
 
 1004 
 
 1038 
 
 1072 
 
 1106 
 
 3 
 
 7 
 
 10 
 
 14 
 
 17 
 
 21 
 
 24 
 
 28 
 
 31 
 
 13 
 
 1139 
 
 1173 
 
 1206 
 
 1239 
 
 1271 
 
 1303 
 
 1335 
 
 1367 
 
 1399 
 
 1430 
 
 3 
 
 6 
 
 10 
 
 13 
 
 IG 
 
 19 
 
 23 
 
 26 
 
 29 
 
 14 
 
 1461 
 
 1492 
 
 1523 
 
 1553 
 
 1584 
 
 1614 
 
 1644 
 
 1673 
 
 1703 
 
 1732 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 15 
 
 1761 
 
 1790 
 
 1818 
 
 1847 
 
 1875 
 
 1903 
 
 1931 
 
 1959 
 
 1987 
 
 2014 
 
 3 
 
 6 
 
 8 
 
 11 
 
 14 
 
 17 
 
 20 
 
 22 
 
 25 
 
 16 
 
 2041 
 
 2068 
 
 2095 
 
 2122 
 
 2148 
 
 2175 
 
 2201 
 
 2227 
 
 2253 
 
 2279 
 
 3 
 
 5 
 
 8 
 
 11 
 
 13 
 
 IG 
 
 18 
 
 21 
 
 24 
 
 17 
 
 2304 
 
 2330 
 
 2355 
 
 2380 
 
 2405 
 
 2430 2455 
 
 2480 
 
 2504 
 
 2529 
 
 2 
 
 5 
 
 7 
 
 10 
 
 12 
 
 15 
 
 17 
 
 20 
 
 22 
 
 18 
 19 
 
 2553 
 
 2788 
 
 2577 
 2810 
 
 2601 
 2833 
 
 2625 
 2856 
 
 2648 
 
 2878 
 
 2672 
 2900 
 
 2695 
 2923 
 
 2718 
 2945 
 
 2742 
 2967 
 
 2765 
 2989 
 
 2 
 
 2 
 
 5 
 
 4 
 
 7 
 7 
 
 9 
 9 
 
 12 
 11 
 
 14 
 13 
 
 16 
 
 IG 
 
 19 
 
 18 
 
 21 
 20 
 
 20 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 2 
 
 4 
 
 6 
 
 8 
 
 11 
 
 13 
 
 15 
 
 17 
 
 19 
 
 21 
 
 3222 
 
 3243 
 
 3263 
 
 3284 
 
 3304 
 
 3324 3345 
 
 3365 
 
 3385 
 
 3404 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 22 
 
 3424 
 
 3444 
 
 3464 
 
 3483 
 
 3502 
 
 3522 
 
 3541 
 
 3560 
 
 3579 
 
 3598 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 15 
 
 17 
 
 23 
 
 3617 
 
 3636 
 
 3655 
 
 3674 
 
 3692 
 
 3711 
 
 3729 
 
 3747 
 
 3766 
 
 3784 
 
 2 
 
 4 
 
 6 
 
 7 
 
 9 
 
 11 
 
 13 
 
 15 
 
 17 
 
 24 
 
 3802 
 
 3820 
 
 3838 
 
 3856 
 
 3874 
 
 3892 
 
 3909 
 
 3927 
 
 3945 
 
 3962 
 
 2 
 
 4 
 
 5 
 
 7 
 
 9 
 
 11 
 
 12 
 
 14 
 
 16 
 
 25 
 
 3979 
 
 3997 
 
 4014 
 
 4031 
 
 4048 
 
 4065 
 
 4082 
 
 4099 
 
 4116 
 
 4133 
 
 2 
 
 3 
 
 5 
 
 7 
 
 9 
 
 10 
 
 12 
 
 14 
 
 15 
 
 26 
 
 4150 
 
 4166 
 
 4183 
 
 4200 
 
 4216 
 
 4232 j 4249 
 
 4265 
 
 4281 
 
 4298 
 
 2 
 
 3 
 
 5 
 
 7 
 
 S 
 
 10 
 
 11 
 
 13 
 
 15 
 
 27 
 
 4314 
 
 4330 
 
 4346 
 
 4362 
 
 4378 
 
 4393 
 
 4409 
 
 4425 
 
 4440 
 
 4456 
 
 2 
 
 3 
 
 5 
 
 6 
 
 8 
 
 9 
 
 11 
 
 13 
 
 14 
 
 28 
 
 4472 
 
 4487 
 
 4502 
 
 4518 
 
 4533 
 
 4548 
 
 4564 
 
 4579 
 
 4594 
 
 4609 
 
 2 
 
 3 
 
 5 
 
 6 
 
 8 
 
 9 
 
 11 
 
 12 
 
 14 
 
 29 
 
 4624 
 
 4639 
 
 4654 
 
 4669 
 
 4683 
 
 4698 
 
 4713 
 
 4728 
 
 4742 
 
 4757 
 
 1 
 
 3 
 
 4 
 
 6 
 
 7 
 
 9 
 
 10 
 
 12 
 
 13 
 
 30 
 
 4771 
 
 4786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 1 
 
 3 
 
 4 
 
 6 
 
 7 
 
 9 
 
 10 
 
 11 
 
 13 
 
 31 
 
 4914 
 
 4928 
 
 4942 
 
 4955 
 
 4969 
 
 4983 
 
 4997 
 
 5011 
 
 5024 
 
 5038 
 
 1 
 
 3 
 
 4 
 
 6 
 
 7 
 
 S 
 
 10 
 
 11 
 
 12 
 
 32 
 
 5051 
 
 5065 
 
 5079 
 
 5092 
 
 5105 
 
 5119 
 
 5132 
 
 5145 
 
 5159 
 
 5172 
 
 1 
 
 3 
 
 4 
 
 5 
 
 7 
 
 8 
 
 9 
 
 11 
 
 12 
 
 33 
 
 5185 
 
 5198 
 
 5211 
 
 5224 
 
 5237 
 
 5250 
 
 5263 
 
 5276 
 
 5289 
 
 5302 
 
 1 
 
 3 
 
 4 
 
 c 
 
 ij 
 
 6 
 
 8 
 
 9 
 
 10 
 
 12 
 
 34 
 
 5315 
 
 5328 
 
 5340 
 
 5353 
 
 5366 
 
 5378 
 
 5391 
 
 5403 
 
 5416 
 
 5428 
 
 1 
 
 3 
 
 4 
 
 5 
 
 G 
 
 8 
 
 9 
 
 10 
 
 11 
 
 35 
 
 5441 
 
 5453 
 
 5465 
 
 5478 
 
 5490 
 
 5502 
 
 5515 
 
 5527 
 
 5539 
 
 5551 
 
 1 
 
 2 
 
 4 
 
 5 
 
 G 
 
 7 
 
 9 
 
 10 
 
 11 
 
 36 
 
 5563 
 
 5575 
 
 5587 
 
 5599 
 
 5611 
 
 5623 
 
 5635 
 
 5647 
 
 5658 
 
 5670 
 
 
 2 
 
 4 
 
 5 
 
 G 
 
 7 
 
 8 
 
 10 
 
 11 
 
 37 
 
 5682 
 
 5694 
 
 5705 
 
 5717 
 
 5729 
 
 5740 
 
 5752 
 
 5763 
 
 5775 
 
 5786 
 
 
 2 
 
 3 
 
 5 
 
 6 
 
 7 
 
 8 
 
 g 
 
 10 
 
 38 
 
 5798 
 
 5809 
 
 5821 
 
 5832 
 
 5843 
 
 5855 
 
 5866 
 
 5877 
 
 5888 
 
 5899 
 
 
 2 
 
 3 
 
 5 
 
 G 
 
 7 
 
 8 
 
 g 
 
 10 
 
 39 
 
 5911 
 
 5922 
 
 5933 
 
 5944 
 
 5955 
 
 5966 
 
 5977 
 
 5988 
 
 5999 
 
 6010 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 7 
 
 8 
 
 e 
 
 10 
 
 40 
 
 6021 
 
 6031 
 
 6042 
 
 6053 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 8 
 
 9 
 
 10 
 
 41 
 
 6128 
 
 6138 
 
 6149 
 
 6160 
 
 6170 
 
 6180 
 
 6191 
 
 6201 
 
 6212 
 
 6222 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 42 
 
 6232 
 
 6243 
 
 6253 
 
 6263 
 
 6274 
 
 6284 
 
 6294 
 
 6304 
 
 6314 
 
 6325 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 43 
 
 6335 
 
 6345 
 
 6355 
 
 6365 
 
 6375 
 
 6385 
 
 6395 
 
 6405 
 
 6415 
 
 6425 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 44 
 
 6435 
 
 6444 
 
 6454 
 
 6464 
 
 6474 
 
 6484 
 
 6493 
 
 6503 
 
 6513 
 
 6522 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 45 
 
 6532 
 
 6542 
 
 6551 
 
 6561 
 
 6571 
 
 6580 
 
 6590 
 
 6599 
 
 6609 
 
 6618 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 46 
 
 6628 6637 
 
 6646 
 
 6656 
 
 6665 
 
 6675 
 
 6684 
 
 6693 
 
 6702 
 
 6712 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 G 
 
 7 
 
 7 
 
 8 
 
 47 
 
 6721 6730 
 
 6739 
 
 6749 
 
 6758 
 
 6767 
 
 6776 
 
 6785 
 
 6794 
 
 6803 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 7 
 
 8 
 
 48 
 
 6812 
 
 6821 
 
 6830 
 
 6839 
 
 6848 
 
 6857 
 
 6866 
 
 6875 
 
 6884 
 
 6893 
 
 1 
 
 2 
 
 3 
 
 4 
 
 4 
 
 ,5 
 
 6 
 
 7 
 
 8 
 
 49 
 
 6902 
 
 6911 
 
 6920 
 
 6928 
 
 6937 
 
 6946 
 
 6955 
 
 6964 
 
 6972 
 
 6981 
 
 1 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 50 
 
 6990 
 
 6998 
 
 7007 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 51 
 
 7076 
 
 7084 
 
 7093 
 
 7101 
 
 7110 
 
 7118 
 
 7126 7135:7143 
 
 7152 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 G 
 
 7 
 
 8 
 
 52 
 
 7160 
 
 7168 
 
 7177 
 
 7185 
 
 7193 
 
 7202 
 
 7210 7218 7226 
 
 7235 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 7 
 
 53 
 
 7243 
 
 7251 
 
 7259 
 
 7267 
 
 7275 
 
 7284 
 
 7292 17300' 7308 
 
 7310 
 
 1 2 
 
 2j 3 
 
 4 
 
 5 
 
 6 
 
 6 
 
 7 
 
 54 
 
 7324 
 
 7332 
 
 7340 
 
 7348 
 
 7356 
 
 7364 
 
 7372 
 
 738073887396 
 
 1 2 
 
 2 3 
 
 4 
 
 5 
 
 6 
 
 6 
 
 7 
 
APPENDIX 
 
 TABLE IV. 
 FOUR PLACE LOGARITHMS. 
 
 103 
 
 Natural 
 numbers. 
 
 O 
 
 I 
 
 2 
 
 3 
 
 4 
 
 '5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 PROPORTIONAL PARTS. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 55 
 
 7404 
 
 74U 
 
 J741S 
 
 7427 
 
 7435 
 
 7443 
 
 7451 
 
 7459 
 
 7466 
 
 7474 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 7 
 
 56 
 
 7482 
 
 749C 
 
 7497 
 
 7505 
 
 7513 
 
 7520 
 
 7528 
 
 7536 
 
 7543 
 
 755 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 7 
 
 57 
 
 7559 
 
 756C 
 
 7574 
 
 7582 
 
 7589 
 
 7597 
 
 7604 
 
 7612 
 
 7619 
 
 7627 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 7 
 
 58 
 
 7634 
 
 7642 
 
 7649 
 
 7657 
 
 7664 
 
 7672 
 
 7679 
 
 7686 
 
 7694 
 
 770 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 7 
 
 59 
 
 7709 
 
 7716 
 
 7723 
 
 7731 
 
 7738 
 
 7745 
 
 7752 
 
 7760 
 
 7767 
 
 7774 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 7 
 
 60 
 
 7782 
 
 7789 
 
 7796 
 
 7803 
 
 7810 
 
 7818 
 
 7825 
 
 7832 
 
 7839 
 
 7846 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 6 
 
 61 
 
 7853|7860 
 
 7868 
 
 7875 
 
 7882 
 
 7889 
 
 7896 
 
 7903 
 
 7910 
 
 7917 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 6 
 
 62 
 
 7924 
 
 7931 
 
 7938 
 
 7945 
 
 7952 
 
 7959 
 
 7966 
 
 7973 
 
 7980 
 
 7987 
 
 1 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 6 
 
 63 
 
 7993 
 
 8000 
 
 8007 
 
 8014 
 
 8021 
 
 8028 
 
 8035 
 
 8041 
 
 8048 
 
 8055 
 
 1 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 64 
 
 8062 
 
 8069 
 
 8075 
 
 8082 
 
 8089 
 
 8096 
 
 8102 
 
 8109 
 
 8116 
 
 8122 
 
 1 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 65 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 1 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 66 
 
 8195 
 
 8202 
 
 8209:8215 
 
 8222 
 
 8228 
 
 8235 
 
 8241 
 
 8248 
 
 8254 
 
 1 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 67 
 
 8261 
 
 8267 
 
 8274 
 
 8280 
 
 8287 
 
 8293 
 
 8299 
 
 8306 
 
 8312 
 
 8319 
 
 1 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 68 
 
 8325 
 
 8331 
 
 8338 
 
 8344 
 
 8351 
 
 8357 
 
 8363 
 
 8370 
 
 8376 
 
 8382 
 
 1 
 
 1 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 69 
 
 8388 
 
 8395 
 
 8401 
 
 8407 
 
 8414 
 
 8420 
 
 8426 
 
 8432 
 
 8439 
 
 8445 
 
 1 
 
 1 
 
 Q 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 70 
 
 8451 
 
 8457 
 
 8463 
 
 8470 
 
 8476 
 
 8482 
 
 8488 
 
 8494 
 
 8500 
 
 8506 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 71 
 
 8513 
 
 8519 
 
 8525:8531 
 
 8537 
 
 8543 
 
 8549 
 
 8555 
 
 8561 
 
 8567 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 5 
 
 72 
 
 8573 
 
 8579 
 
 8585:8591 
 
 8597 
 
 8603 
 
 8609 8615 
 
 8621 
 
 8627 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 2 
 
 5 
 
 5 
 
 73 
 
 8633 
 
 8639 
 
 8645 
 
 8651 
 
 8657 
 
 8663 
 
 8669 
 
 8675 
 
 8681 
 
 8686 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 5 
 
 74 
 
 8692 
 
 8698 
 
 8704 
 
 8710 
 
 8716 
 
 8722 
 
 8727 
 
 8733 
 
 8739 
 
 8745 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 5 
 
 75 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 5 
 
 76 
 
 8808 
 
 881418820 
 
 8825 
 
 8831 
 
 8837 
 
 8842 
 
 8848 
 
 8854 
 
 8859 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 5 
 
 77 
 
 8865 
 
 8871 
 
 8876 
 
 8882 
 
 8887 
 
 8893 
 
 8899 
 
 8904 
 
 8910 
 
 8915 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 78 
 
 8921 
 
 8927 
 
 8932 
 
 8938 
 
 8943 
 
 8949 
 
 8954 
 
 8960 
 
 8965 
 
 8971 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 79 
 
 8976 
 
 8982 
 
 8987 
 
 8993 
 
 8998 
 
 9004 
 
 9009 
 
 9015 
 
 9020 
 
 9026 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 80 
 
 Q031 
 
 9036 
 
 9042 
 
 9047 
 
 9053 
 
 9058 
 
 9063 
 
 9069 
 
 9074 
 
 9079 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 81 
 
 )085 
 
 9090 
 
 909619101 
 
 9106 
 
 9112 
 
 9117 
 
 9122 
 
 9128 
 
 9133 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 82 
 
 9138 
 
 9143 
 
 9149 
 
 9154 
 
 9159 
 
 9165 
 
 9170 
 
 9175 
 
 9180 
 
 9186 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 83 
 
 9191 
 
 9196 
 
 9201 
 
 9206 
 
 9212 
 
 9217 
 
 9222 
 
 9227 
 
 9232 
 
 9238 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 84 
 
 9243 
 
 9248 
 
 9253 
 
 9258 
 
 9263 
 
 9269 
 
 9274 
 
 9279 
 
 9284 
 
 9289 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 85 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 86 
 
 9345 
 
 9350 
 
 9355 
 
 9360 
 
 9365 
 
 9370 
 
 9375 
 
 9380 
 
 9385 
 
 9390 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 5 
 
 87 
 
 9395 
 
 9400 
 
 9405 
 
 9410 
 
 9415 
 
 9420 
 
 9425 
 
 9430 
 
 9435 
 
 9440 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 88 
 
 9445 
 
 9450 
 
 9455 
 
 9460 
 
 9465 
 
 9469 
 
 9474 
 
 9479 
 
 9484 
 
 9489 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 89 
 
 9494 
 
 9499 
 
 9504 
 
 9509 
 
 9513 
 
 9518 
 
 9523 
 
 9528 
 
 9533 
 
 9538 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 90 
 
 9542 
 
 9547 
 
 9552 
 
 9557 
 
 9562 
 
 9566 
 
 9571 
 
 9576 
 
 9581 
 
 9586 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 91 
 
 9590 
 
 9595 
 
 9600 
 
 9605 
 
 9609 
 
 9614 
 
 9619 
 
 9624 
 
 9628 
 
 9633 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 92 
 
 9638 
 
 9643 
 
 9647 
 
 9652 
 
 9657 
 
 9661 
 
 9666 
 
 9671 
 
 9675 
 
 9680 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 93 
 
 9685 
 
 9689 
 
 9694 
 
 9699 
 
 9703 
 
 708 
 
 9713 
 
 9717 
 
 9722 
 
 9727 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 94 
 
 9731 
 
 9736 
 
 9741 
 
 9745 
 
 9750 
 
 754 
 
 9759 
 
 9763 
 
 9768 
 
 9773 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 95 
 
 9777 
 
 9782 
 
 9786 
 
 9791 
 
 9795 
 
 800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 96 
 
 9823 
 
 9827 
 
 9832 
 
 9836 
 
 9841 
 
 845 
 
 9850 
 
 9854 
 
 9859 
 
 9863 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 97 
 
 9868 
 
 9872 
 
 9877 
 
 9881 
 
 9886 
 
 890 
 
 9894 
 
 9899 
 
 9903 
 
 9908 
 
 
 
 1 
 
 1 
 
 2 
 
 2 
 
 3 
 
 3 
 
 4 
 
 4 
 
 98 
 
 9912 
 
 9917 
 
 9921 
 
 9926 
 
 9930 
 
 934 
 
 9939 
 
 9943 
 
 9948 
 
 9952 
 
 
 
 [ 
 
 1 
 
 2 
 
 9 
 
 3 
 
 3 
 
 4 
 
 4 
 
 99 
 
 9956 
 
 9961 
 
 9965 
 
 9969 
 
 9974 
 
 978 
 
 9983 
 
 9987 
 
 9991 
 
 9996 
 
 
 
 ' 
 
 1 
 
 2 
 
 } 
 
 3 
 
 3 
 
 3 
 
 4 
 
104 
 
 APPENDIX 
 
 TABLE V. 
 
 SINES, COSINES, TANGENTS. 
 
 
 Natural. ** 
 
 Logarithmic. 
 
 Sin. 
 
 Cos. 
 
 Tan. 
 
 Sin. 
 
 Cos. 
 
 Tan. 
 
 0.0 
 
 0.0000 
 
 1.0000 
 
 0.0000 
 
 oo 
 
 0.0000 
 
 oo 
 
 0.5 
 
 0.0087 
 
 1.0000 
 
 0.0087 
 
 7.9408 
 
 0.0000 
 
 7.9409 
 
 1. 
 
 0.0175 
 
 0.9998 
 
 0.0175 
 
 8.2419 
 
 9.9999 
 
 8.2419 
 
 1.5 
 
 0.0262 
 
 0.9997 
 
 0.0262 
 
 8.4179 
 
 9.9999 
 
 8.4181 
 
 2. 
 
 0.0349 
 
 0.9994 
 
 0.0349 
 
 8.5428 
 
 9.9997 
 
 8.5431 
 
 2.5 
 
 0.0436 
 
 0.9990 
 
 0.0437 
 
 8.6397 
 
 9.9996 
 
 8.6401 
 
 3. 
 
 0.0523 
 
 0.9986 
 
 0.0524 
 
 8.7188 
 
 9.9994 
 
 8.7194 
 
 4. 
 
 0.0698 
 
 0.9976 
 
 0.0699 
 
 8.8436 
 
 9.9989 
 
 8.8446 
 
 5. 
 
 0.0872 
 
 0.9962 
 
 0.0875 
 
 8.9403 
 
 9.9983 
 
 8.9420 
 
 10. 
 
 0.1736 
 
 0.9848 
 
 0.1763 
 
 9.2397 
 
 9.9934 
 
 9.2463 
 
 15. 
 
 0.2588 
 
 0.9659 
 
 0.2679 
 
 9.4130 
 
 9.9849 
 
 9.4281 
 
 20. 
 
 0.3420 
 
 0.9397 
 
 0.3640 
 
 9.5341 
 
 9.9730 
 
 9.5611 
 
 25. 
 
 0.4226 
 
 0.9063 
 
 0.4663 
 
 9.6259 
 
 9.9573 
 
 9.6687 
 
 30. 
 
 0.5000 
 
 0.8660 
 
 0.5774 
 
 9.6990 
 
 9.9375 
 
 9.7614 
 
 35. 
 
 0.5736 
 
 0.8192 
 
 0.7002 
 
 9.7586 
 
 9.9134 
 
 9.8452 
 
 40. 
 
 0.6428 
 
 0.7660 
 
 0.8391 
 
 9.8081 
 
 9.8843 
 
 9.9238 
 
 45. 
 
 0.7071 
 
 0.7071 
 
 1.0000 
 
 9.8495 
 
 9.8495 
 
 0.0000 
 
 50. 
 
 0.7660 
 
 0.6428 
 
 1.1918 
 
 9.8843 
 
 9.8081 
 
 0.0762 
 
 55. 
 
 0.8192 
 
 0.5736 
 
 1.4281 
 
 9.9134 
 
 9.7586 
 
 0.1548 
 
 60. 
 
 0.8660 
 
 0.5000 
 
 1.7321 
 
 9.9375 
 
 9.6990 
 
 0.2386 
 
 65. 
 
 0.9063 
 
 0.4226 
 
 2.1445 
 
 9.9573 
 
 9.6259 
 
 0.3313 
 
 70. 
 
 0.9397 
 
 0.3420 
 
 2.7475 
 
 9.9730 
 
 9.5341 
 
 . 4389 
 
 75. 
 
 0.9659 
 
 0.2588 
 
 3.7321 
 
 9.9849 
 
 9.4130 
 
 0.5719 
 
 80. 
 
 0.9848 
 
 0.1736 
 
 5.6713 
 
 9.9934 
 
 9.2397 
 
 0.7537 
 
 85. 
 
 0.9962 
 
 0.0872 
 
 11.43 
 
 9.9983 
 
 8.9403 
 
 1.0580 
 
 90. 
 
 1.0000 
 
 0.0000 
 
 00 
 
 0.0000 
 
 00 
 
 00 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
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 This book is DUE on the last date stamped below. 
 
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 i-