REESE XI BRA v UNIVERSITY OF CALIFORNIA DISCUSSION OF THE PRECISION OF MEASUREMENTS. WITH EXAMPLES TAKEN MAINL Y FROM PHYSICS AND ELECTRICAL ENGINEERING. BY SILAS W. HOLMAN, S.B., ASSOCIATE PROFESSOR OF PHYSICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY. FIRST EDITION. FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS, 53 EAST TENTH STREET. 1894. 3 COPYRIGHT, 1892, BY SILAS W. HOLMAN. ROBERT DRTTMMOITO. PERMS Electrotype Printers, Street, m Pearl stree t, New York. New York. PREFACE. THE material presented in this volume is the outcome of several years' teaching of the* ^subject. In a less complete form it was prepared for lecture notes and was printed in pamphlet form, but not published, by the Massachusetts In- stitute of Technology in 1888, having appeared in the Tech- nology Quarterly and in the Electrical Engineer in 1887. In this revised form, the author has felt that it perhaps possessed sufficient completeness and originality to be of in- terest or value to students and teachers, and therefore to merit publication. In venturing to urge the importance of the subject as a course of study for engineers and for students of physics or other pure sciences, the author would suggest the value of the attitude of mind produced by it. One who has in any reason- able degree mastered its methods, although he may never apply them directly, will not only have increased his power to intelligently scrutinize experimental results, but will have acquired a tendency to do so. And it is perhaps not too much to hope that he may acquire a notion of a judicious distribution of effort which, with the best of results to himself, he may carry into quite other matters. SILAS W. HOLM AN. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, BOSTON, September, 1892. CONTENTS. PRECISION OF MEASUREMENTS. PAGE Introductory I DIRECT MEASUREMENTS. Direct Measurements 4 Indirect Measurements 4 Quantities: Independent, Conditioned 5 Sources of Error 5 Errors of Single Observations.- 6 Variable Part 6 Constant Part, Constant Error 7 Elimination of Constant Error , 7 Corrections 8 Example I. A,B,C. Distance by Steel Tape 9 Determinate and Indeterminate Errors 10 Residuals II Accuracy or Error of Result 13 Deviations 14 General Law of Deviations 15 Mean: Best Representative Value 16 Deviation Measure 16 Average Deviation 16 Example II 18 Places of Figures in d.m. ; and Negligible Amounts 20 Best Value of n 22 Other Deviation Measures 23 Special Law of Deviations , 24 Precision Measure of Result 25 To Make Residuals Negligible in P. M ., 26 Criterion 26 Best Value of Residuals: Equal Effects 27 Fractional Deviation, Fractional Precision 29 Mistakes 30 v VI CONTENTS. FACET Criterion for Rejection of Doubtful Observations 30 Weights 31 Meaning of Estimated Accuracy of Direct Result 32 Forms of Problems on Accuracy of Result 33 Data Required to Substantiate Result 36 Planning of Direct Measurement , 36 Solutions of Illustrative Problems in Direct Measurements 37 Example III. Weighing. Balance 37 Example IV. Voltmeter Calibration 41 INDIRECT MEASUREMENTS. Estimate of Accuracy of Indirect Result 45 Error of Method 46 Check Methods 47 Relation between P.M. of Results and of Components 47 Types of Problems 47 General Formulae .... 48 Notation 49 Separate Effects. I, II. Formulae , 49 Resultant Effects. Ill; 1,2. Formulae 50 Equal Effects. Formulae 53 Application to Precision Discussions 54 Formulae for General and Special Functions 55 Simple Functions 56 Separation into Factors which are Functions of Single Components. . . 61 Separation into Groups 63 Critera for Negligibility of 8 in Components 67 Numerical Constants 70 Equal Effects. Demonstration 70 Estimated Precision Measures of Components 72 Components with Special Laws of Deviations 73 Preparation of Functions for Discussion 73 Simplification of Functions 75 Significant Figures 76 Rules for Significant Figures. 1-6 77 Examples V XII 78 Demonstration of Rules 80 Forms of Problems on Accuracy of Result 84 Data Required to Substantiate Result 85 Planning of Indirect Measurement 85 Examples: XIII XVI. Value of g by Simple Pendulum 86 XVII. Calorimeter 88 XVIII. Heat by Incandescent Lamp 89* CONTENTS. Vll XIX. Volume of Sphere 90 XX. Value of g by Simple Pendulum 90 XXI. Cosine Galvanometer 91 XXII. Continuous Calorimeter 94 XXIII. H. P. by Friction Brake 96 XXIV. Specific Resistance 98 BEST MAGNITUDES OF COMPONENTS. Nature of Problems 100 For a Single Component 102 For Two Variable Components 104 Best Ratio. Procedure 104 Best Magnitudes 106 For Several Components 107 Best Ratio 107 Best Magnitudes 108 Approximate Solution by Equal Effects 108 Best Ratio 108 Best Magnitudes 109 Examples: XXV. Best Deflection on Tangent Galvanometer ... no XXVI. Electrical Heating of Conductor in XXVII. Bar for Moment of Inertia 112 XXVIII. Modulus of Elasticity of Wooden Beam 115 XXIX. Specific Resistance of Wire 118 XXX. XXVIII by Another Method 118 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. Example XXXI. Calibration of Voltmeters 120 Example XXXII. Dynamo Efficiency by Stray-Power Method 122 Example XXXIII. Cradle Dynamometer 130 Example XXXIV. Tangent Galvanometer 138 Example XXXV. Electro-static Capacity. Thomson's or Gott's Method 159 Example XXXVI. Magnetometer 160 Example XXXVII. Battery Resistance and E. M. F 161 TABLES. Sines, Cosines, Tangents 166 Constants 166 Squares, Cubes, Reciprocals 167 Logarithms 168 INDEX I7 i PRECISION OF MEASUREMENTS. INTRODUCTORY. AN experimental result whose reliability is unknown is nearly worthless. The grade of accuracy of a measurement must be adapted to the purpose for which the result is desired. The necessary accuracy must be secured with the least possible expenditure of labor. These statements apply no less to the roughest than to the most elaborate work which the engineer is called upon to per- form ; they are no more true of refined scientific research than of every-day engineering and industrial practice. The prin- ciples which underlie these assertions respecting quantitative measurement differ in no essential particular from those which lie at the foundation of all commercial and industrial economy, proved value ; product, labor, and expenditure proportioned to the relative importance of the thing in hand ; results ob- tained with the least effort, and hence with judicious distribu- tion of effort among the various parts of the work. The successful business manager does not hesitate at large expenditure of money or effort in those parts of an under- taking where he perceives them to be necessary, nor does he overlook the importance of economy where expenditure is not essential. Neither does he wait till an enterprise is well under way or completed to determine where the chief points for ex- penditure or economy lie. The wise designer of a structure 2 INTRODUCTORY. devotes his close attention to distributing material to the best advantage : enough, at the best points, and no superfluity. These things are so obvious, and their neglect is so strikingly absurd, that it is the more surprising that the same practice should be so commonly neglected not only in quantitative measurement but in engineering investigations and even in physical research. The engineer, in the measurement of the efficiency or duty of an engine, the efficiency of a dynamo or of a power station ; the physicist in the designing or use of a gas thermometer, in the measurement of an index of refraction, or in the compari- sons of standards of length ; the chemist in analytical investi- gation, or in the experimental test of an industrial plant, can no more afford to omit a preliminary discussion of the precision of the various component measurements entering into his result, than the business man can afford^to estimate and proportion in advance his expenditure in a large undertaking. The one is as essential as the other to complete success. The thoughtful student recognizes early in his experimental work the importance of certain questions which never leave the mind of the experienced observer, namely What accuracy is desired in the result ? What accuracy is therefore necessary in each of the various component measurements from which the result is calculated ? How reliable is the final result when obtained ? The more complicated and indirect the measure- ment, the more difficult it becomes to answer these queries by mere inspection, and hence the greater the necessity for some systematic and rational procedure for reaching the answer. The present volume is the outcome of an effort to establish such a procedure which, while being sufficiently general, shall not be too laborious in its operation. It is intended to be applicable to quantitative measurements of all kinds, whether in engineering or pure science. The illustrative examples throughout the text are taken chiefly from physics and elec- trical engineering, because the students as well as the problems with which the author has been called upon to deal have been chiefly in those subjects. The examples are for the most part IN 7 'ROD UC TOR Y. 3 so fully explained or so simple that they will be easily intelli- gible to students in other lines. The processes of the differen- tial calculus have been used, because without them the methods would necessarily be cumbrous, and also because a large and increasing proportion of those who deal at all with such a subject are amply competent to follow or make such simple differentiations as are required. It is, however, to be noted that the majority of the methods and formulae herein de- veloped can be utilized without any employment whatever of the calculus, so that they may be applied by one who has forgotten his earlier knowledge of that subject or who has never become acquainted with it. Attention is particularly directed in this connection to the rules for significant figures. DIRECT MEASUREMENTS. Direct Measurements. All quantitative work of course involves measurements. These may be separated into two classes, viz. direct and indirect. Direct measurements are those made by methods and instruments whose indications give directly the quantity sought ; e.g. measurements of dis- tance by a scale, of weights (or masses) by an equal-arm bal- ance, of resistance by a Wheatstone bridge, etc. The direct readings, in such cases, may or may not require corrections. The fact that a correction is necessary, that is, that the directly observed value must be more or less modified to remove the effect of known sources of error, does not render the measure- ment indirect. Indirect Measurements are those in which the quantity measured is not given directly by observation or readings taken, but must be calculated from them. Thus in an indirect measurement the quantity sought is a function of one or more quantities which are directly measured and which may be called the component quantities. For instance, the specific gravity of a substance is ordinarily found by measuring its weight in air and its loss of weight in water. Each of these is or may be a direct measurement, but the desired specific gravity is found from them by calculation, viz. by dividing one by the other, and is therefore indirect. To measure a 4 SOURCES OF ERROR IN DIRECT MEASUREMENTS. 5 constant current C, we may pass it through a tangent galva- nometer of factor k and observe the deflection produced : then C = k tan 0. Here C is indirectly measured, being cal- culated from the directly observed deflection by the func- tion indicated by the right-hand member of the expression. Similarly the measurement of g by a simple pendulum, of the E. M. F. of a battery by the two-deflection method or by the PoggendorfT method, of the index of refraction of a prism, of the efficiency of a dynamo or motor, in fact the great majority of physical measurements, are indirect. Quantities may be either independent or conditioned. That is, two or any number of quantities to be measured may be wholly independent, so that the magnitude of one is in no way predetermined by any relations to the others ; or they -may be conditioned so that, for instance, the magnitude oi two out of three being given, that of the third is thereby pre determined. Thus a constant current flowing through a given coil of wire might be anything whatever, according to the potential used, so that measurements of current and of resist- ance at any instant would be independent. But if the poten- tial difference at the ends of the coil were measured simulta- neously with the current and resistance, then the three, current, resistance, and potential, would be conditioned by Ohm's law. The numerical values obtained in the measurement of condi- tioned quantities contain, of course, errors not controlled by these conditions, so that these values fail to fulfil the condi- tions, and require adjustment. Sources of Error in Direct Measurements. All processes of measurement are, of course, fallible. None can give abso- lute accuracy, that is, none can be wholly free from error. The questions with which we have to deal then are only such as relate to the amount or character of the errors occurring, and to their sufficient elimination for the purpose in hand. Inspection of the methods, instruments, and results of any direct measurement will show that the method has some dis- coverable sources of error, that the instruments likewise contain certain inherent sources of error, and finally that however care- O DIRECT MEASUREMENTS. fully the effects of the discoverable sources are removed, some undiscovered or uncorrected sources still remain, since succes- sive equally careful repetitions of the same measurement yield numerical results which are more or less discordant in the last one or two places of significant figures. Example I (A), page 9, The existence of this discordance just referred to proves that the errors from the various sources are not constant, at least that some of them are not, a fact which we know to be true for some of the discoverable sources. And the general rule for the variation doubtless is that under given conditions the error from any given source has a certain average magni- tude about which it varies more or less, being sometimes greater sometimes smaller than that amount. It is therefore reasonable, and will be found convenient, to regard the error from any source as made up of two portions, a constant part, viz. its average value, and a variable part. Of course either of these parts may be wanting in any given instance. Errors of Single Observations. The actual error of any given single observation is obviously the algebraic sum of all the individual errors from the several sources which affect the quantity. As these individual errors have each a constant part and a variable part, so the error of a single observation will be made up of two parts. These will be a constant portion which is the algebraic sum of the constant parts of the individual errors, and a second portion which will vary in different obser- vations and which is, for any observation, the algebraic sum of the variable parts of the individual errors as they existed at the moment when that observation was made. Variable Part. Considering first this part of the error, we can at once see that, if we make a series of observations under sensibly the same conditions and take the average, the result will be partly free from the effect of the variable parts of the error. For each varying error will tend to make the result at one time more or less too large, at another too small, a kind of fluctuation which the process of averaging tends to elimi- nate. That the arithmetical mean removes the variable parts ELIMINA TION OF CONSTANT ERROR. / of the error better than any other function, will be shown more explicitly later. It is easy to see that averaging can never effect a complete removal ; the elimination being, how- ever, more nearly complete (in proportion to Vn) as the num- ber, n, of observations is greater. For the sum total of the positive parts of the variation will naturally be unequal to that of the negative parts, and so long as the number is small the inequality will be considerable, but they will become more and more nearly equal as the number increases. At best, however, there will always remain a residual variable error, and in discussing the correctness of the result this must be taken into account. Example I (B), page 9. Constant Part. Considering next the effect of the con- stant parts of the errors from the various sources, we see that their resultant, viz., the algebraic sum of all these constant parts, will itself necessarily be of the same amount in each and all the single observations taken under the same conditions. The process of averaging a series will therefore do absolutely nothing toward the removal of this resultant of the constant portions. The mean will contain an error of which the con- stant portion will be identical with that of each single observa- tion in the series. The name "constant error " is therefore well applied to this resultant constant portion of the errors. Elimination of Constant Error. Of the constant portions of the individual errors going to make up this resultant con- stant error, some will be positive, some negative, and the magnitudes will be various. They will therefore in part annul one another; that is, the resultant will not in general be as large as the arithmetical sum of all the component parts. If the sum of the positive parts exactly equalled that of the negative parts, the elimination would, of course, be complete, and the constant error of the single observation and mean would be zero. But this condition would naturally be highly exceptional. The constant error is in fact exceedingly difficult of removal, and often proves to be of surprisingly large amount in spite of most painstaking efforts for its elim- 8 DIRECT MEASUREMENTS. ination. For any specific method or apparatus will have its own characteristic set of errors, of which some will be pre- dominant and will determine a constant error more or less large in the results obtained. Another method will on the other hand be characterized by a different set of sources of error and will have a different constant error. Observations taken under diverse conditions with the same method will often also have differing constant errors. And finally, differ- nt observers will show different "personal equations." Thus, results obtained by changing methods, apparatus, observers, or other conditions will materially differ in the sources and amounts of their constant errors. The greater the number and the more complete the variety of the changes, the greater becomes the diversity of the sources of error, and conse- quently the more complete the elimination of their effects by taking a general mean (with due respect to weights) of the various results. It is only -through this repetition by inde- pendent methods that we can gain confidence as to the real accuracy of a result; and the more radically distinct the nature of the methods employed, the more valuable the check. Even a single reliable check greatly enhances the value of a result. It is evident that we can obtain no numerical meas- ure of the amount of the constant error of any result. Such a measure would imply the knowledge of the true result, which is of course always unknown. Example I (C), page 10. Corrections. In as much as each method and apparatus has its own characteristic sources which determine its constant error, it is obvious that in using any method we must do what we can to reduce the effect of those sources to a minimum. For this purpose we must study the method and instruments as thoroughly as possible in advance, to discover all possible sources of error. We must then arrange the work to remove as many as possible of these sources wholly or in part, and we must evaluate the effects of those not removed and eliminate them by the application of the corresponding " corrections " thus determined. EXAMPLE I. 9 Example I. (See pages 6, 7 et seg.) Take as an illustra- tion so simple a measurement as that of the distance between two points by means of a steel tape. (A) There are easily discoverable such sources of error as these : (1) Error in numbering of tape ; (2) Irregular spacing of divisions ; (3) Incorrect unit, i.e., foot not standard in length ; (4) Bends in tape ; (5) Sag of tape ; (6) Stretch of tape (7) Error in setting zero of tape at starting point ; (8) Error of estimation of fraction of division at finishing point ; (9) Temperature not that for which the tape was gradu- ated. Besides these sources there are doubtless many others of greater or less effect, some of which might possibly be dis- covered by further study, but many of which are at present obscure. Successive measurements of the same distance, especially if this be long and if the fraction of an inch to which readings are taken be small, will show discordances of greater or less magnitude. (B) Variable Part. Errors 5, 6, 7, 8, 9 would vary in amount from time to time and between different readings, and would therefore have variable parts. Each would tend to make the single results sometimes larger, at other times smaller, and by irregular amounts. Thus in the average re- sult of a series of observations the variable parts of the error from any single source would in part annul itself. Also in any single observation the surn of the negative variable parts of the errors from all sources would offset in part the sum of the positive variable parts more or less completely, but seldom wholly. (C) Constant Error. The errors i, 2, 3, and 4 would be constant for any given distance; fl also, 5 and 6 will clearly 10 DIRECT MEASUREMENTS. be liable to have some constant portion, as also would 9 under some circumstances. These together will make up the constant error. Some will be of one sign, some of the other, so that they will in part neutralize, but cannot be expected to wholly do so. The separate constant portions are the same in all single observations. Hence the constant error will be the same in each single reading and in the mean result. The part of the constant error due to 5 and 6 may be largely removed by stretching the tape by a spring-balance or other means so that it is always under the same tension. A correction for the amount of sag can then be made, and the stretch will be nearly the same at all times. The error from 9- may be in part removed by measuring the temperature of the tape at various points along its length and correcting for the ex- pansion due to the difference between the observed tempera- ture and that at which the tape is correct. There is liability in this correction to a residual constant error due to uncertainty as to the value of the coefficient of expansion for the metal employed, and due to constant errors in the thermometers. There is liability to variable error from the thermometers being too far apart or improperly located to give the true mean temperature of the tape ; also from the variable errors of the thermometers themselves. It is clear, upon reflection, that the errors in the above measurement, especially the constant errors, are largely peculiar to the method. If the distance were to be measured by a rod, or by a base-line apparatus, or by stadia wires in a telescope, each of the results would be characterized by a different set of errors, of which some might be common to all. The mean of the results of such different methods would be certainly more reliable than the poorest of them, by the natural annulling of the different classes of errors. Determinate and Indeterminate Errors. All sources of error which are discoverable and which may be removed or may have their effects more or less completely allowed for by corrections will here be classed as determinate sources. The corresponding errors will be referred to as determinate errors. RESIDUALS. II Some errors are determinate as to their nature only, others as to sign, others as to both sign and magnitude. On the other hand, all sources which are either undiscover- able, or whose effects cannot be properly determined and allowed for, will be classed as indeterminate sources, and the corresponding errors, as indeterminate errors. This class will contain not only those which are undiscoverable, but also the residuals of determinate errors. Both determinate and inde- terminate sources are inevitably present in every direct, and therefore also in every indirect, measurement. Residuals. In general the processes for the elimination of the determinate errors, whether by the removal of their sources or by corrections, accomplish this object only approxi- mately. There are perhaps a few cases in which the source of a determinate error can be wholly removed, or at least to a far greater extent than is demanded. For instance, in the constant 7t we may retain so many places of figures that the error from rejecting the rest may be utterly insignificant. Ordinarily, however, the source cannot be removed, but its effect can be lessened so as to be small or negligible. For instance, the individual weights of a set can perhaps be adjusted so accurately that their errors are negligible for a purpose in hand ; or the arms of a balance may be made so nearly equal that the error is negligible. But in all such cases there remains an error more or less small which enters into the result and which will be called a residual. A residual may be insignificant, but this requires proof ; and the proof can only be arrived at, in general, by a direct meas- urement of some kind, such as a comparison with a standard, or a measurement of some ratio. For instance, the weights of the set can be assumed to have a negligible error only after each has been weighed against a standard or tested by some equivalent process. This weighing will be made only to a cer- tain grade of accuracy, and will therefore itself leave a residual error. Similarly the ratio of the arms of the balance must be determined by the usual process of balancing and interchang. ing equal masses. Therefore this also is a direct measurement, 12 DIRECT MEASUREMENTS. and will of course be made only to the limit of accuracy fixed by the sensitiveness of the balance, and will leave a residual error. The process of evaluation of a correction is in general also only an approximate one, and consists usually of a direct or indirect measurement carried out only with a certain degree of accuracy leaving a residual error. Thus in the foregoing examples the weights might be adjusted less closely man was demanded for the work in hand, and the corrections to be applied might be evaluated by weighing against standard weights and thus determining the error. But this weighing would be a direct measurement, and would be carried out to an accuracy limited by the sensitiveness of the balance or by some other conditions. A corresponding residual error would therefore be left after the correction was applied. Similarly, the ratio of the balance arms not being close enough, it might be allowed for by measuring the ratio and applying a correc- tion. This again would leave a residual error. Other processes of correction exist, such as the correction for the eccentricity of a circle by reading two verniers 180 apart, and averaging. This is a type of certain mathematical corrections, and these also are usually only approximate, being close enough when the errors are small, but nevertheless leav- ing residual errors. In brief, then, most processes for the elimination of deter- minate errors leave residual errors behind. Also, most such processes involve direct measurements, and the statements already made or to be made respecting direct measurements apply to them. The numerical measure of the residuals will in general therefore be of the nature of precision measures of direct measurements which will presently be discussed. From this it is obvious that if it were necessary in an investigation to work out from the beginning every detail of a research, establishing all standards, ascertaining all correc- tions, developing every process employed, the labor would be enormous as, indeed, it often is. But fortunately the prog- ress of experimental science has provided instruments, pro- ACCURACY OR ERROR OF RESULTS. 13 cesses, methods, and results of known accuracy, which may be appropriated in any desired manner in more complex investi- gations. Accuracy or Error of Results. By the accuracy of a re- sult we mean its freedom from error. The real measure of the accuracy of a result is therefore the error of that result. Thus if we knew that a result had an error of 2 per cent we should say that it was accurate to 2 per cent, or we might say that its accuracy was 98 per cent. The latter phrase, although more exact, is less common and convenient than to say that the accuracy was 2 per cent. Thus if a result were 24.967 metres, and were known to have an error of 0.025 metres, we should say the result was accurate to 0.025 m. or to o.i per cent; or we might say that it had an accuracy #/ o.i per cent, although that phrase would be less precise than to say that it had an accuracy of 99.9 per cent. But it is clear that we can have no numerical measure of the constant error of a result, whether that result be a single observation, a mean of a series by one method, or the mean of results by a large number of methods. For as the error is the amount that the measured result differs from the true value, such a measure necessarily implies that the true value of the quantity is known, which is never the case. Yet it is of the utmost importance that we should be able to form some estimate of the accuracy or of the error of the result, and that this should be expressed numerically, so far as possible. How such an estimate is arrived at will be here indicated, and just what the measure is will be more explicitly stated in a later paragraph. There are only two things upon which this estimate can be based, in the case of the result of a series of observations by a single method, viz. : (1) The degree of care exercised in the study and removal of the determinate errors. (2) The concordance, or rather the discordance, between the single observations of the series. Of the first, we have a partial numerical measure in the 14 DIRECT MEASUREMENTS. measure of the residuals of the determinate errors, but this is only partial. A judgment as to the sources of error which have been overlooked or neglected is essential, but this cannot be given a numerical expression. Of the second, the numerical measure is the " deviation measure " to be presently described. The deviation measure and residuals can be combined to give the " precision measure " which is the final numerical measure to which we are brought in forming our estimate, and of which the significance will be stated later. Thus the most that can be done in forming an estimate of the error of the result of the mean of a series of observations by a single method is this : the precision measure of the result is calculated, giving a partial numerical measure ; and a judgment is formed from an inspection of the method as to whether any constant error comparable with the precision measure probably exists in the result. If results by several different methods, etc., are available, the best representative value (weighted mean) can be obtained from them, and their concordance will give us a further partial indication of the correctness of that value. It becomes necessary, therefore, to discuss the meaning of the terms, and to fix upon certain points respecting deviations and their measure, and the precision measure. Deviations. Suppose any number, n, of direct measure- ments or observations of a quantity to have been made with equal care, and under apparently identical conditions. Let #,, a t , . . . , a n represent the separate results. Let A represent their arithmetical mean or average. Then the differences of these from the mean will be given by d l = a l ~A, d, l =a, t A, ..., d n = a n A. These differences will be called the deviations of the single ob- servations from the mean. They are the effects of the vari- able parts of the errors affecting the measurements. They are not the errors of a lt a 3 , etc. ; for errors are the discrepancies GENERAL LAW OF DEVIATIONS. 15 between observed and true values. But in this as in all cases the true value is unknown, and the deviations are merely the differences from the mean value A, which is selected as being the best representative value, but may differ much from the true value. Thus the deviations measure only the variable part of the errors and give no clue whatever to the constant parts. General Law of Deviations. If the number, n, of obser- vations m the series be very great (to eliminate exceptional irregularities), it is found as the result of the study of actual series of observations that the deviations follow a definite law, both as to sign and magnitude. This law is apparently the same for all kinds of measurements which are affected by a large number of sources of error, and may be called the gen- eral laws of deviations. Special laws arise in certain cases, as will be further indicated. The general law may be approxi- mately stated in words thus: Positive and -negative deviations of any given magnitude occur with equal frequency ; small de- viations are more frequent than large ones ; very large devia- tions occur very seldom. The law is more exactly expressed by the equation y = ke-*\ where y = frequency of occurrence of deviation whose magni- tude has any assigned value x, and where k and h are constants, and e is the base of the Naperian system of logarithms. This expression was deduced by Laplace by an a priori mathematical process as showing the probability of occurrence of an error of any given magnitude when the error was not of simple origin, but was produced by the algebraic combination of a great many independent causes of error, each of which, according to the chance which affects it independently, might produce an error of either sign and of different magnitude. Applied to actual series of observations it is found to sensibly coincide with the distribution of their deviations. This expo- nential equation may then be held as representing the general 1 6 DIRECT MEASUREMENTS. law of distribution of deviations, being in accord both with the theory of probabilities and the results of experience. It is sen- sibly exact when the number of observations is large. When the number is small, the distribution can follow this law only roughly, but no other law would be more closely followed. The approximation with which the series of observations is represented by the law is then greater the larger the number of observations in the series. Mean: Best Representative Value. In a large series of equally careful observations of the same quantity, under the same conditions, the variable parts of the errors will be sensibly eliminated by averaging the results, that is, by the employment of the mean as a representative value. The law of deviations already stated shows that to be true, and as this law has been arrived at by an application of the theory of probabilities and confirmed by the results of specific as well as of general ex- perience, the use of the arithmetical mean as the best repre- sentative value in such a large series can be considered as in accord both with the theory of probabilities and with practical experience. But its employment is, however, justifiable not in in large series only, but in small ones as well. For although the reliability or degree of probability of the mean in a small series will be less than in a larger one, yet the mean has a greater probability even in a very small series than any other representative value which can be indicated. We are accustomed to think of the mean as being more reliable in proportion to the square root of the number of ob- servations in the series, but we must avoid attaching undue weight to this numerical relation when the number of observa tions is very small, as for instance when not exceeding five or ten. A similar caution should be urged respecting all applica- tions of the methods and rules of least squares when n is small, although the use of the methods in such cases is fully justi- fied by the fact that they give the best results obtainable. Deviation Measure, Average Deviation. The magnitudes of the deviations in a given series, although giving no indica- tion as to constant errors, do furnish a measure of the variable DE VIA TIQN ME A SURE, A VERA GE DE VIA TION. 1 7 parts of the errors, since it is to these that they are due. But where the number, n, of observations is not very small, mere inspection does not readily give a definite idea of the magni- tude of the deviations; moreover for many purposes of cal- culation it is necessary to have a single number to represent them. The simplest method of obtaining such a number is to take the arithmetical mean of the deviations without respect to sign, that is, with regard to magnitude only. This quantity will be called the average deviation of the single observation, and will be denoted by a.d. Thus n This, being obviously a measure of the deviations, will be called the deviation measure of the single observation. It gives, at least approximately, the measure of the variations of the resultant indeterminate errors of the individual observations. It shows also that in the given series the observations differ on the average from the mean by this amount ; and we may infer or predict that more observations taken under the same conditions will on the average differ from this mean by about this amount. If we have two series of observations consisting of a different number of observations n 1 and n^ , respectively, all taken under the same conditions and with equal care, then the mean result of the series for which n is greater will be more free from the effects of the variable parts of the errors. The principle of least squares, based upon the general law of de- viations, shows that the reliability in this respect will be in proportion to the square roots of the number of observations respectively, that is, as Vn l : Vn t . Hence we may say that the mean result of a series of observations all made under the same conditions and with equal care is more free from the effect of the variable parts of its errors in proportion to Vn, that is, to the square root of the number of single observations CAUFOv 1 8 DIRECT MEASUREMENTS. from which it is computed. Hence the deviation measure of a mean result would be that of the single observation divided by Vn. Thus, using the average deviation, the deviation measure of the mean result would be This will be called the Average Deviation of the mean. It measures the effect upon the mean result of the average of the variable parts of the errors entering into the single ob- servations, and obviously bears the same relation to a mean result that a.d. does to a single observation. Example II. Suppose 9 separate observations were taken of the distance between two points with the results headed a in the table. The mean result to be used would then be A = 1 6. 2799. The deviations would be found by subtracting A from the values in column a, and are given in column d. The deviation cm. O.006 + 3 9 o + 4 + 13 5 + i 2 9) -43 0.0048 = a.d. . _ 0.0048 , A.D. = ^ = 0.0016 cm. 1/9 EXAMPLE II. 19 measure of the single observation would be the a.d. = 0.0048. The deviation measure of the mean would be the A.D. = 0.0016. This would show us that if we made use of the mean result 16.2299 in any work, the deviation measure to be used would be 0.0016. But if at any other time a single observa- tion only were made of the same distance under apparently identical conditions, and that single result were to be used, the deviation measure which must be used in connection with it would be the a.d., viz. 0.0048. The relative significance of the a.d. and A.D. may be put in another way also. If we wished to compare, as to concord- ance, a number of mean results taken at different times but under similar conditions except as to number of observations, we should use the A.D. of each mean. If we were comparing the relative precision of the single observation in one of these series with that in any other one we should make use of the a.d. The abbreviation d.m. will be occasionally written instead of the full term " deviation measure." It will be understood to denote any deviation measure, viz. the a.d., A.D., or any of those described below, according to the context. The deviation measure is often called the " precision meas- ure," * but the latter term is reserved for another use in these pages. It is essential to note exactly the significance and limita- tions of the deviation measure. It does not tell us that the result, whether a single observation or a mean, is in error by this stated amount (e.g. the a.d. or A.D.), but merely that the variable parts of the errors produce a variation of that average amount in the results. By the law of distribution of these deviations we know that the deviation of any individual observation may be many times the a.d.\ or of a mean result, many times its A.D. In fact that law shows that the chances * This usage was adhered to in the printed Lecture Notes prepared upon this subject, but experience has shown that the change to deviation measure is desirable. 2O DIRECT MEASUREMENTS. that the deviation will assume certain specified magnitudes are those given in this table. ioa 1 a.d. 69. 1 a.d. 43. 2 a.d. II. 3 a.d. 2. 4 a.d. o. I Column second gives the percentage of the whole number of observations which would have a deviation greater than J a.d., a.d, 2 a.d., etc. Thus in any series sufficiently large to fulfil the conditions under which the general law of deviations holds, 43 per cent of the single observations would have a deviation greater than the average, n per cent only (i.e. about one in ten) greater than twice the average, and only one ir? one thousand greater than four times the average. Thus in the foregoing example, where the a.d. was 0.0048, we may say that the chances are 43 to 57, or roughly about even, that any single observation is affected by the variable parts of the errors to an extent of 0.0048 units. The A.D. of the mean of that series is 0.0016, so that we may say of the mean that the chances are nearly even that it is thus affected to the extent of about 0.0016 units. Places of Figures in d.m.; and Negligible Amounts. Ira the numerical value of any deviation measure, two and only two significant figures should be retained ; as was done in the above example. Any single change in the measured quantity, a, due to whatever cause may be regarded as negligible when not exceeding T Vth of the deviation measure of the quantity. Therefore a should be carried out to the place correspond- ing to the last significant figure of the d.m. Similarly any change in the d.m. is negligible when not exceeding ^d.m. The fractional and percentage deviations, d.m. /a and 100 d.m. /a (see page 29), should also contain two and only two significant figures ; and any fractional change is negligible in PLACES OF FIGURES IN d.m. 21 them when not exceeding y^th of their values. Similarly any fractional change in the measured quantity a is negligible when not exceeding y 1 -^ d.m. /a, These statements may be justified as follows : Taking the numbers used in the above example, let 16.2299 denote a mean result of a direct measurement, and 0.0016 its deviation meas- ure. The latter shows that the number 16.2299 ls uncertain by 16 units in the sixth place of significant figures. A change corresponding to -^d.m. would be 2 in this sixth place, already uncertain by 16. It is therefore clear that such a change is immaterial, and may be regarded as negligible. This change of 2 being negligible in the number an equal change would be negligible in the d.m., and, as this is 10 per cent, of that num- ber, a change in d.m. of -^d.m. is negligible. Obviously also the figure corresponding to -fad.m. will always be in the second place of significant figures, so that if we always retain that place and always reject all figures beyond that place in d.m. y we shall never introduce by that process an error exceeding this limit into the d.m. Hence two places of significant figures in d.m. are enough. This limit of -^d.m. as the negligible amount is an arbi- trary selection. A larger or a smaller amount might have been -chosen as the limit, but experience shows this to be both con- venient and suitable in practice. Yet in rather rough work a larger limit may be used, and for such work the d.m. need be retained only to one place when not less than 5 in that place. For instance, if the above example represented rather rough work, the a.d. might be written 0.005 instead of 0.0048, but the A.D. would rarely be written 0.002 instead of 0.0016. By inspection it is easy to see from these statements that the numerical result should in general be carried out to the place corresponding to that of the second significant figure of the d.m. of the quantity. Thus the number 16.2299 should be carried out to the sixth place of figures. This statement is true whether the result is a mean or a single observation, the d.m. being in the first case the A.D., in the second the a.d. In- spection of the data in any case will usually show us what place 22 DIRECT MEASUREMENTS. will correspond to the second of the d.m. even in advance of the exact computation of that quantity. It is obvious that if -fad.m. is negligible, ^d.m./a will be also, for both are the same part of a. Similarly if d.m. must be carried to two places to correspond to this limit, -fad.m./a. must also be carried to two places. The same is, of course, true for the percentage precision. In computing the deviation d by subtracting A from a 1 , etc., it is usually unnecessary to retain for this part of the work more places in A than are given in a lt a^, etc., in the observations. Thus in the foregoing example, to find d l , etc., we use 16.230 instead of 16.2299. If, however, the values of d are very small, the largest value not exceeding perhaps 2 or 3 units in the last place, then it is better to retain the full num- ber of places in A or at least one more than in the values of a. For instance in the example if 16.233 had been the largest and 16.228 the smallest value of a and the mean had been 16.22 995 the deviations would have been formed using 16.2299. It would be useless, however, to retain 16.22993 for this purpose, although it might be proper to retain it for other uses. It is, however, to be noted that when the apparatus gives indications which continually agree within one or two units in the last place of figures obtained by the single observation, it is delusive to hope for much gain in precision by many repetitions. Such cases often occur in practice. They usually show that the indicating part of the apparatus, whatever it may be, is not as sensitive as it might advantageously be made. Thus if in making a weighing of the same object repeatedly by the ordi- nary method, we find that the results agree to one or two units in the last place, e.g. to o.i or 0.2 mgr., this indicates that the balance is delicate enough to have a finer index or to be used by the method of swings. Best Value of n. The question continually arises, how many observations is it worth while to take in order to reduce theA.D. of the mean_? Since A.D. a.d./ Vn, the gain is only in proportion to Vn. But the labor of observing is in direct proportion to n. Thus to double the gain the labor must be O THER DE VIA TIQN MEASURES. 2 3 fourfold, to treble it ninefold, i.e. the labor is as the square of the gain. Obviously then a point would soon be reached where the labor would become excessive in comparison with the gain or with the labor involved in other parts of the work. The limit to the number n of observations to be taken must then be determined by the judgment of the observer as to when the labor becomes excessive in proportion to the gain. In ordi- nary work n = 9 is often a convenient and sufficient number, though a smaller number will frequently suffice. It is rare except in the most careful work, or in work of some special character, that n is made to exceed 25. Other Deviation Measures. Other quantities than the average deviation are also employed as deviation measures. In fact the most common measure is not the average deviation but the so-called " probable error." The relation between the probable error and average deviation is given by the expressions p.e. = o&4 a. ; P.E. = o.Z^A.D., . . . . [3] where p.e. is the probable error of the single observation, and P.E. that of the mean result. The ordinary formulae for com- puting the probable errors from the square root of the sum of the squares of the deviations possess no real advantages over the above, while far more laborious. The probable error, p.e., is merely a deviation of such magnitude that there are, in a large series, just as many deviations greater as less than it ; or in other words, such that in the series there are just as many observations having values lying between^ -{-p.e. and A p.e. as outside those limits ; so that it is an even chance whether any observation taken at random will have a deviation greater or less than p.e. The use of the "probable error" is objectionable partly because of its more artificial character, partly because of the greater labor of computation, but chiefly because the term is seriously misleading. It is, in the first place, not an error at all, but merely a deviation. Neither is it a " probable " value in the ordinary sense, as it is more probable, that is of greater 24 DIRECT MEASUREMENTS. frequency of occurrence, than any given larger deviation, and less probable than any smaller one. Its use leads almost inevitably to a fallacious impression as to the real accuracy of results, and tends to promote negligence as to the constant errors which are of far more serious importance. For a reader meeting a result stated to have a small probable error is liable, unless unusually upon his guard, to receive at once an impres- sion of accurate work and to have his attention diverted from other points upon which the reliability of the work depends to a greater extent. And an observer, with the natural tendency to confidence in his own work, is even more easily misled by the term " probable error." The term "average deviation" tends, on the contrary, rather to call attention to the true character of the quantity; and by the use of "error" solely in connec- tion with constant errors, attention is the more strongly directed upon these. To a competent and experienced observer this discrimination in terms is unimportant, but it is by no means so to the beginner. It may be remarked that the numerical difference between the average deviation and the probable error is negligible in almost all work if we follow the limit already set (viz., ^ a.d. or T V A.I}.). It is also of course true that in all the formulae developed, the probable error may be inserted to replace the average deviation, if desired, a little attention being given to insure consistency. Special Laws of Deviations. Besides the foregoing general law there are other laws which the deviations follow in certain cases. Of these special laws the only one with which we are concerned is that occurring when any deviation between the limits + a and a is equally likely to be obtained, i.e., where all deviations between these limits have an equal frequency. It is easy to see by inspection that the average deviation under this law must be \a. This is the law according to which the deviations occur when tenths of a division are estimated by the eye. With moderate practice, divisions of not less than half a millimeter can be read to tenths with the unaided eye so that the estima- PRECISION MEASURE OF RESULT. 2$ tion shall always give the nearest tenth, that is, so that the ^rror or deviation shall not exceed -f- 0.05 or 0.05 mm. Now as the point to be read is equally likely to lie anywhere along the scale, its actual distance from the estimated tenth is equally likely to be anything within these limits. Thus the &.d. of a single estimation will be 0.025 or ^V tn f a division. Experience demonstrates that this limit is reached without difficulty, and often exceeded where an attempt is made under good conditions to estimate twentieths instead of tenths. Precision Measure of Result. Let d.m. denote the devia- tion measure of the result, viz. the a.d. if the result be a single observation, and A.D. if it be a mean. Let r denote a resid- ual (page n) left by the elimination of a determinate error, r^ , r 2 , . . . , r p being the respective residuals from / determinate -errors. Then the term precision measure, p.m., will be hereafter used to denote the quantity d given by the expression The precision measure of a direct result includes therefore both the deviation measure and the residual effects of all determinate errors, so far as they can be numerically expressed. It is thus the best and only numerical measure obtainable of the accu- racy of that result taken by itself, but it fails, of course, to in- dicate anything more respecting the constant errors than to imply that so far as determinate these have been removed. If the residuals r are all negligible as compared with the d.m., then the precision and deviation measures coincide. The law of accumulation by squares from which the above expres- sion for d is deduced is based on the principle of least squares, and will be further discussed in late sections. The term precision is used intentionally rather than accu- racy in the foregoing paragraphs. A distinction between the denotation of these terms will be maintained. Accuracy will be used only when attention is distinctly directed toward the constant as well as the variable errors. An estimate of the 26 DIRECT MEASUREMENTS. accuracy of a result thus involves a discussion of possible con- stant errors. The precision measure although implying when properly used that no determinate constant errors remain, does not call for a discussion of the constant errors. A result might be precise, and yet contain a large unknown constant error, but it would not then be accurate, in the sense in which these terms are here employed. To Make Residuals Negligible in P.M. One or more of the residuals, r, in the expression ion p.m. may become negligible. Criterion. The criterion is as follows : Any single residual may be regarded as negligible when r = \d.m [5] Any number, ^, of the residuals are simultaneously negligible when the square root of the sum of their squares is \d.m. For instance, r 2 , r s , r^ are simultaneously negligible when y^ + r^ + r: = ^.m ...... [61 A simple though less general criterion for this case is that each neglected residual must not exceed [7] This is based on the assignment of equal effects discussed in the next section. Demonstration. It has been shown that any change which affects the deviation measure by o.i of its amount or less is negligible, and for similar reasons the same is obviously true for the p.m. Suppose first that there is but one residual, r l , what value may r> have consistently with the above limit ? For this case. we shall have, respectively, d 2 = d.m? + r* and ^a.d. This limit is arbitrary and might perhaps be made narrower to advantage. It is not based upon any supposition that an observation with a greater deviation than ^.a.d. necessarily or even presumably contains a mistake. On the contrary, if the law of deviations is followed, observations with a greater de- WEIGHTS. 31 viation than this will sometimes although infrequently occur, the frequency of the deviation ^a.d. being only I in 1000. The basis of any such criterion is rather this : That inasmuch .as the number of observations in the series is always compara- tively small, the large infrequent deviation would have undue influence if allowed to remain ; so that the mean taken after rejecting it is likely to be more reliable than that which would result if it were retained. Something must also be left to the judgment of the ob- server as to the propriety of making a rejection ; and he is -especially entitled to exercise an autocratic power in this re- gard if he has good reason for even suspecting that some ex- cessive or extraordinary cause of error has influenced any given observation. If this is the case the observation ought invariably to be rejected, for one doubtful observation may vitiate a mean by a greater amount than can be compensated by many good ones. There is a tendency, especially among inexperienced ob- servers, to become biassed by the first one or two readings of a series, and to reject, without recording it, any later one which does not closely accord with these, tacitly assuming it to be faulty. This is an essentially vicious practice which cannot be too carefully avoided. Other things being equal the later ob- servations are entitled to greater rather than less weight than the earlier ones, and no result should be rejected without sufficient warrant. Above all things, the integrity of the observer must be beyond question if he would have his results carry any weight, and it is in the matter of the rejec- tion of doubtful or discordant observations that his integrity in scientific or technical work meets its first test. It is of hardly less importance that he should be as far as possible free from bias due either to preconceived opinions or to uncon- scious efforts to obtain concordant results. Weights. Suppose several different independent measure- ments (e.g. by different methods, observers, etc.), to have been made of the same quantity. Let a l , a t , . . . a n denote the results, and p.m^ , p.m.^, ...p.m. n their respective pre- 32 DIRECT MEASUREMENTS. cision measures. And suppose further that it is desired to. find from these results the best representative value. Then if these precision measures give us proper indications of the re- liability of the results, that is if in each case the constant error, so far as discoverable, is negligible compared with the p.m., the weight/ to be assigned to each result is inversely as the square of its p.m. Thus The best representative value will then be the weighted mean viz., A+A + .. The demonstration of this proposition is given in treatises or* Least Squares. Meaning of Estimated Accuracy of Direct Result. This can now be readily defined. When we estimate the accuracy of a result of a direct measurement at a given amount (e.g., if we say that it appears correct to 2 per cent.), we mean that the precision measure of the result does not exceed that amount, and that so far as we can discover there is no con- stant error which is sensible (i.e. not negligible) compared with this/.w. We do not mean that the actual error of the result is of just this amount, for if we did we should correct accordingly. Neither do we mean that this is a more probable value of the error than any other. But using the average deviation as the d.m., we mean that the average effect of all the errors remain- ing, so far as we can discover, is of this amount and may be either -\-or- in sign. This implies that if several results of this kind were to be obtained under the same conditions, the average discrepancy among them would be approximately of this amount. Similarly when we say that a result is desired with an accu- FORMS OF PROBLEMS ON ACCURACY OF RESULT. 33 racy of a stated amount, we mean that the measurement is to be so made that the precision measure of the result shall not exceed the corresponding amount ; and that so far as is dis- coverable the constant error shall not be sensible compared with this. Forms of Problems on Accuracy of Result. Concerning the accuracy of the result of a direct measurement by any single method, problems arise in three different forms. First. To obtain by a proposed method the most accurate result practicable. Second. To obtain a direct measurement of a desired quan- tity and have the result accurate within a specified limit. Third. Given a completed result obtained by a stated method to estimate its accuracy. First. To obtain by a proposed method the most accurate result practicable. To accomplish this the elimination of errors must be carried as far as practicable, i.e. as far as the conditions and the amount of labor which can be devoted to the work will permit. Thus all constant errors as well as the deviation measure must be reduced to the smallest practicable limit. For this purpose, the method, apparatus, and conditions of work must be thoroughly studied to discover, as far as possi- ble, all sources of error, with a view to their removal or to the elimination of their effects. As many as possible of these sources must then be removed by modifying the method, apparatus, or conditions of working. The magnitude of the effects of the remaining determinate sources of error must then be evaluated, i.e. corrections determined for them. Finally, a series of observations must be taken so that their average may reduce the effect of the variable indeterminate errors. To make the result the most accurate practicable with the method, the removals and corrections must be made with suffi- cient exactness to reduce their residuals to negligible amounts, or rather this limit must be approached as closely as can be done without excessive labor. Also the observations must be :r is sfoeing read. Resistance of voltmeter about 17000 ohms. 4 2 DIRECT MEASUREMENTS. Solution. To reach this limit we must be able to get a p.m. of less than cf.2 when all determinate errors are elimi- nated. We must first find the deviation measure. Suppose that several measurements made on a constant voltage of no*. showed an a.d. of O*.o6. Lacking this test we should probably assume about this amount as the a.d., for the following reasons. The deviation would be due to errors of estimation almost wholly. If the index were fine enough, the a.d. of estimating the tenths would be 0^.025 (Special Law of Deviations I, page 21), but with the usual size of index, of deviations, and the un- avoidable parallax 0^.05 to o".i would be a safer assumption. The discoverable sources of instrumental error may be classified as (1) Changes due to change of temperature; (2) Permanent alterations of resistance ; (3) Accidental irregularities in spacing the graduations. (4) Graduation not being correct volts, whether owing to faulty graduation at outset or to change of strength of magnets. Of these, (i) and (3) cannot well be eliminated, but (2) and (4) can be determined, (e.g. by comparison with a Clark cell) at every 10 volts, more or less, along the scale and corrections applied. The points whose errors are thus found will be called the calibrated points, and the corrections, the calibration cor- rections. The precision measure 8 being o*.2 we have and to put in the work to the best advantage we shall make r* = r* = r* r? approx. Thus 0.2' = 0.06' + r 1 ' + r i i + r i '+r 4 ' l o.2 a o.o6 a 0.036 ' *i = r a = r * = r * ~ - = - = 0^.009 ; 4 4 .. r = 0^.09 approx., which is therefore the normal limit for the residuals. SOLUTION OF ILLUSTRATIVE PROBLEMS. 43 (1) By the statement of the makers, the temperature error for the magnetic voltmeter is o.oi per cent, per degree centi- grade. At 1 10* the limit 0^.09 is 0.09/110 = 0.08 per cent approx. This corresponds to a change of 8 C. in the tem- perature of the whole instrument ; for this correction is not for the heating of the coils by the current alone, but is understood to refer to a change of the whole instrument. Apart from the effect due to greater heating of the coil than of the remaining parts, this error might be made negligible by observing the temperature and correcting. This, however, is not practicable and would probably be of doubtful value. (2) An accidental change of resistance of the coils would cause an error proportioned to the change. An error of the limiting amount, that is of 0.08 per cent, would be produced then by an accidental change of 0.0008 X 17000= 14. A change of half this amount can be easily detected by measure- ments on a bridge from time to time, and therefore this source of error can be made negligible. (3) These irregularities must not exceed an average of 0^.09 or practically o.i divisions between the calibrated points. Their amount, however, is practically not determinable, and their effect can be removed only by calibration at many points. (4) The errors from this source must be corrected by cali- bration with Clark cell or otherwise, and with a residual not exceeding o".O9. The Clark cell method is an indirect measure- ment, and would therefore be discussed separately by the methods later given, and will merely be summarized here. The expression for the voltage at the terminals of the voltmeter by one method is where E = E.M.F. of cell at 15 C., a temp, coeff. = 0.00038 for Carhart-Clark cell, t = temp, of cell, R = res. of voltmeter, r = res. between terminals of cell. The results of such a dis- cussion show that, for a result accuracy of o".i in V y the resid. 44 DIRECT MEASUREMENTS. uals for the various component quantities must be as follows : For E, 0^.00056 (just attainable) ; for a, 0.00008 (easily made negligible) ; for /, i.o (easily reduced to o.5, and made neg- ligible) ; for R, 0.04 per cent (attainable) ; for r, 0.04 per cent (attainable). As indicated by the comments in the parenthe- ses, we can do perhaps a little better than o".! in V. But the calibration requires also a reading of the voltmeter when the voltage is Fat its terminals, and the a.d. of this reading is o v .o6. Hence several readings must be taken at each point to be cal- ibrated. If this is done we may expect to get the calibration error with a residual not much exceeding o". i or o". 15. Taking the d.m. and all the residuals into consideration we may then hope to get an accuracy as follows : tf a = o.o6 2 + o.i a + o.i 2 + o.o + o a .i5 = 0.046. .. d o'.2i ; that is, of the prescribed amount. Evidently this limit cannot be reached, however, without great care and the best instru- ments, and it is not to be assumed without experimental proof that the instrument will remain long without a change exceed- ing this amount. INDIRECT MEASUREMENTS. Estimate of Accuracy of Indirect Result. The terms* accuracy and precision are used with the same significance in connection with indirect as with direct measurements. When the estimated accuracy of an indirect result is stated to be of a certain amount it is thereby meant that the pre- cision measure of the result is of that amount, and that, so far as can be discovered, there is no constant error in the result which is sensible (i.e., not negligible) compared with this pre- cision measure. When it is stated that a result is desired with a specified accuracy, it is similarly meant that the precision measure must not exceed that amount, and that no discoverable constant error of an amount not negligible in comparison with this must be left in the result. Thus by stating the accuracy we do not mean that the actual error of the result is of just that amount, for if we did we should correct accordingly. Neither do we mean that this is a more probable value of the error than any other. But using the average deviation as the deviation measure, we mean that, so far as we can discover, the average effect of all the errors remaining is of the stated amount, and may be either positive or negative. This implies that if several results were to be obtained under the same conditions by the same 45 46 INDIRECT MEASUREMENTS. method, apparatus, etc., the average discrepancy amongst them would be approximately of this amount. Thus to be able to estimate the accuracy of an indirect result, we must have data for finding properly the precision measures of its component direct measurements, and we must be able to compute the precision measure of the result from that of the components. Also we must be able to show that due care has been taken in the correction or removal of all the determinate errors of the components, so that none but negli- gible constant errors remain. Finally, we must be able to show that the " error of method " (see next paragraph) is neg- ligible, or if not so to take it into account. The numerical part of the estimate of accuracy will be the P.M. if the " error of method " is negligible, or will be the ^P.MS + R*) if R be the estimated amount of this error. Error of Method. Besides the constant error of the com- ponents there is, in certain cases, another source of error in indirect results. The method adopted may have inherent sources of error ; that is, it may fail to give correct results, however accurate the components, because the result sought does not bear the supposed relation to the components, as expressed by the function from which the result is computed. This may arise through the introduction of some approxima- tion, through the existence of some inexact hypothesis, or from other similar cause. Errors of this sort will be called errors of method. Their amount or the average uncer- tainty arising from them may sometimes be estimated and allowed for. The possibility of their existence should not be overlooked. As an instance, we may cite the case of the " Stray Power Method " of testing dynamos or motors (ex- plained among the final illustrative problems). In this method the assumption is made that the " stray power " is the same when the machine is running under no load as when under full or partial load, certain conditions being fulfilled. The results by this method will be more or less uncertain if this assumption is not strictly true. PRECISION MEASURE OF RESULT AND COMPONENTS. 47 Check methods are of course of the same importance in detecting and eliminating constant errors of indirect as of direct measurements. To obtain the most accurate results and to get clues to constant errors independent results should be obtained by as many different methods as possible. Such checks may be had by changing the methods or instruments for measuring the components, or by changing the indirect process for another. As an instance of the latter way we may take the determination of the commercial efficiency of an electric motor. Several checks on this might be obtained by using the various purely electrical methods, and still other checks by testing on a cradle dynamometer, with a friction brake, etc Relation between the Precision Measure of Result and {Components. Types of Problems. The following three types of problems are the chief ones which arise concerning the re- lation between the precision measure, P.M., of the result and those, p.m., of its components. First. To find the P.M. of the result given the p.m. of each component. Second. To find the best value for the p.m. of each component, for a prescribed P.M. of the result. Third. Given the P.M. of the result, or the p.m. of some of the components, or both, to find the best magnitudes m^, m^, etc., af the components. The third of these, when solvable, enables us to determine in advance the best relative or actual magnitudes of the com- ponents, that is those which will yield the most precise result with the given apparatus. Examples of this occur in such cases as finding the best deflection to employ in using an ordinary tangent galvanometer, the best deflections to use in finding battery E.M.F. or resistance by the two-deflection method, etc. These problems will be again taken up after the first two types have been entirely discussed. The next step will therefore be to establish formulae for the relation between the P.M. of the result and the p.m. of the components for the solution of the first two types of problems. 48 INDIRECT MEASUREMENTS. General Formulce. Let x, y, z, . . . represent a number of independent variables. Then if w is a quantity which is some function /of these, this fact is ordinarily indicated by an ex- pression of the form This expression, and others deducible from it, apply immedi- ately to indirect measurements and their precision discussion. For example, if g were determined by means of a simple pendulum, the process would be an indirect measurement of g- The component quantities directly measured would be /, the length, and /, the time of a single vibration of the pendulum; and the function by which g is deduced from these would be This corresponds to the above general expression thus : g to iv,. / to x, and t to y ; for / and / are independent quantities, and n is a constant. Instead of the ordinary mathematical notation using x, y+ and 3 as in equation (16), it will be much more convenient for this work to employ a special one adapted to the purpose^ We will then use as the general expression where M is any quantity whatever which is a function of the quantities m lt m z , . . . , m n , each of which is directly meas- ured and is wholly independent of the others. A quantity M thus determined will here be called an indi- rect or an indirectly measured quantity. It is often called a derived quantity. SEPARATE EFFECTS. 49 Notation. The following notation will be always em- ployed : M = the final indirect result. m = any component. a,& = constants. n = the number of directly measured components. A = any small finite change (expressed in units) in M. << << << .jjj A T7 = the corresponding fractional change in the result M. - = " " " " " any m. m z/J/and dm will be used instead of A and d when needed for greater clearness. Any change will be considered small which does not ex- ceed a few per cent of the corresponding quantity. f() will be written for brevity in place of /(ni^n*, . . . m n ). Corresponding subscripts will denote corresponding quan- tities. Thus A^ will denote a change in M corresponding to a change d^ in m l , etc., and vice versa. The subscript k will be used to denote a general term. Thus m k will denote any component, and d k or A k the corre- sponding changes in M. Separate Effects. I. What will be the change A l in M corresponding to, or produced by, a change ^ in m l , all other components remaining unchanged? Differentiating /( ) with respect to *#, , all other compo- nents being considered constant, we have as the rate of change of Jf with MI dM _df() dm l dm^ * Passing from the limit to finite changes, we have A dM 50 INDIRECT MEASUREMENTS. the approximation being closer as the changes A^ and d l are smaller. Hence Similarly for a change a in m t , the corresponding change in M will be and so on for all the n components. Example XIII, page 86. II. Given a small change A in M, what change # in any component m alone would correspond to, or would be neces- sary to produce, this change A ? This is evidently merely the converse of I. Consider first m^. By 1 8 we have dm Similarly, and so on for all the n components. The changes A l , z/ 2 , etc., may or may not be equal. Example XIV, page 86. Resultant Effects. III. Suppose that small changes #, in m lt # 3 in m^j and so on, occur simultaneously, what will be the resulting change in M? Here we must distinguish two cases. RESULTANT EFFECTS. 51 i. Where the changes d lt # 2 , etc., are of specified magni- tude and sign. 2. Where we wish to deduce a general expression which will give us the best solution when all that we know respect- ing the tf t , #., , etc., is the following. Each $ may be of either sign, + or , and of a magnitude following a certain law of distribution, i.e. the ' ' ' ' [26] an expression to which reference will occasionally be made. IV. If we were to ask for the resultant effect the question analogous to II for separate effects, it would be, what simul- taneous changes in tf j , # 2 , . . . , d n would produce a specified resultant change A in Ml It is obvious that we might have an infinite number of values of #,, # 2 , etc., which would produce the specified A for any given function f( ). It is therefore necessary to assign some further condition. Equal Effects. For reasons which will appear later, we may advantageously restrict our inquiry to the case where each 6 produces an equal effect with every other on the value of A, i.e., where each d produces in M the same change as every other. This will evidently be the case when in 24 we have [27] as A^ is the change in J/due to . [44] a, b, . . . , k = constants, and may be either + or or of indeterminate sign and of any magnitude. This case therefore includes the preceding. Separate Effects, A, = ad lt 4= &*,, ..., 4 n =kd n . . . [45] Conversely *. = ^i, 6 * = J**> > #*=^ n .. . [46] Resultant Effect, ^ = (aS i y + (&W+...+(te n )> [47] Equal Effects, ad, = to, = . . . = kd n = [48] vn Deduction, j - = a, etc., and substituting these in the general formula gives the above results. 5 8 INDIRECT MEASUREMENTS. M=a.m 1 .m*..... mn [49J a = constant factor. Separate Effects, Conversely Resultant Effect, ' *'* Equal Effects, Example XVII I, page 89. For separate effects the result may be put into words by saying that for any factor the fractional change in the result is equal to the fractional change in the factor. Deduction. - = *.**,. ....*=, etc., and substitut- ing in 33 gives 4 = j- , A = Similarly, 4 = 1 % etc., for separate effects. Substituting these in the general formulae 36 and 38 gives 52 and 53. Separate Effects, M ~ m, 9 M FORMULA FOR GENERAL AND SPECIAL FUNCTIONS. 59 Converse is evident. Resultant Effect Same as 52 since negative signs disappear on squaring. [57] Equal Effects, *!-= -A=A=-^= -4 1581 m l " w a nis m^ ~ ^ n M ' N.B. The signs are of no importance and may be neglected in most precision discussions, for they merely indicate whether a + change in the result is caused by a -f or a change in the component, a fact usually of no interest. Deduction. a; before - - , do UCH-J1C , ' m and so on. Substituting in general formulae gives above re- sults as before. M = am v . .... ............... [59} a = constant factor ; v = constant exponent. Separate Effect, A $ -ITF= ^ ........ [601 M m In words : If the function be a constant power, and he either with or without a constant coefficient, the fractional change in the result is equal to the fractional change in the component multiplied by the exponent of the power. Conversely $ i A Example XIX, page 90. <60 INDIRECT MEASUREMENTS, N.B. As the value of v is unrestricted, this holds for a negative exponent. Thus if v = c, a A d Mam- c = , and -^ = c. . . [62] m c M m Deduction. df(\ M M A 8 - - = avm~ l =. v .*. A = v d and -^-=- = v~. dm m m M m M = a-mS . . . m, n w - ......... . . ..... [63] ^ = constant factor; v, w constant exponents. Separate Effects, Conversely Resultant Effect, ** Equal Effects, m Example XX, page 91. N.B. As v, ID, etc., are unrestricted, any of the exponents may be negative, so that this covers the case where any of the factors are in the denominator. The only difference will be that in separate or equal effects the sign of the exponent will become negative, as shown in formula [62] above. But as FORMULA FOR GENERAL AND SPECIAL FUNCTIONS. 6t the sign is usually of no interest as already explained, we may omit the consideration of it, and in using this formula for factors we may treat a factor in the denominator, i.e., with a negative exponent, precisely as if it were in the numerator, i.e., had a positive exponent. This case evidently includes as special cases all the foregoing functions separable into factors. Demonstration. Obvious from the two preceding. Besides the foregoing simple functions which consist merely of the sum or difference, product, quotient or power of the direct quantities, there are others less simple, for which formulae for equal effects are of much service. Of these the three principal ones are, first, where f() can be separated into- a series of factors each of which is a function of one compo- nent direct measurement and only one ; second, where /() can be separated into two or more terms to be added or subtracted,. each of which is a function of several of the components but has no components common to any two of the functions ; and third, where f( ) can be separated into two or more factors each of which is a function of several of the components and having no component common to any two of the functions. In the second and third case each function comprises a group- of components, and the process will hence be referred to as separation into groups. Of course any of these functions /() could be discussed by the general formulae, the advantage derived from the use of the special ones being that they greatly lessen the work of differentiation and numerical substitution, where f( ) is complicated, as is often the case ; and they also render the solution of the problem clearer and easier to follow. Separation into Factors which are Functions of Single ' Com- ponents. M = Km,). p(m a )-. . . -cr(m) ...... *..:.'. . . [68]. Resultant Effect, MI - - K) p(m,) _+ , ff(tllii) 62 INDIRECT MEASUREMENTS. Equal Effects, _ ____ o-) 4fc M ' These determine the values of - f J-, etc., from the stated 0K) A value of jr~ , and from these values we have to find the corre- <* spending values of or of tf, as the case may require, by the general or special formulae for separate effects. Example XXI, page 91. Deduction. A, = r-. Now Similarly, _ t ) 6, _ - ' a : _ a= etc M ' P( ' By [26] for resultant effect we must have -. . / " h w/> in which substitution of the above values gives [69]. By [38] for equal effects we must have M A A ng values of we obtain the desired expression [70]. from which by substituting the foregoing values of ~ , -j^, etc., FORMULA FOR GENERAL AND SPECIAL FUNCTIONS. 6$ Separation into Groups. M = <|>(mi , . . . , m p ) p(m q , . . . , m s ) . . . where there are p components in the function 0( ), r in the function p(), s in the function (Wi , . . ., m p ) p(mg , . . . , m s ) '; that is to 0.90^ ? The corresponding; change in d* would be A* _ (0.90^)' = ^ 2 - 0.8 1 ^ 2 = o.i9^ 2 ; .*. ^* 8 = o.i9^ a , 4* = 0.44^ ; .*. A k = 4, approximately. The limit A k =. %A is a safer one and preferable for ordinary work where n is small. It corresponds to a change of -$4. This limit of \A is employed in the present demonstration. If instead of a single change A k there are p changes whose joint effect on A is to be considered, then this joint effect would be equal to a single change of the magnitude y A* + A* + . . . + Af. For the effect of these on A' 2 is. equivalent to that of a single change A k such that CRITERIA FOR NEGLIGIBILITY. 69 This furnishes the general limit, the last one given in the cri- terion. A special case of this is, of course, when A a = A b = . . . Ap, so that Whence for A k = A we have This is a more convenient limit to apply practically, but obviously it is not essential that this equality should exist, so that this is an arbitrary or more special criterion. For reasons which will be shown in the section on equal effects, however, this arbitrary limit will not be widely departed from even under the general limit. Corresponding criteria can be given for negligible compo- nents m. Thus cases will be found where the effect of one of the components m k itself upon the result M is so small that the component may be neglected altogether in the computation of M and its measurement may be omitted. This would evidently be the case when the omission of m k did not affect M by an amount exceeding -fad, or better -fad. We should then pro- ceed as follows. Find the value of d k which would be negligible under the criterion A k ~ \A. Then. if m k < d k as thus found, it may be wholly neglected. If it is a question whether several components may be neg- lected, then if / is the number of these, the limit for d k must i A obviously be that corresponding to -- -j=. instead of \a* The opportunity for such omissions occurs most frequently in functions containing correction terms, such as temperature corrections or those in the formula for the tangent galvanom- eter. 70 INDIRECT MEASUREMENTS. Numerical Constants. A large number of functions con- tain numerical constants, either mathematical such as ?r, or physical such as the density of a given substance, etc., whose numerical values are known beyond the requirement of the case. For these we desire to know how many places of significant figures it is necessary to retain in the computation. The error which enters by them is only that due to the rejected figures, e.g., from using 3.142 for n instead of 3. 14 1 59 26$ ... As the number of such constants in any given function rarely exceeds two, we may safely proceed as follows. From the prescribed or computed value of AM calculate what would be a negligible value Sc or dc/c of the constant just as though it were a directly measured component. Reject all places which do not affect the constant beyond this limit, de- parting if at all on the side of safety. Example XXIII, page 96. Equal Effects. Demonstration. In deducing formuLne [27] etc., the condition was arbitrarily imposed that we should A have J, A 9 = . . . = A k = . . . = A n = , in which case Vn the 8 of each component has an equal effect on AM. In what way, and to what extent, this is the best guiding condition when d represents the precision measure may be seen as follows. If the number, n, of components were indefinitely large, and if the in all data, constant factors, products, quotients, and in the result. When the A/M of the result is computed, any places in M retained under the above rule may be rejected if they exceed the requirements of Rule 3. Example XL To be multiplied together: 29.34 with d 0.58, 42 231.6 with d= 1.4, and ^ = 3.14159265. The first number is clearly the least reliable. Its percentage deviation is 0.58 -f- 29. 2.0 per cent. Hence we should use four sig- nificant figures throughout. * It is understood, of course, that this means if 100 \ i per cent and < 10 m ^ per cent, and so on for the others. 8o INDIRE C T ME A S U RE MEN TS. Ordinary Multiplication. Shortened Multiplication. 29.34 1239000 42230 3.142 29-34 42 230. I 239000. 3-142 8802 2478 5868 4956 5868 1239 11736 3717 11736 586 58 9 3717 1239 496 24 1239028.2 3892938. 3 893 ooo. i 239000. 3 893 ooo. Logarithms. 29.34 1.4675 42 230. 4.6256 3.142 0.4972 3 893 ooo. 6.5903 In this shortened multiplication the partial products cannot safely be omitted, until one place beyond that to be retained is reached. The method is sufficiently obvious upon inspec- tion. It saves about one third of the time required for such work. Rule 6. When logarithms are used, retain as many places in the mantissae as there are significant figures retained in the data under Rule 5. The characteristic is not to be considered, as it serves merely to locate the decimal point. Example XII. See under Rule 5. Demonstration of Rules. Rules I, 2 and 3. The reasons for these have already been given at page 18. Briefly stated they are these : Let the place of m corresponding to that of the second significant figure in its # be called the rth place. As m is uncertain by an average amount of $, any change which is I per cent. A similar proof, of course, applies to all cases. A _d' The -=-=. of the result will be ~ , of the least fractionally M > m f A _ precise quantity, by formula [52], so that -.- i per cent. In 6 i d general whatever the value of , an amount - - will be m 10 m d negligible as being insignificant compared with . There- fore --- 7 and -r-j. will be negligible. The smallest value of A corresponding to this limit will be I in the fourth place, viz., when M= 1000; and we must not allow the accumu- RULES FOR SIGNIFICANT FIGURES. 83 lated computation error to exceed this amount. This will be accomplished under the Rule 5. The worst case under this rule is when the first four figures in each factor of the data, constant, intermediate prod- uct or quotient, and the final result, are IOOO., and if it can be .shown that for this the accumulated rejection error does not exceed the limit, the rule will be justified. Let m l , m^, . . . , m n denote the values of the various data, constants, and intermediate products or quotients, and suppose a rejection to be made at each of them, viz., n rejections in all, leaving errors e l , e^ , . . . , e n . Then But when expressed in units in the 4th place, we have under the above conditions m,=m^ = ... m n = 1000. = M, also expressed in units in the 4th place. Therefore as under Rule 4 E* = V ne* approx., . where e = average rejection error = 0.25 in 4th place. Thus for n = 16 = i in 4th place. Thus the rule is justified for the worst case, and is therefore sufficient for all. Here also it is to be remembered that E as thus computed is not a definite but only a most probable value, and is thus a quantity of the same character as 6 or A, and the criterion of I A as a limit of negligibility is properly applied. It is not 84 INDIRECT MEASUREMENTS. asserted that the accumulated rejection error will never exceed E = i in 4th place. On the contrary in 16 rejections an error E of 8 in the 4th place would occur if all the errors e happened to be of the maximum value and in the same direc- tion, an event which would be exceedingly rare. Rules which provided that the maximum error should I // never exceed -r^- would be needlessly wasteful of labor. The limit used above is in accordance with that used elsewhere in this book, and is abundantly close for all work except possibly such as conforms with extraordinary closeness to the premises upon which the method of least squares is built up far more closely than any ordinary physical or technical work. Rule 6. Inspection of a table of logarithms will show that in the worst case, i.e., where m = 9999, a change of I unit in the 4th place of the logarithm will correspond to a change of 2 units in the 4th place of the number, which shows that the accumulated error will be about twice as great in the worst case by logarithms under rule 5 as by numbers under rule 4, But on the average the error is very nearly the same by loga- rithms as by numbers for the same value of n, with the advan- tage in favor of logarithms as reducing n by requiring fewer intermediate operations. Forms of Problems on Accuracy of Result. These are the same as those given for direct measurements at page 33 et seq. The procedures there outlined apply to each of the direct measurements from which the indirect result is made up. It is therefore only necessary to add the following. In an indirect measurement any consideration of the accu- racy of a result involves that of each component direct measure- ment and also involves the discussion of the possible " error of method " (page 46) of the process. The problems present themselves in these forms. First. To obtain by a proposed method the most accurate result practicable. For this purpose the method as a whole, and the method, apparatus and conditions of work in measur- ing each component, must be studied as described at page 33. PLANNING OF INDIRECT MEASUREMENT. 8$ Second. To obtain a measurement of the desired quantity, and have the result accurate within a specified limit. For this the statements made at page 34 apply verbatim, both to the whole method and to the separate components. It is necessary only to add that a preliminary solution by equal effects should be made to determine approximately the best value of the precision measure of each component. Also at the close of the actual observations, it is essential to make a final calculation of the precision measure of the result and to combine with this an estimate, so far as one is possible, of the ** error of method " and of the effect of any constant which may be suspected to exist. Third. Given a completed result obtained by a stated method, to estimate its accuracy. The remarks of the corresponding .section at page 35 apply equally to indirect measurements. Data required to Substantiate Result. The remarks of page 36 apply equally to indirect measurements. Planning of Indirect Measurement. To the statements made under (a) to (a) on page 37 it is only necessary to add that a preliminary precision discussion based on approximate -data should invariably be made before a choice of methods is finally reached, or at least before any experimental work beyond preliminary trials is entered upon. The weak or strong points of the proposed methods are often thus developed, and impor- tant modifications suggested or errors avoided. It is also of .great importance that the plan for the " reduction," i.e., the algebraic and numerical calculations, be thoroughly developed in advance of the measurements. A slight modification of a proposed method may sometimes transform the reduction from a laborious to a much easier one. 86 INDIRECT MEASUREMENTS. EXAMPLES. Example XIII (see page 50). The value of g is to be measured by a simple pendulum whose time, t, of a single vibration is to be about 2 seconds ; (a) what change (in units) in g would correspond to or be produced by a change of o.i cm. in /? Here corresponds to M,l to m iy t to #z a , and n* is a constant. We have to find the value of -J, corresponding to 61 = o.i cm By [19] cm. .-. J, = 2.4 X o.i = 0.24^. (^) What change (in units) in g would correspond to a change of 0.02 sec. in / ? df() J 400 -^ = - 2^ 2 ^ = - 2 X 3-i 8 X ^r = - 960., the length of a 2-sec. pendulum being about 400. cm. cm. .'. J a = - 960 X 0.02 = - 19-^1- The negative sign indicates that a + change in / produces a change in g. Example XIV (see page 50). In a measurement of g by a 2-seconds pendulum, as in Example XIII, (a) what change in / would produce a change of I-OT, in gl EXAMPLES. 87 By [21] */ A I ** Sl =^/ Tr dg It* ^=- = 2.4, J,= i.o .-. 61= 1.0/2.4 = 0.42 cm. (b) What change in t would produce a change of i.o per cent cm. i.o per cent of g is o.oi^, and as g 980. - a nearly, 9GC ^=0.01 X 980. = 9-8^ By [22] dg , / . f t = - 2*'p r= - 2 X 3-I 8 X JT = ~ .-. dt = 9.87 96a = o.oio sec. which is - - = 0.005 or J per cent of t. Example XV (see page 53). Suppose that in Example XIII the changes 61= o.i cm. and $t= 0.02 sec. were of the nature of deviations and occurred simultaneously, what would be the resultant effect on^-? By [24] and by Example XIII J, = .^ = 0.24-^, and sec. at * sec. f = (o.2 4 ) 2 + (- I 9 .) 2 = 0.058 + 360. = 360. cm. /. A = 1/360. = 19. 88 INDIRECT MEASUREMENTS. Example XVI (see page 54). In the measurement of g by a 2-seconds pendulum, as in the foregoing examples, what changes #/ in / and 6t in t would have a combined effect on g of Ag = 3.0 - a , supposing them to be of the nature of devia- scc tions ? Solving for equal effects by [29] etc., we have = and * = = dl / dt ~j = 2.4, -~ = 960., as before, and A = 3.0 .-. / = A. x _L = 0< 88 cm. 1.4 2.4 tf/ = A X rzr- = 0.0022 sec. Example XVII (see page 57). The rise of temperature A = t^ t l of the water of a continuous calorimeter is to be measured by the reading of two thermometers. (a) What will be the precision of A as affected by /, alone if the/.w. dt, iso.02? By [40], A = <*! (b) What will be the/.w. of A for a/.^. of tf^ = o.O2 and = o.o 3 ? By [42], J a = 0.02' + -3 a = ' 4 + ao 9 ao J 3' EXAMPLES 89 (c) What will be the p.m. necessary in each component for &p.m. of o.oi in A ? By [43], dt l = tf/ 2 = -~p = o.oo7i. Example XVIII (see page 58). An incandescent lamp burns for a time t under a voltage v and with a current c. The quantities c, v, and t are measured in order to determine the amount of heat H produced in this time. H = kcvt, where k = a constant. (a) What p.m. in H would correspond to a precision of o. I per cent in c! 6g = o.i per cent = o.ooi. By [50] W = = a 01 - AH The fractional precision in H would then be =y- = o.ooi or o.i per cent, whence AH = o.ooi H could be found if desired, if H were known. (fi) What percentage precision in v alone would correspond to 0.5 per cent in HI d. A. : =Jf =0.5 per cent. (c) What would be the fractional precision in H resulting Sc $v from a fractional precision of = o.ooi, - = 0.003, = 0.002 in the components? A\ j = o.ooi 9 + 0.003' + 0.002' = o.oo ooi 4 ; go INDIRECT MEASUREMENTS. (d) What percentage precision in each component would correspond under equal effects to 0.2 per cent in H? dc 6v dt I = - = T = ~7= X 0.002 = 0.0012, c v t 1/ or o.i 2 per cent in each. Example XIX (see page 59). The volume of a sphere is to be computed from its measured diameter D. V (a) If the precision of D is I per cent, what is the precision? of the result ? A d = v=$X o.oi = 0.03, or 3 per cent. (b) What precision would be requisite in D for I per cent in VI d i A =--=$X o.oi = 0.0033 or % per cent. Example XX (see page 60). For the sake of comparison with former examples take the case of the measurement of g with a 2-sec. pendulum. (a) What would be the precision of the result if the frac- tional/.^. of / were o.i per cent, and of t the same? We should separate into the factors n* X / X t~\ Hence by (a) 6l* = 0.00 ooo i + o.oo ooo 4 = o.oo ooo 5 r = o.oo 22, or 0.22 per cent. EXAMPLES. 91 (ft) What would be the values of 31 and 3t necessary under equal effects to give g with a precision of o.ooi per cent ? By [67] 31 3t i j = 2 = - X O.OOI = 0.00 071. 31 /. y = o.oo 071, or 0.071 per cent, st i = - X o.oo 071 = 0.00 036, or 0.036 per cent. To find 31 and 3t, we must know / and t. t is stated to be 2 sec., and as g must be about 980 /must be about 4.0 nu .-. 31 = o.oo 071 X 400. = 0.28 cm. 3t = o.oo 036 X 2.0 = 0.00071 sec. Example XXI (see page 62). The measurement of a cur- rent by a cosine galvanometer affords a good example of a function which can be separated into the product of several functions, each of one component only. For a primary cosine galvanometer the formula may be written tan 2nn cos os where //"=horiz. comp. of earth's field, r = mean radius of coil, n = whole number of turns in coil, = angle of deflec- tion of needle under the current C, co = angle of tip of coils from vertical when is read. The function would be separated as follows : 92 INDIRECT MEASUREMENTS. Of the components, H and r enter as simple factors, n as a factor to the I power, in the function tan 0, and GO in the function - or (cos GO)~ I both of these functions being COS GO factors. For this value of/(), the fractional precision would be, by [69], Suppose a measurement in which the precision of the compo- dH 3r 6n nents was -77-- = o.ooi, = 0.0005, ne gl-> ^0 = O .025, A <$<& = o.O25, what would be the value of -r^-? From [61] we know that - - . We require further n' 1 n , 8 tan , d cos GO the values of - and - . To find these we must have tan cos GO the values of and GO. The data would furnish this, or in a preliminary discussion typical or limiting values would be used. Suppose = 45 and GO = 60. Then by [33] d tan = ^ d0 = sec 8 deb. tan0 dtan sec 8 2#0 tan tan sin 20 " To compute the numerical value of this we must have #0 ex- pressed in terms of n. 7t o.025 = 0.025 X -- = 0.025 X 0.017 = o.oo 043. Hence 2 X 0.00 043 = O.OO 086. tan sin 90 EXAMPLES. 95 Also #(COS GO)' 1 d COS GO (COS Qo)~ l ' COS ^ ' # cos oo = -7 (cos GO) <$GO =. sin GO dco ^ cos G? sin ft? = -- doo = tan G? * o GO. COS Gz? COS GO Substituting gives )~ I T = 1-7 X 0.00043 = 0.00073. Then = o.oo i 2 + o.oo os 2 + o 2 + o.oo o86 2 + o.oo 073* = o.oo ooo 25 = 0.0016. Equal Effects. What fractional precision in each compo- nent would be necessary, dn being negligible, in order that A/M should be 0.2 per cent? For equal effects we must have 3 H dr __ dn~ l _ $ tan _ (cos co)- 1 I A ~Jj~~r~"~n :i "" tan0 (cos c^)- 1 "" y~^M' As before = , and -, r- = tan GO -6 GO. tan sin 20 (cos c;^)- 1 94 INDIRECT MEASUREMENTS. The number of components is n = 4. .'. - --^ % X 0.002 = O.OOIO. 6H .: -fj- = o.ooio ; 1 = O.OOIO' r d tan 2dcb - = o.ooio = T-~. tan sin 20 /. #0 = 0.00050 sin 90 = 0.00050; -r - TT = o.ooio = tan GO. #G?. (COS GJ)~ .:$(& = o.ooio/tan 60 = o.oo 060. o 0.00060 &GO -- O .035. 0.017 Example XXII (see page 65). The following is taken as a simple illustration of the separation into groups. More complex examples will occur in connection with the tangent galvanometer, etc. A continuous water calorimeter is to be tested by trans- forming into heat within it a measured amount of electrical energy and measuring this heat by the calorimeter. For instance some incandescent lamps or a coil of wire carrying a current are placed within the calorimeter, the mean current c through the coil and the mean voltage v at its terminals are measured ; also the mass m of water passing through the EXAMPLES. 95 calorimeter during the measured time r, and the temperature and , of the entering and outflowing water. Then where k is a constant, viz., the heat-equivalent of I watt, calculable from the mechanical equivalent of heat and of the watt. The test consists in ascertaining how closely the experi- mental value of k, viz., _ K ~" CVT agrees with the computed value. What precision is requisite in each component for a test to o.i per cent? The problem may be solved by the general formula, but could not be solved exactly by the simple formula for factors, since one factor (^ /,) contains two components. Applying the formula [77] for separation into groups, we have for equal effects dm i d(t z t,) _ dc~ l _ far'__ dr~ l _ i A dc~ l dc Noting that = , etc., and neglecting signs as of no consequence, we have dm dc dv df I ^ :Z 7 :: V :: - = ^X 0.001 =0.00 040, from which the numerical values of dm, dc, dv, and dr could be computed if m, c, v, and r were given. 2 - = \ 2 X 0.00040 = 0.00056. fa f-i 96 INDIRECT MEASUREMENTS. To find dt l and #/ 2 we must have the value of , t l Suppose this to be given as 10, then 6(t 9 - /J = o.oos6. Now by [43], for equal effects 2 = dt, = ~-^(t, - /,) = o.oo40. V 2 Example XXIII (see page 68). On a cradle dynamometer or on a friction brake the horse-power is given by _ 27iRNW H.P. = , 33000 where R is the radius at which the load P-Fis applied and ./Vis the number of turns per unit of time. N really involves two measured components, viz., a time and a count ; but the count is usually made by mechanical means, in such a way that the error of observation resides wholly in the time. We might therefore well substitute for JV its equivalent expression =, where T is the time of a single rotation, and thus find $ T. But we may just as well proceed to find d N and from that find 6T. (a) Suppose it required to find whether with no other rejec- 3W A tion -jr = s^th per cent would be negligible for -j = O.OOK The negligible limit by the criterion would be, as before, dW _\ A anc | ^th per cent = 0.0005 would not be negligible. EXAMPLES. 97 (b) Suppose it required to find what values of and -^ A would be simultaneously negligible for -j-^ = 0.001. By the criterion this would be when either was <- I i ^ II _ = X O.OOI = 0.00024. ~ 3 Vp M 34/2 (c) Suppose the problem to be, how far must the constant be carried out in order that the rejection error shall be 33000 JH.P. negligible with respect to p o.ooi. The rules for significant figures would require it to be carried to 5 places, and as it is the only term in it in which a rejection would be made, it must be carried to 5 places of significant figures, viz., to 3.1416. Let us apply the rule of this section and see whether a similar result will be reached. As only one rejection is to be made, and as TC enters as a dn direct factor, we shall have negligible when 6n _ i A or dn = 0.0003 n = o.oo 093. Carrying n only to 3. 142, the rejection only makes an error of dn 0.0004, which is within the limit assigned by the criterion. Hence by following the criterion we should use n = 3.142, by the rules for significant figures n = 3.1416, so that if the criterion is reliable the rules are more than suffi- ciently precise in this case. 98 INDIRECT MEASUREMENTS. (d) Suppose the question to be, would the use for TT of dW 3.142 be admissible with a precision of -^ = 0.00030 and a precision of O.I per cent required in the result? By the criterion this would be the case if I A n 0.00041 = -=0.00013; . 0.00033. This would be barely negligible. Example XXIV (see page 76). The expression for the specific resistance per metre-gramme at o c. of a wire may be written in the form I+/ gr TT where m = mass, /= length, r' = resistance of the sample measured on a Wheatstone bridge, ft' = its temperature coef- ficient, t = its temperature at the time of measurement, ft = temp, coeff. of bridge, T= observed temperature of bridge, r = temperature at which it is correct. For the precision discussions this expression may be simpli- fied by using the approximations = I - A^ approx. and These would leave the expression in the form S = -r'-(l +ft T T)(i - /J r r)(i - ft{t) approx., EXAMPLES. 99 which may be still further simplified by using another approxi ination and writing 5 = ~ r' [i + ft T T - /? T r - /?//] approx. These approximations are well within the limit, for the terms fiT, fir, and fi't are all less than o.i, so that the error of the approximation, which is smaller still, is negligible in proportion to I in the parentheses. This final expression affords a good illustration of the method of separation into groups. Counting the temperature coefficients and r y as we should do in a preliminary discussion at least, where there was any question as to the possibility of their being known accurately enough, we have in the [ ] six components, and in all nine. Hence for equal effects by [78] we should have dm _ 3l_dr' _ ! 2, then the components other than the two considered must be treated as constants. The following discussion applies only when m t and m, are independent of each other. The given values of d or d/m constitute what may be called the precision condi- tions. The problem separates naturally into two parts: first, to find the best ratio /,//, ; second, to find the best numerical values of m l and m t , to solve which we must previously deter- mine the best ratio. It may occur that only the best ratio is required. We will consider first the finding of the ratio, be- ginning with the case where 6 and not d/m is given. Best Ratio. The best value of mjm l will be the one which will make A a minimum for a given value of M. The proced- ure must therefore be, first, to obtain a suitable expression for A or A*, and then to find by the calculus the value of mjm l which will make this a minimum. As to what is a suitable expression for A or A* we may readily see several things. First, it must of course be a func- tion of d lt tf a , m lf and ; a . If it is a function of $ 1 and # 3 alone, e.g., A* = d* -\- tf 2 2 , it shows that there is no best value for the ratio, for A is determined independently of m l and m v Second, if it should be found to contain /() as a factor, that factor must be omitted in deducing the minimum. For if we desire to find the ratio which will make A a minimum for any given value of M we must treat M, and therefore /(), as a con- stant. Now, if we write the expression for A* for any/(), for i AY which we can also write an expression for \jTf] by any of our formulas, this expression for A* must be divisible by M*. TWO VARIABLE COMPONENTS. IC>5 Hence it will be simpler to write at once the expression for ( } . where the function is such that we can do so, and proceed \ M i to find the ratio which will make, that expression a minimum. Jf the expression for ^ a has been written and there is any pos- sibility that it contains f\ ) as a factor, the test for it should be made by dividing through by it. Any constant factor of the whole expression may be removed, since it is of no effect upon the determination of the minimum. p. J> If instead of ^ l and tf s we have given - L arid , there can 7 rL j trl n I A \ 2 be no best values for any function for which we can write ( -jr= J -directly in terms of ^Jm^ and # 2 /;;/ a . For from the expression ( -57) = \P L ) + w~) which is the general one for such W/ v mj v mj ' A functions, it is evident that -j^- is determined by the given d d values of and independently of the ratio or values of m j m^ m^ and m^ . But with these data the case A* = d* + <5" a 3 is soluble, since from the data we have 6 l = const. X m^ and ^ 2 const. X w a , which will give us an expression for A in terms of m l and m 9 . The procedure then briefly stated would be: Write the expression for f (m l , m^ , . . . , m n ) showing properly all the measured components. From this, write out, if possible, the (A V expression for t-jjr] . Otherwise write the expression for A* and remove the factor /" 2 () if it occurs. Remove all constant factors. Find by the calculus the ratio mjm l which will make the resulting expression a minimum. The cases for which there is no minimum are where /() = am l -\-bm^, given #, and # 3 ; d # 2 -and where /() = a-mf-mf, given and -. //2j 7?2g To find the minimum, as m^ and m^ are independent, it is necessary only to differentiate successively with respect to m v or, substituting the general values of d l , etc., in which the last term is not needed to find the ratios merely,, but maybe useful in some cases in finding the best magnitudes. Instead of this general formula we may employ the special one for equal effects which corresponds to the given function. To make the solution then we have merely to substitute the values of 6 l , d 2 , etc., in the suitable equal effects equations and solve successively for mjm^ , m 9 /m 1 , mjm l , etc. If the expression d* = A? + . . . + ^n has a factor /( ), each term must have that factor. And as in the solution for best ratios each term is equated to another, these common factors cancel. Therefore all cases are soluble directly from the expression for A and it is not necessary to remove the f actor f( ). A similar inspection shows that the same solution will be arrived at whether we start from the expression for J 2 or for I j in a case where the latter is applicable. Best Magnitudes. These are determined from the best ratios just as by the exact method. Example XXX, page 1 18. IIO BEST MAGNITUDES OF COMPONENTS. EXAMPLES. Example XXV. Best Magnitudes. One Component. Given a tangent galvanometer read by an index moving over a circle graduated in equal parts. Let be any reading, and K the galvanometer factor. Then the current is C = ^-tan 0. . . , f . , , [89] The deviation measure, #0, of a single reading will be the same at all parts of the scale. What would be the best deflec- tion to use? This problem might arise from either of the two following questions. For a given current how should K be proportioned in order that the value of AC for the given value of #0 should be a minimum ? Or if C were variable at will and K were given, what value of would give C with the greatest fractional precision as far as #0 affects it? The two problems have the same solution ; for, in the first, when A C is a minimum A C/C will also be so, as Cis a constant. K may be omitted, although it is a component which must be measured, for no limitation is assigned to it in the statement of the problem, and we are therefore to assume it as determi- nable with any desired precision or with equal precision what- soever its value. Dividing by f( ) to test whether it is a factor, we have AC ^T-sec 2 12 00 - . o . C ^-tan0 sin 0-cos sin 20 EXAMPLES. 1 1 1 which shows that the factor /( ) exists and leaves us, as the expression to be made a minimum, omitting the constant factor 2, i sin 20* It is easy to see by inspection that this is a minimum for = 45 ; but proceeding with the general method, we have d I d , ^. cos 20 -TT- -. - - = -rr(cosec 20) = 2-.-- d sin 20 d(f) sin 2 20 o. .*. cos 20 = o, and = 45, as the best value of 0, answering either requirement. Example XXVI. Best Magnitude. Two Components. The rate h of production of heat in a conductor is to be de- termined by measuring the current c and the voltage v. The instruments available can measure the current with a precision of dc = 0.05 ampere, and the voltage to dv = 0.05 volt. What are the best values of c and v for a rate of heating of 25 calories per second. For kgm.-, deg. C., true volts and ohms we have, at a temperature of 15 C, h = 0.2387 vc. [90] The conditions are dv = 0.05, dc = 0.05, h 25. There are only two variable components, m l = v, m^ c\ and both are to be discussed. As the resistance of the conduc- tor is not specified, they are independent, Expression [90] for f( ) contains all the measured quantities, and is such that we can / A \ a write the expression for f-^j , viz., (MI = (ir) ~ Then _ _ _- dv\M) ~~ 2 7/ ' dc\M 112 BEST MAGNITUDES OF COMPONENTS. Equating to zero simultaneously and solving we have 8*v d*c c* d*c /O.CKV 2 ?- = 2; ' - 3 - = -^:, = ( -) = i. .'. = I. is the best ratio. v To find the best numerical magnitudes we may substitute this ratio and the value of h in the original expression and solve. Thus h = 0.24 v X v 0.24 v* ; c = v = 10. amperes. This problem is purposely made numerically simple so that in- spection may readily show the results to be correct. Example XXVII. Best Magnitudes. Two out of Three- Components, A bar is to be constructed whose moment of inertia may be computed from its measured mass m and its linear dimensions. It is to be a right circular cylinder of height h and diameter d, and is to swing about a transverse central diameter. Both h and d are to be measured with the same micrometer screw, but owing to the uncertainty from rounded edges and other imperfections of the ends 6k = ^dd. The mass m is to be found by weighing, and can be determined so closely^ that ftm/in is negligible. What is the best ratio of d/h ? Here there are three measured components, but one of them, /#, is omitted from the discussion because its 6m/m is negligible. This, moreover, is a function of the other two components, so that unless it or one of the others could be omitted, the problem could not be solved. The expression for the moment of inertia is --(S+S) EXAMPLES. 113 / A \* The function /( ) is such that we cannot write out 1 -^ ) for it by any of the special formulae. Applying the general formula, we have df() _ mh df() _ md ~dh"' ''~6' ~~dd'~ ~8~* This is not divisible by the expression f( ), hence we cannot take out/( ) as a factor. Omitting then the common constant factor w 2 /4, we have as the expression to be made a minimum - 9 16 which will be denoted by [ ]. Then _ ^ h ~~ "9 ' tfd ~~ "9" The best ratio then is d/h = 28, that is, the cylinder should be a very short one, a disc rather than a long cylinder. This result is rather striking, as the first thought might be that 6k being greater than 6d, h should be made greater than d, so that the fractional accuracy of h should be increased. Further consideration and inspection of the formula for /will, however, show that the result arrived at is rational under the conditions of dh /\.dd. Moreover, if the value of A* be computed for this ratio, and then for another ratio quite different, but which makes the value of /the same, the value of A will be found to be greater in the second case. A test of that sort will also 114 BEST MAGNITUDES OF COMPONENTS. show another thing which has elsewhere been noted, namely, that the value of A will change but little for considerable changes in the ratio. This is, however, dependent on the form To find the best numerical magnitudes for h and d, we may have the necessary magnitude conditions given in several ways ; two examples will be taken. First. The most usual form of the problem would be this. To find the best values of h and d for a bar of stated moment of inertia and material ; for example, moment of inertia to be 1500 c. g. s., and the bar to be of brass. Since m = ^nd*kp, where p is the density, we must know p in order to find what m would be. Suppose for the brass p = 8.5. The magnitude conditions would then be I 1500 and p = 8.5 ; and there would be the equation m = %7td*hp conditioning m, h, and d. To find the best value of d we may substitute the magni- tude conditions and the value h d/2% in the full expression for /, viz., ' ...1500 - X 3-1 X d' X X 8.5 .*. d" = 100000. d = 10. cm. ; h = 10/28 == 0.36 cm. Second. It might be required to construct the bar of brass and with a stated mass, e.g., m = 63. gms. The mag- nitude conditions would then be m = 63. gms., p = 8.5, and there would be the same condition equation m = ^xd*kp. Then d and h must fulfil these conditions and have the ratio d/h = 28, simultaneously. We must therefore eliminate d and h successively between 63 = -nd*h X 8.5, and ^ = 28. /. 63 = J X 3.1 X 28V/' X // X 8.5. h^ 0.0122, // = 0.35 cm., d = 2S/i 9 8 cm. EXAMPLES. 115 These results of course agree with those of the first case within the limits of error of two-place computations, the data being equivalent. Example XXVIII. Best Magnitudes. Several Components. The modulus of elasticity E of an unplaned wooden beam 10 ft. long is to be determined by measuring the weight W .at its centre necessary to produce a transverse central deflec- tion v when supported at the ends, and by measuring the mean breadth b, depth h, and the length /. The value of E is known in advance to be about 1.3 X io 6 Ibs. per sq. in., and exami- nation of the beam and measuring apparatus shows that the precision attainable will be dl = 0.5 in., 6b = dh = 0.05 in., dW 3v = 0.002 in., -- = o.ooi. Desired the best magnitudes of the components. The expression for E is The components are connected by no equation of condition among themselves, so that the best ratios might be found for all of them if the precision conditions permitted, were it not for the fact that, the length / is specified. But we can write I ^ \ a the expression for HjjrJ directly by [57], viz., _ M '- : ~ r + 9 ~"~ +9 '" ~~- 8W Now as -j^r is a constant by the conditions, its effect on M will be the same whatever the value of W', hence there will be no best magnitude for W, and it may be omitted from the dis- cussion. This fact enables us to introduce the condition 1 1 6 BEST MAGNITUDES OF COMPONENTS. /= 120 in. Proceeding then with the others to find the best ratios, we have ar "T* ^r = - 2 -F> _ 7l~ - * O "TITS ~~7 dh h dv Hence tf 1 / cJ 2 ^ ^ 3 I &b _ = ^ 2 _-, ? = _._ h dV dV z' 3 i tf 2 */ ^ - I8 ^ = - 2 -3- --' -0.0000018. .-. = 0.012. To find the best magnitudes, /= 120. in. .. b = o.iol = 12. in. ; ^ = 0.22 / = 26. in. ; z; = 0.012 I 1.4 in. Note that as the components are all independent, and as W may have any value as far as the conditions show, we should be obliged to assign a value to some component, if not to /. To see whether these best magnitudes are practicable we should compute W^to see whether it came within the limits of the testing-machine. Transposing and substituting gives 4t*h*vE 4 X 12 X 26 3 X 1.4 X 1.3 X io 6 _ T- ~75c7~ But i looooo Ibs. would be beyond the limits of most ma- chines, so that we should be obliged to use some smaller values EXAMPLES. 117 for b, h, and v, and should wish to maintain the same ratios. Let us then limit the load to W = 100 ooo. Ibs. We must therefore have Wr io 5 X I20 3 ^3 Xio<; 4 X 1.3 X also b o.io ^ 0.012 ^=^ = -45' A = Substituting the latter gives 0.45 X o.o55>fc 6 = 3.3 x io 4 . .-. 0.025/6 5 = 3.3 X io 4 . .-. h = 17. in. ; b 0.45 X 17. = 7.7 in.; z; = 0.055 X 17. = 0.94 in. This result may be checked by substituting the values in the expression for E, which should yield 1.3 X io 6 within one or two units in the second place. To see how much difference there would be in the precision of E under the two cases we may compute A/M for each. For the first values, = o.oo ooo i + o-oo 014 + o.oo ooi 6 + o.oo 003 6 -f- o.oo ooo 3 O 00 020. A w = 0.014. For the second values, i7. / 1.4 = o.oo ooo i + o.oo 014 + 0.00004 2 + o.oo 008 i + 0.00 ooo 3 o.oo 026. Il8 BEST MAGNITUDES OF COMPONENTS. The close agreement of these two values shows that the second set of magnitudes is sensibly as good as the first. Example XXIX. Best Magnitudes. Conditioned Compo- nents. The specific resistance per metre-gramme of a sample of copper wire is to be determined by measuring the resistance, R, and mass, m, of a measured length / of the wire. Given $R = o.ooi ohm, dm o.ooi grm., dl 0.3 mm. In this form the problem is insoluble. For as the material of the wire is stated, the mass is a function of the length, diameter, and specific gravity, and the resistance is also a func- tion of the length, diameter, and resistance per unit volume, and as both specific gravity and resistance per unit volume are constants, R, m, and / are conditioned quantities. This may be expressed in another way by saying that as the material is fixed, then for any given value of m/l, that is, of mass per unit length, there is a fixed value of R/t, that is, of resistance per unit length. We are therefore not at liberty to assign ratios R : m : I on the basis of the precision conditions. It could be solved for R : I under the condition dm negligible. The above form of the precision conditions is not, however, the one which would ordinarily arise in practice. We should usually have given dR/R, and <57//, that is, R and / would each be measurable with a constant fractional precision. And dm/m would be usually negligible, measurements by the balance available being generally far more precise than the measure- ments of either R or /. In this case there would also be no best ratios, since J/J/is a constant, being fixed by the precision conditions independently of the values of the components. The best values would be determined by other limitations of the apparatus. Example XXX. Best Magnitudes. Equal Effects. In il- lustration of this method the preceding example on the deter- mination of E for a wooden beam will be taken. Applying the method we have _ __ __ ~W ~ 3 T ~ ~~ ~b ~ ~~ 3 T - IT* EXAMPLES. 1 19 6W Then for -^ constant, we have for the best ratios dl Sb b i Sb T = -'T' 7 = ~"3'tt dl dh h dh T = - 3 T' 7 = -*7 = 81 dv v I dv The negative signs indicate merely that an increase in that component causes a decrease, or a negative error, in the result. For best magnitudes with /= 120 in. we then have b = 0.033 I 4-O in. ; h o.io /= 12. in.; v = 0.0013 1= 0.16 in. These dimensions would require a load which is small for the capacity of the machine. One objection to the dimensions would be that they are too small to corre- spond to commercial sizes. They would be modified as in the former example. SOLUTIONS OF ILLUSTRATIVE PROBLEMS. Example XXXI. Problem. A voltmeter is to be calibrated by the Poggendorff method, using a Carhart-Clark cell. The calibration at no volts is desired to 0.2 volt, of which error one half is allowable in the cell measurement, the other o. I volt being assigned to the voltmeter. The arrangement used is to connect the voltmeter in series with a battery of over no volts, a controlling water rheostat, and an accurate adjustable resistance r, the Clark cell being connected around this resist- ance. The water rheostat is adjusted until the voltmeter reads no volts, and the resistance r until on closing a key in the cell circuit no deflection occurs on a sensitive galvanome- ter in that circuit, an approximate adjustment being effected at first by a preliminary computation, in order to avoid injury to the cell. Solution. Let R denote the voltmeter resistance; r, the adjustable resistance ; t, the observed temperature of the cell E 1.438 legal volts (1884), the voltage of the cell at 15 C. ; a = o.oo 038, the temperature coefficient of the cell ; then the voltage at the terminals of the voltmeter will be By computation, if V no, R 17 ooo, and / = 15, we find r = 220 ohms approximately, which would be about the amount which we should require to obtain the balance. 120 EXAMPLE, XXXI. 121 To find the accuracy requisite in the 5 components E, a, t, R, and r, we apply the general formula [38]. We will assume t = 20 as a typical value. dV^_R_ _ I7_ooo._ dE~~~-^ "220" 8 ' rrr r> -j-= &(* I 5) = 1.4 X 80 X (20 15) = 560. ; 7 7-7- E> = Ea = 1.4 X 80 X o.oo 038 = 0.042 ; dV E i *-T =M/220==7srJ dV R 80 J/= o.io volts. J F/ 1/;z = o.io/ ^5 = 0.045 volts. = 0.045/80. = 0.00056^. Attainable. There is probably a constant error of more than half of this amount in the absolute value of the ohm. da = O.O45/( 560) = 0.00008 o. Easily made negligible. 6t = 0.045/C- 0.042) = - i.i. = 0.045 X 1 60 = 7.2 ohms = 0.04 %. } As only the = 0.045 X ( 2) = 0.090 ohms = 0.04 %. ) ratio of R : r is required, the necessary precision may be reached; but with German-silver coils a variation of i in temperature would correspond to this amount. Thus the precision measure attainable would be about -y = o.ooo 68 = 0.068 per cent, which is slightly better than the o.i per cent called for, which may therefore presumably be attained. 122 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. XXXII. Efficiency of Electric Generator or Motor by Stray-power Method. Explanation of Method. If in a generator we let L denote the total power losses in the, machine, whether electrical or mechanical, and if we let the mechanical power applied to the machine, i.e., the input, be denoted by 7, and the electrical power available in the circuit outside of the machine, in other words the electrical output, by O, then the " commercial effi- ciency" of the generator is and the total power losses are L=f-0, all power being, of course, expressed in the same unit. Whence 7=0+ A ....... [07] and From this last expression it is obvious that if we have any means of measuring the losses L we may ascertain the effi- ciency from measurements of L and O alone without measur- ing /. Several methods exist by which this can be done, some of which measure L by electrical, others by mechanical methods and still others by a combination of the two. The above formula for E for generators must, of course, be slightly modified to be applicable to motors. Let i be the electrical input or electrical power supplied to the motor, and o its mechanical output. Then its commercial efficiency is [99] EXAMPLE XXXII. 12$ Then, as before, if / denotes the total losses in the machine,. we have / = i o ; .'. o = i /, . . . . \_1OO] and whence by measuring i and / we may calculate e. As for the generator so for the motor there are several methods by which / may be measured electrically or mechanically. Of the electrical methods for generators and motors the simplest is the *' Stray-power" method. This is applicable to a wide variety of machines, and forms perhaps the best avail- able method for general technical testing. In common with all methods which measure the losses, it possesses an important advantage over those methods which measure both o and i. For inspection of the expression for e will show that an error of i per cent in the losses corresponds only to about o.i per cent in e since / is only about one tenth of t, and a similar statement is true for a generator. The discussion will also show that the number of components to be measured becomes so small that this fact in combination with the above enables the method to give a considerably greater precision in the effi- ciency than is required for any of the components. Briefly explained the method is as follows: The losses of power in a dynamo (either motor or gener- ator) may be divided into, 1st, the loss, A, due to the armature resistance ; 2d, that, F, due to the field resistance ; 3d, the sum total of the losses due to all other sources, viz., Foucault cur- rents, hysteresis, bearing friction, air resistance, etc. The third group has been termed the " stray power," and will be denoted by SP. For a stated condition of running, that is, at a specified voltage, current, and speed, each of these losses has a fixed value provided that the condition of steady temperature of the machine has been reached. 124 SOLUTION'S OF ILLUSTRATIVE PROBLEMS. The first two losses, A and F t can be calculated if the fol- lowing quantities are known, viz., c a = current in armature, c f " " field coils ; r a = resistance of armature ; r f = " " field coils, or Vf voltage between field terminals. For A = c a *r a , F=c/r f , or F = where either c f or v f may be used as is most convenient in any given case. In what follows let us for simplicity suppose that we have a simple shunt-wound motor. The stray power cannot be directly measured, but must be indirectly determined by difference. We have, of course, L = A + F+ SP, [102] Thus if L, A, and F are measured under any given condi- tion, SP for that condition can then be deduced. The stray-power method rests upon this latter fact. It also involves the facts that the SP does not change widely between full load and no load on most types of machine, that it is of sensibly the same amount for the same machine whether acting as a generator or motor, and that its change with slightly differ- ent speeds of the machine may be allowed for approximately. The procedure consists in running the motor with no load under as nearly as possible its rated voltage and speed, and measuring the actual voltage v, current c, and speed s. The armature resistance r at the field resistance yy, and the current Cf or the voltage v f between the field terminals, for the same conditions must also be measured or otherwise ascertained, except as shown below. Then the total loss of power under this condition is /= cv t STRAY-POWER METHOD. 12$ and the stray power is sp = / _ (a +/) = cv - c?r a - cfr f . . [103) Let SP denote the stray power of the machine under any specified load at rated speed 6" and voltage V. Then it is as- sumed, first, that SP = sp sensibly if 5 = s, and, second, that if s differs from S, then SP=-.SJ>. -I ..... [104] Both assumptions are fairly well supported by the com- parison of results of experimental tests of the same machine by different methods, but neither is exact, nor is either reliable for machines of inferior design. The assumptions may proba- bly be regarded as introducing an error of less than one half of one per cent into E under ordinary conditions of testing. Hence the efficiency of the given machine under the specified load may now be calculated. Let V be the rated voltage, C the current corresponding to the specified load, and 5 the rated speed ; then we have l=A +F +SP Here the capitals denote the quantities for the specified load, and the small letters for the measurement with no load, i.e., when the stray power is being measured. The quantities C, V, and 5 are not measured, they are merely specified amounts. The quantities R a and R f must be determined by measurement or otherwise for the specified condition, and likewise C f , except as shown below. There are therefore at most nine quantities, viz., R a , R f , s, c, v, c a , r a) c f , r f , to be measured. But the precision discussion will show that this number can, in prac- tice, be considerably reduced. 126 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. The foregoing formula applies to a motor. A generator would be treated in precisely the same way ; that is, it would be run as a motor under no load and c, v t etc., measured. Then to obtain an expression for its efficiency as a generator at any specified rate of output C and Fwe should have merely to substitute the values of c, v, C, F, etc., in the expression [98] for E. For the purposes of the precision discussion, however, we should not employ that expression, but should simplify it thus : approx. = i approx. \_106~] This expression could not properly be used to compute E, but is exact enough for the precision discussion. Thus we should have for the generator which is identical in form with that for the motor. In either case the input or output corresponding to CV may be anything we choose, e.g., half load, three-quarters load, full load, etc. This will be more fully perceived in the example. Problem. A shunt-wound motor is to be tested for com- mercial efficiency e by the Stray-power Method. It is rated at 220 volts, 40 amperes, and 1200 revolutions per minute, and is stated by the maker to have an armature resistance of 0.14 ohm and a field resistance of 130 ohms. The precision desired in the value of e for full rated load is Ae/e = 0.25 per cent. Required to find by the precision discussion : (a) Precision necessary in the measured components. (b) Whether any of the components can be wholly omitted. (c) How closely to their normal running temperature the field coils and the armature must be when measured. (d) Whether any of the results of (a) could be applied to <)ther motors ; and if so, under what conditions. STRAY-POWER METHOD. \2J Solution. For the solution we require approximate values of c, v, r a , /y, R a , and R f . The better way would be to make a preliminary run and measure these quantities roughly. But it is usually more convenient to make the discussion in advance of any trial. We may do so in this case as follows : Assume that r a = R a = 0.14 ohms, also that r f =R f = 130. ohms, the values stated by the maker. Both of these assumptions will be proved to be close enough by the results of the discussion. We are obliged further to assume a value of e in order to deduce a value for c. This we can usually also do closely enough for the preliminary discussion from inspection of the machine. Suppose that in the present case we estimate the efficiency to be about 88. per cent at full load. Then if v = 220., c must be 0.12 X 40. 4.8 amp., since 100 88 = 12 per cent of the power and therefore of the current applied under normal voltage. The expression above deduced for e must be slightly modi- fied to meet this case, for c a and c f are not here measured, but v v c f = , and c a c -- . r f r f The expression for e, then, containing all the components to be measured properly expressed for the precision discussion is The components to be measured in the test are thus seven, viz., R a , R fy s, c, v, r a , and r f , so that n = 7. We have there- fore to apply the general formula [37] to these. The following simplification may be made. As s may easily be made within a few per cent of 5 in the run we may regard S/s as unity in all differentiations except that with respect to s. The values of C and Fare not measured values, but simply stated to define the condition at which the efficiency is to be computed. Not being measured they are not subject to errors of measurement, 128 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. and are therefore not to be differentiated. Proceeding then with the differentiation and substituting numerical values gives dR a = ~ ~CV\ \ C ~~Tl)\ == ~^ dR t ~ CV r Rf R; */ r 3.1 x io de i SP 3> de i de __ j / ^\^ _2v\ I ^ z " CFr 2 ^' r)r f r f \ ~ 6.3 X io 3 ' ^ JL if -V i L ^r a ~ " g^lr; V f 9.1 X io" ^L _2_j J r ^\^JL \ _ T _ ^ ' " CV \ ' r/r; T r/ ) 3.0 x io 3 ' To find the numerical values of dR a , etc., by [37], we must have Ac. Now Ae/e O.CXD25 and ^ = 0.88; .'. ^=0.0025 X O.88 0.0022. Hence Ae/ Vn = O.OO22/ 1/7 = 0.0022/2.7^ = 0.00081 ; .'. (^) <5^ a = - 8. X io-4 x 6.3 = - 0.0050 ohms = - 3.6 % dR f = + " X 3-1 X 103 = 4- 2 -5 ohms = -f 1.9 % 8s =+ " X 1.5 X io 4 =-j- 12. r. p.m. = + i.o fo = " X 4.0 X io 1 = 0.032 amperes = 0.67 % 8v = " X 6.3 X io 3 = 5.0 volts = 2.3 % dr a =4- " X 9- 1 X io* = 4- 0.73 ohms = 4- 500. % 8r/ = " X 3-0 X io 3 = 2.4 ohms = 1.8 % (b) (c) Inspecting these values the following points may be noted, (i) The armature resistance, r a , in the run under no load is entirely negligible. A single measurement, viz., R a9 with the armature at the temperature of full load is all that is needed ; hence we might use n 6 instead of n = 7. (2) The STRAY-POWER METHOD. I2Q armature coils being of copper will change in resistance by 0.4 per cent per degree centigrade. The maximum allowable change is 3.6 per cent, which corresponds to 3.6/0.4 9. C. Some care is therefore essential that the armature is fully warmed up to its normal state when R a is measured. (3) The values of dR f and 6r f are nearly equal and of opposite sign, and obviously R f = r f very nearly. Hence if the same numer- ical value be used for both, any error in that value will be nearly eliminated, so nearly in fact that a large error will be admissible. We may see how large by substituting r f for R f> and then differentiating e with respect to r f . This gives de_ j_ ( ( _ V\VR^ _ V^ dr f ~ ~ CV\ 2 \ L r f > rf ~ rf + L~ 2 r~ V77+r/J } = ~ 6.8 X io 4 ' .-. 6'r f = 8 X io~ 4 X 6.8 X io 4 54. ohms 42. %. This clearly shows that one rough measurement of the field resistance is sufficient. The temperature may be 42./O.4 105 C. from normal, that is, the cold resistance would be near enough. We might even use the stated resistance with- out measurement. Hence also the error from this source can be easily rendered of negligible amount, and we may use n = 5 instead of n = 7 as above, so that Ae/ Vn would become 0.0022/2.2=0.0010 if the discussion were to be revised. The values admissible for dv, dc, etc., would then be corre- spondingly increased. The three sections of this paragraph answer questions (b) and (c) of the problem. (d) The results above obtained could be applied without material error to any motor not differing from this one by more than about io per cent in the rated values of F, C, R a , and 5, or by more than 40 per cent in R f . Note that if the efficiency for other than full load is to be found with the same or any given precision, a solution similar to the above must be made by substituting the corresponding values of C and R a for the desired load, e.g., 20 amp. (or more exactly 22 amp.) and about 0.14 ohms for half load. 130 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. Example XXXIII. Cradle Dynamometer. The princi- ple and operation of the Brackett cradle dynamometer are so well known that a brief description is sufficient. The cradle consists of a platform suspended at each end from a horizontal knife-edge, both being in the same line. The dynamo (gen- erator or motor) is placed securely upon the platform and adjusted until the axis of rotation of its shaft is coincident with the line of the knife-edges. The centre of gravity of the whole system is then raised or lowered by means of auxiliary weights until just below the axis. The system then oscillates slowly and sensitively like a balance. When the dynamo is in operation the mechanical power applied to or given out by its pulley tends, through the magnetic reaction between the arma- ture and fields, to rotate the machine and therefore the whole system. This tendency is counterbalanced by a weight hung upon a horizontal lever arm projecting from the dynamometer at right angles to the axis. The weight or its distance or both are adjusted until a balance is obtained when this weight w into its horizontal distance / from the axis gives the rotary moment of the system, and therefore that applied to or given out by the dynamo. From this and the speed of the dynamo the power can be computed. Problem. The commercial efficiency at full load of a certain generator is to be measured by a cradle dynamometer. The dynamo is rated at 75 volts, 60 amperes, and 1400 rev. per min., and it has probably an efficiency of about 90 per cent. The diameter of its pulley is 2R = 10 inches. The dynamo and dynamometer together weigh about 3000 Ibs. The length of the arm to carry the dynamometer weights is / = 3.4 ft. Required in advance of the test a precision discussion of the proposed measurement and of the dynamometer. The value of E is desired to one per cent. Solution. First. Precision necessary in each measured com- ponent. The expression for the efficiency in terms of the measured components is E = CV 33000 / 746 2 nln w CRADLE DYNAMOMETER. 131 The measured quantities are C, V, /, n, and w ; .*. n = 5. By [53] we have for equal effects <$C _8V _ 61 _ dn _ dw lC = ~~ V'~ ' ~T'~ ' ~n ~~ ~~^ i AE I = T n ^-^X 0.01 = 0.0045. Each of these components then must be measured to about 0.45 per cent. By the rules for significant figures the constant n must be carried to 5 places, i.e., 3.1416 must be used. This is, however, in excess of the strict requirement in this case, as will almost always be true when those rules are applied, since they are framed to cover the worst possible case. Applying the cri- terion for constants p. 70, we have, to be negligible, O.OOI5 J /. etc. /j = H. By the conditions of the problem then dh 8C _ = _ =0.001. The measurement of H with this accuracy is difficult and laborious ; moreover, the diurnal and local fluctuations are of about this order of magnitude, though the latter may be much larger. It is hardly practicable in any ordinary laboratory work to depend upon H as constant within about twice this limit, or 0.2 per cent, and this only under favorable conditions. By a good magnetometer it may be measured to less than O.2 per cent. In merely relative measurements of C the absolute value of H need not be known and the effect only of variations in H enters. Pi J^ /*** / = s. We must have -~- = o.ooi. As r = 20 cm., s o .s 27tr = 1 20. cm. /. ds = 0.001$ = 0.12 cm. The error in r involves not only errors in the measurement of s, but irregu- larity in the distribution of the convolutions of the coil. The measurement of s can doubtless be made closer than 0.12 cm., but the uncertainty with respect to irregular winding owing to varying tension, to varying thickness of insulation, etc., will probably not be much less than that amount. We may prob- ably count on this limit as about what is practically attainable in a good coil carefully wound into a channel. If the coil is not a true circle but is more or less elliptical, the expression above .given for G will not be exact. To find the amount of error for a given ellipticity it would be necessary to deduce an expression for the field at the centre of an ellip- tical circuit which cannot readily be done. It is easy to see, however, that the field does not change materially for slight TANGENT GALVANOMETER. 141 eccentricity, for if a circular circuit be gradually deformed inta an ellipse the flattened sides approach the centre at first at sensibly the same rate as that at which the bulging ends recede. / 3 = n. This is necessarily a whole number and not subject to error except through mistake in counting. A mistake of one turn in a thousand would correspond to the assigned limit of dC/C, but as the value of n will seldom be as large as 1000, no mistake is allowable. / 4 = tan 0. We will make the solution for == 45, but the result will apply without sensible error to any deflection, between 30 and 60 as shown later. d tan SC = -~ = o.ooi ; tan 45 = 1.0 ; tan C .'. d tan = o.ooi From this we have to determine the corresponding value of #0 by [34]- , , tan oq> = o tan dftan = sec = cos 2 0' .-. = o.ooi cos* = o.ooi x 0.50 = o.oo 050 in radian measure. The value of is a mean of four readings in which the tenths of a degree are estimated by the eye. These estima- tions if properly made will always give the nearest tenth. The extreme error of estimation will therefore be -f- O.O5 and O.O5, and the error will be equally likely to have any value between these limits. The average error of a single estima- tion will be (page 21) o.O25, and of the mean of four will be O.025/ 1/4 0.OI3, which is negligible compared with the value 142 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. o.O3 above deduced. But there are other sources of error in 0, viz., irregularities in graduation, eccentricity, parallax, tor- sion, coils out of vertical plane, coils out of magnetic meridian. Of these the first is partly eliminated by the reading at four different points on the circle, the second by reading both ends of the index, the third by bringing the eye when reading into a position where the index just covers its reflection in the mirror; the fourth, fifth, and sixth are corrected for by the cor- responding correction terms which are separately discussed below. The residual errors from these four corrections are classed separately and therefore do not need to be regarded as augmenting #0. The errors from the first three sources may obviously be of any amount according to construction of instrument, but need not exceed the limit of o.O3 which may be regarded as practically attainable. / 6 = f I -| ^ f). This is the correction term for finite dimensions of the rectangular coil section, and is suffi- ciently close where the depth does not exceed one tenth of the radius. It is in reality a correction to the value of G. 2b = breadth, 2d = depth. The term involves r and therefore the measured components, but as the () differs from unity by only one or two per cent at most it may be omitted in dis- cussing ds, as was done above, and r may now be treated as a constant. This factor contains then two measured quantities 2b and 2d. The value of djf \/f \ corresponding to equal effects for each must then be o.ooio V2 00014, corresponding to which we have to find the values of d(zb] and 8(2d). As/ 6 is sensibly = I we have tf/ o.ooi4/ 6 = 0.0014. By [45] and [34] 6(2b) = 26b = 2*f> 1^ = O.0028/ ^ 00028/ r = i.i cm. TANGENT GALVANOMETER. 143 It is obvious that this limit is needlessly large. A negligible amount would be one third of this, viz., 0.37 cm., and we can easily measure the depth much closer than this. If then the breadth measurement be made to 4 or better to I or 2 mm., its residual error will be negligible. Similarly for the depth measurement dd I 2d = 0.0028 / 5 = i.o cm. Therefore if the depth be measured to 6 mm., or better to I or 2 mm., the residual error will be entirely negligible. The correction itself would be negligible when the coil section was such that 2b < 0.4 cm. and 2d < 0.6 cm. Inspection of the form of the correction shows that if the coil section be so designed that the correction will vanish. Solving gives F 2 2b 5 If the coil be wound to these relative dimensions then the correction may be omitted. Obviously the dimensions must be adhered to within the limits 8(2b] = 4 mm. and d(2d) 6 mm. in this case, or within limits having the proportion of 6(26) : d(2d) 2:3 in any case. These limits must be con- sidered not merely as referring to the outside dimensions of the coils, but to the density of winding as well. If the wind- ing is not in " square order," but is more dense in the depth than in the breadth of the coil, the correction by the above formula with respect to the breadth will be too great relatively 144 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. to the depth about in the proportion of the relative densities,, and a corresponding allowance must be made. /, = II - r-j -- --J sin 2 0j. The expression for G gives the field at the centre of the coil due to a unit (c. g. s.) of current in the coil. The needle has, however, a finite length 2l, and its poles therefore lie in a field which when is nearly zero is slightly less intense than at the centre, and which increases with / 6 = -^ = o.ooio approx. /6 dl Inspection shows that the value of / is greatest for = 60 if the galvanometer is used (for reasons later stated) only between 30 and 60. We will therefore solve for the worst case. Substituting gives 6(21} = 0.0020 / - - - = 0.40 cm. 200 TANGENT GALVANOMETER. 145 Greater accuracy than this is easily attainable. 0.40/3 = 0.13 cm. would be negligible, and this can be reached. As the needle can never be made as short as 0.13 cm. on account of torsion, the correction itself at 60 can never be negligible. But the length can be measured accurately enough so that the residual error shall be negligible. It should be noted that 2/ the pole distance is about 0.85 of the total length of the needle if this is a thin rectangular prism. The .correction obviously vanishes when 3 I" 15 ^ , - r = ~ T sm 0> 4 r* 4 r* which solved for

- A) = 1+tan0 . tanJ - rq-j 154 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. tan (0 -j- A) tan (0 A) = Now 2 0, = 2 A. .- = 2 sin GO = 200 approx. = 2 0, , Q.E.D. /ii. For inclination of mirror to horizontal. In the gal- vanometer here considered the graduated circle is supposed to be secured to the surface of the mirror and the mirror to be placed horizontally below the needle with its index. Since the index always covers its image when a reading is taken, the plane of sight, i.e., the plane containing the eye and the index, is always perpendicular to the mirror. But the index and needle necessarily deflect about a vertical axis. If, therefore,, the mirror and circle be not horizontal but inclined, the angle read off upon it will not be the true angle swept through by the needle, but an oblique projection of that angle, It is, then, essential to know how nearly horizontal the mirror must be to avoid sensible error from this cause. In Fig. 4 let the circle I'A'J'A" show the graduated circle J FIG. 4. in horizontal projection and MN its vertical projection. Sup- pose the needle to be at A. Then if index and circle are both TANGENT GALVANOMETER. 15$ horizontal, the circle would be as shown in Fig. 4, and the plane swept through by the index would be shown by a hori- zontal line through A. But if the mirror were inclined, then the plane of the index would still be horizontal, but that of the mirror would be inclined to it. The result as far as angular readings are concerned, however, would be the same as though the mirror remained horizontal and the plane swept through by the index were inclined. It is more convenient to represent the latter in the drawing. Therefore let IAJ represent the vertical projection of the plane of the index inclined to the mirror at an angle AIL = h. First. Suppose the index when undeflected to stand in the direction A' A" . In vertical projection it will appear as a point at A. Let it be deflected through an angle whose true value is 0, but which is read on the circle as 0' = A' OB'. What is the relation between and 0'? At A' draw a tan- gent A'B', and prolong B"OB' to intersect this at B '. Then ,, A'ff tan0 - Project B f upward to B on IJ. Then AB for AB is the true length of the tangent at A cut off by the pointer and shown in horizontal projection in A'B', and OA is shown in its true length. Therefore tan AB AB i tan ~~ A'B' ~RD~ cos h '' tan = tan / ' and the correction factor for the inclination of the mirror is, in this case, I/cos h, which is the same as though the coils were inclined as in the cosine galvanometer. 156 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. Second. Suppose that the zero position of the index were /y, IJ. Let the needle be deflected through any angle shown in projection upon the mirror as 0' = I' OK'. Thus I'K' tan 0' == . As the tangent line of which I'K' is the projection is parallel to the mirror it appears in this projection at its full length and in the vertical projection as a point at /. Then I'K' tan = -F, since AT is the true length of the side of the angle which appears as OF' in the mirror projection. Therefore tan I'O IL Hence the correction factor for the inclination of the mirror is in this case cos h. The first of these cases is where the mirror is tipped about .a horizontal line through the zero points ; the second, where the tipping is about a line through the 90 points. For a tip- ping about any other line the effect could be ascertained by resolving it into two parts, one with reference to each of the above positions. The effect will be intermediate between the two above extremes, so that it need not be further considered. What value of h will produce the limiting negligible error? As h is very small we may write cos h = I x, where x is a .small fraction. Then for the first case i i - 7 = -- i 4- x approx. cos h i x For the second case we have simply cos h = i x. TANGENT GALVANOMETER. 157 As these enter as direct factors we must have for negligibility :r^ o.oo 03 3. .. cos h = i x = 0.99 967 ; This error can be rendered negligible without difficulty by due care in the original construction of the instrument and by proper levelling at the time of use. The most convenient method for the latter is to have a plumb-line hanging from a, marked point, and arranged to be brought over a reference point on a plate attached to the lower part of the coil, pains being taken to see, once for all, that the mirror is level and the coil vertical when the line so indicates. It should be noted that neither reversal nor reading both ends of the needle tends to eliminate this error. Summary. The following table gives a summary of the re- sults. By bringing the adjustments or by measuring the quan- No. HF- Are Required: I O.OO2 8 If /H- 0.002 Earth's field. 2 O.OOI 8s /s = 0.001 Circumference of coil. 3 0.000 8n = zero Turns in coil. 4 O.OOI 8

<6 dc 264, C sin 2< C sin 2< o or 90 a 25 or 65 0.00131 5 or 85 0.00589 30 or 60 .00116 10 or 80 .00292 35 or 55 .00107 15 or 75 .00200 40 or 50 .00102 20 or 70 .00156 45 or 45 .00100 Combining all the considerations, then, we see that for the very best work 45 is preferable ; that any deflection be- tween 30 and 45 is, however, nearly as good as 45, and that any deflection between 30 and 60 is but slightly inferior to these. For most work then it is indifferent as far as dC/C is concerned what deflection we use between the limits of 30 and 60. For work somewhat inferior in accuracy we may use in- differently any angle between 20 and 70, but should rarely go outside those limits. Example XXXV. Electro- Static Capacity. Thomson's or Coifs Method. Description of method in Physical Labora- tory Notes or in Kempe's Handbook of Electrical Testing. The formula for the method is '=* [125] The battery power used is assumed to be sufficient to enable a change equal to the smallest coil in the resistance-box to be perceived, hence <5R = SR X . The charges in the condensers are greatest when R -)- R x is as large as possible, namely, the total resistance r in the box. By formula [52] F. - \R This will be a minimum when the last parenthesis is so. As R l6o SOLUTIONS OF ILLUSTRATIVE PROBLEMS. and R x are independent we must differentiate with respect to each successively and equate the coefficients ( _i_ * \ J: ^ ( \ 2 Hence The best ratio then is R/R* i, and therefore F/F X = I or F=F X ; that is, it is best to use a known condenser of as nearly as possible the value of the unknown. Example XXXVI. Magnetometer. In measuring M -f- H by the magnetometer, the deflecting magnet is placed succes- sively at two distances, r l and r a from the needle, producing- deflections 0j and 3 . And M r: tan 0, - rf tan 0, = - ~ Desired, the best ratio of r 1 : r^ ; i.e., that which will make M A -. a minimum. The measurements are such that dr l and H dr^ may be considered negligible, and $ tan 0^^ tan 2 ; thus these are the precision conditions. There are no magnitude conditions which bear on this problem. Then and 2 ) H ( M \ ~ r * \H / ~ r* - T' M To make A - a minimum, the fraction in the second H member must be a minimum ; and we wish, therefore, to find BATTERY RESISTANCE AND E. M. F. l6l the value of r l : r 2 , which will make it so. Writing, then, r, : r 9 = n, or r l nr^ , and substituting gives r: + r^ nr? + r^ 6 n" + i ~~ in which we have to find the value of n to produce a mini- mum. Where the two variables are independent, as r, and r 2 in this case, the following proposition may be often of service. Let u = f(x, y) where x and y are independent variables and /is such a function that it may be separated so that u = Then the value of x : y, which makes a { ] a minimum, is \yl the same as will make u a minimum. For x cannot be ex- pressed as a function of , and, therefore, p (x) does not enter into the determination of x \ y for the minimum. The same is, of course, true if the separation be made into [1*8] Then in the problem in hand we have to find the value of n w + I n, which will make -r- ^ rr a minimum. d n 10 I Example XXXVII. Battery Resistance and E. M. F. In the ordinary method of measuring the resistance B of a battery, the currents c^ and c^ produced by the battery through two known external resistances r l and r 2 are observed. Let p^ and p 2 represent the total resistances, including battery, leads, 1 62 SOLUTIONS OF ILLUSTRATIVE PROBLEMS. galvanometer, and rheostat ; and let E denote the battery E. M. F. Then the formula for the method is [129} Desired the best ratio of c l : c y Following the procedure of page 105, the expression for AB must first be obtained. _ a (p, - p,} (c, - c$ - But as c is a function of p and the constant E the solution may be made either by writing E/c for p or E/p for c. The latter will be done. Then f=_^^. Similar,^ d 4=-J^ Then This expression will serve for a galvanometer for which dc is a constant, e.g., a reflecting galvanometer, or one whose scale is uniform and proportional to the current, such as a Weston ammeter. For a tangent galvanometer or any instrument for which dc/c is a constant, the expression must be modified to change d*c to 8*c/c*. This may be done by substituting in the second member E/c^ for p l and E/c t for p a . This gives BATTERY RESISTANCE AND E. M. F. 163 1st. For the reflecting or the Weston galvanometer to find the best ratio x = cjc^ substitute in [130] the equivalent of x = viz., x = Pa/Pi. This gives ^ , (*- Hence for a minimum r i *w for which by approximate solution x = 2.2. The best ratio is then Pa/Pi = 2.2 or cjc^ = 2.2 or c l = 2.2 c^. To find the best external resistances corresponding to this best ratio we may substitute x 2.2 in [132], giving =2ov1 Pi AB = \/2^^ = ' P 8 - Pi c From the latter it will be seen that AB diminishes as p 2 in- creases and as p 1 diminishes. Hence /? 2 should be as large and />, as small as conditions of range of galvanometer and of polarization will permit. We should therefore use by pref- erence deflections of 60 and 30 or of 70 and 20 on the tangent galvanometer. The necessary precision and sensitiveness of galvanometer would be determined as follows. Using 60 and 30, c l : c^ = 3 : I approx. Then C < 2 p l Therefore we must use a galvanometer of a precision at least equal to this value of dc/c, and of such a "factor" that with the smallest value of pJB admissible on account of polariza- tion, the deflection will be about 60. The value of p 2 must r of course, be such as to make the second deflection about 30. If instead of taking two deflections on one galvanometer we take the second deflection or a much more sensitive galva- nometer than the first, but equally precise, that is one which BA TTER Y RESISTANCE AND E. M. F. l6$ will measure a very much smaller current but with equal precision dc/c, we may then make p, many times as great as p l and thus improve the conditions of working. This is really what we do in using a potential galvanometer and a current galvanometer in combination as in the method described in the Physical Laboratory Notes. E. M. F. By similar demonstrations we may show that for measuring the electromotive force of the battery the fol- lowing points. 1st. For a galvanometer for which dc is a constant the best ratio of cjc^ is sensibly the same as for measuring B ; also that, even using this ratio, dE increases with /o, , so that p t should be made as small as is consistent with polarization, just as in measuring B. 2d. For a galvanometer where dc/c is constant we must make p 2 / 'p l as large as possible, making p l large enough to avoid polarization. Of course it is clear that a potential galvanometer is preferable to the two-deflection method for measuring E. SINES, COSINES, TANGENTS. NATURAL. LOGARITHMIC. Sine. Cos. Tan. Sine. Cos. Tan. 0.0 0.0000 I.OOOO 0.0000 CO o.oooo CO 0.5 0.0087 I.OOOO 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 1O. 0.1736 0.9848 0.1763 9.2397 9-9934 9.2463 20. 0.3420 0.9397 0.3640 9-5341 9.9730 9.5611 30. 0.5000 0.8660 0.5774 9.6990 9-9375 9.7614 40. 0.6428 0.7660 0.8391 9.8081 9.8843 9.9238 45. 0.7071 0.7071 I.OOOO 9.8495 9.8495 o.oooo 50. o. 7660 0.6428 1.1918 9.8843 9.8081 0.0762 60. 0.8660 0.5000 1.7321 9-9375 9.6990 0.2386 70. 0.9397 0.3420 2.7475 9.9730 9-5341 0.4389 80. 0.9848 0.1736 5-6713 9-9934 9.2397 0.7537 90. I.OOOO o.oooo 00 0.0000 CO CO CONSTANTS. I metre in inches (U. S. C. S., 1892) I inch in millimetres i kilogramme in pounds avoirdupois (U. S. legal) I pound avoirdupois in kilogrammes (U. S. legal). e = base of Naperian logarithms i/e = modulus of common logarithms Radius is equal in length to an arc of Arc of i in terms of radius Watts per horse-power (see p. 132) Small calories per second per watt (see p. in). . . . = 39-3700 = 25.40 05 = 2. 2O 460 = 0.45 359 7 = 2.71 828 18 = 0.43 429 45 = 57- 29 578 = o.oi 745 329 = 746. = 0.23 87 166 SQUARES, CUBES, RECIPROCALS. JVb. Square. Cube. Recip. No. Square. Cube. Recip. 1.0 I.OO I.OO I.OO 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. I54 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.O 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 n.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 53L .123 3.7 13.7 50.7 .270 8.2 67.2 55L .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- ."5 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. .III 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 in. .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. .IO4 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. .101 167 LOGARITHMS. 1 2 3 4 5 6 7 8 9 1.0 oooo 0043 0086 0128 0170 0212 0253 0294 0334 0374 1.1 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 1.2 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 1.3 1139 H73 1206 1239 1271 1303 1335 1367 T 399 1430 1.4 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 1.5 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 1.6 2041 2068 2095 2122 2148 2175 22OI 2227 2253 2279 1.7 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 1.8 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 1.9 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 2.0 3010 3032 3054 3075 3096 3H8 3139 3160 3181 3201 2.1 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 2.2 3424 3444 3464 3483 3502 3522 354i 356o 3579 3598 2.3 3617 3636 3655 3674 3692 37" 3729 3747 3766 3/84 2.4 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 2.5 3979 3997 4014 4031 4048 4065 4082 4099 4116 4i33 2.6 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 2.7 4314 4330 4346 4362 43/8 4393 4409 4425 4440 4456 2.8 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 2.9 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 3.0 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 3.1 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 3.2 5052 5065 5079 5092 5105 5JI9 5132 5145 5J59 5172 3.3 5i85 5198 5211 5224 5237 5250 5263 5276 5289 5302 3.4 5315 5328 5340 5353 5366 5378 539 1 5403 5416 5428 3.5 5441 5453 5465 5478 549 5502 5515 5527 5539 5551 3.6 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 3.7 5682 5694, 5705 5/17 5729 5740 5752 5763 5775 5786 3.8 5798 58o? 5821 5832 5843 5855 5866 5877 5888 5899 3.9 59" 5922 5933 5944 5955 5966 5977 5988 5999 6010 4.0 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 4.1 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 4.2 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 4.3 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 4.4 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 4.5 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 4.6 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 4.7 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 4.8 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 4.9 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 5.0 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 6.1 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 5.2 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 5.3 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 5.4 7324 7332 7340 7348 7356 7364 7372 738o 7388 7396 168 LOGARITHMS. 1 2 3 4 5 6 7 8 9 5.5 7404 7412 7419 7427 7435 7443 745i 7459 7466 7474 5.6 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 5.7 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 5.8 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 5.9 7709 7716 7723 773i 7738 7745 7752 7760 7767 7774 6.0 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 6.1 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 6.2 7924 793i 7938 7945 7952 7959 7966 7973 7980 7987 6.3 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 6.4 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 6.5 8129 .8136 8142 8149 8156 8162 8169 8176 8182 8189 6.6 8i95 8202 8209 8215 8222 8228 8235 8241 8248 S254 6.7 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 6.8 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 6.9 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 7.0 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 7.1 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 7.2 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 7.3 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 7.4 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 7.5 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 7.6 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 7.7 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 7.8 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 7.9 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 8.0 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 8.1 9085 9090 9096 9101 9106 9112 9117 9122 9128 9 r 33 8.2 9138 9M3 9149 9 T 54 9J59 9165 9170 9!75 9180 9186 8.3 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 8.4 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 8.5 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 8.6 9345 9350 9355 9360 9365 9370 9375 938o 9385 9390 8.7 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 8.8 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 8.9 9494 9499 9504 9509 9513 95i8 9523 9528 9533 9538 9.0 9542 9547 9552 9557 9562 9566 9571 9576 958i 9586 9.1 959 9595 9600 9605 9609 9614 9619 9624 9628 9633 9.2 9638 9643 9647 9652 9 6 57 9661 9666 9671 9675 9680 9.3 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 9.4 9731 9736 9741 9745 9750 9754 9759 9764 9768 9773 9.5 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 9.6 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 9.7 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 9.8 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 9.9 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 169 INDEX. PACK Accuracy 13. 25, 45 of method. See " Error of method " 46 of result 13, 35, 45, 84 Estimation of 13, 32, 35, 45, 86 Importance of estimate of I, 36 Forms of Problems on 33 A. D. See average deviation Application of general formulae to precision discussions 54 Average 16 Average deviation 16 Advantage of, over other deviation measures 24 Examples of. Example II 18 of single observation 16 of mean result 18 Significance of 19 Balance, Weighing by an equal arm. Example III 37 Battery resistance and E. M. F. Example XXXVII 161 Beam, Modulus of elasticity of. Examples XXVIII, XXX 115, 118 Best distribution of labor I, 33, 36, 70 Best magnitudes of components ... 47, 100 Best ratio of components. See " Best magnitudes " , 47, 100 Best representative value 14, 16 Best value of n for a series of observations 20 Best value of precision measure of components 47 Best value of residuals 27 Calibration of voltmeter. Example XXXI c 120 Calorimeter. Examples XVII, XXII 88, 94 Capacity, Electro-static. Thomson's and Gott's methods. Example XXXV. 159 Check methods and results 8, 47 Clark cell, Calibration of voltmeter by. Example XXXI 120 Collective effects 50 Combined effects 50 171 172 INDEX. PAGE Components, Best ratio of. See " Best magnitudes" 47, 100 Criteria for negligibility of, or of d in 67 Precision measure of, as related to that of result 47 To find best magnitudes of, Single component 102 Two variable components 104 Several components 107, 108 To find best value of precision measure of 47 Constant error 7 Constants, Rejection of places of figures in 70 Table of 166 746 watts = i horse-power 132 Corrections 8-12 Cosine galvanometer. Example XXI 91 Cosines, Table of 116 Cradle dynamometer, Example XXVI 96 Efficiency of dynamo by. Example XXXIII 130 Criteria for negligibility of components , 67 d in components 67 residuals 26 rejection of doubtful observations 30 Cubes, Table of 167 Data required to substantiate results 36, 85 Determinate errors 10 Deviations 14 Frequency of 20 General law of 15 Special law of 24 Deviation measure 14, 16, 23 Fractional 29 Negligible amounts in . . 40 of mean result 18 of single observations 17 Significance of 19 Significant figures in 17 Direct measurements 4 Planning of 36 Discordance of observations 6 Doubtful observations 30 Criterion for rejection of 30 INDEX. 17$ Dynamo, Efficiency of ............................................... 122 by cradle dynamometer. Example XXXIII ....... 130 by stray-power method. Example XXXII ........ 122 Efficiency of dynamo .................................. ............... 122 by cradle dynamometer. Example XXXIII ........ 130 by stray-power method. Example XXXII ......... 122. E. M. F. and resistance of battery. Example XXXVII ................ 161 Electro-static capacity. Thomson's and Gott's methods. Example XXXV. 159, Elimination of constant error .......................................... 7 Equal effects, Application to best magnitudes of components ............. 108 Demonstration .......................................... 70 General formulae ........................................ 53 Special formulae, following general formulae. See also/"() in this index. Error of method ........................ i ............................. 46 of result .................................................. 13, 35, 45 of single observation ............................................ 6 Errors, Constant ................... , ................................. 7 Constant part of ............................................... 7 Determinate .................................... . ........... 10 Indeterminate ............... . ................................. 10 Variable part of ................... .......................... 6 Estimated precision measure of component ............................. 72 Estimation of accuracy or error of result .......................... 13, 32, 45 direct measurement ...................................... 13, indirect measurement .................................... 45 Examples. See table of contents. Factors, separation of functions into ....................... 58, 60, 61, 64, 76 Forms of problems or accuracy of result ............................. 33, 84 Formulae for general and special functions. See/"() in this index ........ 55 Frequency of deviations ............. . ................................. 20 Friction brake. Example XXIII ...... . .............................. 96 /(i ,ma, - , MH) ............................................ 55 Wj /w 2 . . . m n ......................................... 56 am, -\- bm-t + + km n ........................ . ................ 57 a-m^mi* . . . *m n ............................................. 58 5 = a>m v ....................................................... . . 59, 1/4 INDEX. . ................ 60 . ......................... 61 = (p(m lf . . . , mp) p(m q , . . , m s ) etc ......................... 63 =