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IRLF THE THEORY OF MEASUREMENTS McGraw-Hill BookCompaiiy Electrical World Engineering Record Railway Age Gazette Signal Engineer Electric Railway Journal Engineering News American Machinist AraericanEngjneer Coal Age Metallurgical and Chemical Engineering Power THE THEORY OF MEASUREMENTS BY A. DE FOREST PALMER, PH.D. Associate Professor of Physics in Brown University. McGRAW-HILL BOOK COMPANY 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, E.G. 1912 COPYRIGHT, 1912, BY THE McGRAW-HILL BOOK COMPANY Stanbopc jjbress H.GILSON COMPANY BOSTON, U.S.A. PREFACE. THE function of laboratory instruction in physics is twofold. Elementary courses are intended to develop the power of discrimi- nating observation and to put the student in personal contact with the phenomena and general principles discussed in textbooks and lecture demonstrations. The apparatus provided should be of the simplest possible nature, the experiments assigned should be for the most part qualitative or only roughly quantitative, and emphasis should be placed on the principles illustrated rather than on the accuracy of the necessary measurements. On the other hand, laboratory courses designed for more mature students, who are supposed to have a working knowledge of fundamental principles, are intended to give instruction in the theory and practice of the methods of precise measurement that underlie all effective research and supply the data on which practical engineering enterprises are based. They should also develop the power of logical argument and expression, and lead the student to draw rational conclusions from his observations. The instruments provided should be of standard design and efficiency in order that the student may gain practice in making adjustments and observations under as nearly as may be the same conditions that prevail in original investigation. Measurements are of little value in either research or engineering applications unless the precision with which they represent the measured magnitude is definitely known. Consequently, the stu- dent should be taught to plan and execute proposed measurements within definitely prescribed limits and to determine the accuracy of the results actually attained. Since the treatment of these matters in available laboratory manuals is fragmentary and often very inadequate if not misleading, the author some years ago under- took to impart the necessary instruction, in the form of lectures, to a class of junior engineering students. Subsequently, textbooks on the Theory of Errors and the Method of Least Squares were adopted but most of the applications to actual practice were still given by lecture. The present treatise is the result of the experi- 257860 VI PREFACE ence gained with a number of succeeding classes. It has been prepared primarily to meet the needs of students in engineering and advanced physics who have a working knowledge of the differ- ential and integral calculus. It is not intended to supersede but to supplement the manuals and instruction sheets usually employed in physical laboratories, Consequently, particular instruments and methods of measurement have been described only in so far as they serve to illustrate the principles under discussion. The usefulness of such a treatise was suggested by the marked tendency of laboratory students to carry out prescribed work in a purely automatic manner with slight regard for the significance or the precision of their measurements. Consequently, an endeavor has been made to develop the general theory of measurements and the errors to which they are subject in a form so clear and concise that it can be comprehended and applied by the average student with the prescribed previous training. To this end, numerical ex- amples have been introduced and completely worked out whenever this course seemed likely to aid the student in obtaining a thorough grasp of the principles they illustrate. On the other hand, inherent difficulties have not been evaded and it is not expected, or even desired, that the student will be able to master the subject without vigorous mental effort. The first seven chapters deal with the general principles that underlie all measurements, with the nature and distribution of the errors to which they are subject, and with the methods by which the most probable result is derived from a series of discordant measurements. The various types of measurement met with in practice are classified, and general methods of dealing with each of them are briefly discussed. Constant errors and mistakes are treated at some length, and then the unavoidable accidental errors of observation are explicitly defined. The residuals corresponding to actual measurements are shown to approach the true accidental errors as limits when the number of observations is indefinitely increased and their normal distribution in regard to sign and mag- nitude is explained and illustrated. After a preliminary notion of its significance has been thus imparted, the law of accidental errors is stated empirically in a form that gives explicit representation to all of the factors involved. It is then proved to be in conformity with the axioms of accidental errors, the principle of the arithmetical ij and the results of experience. The various characteristic PREFACE vii errors that are commonly used as a measure of the accidental errors of given series of measurements are clearly denned and their signifi- cance is very carefully explained in order that they may be used intelligently. Practical methods for computing them are developed and illustrated by numerical examples. Chapters eight to twelve inclusive are devoted to a general dis- cussion of the precision of measurements based on the principles established in the preceding chapters. The criteria of accidental errors and suitable methods for dealing with constant and systematic errors are developed in detail. The precision measure, of the result computed from given observations, is defined and its significance is explained with the aid of numerical illustrations. The proper basis for the criticism of reported measurements and the selection of suitable numerical values from tables of physical constants or other published data is outlined ; and the importance of a careful estimate of the precision of the data adopted in engineering and scientific practice is emphasized. The applications of the theory of errors to the determination of suitable methods for the execution of proposed measurements are discussed at some length and illustrated. In chapter thirteen, the relation between measurement and re- search is pointed out and the general methods of physical research are outlined. Graphical methods of reduction and representation are explained and some applications of the method of least squares are developed. The importance of timely and adequate publication, or other report, of completed investigations is emphasized and some suggestions relative to the form of such reports are given Throughout the book, particular attention is paid to methods of computation and to the proper use of significant figures. For the convenience of the student, a number of useful tables are brought together at the end of the volume. A. DE FOREST PALMER. BROWN UNIVERSITY, July, 1912. CONTENTS. PAGE PREFACE v CHAPTER I. GENERAL PRINCIPLES 1 Introduction Measurement and Units Fundamental and Derived Units Dimensions of Units Systems of Units in Gen- eral Use Transformation of Units. CHAPTER II. MEASUREMENTS 11 Direct Measurements Indirect Measurements Classification of Indirect Measurements Determination of Functional Relations Adjustment, Setting, and Observation of Instruments Record of Observations Independent, Dependent, and Conditioned Measurements Errors and the Precision of Measurements Use of Significant Figures Adjustment of Measurements Discus- sion of Instruments and Methods. CHAPTER III. CLASSIFICATION OF ERRORS 23 Constant Errors Personal Errors Mistakes Accidental Errors Residuals Principles of Probability. CHAPTER IV. THE LAW OF ACCIDENTAL ERRORS 29 Fundamental Propositions Distribution of Residuals Proba- bility of Residuals The Unit Error The Probability Curve Systems of Errors The Probability Function The Precision Constant Discussion of the Probability Function The Proba- bility Integral Comparison of Theory and Experience The Arithmetical Mean. CHAPTER V. CHARACTERISTIC ERRORS 44 The Average Error The Mean Error The Probable Error Relations between the Characteristic Errors Characteristic Errors of the Arithmetical Mean Practical Computation of Characteristic Errors Numerical Example Rules for the Use of Significant Figures. CHAPTER VI. MEASUREMENTS OF UNEQUAL PRECISION 61 Weights of Measurements The General Mean Probable Error of the General Mean Numerical Example. ix x CONTENTS CHAPTER VII. PAGE THE METHOD OF LEAST SQUARES 72 Fundamental Principles Observation Equations Normal Equa- tions Solution with Two Independent Variables Adjustment of the Angles about a Point Computation Checks Gauss's Method of Solution Numerical Illustration of Gauss's Method Con- ditioned Quantities. CHAPTER VIII. PROPAGATION OP ERRORS 95 Derived Quantities Errors of the Function Xi X z X 3 . . . X q Errors of the Function ai-Xi 0:2^2 013X3 =h . . . aqXq Errors of the Function F (Xi, X ? , . . . , Xq) Example Introducing the Fractional Error Fractional Error of the Func- tion aX! n > X X 2 n ' X ... X X q n *. CHAPTER IX. ERRORS OF ADJUSTED MEASUREMENTS 105 Weights of Adjusted Measurements Probable Error of a Single Observation Application to Problems Involving Two Unknowns Application to Problems Involving Three Unknowns. CHAPTER X. DISCUSSION OF COMPLETED OBSERVATIONS 117 Removal of Constant Errors Criteria of Accidental Errors Probability of Large Residuals Chauvenet's Criterion Preci- sion of Direct Measurements Precision of Derived Measurements Numerical Example. CHAPTER XI. DISCUSSION OF PROPOSED MEASUREMENTS 144 Preliminary Considerations The General Problem The Pri- mary Condition The Principle of Equal Effects Adjusted Effects Negligible Effects Treatment of Special Functions Numerical Example. CHAPTER XII. BEST MAGNITUDES FOR COMPONENTS 165 Statement of the Problem General Solutions Special Cases Practical Examples Sensitiveness of Methods and Instruments. CHAPTER XIII. RESEARCH 192 Fundamental Principles General Methods of Physical Research Graphical Methods of Reduction Application of the Method of Least Squares Publication. TABLES 212 INDEX.. 245 LIST OF TABLES. PAGE I. DIMENSIONS OF UNITS 212 II. CONVERSION FACTORS 213 III. TRIGONOMETRICAL RELATIONS 215 IV. SERIES 217 V. DERIVATIVES 219 VI. SOLUTION OF EQUATIONS 220 VII. APPROXIMATE FORMULA 221 VIII. NUMERICAL CONSTANTS 222 IX. EXPONENTIAL FUNCTIONS e x AND e~ x 223 X. EXPONENTIAL FUNCTIONS e* 2 AND e~ xZ 224 XI. THE PROBABILITY INTEGRAL P A 225 XII. THE PROBABILITY INTEGRAL P s 226 XIII. CHAUVENET'S CRITERION 226 XIV. FOR COMPUTING PROBABLE ERRORS BY FORMULAE (31) AND (32). 227 XV. FOR COMPUTING PROBABLE ERRORS BY FORMULA (34) 228 XVI. SQUARES OF NUMBERS 229 XVII. LOGARITHMS; 1000 TO 1409 231 XVIII. LOGARITHMS 232 XIX. NATURAL SINES 234 XX. NATURAL COSINES 236 XXI. NATURAL TANGENTS 238 XXII. NATURAL COTANGENTS 240 XXIII. RADIAN MEASURE. . 242 THE THEOEY OF MEASUREMENTS CHAPTER I. GENERAL PRINCIPLES. i. Introduction. Direct observation of the relative position and motion of surrounding objects and of their similarities and differences is the first step in the acquisition of knowledge. Such observations are possible only through the sensations pro- duced by our environment, and the value of the knowledge thus acquired is dependent on the exactness with which we corre- late these sensations. Such correlation involves a quantitative estimate of the relative intensity of different sensations and of their time and space relations. As our estimates become more and more exact through experience, our ideas regarding the objective world are , gradually modified until they represent the actual condition of things with a considerable degree of precision. The growth of science is analogous to the growth of ideas. Its function is to arrange a mass of apparently isolated and un- related phenomena in systematic order and to determine the in- terrelations between them. For this purpose, each quantity that enters into the several phenomena must be quantitatively deter- mined, while all other quantities are kept constant or allowed to vary by a measured amount. The exactness of the relations thus determined increases with' the precision of the measure- ments and with the success attained in isolating the particular phenomena investigated. A general statement, or a mathematical formula, that ex- presses the observed quantitative relation between the different magnitudes involved in any phenomenon is called the law of that phenomenon. As here used, the word law does not mean 1 2 THE THEORY OF MEASUREMENTS [ART. 2 that the phenomenon must follow the prescribed course, but that, under the given conditions and within the limits of error and the range of our measurements, it has never been found to deviate from that course. In other words, the laws of science are concise statements of our present knowledge regarding phenomena and their relations. As we increase the range and accuracy of our measurements and learn to control the condi- tions of experiment more definitely, the laws that express our results become more exact and cover a wider range of phenomena. Ultimately we arrive at broad generalizations from which the laws of individual phenomena are deducible as special cases. The two greatest factors in the progress of science are the trained imagination of the investigator and the genius of measurement. To the former we owe the rational hypotheses that have pointed the way of advance and to the latter the methods of observation and measurement by which the laws of science have been developed. 2. Measurement and Units. To measure a quantity is to determine the ratio of its magnitude to that of another quan- tity, of the same kind, taken as a unit. The number that expresses this ratio may be either integral or fractional and is called the numeric of the given quantity in terms of the chosen unit. In general, if Q represents the magnitude of a quantity, U the magnitude of the chosen unit, and N the corresponding numeric we have Q = NU, (I) which is the fundamental equation of measurement. The two factors N and U are both essential for the exact specification of the magnitude Q. For example: the length of a certain line is five inches, i.e., the line is five times as long as one inch. It is not sufficient to say that the length of the line is five; for in that case we are uncertain whether its length is five inches, five feet, or five times some other unit. Obviously, the absolute magnitude of a quantity is independent of the units with which we choose to measure it. Hence, if we adopt a different unit U', we shall find a different numeric N' such that Q = N'U', (II) and consequently NU = N'U', ART. 2] ' GENERAL PRINCIPLES 3 or $-^- (HI) Equation (III) expresses the general principle involved in the transformation of units and shows that the numeric varies in- versely as the magnitude of the unit; i.e., if U is twice as large as U', N will be only one-half as large as N'. To take a con- crete example: a length equal to ten inches is also equal to 25.4 centimeters approximately. In this case N equals ten, N' equals 25.4, U equals one inch, and U r equals one centi- N f meter. The ratio of the numerics -^ is 2.54 and hence the inverse ratio of the units -, is also 2.54, i.e., one inch is equal to 2.54 centimeters. Equation (III) may also be written in the form (IV) which shows that the numeric of a given quantity relative to the unit U is equal to its numeric relative to the unit U' multiplied w by the ratio of the unit U f to the unit U. The ratio -jj is called the conversion factor for the unit U f in terms of the unit U. It is equal to the number of units U in one unit U', and when multiplied by the numeric of a quantity in terms of U' gives the numeric of the same quantity in terms of U. The con- version factor for transformation in the opposite direction, i.e., from U to U', is obviously the inverse of the above, or -== In general, the numerator of the conversion factor is the unit in which the magnitude is already expressed and the denominator is the unit to which it is to be transformed. For example: one inch is approximately equal to 2.54 centimeters, hence the numeric of a length in centimeters is about 2.54 times its numeric in inches. Conversely, the numeric in inches is equal to the numeric in centimeters divided by 2.54 or multiplied by the reciprocal of this number. In so far as the theory of mensuration and the attainable accuracy of the result are concerned, measurements may be made in terms of any arbitrary unite and, in fact, the adoption oisuch 4 THE THEORY OF MEASUREMENTS [ART. 3 units is frequently convenient when we are concerned only with relative determinations. In general, however, measurements are of little value unless they are expressed in terms of generally accepted units whose magnitude is accurately known. Some such units have come into use through common consent but most of them have been fixed by government enactment and their per- manence is assured by legal standards whose relative magnitudes have been accurately determined. Such primary standards, pre- served by various governments, have, in many cases, been very carefully intercompared and their conversion factors are accu- rately known. Copies of the more important primary standards may be found in all well-equipped laboratories where they are preserved as the secondary standards to which all exact measure- ments are referred. Carefully made copies are, usually, sufficiently accurate for ordinary purposes, but, when the greatest precision is sought, their exact magnitude must be determined by direct comparison with the primary standards. The National Bureau of Standards at Washington makes such comparisons and issues certificates showing the errors of the standards submitted for test. 3. Fundamental and Derived Units. Since the unit is, neces- sarily, a quantity of the same kind as the quantity measured, we must have as many different units as there are different kinds of quantities to be measured. Each of these units might be fixed by an independent arbitrary standard, but, since most measur- able quantities are connected by definite physical relations, it is more convenient to define our units in accordance with these relations. Thus, measured in terms of any arbitrary unit, a uniform velocity is proportional to the distance described in unit time; but, if we adopt as our unit such a velocity that the unit of length is traversed in the unit of time, the factor of pro- portionality is unity and the velocity is equal to the ratio of the space traveled to the elapsed time. Three independently defined units are sufficient, in connection with known physical relations, to fix the value of most of the other units used in physical measurements. We are thus led to distinguish two classes of units; the three fundamental units, defined by independent arbitrary standards, and the derived units, fixed by definite relations between the fundamental units. The .magnitude, and to some extent the choice, of the fundamental ART. 4] GENERAL PRINCIPLES 5 units is arbitrary, but when definite standards for each of these units have been adopted the magnitude of all of the derived units is fixed. For convenience in practice, legal standards have been adopted to represent some of the derived units. The precision of these standards is determined by indirect comparison with the standards representing the three fundamental units. Such comparisons are based on the known relations between the fundamental and de- rived units and are called absolute measurements. The practical advantage gained by the use of derived standards lies in the fact that absolute measurements are generally very difficult and require great skill and experience in order to secure a reasonable degree of accuracy. On the other hand, direct comparison of derived quantities of the same kind is often a comparatively simple matter and can be carried out with great precision. 4. Dimensions of Units. The dimensions of a unit is a mathematical formula that shows how its magnitude is related to that of the three fundamental units. In writing such formulae, the variables are usually represented by capital letters inclosed in square brackets. Thus, [M], [L] and [T]- represent the dimen- sions of the units of mass, length and time respectively. Dimensional formulae and ordinary algebraic equations are essentially different in significance. The former shows the rela- tive variation of units, while the latter expresses a definite mathe- matical relation between the numerics of measurable quantities. Thus if a point in uniform motion describes the distance L in the time T its velocity V is defined by the relation V = Y (V) Since L and T are concrete quantities of different kind, the right- hand member of this equation is not a ratio in the strict arithmet- ical sense; i.e., it cannot be represented by a simple abstract num- ber. Hence, in virtue of the definite physical relation expressed by equation (V), we are led to extend our idea of ratio to include the case of concrete quantities. From this point of view, the ratio of two quantities expresses the rate of change of the first quantity with respect to the second. It is a concrete quantity of the same kind as the quantity it serves, to define. As an illustration, con- sider the meaning of equation (V). Expressed in words, it is " the 6 THE THEORY OF MEASUREMENTS [ART. 4 velocity of a point, in uniform motion, is equal to the time rate at which it moves through space." If we represent the units of velocity, length, and time by [7], [L], and [T\, respectively, and the corresponding numerics by v, I, and t, we have by equation (I), article two, F = v(V], L = l(L], T = t[T], and equation (V) becomes w-m-i' or [V][T] t Since, by definition, [V] and |~l are quantities of the same kind, their ratio can be expressed by an abstract number k and equation (VI) may be written in the form v = kl, (VII) which is an exact numerical equation containing no concrete quantities. The numerical value of the constant k obviously depends on the units with which L, T, and V are measured. If we define the unit of velocity by the relation ryi-M [TV or, as it is more often written, [F] = [L!T-'] f (VIII) k becomes equal to unity and the relation (VII) between the numerics of velocity, length, and time reduces to the simple form The foregoing argument illustrates the advantage to be gained by defining derived units in accordance with the physical rela- tions on which they depend. By this means we eliminate the often incommensurable constants of proportionality such as k would be if the unit of velocity were defined in any other way than by equation (VIII). ART. 5] GENERAL PRINCIPLES 7 The expression on the right-hand side of equation (VIII) is the dimensions of the unit of velocity when the units of length, mass, and time are chosen as fundamental. The dimensions of any other units may be obtained by the method outlined above when we know the physical relations on which they depend. The form of the dimensional formula depends on the units we choose as fundamental, but the general method of derivation is the same in all cases. As an exercise to fix these ideas the student should verify the following dimensional formulae: choosing [M], [L], and [T] as fundamental units, the dimensions of the units of area, acceleration, and force are [L 2 ], [LT~ 2 ], and [MLT~ 2 ] respectively. As an illustration of the effect of a different choice of fundamental units, it may be shown that the dimensions of the unit of mass is [FL^T 2 ] when the units of length [L], force [F], and time [T] are chosen as fundamental. The dimensions of some important derived units are given in Table I at the end of this volume. 5. Systems of Units in General Use. Consistent systems of units may differ from one another by a difference in the choice of fundamental units or by a difference in the magnitude of the particular fundamental units adopted. The systems in common use illustrate both types of difference. Among scientific men, the so-called c.g.s. system is almost universally adopted, and the results of scientific investigations are seldom expressed in any other units. The advantage of such uniformity of choice is obvious. It greatly facilitates the com- parison of the results of different observers and leads to general advance in our knowledge of the phenomena studied. The units of length, mass, and time are chosen as fundamental in this system and the particular values assigned to them are the centi- meter for the unit of length, the gram for the unit of mass, and the mean solar second for the unit of time. The units used commercially in England and the United States of America are far from systematic, as most of the derived units are arbitrarily defined. So far as they follow any order, they form a length-mass-time system in which the unit of length is the foot, the unit of mass is the mass of a pound, and the unit of time is the second. This system was formerly used quite extensively by English scientists and the results of some classic investigations are expressed in such units. English and American engineers find it more convenient to use 8 THE THEORY OF MEASUREMENTS [ART. 6 a system in which the fundamental units are those of length, force, and time. The particular units chosen are the foot as the unit of length, the pound's weight at London as the unit of force, and the mean solar second as the unit of time. We shall see that this is equivalent to a length-mass-time system in which the units of length and time are the same as above and the unit of mass is the mass of 32.191 pounds. 6. Transformation of Units. When the relative magnitude of corresponding fundamental units in two systems is known, a result expressed in one system can be reduced to the other with the aid of the dimensions of the derived units involved. Thus: let A c represent the magnitude of a square centimeter, A t the magnitude of a square inch, N c the numeric of a given area when measured in square centimeters, and Ni the numeric of the same area when measured in square inches; then, from equation (IV), article two, we have But if L c is the magnitude of a centimeter and LI that of an inch, Ai is equal to Lf, and therefore Hence, the conversion factor -p for reducing square centimeters A-i to square inches is equal to the square of the conversion factor for reducing from centimeters to inches. Now the dimensions Li of the unit of area is [L 2 ], and we see that the conversion factor for area may be obtained by substituting the corresponding con- version factor for lengths in this dimensional formula. This is a simple illustration of the general method of transformation of units. When the fundamental units in the two systems differ in magnitude, but not in kind, the conversion factor for correspond- ing derived units in the two systems is obtained by replacing the fundamental units by their respective conversion factors in the dimensions of the derived units considered. It should be noticed that the fundamental units in the c.g.s. system are those of length, mass, and time, while on the engineer's system they are length, force, and time. In the latter system, ART. 6] GENERAL PRINCIPLES 9 force is supposed to be directly measured and expressed by the dimensions [F]. Consequently the dimensions of the unit of mass are [FL~ 1 T 2 ], and the unit of mass is a mass that will acquire . a velocity of one foot per second in one second when acted upon by a force of one pound's weight. For the sake of definiteness, the unit of force is taken as the pound's weight at London, where the acceleration due to gravity (g) is equal to 32.191 feet per second per second. Otherwise the unit of force would be variable, depending on the place at which the pound is weighed. From Newton's second law of motion we know that the relation between acceleration, mass, and force is given by the expression / = ma. For a constant force the acceleration produced is inversely pro- portional to the mass moved. Now the mass of a pound at London is acted upon by gravity with a force of one pound's weight, and, if free, it moves with an acceleration of 32.191 feet per second per second. Hence a mass equal to that of 32.191 pounds acted upon by a force of one pound's weight would move with an acceler- ation of one foot per second per second, i.e., it would acquire a velocity of one foot per second in one second. Hence the unit of mass in the engineer's system is 32.191 pounds mass. This unit is sometimes called a slugg, but the name is seldom met with since engineers deal primarily with forces rather than masses, and are W content to write for mass without giving the unit a definite 7 name. This is equivalent to saying that the mass of a body, expressed in sluggs, is equal to its weight, at London, expressed in pounds, divided by 32.191. After careful consideration of the foregoing discussion, it will be evident that the engineer's length-force-time system is exactly equivalent to a length-mass-time system in which the unit of length is the foot, the unit of mass is the slugg or 32.191 pounds' mass, and the unit of time is the mean solar second. In the latter system the fundamental units are of the same kind as those of the c.g.s. system. Hence, if the conversion factor for the unit of mass is taken as the ratio of the magnitude of the slugg to that of the gram, quantities expressed in the units of the engineer's system may be reduced to the equivalent values in the c.g.s. system by the method described at the beginning of this article. 10 THE THEORY OF MEASUREMENTS [ART. 6 When, as is frequently the case, the engineer's results are expressed in terms of the local weight of a pound as a unit of force in place of the pound's weight at London, the result of a transformation of units, carried out as above, will be in error by a factor equal to the ratio of the acceleration due to gravity at London and at the location of the measurements. Unless the local gravitational acceleration is definitely stated by the observer and unless he has used his length-force-time units in a consistent manner, it is impossible to derive the exact equivalent of his results on the c.g.s. system. CHAPTER II. MEASUREMENTS. IN article two of the last chapter we defined the term " measure- ment " and showed that any magnitude may be represented by the product of two factors, the numeric and the unit. The object of all measurements is the determination of the numeric that ex- presses the magnitude of the observed quantity in terms of the chosen unit. For convenience of treatment, they may be classified according to the nature of the measured quantity and the methods of observation and reduction. 7. Direct Measurements. The determination of a desired numeric by direct observation of the measured quantity, with the aid of a divided scale or other indicating device graduated in terms of the chosen unit, is called a direct measurement. Such measurements are possible when the chosen unit, together with its multiples and submultiples, can be represented by a material standard, so constructed that it can be directly applied to the measured quantity for the purpose of comparison, or when the unit and the measured magnitudes produce proportional effects on a suitable indicating device. Lengths may be directly measured with a graduated scale, masses by comparison with a set of standard masses on an equal arm balance, time intervals by the use of a clock regulated to give mean solar time, and forces with the aid of a spring balance. Hence magnitudes expressible in terms of the fundamental units of either the c.g.s. or the engineer's system may be directly measured. Many quantities expressible in terms of derived units, that can be represented by material standards, are commonly determined by direct measurement. As illustrations, we may cite the deter- mination of the volume of a liquid with a graduated flask and the measurement of the electrical resistance of a wire by comparison with a set of standard resistances. 8. Indirect Measurements. The determination of a desired numeric by computation from the numerics of one or more 11 12 THE THEORY OF MEASUREMENTS [ART. 9 directly measured magnitudes, that bear a known relation to the desired quantity, is called an indirect measurement. The relation between the observed and computed magnitudes may be expressed in the general form y = Ffa, Xz, x 3 , . . . a, b, c . . . ), where y, x t , x 2 , etc., represent measured or computed magnitudes, or the numerics corresponding to them, a, b, c, etc., represent constants, and F indicates that there is a functional relation between the other quantities. This expression is read, y equals some function of xi, x*, etc., and a, b, c, etc. In any particular case, the form of the function F and the number and nature of the related quantities must be known before the computation of the unknown quantities is undertaken. Most of the indirect measurements made by physicists and engineers fall into one or another of three general classes, char- acterized by the nature of the unknown and measured magnitudes and the form of the function F. 9. Classification of Indirect Measurements. I. In the first class, y represents the desired numeric of a magni- tude that is not directly measured, either because it is impossible or inconvenient to do so, or because greater precision can be at- tained by indirect methods. The form of the function F and the numerical values of all of the constants a, 6, c, etc., appearing in it, are given by theory. The quantities xi, Xz, etc., represent the numerics of directly measured magnitudes. In the following pages indirect measurements belonging to this class will sometimes be referred to as derived measurements. As an illustration we may cite the determination of the density s of a solid sphere from direct measurements of its mass M and its diameter D with the aid of the relation M = F^' Comparing this expression with the general formula given above, we note that s corresponds to y, M to xi, D to x a , J to a, TT to 6, and that F represents the function y^^. The form of the func- ART. 9] MEASUREMENTS 13 tion is given by the definition of density as the ratio of the mass to the volume of a body and the numerical constants and w are given by the known relation between the volume and diameter of a sphere. II. In the second class of indirect measurements, the numerical constants a, b, c, etc., are the unknown quantities to be computed, the form of the function F is known, and all of the quantities y, Xi, x z , etc., are obtained by direct measurements or given by theory. The functions met with in this class of measurements usually represent a continuous variation of the quantity y with respect to the quantities x\, x 2 , etc., as independent variables. Hence the result of a direct measurement of y will depend on the particular values of Xi, x 2 , etc., that obtain at the time of the measurement. Consequently, in computing the constants a, b, c, etc., we must be careful to use only corresponding values of the measured quantities, i.e., values that are, or would be, obtained by coincident observations on the several magnitudes. Every set of corresponding values of the variables y, Xi, x 2 , etc., when used in connection with the given function, gives an algebraic relation between the unknown quantities a, b, c, etc., involving only numerical coefficients and absolute terms. When we have obtained as many independent equations as there are unknown quantities, the latter may be determined by the usual algebraic methods. We shall see, however, that more precise results can be obtained when the number of independent measurements far exceeds the minimum limit thus set and the computation is made by special methods to be described hereafter. The determination of the initial length L and the coefficient of linear expansion a of a metallic bar from a series of measurements of the lengths L t corresponding to different temperatures t with the aid of the functional relation L t = Lo (1 + at) is an example of the class of measurements here considered. Such measurements are sometimes called determinations of empirical constants. 14 THE THEORY OF MEASUREMENTS [ART. 9 III. The third class of indirect measurements includes all cases in which each of a number of directly measured quantities yi, y*, y s , etc., is a given function of the unknown quantities Xi, x 2 , X B , etc., and certain known numerical constants a, 6, c, etc. In such cases we have as many equations of the form y 1 = FI (xi, x 2 , 3 , . . . a, 6, c, . . . ), 2/2 = F 2 (xi, z 2 , $t, . . . a, M, . . . )> as there are measured quantities yi, y 2 , etc. This number must be at least as great as the number of unknowns Xi, x 2 , etc., and may be much greater. The functions F lt F 2 , etc., are frequently dif- ferent in form and some of them may not con- tain all of the un- knowns. The numeri- cal constants, appearing in different functions, are generally different. But the form of each of the functions and the values of all of the constants must be known before a solu- tion of the problem is possible. Problems of this type are frequently met with in astronomy and geod- esy. One of the simplest is known as the adjustment of the angles about a point. Thus, let it be required to find the most probable values of the angles Xi, x 2 , and x 3 , Fig. 1, from direct measurements of yi, y 2 , y 3) . . . y & . In this case the general equations take the form FIG. ART. 11] MEASUREMENTS 15 2/i = xi, 2/2 = xi + x 2 , 2/4 = X 2 , 2/5 = 2 2/6 = , and all of the numerical constants are either unity or zero. The solution of such problems will be discussed in the chapter on the method of least squares. 10. Determination of Functional Relations. When the form of the functional relation between the observed and unknown magnitudes is not known, the solution of the problem requires something more than measurement and computation. In some cases a study of the theory of the observed phenomena, in con- nection with that of allied phenomena, will suggest the form of the required function. Otherwise, a tentative form must be assumed after a careful study of the observations themselves, generally by graphical methods. In either case the constants of the assumed function must be determined by indirect measurements and the results tested by a comparison of the observed and the computed values of the related quantities. If these values agree within the accidental errors of observation, the assumed function may be adopted as an empirical representation of the phenomena. If the agreement is not sufficiently close, the form of the function is modified, in a manner suggested by the observations, and the process of computation and comparison is repeated until a satis- factory agreement is obtained. A more detailed treatment of such processes will be found in Chapter XIII. 11. Adjustment, Setting, and Observation of Instruments. Most of the magnitudes dealt with in physics and engineering are determined by indirect measurements. But we have seen that all such quantities are dependent upon and computed from directly measured quantities. Consequently, a study of the methods and precision of direct measurement is of fundamental importance. In general, every direct measurement involves three distinct operations. First: the instrument adopted is so placed that its 16 THE THEORY OF MEASUREMENTS [ART. 12 scale is in the proper position relative to the magnitude to be measured and all of its parts operate smoothly in the manner and direction prescribed by theory. Operations of this nature are called adjustments. Second: the reference line of the instru- ment is moved, or allowed to move, in the manner demanded by theory, until it coincides with a mark chosen as a point of reference on the measured magnitude. We shall refer to this operation as a setting of the instrument. Third: the position of the index of the instrument, with respect to its graduated scale, is read. This is an observation. As an illustration, consider the measurement of the normal distance between two parallel lines with a micrometer microscope. The instrument must be so mounted that it can be rigidly clamped in any desired position or moved freely in the direction of its optical axis without disturbing the direction of the micrometer screw. The following adjustments are necessary: the axis of the micrometer screw must be made parallel to the plane of the two lines and perpendicular to a normal plane through one of them; the eyepiece must be so placed that the cross-hairs are sharply defined; the microscope must be moved, in the direction of its optical axis, until the image of the two lines, or one of them if the normal distance between them is greater than the field of view of the microscope, is in the same plane with the cross-hairs. The latter adjustment is correct when there is no parallax between the image of the lines and the cross-hairs. The setting is made by turning the micrometer head until the intersection of the cross- hairs bisects the image of one of the lines. Finally the reading of the micrometer scale is observed. A similar setting and ob- servation are made on the other line and the difference between the two observations gives the normal distance between the two lines in terms of the scale of the micrometer. 12. Record of Observations. In the preceding article, the word "observation" is used in a very much restricted sense to indicate merely the scale reading of a measuring instrument. This restriction is convenient in dealing with the technique of measurement, but many other circumstances, affecting the accu- racy of the result, must be observed and taken into account in a complete study of the phenomena considered. There is, however^ little danger of confusion in using the word in the two different senses since the more restricted meaning is in reality only a ART. 13] MEASUREMENTS 17 special case of the general. The particular significance intended in any special case is generally clear from the context. The first essential for accurate measurements is a clear and orderly record of all of the observations. The record should begin with a concise description of the magnitude to be measured, and the instruments and methods adopted for the purpose. Instru- ments may frequently be described, with sufficient precision, by stating their name and number or other distinguishing mark. Methods are generally specified by reference to theoretical treatises or notes. The adjustment and graduation of the instruments should be clearly stated. The date on which the work is carried out and the location of the apparatus should be noted. Observations, in the restricted sense, should be neatly arranged in tabular form. The columns of the table should be so headed, and referred to by subsidiary notes, that the exact significance of all of the recorded figures will be clearly understood at any future time. All circumstances likely to affect the accuracy of the measurements should be carefully observed and recorded in the table or in suitably placed explanatory notes. Observations should be recorded exactly as taken from the instruments with which they are made, without mental computa- tion or reduction of any kind even the simplest. For example: when a micrometer head is divided into any number of parts other than ten or one hundred, it is better to use two columns in the table and record the reading of the main scale in one and that of the micrometer head in the other than to reduce the head reading to a decimal mentally and enter it in the same column with the main scale reading. This is because mistakes are likely to be made in such mental calculations, even by the most expe- rienced observers, and, when the final reduction of the observations is undertaken at a future time, it is frequently difficult or impos- sible to decide whether a large deviation of a single observation from the mean of the others is due to an accidental error of obser- vation or to a mistake in such a mental calculation. 13. Independent, Dependent, and Conditioned Measure- ments. Measurements on the same or different magnitudes are said to be independent when both of the following specifications are fulfilled: first, the measured magnitudes are not required to satisfy a rigorous mathematical relation among themselves; second, the same observation is not used in the computation of 18 THE THEORY OF MEASUREMENTS [ART. 14 any two of the measurements and the different observations are entirely unbiased by one another. When the first of these specifications is fulfilled and the second is not, the measurements are said to be dependent. Thus, when several measurements of the length of a line are all computed from the same zero reading of the scale used, they are all dependent on that observation and any error in the position of the zero mark affects all of them by exactly the same amount. When the position of the index relative to the scale of the measuring instrument is visible while the settings are being made, there is a marked tendency to set the instrument so that successive observations will be exactly alike rather than to make an independent judgment of the bisection of the chosen mark in each case. The observations, corresponding to settings made in this manner, are biased by a preconceived notion regarding the correct position of the index and the measure- ments computed from them are not independent. The impor- tance of avoiding faulty observations of this type cannot be too strongly emphasized. They not only vitiate the results of our measurements, but also render a determination of their precision impossible. Measurements that do not satisfy the first of the above speci- fications are called conditioned measurements. The different determinations of each of the related quantities may or may not be independent, according as they do or do not satisfy the second specification, but the adjusted results of all of the measurements must satisfy the given mathematical relation. Thus, we may make a number of independent measurements of each of the angles of a plane triangle, but the mean results must be so adjusted that the sum of the accepted values is equal to one hundred and eighty degrees. 14. Errors and the Precision of Measurements. Owing to unavoidable imperfections and lack of constant sensitiveness in our instruments, and to the natural limit to the keenness of our senses, the results of our observations and measurements differ somewhat from the true numeric of the observed magnitude. Such differences are called errors of observation or measurement. Some of them are due to known causes and can be eliminated, with sufficient accuracy, by suitable computations. Others are apparently accidental in nature and arbitrary in magnitude. Their probable distribution, in regard to magnitude and frequency ART. 15] MEASUREMENTS 19 of occurrence, can be determined by statistical methods when a sufficient number of independent measurements is available. The precision of a measurement is the degree of approximation with which it represents the true numeric of the observed magni- tude. Usually our measurements serve only to determine the probable limits within which the desired numeric lies. Looked at from this point of view, the precision of a measurement may be considered to be inversely proportional to the difference between the limits thus determined. It increases with the accuracy, adaptability, and sensitiveness of the instruments used, and with the skill and care of the observer. But, after a very moderate precision has been attained, the labor and expense necessary for further increase is very great in proportion to the result obtained. A measurement is of little practical value unless we know the precision with which it represents the observed magnitude. Hence the importance of a thorough study of the nature and dis- tribution of errors in general and of the particular errors that characterize an adopted method of measurement. At first sight it might seem incredible that such errors should follow a definite mathematical law. But, when the number of observations is sufficiently great, we shall see that the theory of probability leads to a definite and easily calculated measure of the precision of a single observation and of the result computed from a number of observations. 15. Use of Significant Figures. When recording the nu- merical results of observations or measurements, and during all of the necessary computations, the number of significant figures employed should be sufficient to express the attained precision and no more. By significant figures we mean the nine digits and zeros when not used merely to locate the decimal point. In the case of the direct observation of the indications of instru- ments, the above specification is usually sufficiently fulfilled by allowing the last recorded significant figure to represent the estimated tenth of the smallest division of the graduated scale. For example: in measuring the length of a line, with a scale divided in millimeters, the position of the ends of the line would be recorded to the nearest estimated tenth of a millimeter. Generally, computed results should be so recorded that the limiting values, used to express the attained precision, differ by only a few units in the last one or two significant figures. Thus: 20 THE THEORY OF MEASUREMENTS [ART. 15 if the length of a line is found to lie between 15.65 millimeters and 15.72 millimeters, we should write 15.68 millimeters as the result of our measurement. The use of a larger number of significant figures would be not only a waste of space and labor, but also a false representation of the precision of the result. Most of the magnitudes we are called upon to measure are incommensurable with the chosen unit, and hence there is no limit to the number of significant figures that might be used if we chose to do so; but experienced observers are always careful to express all observa- tions and results and carry out all computations with a number just sufficient to represent the attained precision. The use of too many or too few significant figures is strong evidence of inex- perience or carelessness in making observations and computations. More specific rules for determining the number of significant figures to be used in special cases will be developed in connection with the methods for determining the precision of measurements. The number of significant figure^ in any numerical expression is entirely independent of the position of the decimal point. Thus: each of the numbers 5,769,600, 5769, 57.69, and 0.0005769 is expressed by four significant figures and represents the corre- sponding magnitude within one-tenth of one per cent, notwith- standing the fact that the different numbers correspond to differ- ent magnitudes. In general, the location of the decimal point shows the order of magnitude of the quantity represented and the number of significant figures indicates the precision with which the actual numeric of the quantity is known. In writing very large or very small numbers, it is convenient to indicate the position of the decimal point by means of a positive or negative power of ten. Thus: the number 56,400,000 may be written 564 X 10 5 or, better, 5.64 X 10 7 , and 0.000075 may be written 75 X W~ or 7.5 X 10~ 5 . When a large number of numerical observations or results are to be tabulated or used in computation, a considerable amount of time and space is saved by adopting this method of representation. The second of the two forms, illustrated above, is very convenient in making com- putations by means of logarithms, as in this case the power of ten always represents the characteristic of the logarithm of the corresponding number. In rounding numbers to the required number of significant figures, the digit in the last place held should be increased by one ART. 17] MEASUREMENTS 21 unit when the digit in the next lower place is greater than five, and left unchanged when the neglected part is less than five- tenths of a unit. When the neglected part is exactly five-tenths of a unit the last digit held is increased by one if odd, and left unchanged if even. Thus: 5687.5 would be rounded to 5688 and 5686.5 to 5686. 1 6. Adjustment of Measurements. The results of inde- pendent measurements of the same magnitude by the same or different methods seldom agree with one another. This is due to the fact that the probability for the occurrence of errors of exactly the same character and magnitude in the different cases is very small indeed. Hence we are led to the problem of determining the best or most probable value of the numeric of the observed magnitude from a series of discordant measurements. The given data may be all of the same precision or it may be necessary to assign a different degree of accuracy to the different measure- ments. In either case the solution of the problem is called the adjustment of the measurements. The principle of least squares, developed in the theory of errors that leads to the measure of precision cited above, is the basis of all such adjustments. But the particular method of solution adopted in any given case depends on the nature of the measure- ments considered. In the case of a series of direct, equally pre- cise, measurements of a single quantity, the principle of least squares leads to the arithmetical mean as the most probable, and therefore the best, value to assign to the measured quantity. This is also the value that has been universally adopted on a priori grounds. In fact many authors assume the maximum probability of the arithmetical mean as the axiomatic basis for the develop- ment of the law of errors. The determination of empirical relations between measured quantities and the constants that enter into them is also based on the principle of least squares. For this reason, such deter- minations are treated in connection with the discussion of the methods for the adjustment of measurements. 17. Discussion of Instruments and Methods. The theory of errors finds another very important application in the discussion of the relative availableness and accuracy of different instruments and methods of measurement. Used in connection with a few preliminary measurements and a thorough knowledge of the 22 THE THEORY OF MEASUREMENTS [ART. 17 theory of the proposed instruments and methods, it is sufficient for the determination of the probable precision of an extended series of careful observations. By such means we are able to select the instruments and methods best adapted to the particular purpose in view. We also become acquainted with the parts of the investigation that require the greatest skill and care in order to give a result with the desired precision. The cost of instruments and the time and skill required in carrying out the measurements increase much more rapidly than the corresponding precision of the results. Hence these factors must be taken into account in determining the availableness of a proposed method. It is by no means always necessary to strive for the greatest attainable precision. In fact, it would be a waste of time and money to carry out a given measurement with greater precision than is required for the use to which it is to be put. For many practical purposes, a result correct within one- tenth of one per cent, or even one per cent, is amply sufficient. In such cases it is essential to adopt apparatus and methods that will give results definitely within these limits without incurring the greater cost and labor necessary for more precise deter- minations. CHAPTER III. CLASSIFICATION OF ERRORS. ALL measurements, of whatever nature, are subject to three distinct classes of errors, namely, constant errors, mistakes, and accidental errors. 18. Constant Errors. Errors that can be determined in sign and magnitude by computations based on a theoretical consideration of the method of measurement used or on a pre- liminary study and calibration of the instruments adopted are called constant errors. They are sometimes due to inadequacy of an adopted method of measurement, but more frequently to inaccurate graduation and imperfect adjustment of instruments. As a simple illustration, consider the measurement of the length of a straight line with a graduated scale. If the scale is not held exactly parallel to the line, the result will be too great or too small according as the line of sight in reading the scale is normal to the line or to the scale. The magnitude of the error thus introduced depends on the angle between the line and the scale and can be exactly computed when we know this angle and the circumstances of the observations. If the scale is not straight, if its divisions are irregular, or if they are not of standard length, the result of the measurement will be in error by an amount depending on the magnitude and distribution of these inaccuracies of construction. The sign and magnitude of such errors can gener o1 ly be determined by a careful study and calibration of the scai If M represents the actual numeric of the measured magnitude, M Q the observed numeric, and Ci, C 2 , C 3 , etc., the constant errors inherent in the method of measurement and the instruments used, M = Mo + Ci + C 2 + C 3 + - . (1) The necessary number of correction terms Ci, G' 2 , C z , etc., is determined by a careful study of the theory and practical appli- cation of the apparatus and method used in finding M Q . The magnitude and sign of each term are determined by subsidiary 23 24 THE THEORY OF MEASUREMENTS [ART. 18 measurements or calculated, on theoretical grounds, from known data. Thus, in the above illustration, suppose that the scale is straight and uniformly graduated, that each of its divisions is 1.01 times as long as the unit in which it is supposed to be gradu- ated, and that the line of the graduations makes an angle a with the line to be measured. Under these conditions, the number of correction terms reduces to two: the first, Ci, due to the false length of the scale divisions, and the second, C 2 , due to the lack of parallelism between the scale and the line. Since the actual length of each division is 1.01, the .length of Mo divisions, i.e., the length that would have been observed on an accurate scale, is M l = Mo X 1.01 = Mo + 0.01 Mo = Mo + Ci, ... Ci = + 0.01 Mo. If the line of sight is normal to the line in making the observa- tions, the length M 2 that would have been obtained if the scale had been properly placed is M 2 = MO cos a = MO + Czj /. C 2 =-M (l-cosa)=-2M sin 2 ^ and (1) takes the form M= Mo + 0.01 Mo - 2M sin 2 |> = M (l+0.01-2sin 2 ^Y The precision with which it is necessary to determine the cor- rection terms Ci, C 2 , etc., and frequently the number of these terms that should be employed depends on the precision with which the observed numeric M is determined. If M is measured within one-tenth of one per cent of its magnitude, the several correction terms should be determined within one one-hundredth of one per cent of M , in order that the neglected part of the sum of the corrections may be less than one-tenth of one per cent of M . If any correction term is found to be less than the. above limit, it may be neglected entirely since it is obviously useless to apply a correction that is less than one-tenth of the uncer- tainty of M . In our illustration, suppose that the precision is such that we are sure that M is less than 1.57 millimeters and greater than ART. 19] CLASSIFICATION OF ERRORS 25 1.55 millimeters, but is not sufficient to give the fourth significant figure within several units. Obviously, it would be useless to determine Ci and C% closer than 0.001 millimeter, and if the mag- nitude of either of these quantities is less than 0.001 millimeter our knowledge of the true value of M is not increased by making the corresponding correction. In fact, it is usually impossible to determine the C's with greater accuracy than the above limit, since, as in our illustration, M Q is usually a factor in the correction terms. Hence the writing down of more than the required num- ber of significant figures is mere waste of labor. When considering the availableness of proposed methods and apparatus, it is important to investigate the nature and magni- tude of the constant errors inherent in their use. It sometimes happens that the sources of such errors can be sufficiently elimi- nated by suitable adjustment of the instruments or modification of the method of observation. When this is not possible the conditions should be so chosen that the correction terms can be computed with the required precision. Even when all possible precautions have been taken, it very seldom happens that the sum of the constant errors reduces to zero or that the magni- tude of the necessary corrections can be exactly determined. Moreover, such errors are never rigorously constant, but present small fortuitous variations, which, to some extent, are indistinguish- able from the accidental errors to be described later. A more detailed discussion of constant errors and the limits within which they should be determined will be given after we have developed the methods for estimating the precision of the observed numeric M. 19. Personal Errors. When setting cross-hairs, or any other indicating device, to bisect a chosen mark, some observers will invariably set too far to one side of the center, while others will as consistently set on the other side. Again, in timing a transit, some persons will signal too soon and others too late. With experienced and careful observers, the errors introduced in this manner are small and nearly constant in magnitude and sign, but they are seldom entirely negligible when the highest possible precision is sought. Errors of this nature will be called personal errors, since their magnitude and sign depend on personal peculiarities of the observer. Their elimination may sometimes be effected by a 26 THE THEORY OF MEASUREMENTS [ART. 20 careful study of the nature of such peculiarities and the magnitude of the effects produced by them under the conditions imposed by the particular problem considered. Suitable methods for this purpose are available in connection with most of the investiga- tions in which an exact knowledge of the personal error is essential. Such a study is .frequently referred to as a determination of the "Personal Equation" of the observer. 20. Mistakes. Mistakes are errors due to reading the indi- cations of an instrument carelessly or to a faulty record of the observations. The most frequent of these are the following : the wrong integer is placed before an accurate fractional reading, e.g., 9.68 for 19.68; the reading is made in the wrong direction of the scale, e.g., 6.3 for 5.7; the significant figures of a number are transposed, e.g., 56 is written for 65. Care and strict attention to the work in hand are the only safeguards against such mistakes. When a large number of observations have been systematically taken and recorded, it is sometimes possible to rectify an obvious mistake, but unless this can be done with certainty the offending observation should be dropped from the series. This statement does not apply to an observation showing a large deviation from the mean but only to obvious mistakes. 21. Accidental Errors. When a series of independent meas- urements of the same magnitude have been made, by the same method and apparatus and with equal care, the results generally differ among themselves by several units in the last one or two significant figures. If in any case they are found to be identical, it is probable that the observations were not independent, the instruments adopted were not sufficiently sensitive, the maximum precision attainable was not utilized, or the observations were carelessly made. Exactly concordant measurements are quite as strong evidence of inaccurate observation as widely divergent ones. As the accuracy of method and the sensitiveness of instruments is increased, the number of concordant figures in the result in- creases but differences always occur in the last attainable figures. Since there is, generally, no reason to suppose that any one of the measurements is more accurate than any other, we are led to believe that they are all affected by small unavoidable errors. After all constant errors and mistakes have been corrected, the re- maining differences between the individual measurements and the true ART. 22] CLASSIFICATION OF ERRORS 27 numeric of the measured magnitude are called accidental errors. They are due to the combined action of a large number of inde- pendent causes each of which is equally likely to produce a posi- tive or a negative effect. Probably most of them have their origin in small fortuitous variations in the sensitiveness and adjustment of our instruments and in the keenness of our senses of sight, hearing, and touch. It is also possible that the correla- tion of our sense perceptions and the judgments that we draw from them are not always rigorously the same under the same set of stimuli. Suppose that N measurements of the same quantity have been made by the same method and with equal care. Let ai, a^, 3, . . . a N represent the several results of the independent meas- urements, after all constant errors and mistakes have been elim- inated, and let X represent the true numeric of the measured magnitude. Then the accidental errors of the individual measure- ments are given by the differences, Ai - ai - X, A 2 = a 2 - X, A 3 = a 3 - X } . . . A^ = a N -X. (2) The accidental errors AI, A 2 , . . . A# thus denned are sometimes called the true errors of the observations ai, a 2 , . . . a N . 22. Residuals. Since the individual measurements a\ t a?, . . . a N differ among themselves, and since there is no reason to suppose that any one of them is more accurate than any other, it is never possible to determine the exact magnitude of the numeric X. Hence the magnitude of the accidental errors A i, A 2 , . . . A# can never be exactly determined. But, if x is the most probable value that we can assign to the numeric X on the basis of our measurements, we can determine the differences ri = di x, r z = a 2 x, . . . r N = a N x. (3) These differences are called the residuals of the individual measure- ments dij 02, . . . a N . They represent the most probable values that we can assign to the accidental errors AI, A 2 , . . . A# on the basis of the given measurements. It should be continually borne in mind that the residuals thus determined are never identical with the accidental errors. How- ever precise our measurements may be, the probability that x is exactly equal to X is always less than unity. As the number and precision of measurements increase, the difference between 28 THE THEORY OF MEASUREMENTS [ART. 23 the magnitudes x and X decreases, and the residuals continually approach the accidental errors, but exact equality is never attain- able with a finite number of observations. 23. Principles of Probability. The theory of errors is an application of the principles of probability to the discussion of series of discordant measurements for the purpose of determining the most probable numeric that can be assigned to the measured quantity and making an estimate of the precision of the result thus obtained. A discussion of the fundamental principles of the theory of probability, sufficient for this purpose, is given in most textbooks on advanced algebra, and the student should master them before undertaking the study of the 1 theory of errors. For the sake of convenience in reference, the three most useful propositions are stated below without proof. PROPOSITION 1. If an event can happen in n independent ways and either happen or fail in N independent ways, the prob- ability p that it will occur in a single trial at random is given by the relation n , A . p - r w Also if p' is the probability that it will fail in a single trial at random, p = l_p = !_.. ( 5 ) PROPOSITION 2. If the probabilities for the separate occurrence of n independent events are respectively pi, p%, . . . p n , the prob- ability PS that some one of these events will occur in a single trial at random is given by the relation PS = Pi + Pz + Pz + ' ' ' + P^ (6) PROPOSITION 3. If the probabilities for the separate occurrence of n independent events are respectively pi, p 2 , . . . p n , the probability P that all of the events will occur at the same time is given by the relation P = Pi X P2 X X Pn. (7) CHAPTER IV. THE LAW OF ACCIDENTAL ERRORS. 24. Fundamental Propositions. The theory of accidental errors is based on the principle of the arithmetical mean and the three axioms of accidental errors. When the word " error " is used without qualification, in the statement of these propositions and in the following pages, accidental errors are to be understood. Principle of the Arithmetical Mean. The most probable value that can be assigned to the numeric of a measured magnitude, on the basis of a number of equally trustworthy direct measurements, is the arithmetical mean of the given 'measurements. This proposition is self-evident in the case of two independent measurements, made by the same method with equal care, since one of them is as likely to be exact as the other, and hence it is more probable that the true numeric lies halfway between them than in any other location. Its extension to more than two measurements is the only rational assumption that we can make and is sanctioned by universal usage. First Axiom. In any large number of measurements, positive and negative errors of the same magnitude are equally likely to occur. The number of negative errors is equal to the number of positive errors. Second Axiom. Small errors are much more likely to occur than large ones. Third Axiom. All of the errors of the measurements in a given series lie between equal positive and negative limits. Very large errors do not occur. The foundation of these propositions is the same as that of the axioms of geometry. Namely: they are general statements that are admitted as self-evident or accepted as a basis of argument by all competent persons. Their justification lies in the fact that the results derived from them are found to be in agreement with experience. 25. Distribution of Residuals. It was pointed out in article twenty-two that the true accidental errors, represented by A's, 29 30 THE THEORY OF MEASUREMENTS [ART. 26 cannot be determined in practice, but the residuals, represented by r's, can be computed from the given observations by equation (3). The A's may be considered as the limiting values toward which the r's approach as the number of observations is indefinitely increased. If the residuals corresponding to a very large num- ber of observations are arranged in groups according to sign and magnitude, the groups containing very small positive or negative residuals will be found to be the largest, and, in general, the magni- tude of the groups will decrease nearly uniformly as the magnitude of the contained residuals increases either positively or negatively. Let n represent the number of residuals in any group, and r their common magnitude, then the distribution of the residuals, in regard to sign and magnitude, may be represented graphically by laying off ordinates proportional to the numbers n against abscissae proportional to the corresponding magnitudes r. The points, thus located, will be approximately uniformly distributed about a curve of the general form illustrated in Fig. 2. The number of residuals in each group will increase with the total number of measurements from which the r's are computed. Consequently the ordinates of the curve in Fig. 2 will depend on the number of observations considered as well as on their accuracy. Hence, if we wish to compare different series of measurements with regard to accuracy, we must in some way eliminate the effect of differences in the number of observations. Moreover, we are not so much concerned with the total number of residuals of any given magnitude as with the relative number of residuals of different magnitudes. For, as we shall see, the acuracy of a series of observations depends on the ratio of the number of small errors to the number of large ones. 26. Probability of Residuals. Suppose that a very large number N of independent measurements have been made and that AKF.27J THE LAW OF ACCIDENTAL ERRORS 31 the corresponding residuals have been computed by equation (3). By arranging the results in groups according to sign and magni- tude, suppose we find HI residuals of magnitude n, n 2 of magni- tude r 2 , etc., and n\ of magnitude n, n/ of magnitude r 2 , etc. If we choose one of the measurements at random, the probability that the corresponding residual is equal to r\ is -^ , since there are N residuals and n\ of them are equal to r\. In general, if y\, y 2 , Hi, 2/2', represent the probabilities for the occurrence of residuals equal to n, r 2 , . . . n, r 2 , . . . respectively, When N is increased by increasing the number of measurements, each of the n's is increased in nearly the same ratio since the residuals of the new measurements are distributed in essentially the same manner as the old ones, provided all of the measure- ments considered are made by the same method and with equal care. Consequently, the y's corresponding to a definite method of observation are nearly independent of the number of measure- ments. As N increases they oscillate, with continually decreas- ing amplitude, about the limiting values that would be obtained with an infinite number of observations. Hence the form of a curve, having y's for ordinates and corresponding r's for abscissae, depends on the accuracy of the measurements considered and is sensibly independent of N, provided it is a large number. 27. The Unit Error. The relative accuracy of different series of measurements might be studied with the aid of the corre- sponding y : r curves, but since the y's are abstract numbers, and the r's are concrete, being of the same kind as the measurements, it is better to adopt a slightly different mode of representation. For this purpose, each of the r's is divided by an arbitrary con- stant k, of the same kind as the measurements, and the abstract numbers y^> -^> etc., are used as abscissae in place of the r's. In A/ K the following pages, k will be called the unit error. Its magnitude may be arbitrarily chosen in particular cases, but, when not definitely specified to the contrary, it will be taken equal to the least magnitude that can be directly observed with the instru- ments and methods used in making the measurements. To 32 THE THEORY OF MEASUREMENTS I ART. 28 illustrate: suppose we are measuring a given length with a scale divided in millimeters. By estimation, the separate observations can be made to one-tenth of a millimeter. Hence, in this case we should take k equal to one-tenth of a millimeter. If the residuals are arranged in the order of increasing magni- tude, it is obvious that the successive differences TI r , r? TI etc., are all equal to k. Hence, if the most probable value of the measured quantity, x in equation (3), is taken to the same num- ber of significant figures as the individual measurements, all of the residuals are integral multiples of k and we have k k 28. The Probability Curve. The result of a study of the distribution of the residuals may be arranged as illustrated in the following table, where n is the number of residuals of magnitude r; y is the probability that a single residual, chosen at random, is of magnitude r; N is the total number of measurements, and k is the unit error. r n V r ~k -r p n' p ~N~ -P -n * w -1 "0 no N ri ni N +1 rp n p w +P M Plotting y against ^ we obtain 2 p discrete points as in Fig. 3. When N is large, these points, are somewhat symmetrically dis- tributed about a curve of the general form illustrated by the dotted line. If a larger number of observations is considered, ART. 29] THE LAW OF ACCIDENTAL ERRORS 33 some of the points will be shifted upward while others will be shifted downward, but the distribution will remain approxi- mately symmetrical with respect to the same curve. In general, successive equal increments to N cause shifts of continually de- creasing magnitude; and in the limit, when TV becomes equal to infinity, and the residuals are equal to the accidental errors, the points would be on a uniform curve symmetrical to the y Q ordi- nate. The curve thus determined represents the relation between the magnitude of an error and the probability of its occurrence in a given series of measurements. For this reason it is called the probability curve. 29. Systems of Errors. The coordinates of the probability curve are y and-r-, since it represents the distribution of the true accidental errors AI, A 2 , etc., in regard to relative frequency and magnitude. Since the curve is uniform, it represents not only the errors of the actual observations, but also the distribution of all of the accidental errors that would be found if the sensitive- ness of our instruments were infinitely increased and an infinite number of observations were made, provided only that all of the observations were made with the same degree of precision and entirely independently. All of the errors represented by a curve of this type belong to a definite system, characterized by the magnitude of the maximum ordinate yo and the slope of the curve. Hence, every probability curve represents a definite system of errors. It also represents the accidental errors of a series of measurements of definite pre- cision. Hence, the accidental errors of series of measurements of different precision belong to different systems, and each series is characterized by a definite system of errors. The probability curves A and B in Fig. 4 represent the systems 34 THE THEORY OF MEASUREMENTS [ART. 30 of errors that characterize two series of measurements of different precision. As the precision of measurement is increased it is obvious that the number of small errors will increase relatively to the number of large ones. Consequently the probability of small errors will be greater and that of large ones will be less in the more precise series A than in the less precise series B. Hence, the curve A has a greater maximum ordinate and slopes more rapidly toward the horizontal axis than the curve B. 30. The Probability Function. The maximum ordinate and the slope of the probability curve depend on the constants that appear in the equation of the curve. When we know the form of the equation and have a method of determining the numerical value of the constants, we are able to determine the relative pre- cision of different series of measurements. Since the curve repre- sents the distribution of the true accidental errors, we are also able to compare the distribution of these errors with that of the resid- uals and thus develop workable methods for finding the most probable numeric of the measured magnitude. It is obvious, from an inspection of Figs. 3 and 4, that y is a continuous function of A, decreasing very rapidly as the magni- tude of A increases either positively or negatively and symmetrical with respect to the y axis. Hence, the probability curve sug- gests an equation in the form (9) ART. 31] THE LAW OF ACCIDENTAL ERRORS 35 where e is the base of the Napierian system of logarithms, o> is a constant depending on the precision of the series of measurements considered, and the other variables have been defined above. This equation can be derived analytically from the three axioms of accidental errors, with the aid of several plausible assumptions regarding the constitution of such errors, or from the principle of the arithmetical mean. However, the strongest evidence of its exactness lies in the fact that it gives results in substantial agreement with experience. Consequently, we will adopt it as an empirical relation, and proceed to show that it is in conformity with the three axioms and leads to the arithmetical mean as the most probable numeric derivable from a series of equally good independent measurements of the same magnitude. Equation (9) is the mathematical expression of the law of accidental errors and is often referred to simply as the law of errors. Its right-hand member is called the probability function and, for the sake of convenience, is represented by (A), giving the relations 2/ = 0(A); ^(A)^' 2 ^. (10) 31. The Precision Constant. The curves in Fig. 4 were plotted, to the same scale, from data computed by equation (9). The constant w was taken twice as great for the curve A as for the curve B, and in both cases values of y were computed for suc- cessive integral values of the ratio r-- The maximum ordinate of each of these curves corresponds to the zero value of A and is equal to the value of co used in computing the y's. The curve A, corresponding to the larger value of o>, approaches the hori- zontal axis much more rapidly than the curve B. Obviously, the constant co determines both the maximum ordinate and the slope of the probability curve. But we have seen that these characteristics are proportional to the precision of the measurements that determine the system of errors repre- sented. Hence co characterizes the system of errors consid- ered and is proportional to the precision of the corresponding measurements. Some writers have called it the precision measure, but, as it depends only on the accidental errors and takes no account of the accuracy with which constant errors are avoided or corrected, it does not give a complete statement of the pre- 36 THE THEORY OF MEASUREMENTS [ART. 32 cision. Consequently the term " precision measure " will be re- served for a function to be discussed later, and a; will be called the precision constant in the following pages. When A is taken equal to zero in equation (9), y is equal to co. Hence the precision of measurements, so far as it depends upon accidental errors, is proportional to the probability for the occur- rence of zero error in the corresponding system of errors. In this connection, it should be borne in mind that the system of errors includes all of the errors that would have been found with an infinite number of observations, and that it cannot be restricted to the errors of the actual measurements for the pur- pose of computing o> directly. Indirect methods for computing a> from given observations will be discussed later. 32. Discussion of the Probability Function. Inspection of the curves in Fig. 4, in connection with equation (9), is sufficient to show that the probability function is in agreement with the first two axioms. Since y is an even function of A, positive and nega- tive errors of the same magnitude are equally probable, and conse- quently equally numerous in an extended series of measurements. Hence the first axiom is fulfilled. Since A enters the function only in the negative exponent, the probability for the occurrence of an error decreases very rapidly as its magnitude increases either positively or negatively. Hence small errors are much more likely to occur than large ones and the second axiom is fulfilled. Since the function </> (A) is continuous for values of A ranging from minus infinity to plus infinity, it is apparently at variance with the third axiom. For, if all of the errors lie between definite finite limits L and + L, (A) should be continuous while A lies between these limits and equal to zero for all values of A outside of them. But we have no means of fixing the limits -f- L and L, in any given case; and we note that 0(A) becomes very small for moderately large values of A. Hence, whatever the true value of L may be, the error involved in extending the limits to oo and +00 is infinitesimal. Consequently, </>(A) is in sub- stantial agreement with the third axiom provided it leads to the conclusion that all possible errors lie between the limits oo and + oo . This will be the case if it gives unity for the probability that a single error, chosen at random, lies between oo and -f oo . For, if all of the errors lie between these limits, the probability considered is a certainty and hence is represented by unity. ART. 33] THE LAW OF ACCIDENTAL ERRORS 37 33. The Probability Integral. The accidental errors, corre- sponding to actual measurements, may be arranged in groups ac- cording to their magnitude in the same manner that the residuals were arranged in article twenty-eight. When this is done the errors in succeeding groups differ in magnitude by an amount equal to the unit error k t since k is the least difference that can be determined with the instruments used in making the obser- vations. Hence, if A p is the common magnitude of the errors in the pth group, -A = A (P+2) -A (p+i) or, expressing the same relation in different form, where a- is an indeterminate quantity that enters each of the equations because we do not know the actual magnitude of the A's. FIG. 5. Let the probability curve in Fig. 5 represent the system of errors to which the errors of the actual measurements belong. Then the ordinates y p , 2/( p +i), 2/( P +2), 2/(p+a) represent the probabilities of the errors A p , A( p +i>, . . . A( p + e ) respectively. Since the errors of the actual measurements satisfy the relation (i), none of them correspond to points of the curve lying between the ordinates y p , 2/( P + i), . . . 2/( P +). Hence, in virtue of equa- tion (6), article twenty-three, if we choose one of the measure- ments at random the probability that the magnitude of its error lies between A p and A( P + Q ) is 2/CP+8)- 38 THE THEORY OF MEASUREMENTS [ART. 33 Multiplying and dividing the second member by q, where y pq is written for the mean of the ordinates between y p and 2/(p+ fl ). From equation (i) & Hence, In the limit, when we consider the errors of an infinite number of measurements made with infinitely sensitive instruments, every point of the curve represents the probability of one of the errors of the system. Consequently, for any finite value of q, Ihe inter- val between the ordinates y p and y( P +q> is infinitesimal, and all of the ordinates between these limits may be considered equal. Hence, in the limit, p = , y pq = 2/ A = and (iii) reduces to =* (A) , (11) where y% +d * represents the probability that the magnitude of a single error, chosen at random, is between A and A + dA. By applying the usual reasoning of the integral calculus, it is evident that the expression rf = I /% (A) JA, (12) /t i/ a represents the probability that the magnitude of an error, chosen at random, lies between the limits a and b. The integral in this expression also represents the area under the probability curve between the ordinates at T and T. Consequently the probability in question is represented graphically by the shaded area in Fig. 6. The probability that an error, chosen at random, is numerically less than a given error A is equal to the probability that it lies ART. 33] THE LAW OF ACCIDENTAL ERRORS 39 between the limits A and -J-A. Hence, if we designate this probability by PA, A A since (A) is an even function of A. Introducing the complete expression for (A) from equation (10) we obtain A 2 k jo For the sake of simplification, put 2 A 2 then /Y'ett, Jo (13) which is an entirely general expression for the probability PA, applicable to any system of errors when we know the correspond- ing values of the constants o> and k. A series of numerical values of the right-hand member of (13), corresponding to successive values of the argument t, is given in Table XI, at the end of this volume. Obviously, this table may be used in computing the probability PA corresponding to any system of errors, since the characteristic constants o> and k appear only in the limit of the integral. Whatever the values of the constants w and k, the limit vVw T 40 THE THEORY OF MEASUREMENTS [ART. 34 becomes infinite when A is equal to infinity. Hence, in every system of errors, * dt = l ) (13a) where the numerical value is that given in Table XI, for the limit t equals infinity. Consequently the probability function (A) leads to the conclusion that all of the errors in any system lie between the limits <x> and +00, and, therefore, it fulfills the condition imposed by the third axiom as explained in the last paragraph of article thirty-two. 34. Comparison of Theory and Experience. Equation (13) may be used to compare the distribution of the residuals actually found in any series of measurements with the theoretical distri- bution of the accidental errors. If N equally trustworthy meas- urements of the same magnitude have been made, all of the N corresponding accidental errors belong to the same system, and the probability that the error of a single measurement is numer- ically less than A is given by PA in equation (13). Consequently, if N is sufficiently large, we should expect to find # A = NP* (iv) errors less than A. For, if we consider only the errors of the actual measurements, the probability that one of them is less than A is equal to the ratio of the number less than A to the total number. In the same manner, the number less than A 7 should be Hence, the number lying between the limits A and A' should be N* = N* - N*. (v) These numbers may be computed by equation (13) with the aid of Table XI, when we know N and the value of the expression V^co corresponding to the given measurements. The number, N r r , of residuals lying between the limits r equals A and r' equals A' may be found by inspecting the series of residuals computed from the given measurements by equation (3), article twenty-two. If N is large and the errors of the given measurements satisfy the theory we have developed, the numbers N% and N r r ' should ART. 34] THE LAW OF ACCIDENTAL ERRORS 41 be very nearly equal, since in an extended series of measurements the residuals are very nearly equal to the accidental errors. The following illustration, taken from Chauvenet's "Manual of Spherical and Practical Astronomy," is based on 470 obser- vations of the right ascension of Sirius and Altair, by Bradley. The errors of these measurements belong to a system character- ized by a particular value of the ratio T that has been computed, by a method to be described later (articles thirty-eight and forty- two), and gives the relation VTTCO k = 1.8086. Consequently, to find the theoretical value of PA, corresponding to any limit A, we take t equal to 1.8086 A in equation (13) and find the corresponding value of the integral by interpolation from Table XL The third column of the following table gives the values of PA corresponding to the chosen values of A in the first column and the computed values of t in the second column. The fourth column gives the corresponding values of N&. computed by equa- tion (iv), taking N equal to 470. The sixth column, computed by equation (v), gives the number, Nj[, of errors that should lie between the limits A and A' given in the fifth. The seventh column gives the number of residuals actually found between the same limits. A t ^A ^A Limits A A' < N r // 0.1 0.1809 0.2019 95 0.0-0.1 95 94 0.2 0.3617 0.3910 184 0.1-0.2 89 88 0.3 0.5426 0.5571 262 0.2-0.3 78 78 0.4 0.7234 0.6937 326 0.3-0.4 64 58 0.5 0.9043 0.7990 376 0.4-0.5 50 51 0.6 1.0852 0.8751 411 0.5-0.6 35 36 0.7 1.2660 0.9266 436 0.6-0.7 " 25 26 0.8 1.4469 0.9593 451 0.7-0.8 15 14 0.9 1.6277 0.9787 460 0.8-0.9 9 10 1.0 1.8086 0.9895 465 0.9-1.0 5 7 00 GO 1.0000 470 l.O-oo 5 8 Comparison of the numbers in the last two columns shows very good agreement between theory, represented by N%, and expe- 42 THE THEORY OF MEASUREMENTS [ART. 35 rience, represented by N r r f , when we remember that the theory assumes an infinite number of observations and that the series considered is finite. Numerous comparisons of this nature have been made, and substantial agreement has been found in all cases in which a sufficient number of independent observations have been considered. In general, the differences between N% and N^' decrease in relative magnitude as the number of obser- vations is increased. 35. The Arithmetical Mean. In article twenty-four it was pointed out, as one of the fundamental principles of the theory of errors, that the arithmetical mean of a number of equally trust- wor^hy direct measurements on the same magnitude is the most probable value that we can assign to the numeric of the measured magnitude. In order to show that the probability function (A) leads to the same conclusion, let eft, a 2 , . AT represent the given measurements, and let x represent the unknown numeric of the measured magnitude. If the actual value of this numeric is X, the true accidental errors of the given measurements are Ai = ai X, A 2 = 02 X, . . . AAT = ax X, (2) and all of them belong to the same system, characterized by a particular value .of the precision constant co. The probability that one of the errors of this system, chosen at random, is equal to an arbitrary magnitude A p is given by the relation Since we cannot determine the true value X, the most probable value that we can assign to x is that which gives a maximum probability that N errors of the system are equal to the N resid- uals TI = ai x, r z = a 2 x, . . . r N = a N x. (3) This is equivalent to determining x, so that the residuals are as nearly as possible equal to the accidental errors. If 2/1, 2/2, ... VN represent the probabilities that a single error of the system, chosen at random, is equal to r\, r 2 , . . . r N respec- tively, 2/i = (n), 2/2 = (r 2 ), . . . y N = Hence, if P is the probability that N of the errors chosen together ART. 35] THE LAW OF ACCIDENTAL ERRORS 43 are equal to n, r 2 , . . . r N respectively, we have, by equation (7), article twenty-three, P = 2/1 X 2/2 X ... X y N Since the exponent in this expression is negative and -^ is con- K stant, the maximum value of P will correspond to the minimum value of (ri 2 + r 2 2 + . . . -f ?W 2 ). Hence the most probable value of x is that which renders the sum of the squares of the residuals a minimum. In the present case, the r's are functions of a single independent variable x. Consequently the sum of the squares of the r's will be a minimum when x satisfies the condition -f-(ri 2 + r 2 2 + . ... +/VO =0. (JJU Substituting the expression for the r's in terms of x from equation (3) this becomes (a, - xY + (a 2 - xY + . . . + (a* - z) 2 = 0. dx( ) Hence, (i - x) + (a 2 - x) + . . . + (a N - x) = 0, (14) ai -f 2 + + AT and x = jy- Consequently, if we take x equal to the arithmetical mean of the a's in (3), the sum of the squares of the computed r's is less than for any other value of x. Hence the probability P that N errors of the system are equal to the N residuals is a maximum, and the arithmetical mean is the most probable value that we can assign to the numeric X on the basis of the given measurements. Equation (14) shows that the sum of the residuals, obtained by subtracting the arithmetical mean from each of the given measurements, is equal to zero. This is a characteristic property of the arithmetical mean and serves as a useful check on the computation of the residuals. The argument of the present article should be regarded as a justification of the probability function 0(A) rather than as a proof of the principle of the arithmetical mean. As pointed out above, this principle is sufficiently established on a priori grounds and by common consent. CHAPTER V. CHARACTERISTIC ERRORS. SEVERAL different derived errors have been used as a measure of the relative accuracy of different series of measurements. Such errors are called characteristic errors of the system, and they de- crease in magnitude as the accuracy of the measurements, on which they depend, increases. Those most commonly employed are the average error A , the mean error M, and the probable error E, any one of which may be used as a measure on the relative accuracy of a single observation. 36. The Average Error. The average error A of a single observation is the arithmetical mean of all of the individual errors of the system taken without regard to sign. That is, all of the errors are taken as positive in forming the average. Hence, if N is the total number of errors, ! _ ~N~ "W where the square bracket [ ] is used as a sign of summation, and the ~~ over the A indicates that, in taking the sum, all of the A's are to be considered positive. In accordance with the usual practice of writers on the theory of errors, the square bracket [ ] will be used as a sign of summa- tion, in the following pages, in place of the customary sign S. This notation is adopted because it saves space and renders com- plicated expressions more explicit. In equation (15) all of the errors of the system are supposed to be included in the summation. Hence, both [A] and N are infinite and the equation cannot be applied to find A directly from the errors of a limited number of measurements. Conse- quently we will proceed to show how the average error can be derived from the probability function, and to find its relation to the precision constant co. A little later we shall see how A can be computed directly from the residuals corresponding to a limited number of measurements. 44 ART. 36] CHARACTERISTIC ERRORS 45 If yd is the probability that the magnitude of a single error, chosen at random, lies between A and A + dA, and rid is the num- ber of errors between these limits, and consequently n d = Ny d = N4> (A) ^ (16) in virtue of equation (11), article thirty-three, where A represents the mean magnitude of the errors lying between A and A + dA. Hence, the sum of the errors between these limits is and the sum of the errors between A = a and A = b is N Substituting the complete expression for </>(A) from equation (10) this becomes Hence, the sum of the positive errors of the system is Nu / -*, -; I Ae kz dA, k Jo and the sum of the negative errors is Nu r k J -<* These two integrals are obviously equal in magnitude and opposite in sign. Consequently the sum of all of the errors of the system taken without regard to sign is Ae -^A (17) 7TCO 46 THE THEORY OF MEASUREMENTS [ART. 37 Hence from equation (15), ~ N and introducing the numerical value of IT, A =0.3183-- (19) CO 37. The Mean Error. The mean error M of a single meas- urement in a given series is the square root of the mean of the squares of the errors in the system determined by the given measurements. Expressed mathematically A^ + A^-f-.* + A^_[A1 N ' N This equation includes all of the errors that belong to the given system. Hence, as pointed out in article thirty-six, in regard to equation (15), it cannot be applied directly to a limited series of measurements. By equation (16) the number of errors with magnitudes between the limits A and A + dA is equal to , . Consequently /c the sum of the squares of the errors between these limits is equal #A 2 4>(A)dA k in the last article, to - .; . Hence, by reasoning similar to that employed (21) / / 2N r A% -, * since the integrand is an even function of A. Integrating by parts, 7TCO The first term of the second member of this equation reduces to AKT.38] CHARACTERISTIC ERRORS 47 zero when the limits are applied. Putting t 2 for in the K second term, [Al-^P^a- (22) TT^CO 2 Jo 2 7TC0 2 in virtue of equation (13a). Hence, N 2* and M = = 0.3989-- CO (23) 38. The Probable Error. The probable error E of a single measurement is a magnitude such that a single error, chosen at random from the given system, is as likely to be numerically greater than E as less than E. In other words, the probability that the error of a single measurement is greater than E is equal to the probability that it is less than E. Hence, in any extended series of measurements, one-half of the errors are less than E and one-half of them are greater than E. The name " probable error," though sanctioned by universal usage, is unfortunate; and the student cannot be too strongly cautioned against a common misinterpretation of its meaning. The probable error is NOT the most probable magnitude of the error of a single measurement and it DOES NOT determine the limits within which the true numeric of the measured magnitude may be expected to lie. Thus, if x represents the measured numeric of a given magnitude Q and E is the probable error of x, it is customary to express the result of the measurement in the form Q = x E. This does not signify that the true numeric of Q lies between the limits x E and x + E, neither does it imply that x is probably in error by the amount E. It means that the numeric of Q is as likely to lie between the above limits as outside of them. If a new measurement is made "by the same method and with equal care, the probability that it will differ from x by less than E is equal to the probability that it will differ by more than E. 48 THE THEORY OF MEASUREMENTS [ART. 38 In article thirty-three it was pointed out that the probability that an error, chosen at random from a given system, lies between the limits A = a and A = b is represented by the area under the probability curve between the ordinates corresponding to the limiting values of A. Hence, the probability that the error of a single measurement is numerically less than E may be represented by the area under the probability curve between the ordinates y- E and y+ E , in Fig. 7, and the probability that it is greater than E by the sum of the areas outside of these ordinates. Since these two FIG. 7. probabilities are equal, by definition, the ordinates correspond- ing to the probable error bisect the areas under the two branches of the probability curve. Since the probability that the error of a single measurement is less than E is equal to the probability that it is greater than E and the probability that it is less than infinity is unity, the probability that it is less than E is one-half. Consequently, putting A equal to E in equation (13), article thirty-three, Pw = ~ rw T" 1 e-dt - 2- \J From Table XI, PA = 0.49375 for the limit t = 0.47, PA = 0.50275 for the limit t = 0.48, and by interpolation, P E = 0.50000 for the limit t = 0.47694. Hence, equation (24) is satisfied when (24) = 0.47694, ART. 39] and we have CHARACTERISTIC ERRORS E 0.47694 k VTT w = 0.2691 - CO 49 (25) 39. Relations between the Characteristic Errors. Elimina- k ting- from equations (18), (23), and (25), taken two at a time, we obtain the relations (26") E = 0.4769 VTT -A = 0.8453 -A, E = 0.4769 V2 M = 0.6745 M,. which express the relative magnitudes of the average, mean, and probable errors. These relations are universally adopted in com- MAE k k k FIG. 8. puting the precision of given series of measurements, and they should be firmly fixed in mind. The three equations from which the relations (26) are derived may be put in the form A = 0.3183 k co M _ 0.3989 k co E = 0.2691 k co The probability curve in Fig. 8 represents the distribution of the errors in a system characterized by a particular value of co, (27) 50 THE THEORY OF MEASUREMENTS [ART. 39 determined by a given series of measurements. The ordinates AM A E VA> VM> an d Us correspond to the abscissae -^> -jp and -"& > com " puted by the above equations. Consequently, y A represents the probability that the error of a single measurement is equal to +A, y M the probability that it is equal to +M, and y E the prob- ability that it is equal to +E. In like manner y- A , y- M , and y~ E represent the respective probabilities for the occurrence of errors equal to A, M, and E. A curve of this type can be constructed to correspond to any given series of measurements, and in all cases the relative loca- tion of the ordinates y A , y M) and y E will be the same. It was pointed out in the last article that the ordinates y E and y- E bisect the areas under the two branches of the curve. Consequently, in an extended series of measurements, somewhat more than one- half of the errors will be less than either the average or the mean error. Moreover, it is obvious from Fig. 8 that an error equal to E is somewhat more likely to occur than one equal to either A or M. Since each of the characteristic errors A, M, and E, bears a constant relation to the precision constant co, any one of them might be used as a measure of the precision of a single measure- ment in a given series, so far as this depends on accidental errors. The probable error is more commonly employed for this purpose on account of its median position in the system of errors deter- mined by the given measurements. It is interesting to observe that the ordinate y M corresponds to a point of inflection in the probability curve. By the ordinary method of the calculus we know that this curve has a point of inflection corresponding to the abscissa that satisfies the relation Substituting the complete expression for y Hence, ART. 40] CHARACTERISTIC ERRORS 51 is the abscissa of the point of inflection. Comparing this with equation (23) we see that and consequently that the ordinates y M and y- M meet the prob- ability curve at points of inflection. 40. Characteristic Errors of the Arithmetical Mean. Equa- tion (23) may be put in the form CO 2 1 where M is the mean error of a single measurement in a series corresponding to the unit error k and the precision constant w. Consequently the probability function, "***& y = we k y corresponding to the same series may be put in the form y = ae 2M *. (i) If A i, A 2 , . . . AJV are the accidental errors of N direct measure- ments in the same series, the probability P that they all occur in a system characterized by the mean error M is equal to the product of the probabilities for the occurrence of the individual errors in that system. Hence, If the individual measurements are represented by a\ t 0,2, . . . a N , and the true numeric of the measured quantity is X, Ai = ai - X; A 2 = a z - X\ . . . A# = a N - X, and, if x is the arithmetical mean of the measurements, the corre- sponding residuals are n = ai x', r z = 2 x; . . . r N = a N x. Consequently, if the error of the arithmetical mean is 5, X - x = 5, and Ai = n - 5; A 2 = r 2 - 5; . . . A# = r N 8. Squaring and adding, [A 2 ] = [r 2 ]-25M+ATS 2 ; (28) 52 THE THEORY OF MEASUREMENTS [ART. 40 since [r] Is equal to zero in virtue of equation (14), article thirty- five. When this value of [A 2 ] is substituted in (ii), the resulting value of P is the probability that the arithmetical mean is in error by an amount 6. For, as we have seen in article thirty-five, the minimum value of [r 2 ] occurs when x is taken equal to the arithmetical mean. Consequently, P is a maximum when <5 is equal to zero and decreases in accordance with the probability function as 5 increases either positively or negatively. We do not know the exact value of either X or 5; but, if y a is the probability that the error of the arithmetical mean is equal to an arbitrary magnitude 5, the foregoing reasoning leads to the relation 2M2 But the arithmetical mean is equivalent to a single measurement in a series of much greater precision than that of the given meas- urements. Hence, if o> a is the precision constant correspond- ing to this hypothetical series and M a is the mean error of the arithmetical mean, we have by analogy with (i) a* y a = w a e 2 M 2 . (iv) Equations (iii) and (iv) are two expressions for the same prob- ability and should give equal values to y a whatever the assumed value of 5. This is possible only when 2M , and 1 N ~ 2M 2 Hence, M M M a = = VN Consequently, the mean % error of the arithmetical mean is equal to the mean error of a single measurement divided by the square root of the number of measurements. Since the average, mean, and probable errors of a single meas- urement are connected by the relations (26), the corresponding Art. 41] CHARACTERISTIC ERRORS 53 errors of the arithmetical mean, distinguished by th.e subscript a, are given by the relations 4 = -4=; M a = -^=; E a = -?j=. (29) VN VN VN 41. Practical Computation of Characteristic* Errors. As pointed out in article thirty-seven, the square of the mean error [A 2 1 M is the limiting value of the ratio ^rp when both members become infinite, i.e., when all of the errors of the given system are considered. But the errors of the actual measurements fall into groups, as explained in article thirty-three, and the errors in succeeding groups differ in magnitude by a constant amount k, depending on the nature of the instruments used in making the observations. Consequently, the ordinates, of the probability curve, corresponding to these errors are uniformly distributed along the horizontal axis. Hence, if we include in [A 2 ] only the errors of the actual measurements, the limiting value of the ratio fA 2 l L -^- when N is indefinitely increased will be nearly the same as if all of the errors of the system were included. Since the ratio approaches its limit very rapidly as N increases, the value of M can be determined, with sufficient precision for most practical purposes, from a somewhat limited series of measurements. If we knew the true accidental errors, the mean error could be computed at once from the relation (v) and, since the residuals are nearly equal to the accidental errors when N is very large, an approximate value can be obtained by using the r's in place of the A's. A better approximation can be obtained if we take account of the difference between the A's and the r's. From equation (28) [A 2 ] = [r 2 ] + AT5 2 , (vi) where 6 is the unknown error of the arithmetical mean. Probably the best approximation we can make to the true value of 8 is to set it equal to the mean error of the arithmetical mean. Hence, from the second of equations (29) 54 THE THEORY OF MEASUREMENTS [ART. 41 Consequently, (vi) becomes NM 2 = [r 2 ] + and we have (30) Thus the square of the mean error of a single measurement is equal to the sum of the squares of the residuals divided by the number of measurements less one. Combining (30) with the third of equations (26), article thirty- nine, we obtain the expression E = 0.6745 V^rj < 31 ) for the probable error of a single measurement. Hence, by equa- tions (29), the mean error M a and the probable error E a of the arithmetical mean are given by the relations and * = - (32) When the number of measurements is large, the computation of the probable errors E and E a by the above formulae is some- what tedious, owing to the necessity of finding the" square of each of the residuals. In such cases a sufficiently close approx- imation for practical purposes can be derived from the average error A with the aid of equations (26). The first of these equa- tions may be written in the form [A3 = T [A] 2 N 2 N 2 ' If we assume that the distribution of the residuals is the same as that of the true accidental errors, a condition that is accurately fulfilled when N is very large, we can put N Consequently, ART. 41] CHARACTERISTIC ERRORS 55 When the mean error M is expressed in terms of the A's, equation (30) becomes [A 2 ]_ M N ' N-l' or [Ag = N [Sp. [r 2 ] tf- 1 [r]2 ' Consequently [A? [r? and, since this ratio is equal to A 2 , we have == and A = - X (33) -1) NVN-1 Combining this result with the second of equations (26) and the third of (29), we obtain E = 0.8453 . ^ ; E a = 0.8453 - ^ . (34) VN(N-1)' NVN-1 The above formulae for computing the characteristic errors from the residuals have been derived on the assumption that the true accidental errors and the residuals follow the same law of dis- tribution. This is strictly true only when the number of measure- ments considered is very large. Yet, for lack of a better method, it is customary to apply the foregoing formulas to the discussion of the errors of limited series of measurements and the results thus obtained are sufficiently accurate for most practical purposes. When the highest attainable precision is sought, the number of observations must be increased to such an extent that the theo- retical conditions are fulfilled. The choice between the formulae involving the average error A and those depending on the mean error M is determined largely by the number of measurements available and the amount of time that it is worth while to devote to the computations. When the number of measurements is very large, both sets of formulae lead to the same values for the probable errors E and E a , and much time is saved by employing those depending on A. For limited series of observations a better approximation to the true values of these errors is obtained by employing the formulae in- volving the mean error. In either case the computation may be 56 THE THEORY OF MEASUREMENTS [ART. 42 facilitated by the use of Tables XIV and XV at the end of this volume. These tables give the values of the functions 0.6745 0.8453 0.8453 0.6745 VN(N-1)' and NVN-l' corresponding to all integral values of N between two and one hundred. 42. Numerical Example. The following example, represent- ing a series of observations taken for the purpose of calibrating the screw of a micrometer microscope, will serve to illustrate the practical application of the foregoing methods. Twenty inde- pendent measurements of the normal -distance between two parallel lines, expressed in terms of the divisions of the micrometer head, are given in the first and fourth columns of the following table under a. a r r i a r r 2 194.7 +0.53 0.2809 194.3 +0.13 0.0169 194.1 -0.07 0.0049 194.3 +0.13 0.0169 194.3 +0.13 0.0169 194.0 -0.17 0.0289 194.0 -0.17 0.0289 194.4 +0.23 0.0529 193.7 -0.47 0.2209 194.5 +0.33 0.1089 194.1 -0.07 0.0049 193.8 -0.37 0.1369 193.9 -0.27 0.0729 193.9 -0.27 0.0729 194.3 +0.13 0.0169 193.9 -0.27 0.0729 194.3 +0.13 0.0169 194.8 +0.63 0.3969 194.4 +0.23 0.0529 193.7 -0.47 0.2209 194.17 5.20 1.8420 .r [r 2 ] Since the observations are independent and equally trust- worthy, the most probable value that we can assign to the numeric of the measured magnitude is the arithmetical mean x; and we find that x is equal to 194.17 micrometer divisions. Subtracting 194.17 from each of the given observations we obtain the residuals in the columns under r. The algebraic sum of these residuals is equal to zero as it should be, owing to the properties of the arith- metical mean. The sum without regard to sign, [r], is equal to 5.20. Squaring each of the residuals gives the numbers in the columns under r 2 and adding these figures gives 1.8920 for the sum of the squares of the residual [r 2 ]. Taking N equal to twenty, in formulae (33) and (34), we find the average and probable errors ART. 42] CHARACTERISTIC ERRORS 57 = =b 0.267; A a = Ar ^ = 0.0596, NVN-l E = 0.8453 7== = 0.226; # = 0.8453 ^-^ = = 0.0504, where the numerical results are written with the indefinite sign since the corresponding errors are as likely to be positive as nega- tive. When formulae (30), (31), and (32) are employed we obtain the mean errors, and the probable errors E = 0.6745 The values of the probable errors E and jEk, computed by the two methods, agree as closely as could be expected with so small a number of observations. Probably the values d= 0.210 and 0.047, computed from the mean errors M and M a , are the more accurate, but those derived from the average errors A and A a are sufficiently exact for most practical purposes. An inspection of the column of residuals is sufficient to show that eleven of them are numerically greater, and nine are numerically less than either of the computed values of E. Consequently, both of these values fulfill the fundamental definition of the probable error of a single measurement as nearly as we ought to expect when only twenty observations are considered. If we use D to represent the measured distance between the parallel lines, in terms of micrometer divisions, we may write the final result of the measurements in the form D = 194.170 =t 0.047 mic. div. This does not mean that the true value of D lies between the specified limits, but that it is equally likely to lie between these limits or outside of them. Thus, if another and independent series of twenty measurements of the same distance were made 58 THE THEORY OF MEASUREMENTS [ART. 43 with the same instrument, and with equal care, the chance that the final result would lie between 194.123 and 194.217 is equal to the chance that it would lie outside of these limits. Equation (25), article thirty-eight, may be written in the form -co 0.4769 Taking E equal to 0.210, we find that v = 2.271 k for the particular system of errors determined by the above meas- urements. Consequently, the probability for the occurrence of an error less than A in this system is, by equation (13), article thirty- three, 2.271.A and, since there are twenty measurements, we should expect to find 20 PA errors numerically less than any assigned value of A. The values of PA, corresponding to various assigned values of A, can be easily computed with the aid of Table XI and applied, as explained in article thirty-four, to compare the theoretical distribution of the accidental errors with that of the residuals given under r in the above table. Such a comparison would have very little significance in the present case, however it resulted, since the number of observations considered is far too small to fulfill the theoretical requirements. But it would show that, even in such extreme cases, the deviations from the law of errors are not greater than might be expected. The actual comparison is left as an exercise for the student. 43. Rules for the Use of Significant Figures. The funda- mental principles underlying the use of significant figures were explained in article fifteen. General rules for their practical ap- plication may be stated in terms of the probable error as follows: All measured quantities should be so expressed that the last recorded significant figure occupies the place corresponding to the second significant figure in the probable error of the quantity considered. The number of significant figures carried through the compu- ART. 43] CHARACTERISTIC ERRORS 59 tations should be sufficient to give the final result within one unit in the last place retained and no more. For practical purposes probable errors should be computed to two significant figures. The example given in the preceding article will serve to illus- trate the application of these rules. The second significant figure in the probable error of the arithmetical mean occupies the third decimal place. Consequently, the final result is carried to three decimal places, notwithstanding the fact that the last place is occupied by a zero. It would obviously be useless to carry out the result farther than this, since the probable error shows that the digit in the second decimal place is equally likely to be in error by more or less than .five units. If less significant figures were used, the fifth figure in computed results might be vitiated by more than one unit. In order to apply the rules to the individual measurements, it is necessary to make a preliminary series of observations, under as nearly as possible the same conditions that will prevail during the final measurements, and compute the probable error of a single observation from the data thus obtained. Then, if possible, all final measurements should be recorded to the second significant figure in this probable error and no farther. It sometimes happens, as in the above example, that the graduation of the measuring instruments used is not sufficiently fine to permit the attainment of the number of significant figures required by the rule. In such cases the observations are recorded to the last attainable figure, .or, if possible, the instruments are so modified that they give the required number of figures. Thus, in the example cited, the second significant figure in the probable error of a single measure- ment is in the second decimal place, but the micrometer can be read only to one-tenth of a division. Hence the individual measurements are recorded to the first instead of the second decimal place. In this case the accuracy attained in making the settings of the instrument was greater than that attained in making the readings, and an observer, with sufficient experience, would be justified in estimating the fractional parts to the nearest hundredth of a division. A better plan would be to provide the micrometer head with a vernier reading to tenths or hundredths of a division. In the opposite case, when the accuracy of setting is less than the attainable accuracy of reading, it is useless to record 60 THE THEORY OF MEASUREMENTS [ART. 43 the readings beyond the second significant figure in the probable error of a single observation. For the purpose of computing the residuals, the arithmetical mean should be rounded to such an extent that the majority of the residuals will come out with two significant figures. This greatly reduces the labor of the computations and gives the calcu- lated characteristic errors within one unit in the second significant figure. CHAPTER VI. MEASUREMENTS OF UNEQUAL PRECISION. 44. Weights of Measurements. In the preceding chapter we have been dealing with measurements of equal precision, and the results obtained have been derived on the supposition that there was no reason to assume that any one of the observations was better than any other. Under these conditions we have seen that the most probable value that we can assign to the numeric of the measured magnitude is the arithmetical mean of the individual observations. Also, if M and E are the mean and probable errors of a single observation, M a and E a the mean and probable errors of the arithmetical mean, and A/" the number of observations, we have the relations # = 0.6745 M; ' E a = 0.6745 M n , M E v (35) The true numeric X of the measured magnitude cannot be exactly determined from the given observations, but the final result of the measurements may be expressed in the form X = x E a , which signifies that X is as likely to lie between the specified limits as outside of them. Now suppose that the results of m independent series of meas- urements of the same magnitude, made by the same or different methods, are given in the form X = xi E lt X = x% it EZ, X = x m d= E m . 61 62 THE THEORY OF MEASUREMENTS [ART. 44 What is the most probable value that can be assigned to X on the basis of these results? Obviously, the arithmetical mean of the x's will not do in this case, unless the E's are all equal, since the x's violate the condition on which the principle of the arithmetical mean is founded. If we knew the individual observations from which each of the x's were derived, and if the probable error of a single observation was the same in each of the series, the most probable value of X would be given by the arithmetical mean of all of the individual observations. Generally we do not have the original observations, and, when we do, it frequently happens that the probable error of a single observation is different in the differ- ent series. Consequently the direct method is seldom applicable. The E's may differ on account of differences in the number of observations in the several series, or from the fact that the prob- able error of a single observation is not the same in all of them, or from both of these causes. Whatever the cause of the difference, it is generally necessary to reduce the given results to a series of equivalent observations having the same probable error before taking the mean. For it is obvious that a result showing a small probable error should count for more, or have greater weight, in determining the value of X than one- that corresponds to a large probable error, since the former result has cost more in time and labor than the latter. The reduction to equivalent observations having the same probable error is accomplished as follows: m numerical quanti- ties wi, w 2 , . . . w m , called the weights of the quantities Xi, x 2 , . . . x m , are determined by the relations E* E a 2 E* W ^E?> W *=Ef'> ' ' Wm =E^' (36) where E a is an arbitrary quantity, generally so chosen that all of the w's are integers, or may be placed equal to the nearest integer without involving an error of more than one or two units in the second significant figure of any of the E's. In the following pages E 8 will be called the probable error of a standard observa- tion. Obviously, the weight of a standard observation is unity on the arbitrary scale adopted in determining, the w's; for, by equations (36), ART. 45] MEASUREMENTS OF UNEQUAL PRECISION 63 Such an observation is not assumed to have occurred in any of the series on which the x's depend, but is arbitrarily chosen as a basis for the computation of the weights of the given results. By comparing equations (35) and (36), we see that E\ is equal to the probable error of the arithmetical mean of w\ standard observations. But it is also the probable error of the given result XL Consequently x\ is equivalent to the arithmetical mean of wi standard observations. Similar reasoning can be applied to the other E's and in general we have Xi = mean of w\ standard observations, x 2 = mean of w 2 standard observations, x m = mean of w m standard observations. (i) The weights Wi, w 2} . . . w m are numbers that express the rela- tive importance of the given measurements for the determination of the most probable value of the numeric of the measured mag- nitude. Each weight represents the number of hypothetical standard observations that must be combined to give an arith- metical mean with a probable error equal to that of the given measurement. 45. The General Mean. From equations (i) it is obvious that = the sum of Wi standard observations, = the sum of w z standard observations, w m x m = the sum of w m standard observations, and, consequently, -f + w m x m is equal to the sum of w\ + ^2 + . . + W TO standard observa- tions. Since the probable error E 8 is common to all of the standard observations, they are equally trustworthy and their arithmetical mean is the most probable value that we can assign to the numeric X on the basis of the given data. Representing this value of X Q we have _ WiXi + W 2 X 2 + * + W m X m X Q( _V Wl+W2 + . . . + Wm The products W&1, etc., are called weighted observations or meas- 64 THE THEORY OF MEASUREMENTS [ART. 45 urements, and x is called the general or weighted mean. The weight W Q of X Q is obviously given by the relation wo = wi + w 2 + - + w m , (38) since X Q is the mean of w standard observations. Equation (37) for the general mean can be established inde- pendently from the law of accidental errors in the following manner: Let coi, o> 2 , . . . w m represent the precision constants correspond- ing to the probable errors EI, E z , E m , and let w s be an arbitrary quantity connected with the arbitrary quantity E 8 by the relation # 8 = 0.2691 - fc> Then, by equations (25) and (36), i 2 C0 2 2 CO TO 2 Wl = ~^> W2 = l^> *- IF- (39) If XQ is the most probable value of the numeric X, the residuals corresponding to the given aj's are ri = xi XQ', r 2 = x z XQ', . . . r m = x m x . The probability that the true accidental error of x\ is equal to r\ s in virtue of equations (39). Similarly, if 2/1, 2/2, Vm are the probabilities that r\, r 2 , . . . r m are the true accidental errors of x m} OJ.2 T-TT 2/2 = co 2 e Hence, if P is the probability that all of the r's are simultaneously equal to true accidental errors, we have w z -Tr-- P = (wio> 2 . . . ov)e and the most probable value of X is that which renders P a maximum. Obviously, the maximum value of P occurs when ART. 45] MEASUREMENTS OF UNEQUAL PRECISION 65 (wirf + w 2 r 2 2 + . . . + w m r m 2 ) is a minimum. Consequently the most probable value X Q is given by the relation ^T (wiri 2 + w 2 r 2 2 + + w m r m 2 ) = 0. Substituting the values of the r's and differentiating this becomes Wi (Xi XQ) + W 2 (X 2 XQ) + W m (x m XQ) = 0. Hence, WiXi + W 2 X 2 + + W m X m XQ ; : : j as given above. If we multiply or divide the numerator and denominator of equation (37) by any integral or fractional constant, the value of #o is unaltered. Hence, from (36), it is obvious that we are at liberty to choose any convenient value for E a) whether or not it gives integral values to the w's. Equations (36) also show that the weights of measurements are inversely proportional to the squares of their probable errors and consequently we may take #! 2 E? EJ w 2 = wi-^-', w 3 = w 1 ^-; . . . w m = wi-^-- (40) Etf 1 &m Hence, if we choose, we can assign any arbitrary weight to one of the given measurements and compute the weights of the others by equation (40). The foregoing methods for computing the weights w\, w 2 , etc., are applicable only when the given measurements x\, x 2 , etc., are entirely free from constant errors and mistakes. When this condition is not fulfilled the method breaks down because the errors of the x's do not follow the law of accidental errors. In such cases it is sometimes possible to assign weights to the given measurements by combining the given probable errors with an estimate of the probable value of the constant errors, based on a thorough study of the methods by which the x's were obtained. Such a procedure is always more or less arbitrary, and requires great care and experience, but when properly applied it leads to a closer approximation to the true numeric of the measured magni- tude than would be obtained by taking the simple arithmetical mean of the x's. Since it involves a knowledge of the laws of propagation of errors and of the methods for estimating the pre- 66 THE THEORY OF MEASUREMENTS [ART. 46 cision attained in removing constant errors and mistakes, it can- not be fully developed until we take up the study of the under- lying principles. 46. Probable Error of the General Mean. When the given x's are free from constant errors and the E's are known, the weights of the individual measurements are given by (36), and the weight W of the general mean is given by (38). Consequently, if E is the probable error of the general mean, we have by analogy with equations (36) 1*0=14 and #0=-- (41) If we choose, E may be expressed in terms of any one of the E's in place of E 8 . Thus, let E n and w n be the probable error and the weight of any one of the x's, then by (36) E > W and eliminating E a between this equation and (41) we have (42) When the weights are assigned by the method outlined in the last paragraph of the preceding article, or when, for any reason, the w's are given but not the E's, (41) and (42) cannot be applied until E a or E n has been derived from the given x's and w's. If the number of given measurements is large, the value of E 8 corre- sponding to the given weights can be computed with sufficient precision by the application of the law of errors as outlined below. If the number of given measurements is small, or if constant errors and mistakes have not been considered in assigning the weights, the following method gives only a rough approximation to the true value of E s , and consequently of E Q) since the condi- tions underlying the law of errors are not strictly fulfilled. It will be readily seen that while E 8 may be arbitrarily assigned for the purpose of computing the weights, when the E's are given, its value is fixed when the weights are given. Let xi, z 2 , . . . x m represent the given measurements and Wi, ^ 2 , ... w m , the corresponding weights. Then, if o? 8 repre- ART. 46] MEASUREMENTS OF UNEQUAL PRECISION 67 sents the precision constant of a standard observation, and wi that of an observation of weight w\, we have by (39) Consequently, if 2/ A is the probability that the error of x i is equal to A, and, by equation (11), article thirty-three, the probability that the error of x\ lies between the limits A and A + dA is Now, WiA 2 is the weigh ted square of the error A, and in the follow- ing pages the product VwA will be called a weighted error. Hence, if we put d = VwjA, and dd = Vw { dA, we have for the probability that the weighted error of Xi lies between the limits 5 and d -\- dd Since the same result would have been obtained if we had started with any other one of the x's and w's, it is obvious that this equa- tion expresses the probability that any one of the x's, chosen at random, is affected by a weighted error lying between the limits 5 and d + dd. But, if rid is the number of #'s affected by weighted errors lying between these limits, and m is the total number of as's, we have also or Hence, the sum of the squares of the weighted errors lying between 5 and 5 -f- dd is given by the relation S2 u s - TO- ,* , = m8 2 -re dS, = " m 68 THE THEORY OF MEASUREMENTS [A RT . 46 and, by the method adopted in articles thirty-six and thirty-seven, we have [g] = 2 a), r m A: Jo where [5 2 ] is supposed to include all possible weighted errors between the limits plus and minus infinity. Introducing the values of the S's in terms of the w's and A's this becomes m m which is an exact equation only when the number of measure- ments considered is practically infinite. If M 8 is the mean error of a standard observation, we have from equation (23) Hence, from equation (26) . = 0.6745 Now, we do not know the true value of the A's and the number of given measurements is seldom sufficiently large to fulfill the con- ditions underlying this equation. But we can compute the gen- eral mean X Q and the residuals Ti = Xi XQ] r 2 = X 2 XQ] . . . T m = X m X , and, by a method exactly analogous to that of article forty-one, it can be shown that the best approximation that we can make is given by the relation [wr 2 ] m m 1 Hence, as a practicable formula for computing E 8 , we have E a = 0.6745 V-T' (43) ~ m 1 and consequently E is given by the relation Eo = 0.6745V... r ,,' in virtue of equation (41). ART. 47] MEASUREMENTS OF UNEQUAL PRECISION 69 When the probable errors of the given measurements are known, and the weights are computed by equation (36), the value of E 8 computed by equation (43) will agree with the value arbi- trarily assigned, for the purpose of determining the w's, provided the x's are sufficiently numerous and free from constant errors and mistakes. The number of measurements considered is seldom sufficient to give exact agreement, but a large difference between the assigned and computed values of E 8 is strong evidence that constant errors have not been removed with sufficient pre- cision. On the other hand, satisfactory agreement may occur when all of the x's are affected by the same constant error. Con- sequently such agreement is not a criterion for the absence of constant errors, but only for their equality in the different meas- urements. 47. Numerical Example. As an illustration of the applica- tion of the foregoing principles, consider the micrometer measure- ments given under x in the following table. They represent the results of six series of measurements similar to that discussed in article forty-two, the last one being taken directly from that article. The probable errors, computed as in article forty-two, are given under E. They differ partly on account of differences in the number of observations in the several series, and partly from the fact that the individual observations were not of the same precision in all of the series. The squares of the probable errors multiplied by 10 4 are given under E 2 X 10 4 to the nearest digit in the last place retained. It would be useless to carry them out further as the weights are to be computed to only two signifi- cant figures. X E E* X 10* w ^5? w 194.03 0.066 44 11 0.066 193.79 0.12 144 3 0.127 194.15 0.091 83 6 0.090 193.85 0.11 121 4 0.110 194.22 0.099 98 5 0.098 194.17 0.047 22 22 0.047 Taking E a equal to 0.22 gives E 8 * X 10 4 equal to 484, and by applying equation (36), we obtain the weights given under w to the nearest integer. Inverting the process and computing the 70 THE THEORY OF MEASUREMENTS [ART. 47 E's from the assigned w's and E 8 gives the numbers in the last column of the table. Since these numbers agree with the given E's within less than two units in the second significant figure, we may assume that the approximation adopted in computing the w's is justified. If the agreement was less exact and any of the differences exceeded two units in the second significant figure, it would be necessary to compute the w's further, or, better, to adopt a different value for E 8 , such that the agreement would be suffi- cient with integral values of the w's. For the purpose of computation, equation (37) may be written in the form X Q = C + - C) + w, (x 2 - C) + W m (X m C) where C is any convenient number. In the present case 193 is chosen, and the products w (x 193) are given in the first column of the following table. w (x - 193) T r2 X 10< wr* X 10< 11.33 -0.065 42 462 2.37 -0.305 930 2790 6.90 +0.055 30 180 3.40 -0.245 600 2400 6.10 +0.125 156 780 25.74 +0.075 56 1232 55.84 7844 Substitution in the above equation for the general mean gives and this is the most probable value that we can assign to the numeric of the measured magnitude on the basis of the given measurements. By equation (38) the weight, w , of the general mean is 51. Hence equation (41) gives 0.22 /= V51 0.031 for the probable error of x . Selecting the first measurement ART. 47] MEASUREMENTS OF UNEQUAL PRECISION 71 since its weight corresponds exactly to its probable error, equa- tion (42) gives Eo = 0.066 i/ = 0.031. 51 If the second, third, or fifth measurement had been chosen, the results derived by the two formulae would not have been exactly alike; but the differences would amount to only a few units in the second significant figure, and consequently would be of no prac- tical importance. However, it is better to proceed as above and select a measurement whose weight corresponds exactly with its probable error as shown by the fifth column of the first table above. The residuals, computed by subtracting x from each of the given measurements, are given under r in the second table; and their squares multiplied by 10 4 are given, to the nearest digit in the last place retained, under r 2 X 10 4 . The last column of the table gives the weighted squares of the residuals multiplied by 10 4 . The sum, [wr 2 ], is equal to 0.784. Hence by equation (43) E 8 = 0.6745 1/ ' 784 = =t 0.27, o and by equation (44) JB, = 0.6745 J^- = 0.037. 51 X o These results agree with the assumed value of E 8 and the pre- viously computed value of E as well as could be expected when so small a number of measurements are considered. Conse- quently we are justified in assuming that the given measurements are either free from constant errors or all affected by the same constant error. In practice the second method of computing E Q is seldom used when the probable errors of the given measurements are known, since its value as an indication of the absence of constant errors is not sufficient to warrant the labor involved. When the prob- able errors of the given measurements are not known it is the only available method for computing EQ and it is carried out here for the sake of illustration. CHAPTER VII. THE METHOD OF LEAST SQUARES. 48. Fundamental Principles. Let Xi, X 2 , . . . X g , and FI, Y 2 , . . . Y n represent the true numerics of a number of quan- tities expressed in terms of a chosen system of units. Suppose that the quantities represented by the Y's have been directly measured and that we wish to determine the remaining quantities indirectly with the aid of the given relations YZ = FZ (Xl, Xz, . . . X q ), Y n = F n (Xi,Xz, . . X q ). (45) The functions FI, F 2 , . . . F n may be alike or different in form and any one of them may or may not contain all of the X's, but the exact form of each of them is supposed to be known. If the F's were known and the number of equations were equal to the number of unknowns, the X's could be derived at once by ordinary algebraic methods. The first condition is never ful- filled since direct measurements never give the true value of the numeric of the measured quantity. Let s i; s 2 , . . . s n represent the most probable values that can be assigned to the F's on the basis of the given measurements. If these values are substituted for the F's in (45), the equations will not be exactly fulfilled and consequently the true value of the X's cannot be determined. The differences Fi(Xi,X Z) . . . X q )-si = k Fz(Xi,Xz, . . . Xq)-s 2 = k *, . . . X q )-s n = A n (46) represent the true accidental errors of the s's. Let Xi, Xz, . . . x q represent the most probable values that we can assign to the X's on the basis of the given data. Then, since 72 ART. 48] THE METHOD OF LEAST SQUARES 73 the s's bear a similar relation to the Y's } equations (45) may be written in the form Fi (Xi, X 2) . . . X q ) = S b F 2 (xi, x 2) . . . x q ) = s 2} F n (xi, x 2} . . . x q ) = s n , (47) where the functions F i} F 2 , etc., have exactly the same form as before. When the number of s's is equal to the number of x's, these equations give an immediate solution of our problem by ordinary algebraic methods; but in such cases we have no data for determining the precision with which the computed results represent the true numerics Xi, X 2) etc. Generally the number of s's is far in excess of the number of unknowns and no system of values can be assigned to the x's that will exactly satisfy all of the equations (47). If any assumed values of the x's are substituted in (47), the differences ^1 (Xi, X 2) . . . X q ) Si = 7*1, F 2 (xi, x 2) . . . x q ) - s 2 = r 2 , F n (Xi, X 2 , . . . X q ) - S- n = T n represent the residuals corresponding to the given s's. ^Obviously, f the most probable values that we can assign to the x's will be those that give a maximum probability that these residuals are equal to the true accidental errors AI, A 2 , etc. If the s's are all of the same weight, the A's all correspond to the same precision constant co. Consequently, as in article thirty- five, the probability that the A's are equal to the r's is and this is a maximum when ri 2 + r 2 2 + . . . + r n 2 = [r 2 ] = a minimum. (49) Hence, as in direct measurements, the most probable values that we can assign to the desired numerics are those that render the sum of the squares of the residuals a minimum. For this reason the process of solution is called the method of least squares. 74 THE THEORY OF MEASUREMENTS [ART. 49 Since the r's are functions of the q unknown quantities x i} x 2) etc., the conditions for a minimum in (49) are provided the x's are entirely independent in the mathematical sense, i.e., they are not required to fulfill any rigorous mathe- matical relation such as that which connects the three angles of a triangle. The equations (47) are not such conditions since the functions F i} F 2 , etc., represent measured magnitudes and may take any value depending on the particular values of the x's that obtain at the time of the measurements. When the r's are re- placed by the equivalent expressions in terms of the x's and s's as given in (48), the conditions (50) give q, and only g, equations from which the x's may be uniquely determined. If the weights of the s's are different, the A's correspond to different precision constants coi, 0)2, . . . , co n given by the rela- tions where w a is the precision constant corresponding to a standard measurement, i.e., a measurement of weight unity; and wi, w 2 , . . . , w n are the weights of the s's. Under these conditions, as in article forty-five, the most probable values of the re's are those that render the sum of the weighted squares of the residuals a minimum. Thus, in the case of measurements of unequal weight, the condition (49) becomes wiri 2 f w 2 2 + + MV 2 = [wr 2 ] = a minimum, (51) and conditions (50) become A M = ; ^M = 0; ... A M = . (52) 49. Observation Equations. The equations (50) or (52) can always be solved when all of the functions FI, F 2) . . . F n are linear in form. Many problems arise in practice which do not satisfy this condition and frequently it is impossible or incon- venient to solve the equations in their original form. In such cases, approximate values are assigned to the unknown quantities and then the most probable corrections for the assumed values are computed by the method of least squares. Whatever the form ART. 50] THE METHOD OF LEAST SQUARES 75 of the original functions, the relations between the corrections can always be put in the linear form by a method to be described in a later chapter. When the given functions are linear in form, or have been reduced to the linear form by the device mentioned above, equa- tions (47) may be written in the form + to + + to + + piX q = si, = s 2 , p n x q = (53) where the a's, 6's, etc., represent numerical constants given either by theory or as the result of direct measurements. These equa- tions are sometimes called equations of condition; but in order to distinguish them from the rigorous mathematical conditions, to be treated later, it is better to follow the German practice and call them observation equations, "Beobachtungsgleichungen." By comparing equations (47), (48), and (53), it is obvious that the expressions + to + CiX 3 + -f to + c 2 x 3 + b n x c n x 3 s 2 = r 2 , p n x q - s n = r n (54) give the resi'duals in terms of the unknown quantities x\, x z , etc., and the measured quantities si, s 2 , etc. 50. Normal Equations. In the case of measurements of equal weight, we have seen that the most probable values of the unknowns x\, x 2 , etc., are given by the solution of equations (50) provided the x's are independent. Assuming the latter condition and performing the differentiations we obtain the equations dr, dr. dr 3 dx t (0 76 THE THEORY OF MEASUREMENTS [ART. 50 Differentiating equations (54) with respect to the x's gives dri _ dr 2 _ ~dx\ ~ ai ' dxi ~~ dx c = a n , = b n , dr 2 and hence equations (i) become r 2 a 2 + i + r 2 6 2 + . . drn ' dx q + r n a n = 0, + r n b n = 0, (ii) (iii) - . . . + r n p n = 0. Introducing the expressions for the r's in terms of the x's from equations (54) and putting [aa] = didi -{- a 2 a 2 -|- a 3 a 3 ~h ~h d n d n} w> [as] = diSi + a 2 s 2 + a 3 s 3 + [bd] = bidi + 6 2 a 2 + b s d s + [66] = &!&! + 6 2 6 2 + 6 3 6 3 + [be] = 6iCi + 6 2 c 2 + 6 3 c 3 + a n s n , 6 n a n = [ab]j b n b n , 6 n c n (55) equations (iii) reduce to [aa] x-i + [ab] x z + [ac] x 3 [ac] [be] x 2 + [cc] x 3 [bp]x q =[bs], [CP] X* = N, (56) giving us q, so-called, normal equations from which to determine the q unknown x's. Since the normal equations are linear in form and contain only numerical coefficients and absolute terms, they can always be solved, by any convenient algebraic method, provided they are entirely independent, i.e., provided no one of them can be ob- tained by multiplying any other one by a constant numerical ART. 50] THE METHOD OF LEAST SQUARES 77 factor. This condition, when strictly applied, is seldom violated in practice; but it occasionally happens that one of the equations is so nearly a multiple or submultiple of another that an exact solution becomes difficult if not impossible. In such cases the number of observation equations may be increased by making additional measurements on quantities that can be represented by known functions of the desired unknowns. The conditions under which these measurements are made can generally be so chosen that the new set of normal equations, derived from all of the observation equations now available, will be so distinctly independent that the solution can be carried out without difficulty to the required degree of precision. By comparing equations (53) and (56), it is obvious that the normal equations may be derived in the following simple manner. Multiply each of the observation equations (53) by the coefficient of xi in that equation and add the products. The result is the first normal equation. In general, q being any integer, multiply each of the observation equations by the coefficient of x q in that equation and add the products. The result is the gth normal equation. The form of equations (56) may be easily fixed in mind by noting the peculiar symmetry of the coefficients. Those in the principal diagonal from left to right are [aa], [66], [cc], etc., and coefficients situated symmetrically above and below this diagonal are equal. When the given measurements are not of equal weight, the observation equations (53), and the residual equations (54) remain unaltered, but the normal equations must be derived from (52) in place of (50). Since the weights Wi, w 2 , etc., are independent of the x's, if we treat equations (52) in the same manner that we have treated (50), we shall obtain the equations * + w n r n a n = 0, '. .4 Wn&n = 0, (iv) + Wtfzpz + ' ' ' + W n r n p n = 0, in place of equations (iii). Hence, if we put [iWia] = Wididi -f~ WzClzCLz ~\~ ' ' ' ~\~ W n d n CLnj (57) [was] = WidiSi + w&zSz + + w n a n s nj ' -\-WnpnPn, 78 THE THEORY OF MEASUREMENTS [ART. 51 the normal equations become [waa] xi + [wab] x 2 + [wac] z 3 [wab] Xi + [wbb] x z + [wbc] x z [wac] X! + [wbc] x 2 + [wcc] x z + [wap] x q = [was], + [wbp] x q = [wbs], + [wcp] x q = [wcs], (58) [wap]xi + [wbp]x 2 + [wcp]x$ + + [wpp]x q = [wps]. These equations are identical in form with equations (56), and they may be solved under the same conditions and by the same methods as those equations. Consequently, in treating methods of solution, we shall consider the measurements to be of equal weight and utilize equations (56). All of these methods may be readily adapted to measurements of unequal weight by substitut- ing the coefficients as given in (57) for those given in (55). 51. Solution with Two Independent Variables. When only two independent quantities are to be determined the observation equations (53) become " = s, and the normal equations (56) reduce to [aa] Xi + [ab] x 2 = [as], [ab] X! + [bb] x 2 = [bs]. Solving these equations we obtain [bb] [as] - [ab] [bs] [aa] [bb] - [ab] 2 _ [aa] [bs] [ab] [as] [aa] [bb] - [ab] 2 As an illustration, consider the determination of the length Z/ at C., and the coefficient of linear expansion a of a metallic bar from the following measurements of its length L t at temper- ature t C. (56a) (59) t L t C. 20 mm. 1000.36 30 1000.53 40 1000.74 50 1000.91 60 1001.06 Ara.51] THE METHOD OF LEAST SQUARES 79 or Within the temperature range considered, L t and t are connected with LO and a by the relation L t = Lo (1 + at), L t = Lo + L at, (v) and a set of observation equations might be written out at once by substituting the observed values of L t and t in this equation. But the formation of the normal equations and the final solution is much simplified when the coefficients and absolute terms in the observation equations are small numbers of nearly the same order of magnitude. To accomplish this simplification, the above func- tional relation may be written in the equivalent form and if we put it becomes L t - 1000 = Lo - 1000 + WL<xx L t - 1000 = s; JQ = 6, LO 1000 = Xi] 10 LOCK = Xz, Xi -J- 6^2 = s. (vi) Using this function, all of the a's in equation (53a) become equal to unity and the 6's and s's may be computed from the given observations by equations (vi). the observation equations are xi + 2 z 2 = xi + 3 x 2 = Hence, in the present case, .36, .53, x l +x 2 = .74, zi + 5z 2 = .91, Xl + 6x 2 = 1.06. For the purpose of forming the normal equations, the squares and products of the coefficients and absolute terms are tabulated as follows : Obs. aa ab as bb bs 1 2 0.36 4 0.72 2 3 0.53 9 1.59 3 4 0.74 16 2.96 4 5 0.91 25 4.55 5 6 1.06 36 6.36 5 20 3.60 90 16.18 [aa] W [as] [bb] [bs] Substituting these values of the coefficients in (56a) gives the normal equations 80 THE THEORY OF MEASUREMENTS [ART. 51 = 3.60, = 16.18, and by (59) we have _ 90 X 3.60 - 20 X 16.18 5 X 90 - 400 5 X 16.18 - 20 X 3.60 = 0.008, = 0.178. 5 X 90 - 400 From these results, with the aid of relations (vi), we find Lo = xi + 1000 = 1000.008, L a = ^ = 0.0178, 0.0178 = 0.0000178, and finally L t = 1000.008 (1 + 0.0000178 1) millimeters. (vii) The differences between the values of L t computed by equation (vii), and the observed values give the residuals. But they can be more simply determined by using the above values of x\ and Xz in the observation equations and taking the difference between the computed and observed values of s. Thus, if s' represents the computed value and r the corresponding residual s' = 0.008 + 0.178 6, and r = s f s. With the values of s and 6 used hi the observation equations we obtain the residuals as tabulated below: s' 8 r 7-2 X 10* 0.364 0.542 0.720 0.898 1.076 0.36 0.53 0.74 0.91 1.06 +0.004 +0.012 -0.020 -0.012 +0.016 0.16 1.44 4.00 1.44 2.56 [r 2 ] = 9.60XlO~ 4 Since the above values of x\ and x 2 were computed by the method of least squares, the resulting value of [r 2 ], i.e., .000960, should be less than that obtainable with any other values of x\ and x%. That this is actually the case may be verified by carrying out the computation with any other values of x\ and x z . ART. 52] THE METHOD OF LEAST SQUARES 81 52. Adjustment of the Angles About a Point. As an illus- tration of the application of the method of least squares to the solution of a problem involving more than two unknown quanti- ties, suppose that we wish to determine the most probable value of the angles AI, AZ, and A 3 , Fig. 9, from a series of independent measurements of equal weight on the angles Mi, M 2 , . . . M 6 . If the given measurements were all exact, the equations AI = Mi; AZ = M 2 ; A 3 = M 3 ; AI-\- AZ = M 4 ; AI + AZ -{-As = MS; and Az -\- As = Me, would all be fulfilled identically. In practice this is never the case and it becomes necessary to adjust the values of the A's so that the sum of the squares of the discrepancies will be a minimum. The adjustment may be ef- fected by adopting the above equations as ob- servation equations and proceeding at once to the solution for the A's by the method of least squares. But the ob- served values of the M's usually involve so many significant figures that the computation would be tedious. It is better to adopt approximate values for the A's and then compute the necessary corrections by the method of least squares. For this purpose, suppose we adopt MI, M 2 , and M 3 as approxi- mate values of A\, A 2 , and A s respectively and let xi, Xz, and x 3 represent the corrections that must be applied to the M's in order to give the most probable values of the A's. Then, putting AI = MI + xi, AZ = MZ + Xz, and A 3 = M 3 + # 3 , (viii) the above equations become FIG. 9. 82 THE THEORY OF MEASUREMENTS [ART. 52 + x 2 = 0, = 0, = 0, = M 4 - (Af ! + M 2 ), To render the problem definite, suppose that the following values of the M's have been determined with an instrument read- ing to minutes of arc by verniers: Mi = 10 49'.5, M 4 = 45 24'.0, M 2 = 34 36'.0, M 6 = 60 53'.5, M 3 = 15 25'.5, M 6 = 50 O'.O. Substituting these values in the above equations we obtain xi = 0, x 2 = 0, 2'.5, Adopting these as our observation equations and comparing with (53) we obtain the coefficients and absolute terms tabulated below: Oba. a b c s 1 1 2 1 3 1 4 1 1 -1.5 5 1 1 1 2.5 6 1 1 -1.5 The squares and products of the coefficients and absolute terms may be tabulated, for the purpose of forming the normal equations, as follows : M ab ac as 66 be bs cc cs 1 1 1 1 1 1 -1.5 2.5 1 1 1 1 1 2 [be] -1.5 2.5 -1.5 1 1 1 2.5 -1.5 3 [aa] [ab] 1 [ac] 1 fas] 4 [66] -0.5 [6s] 3 [cc] 1 [cs] ART. 53] THE METHOD OF LEAST SQUARES 83 Substituting these values in (56) the three normal equations become -0.5, 1 xi -f 2 x 2 + 3 z 3 = 1, and solution by any method gives xi = 0.625; x 2 = - 0.75; x 3 = 0.625. With these results together with the given values of MI, Mz, and M 3 we obtain from equations (viii) A! = 10 50M25, A 2 = 34 35'.25, A 3 = 15 26M25. In a problem so simple as the present the normal equations are generally written out at once from the observation equations by the rule stated in article fifty, without taking the space and time to tabulate the coefficients, etc. But, until the student is thor- oughly familiar with the process, it is well to form the tables as a check on the computations and to make sure that none of the coefficients or absolute terms have been omitted. For this reason the tabulation has been given in full above and the student is advised to carry out the formation of the normal equations by the shorter method as an exercise. 53. Computation Checks. When the number of unknowns is greater than two and a large number of observation equations are given with coefficients and absolute terms involving more than two significant figures, the formation of the normal equations is the most tedious and laborious part of the computations. It is, therefore, advantageous to devise a means of checking the com- puted coefficients and absolute terms in the normal equations before we proceed to the final solution. For this purpose compute the n quantities t\ t ^2, ... t n by the equations ai + &i -f ci + - + pi = ti,~ 02 + &2 4- c 2 -f + pz = h, On + & + C n -f - + p n = (60) 84 THE THEORY OF MEASUREMENTS [ART. 54 where the a's, b's, etc., are the coefficients in the given observa- tion equations. Multiply the first of equations (60) by Si, the second by s 2 , etc., and add the products. The result is [as] + [bs] + [cs] + + \ps] = [ts]. (61) In the same way, multiplying by the a's in order and adding, then by the b's in order and adding, etc., we obtain the following rela- tions [aa] + [db] + [ac] + .-. + [ap] = [at], [ab] + [bb] + [be] + + [bp] = [bt], [ac] + [be] + [cc] + + [cp] = [ct], (62) [ap] + \bp] + [ep] + . . . + \pp] = \pt]. If the absolute terms in the normal equations have been accu- rately computed, equation (61) reduces to an identity. If the coefficients have been accurately computed equations (62) all become identities. Hence (61) is a check on the computation of the absolute terms and equations (62) bear the same relation to the coefficients. The extra labor involved in computing the quan- tities [ts] t [at], . . . , [pt] is more than repaid by the added confi- dence in the accuracy of the normal equations. When all attainable significant figures are retained throughout the computations, the checks (61) and (62) should be identities. In practice the accuracy of the measurements is seldom sufficient to warrant so extensive a use of figures, and, consequently, the squares and products, aa, ab, . . . as, at, etc., are rounded to such an extent that the computed values of the x's will come out with about the same number of significant figures as the given data. Judgment and experience are necessary in determining the number of significant figures that should be retained in any particular problem and it would be difficult to state a general rule that would not meet with many exceptions. When the computed coefficients and absolute terms are rounded, as above, the checks may not come out absolute identities, but they should not be accepted as satisfactory when the discrepancy is more than two units in the last place retained. 54. Gauss's Method of Solution. When the normal equa- tions (56) are entirely independent, they may be solved by any of the well-known methods for the solution of simultaneous linear equations and lead to unique values of the unknown quan- ART. 54] THE METHOD OF LEAST SQUARES 85 titles xi, x 2) etc. Gauss's method of substitution is frequently adopted for this purpose since it permits the computation to be carried out in symmetrical form and provides numerous checks on the accuracy of the numerical work. The general principles of the method will be illustrated and explained by completely working out a case in which there are only three unknowns. Since the process of solution is entirely symmetrical, it can be easily extended for the determination of a larger number of unknowns, but too much space would be required to carry through the more general case here. When only three unknowns are involved, the normal equations (56) and the check equations (60) and (61) may be completely written out in the following form, the computed quantities and equations being placed at the left, and the checks at the right. [aa] xi + [ab] x 2 + [ac] x 3 = [as]. [aa] + [ab] + [ac] = [at]. [ab] xi + [bb] x 2 + [be] x 3 = [bs]. [ab] + [bb] + [be] = [bt]. [ac] xi + [be] x 2 + [cc] x 3 = [cs]. [ac] + [be] + [cc] = [ct]. [as] + [bs]+[cs] =[st].\ Solve the first equation on the left for xi y giving [as] [ab] [ac] Xi = 7 7 f 1 X 2 f 1 X$. [aa] [aa] [aa\ Compute the following auxiliary quantities: (63) [56] _ P4 [ &] = [bb 1], [bt] - pi M = [^ ' 1L L aa] [a61 [6c]- M L " M ~ M = [6s ' 1] ' M ~ N1 = [st As a check on these computations we notice that [bb 1] + [be - 1] = [bb] + [be] - |^| ([ab] + [c]), [aaj = [bt] - lab] - ([at] - [aa]), 86 THE THEORY OF MEASUREMENTS [ART. 54 In a similar way we may show that we should have [6c-l] + [cc-l] = [cM] and [6s- 1] + [cs- 1] = [st- 1]. Substituting (64) in the last two of (63) and placing the above checks to the right, we have the equations [bb -I]x 2 + [be 1] x s = [bs 1], [bb 1] + [be- 1] = [fa 1], [be -I]x 2 + [cc 1] z 3 = [cs 1], [be 1] + [cc- 1] = [ct 1], (65) [6s-l] + [cs.l] = [s*. 1],. which show the same type of symmetry as (63), but contain only two unknown quantities. Solve the first of (65) for x 2 giving __ * 2 ~[6&.l] [bb-lf 3 ' and compute the following auxiliaries: [<*!] - [l^jlfc-1] = [-2], [cM] - l ~^ } [bt. 1} = let- 2], (cs 1} - |^jj [bs 1} - [cs 2], [st 1] - |^|j (bt 1] = [* 2}. By a method similar to that used above we can show that we should have [cc 2] = [ct 2] and [cs 2] = [st 2]. Hence, substituting (66) in the last of (65), we have [cc 2] x 3 = [cs 2], [cc 2] = [ct 2], [cs.2] = N-2], and consequently [cs 2] _. *"fc^t' (67) Having determined the value of x 3 from (67), x% may be cal- culated from (66), and then Xi from (64). A very rigorous check on the entire computation is obtained as follows: using the computed values of Xi, x z , and z 3 in equations (54), derive the residuals (68) - s 2 , T n = d n Xi ~|- O n X 2 ~\- C n Xs S n , and then form the sums [rr] = n 2 + r 2 2 + r 3 2 + - - - + r n 2 , [SS] = Si 2 + S 2 2 + S 3 2 + + n 2 . ART. 55] THE METHOD OF LEAST SQUARES 87 If the computations are all correct, the computed quantities will satisfy the relation W = M-[a S ]-M[6 S .l]- [cs . 2] . (69) To prove this, multiply the first of (68) by ri, the second by r%, etc., and add the products. The result is [rr] = [ar] Xi + [br] x 2 -f [cr] 3 - [sr]. But from equations (iii), article fifty, [ ar ] = [br] = [cr] = 0, consequently [rr]=- []. (70) Multiply each of equations (68) by its s; add, taking account of (70), and we obtain [rr] = [ss] - [as] Xi - [6s] x z - [cs] x z . Eliminating x\, X 2 , and z 3 , in succession with the aid of (64), (66), and (67) we find [rr] = [ss] - [as] - [6s 1] x 2 - [cs 1] x 9 , and finally r i r i l as ] r i [& s * 1] n n t cs ' 2] r Ol [rr] = M ~ y M - I667i] [6s ' 1] - RT2] [cs ' 2] ' which is identical with (69). 55. Numerical Illustration of Gauss's Method. The fore- going methods are most frequently used for the adjustment of astronomical and geodetic observations, and their application to particular problems is fully discussed in practical treatises on such observations. The physical problems, to which they are applicable, usually involve the determination of an empirical relation between mutually varying quantities. Such problems will be discussed at some length in Chapter XIII, and the corre- sponding observation equations will be developed. It would require too much space to carry out the complete dis- cussion of such a problem, in this place, with all of the observa- tions made in any actual investigation. But, for the purpose of illustration, the most probable values of xi, Xz, and x 3 will be 88 THE THEORY OF MEASUREMENTS [ART. 55 derived, from the following typical observation equations, by Gauss's method of solution: + 2x 2 + 0.4z 3 = + 4x 2 + 1.6x3 = + 6 x 2 + 3.6 z 8 = + 8x 2 + 6.4x 3 = +10x 2 +10.0^3 = 0.24, - 1.18, - 1.53, - 0.69, 1.20, 4.27. Since the coefficient of xi is unity in each of these equations, the products aa, ab, aCj as, and at are equal to a, 6, c, s, and t, respectively. Consequently the first five columns of the follow- ing table show the coefficients, absolute terms, and check terms (t = a + b + c) of the observation equations as well as the squares and products indicated at the head of the columns. The sums [aa], [ab], etc., are given at the foot of the columns and the checks, by equations (61) and (62), are given below the tables. In the present case, the coefficients are expressed by so few signifi- cant figures that it is not necessary to round the computed products and consequently the checks come out identities. aa ab ac as at bb be 2 4 6 8 10 0.0 0.4 1.6 3.6 6.4 10.0 0.24 -1.18 -1.53 -0.69 1.20 4.27 1.0 3.4 6.6 10.6 15.4 21.0 4 16 36 64 100 0.0 0.8 6.4 21.6 51.2 100.0 6 M 30 [ab] 22.0 M 2.31 [as] 58.0 M 220 m 180.0 [be] Check: [ aa] + [ab] + [c ic] = 58.0. bs cc cs bl ct st 0.00 -2.36 -6.12 -4.14 9.60 42.70 0.00 0.16 2.56 12.96 40.96 100.00 0.00 -0.472 -2.448 -2.484 7.680 42.700 0.0 6.8 26.4 63.6 123.2 210.0 0.00 1.36 10.56 38.16 98.56 210.00 0.24 - 4.012 -10.098 - 7.314 18.480 89.670 39.68 [bs] 156.64 [cc] 44.976 [cs] 430.0 M 358.64 [ct] 86.966 [st] Checks: [ab] + [66] + [be] = 430.0 [ac] + [be] + [cc] = 358.64 M + [6s] + [cs]= 86.966 ART. 55] THE METHOD OF LEAST SQUARES 89 The normal equations and their checks might now be written out in the form of equations (63), but, since the coefficients and other data necessary for their solution are all tabulated above, it is scarcely worth while to repeat the same data in the form of equations. The computation of the auxiliaries [bb 1], [be 1], etc., and the final solution for x i} x 2) and # 3 by logarithms is best carried out in tabular form as illustrated on pages 90 and 91. The meaning of the various quantities appearing in these tables, and the methods by which they are computed, will be readily under- stood by comparing the numerical process with the literal equa- tions of the preceding article. When the letter n appears after a logarithm it indicates that the corresponding number is to be taken negative in all computations. The computation of the residuals by equations (68) and the final check by (69) is carried out in the following table, where Scale, is written for the value of the expression axi + bx 2 + cxs, when the computed values of x\, x 2 , and x 3 are used and s bs. is the corresponding value of s in the observation equations. Thus +- Si = Si calc. ~ Si obs.- I* S ObB. r *Xio. ss 0.245 -1.195 -1.512 -0.709 1.215 4.264 0.24 -1.18 -1.53 -0.69 1.20 4.27 +0.005 -0.015 +0.018 -0.019 +0.015 -0.006 25 225 324 361 225 36 0.0576 1.3924 2.3409 0.4761 1.4400 18.2329 .001196 [rr] 23.9399 [as] r , [6s 1] , .,, [cs 2] :s-2] 52 = 23.9387 0.0012 [aa\ [oo i\ [cc A\ 0.8893 + 11.3042 + 11.74 Final check by (69): [rr] Since the checks are all satisfactory, we are justified in assum- ing that the computations are correct. Hence the most probable values of the unknowns, derivable from the given observation equations, are xi = 0.245; x 2 = - 1.0003; z 3 = 1.4022, 90 THE THEORY OF MEASUREMENTS [ART. 55 jfl 3 r [ is co ^^ O , ^ _|- Oi 1 ^ g rH - , IQ r^^ 4 1 i CO 00 ^ 1-H o c^ o o HII II II t^ CO O CD CO ^^ t^ CD 1>- GO CO <M O HII II II H" "e "e II II II ^r " ^^ tj ~Q O r ~ l " "e o ^ I '|^ be * bfi ^ bC ^ c\j d II-H OO OO <N >O CO CD CD O i i Oi i i 00 CO rH rH (M II II II II 05 ""ST 5P ^H 1 o o -2 . CD GO O CD b bfi -2 -2 -STe-i 3 1111 i - I CO 1C <M i i O 0^00 dodo S <N ^^ CD CD CD t^* CD Oi O O >O r-< i 1 OO !> II II II II H II II II II II II o W 53 5" o^ 0^ bfi M mi^ ffi 1 M bD * 00 s^ 2 a! se H H H ii H H n 11111 1 1 1? -2 J2 ^l^ 1 co d II II 11 bO ART. 55] THE METHOD OF LEAST SQUARES 91 03 M <N CY^ <M CO (0 O Oi Ji 3 ^ 6 10 " CO CO co CO i ( CO CO CO CO CO 1 CO OS T-( OS CO ^ ^ CO o c^i o d o 10 ^ T-H CO OO CO 10 II II II II II II II II II II 8 1 1 T 1 | i fffl ifL rSl "o S -^ "S "S S - "5 ^ ^ 8 8 ' ' '~~ ' c d d H r-i d bfi ^ ^r 03 ' 1- _ ( ' ^ ( Q^ 1 j3 B . ^ s CO rO II II II II || I M r-O -O *T? O b- b- l>. ^ (N 1 " ' co Uo C3 ^ CO co OS CO tO "^t 1 T 1 CO ill jjO o3 bfi ^O ?7 7 CO T-I OO 00 CO <M II II II II d d i-i II II II T- ' 1 ' T-H T-l TH TH ^ (M <M H H =0 =0 oo -O *O "O ^j ^^ n i 2 bfi bfi ' T ' c3 bfi -2 TH II O -is bfi O Q g 1 g 1 00 8 OS OO O OS : d- iO ii O IO d II II II II II II II II T i T i ^n" 1 i i ^ ^ ^o ^ cT e o co ^ *o O t ^ -O -O ^J o 1 o O "O *^ O 8 | 2 8 jo^g bfi bfi _0 jD ^T^T bO ^O CO CO C^ !> ^H C*l CO * ^L ^^ ^^ *-O ^^ O ^D I^H jO^ -O^ 1 1 II II II II w "17 ^ ^ ^HH O TH' W) o3 bfi o3 II II Tf . 92 THE THEORY OF MEASUREMENTS [ART. 56 and the corresponding empirical relation becomes s = 0.245 a - 1.0003 6 + 1.4022 c. A small number of observation equations with simple coefficients have been chosen, in the above illustration, partly to save space and partly in order that the computations may be more readily followed. In practice it would seldom be worth while to apply the method of least squares to so small a number of observations or to adopt Gauss's method of solution with logarithms when the normal equations are so simple. When the number of observa- tions is large and the coefficients involve more than three or four significant figures, the method given above will be found very convenient on account of the numerous checks and the symmetry of the computations. In order to furnish a model for more complicated problems, the process has been carried out completely even in the parts where the results might have been foreseen without the use of logarithms. 56. Conditioned Quantities. When the unknown quantities, Xi, Xz, etc., are not independent in the mathematical sense, the foregoing method breaks down since the equations (50) no longer express the condition for a minimum of [rr]. In such cases the number of unknowns may be reduced by eliminating as many of them as there are rigorous mathematical relations to be fulfilled. The remaining unknowns are independent and may be deter- mined as above. The eliminated quantities are then determined with the aid of the given mathematical conditions. For the purpose of illustration, consider the case of a single rigorous relation between the unknowns, and let the correspond- ing mathematical condition be represented by the equation 0(x lt x,, . . . , x q ) =0. (71) As in the case of unconditioned quantities, the observation equa- tions (53) are + C&s + + piX q = Si, c n x 3 p n x q = The solution of (71) for x\, in terms of Xz, x s , . . ., x q , may be written in the form xi=f(xz,x a , ..*,*) (72) ART. 56] THE METHOD OF LEAST SQUARES 93 Introducing this value of xi, equations (53) become + ClX* + * + PlX q = Si, + C 2 X 3 + + P2Z a = S 2 , 4- c n z 3 + + p n x q = s. Since the form of 6 is known, that of / is also known. Hence, by collecting the terms in x%, x S} etc., and reducing to linear form, if necessary, we have bixz + ci'x s + + p\x q = s/, The x's in these equations are independent, and, consequently, they may be determined by the methods of the preceding articles. Using the values thus obtained in (71) or (72) gives the remaining unknown x\. The #'s, thus determined, obviously satisfy the mathematical condition (71) exactly, -and give the least magnitude to the quantity [rr] that is consistent with that condition. They are, consequently, the most probable values that can be assigned on the basis of the given data. As a very simple example, consider the adjustment of the angles of a plane triangle. Suppose that the observed values of the angles are si = 60 1'; s 2 = 59 58'; s 3 = 59 59'. The adjusted values must satisfy the condition xi + x 2 + x* = 180, or xi = 180 - x 2 - x 3 . Eliminating Xi from the observation equations, xi = Si' t Xz = s 2 ; and x s s 3 ; and substituting numerical values we have x z +x 3 = 119 59', x 2 = 59 58', x 3 = 59 59'. The corresponding normal equations are 2z 2 + z 3 = 179 57', = 179 58', 94 THE THEORY OF MEASUREMENTS [ART. 56 from which we find x 2 = 59 58'.7 and x s = 59 59'.7. Then, from the equation of condition, xi = 60 1'.6. When there are two relations between the unknowns, expressed by the equations 01 (xi, x t , . . . , x q ) = 0, 02 (xi, x 2 , . . . , x q ) = 0, they may be solved simultaneously for xi and x 2 , in terms of the other x's, in the form xi = fi(x 3 , xt, . . . , x q ), x z = /2(z3, $,..., x q ). Using these in the observation equations (53) we obtain a new set of equations, independent of x\ and x* t that may be solved as above. It will be readily seen that this process can be extended to include any number of equations of condition. When the number of conditions is greater than two, the compu- tation by the above method becomes too complicated for practical application and special methods have been devised for dealing with such cases. The development of these methods is beyond the scope of the present work, but they may be found in treatises on geodesy and practical astronomy in connection with the prob- lems to which they apply. CHAPTER VIII. PROPAGATION OF ERRORS. 57. Derived Quantities. In one class of indirect measure- ments, the desired numeric -X" is obtained by computation from the numerics Xi, X z , etc., of a number of directly measured mag- nitudes, with the aid of the known functional relation X = F(X 1 ,X i , . . . ,X q ). We have seen that the most probable value that we can assign to the numeric of a directly measured quantity is either the arith- metical mean of a series of observations of equal weight or the general mean of a number of measurements of different weight. Consequently, if x\, Xz, . . , x q represent the proper means of the observations on Xi, X 2 , . . . , X q the most probable value x that we can assign to X is given by the relation x = F (xi, x z , . . . , x q ) where F has the same form as in the preceding equation. Obviously, the characteristic errors of x cannot be easily deter- mined by a direct application of the methods discussed in Chapters V and VI, as this would require a separate computation of x from each of the individual observations on which Xi, Xz, etc., depend. Furthermore, it frequently happens that we do not know the original observations and are thus obliged to base our computa- tions on the given mean values, x\, Xz, etc., together with their characteristic errors. Hence it becomes desirable to develop a process for computing the characteristic errors of x from the corresponding errors of Xij xz, etc. For this purpose we will first discuss several simple forms of the function F and from the results thus obtained we will derive a general process applicable to any form of function. 58. Errors of the Function Xi Xz X 3 =t . . . X q . Suppose that the given function is in the form X = Xi + X 2 , or X = Xi - X 2 . These two cases can be treated together by writing the function in the form X = X\ db Xz, 95 96 THE THEORY OF MEASUREMENTS [ART. 58 and remembering that the sign indicates two separate problems rather than, as usual, an indefinite relation in a single problem. If the individual observations on Xi are represented by ai, a 2 , . . . , a n , and those on X 2 by 61, 6 2 , . . . , b n , we have n n and the most probable value of X is given by the relation x = Xi xz. From the given observations we can calculate n independent values of X as follows : Ai = ai &i, A 2 = az d= 6 2 , . . . , A n = a w db 6 n , and it is obvious that the mean of these is equal to x. The true accidental errors of the a's are Aai = oi Xi, Aa 2 = a z Xi, . . . , Aa n = a n Zi; those of the 6's are Ah = 61 - Z 2 , A6 2 = 6 2 - Z 2 , . . . , A6 n = b n - X 2 ; and those of the A's are ^A l =A 1 -X ) &A 2 =A 2 -X, . . . , &A n =A n -X. We cannot determine these errors in practice, since we do not know the true value of the X's, but we can assume them in literal form as above for the purpose of finding the relation between the characteristic errors of the x's. Combining the equations of the preceding paragraph with the given functional relation, we have AA X = (ai 60 - (Zi Z 2 ) = (a! - ZO (61 - Xz) = Aai A&i, and similar expressions for the other A A's. Consequently (AAO 2 = (AaO 2 d= 2 AaiA&i + (A6i) 2 , (AA 2 ) 2 = (Aa 2 ) 2 d= 2 Aa 2 A6 2 (AA n ) 2 = (Aa n ) 2 2 ka n tU) n Adding these equations, we find [(AA) 2 ] = [(Aa) 2 ] 2 [AaA6] + [(A6) 2 ]. ART. 58] PROPAGATION OF ERRORS 97 Since A a and A b are true accidental errors, they are distributed in conformity with the three axioms stated in article twenty-four. Consequently equal positive and negative values of Aa and A6 are equally probable and the term [AaA6] would vanish if an infinite number of observations were considered. In any case it is negligible in comparison with the other terms in the above equation. Hence, on dividing through by n, we have [(AA)1 = [(Aa)l [(A6)*]_ n n n and by equation (20), article thirty-seven, this becomes M A 2 = M a 2 + M b 2 , (73) where M A is the mean error of a single A, M a that of a single a, and M b that of a single b. Since x, xi, and z 2 are the arithmetical means of the A's, a's, and 6's, respectively, their respective mean errors, M , MI, and M 2 , are given by the relations M 2 M 2 Tlf i 2 M* = ^, itf-=, and M, = ^- n n n in virtue of equations (29), article forty. Consequently, by (73) M 2 = Mi 2 + M 2 2 , or M = VMi 2 + M 2 2 . (74) Since the mean and probable errors, corresponding to the same series of observations, are connected by the constant relation (26), article thirty-nine, we have also + Ef, (75) where E, EI, and E z are the probable errors of x, x\, and #2, respectively. It should be noticed that the ambiguous sign does not appear in the expressions for the characteristic errors. The square of the error of the computed quantity is equal to the sum of the squares of the corresponding errors of the directly measured quan- tities; whether the sign in the functional relation is positive or negative. Thus the error of the sum of two quantities is equal to the corresponding error of the difference of the same two quan- tities. Now suppose that the given functional relation is in the form X = Xi d= X 2 X t . 98 THE THEORY OF MEASUREMENTS [ART. 59 The most probable value of X is given by the relation x = xi x z x 3y where the notation has the same meaning as in the preceding case. Represent x\ x z by x p , then a; = x p =t z 3 , and, by an obvious extension of the notation used above, we have M P 2 = Mi 2 + M 2 2 , M z = M P 2 + M 3 2 = Mi 2 + M 2 2 + M 3 2 . Passing to the more general relation X = Xi X 2 X 3 - - - X,, we have a; = 1 db # 2 x 3 z fl , and, by repeated application of the above process, M 2 = M M 2 MJ + - - + M 3 2 , ) + -E- Thus the square of the error of the algebraic sum of a series of terms is equal to the sum of the squares of the corresponding errors of the separate terms whatever the signs of the given terms may ba 59. Errors of the Function a\Xi =t 0:2^2 db a s X 3 =b - a q X q . Let the given functional relation be in the form X = where a\ is any positive or negative, integral or fractional, con- stant. The most probable value that we can assign to X on the basis of n equally good independent measurements of X is x = aiXi, where Xi is the arithmetical mean of the n direct observations ai, a 2 , a s , . . . , a n . The n independent values of X obtainable from the given obser- vations are AI ami, Az aids, . . . , A n = a\a n . The accidental errors of the a's and A's are Aai = a\ Xij Aa 2 = a 2 X\ t . . . , Aa n = a n X\, and A4i = Ai - X, A^ 2 = A t -X, . . . , AA n = A n -X. ART. 60] PROPAGATION OF ERRORS 99 Combining these equations we find and similar expressions for the other AA's. Consequently (AAO 2 = ai 2 (Aax) 2 , and [(AA) 2 ] = ai [(Aa) 2 ]. If M and Af i are the mean errors of x and xi t respectively, and Jf,..I3. Hence M 2 = onWi 2 , (77) and, since the probable error bears a constant relation to the mean error, E 2 = a^! 2 . (78) When the given functional relation is in the more general form X = aiXi =b 0:2^2 =b 0.3X3 =b otqXqj we have x = where the a;'s are the most probable values that can be assigned to the X's on the basis of the given measurements. Applying (77) and (78) to each term of this equation separately and then applying (76) we have t E 2 = where the ATs and E's represent respectively the mean and prob- able errors of the x's with corresponding subscripts. 60. Errors of the Function F (X l} X 2 , . . . , X q ). We are now in a position to consider the general functional relation X = F (Xi, Xz, . . . , X q ), where F represents any function of the independently measured quantities Xi, X 2 , etc. Introducing the most probable values of the observed numerics, the most probable value of the computed numeric is given by the relation x = F fa, x 2) . . . , Xq). (80) This expression may be written in the form & l ), (Z 2 -f-5 2 .. . . . , (* + ,)!, 0) 100 THE THEORY OF MEASUREMENTS [ART. 60 where the I's represent arbitrary constants and the.S's are small corrections given by relations in the form Obviously, the errors of the 5's are equal to the errors of the corre- sponding x's. For, if Mi, Ms, and MI are the errors of Xi, 5i, and Zi, respectively, we have by equation (74) M s * = Mi 2 + Mf. But MI is equal to zero, because I is an arbitrary quantity and any value assigned to it may be considered exact. Consequently Mi 2 . (ii) Since the I's are arbitrary, they may be so chosen that the squares and higher powers of the-5's will be negligible in compari- son with the 8's themselves. Hence, if the x's are independent, (i) may be expanded by Taylor's Theorem in the form dF d , \ ** where = F (z, z, . . . , x) = > and the other differential coefficients have a similar significance. When the observed values of the x's are substituted in these coefficients, they become known numerical constants. The mean error of F (li, Z 2 , . . . , l q ) is equal to zero, since it is a function of arbitrary constants; and the mean errors of the 5's are equal to the mean errors of the corresponding x's by (ii). Consequently, if M, Mi, M 2 , . . . , M q represent the mean errors of x, Xi, x z , . . . , x q , respectively, we have by equation (79) /dF - . V , fdF , , V , = F~ MI ) + brr^ 2 ) + \dxi I \dx 2 I N ~~ , , . (OL) where the E's represent the probable errors of the x's with corre- sponding subscripts. Equations (81) are general expressions for the mean and prob- able errors of derived quantities in terms of the corresponding errors of the independent components. Generally x\ t x 2 , etc., ART. 61] PROPAGATION OF ERRORS 101 represent either the arithmetical or the general means of series of direct observations on the corresponding components, and EI, E z , etc., can be computed by equations (32) or (41). In some cases, the original observations are not available but the mean values together with their probable errors are given. For the purpose of computing the numerical value of the differ- r\Tj1 r\Tj1 ential coefficients -r ; > etc., the given or observed values of oXi 0X2 the components x i} x 2) etc., may generally be rounded to three significant figures. This greatly reduces the labor of computa- tion and does not reduce the precision of the result, since the E's and M's are seldom given or desired to more than two significant figures. 61. Example Introducing the Fractional Error. The prac- tical application of the foregoing process is illustrated in the follow- ing simple example: the volume V of a right circular cylinder is computed from measurements of the diameter D and the length L, and we wish to determine the probable error of the result. In this case, V corresponds to x, D to xi, L to x 2) and the functional relation (80) becomes Also, if EV, E D , and EL are the probable errors of V, D, and L, respectively, the second of equations (81) becomes where sv and dV d /I \ 1 n2 -r^F- = ^F \ -7 TTL) L ] = -TrD*. dL dL\4 / 4 Hence The computation can be simplified by introducing the frac- TTT tional error -^~- Thus, dividing the above equation by we have ^ =4 ^! + ^ T7"O 7~^9 I T O 102 THE THEORY OF MEASUREMENTS [ART. 62 or, writing PV, PD, and PL for the fractional errors, Py 2 = 4 Pz> 2 + P L \ P V and finally E v = FP F = V A similar simplification can be effected, in dealing with many other practical problems, by the introduction of the fractional errors. Consequently it is generally worth while to try this ex- pedient before attempting the direct reduction of the general equation (81).- In order to render the problem specific, suppose that D = 15.67 0.13 mm., L = 56.25 d= 0.65 mm., then V = 10848 PD = = = P L = ^ = ^ = .0116; Pz, 2 = 135 X 10- 6 , = 0.020, E v = VTV = 220 mm Hence 7= 10.85 0.22 cln. 3 62. Fractional Error of the Function aX^ 1 X Z 2 U2 X X a n5 .- Suppose the given relation is in the form X = F(X l ) =aXi where a and n are constants and the =fc sign of the exponent n is used for the purpose of including the two functions aXi +n and aX-r^ in the same discussion. In this case equation (80) becomes x = axi n , and the second of (81) reduces to But _=_ Consequently ART. 62] PROPAGATION OF ERRORS 103 If P and PI are the fractional errors of x and xi, respectively, we have E* - Hence i P = nP,. (82) If we replace n by in the above argument, (80) becomes _ x = aXi m , and we find m Hence the fractional error of any integral or fractional power of a measured numeric is equal to the fractional error of the given numeric multiplied by the exponent of the power. If the given function is in the form of a continuous product X = aX l X X, X X X qt (80) becomes x = axi X x 2 X X x q . dF Hence = ax z X x 3 X X x g , ox\ I dF 1 and - = Hence, by (81), JP _ Ei 2 EJ Eg 2 r z ~ 7~2 ~f~ ~~2 ~r T > Js JL>1 JU2 Lq and, if P, PI, P 2 , . . . , P q represent the fractional errors of the #'s with corresponding subscripts, Combining the above cases we obtain the more general rela- tion X = aXi 1 X Xz 2 X * * X X q , and the corresponding expression for (80) is Applying (82) to each factor separately and then applying (83) to the product, we find f - - - +nfPf. (84) 104 THE THEORY OF MEASUREMENTS [ART. 62 For the sake of illustration and to fix the ideas this result may be compared with the example of the preceding article. If we put x = V, Xi = D, HI = 2, x 2 = L, n 2 = 1, a = -7 , P = Py, PI = PD, and PZ = PL the above expression for x becomes V = %TrD 2 L, and (84) becomes Occasionally it is convenient to express the probable error in the form of a percentage of the measured magnitude. If E and p are respectively the probable and percentage errors of x, p= 100 - = 100 P. (85) x Consequently (84) may be written in the form P 2 = niW + n 2 2 p 2 2 + + nfp*, (84a) where pi, p 2 , . . . , p q are the percentage errors of Xi, x 2 , . . . , x q , respectively CHAPTER IX. ERRORS OF ADJUSTED MEASUREMENTS. WHEN the most probable values of a number of numerics Xi, X 2 ,etc., are determined by the method of least squares, the results Xi, x 2 ,etc., are called adjusted measurements of the quan- tities represented by the X's. In Chapter VII we have seen how the x's come out by the solution of the normal equations (56) or (58), and how these equations are derived from the given obser- vations through the equations (53). In the present chapter we will determine the characteristic errors of the computed x's in terms of the corresponding errors of the direct measurements on which they depend. 63. Weights of Adjusted Measurements. When there are q unknowns and the given observations are all of the same weight, the normal equations, derived in article fifty, are [aa] Xi + [ab] x 2 + [ac] x 3 + - + [ap] x q = [as], [db] x, + [66] x 2 + [6c] *,+ + [bp] x q = [bs], (56) [ap] xi + [bp] x z + [cp] x 3 + + [pp] x q = [ps]. Since these equations are independent, the resulting values of the x's will be the same whatever method of solution is adopted. In Chapter VII Gauss's method of substitution was used on account of the numerous checks it provides. For our present purpose the method of indeterminate multipliers is more convenient as it gives us a direct expression for the x's in terms of the measured s's. Obviously this change of method cannot affect the errors of the computed quantities. Multiply each of equations (56) in order by one of the arbitrary quantities AI, A 2 , . . . , A q and add the products. The result- ing equation is (86) + ([db] A 1 + [bb] A, + + [bp] A q ) x 2 + > = [as] A l + [6s] A 2 + + [ps] A q . 105 [ob] A, + [66] A* + + [6p] A q = 0, 106 THE THEORY OF MEASUREMENTS [ART. 63 Since the A's are arbitrary and q in number, they can be made to satisfy any q relations we choose without affecting the validity of equation (86). Hence, if we determine the A's in terms of the coefficients in (56) by the relations (g7) equation (86) gives an expression for x\ in the form xi = [as] Ai + [&*] 4i +!-'+ \ps]A t . (88) If we repeat this process q times, using a different set of multipliers each time, we obtain q different equations in the form of (86). In each of these equations we may place the coefficient of one of the x's equal to unity and the other coefficients equal to zero, giv- ing q sets of equations in the form of (87) for determining the q sets of multipliers. Representing the successive sets of multipliers by A's, B's, C"s, etc., we obtain (88), and the following expressions for the other x's : x 2 = [as] Bi + [bs] ft +...;+ \p 8 ] B q , x 3 = [as] Ci + [6s] C 2 + + \ps] C q , x q = [as] P! + [6s] P 2 + + \ps] P q . From equations (87), it is obvious that the A's do not involve the observations Si, s 2 , etc. Consequently (88) may be expanded in terms of the observations as follows: Xi = ctiSi + a z s 2 -f + ctgS q , (89) where the a's depend only on the coefficients in the observation equations (53) and are independent of the s'a. Since we are con- sidering the case of observations of equal weight, each of the s's in (89) is subject to the same mean error M 8 . Her e, if MI is the mean error of Xi, we have by equations (79), article fifty-nine, Mx 2 = ai 2 M s 2 + 2 2 M a 2 + - + a n 2 M, 2 = M M, 2 . But, if Wi is the weight of x\ in comparison with that of a single s, we have by (36), article forty-four, Wl w i (90) Mi 2 [act] ART. 63] ERRORS OF ADJUSTED MEASUREMENTS 107 since the ratio of the mean errors of two quantities is equal to the ratio of their probable errors. Comparing equations (88) and (89), with the aid of equations (55), article fifty, we see that biA 2 + +piA q , (i) a n = p n A q . Multiply each of these equations by its a and add the products, then multiply each by its b and add, and so on until all of the coefficients have been used as multipliers. We thus obtain the q sums [aa], [ba], . . . , [pa], and by taking account of equations (87) we have [aa] = 1, > [ba] = [ca] = . = [pa] = 0. ) Hence, if we multiply each of equations (i) by its a and add the products, we have [aa] = A i. Consequently equation (90) becomes A l (91) The weights of the other x's may be obtained, by an exactly similar process, from equations (88a). The results of such an analysis are as follows: M M a 2 P t (91a) Obviously the coefficients of the sums [as], [bs], etc., in equa- tions (88) and (88a) do not depend upon the particular method by which the normal equations are solved, since the resulting values of the x's must be the same whatever method is used. Conse- quently, if the absolute terms [as], [bs], . . . , [ps] are kept in literal form during the solution of the normal equations by any method whatever, the results may be written in the form of equations 108 THE THEORY OF MEASUREMENTS [ART. 64 (88) and (88a); and the quantities AI, B 2) etc., will be numerical if the coefficients [aa], [ab], . . . , [bb], . . . , [pp] are expressed numerically. Hence, in virtue of (91) and (91 a), we have the following rule for computing the weights of the z's. Retain the absolute terms of the normal equations in literal form, solve by any convenient method, and write out the solution in the form a?i = [as] A! + [bs] A 2 + [cs] A 3 + - + \ps] A qt x 2 = [as] B l + [bs] B 2 + [cs] B 3 + - - - + \ps] B q , x q = [as] P 1 + [bs] P 2 + [cs] P, + - - + [ps] P q . Then the weight of x\ is the reciprocal of the coefficient of [as] in the equation for x\, the weight of x 2 is the reciprocal of the co- efficient of [bs] in the equation for x%, and in general the weight of x q is the reciprocal of the coefficient of [ps] in the equation for x q . As an aid to the memory, it may be noticed that the coefficients AI, B 2 , Cs, . . . , P q , that determine the weights, all lie in the main diagonal of the second members of the above equations. When the number of unknowns is greater than two, the labor of computing all of the A's, B's, etc., would be excessive, and conse- quently it is better to determine the x's by the methods of Chap- ter VII. The essential coefficients AI, B 2 , C 3 , . . . , P q can be determined independently of the others by the method of deter- minants as will be explained later. If the given observations are not of equal weight, the weights of the x's may be determined by a process similar to the above, starting with normal equations in the form of (58), article fifty. The result of such an analysis can be expressed by the rule stated above if we replace the sums [as], [bs], . . . , [ps] by the weighted sums [was], [wbs], . . . , [wps], the notation being the same as in article fifty. 64. Probable Error of a Single Observation. By definition, article thirty-seven, the mean error M 8 of a single observation is given by the expression _ Af + A^+.-.+A.' _ [AA] , (iii) n n where the A's represent the true accidental errors of the s's. When the number of observations is very great, the residuals given ART. 64] ERRORS OF ADJUSTED MEASUREMENTS 109 by equations (54) may be used in place of the A's without causing appreciable error in the computed value of M 8 . But, in most practical cases, n is so small that this simplification is not admis- sible and it becomes necessary to take account of the difference between the residuals and the accidental errors. Let Ui, u 2 , . . . , u q represent the true errors of the x's ob- tained by solution of the normal equations (56). Then the true accidental error of the first observation is given by the relation Ol (Xi + Ui) + 61 (X 2 + U 2 ) + + Pl (X q + U q ) - Si = Ai. But, by the first of equations (54), aiXi + 6ix 2 -f cix s + + pix q si = ri, where r\ is the residual corresponding to the first observation. Combining these equations and applying them in succession to the several observations, we obtain the following expressions for the A's in terms of the r's: ri + aiui + biu 2 + CiU 3 + - + piUq = Ai, A 2 , ,.* + b n u 2 + c n u 3 + + p n u q = A n . Multiply each of these equations by its r and add; the result is [rr] + [ar] HI + [br] u 2 + [cr] u 3 + + [pr] u q = [Ar]. But by equations (iii), article fifty, [ar] = [br] = [cr] = = for] = 0, (v) and, consequently, [rr] = [Ar]. (vi) Multiply each of equations (iv) by its A and add. Then, taking account of (vi), we have [rr] + [aA] Ul + [6A] u 2 + + [pA] u q = [AA]. (vii) In order to obtain an expression for the u's in terms of the A's, multiply each of equations (iv) by its a and add, then multiply by the b's in order and add, and so on with the other coefficients. The first term in each of these sums vanishes in virtue of (v), and we have [aa] ui + [ab] w 2 + + [ap] u q = [aA], [db] Ul + [bb] u, + + \bp] u q = [6A], lap] ui + [bp] u 2 + - - - + [pp] u q = (viii) 110 THE THEORY OF MEASUREMENTS [Am. 64 These equations are in the same form as the normal equations (56) with the z's replaced by u's and the s's by A's. Hence any solu- tion of (56) for the x's may be transposed into a solution of (viii) for the u's by replacing the s's by A's without changing the coeffi- cients of the s's. Consequently, by (89), we have and similar expressions for the other u's. The coefficients of the u's in (vii) expand in the form [aA] = aiAi + a 2 A 2 + + a n A n . Hence [aA] ui = aiaiAi 2 + a 2 2 A 2 2 +.+ a n a n A n 2 , Since positive and negative A's are equally likely to occur, the sum of the terms involving products of A's with different subscripts will be negligible in comparison with the other terms. The sum of the remaining terms cannot be exactly evaluated, but a suffi- ciently close approximation is obtained by placing each of the A 2 's equal to the mean square of all of them, - - -* Consequently, as the best approximation that we can make, we may put n But, by equations (ii), [aa] is equal to unity. Hence [aA] - M. iv Since there is nothing in the foregoing argument that depends on the particular u chosen, the same result would have been obtained with any other u. .Consequently, in equation (vii), each term that involves one of the u's must be equal to - - !i and, since there tv are q such terms, the equation becomes Hence, by equation (iii), and ART. 64] ERRORS OF ADJUSTED MEASUREMENTS 111 where the r's represent the residuals, computed by equations (54) ; n is the number of observations ; and q is the number of unknowns involved in the observation equations (53). In the case of direct measurements, the number of unknowns is one, and (92) reduces to the form already found in article forty-one, equation (30), for the mean error of a single observation. When the observations are not of equal weight, the mean error M 8 of a standard observation, i.e. an observation of weight unity, is given by the expression 2 = n where the w's are the weights of the individual observations. Starting with this relation in place of (iii) and making correspond- ing changes in other equations, an analysis essentially like the preceding leads to the result Ma = ^'^-, (93) T n q which reduces to the same form as (92) when the weights are all unity. Introducing the constant relation between the mean and probable errors, we have the expressions E 8 = 0.6741/-M- , (94) V n q for the probable error of a single observation in the case of equal weights, and E 8 = 0.674\/-^i, (95) V n q for the probable error of a standard observation in the case of different weights. Finally, if M k , E k , and w k represent the mean error, the probable error, and the weight of x k , any one of the unknown quantities, we may derive the following relations from the above equations by applying equations (36), article forty-four: M s - = 7= V ' A/in. T n o (96) 112 THE THEORY OF MEASUREMENTS [ART. 65 when the weights of the given observations are equal, and M k = -^= = L Y/-^-> v Wk vWk n ~ Q E, 0.674 Ek = / - = (97) ~ 2 when the weights of the given observations are not equal. 65 . Application to Problems Involving Two Unknowns . When the observation equations involve only two unknown quantities, the solution of the normal equations is given by (59), article fifty-one, in the form _ [66] [as] - [ab] [bs] [aa] [bb] - [ab] 2 ' _ [aa] [bs] [ab] [as] [aa] [bb] - [ab] 2 By the rule of article sixty-three, the weight of Xi is equal to the reciprocal of the coefficient of [as] in the equation for Xi, and the weight of #2 is equal to the reciprocal of the coefficient of [bs] in the equation for x 2 . Hence, by inspection of the above equations, we have [aa] [bb] - [ab] 2 _ W 2 = [bb] [aa] [bb] - [ab] 2 [aa] (98) Since there are only two unknown quantities, and the observa- tions are of equal weight, equation (92) gives the mean error of a single observation when q is taken equal to two. Hence (99) where n is the number of observation equations and [rr] is the sum of the squares of the residuals that are obtained when the computed values of Xi and Xz are substituted in equations (53a), article fifty-one. Combining equations (98) and (99) with (96), we obtain the following expressions for the probable errors of Xi and x 2 : 0.674 E 2 = 0.674 v/ v/ [66] [aa][bb] - [ab] 2 n-2 [aa] [rr> [aa] [bb] - [ab] 2 n-2 (100) ART. 65] ERRORS OF ADJUSTED MEASUREMENTS 113 For the purpose of illustration, we will compute the probable errors of the values of x\ and x 2 obtained in the numerical prob- lem worked out in article fifty-one. Referring to the numerical tables in that article, we find [aa] = 5; [ab] = 20; [bb] = 90; n = 5; [rr] = 9.60 X 1Q- 4 . Hence, by equations (100), *' V / 5X90-400 By equations (vi), article fifty-one, the length L of the bar at C., and the coefficient of linear expansion a are given by the relations L = iooo + si; a = -L.*. 10 -L70 Since L is equal to #1 plus a constant, its probable error is equal to that of Xi by the argument underlying equation (ii), article sixty. Hence EL. = E! = =fc 0.016. To find the probable error of a, we have by equations (81), article sixty, But, since L is very large in comparison with x 2 , the second term on the right-hand side is negligible in comparison with the first. Consequently, without affecting the second significant figure of the result, we may put = Ei X 10- 4 = =fc 0.038 X 10- 5 . Hence the final results of the computations in article fifty-one may be more comprehensively expressed in the form L Q = 1000.008 db 0.016 millimeters, a = (1.780 db 0.038) X 10~ 5 , 114 THE THEORY OF MEASUREMENTS [AET. 66 when we wish to indicate the precision of the observations on which they depend. 66. Application to Problems Involving Three Unknowns. The normal equations, for the determination of three unknowns, take the form [aa] Xi + [ah] x 2 + [ac] x 3 = [as], [ac] xi + [be] x 2 + [cc] x 3 = [cs]. Solving by the method of determinants and putting we have [as] x 2 = [as [as] Hence, by the rule of article sixty-three, D Wl [bb][cc] -[be] 2 ' = D 2 ~~ [aa] [cc] [ac] 2 ' D [aa][bb]-[ab]*' [aa] [ab] [ac] [ab] [ac [66] [be [be] [cc ] = A [bb] [be] [be] [cc] 1 J [be] [cc] [06] [ac] 4 -[cs] [06] [ac] [bb] [be] t D D D [ac] [cc] [06] [6c] - + [6s] [aa] [ac] [ac] [cc] - -[cs] [ab] [be] [aa] [ac] , D D D [ah] [66] [ac] [6c] + N- [ac] [be] [aa] [ab] + [cs] [aa] [ab] [ab] [bb] D D D w s = (ix) (x) The determinant D can be eliminated from equations (x), if we can obtain an independent expression for any one of the w's. The solution of the normal equations by Gauss's Method in article fifty-four led to the result - X3 ~ [cc'2] ART. 66] ERRORS OF ADJUSTED MEASUREMENTS 115 The auxiliary [cc 2] is independent of the absolute terms [as], [6s], and [cs]. The auxiliary [cs 2] may be expanded as follows: [oc] r , [6cl] ( , [ab] [6cl] ( , ~ PTTJ \ M - Hence the coefficient of [cs] in the above expression for x$ is r - ~y, and, consequently, the weight of x$ is equal to [cc2]. [CC ZJ Substituting this value for w s in the third of equations (x) and eliminating D from the other two we have [aa] [bb 1] [66 (101) w 3 = [cc 2], where the auxiliary quantities [66 1], [cc 1], and [cc 2] have the same significance as in article fifty-four. The weights of the x's having been determined by equations (101), their probable errors may be computed by equations (96). In the present case q is taken equal to three, since there are three unknowns, and the r's are given by equations (68). In the numerical illustration of Gauss's Method, worked out in article fifty-five, we found the following values of the quantities appearing in equations (96) and (101): [aa] = 6; [66] = 220; [6c] = 180; [cc] = 157; [66 1] = 70; [cc 1] = 76.0; [cc 2] = 5.97; [rr] = 0.00120; n = 6; q = 3. These values have been rounded to three significant figures, when necessary, since the probable errors of the #'s are desired to only two significant figures. Substituting in equations (101) we have Wl = 6X7 _ 2 5.97 -1.17, 220 X 157 - 180 70 ^2 = y^5.97 = 5.50, w 3 = 5.97, 116 THE THEORY OF MEASUREMENTS [ART. 66 From equation (94) \E. and, by equations (96), a = 0.674 1/ ' 0012 = 0.0135, 0.0135 . O.UUoo. Consequently the precision of the measurements, so far as it depends on accidental errors, may be expressed by writing the computed values of the x's in the form xi = 0.245 0.012, X2 =- 1.0003 0.0057, z 3 = 1.4022 0.0055. Since the last significant figure in each of the x's occupies the same place as the second significant figure in the corresponding prob- able error, it is evident that the proper number of figures were retained throughout the computations in article fifty-five. CHAPTER X. DISCUSSION OF COMPLETED OBSERVATIONS. 67. Removal of Constant Errors. The discussion of acci- dental errors and the determination of their effect on the result computed from a given series of observations, as carried out in the preceding chapters, are based on the assumption that the meas- urements are entirely free from constant errors and mistakes. Hence the first matter of importance, in undertaking the reduction of observations, is the determination and removal of all constant errors and mistakes. Also, in criticizing published or reported results, judgment is based very largely on the skill and care with which such errors have been treated. In the former case, if suit- able methods and apparatus have been chosen and the adjust- ments of instruments have been properly made, sufficient data is usually at hand for determining the necessary corrections within the accidental errors. In the latter case we must rely on the dis- cussion of methods, apparatus, and adjustments given by the author and very little weight should be given to the reported measurements if this discussion is not clear and 'adequate. No evidence can be obtained from the observations themselves regarding the presence or absence of strictly constant errors. The majority of them are due to inexact graduation of scales, imperfect adjustment of instruments, personal peculiarities of the observer, and faulty methods of manipulation. They affect all of the observations by the same relative amount. Their detec- tion and correction or elimination depend entirely on the judg- ment, experience, and care of the observer and the computer. When the same magnitude has been measured by a number of different observers, using different methods and apparatus, the probability that the constant errors have been the same in all of the measurements is very small. Consequently if the corrected results agree, within the accidental errors of observation, it is highly probable that they are free from constant errors. This is the only criterion we have for the absence of such errors and it 117 118 THE THEORY OF MEASUREMENTS [ART. 67 breaks down in some cases when the measured magnitude is not strictly constant. Sometimes constant errors are not strictly constant but vary progressively from observation to observation owing to gradual changes in surrounding conditions or in the adjustment of instru- ments. The slow expansion of metallic scales due to the heat radiated from the body of the observer is an illustration of a progressive change. Such variations are usually called systematic errors. They may be corrected or eliminated by the same methods that apply to strictly constant errors when adequate means are provided for detecting them and determining the magnitude of the effects produced. When their range in magnitude is compara- ble with that of the accidental errors, their presence can usually be determined by a critical study of the given observations and their residuals. But, if they have not been foreseen and provided for in making the observations, their correction is generally difficult if not impossible. In many cases our only recourse is a new series of observations taken under more favorable conditions and accom- panied by adequate means of evaluating the systematic errors. A general discussion of the nature of constant errors and of the methods by which they are eliminated from single direct observa- tions was given in Chapter III. These processes will now be con- sidered a little more in detail and extended to the arithmetical mean of a number of direct observations. Let a\ t d 2 , a s , . . . , a n represent a series of direct observations after each one of them has been corrected for all constant errors. Then the most prob- able value that can be assigned to the numeric of the measured magnitude is the arithmetical mean x = q i + fl2 + +a n /jx IV Now suppose, that the actual uncorrected observations are 01, o 2 , o 3 , , o n , then ai = 01 + cj + cj' + cj" + + ci<*> = 01 + [cj, a 2 = o 2 + cj + c 2 " + cj" + + c 2 ("> = o 2 + C*n = O n + C n ' + C" + C n '" + + cj* = O n + [c where the c's represent the constant errors to be eliminated and may be either positive or negative. There are as many c's in each equation as there are sources of constant error to be consid- ART. 67] DISCUSSION OF COMPLETED OBSERVATIONS 119 ered. Usually, when all of the observations are made by the same method and with equal care, the number of c's is the same in all of the equations. Substituting (ii) in (i) J . = 0l + 02+ +. [Cj + [cj+ - - +[ftj n n ' When there are no systematic errors Cl = Cz = C 3 ' = Cl " = C 2 " = C," = = Cn " = C ", = C 3 ' = * = Cn Consequently [ci] = [c z ] = [c 3 ] = = [c n ] = [c], (iv) and we have x = + [c] n = Om + c' + c" + c"' + -f c<>, (102) where o m is written for the mean of the actual observations. Hence, when all of the observations are affected by the same con- stant errors, the corrections may be applied to the arithmetical mean of the actual observations and the resulting value of x will be the same as if the observations were separately corrected before taking the mean. The residuals corresponding to the corrected observations ai, a 2 , a 3 , . . . , a n are given by equations (3), article twenty-two. Replacing x and the a's by their values in terms of o m and the o's as given in (102) and (ii), and taking account of (iv), equations (3) become ri = di X = Oi+ [Ci] - Om- [C\ = 01 - O m , r 2 = a 2 x = o 2 + [c 2 ] o m [c] = o 2 o m , (103) r n = a n - X = O n + [C n ] -Om- [c] = O n - O m . Consequently, when there are no systematic errors, the residuals computed from the o's and o m will be identical with those com- puted from the a's and x. Hence, if the uncorrected observations are used in computing the probable error of x, by the formula / W E = 0.674\/ / J 1X > V n (n 1) 120 THE THEORY OF MEASUREMENTS [ART. 67 the result will be the same as if the corrected observations had been used; and, as pointed out above, the observations and their corresponding residuals give no evidence of the presence of strictly constant errors. When the constant errors affecting the different observations are different or when any of them are systematic in character, equation (iv) no longer holds, and, consequently, the simplifica- tion expressed by (102) is no longer possible. In the former case the observations should be individually corrected before the mean is taken. The same result might be obtained from equation (iii), but the computation would not be simplified by its use. In the latter case the several observations are affected by errors due to the same causes but varying progressively in magnitude in response to more or less continuous variations in the conditions under which they are made. In equations (ii) the c's having the same index may be con- sidered to be due to the same cause, but to vary in magnitude from equation to equation as indicated by the subscripts. The arithmetical means of the errors due to the same causes are , _ Ci' + C 2 ' + + C n ' Cm '~ ~ _ Cm - n and the mean of the observations is 01 + 02 + ' ' ' O m = n Substituting (ii) in (i) and taking account of the above relations we have X = O m + C m ' + C m " + ' ' ' + C w <> . (104) Hence, in the case of systematic errors, the most probable value of the numeric of the measured magnitude may be obtained from the mean of the uncorrected observations by applying mean cor- rections for the systematic errors. When all of the errors are strictly constant equation (104) becomes identical with (102) because all of the errors having the same index are equal. Obvi- ART. 68] DISCUSSION OF COMPLETED OBSERVATIONS 121 ously it also holds when part of the c's are strictly constant and the remainder are systematic. If we use the value of x given by (104) in place of that given by (102) in the residual equations (103), the c's will not cancel. Hence, if any of the constant errors are systematic in nature,. the residuals computed from the o's and o m will be different from those computed from the a's and x; and, consequently, they will not be distributed in accordance with the law of accidental errors. In practice it is generally advisable to correct each of the ob- servations separately before taking the mean rather than to use equation (104), since the true residuals are required in computing the probable error of x, and they cannot be derived from the un- corrected observations. Whenever possible the conditions should be so chosen that systematic errors are avoided and then the necessary computation can be made by equations (102) and (103). 68. Criteria of Accidental Errors. We have seen that the residuals computed from observations affected by systematic errors do not follow the law of accidental errors. Hence, if it can be shown that the residuals computed from any given series of obser- vations are distributed in conformity with the law of errors, it is probable that the given observations are free from systematic errors or that such errors are negligible in comparison with the accidental errors. Observations that satisfy this condition may or may not be free from strictly constant errors, but necessary corrections can be made by equation (102) and the probable error of the mean may be computed from the residuals given by equation (103). Systematic errors should be very carefully guarded against in making the observations, and the conditions that produce them should be constantly watched and recorded during the progress of the work. After the observations have been completed they should be individually corrected for all known systematic errors before taking the mean. The strictly constant errors may then be removed from the mean, but before this is done it is well to compute the residuals and see if they satisfy the law of accidental errors. If they do not, search must be made for further causes of systematic error in the conditions surrounding the measure- ments and a new series of observations should be made, under more favorable conditions, whenever sufficient data for this pur- pose is not available. 122 THE THEORY OF MEASUREMENTS [ART. 68 Residuals, when sufficiently numerous, follow the same law of distribution as the true accidental errors. Consequently system- atic errors and mistakes might be detected by a direct comparison of the actual distribution with the theoretical, as carried out in article thirty-four, provided the number of observations is very large. However, in most practical measurements, the residuals are not sufficiently numerous to fulfill the conditions underlying the law of errors, and a considerable difference between their actual and theoretical distribution is quite as likely to be due to^ this fact as to the presence of systematic errors. Whatever the number of observations, a close agreement between theory and practice is strong evidence of the absence of such errors but it is seldom worth while to carry out the comparison with less than one hundred residuals. When the residuals are numerous and distributed in the same manner as the accidental errors, the average error of a single observation, computed by the formula Vn(n- 1)' and the mean error, computed by the formula satisfy the relation M = 1.253 A. Also the formulae E = 0.8453 A and E = 0.6745 M give the same value for the probable error of a single observation. When the number of observations is limited, exact fulfillment of these relations ought not to be expected, but a large deviation from them is strong evidence of the presence of systematic errors or mistakes. Unless the number of observations is very small, ten or less, the relations should be fulfilled within a few units in the second significant figure, as is the case in the numerical example worked out in article forty-two. Obviously the arithmetical mean is independent of the order in which the observations are arranged in taking it, but the order of the residuals in regard to sign and magnitude depends on the order of the observations. When there are systematic errors and the observations are arranged in the order of progression of their ART. 68] DISCUSSION OF COMPLETED OBSERVATIONS 123 cause, the residuals will gradually increase or decrease in absolute magnitude in the same order; and, if the systematic errors are large in comparison with the accidental errors, there will be but one change of sign in the series. Thus, if the temperature is gradually rising while a length is being measured with a metallic scale and the observations are arranged in the order in which they are taken, the first half of them will be larger than the mean and the last half smaller, except for the variations caused by accidental errors. For the purpose of illustration, suppose that the observa- tions are 1001.0; 1000.9; 1000.8; 1000.7; 1000.6; 1000.5; 1000.4. The mean is 1000.7 and the residuals + .3; +.2; +.1; .0; -.1; -.2; -.3 decrease in absolute magnitude from left to right, i.e., in the order in which the observations were made. There are five cases in which the signs of succeeding residuals are alike and one in which they are different; the former cases will be called sign-follows and the^latter a sign-change. This order of the residuals in regard to magnitude and sign is typical of observations affected by sys- tematic errors when they are arranged in conformity with the changes in surrounding conditions. Since such changes are usually continuous functions of the time, the required arrangement is generally the order in which the observations are taken. Such extreme cases as that illustrated above are seldom met with in practice owing to the impossibility of avoiding accidental errors of observation and the complications they produce in the sequence of residuals. Generally the systematic errors that are not readily discovered and corrected before making further re- ductions are comparable in magnitude with the accidental errors. Consequently they cannot control the sequence in the signs of the residuals but they do modify the sequence characteristic of true accidental errors. In any extended series of observations there should be as many negative residuals as positive ones, since positive and negative errors are equally likely to occur. After any number of observations have been made, the probability that the residual of the next obser- vation will be positive is equal to the probability that it will be nega- tive, since the possible number of either positive or negative errors is infinite. Consequently the chance that succeeding residuals 124 THE THEORY OF MEASUREMENTS [ART. 69 will have the same sign is equal to the chance that they will have different signs. Hence, if the residuals are arranged in the order in which the corresponding observations were made, the number of sign-follows should be equal to the number of sign-changes. The residuals, computed from limited series of observations, seldom exhibit the theoretical sequence of signs exactly because they are not sufficiently numerous to fulfill the underlying condi- tions. Nevertheless, a marked departure from that sequence suggests the presence of systematic errors or mistakes and should lead to a careful scrutiny of the observations and the conditions under which they were made. If the disturbing causes cannot be detected and their effects eliminated, it is generally advisable to repeat the observations under more favorable conditions. The numerical example, worked out in article forty-two, may be cited as an illustration from practice. The observations were made in the order in which they are tabulated, beginning at the top of the first column and ending at the bottom of the fourth column. In the second and fifth columns we find ten positive and ten negative residuals. The number of sign-follows is ten and the number of sign-changes is nine. This is rather better agreement with the theoretical sequence of signs than is usually obtained with so few residuals. It indicates that the observations were made under favorable conditions and are sensibly free from systematic errors but it gives no evidence whatever that strictly constant errors are absent. Although the foregoing criteria of accidental errors are only approximately fulfilled when the number of observations is lim- ited, their application frequently leads to the detection and elimi- nation of unforeseen systematic errors. The first method is rather tedious and of little value when less than one hundred obser- vations are considered, but the last two methods may be easily carried out and are generally exact enough for the detection of systematic errors comparable in magnitude with the probable error of a single observation. 69. Probability of Large Residuals. In discussing the dis- tribution of residuals in regard to magnitude, the words large and small are used in a comparative sense. A large residual is one that is large in comparison with the majority of residuals in the series considered. Thus, a residual that would be classed as large in a series of very precise observations would be considered small in ART. 69] DISCUSSION OF COMPLETED OBSERVATIONS 125 dealing with less exact observations. Consequently, in expressing the relative magnitudes of residuals, it is customary to adopt a unit that depends on the precision of the measurements considered. The probable error of a single observation is the best magnitude to adopt for this purpose, since it is greater than one-half of the errors and less than the other half. If we represent the relative magnitude of a given error by S, the actual magnitude by A, and the probable error of a single observation by E, S = |- (105) The relative magnitudes of the residuals may be represented in the same way by replacing the error A by the residual r. It is obvious that values of S less than unity correspond to small re- siduals and values greater than unity to large residuals in any series of observations. In equation (13), article thirty-three, the probability that an error chosen at random is less than a given error A is expressed by the integral */~ A o / v j PA = -^= e-*dt. (13) V-n-Jo Equation (25), article thirty-eight, may be put in the form V ** k & = 7= -> VTT a? where $ is written for the numerical constant 0.47694. Hence, introducing (105), and (13) becomes P 8 = 'eft. (106) Obviously this integral expresses the probability that an error chosen at random is less than S times the probable error of a single observation. It is independent of the particular series to which the observations belong and its values, corresponding to a series of values of the argument S, are given in Table XII. Since all of the errors in any system are less than infinity, Poo is equal to unity. Hence the probability that a single error, 126 THE THEORY OF MEASUREMENTS [ART. 69 chosen at random, is greater than S times E is given by the rela- tion Qs = 1 - Pa- (V) Now the residuals, when sufficiently numerous and free from systematic errors and mistakes, should follow the same distri- bution as the accidental errors. Hence, if n s is the number of residuals numerically greater than SE and N is the total number in any series of observations, we should have Qs = T?" (vi) Since the numerical value of P 8 , and consequently that of Q 8 depends only on the limit S and is independent of the precision of the particular series of measurements considered, the ratio jj. > corresponding to any given limit S, should be the same in all cases. Consequently, if N observations have been made on any magnitude and by any method whatever, n 8 of them should corre- spond to residuals numerically greater than SE. Conversely, if we assign any arbitrary number to n a , equation (vi) defines the number of observations that we should expect to make without exceeding the assigned number of residuals greater than SE. Hence, if N a is the number of observations among which there should be only one residual greater than S times the probable error of a single observation, we have, by placing n s equal to one in (vi), and substituting the value of Q 8 from (v), *--r^>r (107) The fourth column of the following table gives the values of N a , to the nearest integer, corresponding to the integral values of the limit S given in the first column. The values of P 8 in the second column are taken from Table XII, and those of Q 8 in the third column are computed by equation (v). S P. e. N s 1 0.50000 0.50000 2 2 0.82266 0.17734 6 3 0.95698 0.04302 23 4 0.99302 0.00698 143 5 0.99926 0.00074 1351 ART. 70] DISCUSSION OF COMPLETED OBSERVATIONS 127 To illustrate the significance of this table, suppose that 143 direct observations have been made on any magnitude by any method whatever. The probable error E of a single observation in this series may be computed from the residuals by equation (31) or (34). Then, if the residuals follow the law of errors, not more than one of them should be greater than four times as large as E. If the number of observations had been 1351, we should expect to find one residual greater than five times E, and on the other hand if the number had been only twenty-three, not more than one residual should be greater than three times E. Although the probability for the occurrence of large residuals is small, and very few of them should occur in limited series of observations, their distribution among the observations, in respect to the order in which they occur, is entirely fortuitous. A large residual is as likely to occur in the first, or any other, observation of an extended series as in the last observation. Con- sequently the limited series of observations, taken in practice, frequently contain abnormally large residuals. This is not due to a departure from the law of errors, but to a lack of sufficient observations to fulfill the theoretical conditions. In such cases there are not enough observations with normal residuals to balance those with abnormally large ones. Consequently a closer approxi- mation to the arithmetical mean that would have been obtained with a more extended series of observations is obtained when the abnormal observations are rejected from the series before taking the mean. Observations should not be rejected simply because they show large residuals, unless it can be shown that the limit set by the theory of errors, for the number of observations considered, is exceeded. This can be judged approximately by comparing the residuals of the given observations with the numbers given in the first and last columns of the above table, but a more rigorous test is obtained by applying Chauvenet's Criterion, as explained in the following article. 70. Chauvenet's Criterion. The probability that the error of a single observation, chosen at random, is less than SE is expressed by P a in equation (106). Now, the taking of N inde- pendent observations is equivalent to N selections at random from the infinite number of possible accidental errors. Hence, by equation (7), article twenty-three, the probability that each of 128 THE THEORY OF MEASUREMENTS [ART. 70 the N observations in any series is affected by an error less than SE is equal to P N . Since all of the N errors must be either greater or less than SE } the probability that at least one of them is greater than this limit is equal to 1 P 8 N . Placing this probability equal to one-half, we have i - P." = i, or P. - (1 - (vii) If the limit S is determined by this equation, there is an even chance that at least one of the N observations is affected by an error greater than SE. Expanding the second member of (vii) by the Binomial Theorem 11 N -I I (N- l)(2N-l) 1 N 2 1-2-N 2 4 1-2- 3- N* 8 1-2-3 . . . K-N K The terms of this series decrease very rapidly and all but the first are negative. Consequently the sum of the terms beyond the second is small in comparison with the other two; and, whatever the value of N, (1 %) N is nearly equal to, but always slightly less than, - ^-^ - . Since P 8 and S increase together, the limit T determined by the relation 2N-1 2N (108) is slightly greater than the limit S determined by (vii). Hence, if N independent direct observations have been made, the prob- ability against the occurrence of a single error greater than A r = TE (109) is greater than the probability for its occurrence. Consequently, if the given series contains a residual greater than A r , the prob- able precision of the arithmetical mean is increased by excluding the corresponding observation. ART. 70] DISCUSSION OF COMPLETED OBSERVATIONS 129 Equations (108) and (109) express Chauvenet's Criterion for the rejection of doubtful observations. In applying them, the prob- able error E of a single observation is first computed from the residuals of all of the observations by either equation (31) or the first of equations (34) with the aid of Table XIV or XV. If any of the residuals appear large in comparison with the computed value of E, PT is determined from (108) by placing N equal to the number of observations in the given series. T is then obtained by interpolation from Table XII, and finally A r is computed by (109). If one or more of the residuals are greater than the com- puted A r , the observation corresponding to the largest of them is excluded from the series and the process of applying the criterion is repeated from the beginning. If one or more of the new residuals are greater than the new value of A r , the observation correspond- ing to the largest of them is rejected. This process is repeated and observations rejected one at a time until a value of A r is ob- tained that is greater than any of the residuals. When more than one residual is greater than the computed value of Ay, only the observation corresponding to the largest of them should be rejected without further study. The rejection of a single observation from the given series changes the arith- metical mean, and hence all of the residuals and the value of E computed from them. If r and r' are the residuals corresponding to the same observation before and after the rejection of a more faulty observation, and if A r and A r ' are the corresponding limiting errors, it may happen that r' is less than A/, although r is greater than Ay. Hence the second application of the criterion may show that a given observation should be retained notwith- standing the fact that its residual was greater than the limiting error in the first application, provided an observation with a larger residual was excluded on the first trial. To facilitate the computation of Ay, the values of T corre- sponding to a number of different values of N have been interpolated from Table XII and entered in the second column of Table XIII. For the purpose of illustration, suppose that ten micrometer settings have been made on the same mark and recorded, to the nearest tenth of a division of the micrometer head, as in the first column of the following table. 130 THE THEORY OF MEASUREMENTS [ART. 71 Obs. r r' 2.567 +0.0118 2.559 +0.0038 +0.0051 2.556 +0.0008 +0.0021 2.552 -0.0032 -0.0019 2.551 -0.0042 -0.0029 2.553 -0.0022 -0.0009 2.555 -0.0002 +0.0011 2.548 -0.0072 -0.0059 2.554 -0.0012 +0.0001 2.557 +0.0018 +0.0031 x =2.5552 [r] = 0.0364 [r>] = 0.0231 z'=2.5539 # = 0.0032 #' = 0.0023 IF = 2. 91 T' = 2.84 Ar = 0.0093 A/ = 0.0065 The residuals, computed from the mean x, are given under r. The probable error E } computed from [r] by the first of equations (34), with the aid of Table XV, is 0.0032. The value of T corre- sponding to ten observations is 2.91 from Table XIII, and the limiting error Ay is equal to 0.0093. Since this is less than the residual 0.0118, the corresponding observation (2.567) should be rejected from the series. The mean of the retained observations, xi, is 2.5539, and the corresponding residuals are given under r' in the third column of the above table. The new value of the limiting error (A/), com- puted by the same method as above, is 0.0065. Since none of the new residuals are larger than this, the nine observations left by the first application of the criterion should all be retained. 71. Precision of Direct Measurements. The first step in the reduction of a series of direct observations is the correction of all known systematic errors and the test of the completeness of this process by the criteria of article sixty-eight. In general, the systematic errors represent small variations of otherwise constant errors; and, in making the preliminary corrections, it is best to consider only this variable part, i.e., the corrections are so applied that all of the corrected observations are left with exactly the same constant errors. Thus, suppose that the temperature of a scale is varying slowly during a series of observations, and is never very near to the temperature at which the scale is standard. It is better to correct each observation to the mean temperature of the scale and leave the larger correction, from mean to standard ART. 71] DISCUSSION OF COMPLETED OBSERVATIONS 131 temperature, until it can be applied to the arithmetical mean in connection with the corrections for other strictly constant errors. This is because the systematic variations in the length of the scale are so small that the unavoidable errors in the observed temperatures and the adopted coefficient of expansion of the scale can produce no appreciable effect on the corrections to mean temperature. The effect of these errors on the larger correction from mean to standard temperature is more simply treated in connection with the arithmetical mean than with the individual observations. Let 01, 02, . . . , o n represent a series of direct observations corrected for all known systematic errors and satisfying the criteria of accidental errors. We have seen that the most prob- able value that we can assign to the numeric of the measured mag- nitude, on the basis of such a series, is given by the relation x = o m + c'+c"+ - +cfe>, (102) where o m is the arithmetical mean of the o's, and the c's represent corrections for strictly constant errors. If the c's could be deter- mined with absolute accuracy, or even within limiting errors that are negligible in comparison with the accidental errors of the o's, the only uncertainty in the above expression for x would be that due to the accidental error of o m . Hence, by equations (103), if E x and E m are the probable errors of x and o m , respectively, we should have *. = *_ = 0.674 Vy '.' (HO) . . If we follow the usual practice and regard the probable error of a quantity as a measure of the accidental errors of the observations from which it is directly computed, equation (110) still holds when the accidental errors of the c's are not negligible; but, as we shall see, E x is no longer a complete measure of the precision of x in such cases. In practice each of the c's must be computed, on theoretical grounds, from subsidiary observations with the aid of physical constants that have been previously determined by direct or indirect measurements. For the sake of brevity the quantities on which the c's depend will be called correction factors. Since all of them are subject to accidental errors, the computed c's are affected by residual errors of indeterminate sign and magnitude. 132 THE THEORY OF MEASUREMENTS [ART. 71 When the probable errors of the correction factors are known the probable errors of the c's may be computed by the laws of propa- gation of errors with the aid of the correction formulae by which the c's are determined. Equation (102) gives x as a continuous sum of o m and the c's. Consequently, if we represent the probable errors of the c's by Ei t E 2 , . . . , E q , respectively, we have by equation (76), article fifty-eight, R x 2 = E m * + Ei* + +E q *, (111) wnere R x is the resultant probable error of x due to the correspond- ing errors of o m and the c's. To distinguish R x from the probable error E X) which depends only on the accidental error of o m , we shall call it the precision measure of x. Although equation (111) is simple in form, the separate compu- tation of the E'SJ from the errors of the correction factors on which they depend, is frequently a tedious process. Moreover several of the c's may depend on the same determining quantities. Con- sequently the computation of x and R x is frequently facilitated by bringing the correction factors into the equation for x explicitly, rather than allowing them to remain implicit in the c's. Thus, if a, )8, . . . , p represent the correction factors on which the c's depend, equation (102) may be put in the form x = F(o m ,a,0, . . . , P). (112) Hence, by equation (81), article sixty, where E a , Ep, etc., are the probable errors of a, ft, etc. For example, suppose that o m represents the mean of a num- ber of observations of the distance between two parallel lines expressed in terms of the divisions of the scale used in making the measurements. Let t\ represent the mean temperature of the scale during the observations; L the mean length of the scale divisions at the standard temperature U, in terms of the chosen unit; a the coefficient of expansion of the scale; and ft the angle between the scale and the normal to the lines. Then, if the individual observations have been corrected to mean temperature ti before computing the mean observation o m , the best approxima- ART. 71] DISCUSSION OF COMPLETED OBSERVATIONS 133 tion that we can make to the true distance between the lines is given by the expression x = o m L\l]+a(ti - t ) I , in which the correction factors L, a, /?, fa, and to appear explicitly , as in the general equation (112). A more detailed discussion of this example will be found in article seventy-three. If we represent the separate effects of the errors E m , E a , . . . , E p on the error R x by D m , D a , D$, . . . , D PJ respectively, we have *-*/ D - - S E *-> :.:i ' D > * T P E < m > and (113) becomes R* 2 = D m * + D a 2 + Df + - - - + D P 2 . (115) In some cases the fractional effects _Drn, _D. . _D, m ~ x ' a ~ x ' ' ' ' p ~ x can be more easily computed numerically than the corresponding D's. When this occurs, the fractional precision measure is first computed and then R x is determined by the relation R x = x-P x . (117) While equations (112) to (117) are apparently more complicated than (102) and (111), they generally lead to more simple numerical computations. Moreover the probable errors of some of the correction factors are frequently so small that they produce no appreciable effect on R x . When either equation (115) or (116) is used, such cases are easily recognized because the corresponding D's or P's are negligible in comparison with D m or P m . Obvi- ously the same condition applies to the E's in equation (111), but the numerical computation of either the D's or the P's is generally more simple than that of the E's in (111) because approximate values of o m and the correction factors may be used in evaluat- ing the differential coefficients in (114). The allowable degree of approximation, the limit of negligibility of the D's, and some other 134 THE THEORY OF MEASUREMENTS [ART. 71 details of the computation will be discussed more extensively in the next article. If the true numeric of the measured magnitude is represented by Xj the final result of a series of direct measurements may be expressed in the form X = xR x , (118) where x is the most probable value that can be assigned to X on the basis of the given observations, and R x is the precision measure of x. In practice x may be computed by either equation (102) or (112), or the arithmetical mean of the individually corrected observations may be taken, and R x is given by equations (111), (115), or (117), the choice of methods depending on the nature of the given data and the preference of the computer. The exact significance of equation (118) should be carefully borne in mind, and it should be used only when the implied condi- tions have been fulfilled. Briefly stated, these conditions are as follows : 1st. The accidental errors of the observations on which x depends follow the general law of such errors. 2nd. A careful study of the methods and apparatus used has been made for the purpose of detecting all sources of constant or systematic errors and applying the necessary corrections. 3rd. The given value of x is the most probable that can be computed from the observations after all constant errors, system- atic errors, and mistakes have been as completely removed as possible. 4th. The resultant effect of all sources of error, whether acci- dental errors of observation or residual errors left by the correc- tions for constant errors, is as likely to be less than R x as greater than R x . The expressions in the form X = x E x , used in preceding chapters, are not violations of the above principles because, in those cases, we were discussing only the effects of accidental errors and the observations were assumed to be free from all con- stant errors and mistakes. Such ideal conditions never occur in practice. Consequently R x should not be replaced by E x in expressing the result of actual measurements in the form of equa- tion (118), unless it can be shown by equation (115), and the given data that the sum of the squares of the D's corresponding to all of the correction factors is negligible in comparison with Z) m 2 . ART. 72] DISCUSSION OF COMPLETED OBSERVATIONS 135 In the latter case E x and R x are identical as may be easily seen by comparing equations (110), (111), and (115). 72. Precision of Derived Measurements. When a desired numeric Z is connected with the numerics Xi, X 2 , . . . , X q of a number of directly measured magnitudes by the relation XQ = F (Xi, X%, . . . , X q ), the most probable value that we can assign to X Q is given by the expression x = F(x 1 ,xt, . . . , x q ), (119) where the x's are the most probable values of the X's with corre- sponding subscripts. Each of the component x's, together with its precision measure, can be computed by the methods of the pre- ceding article. The precision measure of X Q may be computed with the aid of equation (81), article sixty, by replacing the E's in that equation by the R's with corresponding subscripts. Sometimes the numerical computations are simplified and the discussion is clarified by bringing the direct observations and the correction factors explicitly into the expression for XQ. If o a , Ob, . . . , Op are the arithmetical means of the direct observa- tions, after correction for systematic errors, on which Xi, x z , . . . , x q respectively depend, and a, /?, . . . , p are the correction factors involved in the constant errors of the observations, equa- tion (119) may be put in the form x = d (o a , o b , . . . , o p , a, j8, . . . , p). (120) The function 6 is always determinable when the function F in (119) is given and the correction formulae for the constant errors are known. Representing the precision measure of XQ by R , and adopting an obvious extension of the notation of the preceding article, we have, by equation (81), Introducing the separate effects of the E's, *-*' ' ' ' = *=l^' (121) becomes *' ' ' ' ; '-*- (122) . (123) 136 THE THEORY OF MEASUREMENTS [ART. 72 The fractional effects of the E's are P _. . P =5*. P = ^. . P _A? ^ " XQ ' ' p x ' a Z ' p " X Q ' and the fractional precision measure of x is given by the relation XQ When the numerical computation of the P's is simpler than that of the D's, PO is first computed by equation (124) and then RQ is determined by the relation #o = z Po. (125) The expression of the final result of the observations and com- putations in the form XQ = XQ RQ has exactly the same significance with respect to X Q , XQ, and R Q that (118) has with respect to X, x, and R x . It should not be used until all of the underlying conditions have been fulfilled as pointed out in the preceding article. Confusion of the precision measure R with the probable error E 0) and insufficient rigor in eliminating constant errors have led many experimenters to an entirely fictitious idea of the precision of their measurements. When the correction factors are explicitly expressed in the reduction formulae, as in equations (112) and (120), the only difference between the expressions for direct and derived measure- ments is seen to lie in the greater number of directly observed quantities, o a , o&, etc., that appear in the latter equation. The same methods of computation are available in both cases and the following remarks apply equally well to either of them. For practical purposes, the precision measure R is computed to only two significant figures and the corresponding x is carried out to the place occupied by the second significant figure in R. The reasons underlying this rule have been fully discussed in article forty-three, in connection with the probable error, and need not be repeated here. In computing the numerical value of the differential coefficients in equations (113), (114), (121), and (122), the observed components, o m , o a , o&, etc., and the correc- tion factors, a, , etc., are rounded to three significant figures, and those that affect the result by less than one per cent are neg- lected. This degree of approximation will always give R within ART. 72] DISCUSSION OF COMPLETED OBSERVATIONS 137 one unit in the second significant figure and usually decreases the labor of computation. Generally the components o m , o a , o b , etc., represent the arith- metical means of series of direct observations that have been corrected for systematic errors. In such cases the corresponding probable errors E mt E a , Eb, etc., can be computed, by equations in the form of (110), from the residuals determined by equations in the form of (103), with the aid of the observations on which the o's depend. If the observations are sufficiently numerous, the computation of the .27's.may be simplified by using formulae depending on the average error in the form E = 0.845 fl=> (34) n Vn 1 where [f] is the sum of the residuals without regard to sign and n is the number of observations. If the observations on which any of the o's depend are not of equal weight, the general mean should be used in place of the arithmetical mean and the corresponding probable errors should be computed by equations (41), (42), or (44), depending on the circumstances of the observations. The o's in equation (120) are supposed to represent simultane- ous values of the directly observed magnitudes. When any of these quantities are continuous functions of the time, or of any other independent variables, it frequently happens that only a single observation can be made on them that is simultaneous with the other components. In such cases this single observation must be used in place of the corresponding o in (120), and its probable error must be determined for use in equation (122). For the latter purpose, it is sometimes possible to make an auxil- iary series of observations under the same conditions that pre- vailed during the simultaneous measurements except that the independent variables are controlled. The required E may be assumed to be equal to the probable error of a single observation in the auxiliary series. Consequently it may be computed by formulae in the form, E = 0.674* /W E = 0.845 n- I or [r] 138 THE THEORY OF MEASUREMENTS [ART. 72 where n is the number of auxiliary observations, and the r's are the corresponding residuals. In some cases this simple expedient is not available; and approximate values must be assigned to the E's on theoretical grounds, depending on the nature of the meas- urements; or more or less extensive experimental investigations must be undertaken to determine their values more precisely. Such investigations are so various in character and their utility depends so much on the skill and ingenuity of the experimenter, that a detailed general discussion of them would be impossible. They may be illustrated by the following very common case. Suppose that one of the components in equation (120) repre- sents the gradually changing temperature of a bath. In com- puting x Q we must use the thermometer reading o t taken at the time the other components are observed. The errors of the fixed points of the thermometer and its calibration errors enter the equation among the correction factors a, /?, etc., and do not con- cern us in the present discussion. In order to determine the probable error of o t , the temperature of the bath may be caused to rise uniformly, through a range that includes o t , by passing a constant current through an electric heating coil, or the bath may be allowed to cool off gradually by radiation. In either case the rate of change of temperature should be nearly the same as prevailed when o t was observed. A series of corresponding obser- vations of the time T and the temperature t are made under these conditions, and the empirical relation between T and t is determined graphically or by the method of least squares. The probable error of o t may be assumed to be equal to the probable error of a single observation of t in this series, and may be com- puted by equation (94), article sixty-four. Some of the correction factors a, ft, etc., appearing as com- ponents in equations (112) and (120), represent subsidiary obser- vations, and some of them represent physical constants. The subsidiary observations may be treated by the methods outlined above. When the highest attainable precision is desired, the physical constants, together with their probable errors, must be determined by special investigation. In less exact work they may be taken from tables of physical constants. Such tabular values seldom correspond exactly to the conditions of the experi- ments in hand and their probable errors are seldom given. Generally a considerable range of values is given, and, unless ART. 72] DISCUSSION OF COMPLETED OBSERVATIONS 139 there is definite reason in the experimental conditions for the selection of a particular value, the mean of all of them should be adopted and its probable error placed equal to one-half the range of the tabular values. The deviations of the tabular values from the mean are due more to differences in experimental conditions and in the material treated than to accidental errors. Conse- quently a probable error calculated from the deviations would have no significance unless these differences could be taken into account. The selection of suitable values from tables of physical constants requires judgment and experience, and the general statements above should not be blindly followed. In many cases the original sources of the data must be consulted in order to determine the values that most nearly satisfy the conditions of the experiments in hand. In good practice the conditions of the experiment are usually so arranged that the D's, in equation (123), corresponding to the direct observations o a , o&, etc., are all equal. None of the D's corresponding to correction factors should be greater than this limit, but it sometimes happens that some of them are much smaller. Since R is to be computed to only two significant figures, any single D which is less than one-tenth of the average of the other D's may be neglected in the computation. If the sum of any number of D's is less than one-tenth of the average of the remaining D's they may all be neglected. A somewhat more rigorous limit of rejection can be developed for use in plan- ning proposed measurements, but it is scarcely worth while in the present connection since the correction factors and all other quantities must be taken as they occurred in the actual measure- ments, and negligible D's are very easily distinguished by inspec- tion after a little experience. After #o has been determined, x may be computed by either equation (119) or (120). If (119) is used the x's must first be determined by (102) or (112). Sometimes the computation may be facilitated by using a modification of (120), in which some of the correction factors appear explicitly while others are allowed to remain implicit in the z's to which they apply. Such cases cannot be treated generally, but must be left to the ingenuity of the computer. Whatever formula is used, the observed quanti- ties and the correction factors should be expressed by sufficient significant figures to give the computed X Q within a few units in 140 THE THEORY OF MEASUREMENTS [ART. 73 the place occupied by the second significant figure of R . Occa- sionally the total effect of one or more of the correction factors is less than this limit and may be neglected in the computation. For f$ W 7? a single factor, say a, this is the case when a is less than ~ 73. Numerical Example. The following illustration repre- sents a series of measurements taken for the purpose of cali- brating the interval between the twenty-fifth and seventy-fifth graduations on a steel scale supposed to be divided in centimeters. The observations were made with a cathetometer provided with a brass scale and a vernier reading to one one-thousandth of a division. One division of the level on this instrument corre- sponds to an angular deviation of 3 X 10~ 4 radians, and the ad- justments were all well within this limit. The steel scale was placed in a vertical position with the aid of a plumb-line, and, since a deviation of one-half, millimeter per meter could have been easily detected, the error of this adjustment did not exceed 5 X 10~ 4 radians. Consequently the angle between the two scales was not greater than 8 X 10~ 4 radians, and it may have been much smaller than this. The temperature of the scales was determined by mercury in glass thermometers hanging in loose contact with them. The probable error of these determinations was estimated at five-tenths of a degree centigrade, due partly to looseness of contact and partly to an imperfect knowledge of the calibration errors of the thermometers. Twenty independent observations, when tested by the last two criteria of article sixty-eight, showed no evidence of the pres- ence of systematic errors or mistakes. Consequently the mean o m , in terms of cathetometer scale divisions, and its probable error E m were computed before the removal of constant errors. The following numerical data represents the results of the obser- vations and the known calibration constants of the cathetometer. Mean temperature of the steel scale, T 20 0.5 C. Mean temperature of the brass scale, ti 21.3 =t 0.5 C. Mean of twenty observations on the measured interval in terms of brass scale divisions, o m . . 50.0051 db 0.0015 scale div. Mean length, at standard temperature, of the brass scale divisions in the interval used, S. . 0.999853 d= 0.000024 cm. Standard temperature of brass scale, t 15.0 C. Coefficient of linear expansion of brass scale, a. (182 12) X 10~ 7 . Angle between two scales, /3, less than 8 X 10- 4 rad. ART. 731 DISCUSSION OF COMPLETED OBSERVATIONS 141 The most probable value that can be assigned to the measured interval is given by the expression Since ft is a very small angle, -- - may be treated by the approxi- COS p mate formulae of Table VII, and the above expression becomes where t = fa-to. The quantity S (1 -f- at) is very nearly equal to unity. Hence, neglecting small quantities of the second and higher orders, the correction due to the angle ft is < 0.000016. Since this is less than two per cent of the probable error of o m , it is negligible in comparison with the accidental errors of observation. Consequently the precision of x is not increased by retaining the term involving ft, and we may put x = OmS (1 + at). (a) The probable error of t Q is zero, because the accidental errors of the temperature observations, made during the calibration of the brass scale, are included in the probable errors of S and a com- puted by the method of article sixty-five. Consequently the probable error of t is equal to that of fa, and we have t = 6,3 0.5 C. In the present case equation (115) is the most convenient for computing the precision measure ,.R X of x. Only two significant figures are to be retained in the separate effects computed by equation (114). Consequently the factor (1 + at) may be taken equal to unity, and the numerical values of o m and S may be rounded to three significant figures for the purpose of this com- putation. Thus, taking o m equal to 50.0, S equal to 1.00, and the other data as given above, we have 142 THE THEORY OF MEASUREMENTS [ART. 73 D m = -E m = S(l+ at) E m = 1 X E m = 0.0015. oo m D,= ~ Q E t =o m (l + at) E,= 50 X E a = 0.0012. do D a =~E a = OmStE a = 50 X 6.3 X E a = 0.00038. da m =50 X 182 X 10~ 7 X E t = 0.00046. ot D m 2 = 225.0 X 10~ 8 A, 2 = 144.0 X 10~ 8 Z> 2 = 14.4 X 10~ 8 A 2 = 21.2 X 10~ 8 [D 2 ] = 404.6 X 10~ 8 Hence, by equation (115), R x *= [D 2 ] = 404.6 X 10- 8 , JB X = V404.6 X 10- 8 = 0.0020. For the purpose of computing x, it is convenient to put the given data in the form Om = 50 (1+0.000102), S = 1- 0.000147, at = 0.000115. Then, by equation (a), x = 50 (1 + 0.000102) (1 - 0.000147) (1 + 0.000115), and by formula 7, Table VII, x = 50 (1 + 0.000102 - 0.000147 + 0.000115) = 50 (1 + 0.00007) = 50.0035. This method of computation, by the use of the approximate formulae of Table VII, gives x within less than one unit in the last place held, and is much less laborious than the use of logarithms. Since the length S of the cathetometer scale divisions is given in centimeters, the computed values of x and R x are also expressed in centimeters and our uncertainty regarding the true distance L between the twenty-fifth and the seventy-fifth graduations of the steel scale is definitely stated by the expression L = 50.0035 d= 0.0020 centimeters, at the temperature T r = 20.00.5C. ART. 73] DISCUSSION OF COMPLETED OBSERVATIONS 143 The above discussion shows that the precision of the result would not have been materially increased by a more accurate determination of T, fa, and a, since the effects of the errors of these quantities are small in comparison with that of the errors of o m and S. The probable error of o m might have been reduced by making a larger number of observations and taking care to keep the instrument in adjustment within one-tenth of a level division or less. But the given value of E m is of the same order of magnitude as the least count of the vernier used, and, since each observation represents the difference of two scale readings, it would not be decreased in proportion to the increased labor of observation. Moreover, the terms D m and D 8 in the above value of R x are nearly equal in magnitude, and it would not be worth while to devote time and labor to the reduction of one of them unless the other could be reduced in like proportion. CHAPTER XI. DISCUSSION OF PROPOSED MEASUREMENTS. 74. Preliminary Considerations. The measurement of a given quantity may generally be carried out by any one of several different, and more or less independent, methods. The available instruments usually differ in type and in functional efficiency. A choice among methods and instruments should be determined by the desired precision of the result and the time and labor that it is worth while to devote to the observations and reductions. Since the labor of observation and the cost of instruments in- crease more rapidly than the inverse square of the precision measure of the attained result, a considerable waste of time and money is involved in any measurement that is executed with greater precision than is demanded by the use to which the result is to be put. On the other hand, if the precision attained is not sufficient for the purpose in hand, the measurement must be repeated by a more exact method. Consequently the labor and expense of the first determination contributes very little to the final result and the waste is quite as great as in the preceding case. Sometimes the expense of a second determination is avoided by using the inexact result of the first, but such a saving is likely to prove disastrous unless the uncertainty of the adapted data is duly considered. In general the greatest economy is attained by so planning and executing the measurement that the result is given with the desired precision and neglecting all refinements of method and apparatus that are not essential to this end. While these con- siderations have greater weight in connection with measurements carried out for practical purposes they should never be neglected in planning investigations undertaken primarily for the advance- ment of science. In the former case the cost of necessary measure- ments may represent an appreciable fraction of the expense of a proposed engineering enterprise and must be taken into account in preparing estimates. In the latter case there is no excuse for burdening the limited funds available for research with the expense 144 ART. 75] DISCUSSION OF PROPOSED MEASUREMENTS 145 of ill-contrived and haphazard measurements. The precision requirements may be, and indeed usually are, quite different in the two cases, 'but the same process of arriving at suitable methods applies to both. 75. The General Problem. In its most general form the problem may be stated as follows : Required the magnitude of a quantity X within the limits R, X being a function of several directly measured quantities X\, X 2 , etc. ; within what limits must we determine the value of each of the components X\, X z , etc.? In discussing this problem, all sources of error both constant and accidental must be taken into account. For this purpose the various methods available for the measurement of the several components are considered with regard to the labor of execution and the magnitude of the errors involved as well as with regard to the facility and accuracy with which constant errors can be removed. After such a study, certain definite methods are adopted pro- visionally, and examined to determine whether or not the re- quired precision in the final result can be attained by their use. As the first step in this process, the function that gives the rela- tion between X and the components, Xi, X 2 , etc., is written out in its most complete form with all correction factors explicitly represented. Thus, as in article seventy-two, the most probable value of the quantity X may be expressed in the form X Q = 0(o a ,o bj . . . , p ,a,/3, . . . , p), (120) where the o's represent observed values of X\ t X 2 , etc., and a, /3, . . . , p, represent the factors on which the corrections for con- stant errors depend as pointed out in connection with equation (112), article seventy-one. The form of the function 0, and the nature and magnitude of the correction factors appearing in it, will depend on the nature of the proposed methods of measurement. Since all detectable constant errors are explicitly represented by suitable correction factors, all of the quantities appearing in the function may be treated as directly measured components subject to accidental errors only. Hence the problem reduces to the determination of the probable errors within which each of the components must be determined in order that the computed value of XQ may come out with a precision measure equal to the given magnitude R Q . If all of the components can be determined within the limits set 146 THE THEORY OF MEASUREMENTS [ART. 76 by the probable errors thus found, without exceeding the limits of time and expense imposed by the preliminary considerations, the provisionally adopted methods are adequate for the purpose in hand and the measurements may be carried out with con- fidence that the final result will be precise within the required limits. When one or more of the components cannot be deter- mined within the limits thus set without undue labor or expense, the proposed methods must be modified in such a manner that the necessary measurements will be feasible. 76. The Primary Condition. The present problem is, to some extent, the inverse of that treated in articles seventy-one and seventy-two. In the latter case the given data represented the results of completed series of observations on the several component quantities appearing in the function 0, together with their respective probable errors. The purpose of the analysis was the determination of the most probable value XQ that could be assigned to the measured magnitude and the precision measure of the result. In the present case approximate values of x and the components in 6 are given, and the object of the analysis is the determination of the probable errors within which each of the components must be measured in order that the value of XQ, computed from the completed observations, may come out with a precision measure equal to a given magnitude R . If D , Db, . . . , D p , D a) Dp, . . . , D p represent the separate effects of the probable errors E a , Eb, . . . , E p , E a , Ep, . . . , E p of the components o aj o b , . . . , o p , a, /3, . . . , p, respec- tively, we have, as in article seventy-two, and the primary condition imposed on these quantities is given by the relation #o 2 = Da 2 + ZV + - + ZV + ZV + iy + - - . +D P 2 . (123) The precision measure R and approximate values of the com- ponents are given by the conditions of the problem and the pro- posed methods of measurement. The E's, and hence also the D's, are the unknown quantities to be determined. Conse- quently there are as many unknowns in equation (123) as there are different components in the function 0. Obviously the problem is indeterminate unless some further conditions can be imposed ART. 77] DISCUSSION OF PROPOSED MEASUREMENTS 147 on the D's; for otherwise it would be possible to assign an infinite number of different values to each of the D's which, by proper selection and combination, could be made to satisfy the primary condition (123). 77. The Principle of Equal Effects. An ideal condition to impose on the D's would specify that they should be so determined that the required precision in the final result X Q would be attained with the least possible expense for labor and apparatus. Un- fortunately this condition cannot be put into exact mathematical form since there is no exact general relation between the difficulty and the precision of measurements. However, it is easy to see that the condition is approximately fulfilled when the measure- ments are so made that the D's are all equal to the same magnitude. For, the probable error of any component is inversely proportional to the square root of the number of observations on which it depends and the expense of a measurement increases directly with the number of observations. Consequently the expense W a of the component o a is approximately proportional to 7^-5 or, &a n/j 1 since r is constant, to -^ 9 . Similar relations hold for the other do a D a 2 components. Hence, as a first approximation, we may assume that A2 A2 A2 A2 where W is the total expense of the determination of x , and A is a constant. By the usual method of finding the minimum value of a function of conditioned quantities, the least value of W con- sistent with equation (123) occurs when the D's satisfy (123) and also fulfill the relations _ dD a "* ^ dD a = ML + ***?- o dD b ^ * dD b - = SD * ^ dD 148 THE THEORY OF MEASUREMENTS [ART. 77 where K is a constant. Introducing the expressions for R<? and W in terms of the D's, differentiating, and reducing, we have and by equation (123) where AT is the number of D's in (123) or the equal number of components in the function 6. Consequently equation (123) is fulfilled and the condition of minimum expense is approximately satisfied when the components are so determined that the separate effects of their probable errors satisfy the relation D a = D b = - . - = D a = Dp = = -. (127) Equation (127) is the mathematical expression of the principle of equal effects. It does not always express an exact solution of the problem, since A is seldom strictly constant; but it is the best approximation that we can adopt for the preliminary com- putation of the D's and E's. The results thus obtained will usually require some adjustment among themselves before they will satisfy both the preliminary considerations and the primary condition (123). We shall see that the necessary adjustment is never very great; and, in fact, that a marked departure from the condition of equal effects is never possible when equation (123) is satisfied. Combining equations (122) and (127), we find E ^ . ^ - E ^ . ^ VAT " de ' a VN ' de ' da w Ro i . * = ~~7= ' ~^7T > VN <&' VN y, do b 5/3 (128) Hence, if the final measurements are so executed that the probable errors of the several components are equal to the corresponding values given by equations (128), the final result XQ, computed by equation (120), will come out with a precision measure equal to ART. 78] DISCUSSION OF PROPOSED MEASUREMENTS 149 the specified R Q , and the condition of equal effects (127) will be fulfilled. In computing the E's by equation (128), R Q is taken equal to the given precision measure of X Q and N is placed equal to the J/3 number of components in the function 0. The derivatives T do a etc., are evaluated with the aid of approximate values of the components obtained by a preliminary trial of the proposed methods or by computation, on theoretical grounds, from an approximate value of XQ and a knowledge of the conditions under which the measurements are to be made. Since only two sig- nificant figures are required in any of the E's, the adopted values of the components may be in error by several per cent, without affecting the significance of the results. Moreover, any number of components, whose combined effect on any derivative is less than five per cent, may be entirely neglected in computing that derivative. Consequently the function frequently may be sim- plified very much for the purpose of computing the derivatives and this simplification may take different forms in the case of differ- ent derivatives. No more than three significant figures should be retained at any step of the process and sometimes the required pre- cision can be attained with the approximate formulae of Table VII. Since equation (127) is an approximation, the E's derived from equations (128) are to be regarded as provisional limits for the corresponding components. If all of them are attainable, i.e., if all of the components can be determined within the provisional limits, without exceeding the limit of expense set by the prelim- inary considerations, the solution of the problem is complete and the proposed methods are suitable for the work in hand. 78. Adjusted Effects. Generally some of the E's given by (128) will be unattainable in practice while others will be larger than a limit that can be easily reached. In other words, it will be found that the labor involved in determining some of the components within the provisional limit is prohibitive while other components can be determined with more than the pro- visional precision without undue labor. In such a case the pro- visional limits are modified by increasing the E's corresponding to the more difficult determinations and decreasing the E's that correspond to the more easily determinable components in such a way that the combined effects satisfy the condition (123). 150 THE THEORY OF MEASUREMENTS [ART. 78 The maximum allowable increase in a single E is by the factor . For, taking E a for illustration, B0 a and consequently Hence (123) cannot be satisfied unless all of the rest of the D's are negligibly small. For example, if there are nine components, VN is equal to three. Consequently no one of the E's can be increased to more than three times the value given by the condi- tion of equal effects if (123) is to be satisfied. When, as is fre- quently the case, the number of components is less than nine, or when more than one of the E's is to be increased, the limit of allowable adjustment is much less than the above. The extent to which any number of E's may be increased is also limited by the difficulty, or impossibility, of reducing the effects of the remaining E's to the negligible limit. If the probable errors given by equations (128) can be modified, to such an extent that the corresponding measurements become feasible, without violating the condition (123), the proposed methods are suitable for the final determination of XQ. Other- wise they must be so modified that they satisfy the conditions of the problem or different methods may be adopted provisionally and tested for availability as above. Sometimes it will be found that the proposed methods are capable of greater precision than is demanded by equations (128). In such cases the expense of the measurements may be reduced without exceeding the given precision measure of XQ by using less precise methods. But such methods should never be finally adopted until their feasibility has been tested by the process out- lined above. A discussion on the foregoing lines not only determines the practicability of the proposed methods, but also serves as a guide in determining the relative care with which the various parts of the work should be carried out. For, if the final result is to come out with a precision measure R Q , it is obvious that all adjustments and measurements must be so executed that each of the com- ART. 79] DISCUSSION OF PROPOSED MEASUREMENTS 151 ponents is determined within the limits set by equations (128), or by the adjusted E's that satisfy (123). 79. Negligible Effects. In the preceding article it was pointed out that the availableness of proposed methods of meas- urement frequently depends on the possibility of so adjusting the E's given by equations (128) that they are all attainable and at the same time satisfy the primary condition (123). Generally this cannot be accomplished unless some of the E's can be reduced in magnitude to such an extent that their effect on the precision measure R is negligible. On account of the meaning of the precision measure, and the fact that it is expressed by only two significant figures, it is obvi- ous that any D is negligible when its contribution to the value of 73 #0 is less than y^. Thus, if Ri is the value of the right-hand member of equation (123), when D a is omitted, D a is negligible provided or 0. Squaring gives 0.81 Bo 2 < #i 2 , and by definition R<? - RS = D*. Consequently 0.81 #o 2 < #o 2 - D*, and Z> a 2 <0.19# 2 , or D a < 0.436 #o. Hence, if D a is less than 0.436 # , it will contribute lees than ten per cent of the value of R Q . Since the true error of x is as likely to be greater than R as it is to be less than R Q , a change of ten per cent in the value of R Q can have no practical importance. Consequently D a is negligible when it satisfies the above condi- tion. However, the constant 0.436 is somewhat awkward to handle, and if D a is very nearly equal to the limit 0.436 RQ, the propriety of omitting it is doubtful. These difficulties may be avoided by adopting the smaller and more easily calculated limit of rejection given by the condition D = R Q . (129) 152 THE THEORY OF MEASUREMENTS [ART. 79 This limit corresponds to a change of about six per cent in the value of Ro given by equation (123), and is obviously safe for all practical purposes. Since the above reasoning is independent of the particular D chosen, the condition (129) is perfectly general and applies to any one of the D's in equation (123). When two or more of the D's satisfy (129) independently, any one of them may be neglected, but all of them cannot be neg- lected without further investigation for otherwise the change in Ro might exceed ten per cent. This would always happen if all T~) of the D's considered were very nearly equal to the limit ~^- o However, by analogy with the above argument, it is obvious that any q of the D's are simultaneously negligible when + D 2 2 + . . . + D 3 2 == Jflo, (130) where the numerical subscripts 1, 2, . . . , q are used in place of the literal subscripts occurring in equation (123) in order to render the condition (130) entirely general. Thus DI may corre- spond to any one of the D's in (123), D 2 to any other one, etc. By applying the principle of equal effects, the condition (130) may be reduced to the simple form D, = D 2 = ... = D q = - ^ (131) 3 Vg If some of the D's in (131) can be easily reduced below the limit p j=. , the others may exceed that limit somewhat without violating 3 V q the condition (130). However, equation (131) generally gives the best practical limit for the simultaneous rejection of a number of D's, and all departures from it should be carefully checked by (130). To illustrate the practical application of the foregoing discussion, suppose that the practicability of certain proposed methods of measurement is to be tested by the principle of equal effects developed in article seventy-seven. Let there be N components in the function 0, and suppose that q of them, represented by ai, 2, . . . , a q , can be easily determined with greater precision than is demanded by equations (128), while the measurement of the remaining N q components within the^limits thus set would be very difficult. Obviously some adjustment of the E's given by (128) is desirable in order that the labor involved in the various parts of the measurement may be more evenly balanced. ART. 79] DISCUSSION OF PROPOSED MEASUREMENTS 153 The greatest possible increase in the E's corresponding to the N q difficult components will be allowable when the E's of the q easy components can be reduced to the negligible limit. To determine the necessary limits, R is taken equal to the given precision measure of XQ, and the negligible D's corresponding to the q easy components are determined by equation (131). Then by equations (122), the corresponding E's will be negligible when E!=Z -^ 3 Vq 1 1 If dai E 2 = -^L< 1 w (132) A r J_^ 6^ da q If these limits can be attained with as little difficulty as the pre- viously determined E's of the N q remaining components, the corresponding D's may be omitted from equation (123) during the further discussion of precision limits. Since q of the D's have disappeared, the others may be some- what increased and still satisfy the primary condition (123). The corresponding new limits for the E's of the difficult components may be obtained from equations (127) and (128) by replacing N by N q. If these new limits together with the negligible limits given by equations (132) can all be attained, without exceeding the expense set by the preliminary considerations, the proposed methods may be considered suitable for the final deter- mination of XQ with the desired precision. Otherwise new methods must be devised and investigated as above. Equations (132) may also be used to determine the extent to which mathematical constants should be carried out during the computations. For this purpose the components i, 0% , , or part of them, represent the mathematical constants appearing in the function 8. The corresponding E's, determined by equa- tions (132), give the allowable limits of rejection in rounding the numerical values of the constants for the purpose of simplifying 154 THE THEORY OF MEASUREMENTS [ART. 79 the computations. Thus, suppose that the volume of a right circular cylinder of length L and radius a is to be computed within one-tenth of one per cent, how many figures should be retained in the constant TT? In this case n / \ 17 9 T (Oa , , , ) = y = *<, RQ = 0.001 V = 0.001 7ra 2 L, 60 6V = 0.00105. 0.001 7T If TT is taken equal to 3.142 the error due to rounding is 0.00041 . Since this is less than the negligible limit E r , four significant figures in TT are sufficient for the purpose in hand. It sometimes happens that the total effect of one or more of the components in the function 0, on the computed value of x , is negligible in comparison with RQ. This will obviously be the case when 60 RQ a^ a ^ IF' for a single component a or when KM \ 2 -L-/ de z~~ a i) + (^~~ dai I \da2 da for q components. Thus, on the principle of equal effects, the components i, <* 2 , , <* 3 will be simultaneously negligible when they satisfy the conditions 1 RQ 1 * 155 i (133) RQ 1 daz 7"> 1 \7^'~d0~ Such cases frequently arise in connection with the components that represent correction factors. ART. 80] DISCUSSION OF PROPOSED MEASUREMENTS 155 80. Treatment of Special Functions. During the foregoing argument, it has been assumed that the function 6 in equation (120) is expressed in the most general form consistent with the pro- posed methods of measurement. Such an expression involves the explicit representation of all directly measured quantities, and all possible correction factors. Part of the latter class of com- ponents represent departures of the proposed methods from the theoretical conditions underlying them, and others depend upon inaccuracies in the adjustment of instruments. In practice it frequently happens that the general function is very compli- cated, and consequently that the direct discussion of precision as above is a very tedious process. Under these conditions it is desirable to modify the form of the function in such a manner as to facilitate the discussion. Sometimes the general function 9 can be broken up into a series of independent functions or expressed as a continuous product of such functions. Thus, it may be possible to express 6 in the form XQ = 6 (o a , o b , . . ., a, |8, . . .) = /i(ai,a 2 , . . . )/ 2 (&i,& 2 , . . . )/ 3 (ci,c 2 , . . . or in the form XQ = d (O a , O b) . (134) (135) = /i(ai,a 2 , . . . ) X/2(&i,&2, . ) X/ 3 (ci,c 2 , . . . X ... X / (mi, m 2) . . . ), where the a's, &'s, . . . , and m's represent the same components, o a , o b , . . . , a, 0, . . . , that appear in 6 by a new and more general notation. The functions /i, / 2 , . . . , f n may take any form consistent with the problem in hand, but the precision dis- cussion will not be much facilitated unless they are independent in the sense that no two of them contain the same or mutually dependent variables. Sometimes the latter condition is imprac- ticable and it becomes necessary to include the same component in two or more of the functions. Under such conditions the expan- sion has no advantage over the general expression for 0, unless the effect of the errors of each of the common components can be rendered negligible in all but one of the functions. It is scarcely necessary to point out that equations (134) and (135) represent different problems, and that if it were possible to expand 156 THE THEORY OF MEASUREMENTS [ART. 80 the same function in both ways, the component functions /i, /2, , fn would be different in the two cases. For the sake of convenience let /I (Oi, 2, ) = 2 /2 (6l, 6 2 , . . . ) = ^2 jfn (Wi,m 2 ,. . . ) = 2 Then equation (134) may be written in the form X = Zi 2 2! 3 . . . d= 2, (137) and (135) may be put in the form x = z l Xz z Xz 3 X . . . Xz n . (138) First consider the case in which the function representing the proposed methods of measurement has been put in the form of (137). Since the precision measure follows the same laws of propagation as the probable error, the discussion given in article fifty-eight leads to the relation # 2 = 7^2 + # 2 2 + R f + _ m + Rn 2 } ( 139) where RQ is the precision measure of x , and each of the other R's represents the precision measure of the z with corresponding sub- script. Hence, by the principle of equal effects, provisional values of the R's may be obtained from the relation R, = R 2 = R, = . . . = R n = A . ( 140 ) The R's having been determined by (140), the corresponding probable errors of the a's, 6's, etc., may be computed by the methods of the preceding articles with the aid of equations (136). If the provisional limits of precision thus found are not all attain- able with approximately equal facility, the conditions of the problem may be better satisfied by moderately adjusted relative values of the probable errors as pointed out in article seventy- eight. Obviously the adjusted values must satisfy equation (139) if the value of x computed by (137) is to come out with a pre- cision measure equal to the given R . When the function representing the proposed methods can be put in the form of (138) the computation is facilitated by intro- ducing the fractional errors P = ; Pl = ! ; P 2 = f2;...; P n = f" (141) XQ Zi Zz Z n ART. 81] DISCUSSION OF PROPOSED MEASUREMENTS 157 For, by the argument underlying equation (83), article sixty-two, Po 2 = Pi 2 + P 2 2 + Pa 2 + . . . + P 2 , (142) and, by the principle of equal effects, provisional values of the P's are given by the relation Pi = P 2 = P 3 = . . . = P = *=. (143) Vn Since RQ and approximate values of the components are given, PO can be computed with sufficient accuracy with the aid of (138) and the first of (141). Consequently provisional fractional limits for the components can be determined by (143), and the corresponding precision measures by the last n of equations (141). Beyond this point the problem is identical with the preceding case, except that the adjusted limits of precision must satisfy (142) in place of (139). The methods developed in the preceding articles are entirely general and applicable to any form of the function 6, but they frequently lead to complicated computations. In the present article we have seen how the discussion can be simplified when the function can be put in either of the particular forms represented by (134) and (135). Many of the problems met with in practice cannot be put in either of these special forms, but it frequently happens that the treatment of the functions representing them can be simplified by a suitable modification or combination of the above general and particular methods. The general ideas under- lying all discussions of the necessary precision of components have been discussed above with sufficient fullness to show their nature and significance. Their application to particular prob- lems must be left to the ingenuity of the observer and computer. 81. Numerical Example. As an illustration of the fore- going methods, suppose that the electromotive force of a battery is to be determined, and that the precision measure of the result is required to satisfy the condition R = 0.0012 volts, (i) T-> within the limits T?!>i- e -> #o must lie between 0.0011 and =b 0.0013 volt. Preliminary considerations demand that the expense of the work shall be as low as is consistent with the required precision. 158 THE THEORY OF MEASUREMENTS [ART. 81 The given conditions are most likely to be fulfilled by some form of potentiometer method. Suppose that the arrangement of apparatus illustrated in Fig. 10 is adopted provisionally; and, to simplify the discussion, suppose that the various parts of the apparatus are so well insulated that leakage currents need not be considered. The generality of the problem is not appreciably affected by the latter assumption since the specified condition can be easily satisfied in practice within negligible limits. With what precision must the several components and correction factors be determined in order that equation (i) may be satisfied? -T&Z FIG. 10. L e t V = e.m.f. of tested battery BI, Et = e.m.f. of Clark cell B 2 at time of observation, t = temperature of Clark cell at time of observation, Ri = resistance between 1 and 2, Rz = resistance between 1 and 3, / = current in circuit 1, 2, 3, B 3 , 1 when the key K is open, 5i = algebraic sum of thermo e.m.f.'s in the circuit 1, 2, 6, G, 1 when K is closed to 6, 2 = algebraic sum of thermo e.m.f. 's in the circuit 1, 3, a, G, 1 when K is closed to a, Ei5 e.m.f. of Clark cell at temperature 15 C., a. = mean temperature coefficient of Clark cell in the neighborhood of 20 C. ART. 81] DISCUSSION OF PROPOSED MEASUREMENTS 159 When the sliding contacts 2 and 3 are so adjusted that the galvanometer G shows no deflection on closing the key K to either a or 6, RI RZ Consequently F = (^+6 2 )|- 1 -5 1 . (ii) -fi/2 But (in) Hence F = -B 16 !l-a-15)jf- 1 + 2 f- 1 -8 1 . (iv) KZ n>z The resistances RI and # 2 are functions of the temperature; but, since they represent simultaneous adjustments with the cells BI p and Bz and are composed of the same coils, the ratio ~ is inde- KZ pendent of the temperature. Thus, if R t ' and R t " represent the resistances of the used coils at t C., and ft is their temperature coefficient, RS Ri(l+ fit) Ri whatever the temperature t at which the comparison is made. This advantage is due to the particular method of connection and adjustment adopted, and is by no means common to all forms of the potentiometer method. Under the conditions specified above, equation (iv) may be adopted as the complete expression for the discussion of precision. It corresponds to equation (120) in the general treatment of the problem. Suppose that the following approximate values of the components, which are sufficiently close for the determination of the capabilities of the method, have been obtained from the normal constants of the Clark cell and a preliminary adjustment of the apparatus or by computation from a known approximate value of V: #15 = 1.434 volts; a = 0.00086; t = 20 C.; Ri = 1000 ohms; R 2 = 1310 ohms; V = 1.1 volts. The thermoelectromotive forces 5i and 5 2 are to some extent due to inhomogeneity of the wires used in the construction of the instruments and connections. For the most part, however, (v) 160 THE THEORY OF MEASUREMENTS [ART. 81 they arise from the junctions of dissimilar metals in the circuits considered. Suppose that the resistances R\ and #2 are made of manganin, the key K of brass, and that the copper used in the galvanometer coil and the connecting wires is thermoelectrically different. Both 5i and 5 2 would represent the resultant action of at least six thermo-elements in series. While these effects can- not be accurately specified in advance, their combined action would not be likely to be greater than twenty-five microvolts per degree difference in temperature between the various parts of the apparatus, and it might be much less than this. Obviously 5i and 6 2 are both equal to zero when the temperature of the appa- ratus is uniform throughout. By equations (133), article seventy-nine, the correction terms depending on thermoelectric forces will be negligible in compar- ison with the given precision measure R , when 5i and 62 satisfy the conditions . 1 #o 1 , - 1 flo 1 ' l *3'vT5E ^s'vTE' ddi dd 2 In the present case Ro = 0.0012 volt; q = 2; dV . dV R l sE*-- 1 ' and srsr Consequently the above conditions become - 5^i? . _L _ 0.00028 volt = 280 microvolts, 3 v 2 1 _L - 0.00037 volt = 370 microvolts. 0.76 From the above discussion of the possible magnitude of the thermo- electromotive forces in the circuits considered, it is obvious that these limits correspond to temperature differences of approxi- mately ten degrees between the various parts of the apparatus. Since the temperature of the apparatus can be easily maintained uniform within five degrees, the last two terms in equation (iv) are negligible within the limits of precision set in the present problem. Hence, for the determination of the required precision of the remaining components, the functional relation (iv) may be taken in the form (vi) ART. 81] DISCUSSION OF PROPOSED MEASUREMENTS 161 By equation (123), article seventy-six, the primary condition for determining the necessary precision of the components is R<? = 144 X 10~ 8 = Z>! 2 + > 2 2 + D 3 2 + > 4 2 + D<?, (vii) where dV 67 67 67 (viii) and EI, EZ, E 3 , E^ E$ are the required probable errors of EI$, a, t, Ri, and Rz, respectively. For the preliminary determination of the jE"s by the principle of equal effects, equation (127), article seventy-seven, becomes = 0.00054. (ix) VN V5 Neglecting all factors that do not affect the differential coefficients by more than one unit in the second significant figure and adopt- ing the approximate values of the components given in (v), 67 R! 1000 n _ j- = p- = T^ = 0.76, d-Cns /t2 lolU =- E 15 a = - - 0.00094, it/2 = Eu = 0.0011, it 2 (x) Hence, by combining (viii) and (ix), or directly from equations (128), article seventy-seven, , 0.00054 (xi) E z 0.76 0.00054 ZEI V7.V7UV/I J. VWftVj n nnnoQS v*y (b) (c) (d) (e) 5.5 0.00054 = 0.57 C. = =b 0.49 ohm = db 0.65 ohm. 0.00094 0.00054 - o.oon 0.00054 0.00083 162 THE THEORY OF MEASUREMENTS [ART. 81 In practice the attainableness of these limits might be deter- mined experimentally; but in the present case, as in most practical problems, general considerations based on theory and previous experience lead to equally trustworthy results. In the first place, it is obvious that the temperature of the Clark cell can be easily determined closer than 0.6 C. Consequently the limit (c) is easily attainable and might possibly be reduced to a negligible quantity. The constants of the normal Clark cell are known well within the limits (a) and (b). But it requires very careful treatment of the cell to keep Ei 6 constant within the limit (a), and new cells, unless they are set up with great care and skill, are likely to vary among themselves and from the normal cell by more than 0.0007 volt. Consequently the limit (a) is somewhat smaller than is desirable in practical work of the precision considered in the present problem. On the other hand, the limit (b) is very rarely exceeded by either old or new cells unless they are very care- lessly constructed and handled. Hence E 2 could probably be reduced to the negligible limit. With a suitable galvanometer, the nominal values of the resist- ances Ri and R% can be easily adjusted within the limits (d) and (e). But EI and E 5 must be considered practically as the pre- cision measures of R i and R 2 . They include the calibration errors of the resistances, the errors due to leakage between the terminals of individual coils, and the errors due to nonuniformity of temperature as well as the errors of setting of the contacts 2 and 3, Fig. 10. The resultant of these errors can be reduced below the limits (d) and (e), but in the present case it would be convenient to have somewhat larger limits in order to reduce the expense of construction and calibration. Hence, while all of the E's given by equations (xi) are within attainable limits, the preliminary consideration of minimum expense would be more likely to be fulfilled if the limits (a), (d), and (e) were somewhat larger. Obviously the magnitude of these limits can be increased without violating the primary con- dition (vii) provided a corresponding decrease in the magnitudes of the limits (b) and (c) is possible. By equation (131), article seventy-nine, the separate effects D 2 and DZ will be simultaneously negligible if n n 1 #o 1 0.0012 1/2 = DZ = 7^ = ~ ;= ^ 3 Vq 3 V2 ABT.SI] DISCUSSION OF PROPOSED MEASUREMENTS 163 Hence, by equations (132), the errors of a and t will be negligible when 0.00028 E 2 = ^~ =; 0.000051, (b') and 00028 Ets mm ^o-3oc. (C ') Since these limits can be reached with much greater ease than the limits (a), (d), and (e), they may be adopted as final specifica- tions and the corresponding Z)'s may be omitted during the deter- mination of new limits for the components E 15) R 1} and R%. Under these conditions, equation (ix) becomes Hence the largest allowable limits for the errors of EM, Ri, and RZ are OOOfiQ ~ = 0.00091 volt, (a') = .63 ohm, (d') While these limits cannot be quite so easily attained as (b') and (c'), they cannot be increased without violating the primary con- dition (vii). Consequently they satisfy the condition of minimum expense, so far as the proposed method is concerned, and may be adopted as final specifications. The fractional errors corresponding to the specified precision measure of V and the above limiting errors of the components Po = Y = 0.0011 = 0.11%, Pi = ~ = db 0,00063 = 0.063%, P 2 = ^ = 0.059 = =t 5.9%, P 3 = y = 0.015 = db 1.5%, P 4 = ~ = =fc 0.00063 = d= 0.063%, P 5 = f- 5 = 0.00063 = 0.063%. 164 THE THEORY OF MEASUREMENTS [ART. 81 Consequently in order to obtain a value of V that is exact within 0.11 per cent by the proposed method, a must be determined within 5.9 per cent, t within 1.5 per cent, and E.-&, Ri, and R 2) each within 0.063 per cent. These limits are all attainable in practice under suitable conditions, as pointed out above. Hence the pro- posed method is practicable. If the final measurements are so devised and executed that the above conditions are fulfilled, the precision of the result computed from them will be within the specified limits and the expense of the work will be reduced to the lowest limit compatible with the proposed method. The desired result might be obtained at less expense by some other method, but a decision on this point can be reached only by comparing the precision requirements and practicability of various methods with the aid of analyses similar to the above. CHAPTER XII. BEST MAGNITUDES FOR COMPONENTS. 82. Statement of the Problem. The precision of a derived quantity depends on the relative magnitudes and precision of the components from which it is computed, as explained in Chapter VIII. Thus, if the derived quantity X Q is given in terms of the components x\, x^ . . . , x q by the expression x = F (xi, x 2 , . . . , x g ), (144) the probable error of X Q is given by the expression E Q * = SSEJ + S 2 2 E 2 Z + + S q *E q 2 , (145) where the E's represent the probable errors of the x's with corre- sponding subscripts, and AF AF AF *-& *-&'< (146) The error E, corresponding to any directly measured com- ponent, is generally, but not always, independent of the absolute magnitude of that component so long as the measurements are made by the same method and apparatus. For example: the probable error of a single measurement with a micrometer caliper, graduated to 0.01 millimeter, is approximately equal to 0.004 millimeter, whatever the magnitude of the object measured so long as it is within the range of the instrument. Hence, when the methods and instruments to be used in measuring each of the components are known in advance, the probable errors EI, E 2 , etc., can be determined, at least approximately, by preliminary measurements on quantities of the same kind as the components but of any convenient magnitude. Under these conditions the E's on the right-hand side of equation (145) may be treated as known constants, and, since the S's are expressible in terms of Xi, x z , etc., by equations (146), the value of E corresponding to the given methods cannot be changed without a simultaneous change in the relative or absolute magnitudes of the components. 165 166 THE THEORY OF MEASUREMENTS [ART. 82 Since equation (144) must always be fulfilled, and since the value of XQ is usually fixed by the conditions of the problem, a change in the magnitudes of the re's is not always possible. But it frequently happens that the form of the function F is such that the relative magnitudes of the components can be changed through somewhat wide limits and still satisfy equation (144). Thus, if a cylinder is to have a specified volume, it may be made long and thin, or short and thick, and have the same volume in either case. Consequently it is sometimes possible to select magnitudes for the components that will give a minimum value of E and at the same time satisfy equation (144). The problem before us may be briefly stated as follows : Having given definite methods and apparatus for the measurement of the components of a derived quantity reo, what magnitudes of the components will give a minimum value to the probable error EQ of XQ and at the same time satisfy the functional relation (144)? It can be easily seen that a practical solution of this problem is not always possible. In the first place the form of the function F may be such as to admit of but a single system of magnitudes of the components, and consequently the value of EQ is definitely fixed by equation (145). In some cases there are no real values of the re's that will satisfy both (144) and the conditions for a minimum of EQ. When values can be found that satisfy the mathematical conditions they are not always attainable in prac- tice. Finally the probable errors Ei, E 2 , etc., may not be inde- pendent of the magnitudes of the corresponding components or it may be impossible to determine them in advance of the final measurements. When the E's are not independent of the re's it sometimes happens that the fractional errors Pi = ? ; p * = ? '> p * = ? (147) 3/1 it/2 Xq are constant and determinable in advance. In such cases the problem may be solvable by putting (145) in the equivalent form Ef = SfPfy? + SfPfxf +!>+ S q *P q *x q *, (148) expressing the S's in terms of the components by equations (146), and determining the values of the re's that will render (148) a minimum subject to the condition (144). ART. 83] BEST MAGNITUDES FOR COMPONENTS 167 When a practicable solution of the problem is possible, it is obvious that the results thus obtained are the best magnitudes that can be assigned to the components, and that they should be adopted as nearly as possible in carrying out the final measure- ments from which X Q is to be computed. 83. General Solutions. The general conditions for a mini- mum or a maximum value of E Q 2 , when XQ is treated as a constant and the variables are required to satisfy the relation (144), but are otherwise independent, are dF ^ A = U, 0) where K is an arbitrary constant. By introducing the expressions (145) and (146), transposing and dividing by two, equations (i) become Sl gtf 1 . + S ,g^ + ... o O&1 ET 2 _j_ O 0O2 pi 2 i 1 dx 2 2 ^2 (149) When the S's have been replaced by x's with the aid of equa- tions (146), the q equations (149), together withj(144), are theoreti- cally sufficient for the determination of all of the q + 1 unknown quantities Xi, x 2 , . . . , x q , and K. However, in some cases a practicable solution is not possible, and in others the components or their ratios come out as the roots of equations of the second or higher degree. The zero, infinite, and imaginary roots of these equations have no practical significance in the present discussion and need not be considered. Some of the real roots correspond to a maximum, some to a minimum, and others to neither a maximum nor a minimum value of E Z . In most cases the roots that corre- spond to a minimum of E 2 can be selected by inspection with the 168 THE THEORY OF MEASUREMENTS [ART. 83 aid of equation (145), but it is sometimes necessary to apply the well-known criteria of the calculus. Dividing equation (145) by x Q 2 and putting XQ dX 2 ' q XQ XQ dX q XQ XQ dXi gives the expression PZ = EI X 2 XQ (150) + T*E* (151) for the fractional error of XQ. Since XQ is a constant in any given problem the maxima and minima of P 2 correspond to the same values of the components as those of E Q 2 . Sometimes the form of the function F is such that the expression (151), when expanded in terms of the x's, is much simpler than (145). In such cases it is much easier to determine the minima of P 2 than of E 2 . For this purpose the equations of condition (i) may be put in the form 6X1 XQ dXi KdF_ XQ 6X2 dx, (152) , q XQ dX q and by substitution and transposition we have dTi dT% dT g 1 dxi 2 dxi 2 q dxi dT< (153) When the components are required to satisfy the condition (144) and a given constant value is assigned to XQ, equations (153) lead to exactly the same results as equations (149). In fact either of these sets of equations can be derived from the other by purely algebraic methods when the $'s and T's are expressed in terms of the x's. In practice one or the other of the sets will be the simpler, depending on the form of the function F; and the simpler form ART. 83] BEST MAGNITUDES FOR COMPONENTS 169 can be more easily derived by direct methods as above than by algebraic transformation. In some problems the magnitude of one or more of the com- ponents in the function F can be varied at will and determined with such precision that their probable errors are negligible in comparison with those of the other components. Variables that fulfill these conditions will be called free components. Since any convenient magnitude can be assigned to them, their values can always be so chosen that the condition (144) will be fulfilled whatever the values of the other components. Consequently the latter components may be treated as independent variables in determining the minima of E Q 2 or P Q 2 . Under these conditions the E's corresponding to the free com- ponents can be placed equal to zero, and either E 2 or P 2 can sometimes be expressed as a function of independent variables only by eliminating the free components from the S's or the T's with the aid of equation (144). When this elimination can be effected, the minimum conditions may be derived from equations (149) or (153), as the case may be, by placing K equal to zero and omitting the equations involving derivatives with respect to the free components. This is evident because the remaining com- ponents are entirely independent, and consequently the partial derivatives of E Q 2 or P 2 with respect to each of them must vanish when the values of the variables correspond to the maxima or minima of these functions. When the elimination cannot be accomplished, neither equations (149) nor (153) will lead to con- sistent results and the problem is generally insolvable. In practice it frequently happens that the free components are factors of the function F, and are not included in any other way. Under these conditions they do not occur in the T's corresponding to the remaining components, since the form of equations (150) is such that they are automatically eliminated. Consequently, in this case, the conditions for a minimum are given at once by equations (153) when K is taken equal to zero, since the derivatives with respect to the free components all vanish and the correspond- ing E's are negligible. It is scarcely necessary to point out that the remarks in the paragraph following equations (149), except for obvious changes in notation, apply with equal rigor to equa- tions (153), whether K is zero or finite. The values of the x's derived from these equations should never be assumed to corre- spond to the minima of P 2 without further investigation. 170 THE THEORY OF MEASUREMENTS [ART. 84 84. Special Cases. Suppose that the relation between the derived quantity XQ and the measured components xi, # 2 , and x s is given in the form XQ = ax?* + bxj 1 * + cxj 1 *, (ii) where a, b, c, and the n's are constants. If the probable errors Ei t E z , and E 3 of the x's with corresponding subscripts are known, and independent of the magnitude of the components, what mag- nitudes of the components will give the least possible value to the probable error E of X Q ? By equations (146), Si = arnxi^-V; S 2 = bn&^'-V; S s = c/W^-D. (iii) Consequently dSi , i\ ( <>\ ^$2 rv ^$3 _. -!(, -I)**-*; ._ =0; = 0, Substituting these results in equations (149) and dividing the first equation by Si, the second by $ 2 , and the third by S S) the conditions for a minimum value of E Q 2 become Efari! (m - 1) xi<*-*> = K, Dividing the second and third of these equations by the first and transposing the coefficients to the second member gives the ratios of the components in the form x 2 (n ^- 2) = EJani (ni-l) T,(tti-2) ~~ EL2Jm n (n n - IV (HI - 1) ~ (n s - These two equations together with (ii) are theoretically sufficient for the determination of the best magnitudes for the three com- ponents xij Xzj and x$] but it can be easily seen, from the form of the equations, that a solution is not practicable for all possible values of the n's. ART. 84] BEST MAGNITUDES FOR COMPONENTS 171 For example, if the n's are all equal to unity, the ratios of the components given by (iv) are both indeterminate, each being equal to ^- Consequently the problem has no solution in this case. This conclusion might have been reached at once by inspecting the value of E Q 2 given by equation (145), when the S's are expressed in terms of the components. Thus, placing the n's equal to unity in equations (iii) and substituting the results in (145), we find Since E<? is independent of the x's it can have no maxima or minima with respect to the components. When each of the n's equals two, equations (iv) are inde- pendent of the x's, and consequently the problem is not solvable. In this case (ii) becomes XQ = and (145) reduces to E 2 = 4 Since these equations differ only in the values of the constant coefficients of the x's, no magnitudes can be assigned to the com- ponents that will give a minimum value to E Q 2 , and at the same time satisfy the equation for XQ. If each of the n's is placed equal to three, equation (ii) takes the form XQ = ax^ + bx 2 * + c#3 3 , (v) and equations (iv) become Xt~bEf' (iv') C# 3 2 In this case the problem can be easily solved when the numerical values of the coefficients and the E's are known. As a very simple illustration, suppose that 7 -f J 77T -TGI ~Ij1 XT' a = o = c = 1, and J^i = & 2 MS &, then, by (iv') and (v), and, by (145) and (iii), 172 THE THEORY OF MEASUREMENTS [ART. 84 Since a decrease in the magnitude of one of the x's involves an increase in that of one or both of the others, in order to satisfy equation (v), and since the fourth power of a quantity varies more rapidly than the third, it is obvious that the minimum value of E 2 will occur when the x's are all equal. Consequently the above solution corresponds to a minimum of E 2 . It can be easily seen that there are many other cases in which equations (ii) and (iv) can be solved, and also some others in which no solution is possible. The extension of the problem to functions in the same form as equation (ii), but containing any number of similar terms, involves only the addition of one equa- tion in the form of (iv) for each added component. Obviously these equations hold for negative as well as positive values of the coefficients and exponents of the x's. As a second example, consider the functional relation x = axi n i X xf*. (vi) In this case the solution is more easily effected by the second method given in the preceding article. By equations (150) Consequently and equations (153) reduce to the simple form ^ES=-K; %Ef = -K, ; (viii) where EI and E 2 are the known constant probable errors of Xi and #2. Eliminating K, we have Consequently the problem is always solvable when n\ and n 2 have the same sign. When they have different signs the solu- tion is imaginary. Hence there are no best magnitudes for the components when the derived quantity is given as the ratio of two measured quantities. ART. 85] BEST MAGNITUDES FOR COMPONENTS 173 The extension of this solution to functions involving any num- ber of factors is obvious. When the exponents of all of the factors have the same sign the problem is always solvable but the best magnitudes thus found may not be attainable in practice. If part of the exponents are positive and others are negative the solution is imaginary. 85. Practical Examples. I. In many experiments the desired result depends directly upon the determination of the quantity of heat generated by an electric current in passing through a resistance coil. Let I represent the current intensity and E the fall of potential between the terminals of the coil. Then the quantity of heat H developed in t seconds may be computed by the relation JH = TEt, where J represents the mechanical equivalent of heat. If H is measured in calories, I in amperes, E in volts, and t in seconds, y is equal to 0.239 calorie per Joule and the above relation becomes H = 0.239 lEt. (ix) Suppose that the conditions of the problem in hand are such that H should be made approximately equal to 1000 calories. Since the resistance of the heating coil is not specified it can be so chosen that 7 and E may have any convenient values that satisfy the relation (ix) when H has the above value. Obviously t can be varied at will, by changing the time of run, and (ix) will not be violated if suitable values are assigned to / and E. If the instruments available for measuring /, E, and t are an ammeter graduated to tenths of an ampere, a voltmeter graduated to tenths of a volt, and a common watch with a seconds hand, what are the best magnitudes that can be assigned to the components, i.e., what magnitudes of /, E, and t will give the computed H with the least probable error? By comparing equations (ix) and (vi), it is easy to see that the present problem is an application of the second special case worked out in the preceding article when a third variable factor Z 3 n 3 is annexed to (vi). H corresponds "to x 0) I to Xi, E to x z , t to # 3 /and all of the n's in (vi) are equal to unity. Consequently 174 THE THEORY OF MEASUREMENTS [ART. 85 the solution can be derived at once from three equations in the form of (viii) if suitable values can be assigned to the probable errors of the components. With the available instruments, the probable errors E i} E e , and E t of /, E, and t, respectively, will be practically independent of the magnitude of the measured quantities so long as the range of the instruments is not exceeded. Under the conditions that usually prevail in such observations the following precision may be attained with reasonable care: E t = 0.05 ampere; E e = 0.05 volt; E t = 1 second. The conditions for a minimum value of the probable error E of H can be derived by exactly the same method that was used in obtaining equations (viii), or these equations may be used at once with proper substitutions as outlined above. Consequently the best magnitudes for the components are given by the simul- taneous solution of (ix) and the following three equations, ^ 2 _ K . ^_ E? ~P = ~ K > ~W~ ~ K > ~P = Eliminating K and substituting the numerical values of the probable errors we have E_E e _. l_E t _ I ~ E<~ L > I~ Ei~ Consequently E = I and t = 20 /. (x) Substituting these results and the numerical value of H in (ix) we have 1000 = 0.239 X 20 X / 3 , and hence I = 5.94 amperes is the best magnitude to assign to the current strength under the given conditions. The corresponding magnitudes for the electro- motive force and time found by (x) are E = 5.94 volts and t = 119 seconds. If the above values of the components and their probable errors are substituted in equation (151), the fractional error of H comes out ART. 85] BEST MAGNITUDES FOR COMPONENTS 175 and the probable error of H is given by the relation E Q = 1000 Po =15 calories. If any other magnitudes for the components, that satisfy equa- tion (ix), are used in place of the above in (151), the computed value of E will be greater than fifteen calories. Consequently the above solution corresponds to a minimum value of E Q . In order to fulfill the above conditions the resistance of the heating coil must be so chosen as to satisfy the relation *- Since our solution calls for numerically equal values of I and E, the resistance R must be made equal to one ohm. It can be easily seen that small variations in the values of the components will produce no appreciable effect on the probable error of H, ^ince the numerical value of E is never expressed by more than two significant figures. Consequently the foregoing discussion leads to the following practical suggestions regarding the conduct of the experiment. The heating coil should be so constructed that the heat developed in the leads is negligible in comparison with that developed between the terminals of the voltmeter. The resistance of the coil should be one ohm. The current strength should be adjusted to approximately six amperes and allowed to flow continuously for about two minutes. Under these conditions the difference in potential between the terminals of the coil will be about six volts. The conditions under which 7, E, and t are observed should be so chosen that the probable errors specified above are not exceeded. If the above suggestions are carried out in practice the value of H computed from the observed values of /, E, and t by equa- tion (ix) will be approximately 1000 calories, and its probable error will be about fifteen calories. A more precise result than this cannot be obtained with the given instruments unless the probable errors of 7, E, and t can be materially decreased by modifying the conditions and methods of observation. II. A partial discussion of the problem of finding the best magni- tudes for the components involved in the measurement of the strength of an electric current with a tangent galvanometer may 176 THE THEORY OF MEASUREMENTS [ART. 85 be found in many laboratory manuals and textbooks. Such dis- cussions are usually confined to a consideration of the error in the computed current strength due to a given error in the observed deflection. On the assumption, tacit or expressed, that the effects of the errors of all other components are negligible it is proved that the effect of the deflection error is a minimum when the deflection is about forty-five degrees. Although the tangent gal- vanometer is now seldom used in practice it provides an instructive example in the calculation of best magnitudes since the general bearings of the problem are already familiar to most students. In order to avoid unnecessary complications, consider a simple form of instrument with a compass needle whose position is observed directly on a circle graduated in degrees. Suppose that the needle is pivoted at the center of a single coil of N turns of wire, and R centimeters mean radius. Under these conditions the current strength I is connected with the observed deflection (f> by the relation where H is the horizontal intensity of a uniform external magnetic field parallel to the plane of the coil. In practice the plane of the coil is usually placed parallel to the magnetic meridian and H is taken equal to the horizontal component of the earth's mag- netism. N is an observed component but it can be so precisely deter- mined by direct counting, during the construction of the coil, that its error may be considered negligible in comparison with those of the other components. Furthermore it can be given any desired value when an instrument is designed to meet special needs, and a choice among a number of different values is possi- ble in most completed instruments. Consequently the quantity x TT may be treated as a free component, represented by A, and the expression for the current strength may be written in the form 7 = A#.tan0. (xi) Comparing this expression with the general equation (144) we note that / corresponds to x 0) H to x\, R to x 2 , and to z 3 . Since A is free, the components H, R, and </> are entirely inde- pendent; and any convenient magnitudes can be made to satisfy ART. 85] BEST MAGNITUDES FOR COMPONENTS 177 (xi) by suitably choosing the number of turns in the coil. Con- sequently, as pointed out in article eighty-three with respect to functions containing a free component as a factor, the conditions for a minimum probable error of / are given by equations (153) with K placed equal to zero. By making the above substitutions for the x's in equations (150) and performing the differentiations we have I/' 7?' oi-r O ^ * V^^X 11 /L bill cp Consequently 0/77 -i z\nn H^TI ol i 1 . o J. 2 f\ OJ. 3 ~ dH = ~H~ 2 ' dH ; ~dH = ' *^/T7 *\ ATT "I fk T7 ?l n ^ 2 _ _ L ^ 3 _ n. dR ~ dR R 2 ' dR ' dTi ^ = n- dTz = 4cos2<?i> d0 60 " 60 sin 2 2 ' and, if the probable errors of H, R, and are represented by E\ 9 EZ, and #3, respectively, equations (153) become If EI and E 2 could be made negligible, as is tacitly assumed in most discussions of the present problem, the first two of equations (xiii) would be satisfied whatever the values of H and R. Conse- quently these components would be free and would be the only independent variable involved in equation (xi). Under these conditions the minimum value of the probable error of 7 corre- sponds to the value of derived from the third of equations (xiii). The general solution of this equation is 0= (2n-l)|> where n represents any integer. But, since values of greater than I are not attainable in practice, n must be taken equal to unity in the present case and consequently the best magnitude for the deflection is forty-five degrees. It is obvious that (xi) can always be satisfied when / has any given value, and is equal to forty-five degrees by suitably choosing the values of the free components 2V, H, and R. 178 THE THEORY OF MEASUREMENTS [ART. 85 If the fractional error of / is represented by P and the T's given by equations (xii) are substituted in (151), H 2 ' R 2 ' sin 2 20 Pi 2 + P 2 2 + Pa 2 , (xiv) = Pi 2 + P 2 2 + P 3 2 , where 2 = : and are the separate effects of the probable errors E\, EZ, and E 3) respectively. If both ends of the needle are read with direct and reversed current so that represents the mean of four observa- tions, EZ should not exceed 0.025 or 0.00044 radians, and it might be made less than this with sufficient care. Consequently, when <j> is equal to forty-five degrees, P 3 = 0.00088. By an argument similar to that given in article seventy-nine it can be proved that PI and P 2 will be simultaneously negligible when they satisfy the condition p l = P 2 = i A = 0.00021. 3V2 Hence, in order that the effects of E\ and E% may be negligible in comparison with that of E 3 , H and R must be determined within about two one-hundredths of one per cent. With an instrument of the type considered it would seldom be possible and never worth while to determine H and R with the precision necessary to fulfill the above condition. In common practice E\ and E 2 are generally far above the negligible limit and it would be necessary to make both H and R equal to infinity in order to satisfy the first two of the minimum conditions (xiii). Hence there is no practically attainable minimum value of P . This conclusion can also be derived directly by inspection of equation (xiv). P 2 decreases uniformly as H and R are increased, and becomes equal to Ps 2 when they reach infinity. Although a minimum value of P is not attainable, the fore- going discussion leads to some practical suggestions regarding the design and use of the tangent galvanometer. For any given values of E\, E 2 , and E 3 , the minimum value of PS occurs when <j> is equal to forty-five degrees. Also PI and P% decrease as H and R increase. Consequently the directive force H and the radius ART. 85] BEST MAGNITUDES FOR COMPONENTS 179 of the coil R should be made as large as is consistent with the conditions under which the instrument is to be used, and the number of turns N in the coil should be so chosen that the observed deflection will be about forty-five degrees. The practical limit to the magnitude of R is generally set by a consideration of the cost and convenient size of the instrument. Moreover when R is increased N must be increased in like ratio in order to satisfy the fundamental relation (xi) without altering the observed deflection or decreasing the value of H. There is an indefinite limit beyond which N cannot be increased with- out introducing the chance of error in counting and greatly in- creasing the difficulty of determining the exact magnitude of R. Above this limit E 2 is approximately proportional to R, and, as can be easily seen by equation (xiv), there is no advantage to be gamed by a further increase in the magnitude of R. H can be varied by suitably placed permanent magnets, but it is difficult to maintain strong magnetic fields uniform and con- stant within the required limits. Even under the most favorable conditions, the exact determination of H is very tedious and involves relatively large errors. Consequently Pi 2 is likely to be the largest of the three terms on the right-hand side of equation (xiv). Under suitable conditions it can be reduced in magnitude by increasing H to the limit at which the value of EI begins to increase. However, such a procedure involves an increased value of N in order to satisfy equation (xi), and consequently it may cause an increase in E 2 owing to the relation between N and R pointed out in the preceding paragraph. In such a case the gain in precision due to a decreased value of PI would be nearly bal- anced by an increased value of P%. In common practice the instrument is so adjusted that H is equal to the horizontal component of the earth's magnetic field at the time and place of observation. Unless H is very carefully determined at the exact location of the instrument, EI is likely to be as large as 0.005 ~5 and, since the order of magnitude Cat, of H is about 0.2 ^r , -Pi will be approximately equal to 0.025. cm Hence both P 2 and P 3 will be negligible in comparison with PI if they satisfy the relation P 2 = P 3 = - ^j= = 0.0059. "3 V2 180 THE THEORY OF MEASUREMENTS [ART. 85 Under ordinary conditions R and < can be easily determined within the above limit. Consequently, in the supposed case, PO = PI = 2.5 per cent, and it would be useless to attempt an improvement in precision by adjusting the values of N, R } and <. With sufficient care in determining H, PI can be reduced to such an extent that it be- comes worth while to carry out the suggestions regarding the design and use of the instrument given by the foregoing theory. But when the value of H is assumed from measurements made in a neighboring location or is taken from tables or charts the per- centage error of / will be nearly equal to that of H regardless of the adopted values of R and <. Under such conditions P Q can- not be exactly determined but it will seldom be less than two or three per cent of the measured magnitude of I. The above problem has been discussed somewhat in detail in order to illustrate the inconsistent results that are likely to be obtained in determining best magnitudes when the effects of the errors of some of the components are neglected. It is never safe to assume that the error of a component is negligible until its effect has been compared with that of the errors of the other components. III. Figure eleven is a diagram of the apparatus and connections commonly used in determining the internal resistance of a bat- tery by the condenser method. G is a ballistic galvanometer, C a condenser, R a known resistance, KI a charge and discharge key, Kz a plug or mercury key, and B a battery to be tested. Let Xi represent the ballistic throw of the galvanometer when the condenser is charged and discharged with the key K 2 open, and x z the corresponding throw when K 2 is closed. Then the internal resistance R Q of the battery may be computed by the relation Ro = R ^L^l. ( XV ) Under ordinary conditions the probable errors of x\ and x^ cannot be made much less than one-half of one per cent of the observed throws when a telescope, mirror, and scale are used. On the other hand the probable error of R should not exceed one-tenth of one per cent if a suitably calibrated resistance is used and the ART. 85] BEST MAGNITUDES FOR COMPONENTS 181 connections are carefully made. When these conditions are ful- filled, it can be easily proved that the effect of the error of R is negligible in comparison with that of the errors of Zi and x 2 . Furthermore any convenient value can be assigned to R, such <T 2 R " !L -A/WVW\AAAA/ B FIG. 11. that (xv) will be satisfied whatever the values of Xi and #2. Con- sequently R may be treated as a free component and the throws Xi and x z as independent variables. For the purpose of determining the magnitudes of the com- ponents R, xij and x z that correspond to a minimum value of the fractional error P of RQ, we have by equations (150) and (xv) Consequently - X 2 ) (xvi) Since x\ and x 2 are independent, K must be taken equal to zero in the minimum conditions (153). Hence, dividing the first two equations by T i} we have 1 xi 1 1 E, 2 -^ = o, = 0, (x,-xz) 2 x 2 x 2 2 (x l -xz) 2 where EI and E 2 are the probable errors of x\ and 2 , respectively. 182 THE THEORY OF MEASUREMENTS [ART. 85 Multiply each of these equations by -- 1 ^ 2 and they as- sume the simple form +- * Since #i 2 and Ez 2 are always positive, it is obvious that there are no values of Xi and x% that will satisfy both of these equations at the same time. Hence, when Xi and x z can be varied inde- pendently, they cannot be so chosen that the fractional error P will be a minimum. However, if Xz is kept constant at any as- signed value, PO will pass through a minimum when Xi satisfies equation (a). On the other hand if any constant value is assigned to Xi the minima and maxima of P will correspond to the roots of equation (b). In practice x\ is the throw of the galvanometer needle due to the electromotive force of the battery when on open circuit; and it is very nearly constant, during a series of observations, when suitable precautions are taken to avoid the effects of polariza- tion. Both Xi and Xz can be varied by changing the capacity of the condenser or the sensitiveness of the galvanometer, but their ratio depends only on the ratio of R to R. Consequently, if any convenient magnitude is assigned to Xi, the root of equa- tion (b) that corresponds to a minimum value of PO gives the best magnitude for the component Xz. Since x\ and x 2 are similar quantities, determined with the same instruments and under the same conditions, E\ is generally equal to EZ. Hence, if we replace the ratio -- by y, equation (b) be- ^-2^-1 = 0^ (b') The only real root of this equation is y = 2.2056. By equations (151) and (xvi) Putting E l = Ez = E and - = y, Xz Pl = y* + y* E* x 2 -l 2 ' ART. 86] BEST MAGNITUDES FOR COMPONENTS 183 Since Xi is necessarily greater than x 2 , y cannot be less than unity. P 2 Under this condition it can be easily proved by trial that -== & approaches a minimum as y approaches the value given above, provided any constant value is assigned to x\. Equation (xv) may be put in the form R Q = R(y- 1), and, by introducing the value of y given by the minimum condi- tion (b')> we have R = 0.83 R . Consequently the greatest attainable precision in the determina- tion of RQ will be obtained when R is made equal to about eighty three per cent of RQ. If R is adjusted to this value Xi and x% will satisfy equation (b), whatever the magnitude of the capacity used, provided the observations are so made that E\ and E% are equal. When the internal resistance of the battery is very low it is sometimes impracticable to fulfill the above theoretical conditions because the errors due to polarization are likely to more than off- set the gain in precision corresponding to the theoretically best magnitudes of the components. In such cases a high degree of precision is not attainable, but it is generally advisable to make R considerably larger than R Q in order to reduce polarization errors. 86. Sensitiveness of Methods and Instruments. The pre- cision attainable in the determination of directly measured com- ponents depends very largely on the sensitiveness of indicating instruments and on the methods of adjustment and observation. The design and construction of an instrument fixes its intrinsic sensitiveness; but its effective sensitiveness, when used as an indi- cating device, depends on the circumstances under which it is used and is frequently a function of the magnitudes of measured quan- tities and other determining factors. Thus; the intrinsic sensi- tiveness of a galvanometer is determined by the number of windings in the coils, the moment of the directive couple, and various other factors that enter into its design and construction. On the other hand its effective sensitiveness as an indicator in a Wheatstone Bridge is a function of the resistances in the various arms of the bridge and the electromotive force of the battery used. An increase in the intrinsic sensitiveness of an instrument may cause an increase or a decrease in its effective sensitiveness, 184 THE THEORY OF MEASUREMENTS [ART. 86 depending on the nature of the corresponding modification in design and the circumstances under which the instrument is used. By a suitable choice of the magnitudes of observed components and other determining factors it is sometimes possible to increase the effective sensitiveness of indicating instruments and hence also the precision of the measurements. On the other hand, as pointed out in Chapter XI, the precision of measurements should not be greater than that demanded by the use to which they are to be put. In all cases the effective sensitiveness of instruments and methods should be adjusted to give a result definitely within the required precision limits determined as in Chapter XI. Consequently the best magnitudes for the quan- tities that determine the effective sensitiveness are those that will give the required precision with the least labor and expense. The methods by which such magnitudes can be determined depend largely on the nature of the problem in hand, and a general treat- ment of them is quite beyond the scope of the present treatise. Each separate case demands a somewhat detailed discussion of the theory and practice of the proposed measurements and only a single example can be given here for the purpose of illustration. Since the potentiometer method of comparing electromotive forces has been quite fully discussed in article eighty-one, it will be taken as a basis for the illustration and we will proceed to find the relation between the effective sensitiveness of the galvanom- eter and the various resistances and electromotive forces involved. Since the directly observed components in this method are the resistances R\ and R%, the effective sensitiveness is equal to the galvanometer deflection corresponding to a unit fractional devia- tion of Ri or R z from the condition of balance. From the discussion given in article eighty-one it is evident that the potentiometer method could be carried out with any conven- ient values of the resistances R\ and R 2 provided they are so ad- 7- justed that the ratio - satisfies equation (ii) in the cited article. tiz The absolute magnitudes of these resistances depend on the electro- motive force of the battery J5 3 and the total resistance of the cir- cuit 1, 2, 3, B 3 , 1 in Fig. 10. The effective sensitiveness of the method, and hence the accuracy attainable in adjusting the con- tacts 2 and 3 for the condition of balance, depends on the above ABT.86] BEST MAGNITUDES FOR COMPONENTS 185 factors together with the resistance and intrinsic sensitiveness of the galvanometer. Since RI and R% are adjusted in the same way and under the same conditions, the effective sensitiveness of the method is the same for both. Consequently only one of them will be considered in the present discussion, but the results obtained will apply with equal rigor to either. The essential parts of the apparatus and connections are illustrated in Fig. 12, which is the same as Fig. 10 with the battery B 2 and its connections omitted. FIG. 12. Let V = e.m.f. of battery BI, E = e.m.f. of battery B 3 , R = resistance between 1 and 2, W = total resistance of the circuit 1, 2, B s , 1, G = total resistance of the branch 1, G, BI, 2, I = current through B 3) r = current through R, g = current through BI and G. When the contact 2 is adjusted to the balance position Consequently = 0, r = 7, and 7=^ = -^ (xvii) This is the fundamental equation of the potentiometer and must be fulfilled in every case of balance. Consequently E must be 186 THE THEORY OF MEASUREMENTS [ART. 86 chosen larger than V because R is a part of the resistance in the circuit 1, 2, B z , 1, and hence is always less than W. Equation (xvii) may then be satisfied by a suitable adjustment of R. By applying Kirchhoff's laws to the circuits 1, G, BI, 2, 1, and 1, 2, B 3) 1, when the contact 2 is not in the balance position, we have Rr-Gg= V, and Rr + (W - R) I = E. But r = I - g. Hence RI-(R + G)g = V, and WI - Rg = E. Eliminating I and solving for g we find WV -RE If D is the galvanometer deflection corresponding to the current g and K is the constant of the instrument g = KD. Most galvanometers are, or can be, provided with interchange- able coils. The winding space in such coils is usually constant, but the number of windings, and hence the resistance, is variable. Under these conditions the resistance of the galvanometer will be approximately proportional to the square of the number of turns of wire in the coils used. For the purpose of the present discussion, this resistance may be assumed to be equal to G since the resist- ance of the battery and connecting wires in branch 1, G, BI, 2, can usually be made very small in comparison with that of the galvanometer. The constant K is inversely proportional to the number of windings in the coils used. Consequently, as a suffi- ciently close approximation for our present purpose, we have T v K = T=> VG where T is a constant determined by the dimensions of the coils, the moment of the directive couple, and various other factors depending on the type of galvanometer adopted. Hence, for any given instrument, ART. 86] BEST MAGNITUDES FOR COMPONENTS 187 VG The quantity -jr is the intrinsic sensitiveness of the galvanometer. It is equal to the deflection that would be produced by unit current if the instrument followed the same law for all values of g. By equation (xix) and (xviii) VG WV-RE T *R*-WR-WG' The variation in D due to a change dR in R is dD VG E(R*-WR-WG) + (WV-RE)(2R-W) dR ' T ' (R*-WR-WGY When the potentiometer is adjusted for a balance, D is equal to zero and WV is equal to RE by equation (xvii). Hence, if d is the galvanometer deflection produced when the resistance R is changed from the balancing value by an amount dR, equation (xx) may be put in the form 1 VVG The fractional change in R corresponding to the total change dR is . I '-f : I Consequently 1 VVO ~' ' is the galvanometer deflection corresponding to a fractional error P r in the adjustment of R for balance. The coefficient of P r in equation (xxi) is the effective sensitiveness of the method under the given conditions. If this quantity is represented by S, equa- tion (xxi) becomes 8 = SP r , 8 I All of the quantities appearing in the right-hand member of this equation may be considered as independent variables since equa- tion (xvii) can always be satisfied, and hence the potentiometer 188 THE THEORY OF MEASUREMENTS [ART. 86 can be balanced, when R, V, and E have any assigned values, if the resistance W is suitably chosen. If d' is the smallest galvanometer deflection that can be defi- nitely recognized with the available means of observation, the frac- tional error P/ of a single observation on R should not be greater 5' than -~ Since the precision attainable in adj usting the potentiom- o eter for balance is inversely proportional to P/, it is directly pro- portional to the effective sensitiveness S. By choosing suitable magnitudes for the variables T, G, R, and E, it is usually possible to adjust the value of S, and hence also of P/, to meet the re- quirements of any problem. From equation (xxii) it is evident that S will increase in magni- tude continuously as the quantities T, R, and E decrease and that it does not pass through a maximum value. The practicable in- crease in S is limited by the following considerations: E must be greater than V, for the reason pointed out above, and its variation is limited by the nature of available batteries. Since E must remain constant while the potentiometer is being balanced alter- nately against V and the electromotive force of a standard cell, as explained in article eighty-one, the battery B 3 must be capable of generating a constant electromotive force during a considerable period of time. In practice storage cells are commonly used for this purpose and E may be varied by steps of about two volts by connecting the required number of cells in series. Obviously E should be made as nearly equal to V as local conditions permit. When the potentiometer is balanced V E If R is reduced for the purpose of increasing the effective sensitive- ness, W must also be reduced in like ratio, and, consequently, the current 7 through the instrument will be increased. The prac- tical limit to this adjustment is reached when the heating effect of the current becomes sufficient to cause an appreciable change in the resistances R and W. With ordinary resistance boxes this limit is reached when 7 is equal to a few thousandths of an ampere. Consequently, if E is about two volts, R should not be made much less than one thousand ohms. Resistance coils made expressly for use in a potentiometer can be designed to carry a much larger ART. 86] BEST MAGNITUDES FOR COMPONENTS 189 current so that R may be made less than one hundred ohms with- out introducing serious errors due to the heating effect of the current. The constant T depends on the type and design of the galva- nometer. In the suspended magnet type it can be varied some- what by changing the strength of the external magnetic field, and in the D'Arsonval type the same result may be attained by chang- ing the suspending wires of the movable coil. The effects of the vibrations of the building in which the instrument is located and of accidental changes in the external magnetic field become much more troublesome as T is decreased, i.e., as the intrinsic sensitive- ness is increased. Consequently the practical limit to the reduc- tion of T is reached when the above effects become sufficient to render the observation of small values of 6 uncertain. This limit will depend largely on the location of the instrument and the care that is taken in mounting it. Sometimes a considerable reduc- tion in T can be effected by selecting a type of galvanometer suited to the local conditions. If the quantities T 7 , R, V, and E are kept constant, S passes through a maximum value when G satisfies the condition *?' It can be easily proved by direct differentiation that this is the case when G = Hence, after suitable values of the other variables have been de- termined as outlined above, the best magnitude for G is given by equation (xxiii). Generally this condition cannot be exactly ful- filled in practice unless a galvanometer coil is specially wound for the purpose; but, when several interchangeable coils are available, the one should be chosen that most nearly fulfills the condition. In some galvanometers T and G cannot be varied independently, and in such cases suitable values can be determined only by trial. Since the ease and rapidity with which the observations can be made increase with T, it is usually advisable to adjust the other variables to give the greatest practicable value to the second factor in S, and then adjust T so that the effective sensitiveness 190 THE THEORY OF MEASUREMENTS [ART. 86 will be just sufficient to give the required precision in the deter- mination of R. As an illustration consider the numerical data given in article eighty-one. It was proved that the specified precision require- ments cannot be satisfied unless R is determined within a frac- tional precision measure equal to 0.00063. Allowing one-half of this to errors of calibration we have left for the allowable error in adjusting the potentiometer P r ' = 0.00031. If a single storage cell is used at B$, E is approximately two volts, and, with ordinary resistance boxes, R should be about one thou- sand ohms, for the reason pointed out above. This condition is fulfilled by the cited data; and, for our present purpose, it will be sufficiently exact to take V equal to one volt. Hence, by equa- tion (xxiii), the most advantageous magnitude for G is about five hundred ohms; and, by equation (xxii), the largest practi- cable value for the second factor in S is ST = V Jf = 0.0224. gf 1-41+0 With a mirror galvanometer of the D'Arsonval type, read by telescope and scale, a deflection of one-half a millimeter can be easily detected. Consequently, if we express the galvanometer constant K in terms of amperes per centimeter deflection, we must take 5' equal to 0.05 centimeter; and, in order to fulfill the specified precision requirements, the effective sensitiveness must satisfy the condition S' 0.05 ~P7~00003l~ Combining this result with the above maximum value of ST we find that the intrinsic sensitiveness must be such that 0.0224 _ 161 Hence the galvanometer should be so constructed and adjusted that G = 500 ohms, and T K = = = 6.2 X lO" 6 amperes per centimeter deflection. ART. 86] BEST MAGNITUDES FOR COMPONENTS 191 D'Arsonval galvanometers that satisfy the above specifications can be very easily obtained and are much less expensive than more sensitive instruments. They are so nearly dead-beat and free from the effects of vibration that the adjustment of the poten- tiometer for balance can be easily and rapidly carried out with the necessary precision. Hence the use of such an instrument reduces the expense of the measurements without increasing the errors of observation beyond the specified limit. CHAPTER XIII. RESEARCH. 87. Fundamental Principles. The word research, as used by men of science, signifies a detailed study of some natural phenomenon for the purpose of determining the relation between the variables involved or a comparative study of different phe- nomena for the purpose of classification. The mere execution of measurements, however precise they may be, is not research. On the other hand, the development of suitable methods of measure- ment and instruments for any specific purpose, the estimation of unavoidable errors, and the determination of the attainable limit of precision frequently demand rigorous and far-reaching research. As an illustration, it is sufficient to cite Michelson's determination of the length of the meter in terms of the wave length of light. A repetition of this measurement by exactly the same method and with the same instruments would involve no research, but the original development of the method and apparatus was the result of careful researches extending over many years. The first and most essential prerequisite for research in any field is an idea. The importance of research, as a factor in the advance- ment of science, is directly proportional to the fecundity of the underlying ideas. A detailed discussion of the nature of ideas and of the conditions necessary for their occurrence and development would lead us too far into the field of psychology. They arise more or less vividly in the mind in response to various and often apparently trivial circumstances. Their inception is sometimes due to a flash of intuition during a period of repose when the mind is free to respond to feeble stimuli from the subconscious. Their development and execution generally demand vigorous and sustained mental effort. Probably they arise most frequently in response to suggestion or as the result of careful, though tentative, observations. A large majority of our ideas have been received, in more or less fully developed form, through the spoken or written dis- course of their authors or expositors. Such ideas are the common 192 ART. 88] RESEARCH 193 heritage of mankind, and it is one of the functions of research to correct and amplify them. On the other hand, original ideas, that may serve as a basis for effective research, frequently arise from suggestions received during the study of generally accepted notions or during the progress of other and sometimes quite differ- ent investigations. The originality and productiveness of our ideas are determined by our previous mental training, by our habits of thought and action, and by inherited tendencies. Without these attributes, an idea has very little influence on the advancement of science. Important researches may be, and sometimes are, carried out by investigators who did not originate the underlying ideas. But, however these ideas may have originated, they must be so thor- oughly assimilated by the investigator that they supply the stim- ulus and driving power necessary to overcome the obstacles that inevitably arise during the prosecution of the work. The driving power of an idea is due to the mental state that it produces in the investigator whereby he is unable to rest content until the idea has been thoroughly tested in all its bearings and definitely proved to be true or false. It acts by sustaining an effective concentra- tion of the mental and physical faculties that quickens his in- genuity, broadens his insight, and increases his dexterity. In order to become effective, an idea must furnish the incentive for research, direct the development of suitable methods of pro- cedure, and guide the interpretation of results. But it must never be dogmatically applied to warp the facts of observation into conformity with itself. The mind of the investigator must be as ready to receive and give due weight to evidence against his ideas as to that in their favor. The ultimate truth regarding phenomena and their relations should be sought regardless of the collapse of generally accepted or preconceived notions. From this point of view, research is the process by which ideas are tested in regard to their validity. 88. General Methods of Physical Research. Researches that pertain to the physical sciences may be roughly classified in two groups: one comprising determinations of the so-called physical constants such as the atomic weights of the elements, the velocity of light, the constant of gravitation, etc.; the other containing investigations of physical relations such as that which connects the mass, volume, .pressure, and temperature of a gas. 194 THE THEORY OF MEASUREMENTS [ART. 88 The researches in the first group ultimately reduce to a careful execution of direct or indirect measurements and a determination of the precision of the results obtained. The general principles that should be followed in this part of the work have been suffi- ciently discussed in preceding chapters. Their application to prac- tical problems must be left to the ingenuity and insight of the investigator. Some men, with large experience, make such appli- cations almost intuitively. But most of us must depend on a more or less detailed study of the relative capabilities of available methods to guide us in the prosecution of investigations and in the discussion of results. In general, physical constants do not maintain exactly the same numerical value under all circumstances, but vary somewhat with changes in surrounding conditions or with lapse of time. Thus the velocity of light is different in different media and in dispersive media it is a function of the frequency of the vibrations on which it depends. Consequently the determination of such constants should be accompanied by a thorough study of all of the factors that are likely to affect the values obtained and an exact specifica- tion of the conditions under which the measurements are made. Such a study frequently involves extensive investigations of the phenomena on which the constants depend and it should be carried out by very much the same methods that apply to the determination of physical relations in general. On the other hand, the exact expression of a physical relation generally involves one or more constants that must be determined by direct or in- direct measurements. Hence there is no sharp line of division between the first and second groups specified above, many re- searches belonging partly to one group and partly to the other. The occurrence of any phenomenon is usually the result of the coexistence of a number of more or less independent antecedents. Its complete investigation requires an exact determination of the relative effect of each of the contributary causes and the develop- ment of the general relation by which their interaction is expressed. A determination of the nature and mode of action of all of the antecedents is the first step in this process. Since it is gen- erally impossible to derive useful information by observing the combined action of a number of different causal factors, it becomes necessary to devise means by which the effects of the several factors can be controlled in such manner that they can be studied ART. 88] RESEARCH 195 separately. The success of researches of this type depends very largely on the effectiveness of such means of control and the accuracy with which departures from specified conditions can be determined. Suppose that an idea has occurred to us that a certain phenome- non is due to the interaction of a number of different factors that we will represent by A, B, C, . . . , P. This idea may involve a more or less definite notion regarding the relative effects of the several factors or it may comprehend only a notion that they are connected by some functional relation. In either case we wish to submit our idea to the test of careful research and to determine the exact form of the functional relation if it exists. The investigation is initiated by making a series of preliminary observations of the phenomenon corresponding to as many vari- ations in the values of the several factors as can be easily effected. The nature of such observations and the precision with which they should be made depend so much on the character of the problem in hand that it would be impossible to give a useful general dis- cussion of suitable methods of procedure. Sometimes roughly quantitative, or even qualitative, observations are sufficient. In other cases a considerable degree of precision is necessary before definite information can be obtained. In all cases the observa- tions should be sufficiently extensive and exact to reveal the gen- eral nature and approximate relative magnitudes of the effects produced by each of the factors. They should also serve to detect the presence of factors not initially contemplated. With the aid of the information derived from preliminary obser- vations and from a study of such theoretical considerations as they may suggest, means are devised for exactly controlling the magnitude of each of the factors. Methods are then developed for the precise measurement of these magnitudes under the con- ditions imposed by the adopted means of control. This process often involves a preliminary trial of several different methods for the purpose of determining their relative availability and pre- cision. The methods that are found to be most exact and con- venient usually require some modification to adapt them to the requirements of a particular problem. Sometimes it becomes necessary to devise and test entirely new methods. During this part of the investigation the discussions of the precision of meas- urements given in the preceding chapters find constant applica- 196 THE THEORY OF MEASUREMENTS [ART. 88 tion and it is largely through them that the suitableness of proposed methods is determined. After definite methods of measurement and means of control have been adopted and perfected to the required degree of pre- cision, the final measurements on the factors, A, B, C, . . . , P, are carried out under the conditions that are found to be most advantageous. Usually two of the factors, say A and B, are caused to vary through as large a range of values as conditions will permit while the other factors are maintained constant at definite observed values. At stated intervals the progress of the variation is arrested and corresponding values of A and B are measured while they are kept constant. From a sufficiently extended series of such observations it is usually possible to make an empirical determination of the form of the functional relation A =/i(); C,Z>, . . . ,P. constant. (i) On the other hand, if the form of the function /i is given as a theoretical deduction from the idea underlying the investigation, the observations serve to test the exactness of the idea and de- termine the magnitudes of the constants involved in the given function. By allowing different factors to vary and making corresponding measurements, the relations A =/ 2 (C); B,D, . . , P, constant, A =/ n (P); ,C,Z>, ., constant, (ii) may be empirically determined or verified. As many functions of this type as there are pairs of factors might be determined, but usually it is not necessary to establish more than one relation for each factor. Generally it is convenient to determine one of the factors as a function of each of the others as illustrated above; but it is not necessary to do so, and sometimes the determination of a different set of relations facilitates the investigation. During the establishment of the relation between any two factors all of the others are supposed to remain rigorously con- stant. Frequently this condition cannot be exactly fulfilled with available means of control, but the variations thus introduced can usually be made so small that their effects can be treated as constant errors and removed with the aid of the relations after- wards found to exist between the factors concerned, For this ART. 88] RESEARCH 197 purpose frequent observations must be made on the factors that are supposed to remain constant during the measurement of the two principal variables. If the variations in these factors are not very small all of the relations determined by the principal measure- ments will be more or less in error and must be treated as first approximations. Usually such errors can be eliminated and the true relations established with sufficient precision, by a series of successive approximations. However, the weight of the final result increases very rapidly with the effectiveness of the means of control and it is always worth while to exercise the care necessary to make them adequate. When the functions involved in equations (i) and (ii), or their equivalents in terms of other combinations of factors, have been determined with sufficient precision, they can usually be com- bined into a single relation, in the form or A=F(B,C,D, . . . ,P), F(A,B,C,D, ,P)=0, (iii) which expresses the general course of the investigated phenomenon in response to variations of the factors within the limits of the observations. Such generalizations may be purely empirical or they may rest partly or entirely on theoretical deductions from well-established principles. In either case the test of their validity lies in the exactness with which they represent observed facts. While an exact empirical formula finds many useful applications in practical problems it should not be assumed to express the true physical nature of the phenomenon it represents. In fact our understanding of any phenomenon is but scanty until we can represent its course by a formula that gives explicit or implicit expression to the physical principles that underlie it. Conse- quently a research ought not to be considered complete until the investigated phenomenon has been classified and represented by a function that exhibits the physical relations among its factors. (i It is scat cely necessary to point out that a complete research as outlined above is seldom carried out by one man and that the underlying ideas very rarely originate at the same time or in the same person. The preliminary relations in the form of equations (i) and (ii) are frequently inspired by independent ideas and worked out by different men. The exact determination of any 198 THE THEORY OF MEASUREMENTS [ART. 89 one of them constitutes a research that is complete so far as it goes. The establishment of the general relation that compre- hends all of the others and the interpretation of its physical signifi- cance are generally the result of a process of gradual growth and modification to which many men have contributed. 89. Graphical Methods of Reduction. After the necessary measurements have been completed and corrected for all known constant errors, the form of the functions appearing in equations (i) and (ii), or other equations of similar type, and the numerical value of the constants involved can sometimes be determined easily and effectively by graphical methods. Such methods are almost universally adopted for the discussion of preliminary obser- vations and the determination of approximate values of the con- stants. In some cases they are the only methods by which the results of the measurements can be expressed. In some other cases the constants can be more exactly determined by an appli- cation of the method of least squares to be described later. Usu- ally, however, the general form of the functions and approximate values of the constants must first be determined by graphical methods or otherwise. Let x and y represent the simultaneous values of two variable factors corresponding to specified constant values of the other factors involved in the phenomenon under investigation. Suppose that x has been varied by successive nearly equal steps through as great a range as conditions permit and that the simultaneous values x and y have been measured after each of these steps while the factors that they represent were kept constant. If all other factors have remained constant throughout these operations, the above series of measurements on x and y may be applied at once to the determination of the form and constants of the functional relation This expression is of the same type as equations (i.) and (ii). Consequently the following discussion applies generally to all cases in which there are only two variable factors. If the sup- posedly constant factors are not strictly constant during the measurements, the observations on x and y will not give the true form of the function in (iv) until they have been corrected for the effects of the variations thus introduced. ART. 89] RESEARCH 199 As the first step in the graphical method of reduction, the observations on x and y are laid off as abscissae and ordinates on accurately squared paper, and the points determined by corre- sponding coordinates are accurately located with a fine pointed needle. The visibility of these points is usually increased by drawing a small circle or other figure with its center exactly at the indicated point. The scale of the plot should be so chosen that the form of the curve determined by the located points is easily recognized by eye. In order to bring out the desired rela- tion, it is frequently necessary to adopt a different scale for ordi- nates and abscissae. Usually it is advantageous to choose such scales that the total variations of x and y will be represented by approximately equal spaces. Thus, if the total variation of y is numerically equal to about one-tenth of the corresponding vari- ation of x, the i/'s should be plotted to a scale approximately ten times as large as that adopted for the x's. In all cases the adopted scales should be clearly indicated by suitable numbers placed at equal intervals along the vertical and horizontal axes. Letters or other abbreviations should be placed near the ends of the axes to indicate the quantities represented. The points thus located usually lie very nearly on a uniform curve that represents the functional relation (iv). Consequently the problem in hand may be solved by determining the equation of this curve and the numerical value of the constants involved in it. Sometimes it is impossible or inadvisable to carry out such a determination in practice and in such cases the plotted curve is the only available means of representing the relation between the observed factors. In all cases the deviations of the located points from the uniform curve represent the residuals of the observations, and, consequently, indicate the precision of the measurements on x and y. The simplest case, and one that frequently occurs in practice, is illustrated in Fig. 13. The plotted points lie very nearly on a straight line. Consequently the functional relation (iv) takes the linear form y = Ax + B, (v) where A is the tangent of the angle a between the line and the positive direction of the x axis, and B is the intercept OP on the y axis. For the determination of the numerical values of the 200 THE THEORY OF MEASUREMENTS [ART. 89 constants A and B, the line should be sharply drawn in such a position that the plotted points deviate from it about equally in opposite directions, i.e., the sum of the positive deviations should be made as nearly as possible equal to the sum of the negative deviations. If this has been carefully and accurately done, the constant B may be determined by a direct measurement of the intercept OP in terms of the scale used in plotting the y's- 0.10 05 25 FIG. 13. 50 75 The constant A may be computed from measurements of the coordinates x\ and 2/1 of any point on the line, not one of the plotted points, by the relation If the position of the line is such that the point P does not fall within the limits of the plotting sheet, the coordinates, Xi, y\ and 2, 2/2, of two points on the line are measured. Since they must satisfy equation (v), 2/i = Axi + B, and 2/2 = Ax 2 + B. Consequently A = and B X 2 The points selected for this purpose should be as widely separated as possible in order to reduce the effect of errors of plotting and ART. 89] RESEARCH 201 measurement. The accuracy of these determinations is likely to be greatest when the vertical and horizontal scales are so chosen that the line makes an angle of approximately forty-five degrees with the x axis. Space may sometimes be saved and the appear- ance of the plot improved by subtracting a constant quantity, nearly equal to B, from each of the y's before they are plotted. Many physical relations are not linear in form. Perhaps none of them are strictly linear when large ranges of variation are con- sidered. Consequently the plotted points are more likely to lie nearly on some regular curve than on a straight line. In such cases the form of the functional relation (iv) is sometimes sug- gested by theoretical considerations, but frequently it must be determined by the method of trial and error or successive approxi- mations. For this purpose the curve representing the observa- tions is compared with a number of curves representing known equations. The equation of the curve that comes nearest to the desired form is modified by altering the numerical values of its constants until it represents the given measurements within the accidental errors of observation. Frequently several different equations and a number of modifications of the constants must be tried before satisfactory agreement is obtained. When the desired relation does not contain more than two inde- pendent constants, it can sometimes be reduced to a linear relation between simple functions of x and y. Thus, the equation y = Be~ Ax , . (vi) represented by the curve in Fig. 14, is frequently met with in physical investigations. By inverting (vi) and introducing ' log- arithms, we obtain the relation log* y = log* B - Ax. Hence if the logarithms of the y's are laid off as ordinates against the corresponding x's as abscissae, the located points will lie very nearly on a straight line if the given observations satisfy the func- tional relation (vi) . When this is the case, the constants A and loge B may be determined by the methods developed during the discussion of equation (v). If logarithms to the base ten are used the above equation becomes ^| log y = logio B - x, 202 THE THEORY OF MEASUREMENTS [ART. 89 where M is the modulus of the natural system of logarithms. In ^ this case the plot gives the values of logio B and -^ from which the constants A and B can be easily computed. When the plotted points do not lie nearer to a straight line than to any other curve, y 10 \ \ 0.5 1.0 1.5 FIG. 14. equation (vi) does not represent the functional relation between the observed factors and some other form must be tried. Many of the commonly occurring forms may be treated by the above method and the process is usually so simple that further illustra- tion seems unnecessary. The curve determined by plotting the x's and y's directly fre- quently exhibits points of discontinuity or sharp bends as at p and q in Fig. 15. Such irregularities are generally due to changes in the state of the material under investigation. The nature- and causes of such changes are frequently determined, or at least suggested, by the location and character of such points. The different branches of the curve may correspond to entirely differ- ent equations or to equations in the same form but with different constants. In either case the equation of each branch must be determined separately. The accuracy attainable by graphical methods depends very largely on the skill of the draughtsman in choosing suitable scales and executing the necessary operations. In many cases the errors ART. 90] RESEARCH 203 due to the plot are less than the errors of observation and it would be useless to adopt a more precise method of reduction. When the means of control are so well devised and effective that the constant errors left in the measurements are less than the errors of plotting it is probably worth while to make the reductions by the method of least squares, as explained in the following article. y 'FiG. 15. 90. Application of the Method of Least Squares. In the case of linear relations, expressible in the form of equation (v), the best values of the constants A and B can be very easily deter- mined by applying the method of least squares in the manner explained in article fifty-one. However, as pointed out in the preceding article, very few physical relations are strictly linear when large variations of the involved factors are considered. Consequently a straight line, corresponding to constants deter- mined as above, usually represents only a small part of the course of the investigated phenomenon. Such a line is generally a short chord of the curve that represents the true relation and conse- quently its direction depends on the particular range covered by the observations from which it is derived. When the measurements are extended over a sufficiently wide range, the points plotted from them usually deviate from a straight line in an approximately regular manner, as illustrated in Fig. 16, 204 THE THEORY OF MEASUREMENTS [ART. 90 and lie very near to a continuous curve of slight curvature. Meas- urements of this type can always be represented empirically by a power series in the form y = A + Bx + Cx* + . - - , (vii) the number of terms and the signs of the constants depending on the magnitude and sign of the curvature to be represented. FIG. 16. Since equation (vii) is linear with respect to the constants A, B, C, etc., they might be computed directly from the observations on x and y by the method of least squares. Usually, however, the computations can be simplified by introducing approximate values of the constants A and B. Thus, let A' and B' represent two numerical quantities so chosen that the line y' = A' + B'x passes in the same general direction as the plotted points, in the manner illustrated by the dotted line in pig. 16. The difference between y and y' can be put in the form y y' = (A A') + MI (B B'} -^ 4- M 2 C + . . . (viii) MI M 2 where Afi, M 2 , etc., represent numerical constants so chosen that *Y* s2 the quantities y - y', -=, etc., are nearly of the same order ART. 90] RESEARCH 205 of magnitude. For the sake of convenience let (ix) and The quantities s, 6, c, etc., may be derived from the observations, with the aid of the assumed constants A', B', MI, M z , etc.; and xi, x z , x S} etc., are the unknowns to be computed by the method of least squares. After the above substitutions, equation (viii) takes the simple form xi + bx 2 + cx 3 + = s, which is identical with that of the observation equations (53), article forty-nine. As many equations of this type may be formed as there are pairs of corresponding measurements on x andj y. The normal equations (56) may be derived from the observation equations thus established, by the methods explained in articles fifty and fifty-three. Their final solution for the unknowns Xi, Xz, xsj etc., may be effected by Gauss's method, developed in article fifty-four and illustrated in article fifty-five, or by any other con- venient method. The corresponding numerical values of the constants A, B, C, etc., may then be computed by equations (ix). These values, when substituted in (vii) , give the required empirical relation between x and y. If a sufficient number of terms have been included in equation (vii), the relation thus established will represent the given measure- ments within the accidental errors of observation. The residuals, computed by equations (54), article forty-nine, and arranged in the order of increasing values of y, should show approximately as many sign changes as sign follows. When this is not the case the observed y's deviate systematically from the values given by equation (vii) for corresponding x's. In such cases the number of terms employed is not sufficient for the exact representation of the observed phenomenon, and a new relation in the same general form as the one already tested but containing more independent constants should be determined. This process must be repeated until such a relation is established that systematically varying differences between observed and computed y's no longer occur. The observation equations used as a basis for the numerical illustration given in article fifty-five were derived from the follow- 206 THE THEORY OF MEASUREMENTS [ART. 90 ing observations on the thermal expansion of petroleum by equa- tions (viii) and (ix), taking A' = 1000; B' = l; M l = 10; and M 2 = 1000. X temperature volume degrees cc. 1000.24 20 1018.82 40 1038.47 60 1059 31 80 1081.20 100 1104.27 The computations carried out in the cited article resulted as follows : xi = 0.245; x 2 = - 1.0003; x 3 = 1.4022. Hence, by equations (ix) A = 1000.245; B = 0.89997; C = 0.0014022, and the functional relation (vii) becomes y = 1000.245 + 0.89997 x + 0.0014022 x\ The residuals corresponding to this relation, computed and tab- ulated in article fifty-five, show five sign changes and no sign follows. Such a distribution of signs sometimes indicates that the observed factors deviate periodically from the assumed functional relation. In the present case, however, the number of observa- tions is so small that the apparent indications of the residuals are probably fortuitous. Consequently it would not be worth while to repeat the computations with a larger number of terms unless it could be shown by independent means that the accidental errors of the observations are less than the residuals corresponding to the above relation. Any continuous relation between two variables can usually be represented empirically by an expression in the form of equation (vii). However, it frequently happens that the physical signifi- cance of the investigated phenomenon is not suggested by such an expression but is represented explicitly by a function that is not linear with respect to either the variable factors or the constants involved. Such functions usually contain more than two inde- pendent constants and sometimes include more than two variable factors. They may be expressed by the general equation y = F(A,B,C,. ,x,z,. . ), (154) ART. 90] RESEARCH 207 where A, B, C, etc., represent constants to be determined and y t x, z, etc., represent corresponding values of observed factors. Sometimes the form of the function F is given by theoretical considerations, but more frequently it must be determined, to- gether with the numerical values of the constants, by the method of successive approximations. In the latter case a definite form, suggested by the graphical representation of the observations or by analogy with similar phenomena, is assumed tentatively as a first approximation. Then, by substituting a number of different corresponding observations on y, x, z, etc., in (154), as many inde- pendent equations as there are constants in the assumed function are established. The simultaneous solution of these equations gives first approximations to the values of the constants A, B, C, etc. Sometimes the solution cannot be effected directly by means of the ordinary algebraic methods, but it can usually be accom- plished with sufficient accuracy either by trial and error or by some other method of approximation. Let A', B' ', C', etc., represent approximate values of the con- stants and let 61, 5 2 , 5 3 , etc., represent their respective deviations from the true values. Then A=A' + 5 1 ; B = B' + d 2 ] C = C' + 5 3 , etc., (155) and (154) may be put in the form y-F\(A' + Sd, (B' + fc), (C" + .) ---- ,*,*, . - . | (x) If the S's are so small that their squares and higher powers may be neglected, expansion by Taylor's Theorem gives y-F(A',B',C', . . . ,x,z, . . dF dF , dF ,,. . .,,,.. By putting y-F(A',B',C', . . . ,x,z, . . . ) = ; (156) and transposing, equation (xi) becomes adi + 65 2 + c5 3 + . . . = s. (157) As many independent equations of this type as there are sets of corresponding observations on y, x, z, etc., can be formed. The absolute term s and the coefficients a, 6, c, etc., in each equation are computed from a single set of observations by the relations 208 THE THEORY OF MEASUREMENTS [ART. 90 (156) with the aid of the approximate values A', B f , C", etc. Since the resulting equations are in the same form as the observation equations (53), the normal equations (56) may be found and solved by the methods described in Chapter VII. The values of $1, 6 2 , 5 3 , etc., thus obtained, when substituted in (155), give second approximations to the values of the constants A, B, C, etc. The accuracy of the second approximations will depend on the assumed form of the function F and on the magnitude of the correc- tions Si, 6 2 , 6 3 , etc. If these corrections are not small, the con- ditions underlying equation (xi) are not fulfilled and the results obtained by the above process may deviate widely from the correct values of the constants; but, except in extreme cases, they are more accurate than the first approximations A', B f , C', etc. Let A", B", C", etc., represent the second approximations. The corresponding residuals, n, r 2 , . . . , r n , may be computed by substituting different sets of corresponding observations on y, x, z, etc., successively in the equation F(A",B",C", . . . ,x,z, . . . )-y = r, (xii) where the function F has the same form that was used in comput- ing the corrections 5i, ^2, 5 3 , etc. If these residuals are of the same order of magnitude as the accidental errors of the observations and distributed in accordance with the laws of such errors, the functional relation y = F(A",B",C", . . . ,x,z, . . . ) (158) is the most probable result that can be derived from the given observations. Frequently the residuals corresponding to the second approxi- mations do not atisfy the above conditions. This may be due to the inadequacy of the assumed form of the function F, to insufficient precision of the approximations A", B", C", etc., or to both of these causes. If the form of the function is faulty, the residuals usually show systematic and easily recognizable deviations from the distribu- tion characteristic of accidental errors. Generally the number of sign follows greatly exceeds the number of sign changes, when the residuals are arranged in the order of increasing y's, and opposite signs do not occur with nearly the same frequency. Sometimes the nature of the fault can be determined by inspecting the order ART. 91] RESEARCH 209 of sequence of the residuals or by comparing the graph correspond- ing to equation (158) with the plotted observations. After the form of the function F has been rectified, by the above means or otherwise, the computations must be repeated from the beginning and the new form must be tested in the same manner as its prede- cessor. This process should be continued until the residuals cor- responding to the second approximations give no evidence that the form of the function on which they depend is faulty. When the residuals, computed by equation (xii), do not suggest that the assumed form of the function F is inadequate, but are large in comparison with the probable errors of the observations, the second approximations are not sufficiently exact. In such cases new equations in the form of (157) are derived by using A", B" , C", etc., in place of A', B f , C', etc., in equations (156). The solution of the equations thus formed, by the method of least squares, gives the corrections 5/, 5 2 ', 5 3 ', etc., that must be applied to A", B", C", etc., in order to obtain the third approximations At tt A n I x / . T>itt ~Dir I <j / . r</n rut \ * t . 4. = A -f- di ; > = n + 62 ; C = C + 03 ; etc. These operations must be repeated until the residuals correspond- ing to the last approximations are of the same order of magnitude as the accidental errors of the observations. Although an algebraic expression, that represents any given series of observations with sufficient precision, can usually be de- rived by the foregoing methods, such a procedure is by no means advisable in all cases. In many investigations, a graphical repre- sentation of the results leads to quite as definite and trustworthy conclusions as the more tedious mathematical process. Conse- quently the latter method is usually adopted only when the former is inapplicable or fails to utilize the full precision of the observa- tions. In all cases the choice of suitable methods and the estab- lishment of rational conclusions is a matter of judgment and experience. 91. Publication. Research does not become effective as a factor in the advancement of science until its results have been published, or otherwise reported, in intelligible and widely acces- sible form. It is the duty as well as the privilege of the investiga- tor to make such report as soon as he has arrived at definite conclusions. But nothing could be more inadvisable or untimely than the premature publication of observations that have not been thoroughly discussed and correlated with fundamental principles. 210 THE THEORY OF MEASUREMENTS [ART. 91 Until an investigation has progressed to such a point that it makes some definite addition to existing ideas, or gives some important physical constant with increased precision, its publication is likely to retard rather than stimulate the progress of science. On the other hand, free discussion of methods and preliminary results is an effective molder of ideas. The form of a published report is scarcely less important than the substance. The significance of the most brilliant ideas may be entirely masked by faulty or inadequate expression. Hence the investigator should strive to develop a lucid and concise style that will present his ideas and the observations that support them in logical sequence. Above all things he should remember that the value of a scientific communication is measured by the importance of the underlying ideas, not by its length. The author's point of view, the problem he proposes to solve, and the ideas that have guided his work should be clearly defined. Theoretical considerations should be rigorously developed in so far as they have direct bearing on the work in hand. But general discussions that can be found in well-known treatises or in easily accessible journals should be given by reference, and the formulae derived therein assumed without further proof whenever their rigor is not questioned. However, the author should always explain his own interpretation of adopted formulae and point out their significance with respect to his observations. Due weight and credit should be given to the ideas and results of other workers in the same or closely related fields, but lengthy descriptions of their methods and apparatus should be avoided. Explicit refer- ence to original sources is usually sufficient. The methods and apparatus actually used in making the re- ported observations, should be concisely described, with the aid of schematic diagrams whenever possible. Well-known methods and instruments should be described only in so far as they have been modified to fulfill special purposes. Detailed discussion of all of the methods and instruments that have been found to be inadequate are generally superfluous, but the difficulties that have been overcome should be briefly pointed out and explained. The precautions adopted to avoid constant errors should be explicitly stated and the processes by which unavoidable errors of this type have been removed from the measurements should be clearly described. The effects likely to arise from such errors should be ART. 91] RESEARCH 211 considered briefly and the magnitude of applied corrections should be stated. Observations and the results derived from them should be reported in such form that their significance is readily intelligible and their precision easily ascertainable. In many cases graphical methods of representation are the most suitable provided the points determined by the observations are accurately located and marked. The reproduction of a large mass of numerical data is thus avoided without detracting from the comprehensiveness of the report. When such methods do not exhibit the full pre- cision of the observations or when they are inapplicable on account of the nature of the problem in hand, the original data should be reproduced with sufficient fullness to substantiate the conclusions drawn from them. In such cases the significance of the obser- vations and derived results can generally be most convincingly brought out by a suitable tabulation of numerical data. An estimate of the precision attained should be made whenever the results of the investigation can -be expressed numerically. Final conclusions should be logically drawn, explicitly stated, and rigorously developed in their theoretical bearings. They express a culmination of the author's ideas relative to the inves- tigated phenomena and invite criticism of their exactness and rationality. Unless they are amply substantiated by the obser- vations and theoretical considerations brought forward in their support, and constitute a real addition to scientific knowledge, they are likely to receive scant recognition. TABLES. The following tables contain formulae and numerical data that will be found useful to the student in applying the principles developed in the preceding chapters. The four figure numerical tables are amply sufficient for the computation of errors, but more extensive tables should be used in computing indirectly measured magnitudes whenever the precision of the observations warrants the use of more than four significant figures. The references placed under some of the tables indicate the texts from which they were adapted. TABLE I. DIMENSIONS OF UNITS. Units. Dimensions. Fundamental. Length, mass, time Length, force, time. Length [L] [M] [T] [LMT-*] m M [L-W] [Llr*\ [LT-i] pNj [LT-*\ [T- 2 ] [LMT~ l ] [L*M] [LW7 7 - 1 ] [L*MT-*] [L-W7 1 - 2 ] [LW7 7 - 2 ] [Lwr- 8 ] [L] [L-iFT*] [T] [F] [V] [If] [L-*FT*] [LL-i] [LT-i] [T- 1 ] [LT-*] [T - 2] [FT] [LFT*] [LFT] [LF] [L-*F] [LF] [LFT- 1 ] Mass Time Force Area Volume Density Angle Velocity, linear Velocity, angular Acceleration, linear Acceleration, angular Momentum Moment of inertia Moment of momentum Torque Pressure Energy, work Power 212 TABLES 213 TABLE II. CONVERSION FACTORS. Length Units. Logarithm. 1 centimeter (cm.) _ = 0. 393700 inch 1 . 5951654 " " = 0. 0328083 foot 2. 5159842 " = 0. 0109361 yard 2. 0388629 1 meter (m.) = 1000 millimeters 3. 0000000 " = 100 centimeters 2. 0000000 " = 10 decimeters. 1.0000000 1 kilometer (km.) = 1000 meters 3. 0000000 = 0. 621370 mile 1. 7933503 " = 3280. 83 feet 3. 5159842 1 inch (in.) = 2. 540005 centimeters 0. 4048346 1 foot (ft.) =12 inches 1.0791812 = 30. 4801 centimeters 1 . 4840158 1 yard (yd.) = 36 inches 1. 5563025 " =3 feet 0. 4771213 " = 91.4402 centimeters 1.9611371 1 mile (ml.) = 5280 feet 3 . 7226339 " = 1760 yards 3. 2455127 = 1609. 35 meters 3.2066497 = 0.868392 knot (U. S.) 1.9387157 Mass Units. 1 gram (g.) = 1000 milligrams 3. 0000000 " = 100 centigrams 2. 0000000 " = 10 decigrams 1. 0000000 = 0.0352740 ounce (av.) 2.5474542 " = 0. 00220462 pound (av.) 3. 3433342 = 0. 000068486 slugg 5. 8355997 1 kilogram (kg.) = 1000 grams 3. 0000000 1 ounce (oz.) (av.) = 28. 3495 grams 1. 4525458 = 0. 062500 pound (av.) 2. 7958800 " =0.0019415 slugg 3.2881455 1 pound (Ib.) (av.) = 16 ounces (av.) 1.2041200 " = 453. 5924277 grams 2. 6566658 = 0.0310646 slugg 2.4922655 1 slugg (sg.) = 32. 191 pounds (av.) 1. 5077345 = 515.06 ounces (av.) 2.7118545 = 14601. 6 grams 4. 1644003 1 short ton (tn.) = 2000 pounds (av.) 3. 3010300 = 907. 185 kilograms 2. 9576958 " =62. 129 sluggs 1 . 7932955 214 THE THEORY OF MEASUREMENTS TABLE II. CONVERSION FACTORS (Concluded}. Force Units. The following gravitational units are expressed in terms of the earth's attraction at London where the acceleration due to gravity is 32.191 ft. /sec. 2 or 981.19 cm./sec Logarithm. 1 dyne = 1 . 01917 milligram's wt 0. 0082469 " = 0. 00101917 gram's wt 3 . 0082469 " =2.2469 X 10- 6 pound's wt 6.3515811 1 gram's wt. = 981.19 dynes 2. 9917531 1 kilogram's wt. = 1000 gram's wt 3. 0000000 = 98. 119 X 10 4 dynes 5.9917531 = 2.20462 pound's wt 0. 3433342 1 pound's wt. =0. 45359 kilogram's wt 1 . 6566658 = 44.506 X 10 4 dynes 5.6484189 1 pound's wt. (local) = 0/32.191 pound's wt. at London. g = local acceleration due to gravity in ft./secT 2 . Mean Solar Time Units. 1 second (s.) = 0. 016667 minute 2. 2218487 " = 0. 00027778 hour 4. 4436975 = 0.000011574 day 5.0634863 1 minute (m.) = 60 seconds 1 . 7781513 " =0.016667 hour 2.2218487 = 0.00069444 day 4.8416375 1 hour (h.) = 3600 seconds 3. 5563025 = 60 minutes 1. 7781513 " = 0. 041667 day 2. 6197888 1 day (d.) = 86400 seconds 4. 9365137 = 1440 minutes 3. 1583625 " =24 hours 1.3802112 1 mean solar unit = 1 . 00273791 sidereal units 0. 0011874 Angle Units. 1 circumference = 360 degrees 2. 5563025 = 2 TT radians 0. 7981799 " = 6.28319 radians 0. 7981799 1 degree () = 0. 017453 radian 2. 2418774 = 60 minutes 1. 7781513 = 3600 seconds 3. 5563025 1 minute (') =2. 9089 X 10- 4 radians 4 . 4637261 = 0.016667 degree 2.2218487 = 60 seconds 1. 7781513 1 second (') = 4.8481 X KH 5 radians 6 . 6855749 = 2. 7778 X 10- 4 degrees 4. 4436975 = 0. 01667 minute 2. 2218487 1 radian = 57.29578 degrees 1. 7581226 = 3437.7468 minutes 3. 5362739 = 206264.8 seconds . . 5. 3144251 TABLES 215 TABLE III. TRIGONOMETRICAL RELATIONS. a 3 . a 5 t <t\, sma = a 777 +T? (!) (2n-l)! cos 2 a = 1 cos 2 a 2 cosec a _ . ce a cos a tan a = 2 sin ^ cos tr 2 2 cot a sec a tan a 1 - = cos a tan a Vl+tan 2 a VI + cot 2 a = sin /3 cos (|8 a) cos /3 sin (/3 a) = cos /3 sin (0 + a) sin /8 cos (/3 + a). l/l a =y cos a 2~~ 2 tan a sin 2 a = 2 sin a cos a = ., 1 + tan 2 a. sin 2 a = 1 cos 2 a = \ (cos 2 a 1). sin (a j8) = sin a cos cos a sin 0. sin a =fc sin /3 = 2 sin (a d= /8) cos |( =F 0). sin 2 a + sin 2 /3 = 1 cos (a + /3) cos (a /8). sin 2 a sin 2 = cos 2 /3 cos 2 a = sin (a + 0) sin (a /8). V 1 + sin a = sin | a + cos a. VI sin a = (sin | a cos \ a). cos cos 2 i a sin 2 | a cot a V 1 + tan 2 a V 1 + cot 2 a sin a cot a 1 = sin a cot a COS ^ a = tan o; cosec a sec a = cos cos (a + /8) + sin sin (a + 0) = cos /? cos ((8 - a) + sin ft sin (0 a). 1 + cos a 216 THE THEORY OF MEASUREMENTS TABLE III. TRIGONOMETRICAL RELATIONS (Continued). cos 2 a = 2 cos 2 a 1 = 1 2 sin 2 a 1 - tan 2 a = cos 2 a sm 2 a. = ^ 1 + tan 2 a cos 2 a. = 1 - sin 2 a = (cos 2 a + 1). cos (a d= 0) = cos a cos T sin a sin 0. cos a + cos = 2 cos 5 (a + 0) cos H 0)- cos a cos = 2 sin (a -f 0) sin | (a 0) . cos 2 a + cos 2 = 1 + cos (a + 0) cos (a - 0). cos 2 a cos 2 = sin 2 sin 2 a = sin (a + 0) sin (a 0). cos 2 a sin 2 = cos (a + 0) cos (a 0) = cos 2 sin 2 a. sin a + cos a = V 1 + sin 2 a. sin a cos a = Vi sin 2 a. sin 2 a + cos 2 a. = 1. sin 2 a cos 2 a: = cos 2 or. tan a = a + | a 3 + - r 2 5 CK 5 + 3^5 a 7 + . . . w > a> TT sin a. sin 2 a 1 cos 2 cos a 1 + cos 2 a sin 2 a V'l cos 2 a _ 4 / 1+ cos 2 a " V cos 2 a VI sin 2 a = Vsec 2 a I tan 2 a = cosec a: Vcosec 2 a-l = cot a 2 cot 2 a cot a sin (a + 0) + sin ( 0) _ cos (a 0) cos (a + 0) cos (a + 0) + cos (a 0) sin (a + 0) - sin (a - 0) 2 tan a 2 cot a 2 1 tan 2 a cot 2 a 1 cot a tan a tan f a. = - ; - = cosec a cot or. 1 + sec a ( . R\ tan a tan _ cos 2 cos 2 a * W * 1 T tan a tan ~ sin 2 =F sin 2 a sin (a 0) tan a tan = cos a cos TABLES 217 TABLE III. TRIGONOMETRICAL RELATIONS (Concluded). Ill 2 cot a. = -- - a j= a 3 ^r-= a 5 TT > a > IT a: 3 45 olo cos a _ sin 2 a _ 1 + cos 2 a sin a ~~ 1 cos 2 a "~ sin 2 a V/ 1 + cos 2 o: _ cos a vl sin 2 a 1 cos 2 a Vl cos 2 a = tan a. + 2 cot 2 a. tan a. _ 1 tan 2 a. _ cot 2 a 1 cot a tan a " 2 tan a 2cota ~^~ cot - a = (1 + sec a) cot a 2<-. v j. | kjv/v; <-*. y vv/u c*. : cosec a cot a 1 =F tan a tan /? cot cot =F 1 cot (a d= 0) = tan a tan cot d= cot a sin TABLE IV. SERIES. Taylor's Theorem. /(*+&)=/(*) + AT (*) + ^/" (*)+;+ ^/W (x) + f(x + h, y + k, where u = f (x, y, z). Maclaurin's Series. /(0) + f /' (0) + !/" (0) + + fj/N (0). 218 THE THEORY OF MEASUREMENTS TABLE IV. SERIES (Concluded}. Binomial Theorem. = xm + rn x ^ ly + m(n^ xm _^ + . . . , * (m - 1) . . . (m - n + 1) ^- y> when m is a positive integer, also when m is negative or fractional and x > y. When x < y and m is fractional or negative the series must be taken in the form (x + y) m = y m + j y m -*x+ v ^ *' y *-'z + m (m - 1) . . . (m - n + 1) n! Fourier's Series. j- / \ It it ""E i t 2 7TX , 3 7TX . / (x) = - 6 + &i cos H &2 cos - + 6 3 cos H C C C . TTX . . 2irX . . STTX , + 01 sin \- a 2 sin h a 3 sin f- c c c where 1 r + c ,/ v WTTX . >m = ~ I / (*) COS - dx, C / c t/ 1 f +c r/ v . m-n-x , m = - \ f(x) sm dx, C / c ^ 2 / c , , , . WTTX , = - I / W sin - " C /o C provided / (x) is single valued, uniform, and continuous, and c > x > c. For values of x lying between zero and c the function may be ex- panded in the form , / x . TTX . . 2-JTX . . 3 TTX , f (x) = 0,1 sin -- \-a-i sin -- H a 3 sin --- (- , where a Also f(x) =^60 + 61 cosy 4-6 2 cos + 6 3 cos 2 r c - / x WTTX , where b m = - I / (x) cos - ax. C JQ C General Series. xloga (x log a) 2 (x log a) 3 (x log a) n ~~ ~~ ~~ ~ .- :>} TABLES 219 TABLE V. DERIVATIVES. U, F, W any functions; a, 6, c constants. dx F 2 S : ***St^T? axx a , log a e. _log a x= , dU a . i at; _ logaC7 . = __ V dx = a x log a. dx d dx ( a dx sm x = cos x. ax . r , . sm aC7 = a cos ac7 ^ , ax ax a l tan x = r = sec 2 x: ax cos 2 x cos x = sm x. ax a -i cot x = . , = cosec 2 x. ax sin 2 x sec x = tan x sec x; oX cosec x = cot x cosec x. ax log sinx = cotx; log cos x = tan x. ox The following expressions for the derivatives of inverse functions hold for angles in the first and third quadrants. For angles in the second and fourth quadrants the signs should be reversed. ax tan- 1 x = = ax i . T- cos- 1 x = dx i 220 THE THEORY OF MEASUREMENTS TABLE VI. SOLUTION OF EQUATIONS. The following algebraic expressions for the roots of equations of the second, third, and fourth degrees are in the form given by Merriman. (Merriman and Woodward, "Higher Mathematics"; Wiley and Sons, 1896.) The Quadratic Equation. Reduce to the form x 2 Then the two roots are x\ = a + a? 6; z 2 = a Va 2 b The Cubic Equation. Reduce to the form = 0. Compute the following auxiliary quantities : B = - a 2 + 6; C = a 3 - f ab + c; Then the three roots are xi=-a + (si + s 2 ), _ x z =-a -Mi+s 2 ) +| V-_3( Sl -s 2 ), x 3 = - a - HSI + s 2 ) - | V- 3 (si - s a ). When B 3 + C 2 is negative the roots are all real but they cannot be de- termined numerically by the above formulae owing to the complex nature of si and s 2 . In such cases the numerical values of the roots can be deter- mined only by some method of approximation. The Quartic Equation. Reduce to the form z 4 + 4az 3 + 66z 2 + 4cz + d = 0. Compute the following auxiliary quantities : g = a*-b; h = 6 3 + c 2 -2abc + dg; fc = |ac - 6 2 - |d; I = I (h + V^TF')* + 1 (h - VF+^)*; u = g + l', v = 2g-l; w = 4u* + 3k - 12gl. Then the four roots are xi = a + ^u + Vy + a u in which the signs are to be used as written provided that 2 a 3 3 ab + c is a negative number; but if this is positive all radicals except Vw are to be changed in sign. The above expressions are irreducible when h z + k* is a negative number. In this case the given equation has either four imaginary roots or four real roots that can be determined numerically only by some method of approxi- mation. TABLES 221 TABLE VII. APPROXIMATE FORMULA. In the following formulae, a, /3, 5, etc., represent quantities so small that their squares, higher powers, and products are negligible in comparison with unity. The limit of negligibility depends on the particular problem in hand. Most of the formulae give results within one part in one million when the variables are equal to or less than 0.001. 1. (l+a) n =l+n; (1 -a) n = 1 - na. 4. 6 l = 1 --' , l = 1 +- ' Vl+ n' Vl -a n 7. 9. (x + a When the angle a, expressed in radians, is small in comparison with unity a first approximation gives 10. sin a = a', sin (x a) = sin x a cos x. 11. cos a = 1; cos (x a) = cos x =F a sin x. 12. tan a = a] tan (x d= a) = tana; ^ The second approximation gives 13. sin a = a -TT ; sin 2 a = a 2 1 ^r- o \ o a 2 14. cos a = 1 -5- ; COS 2 a = 1 a 2 . 3 / o \ 15. tana = a + ^-| tan 2 a = a 2 ( 1 + ^ a 2 V (Kohlrausch, "Praktische Physik.") 222 THE THEORY OF MEASUREMENTS TABLE VIII. NUMERICAL CONSTANTS. Logarithm . Base of Naperian logarithms: e = 2. 7182818 ........ 0. 4342945 Modulus of Naperian log.: M = ^ = 2.30259 ........... 0.3622157 Modulus of common log.: = log e = 0. 4342945 ......... 1. 6377843 Circumference ,.. 1415 9265 . 0. 4971499 Diameter 2?r = 6.28318530 .............. 0.7981799 - =0.3183099 . 1.5028501 7T Tr 2 = 9.8696044 . . ............. 0.9942998 V^ = 1.7724539 ............... 0.2485749 | = 0.7853982 ............... 1.8950899 5 =0.5235988 . 1.7189986 o w = Precision constant; k = Unit error; A = Average error; M = Mean error; E = Probable error. 4p = 0.31831 ................. 1.5028501 ^ = 0.39894 ................. 1.6009101 ^ = 0.26908 ................. 1.4298888 ^ = 1.25331 ................. 0.0980600 A. f = 0.84535 ................. 1.9270387 A. = 0.67449 ................. 1.8289787 TABLES 223 TABLE IX. EXPONENTIAL FUNCTIONS. X logic (e*) e* e* X log 10 (O e' e~' 0.0 0.00000 1.0000 1.000000 5.0 2.17147 148.41 0.006738 0.1 0.04343 1.1052 0.904837 5.1 2.21490 164.02 0.006097 0.2 0.08686 1.2214 0.818731 5.2 2.25833 181.27 0.005517 0.3 0.13029 1.3499 0.740818 5.3 2.30176 200.34 0.004992 0.4 0.17372 1.4918 0.670320 5.4 2.34519 221.41 0.004517 0.5 0.21715 1.6487 0.606531 5.5 2.38862 244.69 0.004087 0.6 0.26058 1.8221 0.548812 5.6 2.43205 270.43 0.003698 0.7 0.30401 2.0138 0.496585 5.7 2.47548 298.87 0.003346 0.8 0.34744 2.2255 0.449329 5.8 2.51891 330.30 0.003028 0.9 0.39087 2.4596 0.406570 5.9 2.56234 365.04 0.002739 1.0 0.43429 2.7183 0.367879 6.0 2.60577 403.43 0.002479 1.1 0.47772 3.0042 0.332871 6.1 2.64920 445.86 0.002243 1.2 0.52115 3.3201 0.301194 6.2 2.69263 492.75 0.002029 1.3 0.56458 3.6693 0.272532 6.3 2.73606 544.57 0.001836 1.4 0.60801 4.0552 0.246597 6.4 2.77948 601.85 0.001662 .5 0.65144 4.4817 0.223130 6.5 2.82291 665.14 0.001503 .6 0.69487 4.9530 0.201897 6.6 2.86634 735.10 0.001360 .7 0.73830 5.4739 0.182684 6.7 2.90977 812.41 0.001231 .8 0.78173 6.0496 0.165299 6.8 2.95320 897.85 0.001114 .9 0.82516 6.6859 0.149569 6.9 2.99663 992.27 0.001008 2.0 0.86859 7.3891 0.135335 7.0 3.04006 1096.6 0.000912 2.1 0.91202 8.1662 0.122456 7.1 3.08349 1212.0 0.000825 2.2 0.95545 9.0250 0.110803 7.2 3.12692 1339.4 0.000747 2.3 0.99888 9.9742 0.100259 7.3 3.17035 1480.3 0.000676 2.4 1.04231 11.023 0.090718 7.4 3.21378 1636.0 0.000611 2.5 1.08574 12.182 0.082085 7.5 3.25721 1808.0 0.000553 2.6 1.12917 13.464 0.074274 7.6 3.30064 1998.2 0.000500 2.7 1 . 17260 14.880 0.067206 7.7 3.34407 2208.3 0.000453 2.8 1.21602 16.445 0.060810 7.8 3.38750 2440.6 0.000410 2.9 1.25945 18.174 0.055023 7.9 3.43093 2697.3 0.000371 3.0 1.30288 20.086 0.049787 8.0 3.47436 2981.0 0.000335 3.1 1.34631 22.198 0.045049 8.1 3.51779 3294.5 0.000304 3.2 1.38974 24.533 0.040762 8.2 3.56121 3641.0 0.000275 3.3 1.43317 27.113 0.036883 8.3 3.60464 4023.9 0.000249 3.4 1.47660 29.964 0.033373 8.4 3.64807 4447.1 0.000225 3.5 1.52003 33.115 0.030197 8.5 3.69150 4914.8 0.000203 3.6 1.56346 36.598 0.027324 8.6 3.73493 5431.7 0.000184 3.7 1.60689 40.447 0.024724 8.7 3.77836 6002.9 0.000167 3.8 1.65032 44.701 0.022371 8.8 3.82179 6634.2 0.000151 3.9 1.69375 49.402 0.020242 8.9 3.86522 7332.0 0.000136 4.0 1.73718 54.598 0.018316 9.0 3.90865 8103.1 0.000123 4.1 .78061 60.340 0.016573 9.1 3.95208 8955.3 0.000112 4.2 .82404 66.686 0.014996 9.2 3.99551 9897.1 0.000101 4.3 .86747 73.700 0.013569 9.3 4.03894 10938. 0.000091 4.4 .91090 81.451 0.012277 9.4 4.08237 12088. 0.000083 4.5 .95433 90.017 0.011109 9.5 4.12580 13360. 0.000075 4.6 .99775 99.484 0.010052 9.6 4.16923 14765. 0.000068 4.7 2.04118 109.95 0.009095 9.7 4.21266 16318. 0.000061 4.8 2.08461 121.51 0.008230 9.8 4.25609 18034. 0.000055 4.9 2.12804 134.29 0.007447 9.9 4.29952 19930. 0.000050 5.0 2.17147 148.41 0.006738 10.0 4.34294 22026. 0.000045 Taken from Glaisher's "Tables of the Exponential Function," Trans. Cambridge Phil. Soc., vol. xiii, 1883. This volume also contains a " Table of the Descending Exponential to Twelve or Fourteen Places of Decimals," by F. W. Newman. 224 THE THEORY OF MEASUREMENTS TABLE X. EXPONENTIAL FUNCTIONS. Value of e x<t and er x<i and their logarithms. X <? log e 2 e~* 2 log e'* z 0.1 1.0101 0.00434 0.99005 1.99566 0.2 1.0408 0.01737 0.96079 1.98263 0.3 1.0942 0.03909 0.91393 1.96091 0.4 .1735 0.06949 0.85214 .93051 0.5 .2840 0.10857 0.77880 .89143 0.6 .4333 0.15635 0.69768 .84365 0.7 .6323 0.21280 0.61263 .78720 0.8 .8965 0.27795 0.52729 .72205 0.9 2.2479 0.35178 0.44486 .64822 1.0 2.7183 0.43429 0.36788 .56571 1.1 3.3535 0.52550 0.29820 .47450 1.2 4.2207 0.62538 0.23693 .37462 1.3 5.4195 0.73396 0.18452 .26604 1.4 7.0993 0.85122 0.14086 .14878 1.5 9.4877 0.97716 0.10540 .02284 1.6 1.2936X10 1.11179 0. 77305 XlO- 1 2.88821 1.7 1.7993X10 1.25511 0. 55576 XlO- 1 2.74489 1.8 2.5534x10 1.40711 0. 39164 XlO- 1 2.59289 1.9 3.6966X10 1.56780 0.27052 XlO- 1 2.43220 2.0 5.4598X10 1.73718 0.18316 XlO- 1 2.26282 2.1 8.2269x10 1.91524 0.12155 XlO- 1 2.08476 22 1.2647X10 2 2.10199 0.79071 XlO- 2 3.89801 2.3 1.9834X10 2 2.29742 0.50417 XlO- 2 3.70258 2.4 3.1735X10 2 2.50154 0.31511 XlO- 2 3.49846 2.5 5.1801X10 2 2.71434 0.19305 XlO- 2 3.28566 2.6 8.6264X10 2 2.93583 0.1 1592 XlO- 2 3.06417 2.7 1.4656X10 3 3.16601 0.68232X10- 3 4.83399 2.8 2.5402X10 3 3.40487 0.39367X10- 3 4.59513 2.9 4.4918X10 3 3.65242 0.22263X10-3 4.34758 3.0 8.1031X10 3 3.90865 0.12341X10- 3 4.09135 3.1 1.4913X10 4 4.17357 0.67055x10-* 5.82643 3.2 2.8001X10 4 4.44718 0.35713 XlO- 4 5.55282 3.3 5.3637X10 4 4.72947 0.18644 XlO- 4 5.27053 3.4 1.0482X10 5 5.02044 0.95403 XlO- 5 6.97956 3.5 2.0898X10 5 5.32011 0.47851 XlO- 5 6.67989 3.6 4.2507X10 5 5.62846 0.23526 XlO- 5 6.37154 3.7 8.8204X10 5 5.94549 0.1 1337 XlO- 5 6.05451 3.8 1.8673X10 6 6.27121 0.53554 XlO- 6 7.72879 3.9 4.0329X10 6 6.60562 0.24796 XlO- 6 ?. 39438 4.0 8.8861X10 6 6.94871 0.11254X10- 6 7.05129 41 1.9975X10 7 7.30049 0.50062 XlO- 7 .69951 4.2 4.5809X10 7 7.66095 0.21830 XlO- 7 S. 33905 4.3 1.0718X10 8 8.03011 0.93302 XlO- 8 9.96989 4.4 2.5582X10 8 8.40794 0.39089 XlO- 8 9.59206 4.5 6.2296X10 8 8.79446 0.16052X10- 8 9.20554 4.6 1.5476X10 9 9.18967 0.64614X10- 9 10.81033 4.7 3.9226X10 9 9.59357 0.25494X10- 9 10.40643 4.8 1.0143X10 10 10.00615 0.98594 XlO- 10 11.99385 4.9 2.6755X10 10 10.42741 0.37376 XlO- 10 11.57259 5.0 7.2005X10 10 10.85736 0.13888 XlO- 10 11.14264 TABLES 225 TABLE XI. VALUES OF THE PROBABILITY INTEGRAL. t P* Diff. t ^A Diff. 1 ^A Diff t ^A Diff. 0.00 0.00000 1 IOC 0.50 0.52050 074 1.00 0.84270 A 1 1.50 0.96611 0.01 0.01128 1 I ! 1 100 0.51 0.52924 o/ 1 O/3f> 1.01 0.84681 411 A AO 1.51 0.96728 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.02256 0.03384 0.04511 0.05637 0.06762 0.07886 0.09008 0.10128 liZo 1128 1127 1126 1125 1124 1122 1120 1 1 1C 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.53790 0.54646 0.55494 0.56332 0.57162 0.57982 0.58792 0.59594 ODD 856 848 838 830 820 810 802 7QO 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 0.85084 0.85478 0.85865 0.86244 0.86614 0.86977 0.87333 0.87680 403 394 387 379 370 363 356 347 O/M 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 0.96841 0.96952 0.97059 0.97162 0.97263 0.97360 0.97455 0.97546 113 111 107 103 101 97 95 91 OA 0.10 0.11 0.12 0.11246 0.12362 0.13476 1 1 J.O 1116 1114 1111 0.60 0.61 0.62 0.60386 0.61168 0.61941 /y^ 782 773 7j 1.10 1.11 1.12 0.88021 0.88353 0.88679 O41 332 326 010 1.60 1.61 1.62 0.97635 0.97721 0.97804 89 86 83 Qf\ 0.13 0.14 0.15 0.16 0.17 0.14587 0.15695 0.16800 0.17901 0.18999 1 1 i. i 1108 1105 1101 1098 1 AQ~ 0.63 0.64 0.65 0.66 0.67 0.62705 0.63459 0.64203 0.64938 0.65663 < D^ 754 744 735 725 71 t\ 1.13 1.14 1.15 1.16 1.17 0.88997 0.89308 0.89612 0.89910 0.90200 OIo 311 304 298 290 oo/i 1.63 .64 .65 .66 .67 0.97884 0.97962 0.98038 0.98110 0.98181 oU 78 76 72 71 f*Q 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.20094 0.21184 0.22270 0.23352 0.24430 0.25502 0.26570 0.27633 0.28690 0.29742 0.30788 0.31828 0.32863 0.33891 0.34913 iuyo 1090 1086 1082 1078 1072 1068 1083 1057 1052 1046 1040 1035 1028 1022 1 A1 K. 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.66378 0.67084 0.67780 . 68467 0.69143 0.69810 0.70468 0.71116 0.71754 0.72382 0.73001 0.73610 0.74210 0.74800 0.75381 < 10 706 696 687 676 667 658 648 638 628 619 609 600 590 581 CT1 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 0.90484 0.90761 0.91031 0.91296 0.91553 0.91805 0.92051 0.92290 0.92524 0.92751 0.92973 0.93190 0.93401 0.93606 0.93807 Zo4 277 270 265 257 252 246 239 234 227 222 217 211 205 201 .68 .69 .70 .71 .72 .73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 0.98249 0.98315 0.98379 0.98441 0.98500 0.98558 0.98613 0.98667 0.98719 0.98769 0.98817 0.98864 0.98909 0.98952 0.98994 Do 66 64 62 59 58 55 54 52 50 48 47 45 43 42 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.35928 0.36936 0.37938 0.38933 0.39921 0.40901 0.41874 lUio 1008 1002 995 988 980 973 Qf?r 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.75952 0.76514 0.77067 0.77610 0.78144 0.78669 0.79184 571 562 553 543 534 525 515 CA*7 .33 .34 .35 .36 .37 .38 .39 0.94002 0.94191 0.94376 0.94556 0.94731 0.94902 0.95067 195 189 185 180 175 171 165 1 JO 1.83 1.84 1.85 1.86 1.87 1.88 1.89 0.99035 0.99074 0.99111 0.99147 0.99182 0.99216 0.99248 41 39 37 36 35 34 32 01 0.40 0.41 0.42839 0.43797 yoo 958 nr:n 0.90 0.91 0.79691 0.80188 oU/ 497 4P.Q .40 .41 0.95229 0.95385 ItW 156 i ^. 1.90 1.91 0.99279 0.99309 ol 30 on 0.42 0.44747 you n/io 0.92 0.80677 "oy A>-r{\ 1.42 0.95538 100 1 AO 1.92 3.99338 zy oo 0.43 0.44 0.45689 0.46623 y4z 934 QOC 0.93 0.94 0.81156 0.81627 479 471 4fi9 1.43 1.44 0.95686 0.95830 148 144 140 1.93 1.94 3.99366 3.99392 28 26 9fi 0.45 0.47548 t/^O QIC 0.95 0.82089 '\j 4CQ 1.45 0.95970 J.TAJ IOC 1.95 3.99418 ^O OK 0.46 0.47 0.48 0.49 0.50 0.48466 0.49375 0.50275 0.51167 0.52050 7 j.o 909 900 892 883 0.96 0.97 0.98 0.99 1.00 0.82542 0.82987 0.83423 0.83851 0.84270 ^too 445 436 428 419 1.46 1.47 1.48 1.49 1.50 0.96105 0.96237 0.96365 0.96490 0.96611 J.OO 132 128 125 121 1.96 1.97 1.98 1.99 2.00 3.99443 3.99466 3.99489 3.99511 3.99532 ^O 23 23 22 21 oo .00000 (Chauvenet, " Spherical and Practical Astronomy.") 226 THE THEORY OF MEASUREMENTS TABLE XII. VALUES OF THE PROBABILITY INTEGRAL. 3 1 2 3 4 5 6 7 8 9 0.0 .00000 .00538 .01076 .01614 .02152 .02690 .03228 .03766 . 04303 .04840 0.1 .05378 .05914 .06451 .06987 .07523 .08059 .08594 .09129 .09663 . 10197 0.2 .10731 .11264 .11796 . 12328 . 12860 . 13391 . 13921 . 14451 . 14980 . 15508 0.3 .16035 . 16562 . 17088 . 17614 . 18138 . 18662 .19185 . 19707 .20229 .20749 0.4 .21268 .21787 .22304 .22821 .23336 .23851 .24364 .24876 .25388 .25898 0.5 .26407 .26915 .27421 .27927 .28431 .28934 .29436 .29936 .30435 .30933 0.6 .31430 .31925 .32419 .32911 .33402 .33892 .34380 .34866 .35352 .35835 0.7 .36317 .36798 .37277 .37755 .38231 .38705 .39178 .39649 .40118 .40586 0.8 .41052 .41517 .41979 .42440 . 42899 . 43357 . 43813 . 44267 .44719 .45169 0.9 .45618 .46064 .46509 .46952 .47393 . 47832 . 48270 . 48605 .49139 .49570 .0 .50000 .50428 .50853 .51277 .51699 .52119 .52537 .52952 .53366 .53778 .1 .54188 .54595 .55001 .55404 .55806 .56205 .56602 . 56998 .57391 .57782 .2 .58171 .58558 .58942 .59325 .59705 .60083 .60460 . 60833 .61205 .61575 .3 .61942 .62308 .62671 .63032 .63391 .63747 .64102 .64454 .64804 .65152 .4 .65498 .65841 .66182 .66521 .66858 .67193 .67526 .67856 .68184 .68510 .5 .68833 .69155 .69474 .69791 .70106 .70419 .70729 .71038 .71344 .71648 .6 .71949 .72249 .72546 .72841 .73134 .73425 .73714 .74000 .74285 .74567 .7 .74847 .75124 .75400 .75674 .75945 .76214 .76481 .76746 .77009 .77270 .8 .77528 .77785 .78039 .78291 .78542 .78790 .79036 .79280 .79522 .79761 .9 .79999 .80235 .80469 .80700 .80930 .81158 .81383 .81607 .81828 .82048 2.0 .82266 .82481 .82695 .82907 .83117 .83324 .83530 .83734 .83936 .84137 2.1 .84335 .84531 .84726 .84919 .85109 .85298 .85486 .85671 .85854 .86036 2.2 .86216 .86394 .86570 .86745 .86917 .87088 .87258 .87425 .87591 .87755 2.3 .87918 .88078 .88237 .88395 .88550 .88705 .88857 .89008 : 89157 .89304 2.4 .89450 .89595 .89738 .89879 .90019 .90157 .90293 .90428 .90562 .90694 25 .90825 .90954 .91082 .91208 .91332 .91456 .91578 .91698 .91817 .91935 2.6 .92051 .92166 .92280 .92392 .92503 .92613 .92721 .92828 .92934 .93038 2.7 .93141 .93243 .93344 .93443 .93541 .93638 .93734 .93828 .93922 .94014 2.8 .94105 .94195 .94284 .94371 .94458 .94543 .94627 .94711 .94793 .94874 2.9 .94954 .95033 .95111 .95187 .95263 .95338 .95412 .95485 .95557 .95628 3 .95698 .96346 96910 .97397 .97817 .98176 .98482 .98743 .98962 .99147 4 .99302 .99431 .99539 .99627 .99700 .99760 .99808 .99848 .99879 .99905 5 .99926 .99943 .99956 .99966 .99974 .99980 .99985 . 99988 .99991 .99993 TABLE XIII. CHAUVENET'S CRITERION. N T N r AT r 3 2.05 13 3.07 23 3.40 4 2.27 14 3.11 24 3.43 5 2.44 15 3.15 25 3.45 6 2.57 16 3.19 30 3.55 7 2.67 17 3.22 40 3.70 8 2.76 18 3.26 50 3.82 9 2.84 19 3.29 75 4.02 10 2.91 20 3.32 100 4.16 11 2.97 21 3.35 200 4.48 12 3.02 22 3.38 500 4.90 TABLES 227 TABLE XTV. FOR COMPUTING PROBABLE ERRORS BY FORMULA (31) AND (32). AT 0.6745 0.6745 AT 0.6745 0.6745 iV VJv^T VN(N-l) iM vim v# (AT- i) 40 0.1080 0.0171 41 0.1066 0.0167 2 0.6745 0.4769 42 0.1053 0.0163 3 0.4769 0.2754 43 0.1041 0.0159 4 0.3894 0.1947 44 0.1029 0.0155 5 0.3372 0.1508 45 0.1017 0.0152 6 0.3016 0.1231 46 0.1005 0.0148 7 0.2754 0.1041 47 0.0994 0.0145 8 0.2549 0.0901 48 0.0984 0.0142 9 0.2385 0.0795 49 0.0974 0.0139 10 0.2248 0.0711 50 0.0964 0.0136 11 0.2133 0.0643 51 0.0954 0.0134 12 0.2029 0.0587 52 0.0944 0.0131 13 0.1947 0.0540 53 0.0935 0.0128 14 0.1871 0.0500 54 0.0926 0.0126 15 0.1803 0.0465 55 0.0918 0.0124 16 0.1742 0.0435 56 0.0909 0.0122 17 0.1686 0.0409 57 0.0901 0.0119 18 0.1636 0.0386 58 0.0893 0.0117 19 0.1590 0.0365 59 0.0886 0.0115 20 0.1547 0.0346 60 0.0878 0.0113 21 0.1508 0.0329 61 0.0871 0.0111 22 0.1472 0.0314 62 0.0864 0.0110 23 0.1438 0.0300 63 0.0857 0.0108 24 0.1406 0.0287 64 0.0850 0.0106 25 0.1377 0.0275 65 0.0843 0.0105 26 0.1349 0.0265 66 0.0837 0.0103 27 0.1323 0.0255 67 0.0830 0.0101 28 0.1298 0.0245 68 0.0824 0.0100 29 0.1275 0.0237 69 0.0818 0.0098 30 0.1252 0.0229 70 0.0812 0,0097 31 0.1231 0.0221 71 0.0806 0.0096 32 0.1211 0.0214 72 0.0800 0.0094 33 0.1192 0.0208 73 0.0795 0.0093 34 0.1174 0.0201 74 0.0789 0.0092 35 0.1157 0.0196 75 0.0784 0.0091 36 0.1140 0.0190 80 0.0759 0.0085 37 0.1124 0.0185 85 0.0736 0.0080 38 0.1109 0.0180 90 0.0713 0.0075 39 0.1094 0.0175 100 0.0678 0.0068 (Merriman, " Least Squares. ") 228 THE THEORY OF MEASUREMENTS TABLE XV. FOR COMPUTING PROBABLE ERRORS BY FORMULAE (34). N 0.8453 0.8453 N 0.8453 0.8453 ^N(N - 1) N^N-1 VN(N - 1) N^W=1 40 0.0214 0.0034 41 0.0209 0.0033 2 0.5978 0.4227 42 0.0204 0.0031 3 0.3451 0.1993 43 0.0199 0.0030 4 0.2440 0.1220 44 0.0194 0.0029 5 0.1890 0.0845 45 0.0190 0.0028 6 0.1543 0.0630 46 0.0186 0.0027 7 0.1304 0.0493 47 0.0182 0.0027 8 0.1130 0.0399 48 0.0178 0.0026 9 0.0996 0.0332 49 0.0174 0.0025 10 0.0891 0.0282 50 0.0171 0.0024 11 0.0806 0.0243 51 0.0167 0.0023 12 0.0736 0.0212 52 0.0164 0.0023 13 0.0677 0.0188 53 0.0161 0.0022 14 0.0627 0.0167 54 0.0158 0.0022 15 0.0583 0.0151 55 0.0155 0.0021 16 0.0546 0.0136 56 0.0152 0.0020 17 0.0513 0.0124 57 0.0150 0.0020 18 0.0483 0.0114 58 0.0147 0.0019 19 0.0457 0.0105 59 0.0145 0.0019 20 0.0434 0.0097 60 0.0142 0.0018 21 0.0412 0.0090 61 0.0140 0.0018 22 0.0393 0.0084 62 0.0137 0.0017 23 0.0376 0.0078 63 0.0135 0.0017 24 0.0360 0.0073 64 0.0133 0.0017 25 0.0345 0.0069 65 0.0131 0.0016 26 0.0332 0.0065 66 0.0129 0.0016 27 0.0319 0.0061 67 0.0127 0.0016 28 0.0307 0.0058 68 0.0125 0.0015 29 0.0297 0.0055 69 0.0123 0.0015 30 0.0287 0.0052 70 0.0122 0.0015 31 0.0277 0.0050 71 0.0120 0.0014 32 0.0268 0.0047 72 0.0118 0.0014 33 0.0260 0.0045 73 0.0117 0.0014 34 0.0252 0.0043 74 0.0115 0.0013 35 0.0245 0.0041 75 0.0113 0.0013 36 0.0238 0.0040 80 0.0106 0.0012 37 0.0232 0.0038 85 0.0100 0.0011 38 0.0225 0.0037 90 0.0095 0.0010 39 0.0220 0.0035 100 0.0085 0.0008 (Merriman, "Least Squares.") TABLES 229 TABLE XVI. SQUARES OP NUMBERS. n i 2 3 4 5 6 7 8 9 Diff. 1.0 1.000 1.020 1.040 1.061 1.082 1.103 1.124 1.145 1.166 1.188 22 1.1 1.210 1.232 1.254 1.277 1.300 1.323 1.346 1.369 1.392 1.416 24 1.2 1.440 1.464 1.488 1.513 1.538 1.563 1.588 1.613 1.638 1.664 26 1.3 1.690 1.716 1.742 1.769 1.796 1.823 1.850 1.877 1.904 1.932 28 1.4 1.960 1.988 2.016 2.045 2.074 2.103 2.132 2.161 2.190 2.220 30 1.5 2.250 2.280 2.310 2.341 2.372 2.403 2.434 2.465 2.496 2.528 32 1.6 2.560 2.592 2.624 2.657 2.690 2.723 2.756 2.789 2.822 2.856 34 1.7 2.890 2.924 2.958 2.993 3.028 3.063 3.098 3.133 3.168 3.204 36 1.8 3.240 3.276 3.312 3.349 3.386 3.423 3.460 3.497 3.534 3.572 38 1.9 3.610 3.648 3.686 3.725 3.764 3.803 3.842 3.881 3.920 3.960 40 2.0 4.000 4.040 4.080 4.121 4.162 4.203 4.244 4.285 4.326 4.368 42 2.1 4.410 4.452 4.494 4.537 4.580 4.623 4.666 4.709 4.752 4.796 44 2.2 4.840 4.884 4.928 4.973 5.018 5.063 5.108 5.153 5.198 5.244 46 23 5.290 5.336 5.382 5.429 5.476 5.523 5.570 5.617 5.664 5.712 48 2.4 5.760 5.808 5.856 5.905 5.954 6.003 6.052 6.101 6.150 6.200 50 25 6.250 6.300 6.350 6.401 6.452 6.503 6.554 6.605 6.656 6.708 52 2.6 6.760 6.812 6.864 6.917 6.970 7.023 7.076 7.129 7.182 7.236 54 27 7.290 7.344 7.398 7.453 7.508 7.563 7.618 7.673 7.728 7.784 56 2.8 7.840 7.896 7.952 8.009 8.066 8.123 8.180 8.237 8.294 8.352 58 2.9 8.410 8.468 8.526 8.585 8.644 8.703 8.762 8.821 8.880 8.940 60 3.0 9.000 9.060 9.120 9.181 9.242 9.303 9.364 9.425 9.486 9.548 62 3.1 9.610 9.672 9.734 9.797 9.860 9.923 9.986 10.05 10.11 10.18 6 3.2 10.24 10.30 10.37 10.43 10.50 10.56 10.63 10.69 10.76 10.82 7 3.3 10.89 10.96 11.02 11.09 11.16 11.22 11.29 11.36 11.42 11.49 7 3.4 11.56 11.63 11.70 11.76 11.83 11.90 11.97 12.04 12.11 12.18 7 3.5 12.25 12.32 12.39 12.46 12.53 12.60 12.67 12.74 12.82 12.89 7 3.6 12.96 13.03 13.10 13.18 13.25 13.32 13.40 13.47 13.54 14.62 7 3.7 13.69 13.76 13.84 13.91 13.99 14.06 14.14 14.21 14.29 14.36 8 3.8 14.44 14.52 14.59 14.67 14.75 14.82 14.90 14.98 15.05 15.13 8 3.9 15.21 15.29 15.37 15.44 15.52 15.60 15.68 15.76 15.84 15.92 8 4.0 16.00 16.08 16.16 16.24 16.32 16.40 16.48 16.56 16.65 16.73 8 4.1 16.81 16.89 16.97 17.06 17.14 17.22 17.31 17.39 17.47 17.65 8 4.2 17.64 17.72 17.81 17.89 17.98 18.06 18.15 18.23 18.32 18.40 9 4.3 18.49 18.58 18.66 18.75 18.84 18.92 19.01 19.10 19.18 19.27 9 4.4 19.36 19.45 19.54 19.62 19.71 19.80 19.89 19.98 20.07 20.16 9 4.5 20.25 20.34 20.43 20.52 20.61 20.70 20.79 20.88 20.98 21.07 9 4.6 21.16 21.25 21.34 21.44 21.53 21.62 21.72 21.81 21.90 22.00 9 4.7 22.09 22.18 22.28 22.37 22.47 22.56 22.66 22.75 22.85 22.94 10 4.8 23.04 23.14 23.23 23.33 23.43 23.52 23.62 23.72 23.81 23.91 10 4.9 24.01 24.11 24.21 24.30 24.40 24.50 24.60 24.70 24.80 24.90 10 5.0 25.00 25.10 25.20 25.30 25.40 25.50 25.60 25.70 25.81 25.91 10 5.1 26.01 26.11 26.21 26.32 26.42 26.52 26.63 26.73 26.83 26.94 10 5.2 27.04 27.14 27.25 27.35 27.46 27.56 27.67 27.77 27.88 27.98 11 5.3 28.09 28.20 28.30 28.41 28.52 28.62 28.73 28.84 28.94 29.05 11 5.4 29.16 29.27 29.38 29.48 29.59 29.70 29.81 29.92 30.03 30.14 11 n 1 2 3 4 5 6 7 8 9 Diff. (Merriman, "Least Squares.") 230 THE THEORY OF MEASUREMENTS TABLE XVI. SQUARES OF NUMBERS (Concluded). n i 2 3 4 5 6 7 8 9 Diff. 5.5 30.25 30.36 30.47 30.58 30.69 30.80 30.91 31.02 31.14 31.25 11 5.6 31.36 31.47 31.58 31.70 31.81 31.92 32.04 32.15 32.26 32.38 11 5.7 32.49 32.60 32 72 32.83 32.95 33.0633.18 33.29 33.41 33.52 12 5.8 33.64 33.76 33.87 33.99 34.11 34.2234.34 34.46 34.57 34.69 12 5.9 34.81 34.93 35.05 35.16 35.28 35.40 35.52 35.64 35.76 35.88 12 6.0 36.00 36.12 36.24 36.36 36.48 36.60 36.72 36.84 36.97 37.09 12 6.1 37.21 37.33 37.45 37.58 37.70 37.82 37.95 38.07 38.19 38.32 12 6.2 38.44 38.56 38.69 38.81 38.94 39.06 39.19 39.31 39.44 39.56 13 6.3 39.69 39.82 39.94 40.07 40.20 40.32 40.45 40.58 40.70 40.83 13 6.4 40.96 41.09 41.22 41.34 41.47 41.60 41.73 41.86 41.99 42.12 13 6.5 42.25 42.38 42.51 42.64 42.77 42.90 43.03 43.16 43.30 43.43 13 6.6 43.56 43.69 43.82 43.96 44.09 44.22 44.36 44.49 44.62 44.76 13 6.7 44.89 45.02 45.16 45.29 45.43 45.56 45.70 45.83 45.97 46.10 14 6.8 46.24 46.38 46.51 46.65 46.79 46.92 47.06 47.20 47.33 47.47 14 6.9 47.61 47.75 47.89 48.02 48.16 48.30 48.44 48.58 48.72 48.86 14 7.0 49.00 49.14 49.28 49.42 49.56 49.70 49.84 49.98 50.13 50.27 14 7.1 50.41 50.55 50.69 50.84 50.98 51.12 51.27 51.41 51.55 51.70 14 7.2 51.84 51.98 52.13 52.27 52.42 52.56 52.71 52.85 53.00 53.14 15 7.3 53.29 53.44 53.58 53.73 53.88 54.02 54.17 54.32 54.46 54.61 15 7.4 54.76 54.91 55.06 55.20 55.35 55.50 55.65 55.80 55.95 56.10 15 7.5 56.25 56.40 "56.55 56.70 56.85 57.00 57.15 57.30 57.46 57.61 15 7.6 57.76 57.91 58.06 58.22 58.37 58.52 58.68 58.83 58.98 59.14 15 7.7 59.29 59.44 59.60 59.75 59.91 60.06 60.22 60.37 60.53 60.68 16 7.8 60.84 61.00 61.15 61.31 61.47 61.62 61.78 61.94 62.09 62.25 16 7.9 62.41 62.57 62.73 62.88 63.04 63.20 63.36 63.52 63.68 63.84 16 8.0 64.00 64.16 64.32 64.48 64.64 64.80 64.96 65.12 65.29 65.45 16 8.1 65.61 65.77 65.93 66.10 66.26 66.42 66.59 66.75 66.91 67.08 16 8.2 67.24 67.40 67.57 67.73 67.90 68.06 68.23 68.39 68.56 68.72 17 8.3 68.89 69.06 69.22 69.39 69.56 69.72 69.89 70.06 70.22 70.39 17 8.4 70.56 70.73 70.90 71.06 71.23 71.40 71.57 71.74 71.91 72.08 17 8.5 72.25 72.42 72.59 72.76 72.93 73.10 73.27 73.44 73.62 73.79 17 8.6 73.96 74.13 74.30 74.48 74.65 74.82 75.00 75.17 75.34 75.52 17 8.7 75.69 75.86 76.04 76.21 76.39 76.56 76.74 76.91 77.09 77.26 18 8.8 77.44 77.62 77.79 77.97 78.15 78.32 78.50 78.68 78.85 79.03 18 8.9 79.21 79.39 79.57 79.74 79.92 80.10 80.28 80.46 80.64 80.82 18 9.0 81.00 81.18 81.36 81.54 81.72 81.90 82.08 82.26 82.45 82.63 18 9.1 82.81 82.99 83.17 83.36 83.54 83.72 83.91 84.09 84.27 84.46 18 9.2 84.64 84.82 85.01 85.19 85.38 85.56 85.75 85.93 86.12 86.30 19 9.3 86.49 86.68 86.86 87.05 87.24 87.42 87.61 87.80 87.98 88.17 19 9.4 88.36 88.55 88.74 88.92 89.11 89.30 89.49 89.68 89.87 90.06 19 9.5 90.25 90.44 90.63 90.82 91.01 91.20 91.39 91.58 91.78 91.97 19 9.6 92.16 92.35 92.54 92.74 92.93 93.12 93.32 93.51 93.70 93.90 19 9.7 94.09 94.28 94.48 94.67 94.87 95.06 95.26 95.45 95.65 95.84 20 9.8 96.04 96.24 96.43 96.63 96.83 97.02 97.22 97.42 97.61 97.81 20 9.9 98.01 98.21 98.41 98.60 98.80 99.00 99.20 99.40 99.60 99.80 20 n l 2 3 4 5 6 7 8 9 Diff. TABLES TABLE XVII. LOGARITHMS; 1000 TO 1409. 231 1 2 3 4 5 6 7 8 9 100 0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 101 0043 0048 0052 0056 0060 0065 0069 0073 0077 0082 102 0086 0090 0095 0099 0103 0107 0111 0116 0120 0124 103 0128 0133 0137 0141 0145 0149 0154 0158 0162 0166 104 0170 0175 0179 0183 0187 0191 0195 0199 0204 0208 105 0212 0216 0220 0224 0228 0233 0237 0241 0245 0249 106 0253 0257 0261 0265 0269 0273 0278 0282 0286 0290 107 0294 0298 0302 0306 0310 0314 0318 0322 0326 0330 108 0334 0338 0342 0346 0350 0354 0358 0362 0366 0370 109 0374 0378 0382 0386 0390 0394 0398 0402 0406 0410 110 0414 0418 0422 0426 0430 0434 0438 0441 0445 0449 111 0453 0457 0461 0465 0469 0473 0477 0481 0484 0488 112 0492 0496 0500 0504 0508 0512 0515 0519 0523 0527 113 0531 0535 0538 0542 0546 0550 0554 0558 0561 0565 114 0569 0573 0577 0580 0584 0588 0592 0596 0599 0603 115 0607 0611 0615 0618 0622 0626 0630 0633 0637 0641 116 0645 0648 0652 0656 0660 0663 0667 0671 0674 0678 117 0682 0686 0689 0693 0697 0700 0704 0708 0711 0715 118 0719 0722 0726 0730 0734 0737 0741 0745 0748 0752 119 0755 0759 0763 0766 0770 0774 0777 0781 0785 0788 120 0792 0795 0799 0803 0806 0810 0813 0817 0821 0824 121 0828 0831 0835 0839 0842 0846 0849 0853 0856 0860 122 0864 0867 0871 0874 0878 0881 0885 0888 0892 0896 123 0899 0903 0906 0910 0913 0917 0920 0924 0927 0931 124 0934 0938 0941 0945 0948 0952 0955 0959 0962 0966 125 0969 0973 0976 0980 0983 0986 0990 0993 0997 1000 126 1004 1007 1011 1014 1017 1021 1024 1028 1031 1035 127 1038 1041 1045 1048 1052 1055 1059 1062 1065 1069 128 1072 1075 1079 1082 1086 1089 1092 1096 1099 1103 129 1106 1109 1113 1116 1119 1123 1126 1129 1133 1136 130 1139 1143 1146 1149 1153 1156 1159 1163 1166 1169 131 1173 1176 1179 1183 1186 1189 1193 1196 1199 1202 132 1206 1209 1212 1216 1219 1222 1225 1229 1232 1235 133 1239 1242 1245 1248 1252 1255 1258 1261 1265 1268 134 1271 1274 1278 1281 1284 1287 1290 1294 1297 1300 135 1303 1307 1310 1313 1316 1319 1323 1326 1329 1332 136 1335 1339 1342 1345 1348 1351 1355 1358 1361 1364 137 1367 1370 1374 1377 1380 1383 1386 1389 1392 1396 138 1399 1402 1405 1408 1411 1414 1418 1421 1424 1427 139 1430 1433 1436 1440 1443 1446 1449 1452 1455 1458 140 1461 1464 1467 1471 1474 1477 1480 1483 1486 1489 (Bottomley, "Four Fig. Math. Tables.") 232 THE THEORY OF MEASUREMENTS * TABLE XVIII. LOGARITHMS. 1 2 3 4 5 6 7 & 9 123 456 789 10 0000 0043 0086 0128 0170 O2I2 0253 0294 0334 0374 4812 17 21 2 5 29 33 37 11 12 13 0414 0792 "39 0453 0828 "73 0492 0864 1206 0531 0899 1239 0569 0934 1271 0607 9 6 9 I33 0645 100^ 1335 0682 1038 1367 0719 1072 1399 0755 1106 1430 4811 3 7io 3 6 10 15 19 23 14 17 21 13 16 10 26 30 34 24 28 31 23 26 29 21 24 27 20 22 25 18 21 24 14 15 16 1461 1761 2041 1492 1790 2068 iffi 2095 1553 1847 2122 1584 1875 2148 i6i<: 1903 2175 164^: 1931 22OI 1673 1959 2227 1703 1987 2253 1732 2014 2279 3 6 9 36 8 3 5 8 12 15 18 ii 14 17 ii 13 16 17 18 19 2304 $1 2330 2577 2810 2355 2601 2833 2380 2625 2856 2405 2648 2878 2430 2672 2900 2455 2695 2923 2480 2718 2945 2504 2742 2967 2529 2765 2989 2 57 2 5 7 247 10 12 15 9 12 14 9 " I 2 17 2O 22 16 19 21 16 18 20 20 3010 3032 3054 375 3096 3"8 3139 3160 3181 3201 24 6 8 ii 13 15 17 19 21 22 23 3222 3424 3617 3243 3444 3636 3263 3464 3655 3284 3483 3674 3304 3502 3692 3324 3522 37" 3345 354i 3729 3365 3560 3747 3385 3579 3766 3404 3598 3784 2 4 6 24 6 2 4 6 8 10 12 8 10 12 7 9 ii 14 16 18 H 15 17 J 3 15 '7 24 25 26 3802 3979 415 3820 3997 4166 3838 4014 4183 3856 403 1 4200 3874 4048 4216 3892 4065 4232 3909 4082 4249 3927 4099 4265 3945 4116 4281 3962 4133 4298 245 235 235 7 9 ii 7 9 10 7 8 10 12 14 16 12 14 15 II 13 15 27 28 29 43H 4472 4624 4330 4487 4639 4346 4502 4654 4362 45 l8 4669 4378 4533 4683 4393 4548 4698 4409 45 6 4 47 J 3 4425 4579 4728 444 4594 4742 445 6 4609 4757 2 3 5 2 3 5 1 3 4 689 689 6 7 9 II 13 14 II 12 \i 10 12 13 30 477 1 4786 4800 4814 4829 4843 4857 4871 4886 4900 i 3 4 6 7 9 10 ii 13 31 32 33 4914 505i 5185 4928 5065 5198 4942 5079 5211 4955 5092 5224 4969 5105 5237 4983 5"9 5250 4997 5i|2 5263 5011 5*45 5276 5024 5159 5289 5 38 5172 5302 3 4 3 4 3 4 6 7 8 HI 10 II 12 9 II 12 9 10 12 34 35 36 5315 544i 55 6 3 5328 5453 5575 5340 5465 5587 5353 5478 5599 5366 5490 5611 5378 5502 5 6 23 5391 554 5 6 35 5403 5527 5647 54i6 5539 5658 5428 555i 5670 3 4 2 4 2 4 !.:; 5 6 7 9 10 ii 9 10 ii 8 10 ii 37 38 39 5682 5798 59" 5 6 94 5809 5922 5705 5821 5933 5717 5832 5944 5729 5843 5955 5740 58 II 5966 5977 5763 5877 5988 577 C 5999 5786 5899 6010 2 3 2 3 2 3 5 6 7 5 6 7 4 5 7 8 9 10 8 9 10 8 9 10 40 6021 6031 6042 6o53 6064 6075 6085 6096 6107 6117 2 3 4 5 6 8 9 10 41 42 43 6128 6232 6335 6138 6243 6345 6149 6253 6355 6160 6263 6365 6170 6274 6375 6180 6284 6385 6191 6294 6395 6201 6304 6405 6212 6314 6415 6222 6 3 2 5 6425 2 3 2 3 2 3 4 5 6 4 5 6 4 5 6 7 8 9 7 8 9 7 8 9 44 45 46 6435 6532 6628 6444 6542 6637 6454 ^6 6464 6561 6656 6474 657 1 6665 6484 6580 6675 6 493 6590 6684 6 53 6599 6513 6609 6702 6522 6618 6712 2 3 I 2 3 I 2 3 4 5 6 4 5 6 456 7 8 9 7 8 9 7 7 8 47 48 49 6721 6812 6902 6730 6821 6911 6739 683O 6920 6749 6839 6928 6758 6848 6937 6767 6857 6946 6776 6866 6955 6785 ?2 5 6964 6794 6884 6972 6803 6893 6981 I 2 3 I 2 3 I 2 3 4 5 5 4 4 5 445 6 7 8 678 678 50 6990 6998 7007 7016 7024 733 7042 7050 7059 7067 I 2 3 3 4 5 678 51 52 53 7076 7160 7243 7084 7168 7251 7093 7177 7259 7101 7185 7267 7110 7193 7275 7118 7202 7284 7126 7210 7292 7i35 7218 7300 7H3 7226 7308 7152 7235 73i6 I 2 3 122 I 2 2 3 4 5 3 4 5 345 678 6 7 7 667 54 7324 7332 7340 7348 735 6 7364 7372 7380 7388 7396 I 2 2 3 4 5 667 * From Bottomley's Four Figure Mathematical Tables, by courtesy of The Macmillan Company. TABLES 233 TABLE XVIII. LOGARITHMS (Concluded). 1 2 3 4 5 6 7 8 9 1 23 456 789 55 7404 .7412 7490 566 ^642 74i9 7427 7435 7443 745i 7459 7466 7474 122 3 4 5 5 6 7 56 57 58 7482 7559 7 6 34 7497 7574 7649 7505 7582 7657 75*3 7589 7664 7520 7597 7672 7528 7604 7679 7536 7612 7686 7543 7619 7694 755i 7627 7701 2 2 2 2 I 2 345 3 4 5 344 5 6 7 5 6 7 5 6 7 59 60 61 7709 7782 7853 7716 7789 7860 7723 7796 7868 773i 7803 7875 7738 7810 7882 7745 7818 7889 Ws 7896 7760 7832 7903 7767 7839 7910 7774 7846 7917 2 2 2 344 344 344 5 6 7 566 5 6 6 62 63 64 7924 7993 8062 7931 8000 8069 7938 8007 8075 7945 8014 8082 7952 8021 8089 79^Q 8096 7966 8035 8102 7973 8041 8109 7980 8048 8116 7987 8055 8122 2 2 2 334 334 334 566 5 5 6 5 5 6 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 2 334 5 5 6 66 67 68 .8195 8261 8325 8202 8267 833i 8209 8274 8338 8215 8280 8344 8222 8287 8351 8228 8293 8357 8235 8299 8363 8241 8306 8370 8248 8312 8376 8254 8319 8382 2 2 2 334 334 334 5 5 6 5 5 \ 4 5 6 69 70 71 8388 8451 8513 8395 8457 8519 8401 8463 8525 8407 8470 8531 8414 8476 8537 8420 8482 8543 8426 8488 8549 8432 8494 8555 8439 8500 8561 8445 8506 8567 2 2 2 234 234 234 4 5 6 4 5 6 4 5 5 72 73 74 ~75~ 8573 8633 8692 8579 8639 8698 8585 8645 8704 8591 8651 8710 8597 8657 8716 8603 866 3 8722 8609 8669 8727 8615 8675 8733 8621 8681 8739 8627 8686 8745 2 2 2 234 234 234 455 455 4 5 5 875i 8756 8762 8768 8774 8779 8785 8791 8797 8802 2 233 4 5 5 76 77 78 8808 8865 8921 8814 8871 8927 8820 8876 8932 8825 8882 8938 8831 8887 8943 8837 8893 8949 8842 8899 8954 8848 8904 8960 8854 8910 8965 8859 8915 8971 2 2 2 233 233 233 4 5 5 4 4 5 4 4 5 445 4 4 5 445 79 80 81 8976 9031 9085 8982 9036 9090 8987 9042 9096 8993 9047 9101 8998 9053 9106 9004 9058 9112 9009 9063 9117 9015 9069 9122 9020 9074 9128 9025 9079 9U3 2 2 2 233 233 233 82 83 84 9138 9191 9243 9H3 9196 9248 9149 9201 9253 9154 9206 9258 9159 9212 9263 9165 9217 9269 9170 9222 9274 9175 9227 9279 9180 9232 9284 9186 9238 9289 2 2 2 233 233 233 4 4 5 445 445 85 9294 9299 9304 9309 93i5 9320 9325 9330 9335 9340 I 2 233 445 86 87 88 9345 9395 9445 935 9400 945 9355 9405 9455 9360 9410 9460 9365 94i5 9465 9370 9420 9469 9375 9425 9474 938o 943 9479 9385 9435 9484 9390 9440 9489 I 2 O 233 223 223 4 4 5 344 344 89 90 91 9494 9542 9590 9499 9547 9595 954 9552 9600 959 9557 9605 95*3 9562 9609 9518 95 66 9614 9523 957i 9619 9528 9576 9624 9533 9628 9538 9586 9633 O O 223 223 223 344 344 344 92 93 94 ~95~ 9638 9685 973i 9643 9689 9736 9647 9694 974i 9652 9699 9745 9657 973 975 9661 9708 9754 9666 97 i 3 9759 9671 9717 9763 9675 9722 9768 9680 9727 9773 O O 223 223 223 344 344 344 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 223 344 96 97 98 9823 9868 9912 9827 9872 9917 9832 9877 9921 9836 9881 9926 9841 9886 9930 9845 9890 9934 9850 9894 9939 9854 9899 9943 9859 9903 9948 9863 9908 9952 O O O 223 223 223 344 344 344 99 995 6 9961 9965 9969 9974 9978 9983 9987 9991 9996 I I 223 334 234 THE THEORY OF MEASUREMENTS * TABLE XIX. NATURAL SINES. 0' 6' 12' 18' 24' SO' 36' 42' 48' 54' 123 4 5 oooo 0017 oo35 0052 0070 0087 0105 OI22 0140 oi57 369 12 15 1 2 3 0175 0349 0523 0192 0366 0541 0209 0384 0558 0227 0401 0576 0244 0419 0593 0262 0436 0610 0279 0454 0628 0297 0471 0645 0314 0488 o663 0332 0506 0680 369 369 369 12 I 5 12 I 5 12 I 5 4 5 6 ~7~ 8 9 0698 0872 1045 7!5 0889 1063 0732 0906 1080 0750 0924 1097 0767 0941 "15 0785 0958 1132 0802 0976 1149 0819 0993 1167 0837 ion 1184 0854 1028 I2OI 369 369 369 12 I 5 12 14 12 14 1219 1392 1564 1236 1409 1582 1253 1426 1599 1271 1444 1616 1288 1461 1633 1305 1478 1650 J 323 1495 1668 1340 \&1 1357 1530 1702 !374 J 547 1719 369 369 369 12 14 12 14 12 14 10 1736 !754 1771 1788 1805 1822 1840 1857 1874 1891 369 12 14 11 12 13 1908 2079 2250 1925 2096 2267 1942 2113 2284 1959 2130 2300 1977 2147 2317 1994 2164 2334 2OII 2181 235 I 2028 2198 2368 2045 2215 2385 2062 2232 2402 369 369 368 II I 4 II 14 II I 4 14 15 16 TT 18 19 2419 2588 2756 2436 2605 2773 2453 2622 2790 2470 2639 2807 2487 2656 2823 2504 2672 2840 2521 2689 2857 2538 2706 2874 2554 2723 2890 257i 2740 2907 368 368 368 II 14 II I 4 II 14 2924 3090 3256 2940 3io7 3272 2957 3123 3289 2974 3 J 4 3305 2990 3156 3322 3007 3i73 3338 3024 3190 3355 3040 3206 337 1 3057 3223 3387 3074 3239 3404 3 6 8 368 3 5 8 II 14 II 14 II 14 20 3420 3437 3453 3469 3486 3502 35i8 3535 3551 35 6 7 3 5 8 II 14 21 22 23 ~24~ 25 26 3584 3746 3907 3600 3762 3923 3616 3778 3939 3633 3795 3955 3 6 49 3811 397 1 3665 3827 3987 3681 3843 4003 3697 3859 4019 37H 3875 4035 3730 3891 405 l 3 5 8 3 5 8 3 5 8 II 14 II 14 II 14 4067 4226 4384 4083 4242 4399 4099 4258 4415 4H5 4274 443i 4131 4289 4446 4147 435 4462 4163 432i 4478 4179 4337 4493 4195 4352 459 4210 4368 4524 3 5 8 3 5 8 3 5 8 II 13 II 13 10 13 27 28 29 4540 4695 4848 4555 4710 4863 457 i 4726 4879 4586 474i 4894 4602 475 6 4909 4617 4772 4924 4633 4787 4939 4648 4802 4955 4664 4818 497 4679 4833 4985 3 5 8 3 5 8 3 5 8 10 13 10 13 10 13 30 5000 5015 53o 545 5060 5075 5090 5 I0 5 5120 5135 3 5 8 10 13 31 32 33 5150 5299 5446 5*65 53H 546i 5180 5329 5476 5195 5344 5490 5210 5358 5505 5225 5373 5519 5240 5388 5534 5255 5402 5548 5270 5417 5563 5284 5432 5577 2 5 7 257 2 5 7 IO 12 10 12 IO 12 34 35 36 5592 5736 5878 5606 575 5892 5621 5764 5906 5635 5779 5920 5650 5793 5934 5664 5807 5948 5678 5821 5962 5693 5835 5976 577 5850 5990 572i 5864 6004 257 2 5 7 2 5 7 IO 12 IO 12 9 12 37 38 39 6018 6157 6293 6032 6170 6307 6046 6184 6320 6060 6198 6334 6074 6211 6347 6088 6225 6361 6101 6239 6374 6115 6252 6388 6129 6266 6401 6143 6280 6414 257 2 5 7 247 9 12 9 ii 9 ii 40 6428 6441 6 455 6468 6481 6494 6508 6521 6534 6 547 247 9 ii 41 42 43 6561 6820 6 574 6704 6833 6587 6717 6845 6600 6730 6858 6613 6743 6871 6626 6756 6884 6639 6769 6896 6652 6782 6909 6665 6794 6921 6678 6807 6934 247 2 4 6 246 9 ii 9 " 8 ii 44 6947 6959 6972 6984 6997 7009 7022 7034 7046 7059 246 8 10 * From Bottomley's Four Figure Mathematical Tables, by courtesy of The Macmillan Company. TABLES TABLE XIX. NATURAL SINES (Concluded). 235 0' 6' 12' 18' 24' 3O' 36' 42' 48' 54' 123 4 5 45 7071 7083 7096 7108 7120 7 J 33 7H5 7i57 7169 7181 246 8 10 46 47 48 7 J 93 73*4 743i 7206 7325 7443 7218 7337 7455 7230 7349 7466 7242 73 61 7478 7254 7373 7490 7266 7385 7501 7278 7396 75i3 7290 7408 7524 7302 7420 7536 246 246 246 8 10 8 10 8 10 49 50 51 7547 7660 7771 7558 7672 7782 757 7683 7793 758i 7694 7804 7593 7705 7815 7604 7716 7826 7 6l 5 7727 7837 7627 7738 7848 7638 7749 7859 7649 7760 7869 2 4 6 246 2 4 5 8 9 7 9 7 9 52 53 54 7880 7986 8090 7891 7997 8100 7902 8007 8111 7912 8018 8121 7923 8028 8131 7934 8039 8141 7944 8049 8151 7955 8059 8161 7965 8070 8171 7976 8080 8181 2 4 5 235 2 3 5 7 9 7 9 7 8 55 8192 8202 8211 8221 8231 8241 8251 8261 8271 8281 2 3 5 7 8 56 57 58 8290 8387 8480 8300 8396 8490 8310 8406 8499 8320 8415 8508 8329 8425 8517 8339 8434 8526 8348 8443 8536 8358 8453 8545 8368 8462 8554 8377 8471 8563 2 3 5 2 3 5 2 3 5 6 8 6 8 6 8 59 60 61 8572 8660 8746 8581 8669 8755 8590 8678 8763 8599 8686 8771 8607 8695 8780 8616 8704 8788 8625 8712 8796 8634 8721 8805 8643 8729 8813 8652 8738 8821 i 3 4 i 3 4 i 3 4 6 7 2 ? 62 63 64 8829 8910 8988 8838 8918 8996 8846 8926 9003 8854 8934 9011 8862 8942 9018 8870 8949 9026 8878 8957 9033 8886 8965 9041 8894 8973 9048 8902 8980 9056 i 3 4 i 3 4 i 3 4 1 I 5 6 65 9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 I 2 4 5 6 66 67 68 9135 9205 9272 9M3 9212 9278 915 9219 9285 9157 9225 9291 9164 9232 9298 9171 9239 934 9178 9245 93" 9184 9252 9317 9191 9259 9323 9198 9265 9330 I 2 3 I 2 3 I 2 3 5 6 4 6 4 5 69 70 71 9336 9397 9455 9342 9403 9461 9348 9409 9466 9354 94i5 9472 936i 9421 9478 9367 9426 9483 9373 9432 9489 9379 9438 9494 9385 9444 9500 939i 9449 955 2 3 2 3 2 3 4 5 4 5 4 5 72 73 74 95 11 95 6 3 9613 95 l6 9568 9617 952i 9573 9622 95 2 7 9578 9627 9532 9583 9632 9537 9588 9636 9542 9593 9641 9548 9598 9646 9553 9603 9650 9558 9608 9655 2 3 2 2 2 2 4 4 3 4 3 4 75 9659 9664 9668 9673 9677 9681 9686 9690 9694 9699 I 2 3 4 76 77 78 9703 9744 9781 9707 9748 9785 9711 975i 9789 9715 9755 9792 9720 9759 9796 9724 97 6 3 9799 9728 9767 9803 9732 977 9806 9736 9774 9810 9740 9778 9813 2 2 2 3 3 3 3 2 3 79 80 81 9816 9848 9877 9820 9851 9880 9823 9854 9882 9826 9857 9885 9829 9860 9888 9833 9863 9890 9836 9866 9893 9839 9869 9895 9842 9871 9898 9845 9874 9900 I 2 O O 2 3 2 2 2 2 82 83 84 9903 9925 9945 9905 9928 9947 9907 9930 9949 9910 9932 995i 9912 9934 995 2 9914 9936 9954 9917 9938 995 6 9919 9940 9957 992i 9942 9959 9923 9943 9960 O O 2 2 I 2 I I 85 9962 9963 9965 9966 9968 9969 9971 9972 9973 9974 001 I I 86 87 88 9976 9986 9994 9977 9987 9995 9978 9988 9995 9979 9989 9996 9980 9990 9996 998i 9990 9997 9982 9991 9997 9983 9992 9997 9984 9993 9998 9985 9993 9998 I O O O O O O I I I I O O 89 9998 9999 9999 9999 9999 I'OOO nearly. rooo nearly. rooo nearly. I'OOO nearly. I'OOO nearly. O O O O O 236 THE THEORY OF MEASUREMENTS * TABLE XX. NATURAL COSINES. O' & 12' 18' 24' 3O' 36' 42' 48' 54' 123 4 5 I '000 I'OOO nearly. rooo nearly. rooo nearly. rooo nearly. 9999 9999 9999 9999 9999 o o o 1 2 3 9998 9994 9986 9998 999 8 9993 9984 9997 9992 9983 9997 9991 9982 9997 9990 9981 9996 9990 9980 9996 9989 9979 9995 9988 9978 9995 9987 9977 000 o o o O O I I I I I 4 5 6 9976 9962 9945 9974 9960 9943 9973 9959 9942 9972 9957 9940 9971 995 6 9938 9969 9954 9936 9968 9952 9934 9966 9951 9932 9965 9949 9930 9963 9947 9928 o o I O I I I I 2 I 2 7 8 9 9925 9903 9877 9923 9900 9874 9921 9898 9871 9919 9895 9869 9917 9893 9866 9914 9890 9863 9912 9888 9860 9910 9885 9857 9907 9882 9854 9905 9880 9851 I O I I I 2 2 2 2 2 2 10 9848 9845 9842 9839 9836 9833 9829 9826 9823 9820 112 2 3 11 12 13 9816 9781 9744 9813 9778 9740 9810 9774 9736 9806 977 9732 9803 9767 9728 9799 9763 9724 9796 9759 9720 9792 9755 9715 9789 9751 9711 9785 9748 9707 112 I I 2 I I 2 2 3 3 3 3 3 14 15 16 973 9659 9613 9699 9655 9608 9694 9650 9603 9690 9646 9598 9686 9641 9593 9681 9636 9588 9677 9632 9583 9673 9627 9578 9668 9622 9573 9664 9617 9568 I I 2 122 122 3 4 3 4 3 4 17 18 19 95 6 3 95 11 9455 9558 955 9449 9553 9500 9444 9548 9494 9438 9542 9489 9432 9537 9483 9426 9532 9478 9421 9527 9472 94i5 95 21 9466 9409 95i6 9461 9403 I 2 3 i 2 3 I 2 3 4 4 4 5 4 5 20 9397 939i 9385 9379 9373 9367 9361 9354 9348 9342 I 2 3 4 5 21 22 23 9336 9272 9205 9330 9265 9198 9323 9259 9191 9317 9252 9184 93" 9245 9178 934 9239 9171 9298 9232 9164 9291 9225 9157 9285 9219 915 9278 9212 9H3 I 2 3 I 2 3 I 2 3 4 5 4 6 5 6 24 25 26 9135 9063 8988 9128 9056 8980 9121 9048 8973 9114 9041 8965 9107 9033 8957 9100 9026 8949 9092 9018 8942 9085 9011 8934 9078 9003 8926 9070 8996 8918 I 2 4 i 3 4 i 3 4 5 6 5 6 5 6 27 28 29 8910 8829 8746 8902 8821 8738 8894 8813 8729 8886 8805 8721 8878 8796 8712 8870 8788 8704 8862 8780 8695 8854 8771 8686 8846 8763 8678 8838 8755 8669 i 3 4 i 3 4 i 3 4 5 7 6 7 6 7 30 8660 8652 8643 8634 8625 8616 8607 8599 8590 8581 1 3 4 6 7 31 32 33 8572 8480 8387 8563 8471 8377 8462 8368 8545 8453 8358 8536 8443 8348 8526 8434 8339 8517 8425 8329 8508 8415 8320 8499 8406 8310 8490 8396 8300 2 3 5 2 3 5 235 6 8 6 8 6 8 34 35 36 8290 8192 8090 8281 8181 8080 8271 8171 8070 8261 8161 8059 8251 8151 8049 8241 8141 8039 8231 8131 8028 8221 8121 8018 8211 8111 8007 8202 8100 7997 2 3 5 2 3 5 235 7 8 7 8 7 9 37 38 39 7986 7880 7771 7976 7869 7760 7965 7859 7749 7955 7848 7738 7944 7837 7727 7934 7826 7716 7923 78i5 775 7912 7804 7694 7902 7793 7683 7891 7782 7672 245 245 246 7 9 7 9 7 9 40 7660 7649 7638 7627 7 6l 5 7604 7593 758i 757 7559 2 4 6 8 9 41 42 43 7547 7431 73H 7536 7420 7302 7524 7408 7290 7513 7396 7278 75 01 73 fl 7266 7490 7373 7254 7478 736i 7242 7466 7349 7230 7455 7337 7218 7443 7325 7206 246 246 2 4 6 8 10 8 10 8 10 44 7'93 7181 7169 7157 7H5 7133 7120 7108 7096 7083 2 4 6 8 10 N.B. Numbers in difference-columns to be subtracted, not added. * From Bottomley'g Four Figure Mathematical Tables, by courtesy of The Macmillan Company. TABLES 237 TABLE XX. NATURAL COSINES (Concluded). O' 6' 12' 18' 24' 30' 36' 42' 48' 54' 123 4 5 45 7071 759 7046 734 7022 7009 6997 6984 6972 6959 246 8 10 46 47 48 6947 6820 6691 6934 6807 6678 6921 6794 6665 6909 6782 6652 6896 6769 6639 6884 6756 6626 6871 6 743 6613 6858 6730 6600 6845 6717 6587 6833 6704 6574 246 2 4 6 247 8 ii 9 u 9 ii 49 50 51 6561 6428 6293 6 547 6414 6280 6534 6401 6266 6521 6388 6252 6508 6374 6239 6494 6361 6225 6481 6347 6211 6468 6334 6198 6 455 6320 6184 6441 6307 6170 247 247 2 5 7 9 ii 9 ii 9 ii 52 53 54 6l 57 6018 5878 6i43 6004 5864 6129 5990 5850 6115 5976 5835 6101 5962 5821 6088 5948 58-07 6074 5934 5793 6060 5920 5779 6046 5906 57 6 4 6032 5892 5750 2 5 7 257 257 9 12 9 12 9 12 55 5736 572i 5707 5693 5678 5664 5650 5635 5621 5606 2 5 7 10 12 56 57 58 5592 5446 5299 5577 5432 5284 55 6 3 54i7 5270 5548 5402 5255 5534 5388 5240 55i9 5373 5225 5505 5358 5210 5490 5344 5 J 95 5476 5329 5180 546i 53H 5 l6 5 2 5 7 2 5 7 257 10 12 10 12 10 12 59 60 61 5 Z 5 5000 4848 5i35 4985 4833 5120 4970 4818 5105 4955 4802 5090 4939 4787 575 4924 4772 5060 4909 475 6 5045 4894 474i 5030 4879 4726 5i5 4863 4710 3 5 8 3 5 8 3 5 8 10 13 10 13 10 13 62 63 64 4695 4540 4384 4679 4524 4368 4664 459 4352 4648 4493 4337 4633 4478 4321 4617 4462 4305 4602 4446 4289 4586 443i 4274 457 1 4415 4258 4555 4399 4242 3 5 8 3 5 8 3 5 8 10 13 10 13 II 13 65 4226 4210 4195 4179 4163 4 J 47 4131 4"5 4099 4083 3 5 8 II 13 66 67 68 4067 3907 3746 405 I 3891 3730 4035 3875 37H 4019 3859 3697 4003 3843 3681 3987 3827 3665 397 1 3811 3 6 49 3955 3795 3633 3939 3778 3616 3923 3762 3600 3 5 8 3 5 8 3 5 8 II 14 II 14 II 14 69 70 71 3584 3420 3256 3567 3404 3239 355i 3387 3223 3535 337i 3206 35i8 3355 3190 3502 3338 3173 3486 3322 3156 3469 3305 3140 3453 3289 3123 3437 3272 3 J 07 3 5 8 3 5 8 3 6 8 II 14 II 14 II 14 72 73 74 3090 2924 2756 374 2907 2740 3057 2890 2723 3040 2874 2706 3024 2857 2689 3007 2840 2672 2990 2823 2656 2974 2807 2639 2957 2790 2622 2940 2773 2605 368 368 368 II 14 II 14 II 14 75 2588 257i 2554 2538 2521 2504 2487 2470 2453 2436 368 II 14 76 77 78 2419 2250 2079 2402 2233 2062 2385 2215 2045 2368 2198 2028 2351 2181 2OII 2334 2164 1994 2317 2147 1977 2300 2130 1959 2284 2113 1942 2267 2096 1925 368 369 3 6 9 II 14 II 14 II 14 79 80 81 1908 1736 i5 6 4 1891 1719 '547 1874 1702 1530 1857 1685 1513 1840 1668 1495 1822 1650 1478 1805 1633 1461 1788 1616 1444 1771 1599 1426 *754 1582 1409 3 6 9 3 6 9 369 12 14 12 14 12 14 82 83 84 1392 1219 1045 1374 I2OI 1028 1357 1184 IOII 1340 1167 0993 1323 1149 0976 1305 1132 0958 1288 i"5 0941 1271 1097 0924 1253 1080 0906 1236 1063 0889 369 369 369 12 I 4 12 I 4 12 14 85 0872 0854 0837 0819 0802 0785 0767 0750 0732 0715 3 6 9 12 I 5 86 87 88 0698 0523 0349 0680 0506 0332 o663 0488 03H 0645 0471 0297 0628 0454 0279 0610 0436 0262 0593 0419 0244 0576 0401 0227 0558 0384 0209 0541 0366 0192 369 369 369 12 15 12 15 12 15 89 oi75 0157 0140 0122 0105 0087 0070 0052 0035 0017 369 12 15 iV.B. Numbers in difference-columns to be subtracted, not added. 238 THE THEORY OF MEASUREMENTS TABLE XXI. NATURAL TANGENTS. O' & 12' 18' 24' 3O' 36' 42' 48' 54' 123 4 5 oooo 0017 0035 0052 0070 0087 0105 OI22 0140 oi57 369 12 14 1 2 3 0175 0349 0524 0192 0367 0542 0209 0384 0559 0227 0402 577 0244 0419 0594 0262 0437 0612 0279 0454 0629 0297 0472 0647 0314 0489 0664 0332 0507 0682 369 369 369 12 I 5 12 15 12 I 5 4 5 6 0699 0875 1051 0717 0892 1069 0734 0910 1086 0752 0928 1104 0769 0945 1122 0787 0963 "39 0805 0981 "57 0822 99 8 "75 0840 1016 1192 0857 1033 I2IO 369 369 369 12 I 5 12 I 5 12 I 5 7 8 9 1228 1405 1584 1246 1423 1602 1263 1441 1620 1281 H59 1638 1299 H77 1655 1317 H95 1673 1334 1512 1691 1352 1530 1709 1370 1548 1727 1388 1566 1745 369 369 369 12 I 5 12 I 5 12 I 5 10 1763 1781 1799 1817 1835 1853 1871 1890 1908 1926 369 12 I 5 11 12 13 1944 2126 2309 1962 2144 2327 1980 2162 2345 1998 2180 2364 2016 2199 2382 2035 2217 2401 2053 2235 2419 2071 2254 2438 2089 2272 2456 2IO7 2290 2475 369 369 369 12 I 5 12 I 5 12 I 5 14 15 16 2493 2679 2867 2512 2698 2886 2 53 2717 2905 2549 2736 2924 2568 2754 2943 2586 2773 2962 2605 2792 2981 2623 2811 3000 2642 2830 3019 2661 2849 3038 369 369 369 12 l6 13 16 13 16 17 18 19 3057 3249 3443 3076 3269 3463 3096 3288 3482 3"5 3307 3502 3134 3327 3522 3i53 3346 354i 3172 3365 356i 3 J 9i 3385 358i 3211 3404 3600 3230 3424 3620 3 6 10 3 6 10 3 6 10 13 16 13 16 13 17 20 3640 3659 3679 3699 37 J 9 3739 3759 3779 3799 3819 3 7 I0 13 17 21 22 23 3839 4040 4245 3859 4061 4265 3879 4081 4286 3899 4101 4307 3919 4122 4327 3939 4142 4348 3959 4163 4369 3979 4183 4390 4000 4204 44" 4O2O 4224 4431 3 7 I0 3 7 I0 3 7 10 13 17 14 17 14 17 24 25 26 4452 4663 4877 4473 4684 4899 4494 4706 4921 45i5 4727 4942 4536 4748 4964 4557 477 4986 4578 479i 5008 4599 4813 5029 4621 4834 5051 4642 4856 573 4 7 10 4 7 ii 4 7 ii 14 18 14 18 15 18 27 28 29 5095 5317 '5543 5"7 5340 5566 5139 5362 5589 5161 5384 5612 5184 5407 5635 5206 5430 5658 5228 5452 5681 5250 5475 5704 5272 5498 5727 5295 5520 575 4 7 ii 4 8 ii 4 8 12 15 18 15 19 15 19 30 '5774 5797 5820 5844 5867 5890 59H 5938 596i 5985 4 8 12 16 20 31 32 33 6009 6249 6494 6032 6273 6519 6056 6297 6544 6080 6322 6569 6104 6346 6594 6128 6371 6619 6152 6395 6644 6176 6420 6669 6200 6445 6694 6224 6469 6720 4 8 12 4 8 12 4 8 13 16 20 16 20 17 21 34 35 36 ' 6 745 7002 7265 6771 7028 7292 6796 7054 7319 6822 7080 7346 6847 7107 7373 6873 7 ! 33 7400 6899 7*59 7427 6924 7186 7454 6950 7212 748i 6976 7239 7508 4 9 13 4 9 13 5 9 H 17 21 18 22 18 23 37 38 39 7536 7813 8098 7563 7841 8127 7590 8156 7618 7898 8185 7646 7926 8214 7673 7954 8243 7701 7983 8273 7729 8012 8302 7757 8040 8332 7785 8069 8361 5 9 H 5 I0 M 5 10 15 18 23 19 24 20 24 40 8391 8421 8451 8481 8511 8541 857i 8601 8632 8662 5 1 '5 20 25 41 42 43 8693 9004 9325 8724 9036 9358 8754 9067 9391 8785 9099 9424 8816 9131 9457 8847 9163 9490 8878 9195 9523 8910 9228 9556 8941 9260 9590 8972 9293 9623 5 10 16 5 " 16 6 ii 17 21 26 21 27 22 28 44 9657 9691 97 2 5 9759 9793 9827 9861 9896 9930 9965 6 ii 17 23 29 * From Bottomley's Four Figure Mathematical Tables, by courtesy of The Macmillan Company. TABLES 239 TABLE XXI. NATURAL TANGENTS (Concluded). 0' 6' 12' 18' 24' 3O' 36' 42' 48' 54' 123 4 5 45 I -0000 0035 0070 0105 0141 0176 O2I2 0247 0283 0319 6 12 18 24 30 46 47 48 1-0355 1-0724 1-1106 0392 0761 "45 0428 0799 1184 0464 0837 1224 0501 0875 1263 0538 0913 1303 0575 0951 1343 0612 0990 1383 0649 1028 1423 0686 1067 1463 6 12 18 6 13 19 7 13 20 25 3i 25 32 26 33 49 50 51 1504 1918 2349 1544 1960 2393 1585 2OO2 2437 1626 2045 2482 1667 2088 2527 1708 2131 2572 1750 2174 2617 1792 2218 2662 1833 2261 2708 1875 2305 2753 7 H 21 7 14 22 8 15 23 28 34 29 36 30 38 52 53 54 2799 3270 3764 2846 3319 3814 2892 3367 3865 2938 34i6 3916 2985 3465 3968 3032 35H 4019 3079 3564 4071 3127 3613 4124 3i75 3663 4176 3222 37i3 4229 8 16 23 8 16 25 9 17 26 3i 39 33 4i 34 43 55 4281 4335 4388 444 2 4496 4550 4605 4659 4715 4770 9 18 27 36 45 56 57 58 4826 5399 6003 4882 5458 6066 4938 5517 6128 4994 5577 6191 5051 5637 6255 5108 5697 6319 5166 mi 5224 5818 6447 5282 5880 6512 5340 594i 6577 10 19 29 10 20 30 II 21 32 38 48 40 50 43 53 59 60 61 6643 7321 8040 6709 739i 8115 6775 7461 8190 6842 7532 8265 6909 7603 8341 6977 7675 8418 7045 7747 8495 7113 7820 8572 7182 7893 8650 725 1 7966 8728 ii 23 34 12 24 36 13 26 38 45 5 6 48 60 5 1 6 4 62 63 64 1-8807 1-9626 2-0503 8887 9711 0594 8967 9797 0686 9047 9883 0778 9128 9970 0872 9210 0057 0965 9292 0145 1060 9375 0233 "55 9458 0323 1251 9542 041; 1348 14 27 41 15 29 44 16 31 47 55 68 58 73 63 78 65 2-1445 1543 1642 1742 1842 1943 2045 2148 2251 2355 1 7 34 5 1 68 85 66 67 68 2-2460 2-3559 2'475 i 2566 3673 4876 2673 3789 5002 2781 3906 5129 2889 4023 5257 2998 4142 5386 3109 4262 5517 3220 4383 5649 3332 454 5782 3445 4627 59i6 18 37 55 20 40 60 22 43 65 74 92 79 99 87 108 69 70 71 2-6051 27475 2-9042 6187 7625 9208 6325 7776 9375 6464 7929 9544 6605 8083 97H 6746 8239 9887 6889 8397 0061 734 8556 0237 7179 8716 0415 7326 8878 0595 24 47 7i 26 52 78 29 58 87 95 "8 104 130 "5 !44 72 73 74 3-0777 3-2709 3-4874 0961 2914 5 I0 5 1146 3122 5339 1334 3332 5576 1524 3544 5816 1716 3759 6059 1910 3977 6305 2106 4197 6554 '23 5 4420 6806 2506 4646 7062 32 64 96 36 72 108 41 82 122 129 161 144 180 162 203 75 3-732I 7583 7848 8118 8391 8667 8947 9232 9520 9812 46 94 139 i 86 232 76 77 78 4-0108 4-33I5 4-7046 0408 3662 7453 0713 4015 7867 IO22 4374 8288 1335 4737 8716 l6 53 5 I0 7 9152 1976 5483 9594 2303 5864 0045 2635 6252 0504 2972 6646 0970 53 107 i 60 62 124 186 73 146 219 214 267 248 310 292 365 79 80 81 5-I446 5-67I3 6-3138 1929 7297 3859 2422 7894 4596 2924 8502 5350 3435 9124 6122 3955 9758 6912 4486 0405 7920 5026 5578 6140 2432 0264 87 175 262 350 437 1066 8548 1742 9395 Difference-columns cease to be useful, owing to the rapidity with which the value of the tangent changes. 82 83 84 r"54 8-1443 9-5H4 2066 2636 9-677 3002 3863 9-845 3962 5126 IO-O2 4947 6427 10-20 5958 7769 10-39 6996 9152 10-58 8062 0579 10-78 9158 2052 10-99 0285 3572 11-20 85 n-43 11-66 11-91 12-16 12-43 12-71 13-00 13-30 13-62 I3-95 86 87 88 14-30 19-08 28-64 14-67 I9-74 30-14 15-06 20-45 31-82 I5-46 21-20 3J69 15-89 22-02 35-8o 16-35 22-90 38-19 16-83 23-86 40-92 I7-34 24-90 44-07 17-89 26-03 47-74 18-46 27-27 52-08 89 57'29 63-66 71-62 81-85 95-49 114-6 143-2 191-0 286-5 573-0 240 THE THEORY OF MEASUREMENTS * TABLE XXII. NATURAL COTANGENTS. O' 6' 12' 18' 24' 30' 36' 42' 48' 54' Difference-columns not useful here, owing to the rapidity with which the value of the cotangent changes. Inf. 573-o 286-5 191-0 143-2 114-6 95'49 81-85 71-62 63-66 1 2 3 57-29 28-64 19-08 52-08 27-27 18-46 4774 26-03 17-89 44-07 24-90 17-34 40-92 23-86 16-83 38-19 22-90 i6'35 35-80 22-02 15-89 33-69 2 1 -2O 31-82 20-45 15-06 19-74 14-67 4 5 6 14-30 ii'43 9-5I44 I3-95 II'2O 3572 13-62 10-99 2052 13-3 10-78 0579 13-00 10-58 9152 12-71 10-39 7769 12-43 10-20 6427 I2'l6 10-02 5126 11-91 9-845 3863 u-66 9-677 2636 7 8 9 8-1443 7'"54 6-3138 0285 0264 2432 9158 9395 1742 8062 8548 1066 6996 7920 0405 5958 6912 97S8 4947 6122 9124 3962 5350 8502 3002 4596 7894 2066 3859 7297 10 5-67I3 6140 5578 5026 4486 3955 3435 2924 2422 1929 123 4 5 11 12 13 4-7046 4-33I5 0970 6646 2972 0504 6252 2635 0045 5864 2303 9594 5483 1976 9152 5107 1653 8716 4737 1335 8288 4374 1022 7867 4015 0713 7453 3662 0408 74 148 222 63 125 i 88 53 107 160 296 370 252 314 214 267 14 15 16 4-0108 J4874 9812 7062 4646 9520 6806 4420 9232 6554 4197 8947 6305 3977 8667 6059 3759 5816 3544 8118 5576 3332 7848 5339 3122 7583 5105 29H 46 93 139 41 82 122 36 72 108 i 86 232 163 204 144 180 17 18 19 3-2709 3-0777 2-9042 2506 595 8878 2305 0415 8716 2106 0237 8556 1910 0061 8397 1716 9887 8239 5 2 4 9714 8083 1334 9544 7929 1146 9375 7776 0961 9208 7625 32 64 96 29 58 87 26 52 78 129 161 "5 *44 104 130 2*7475 7326 7179 734 6889 6746 6605 6464 6325 6187 24 47 7i 95 "8 21 22 23 2-6051 2-475 * 2-3559 5916 4627 3445 5782 454 3332 5649 4383 3220 5517 4262 3109 5386 4142 2998 5257 4023 2889 3906 2781 5002 3789 2673 4876 3673 2566 22 43 65 20 40 60 18 37 55 87 108 79 99 74 92 24 25 26 ~27~ 28 29 2-2460 2-1445 2-0503 2355 1348 0413 2251 1251 0323 2148 "55 0233 2045 1060 0145 1943 0965 0057 1842 0872 9970 1742 0778 988 3 1642 0686 9797 1543 0594 97" 17 34 5i 16 31 47 15 29 44 68 85 63 78 58 73 1-9626 1-8807 1-8040 9542 8728 7966 9458 8650 7893 9375 8572 7820 9292 8495 7747 9210 8418 7675 9128 8341 7603 9047 8265 753 2 8967 8190 7461 8887 8115 739i 14 27 41 i3 26 38 12 24 36 55 68 5 1 64 48 60 30 1-7321 7251 7182 7"3 745 6977 6909 6842 6775 6709 ii 23 34 45 56 31 32 33 1-6643 1-6003 1-5399 6577 5340 6512 5880 5282 6447 5818 5224 6383 mi 6319 5697 5108 6255 5637 5051 6191 5577 4994 6128 5517 4938 6066 5458 4882 II 21 32 10 20 30 10 19 29 43 53 40 5 38 48 34 35 36 1-4826 1-4281 1-3764 4770 4229 3713 4715 4176 3663 4659 4124 3613 4605 4071 3564 4550 4019 35H 4496 3968 3465 4442 3916 4388 3865 3367 4335 3814 3319 9 18 27 9 17 26 8 16 25 36 45 34 43 33 4i 37 38 39 1-3270 1-2799 1-2349 3222 2753 2305 2708 2261 3^27 2662 2218 3079 2617 2174 3032 2572 2131 2985 2527 2088 2938 2482 2045 2892 2437 2OO2 2846 2393 1960 8 16 23 8 15 23 7 14 22 3 1 39 30 38 29 36 40 1-1918 1875 1833 1792 !75o 1708 1667 1626 1585 1544 7 *4 21 28 34 41 42 43 1-1504 1-1106 1-0724 1463 1067 0686 1423 1028 0649 1383 0990 0612 1343 0951 0575 1303 0913 0538 1263 0875 0501 1224 0837 0464 1184 0799 0428 "45 0761 0392 7 13 20 6 13 19 6 12 18 26 33 25 32 25 31 44 1-0355 0319 0283 0247 O2I2 0176 0141 0105 0070 0035 6 12 18 24 30 N.B. Numbers in difference-columns to be subtracted, not added. * From Bottomley's Four Figure Mathematical Tables, by courtesy of The Macmillan Company. TABLES 241 TABLE XXII. NATURAL COTANGENTS (Concluded). O' 6' 12 18' 24' 3O' 36' 42' 48' 54' 123 4 5 45 ro 0-9965 0-9930 0-9896 0-9861 0-9827 0-9793 '9759 0-9725 0-9691 6 ii 17 23 29 46 47 48 9657 9325 9004 9623 9293 8972 9590 9260 8941 955 6 9228 8910 9523 9i95 8878 9490 9163 8847 9457 9131 8816 9424 9099 8785 939i 9067 8754 9358 9036 8724 6 ii 17 5 ii 10 5 10 16 22 28 21 27 21 26 49 50 51 8693 8391 8098 8662 8361 8069 8632 8332 8040 8601 8302 8012 8571 8273 7983 8541 8243 7954 8511 8214 7926 8481 8185 7898 8451 8156 7869 8421 8127 7841 5 10 i5 5 10 15 5 I0 M 20 25 20 24 19 24 52 53 54 7813 7536 7265 7785 7508 7239 7757 748i 7212 7729 $3 7701 7427 7i59 7 6 73 7400 7133 7646 7373 7107 7618 7346 7080 7590 73i9 754 75 6 3 7292 7028 5 9 H 5 9 H 4 9 13 18 23 18 23 18 22 55 7002 6976 6950 6924 6899 6873 6847 6822 6796 6771 4 9 13 I 7 21 56 57 58 >6 745 6494 6249 6720 6469 6224 6694 6445 6200 6669 6420 6176 6644 6395 6152 6619 637 1 6128 6594 6346 6104 6569 6322 6080 6544 6297 6056 6519 6273 6032 4 8 13 4 8 12 4 8 12 17 21 16 20 16 20 59 60 61 6009 '5774 '5543 5985 5750 5520 596i 5727 5498 5938 574 5475 59H 5681 5452 5890 5658 5430 5867 5^35 5407 5844 5612 5384 5820 5589 5362 5797 5566 5340 4 8 12 4 8 12 4 8 ii 16 20 15 !9 15 !9 62 63 64 5317 595 4877 5295 573 4856 5272 505 1 4834 5250 5029 4813 5228 5008 479i 5206 4986 4770 5184 4964 4748 5161 4942 4727 5*39 4921 4706 5"7 4899 4684 4 7 ii 4 7 ii 4 7 ii 15 18 15 18 14 18 65 4663 4642 4621 4599 4578 4557 4536 4515 4494 4473 4 7 10 14 18 66 67 68 "445 2 4245 4040 443i 4224 4020 4411 4204 4000 4390 4183 3979 4369 4163 3959 4348 4142 3939 4327 4122 3919 4307 4101 3899 4286 4081 3879 4265 4061 3859 371 3 7 10 3 7 I0 14 17 14 17 13 17 69 70 71 3839 3640 '3443 3819 3620 3424 3799 3600 3404 3779 358i 3385 3759 356i 3365 3739 354i 3346 3719 3522 3327 3699 3502 3307 3679 3482 3288 3659 3463 3269 3 7 10 3 6 10 3 6 10 13 17 13 17 13 16 72 73 74 3249 3057 2867 3230 3038 2849 3211 3019 2830 3i9i 3000 2811 3172 2981 2792 3153 2962 2773 3134 2943 2754 3"5 2924 2736 3096 2905 2717 2698 3 6 10 369 369 13 16 13 16 13 16 75 2679 2661 2642 2623 2605 2586 2568 2549 2530 2512 369 12 16 76 77 78 2493 2309 2126 2475 2290 2107 2456 2272 2089 2438 2254 2071 2419 2235 2053 2401 2217 2035 2382 2199 2016 2364 2180 1998 2345 2162 1980 2327 2144 1962 369 3 6 9 369 12 15 12 15 12 I 5 79 80 81 1944 1763 1584 1926 '745 1566 1908 1727 1548 1890 1709 1530 1871 1691 1512 1853 1673 H95 1835 l6 55 H77 1817 1638 H59 1799 1620 1441 1781 1602 1423 369 369 369 12 I 5 12 I 5 12 I 5 82 83 84 1405 1228 1051 1388 I2IO 1033 1370 1192 1016 1352 "75 0998 1334 "57 0981 1317 "39 0963 1299 1122 0945 1281 1104 0928 1263 1086 0910 1246 1069 0892 369 369 3 6 9 12 15 12 15 12 15 85 0875 08 57 0840 0822 0805 0787 0769 0752 0734 0717 3 6 9 12 I 5 86 87 88 0699 0524 0349 0682 0507 0332 0664 0489 3i4 0647 0472 0297 0629 0454 0279 0612 0437 0262 0594 0419 0244 0577 0402 0227 0559 0384 0209 0542 0367 0192 369 3 6 9 369 12 15 12 15 12 I 5 89 oi75 0157 0140 0122 0105 0087 OC>7O 0052 0035 0017 369 12 14 N.B. Numbers in difference-columns to be subtracted, not added. 242 THE THEORY OF MEASUREMENTS TABLE XXIII. RADIAN MEASURE. 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 123 4 5 0.0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 369 12 15 1 0.0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 369 12 15 2 0.0349 0367 0384 0401 0419 0436 0454 0471 0489 0506 369 12 15 3 0.0524 0541 0559 0576 0593 0611 0628 0646 0663 0681 369 12 15 4 0.0698 0716 0733 0750 0768J 0785 0803 0820 0838 0855 369 12 15 5 0.0873 0890 0908 0925 0942 0960 0977 0995 1012 1030 369 12 15 6 0.1047 1065 1082 1100 1117 1134 1152 1169 1187 1204 369 12 15 7 0.1222 1239 1257 1274 1292 1309 1326 1344 1361 1379 369 12 15 8 0.1396 1414 1431 1449 1466 1484 1501 1518 1536 1553 369 12 15 9 0.1571 1588 1606 1623 1641 1658 1676 1693 1710 1728 369 12 15 10 0.1745 1763 1780 1798 1815 1833 1850 1868 1885 1902 369 12 15 11 0.1920 1937 1955 1972 1990 2007 2025 2042 2059 2077 369 12 15 12 0.2094 2112 2129J2147 2164 2182 2199 2217 2234 2251 369 12 15 13 0.2269 2286 230412321 2339 2356 2374 2391 2409 2426 369 12 15 14 0.2443 2461 2478 2496 2513 2531 2548 2566 2583 2601 369 12 15 15 0.2618 2635 2653 2670 2688 2705 2723 2740 2758 2775 369 12 15 16 0.2793 2810 2827 2845 2862 2880 2897 2915 2932 2950 369 12 15 17 0.2967 2985 3002 3019 3037 3054 3072 3089 3107 3124 369 12 15 18 0.3142 3159 3176 3194 3211 3229 3246 3264 3281 3299 369 12 15 19 0.3316 3334 3351 3368 3386 3403 3421 3438 3456 3473 369 12 15 20 0.3491 3508 3526 3543 3560 3578 3595 3613 3630 3648 369 12 15 21 0.3665 3683 3700 3718 3735 3752 3770 3787 3805 3822 369 12 15 22 0.3840 3857 3875 3892 3910 3927 3944 3962 3979 3997 369 12 15 23 0.4014 4032 4049 4067 4084 4102 4119 4136 4154 4171 369 12 15 24 0.4189 4206 4224 4241 4259 4276 4294 4311 4328 4346 369 12 15 25 0.4363 4381 4398 4416 4433 4451 4468 4485 4503 4520 369 12 15 26 0.4538 4555 4573 4590 4608 4625 4643 4660 4677 4695 369 12 15 27 0.4712 4730 4747 4765 4782 4800 4817 4835 4852 4869 369 12 15 28 0.4887 4904 4922 4939 4957 4974 4992 5009 5027 5044 369 12 15 29 0.5061 5079 5096 5114 5131 5149 5166 5184 5201 5219 369 12 15 30 0.5236 5253 5271 5288 5306 5323 5341 5358 5376 5393 369 12 15 31 0.5411 5428 5445 5463 5480 5498 5515 5533 5550 5568 369 12 15 32 0.5585 5603 5620 5637 5655 5672 5690 5707 5725 5742 369 12 15 33 0.5760 5777 5794 5812 5829 5847 5864 5882 5899 5917 369 12 15 34 0.5934 5952 5969 5986 6004 6021 6039 6056 6074 6091 369 12 15 35 0.6109 6126 6144 6161 6178 6196 6213 6231 6248 6266 369 12 15 36 0.6283 6301 6318 6336 6353 6370 6388 6405 6423 6440 369 12 15 37 0.6458 6475 6493 6510 6528 6545 6562 6580 6597 6615 369 12 15 38 0.6632 6650 6667 6685 6702 6720 6737 6754 6772 6789 369 12 15 39 0.6807 6824 6842 6859 6877 6894 6912 6929 6946 6964 369 12 15 40 0.6981 6999 7016 7034 7051 7069 7086 7103 7121 7138 369 12 15 41 0.7156 7173 7191 7208 7226 7243 7261 7278 7295 7313 369 12 15 42 0.7330 7348 7365 7383 7400 7418 7435 7453 7470 7487 369 12 15 43 0.7505 7522 7540 7557 7575 7592 7610 7627 7645 7662 369 12 15 44 0.7679 7697 7714 7732 7749 7767 7784 7802 7819 7837 369 12 15 (Bottomley, " Four Fig. Math. Tables.") TABLES TABLE XXIII. RADIAN MEASURE (Concluded). 243 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1 2 3 4 5 45 0.7854 7871 7889 7906 7924 7941 7959 7976 7994 8011 369 12 15 46 0.8029 8046 8063 8081 8098 8116 8133 8151 8168 8186 369 12 15 47 0.8203 8221 8238 8255 8273 8290 8308 8325 8343 8360 369 12 15 48 0.8378 8395 8412 8430 8447 8465 8482 8500 8517 8535 369 12 15 49 0.8552 8570 8587 8604 8622 8639 8657 8674 8692 8709 369 12 15 50 0.8727 8744 8762 8779 8796 8814 8831 8849 8866 8884 369 12 15 51 0.8901 8919 8936 8954 8971 8988 9006 9023 9041 9058 369 12 15 52 0.9076 9093 9111 9128 9146 9163 9180 9198 9215 9233 369 12 15 53 0.9250 9268 9285 9303 9320 9338 9355 9372 9390 9407 369 12 15 54 0.9425 9442 9460 9477 9495 9512 9529 9547 9564 9582 369 12 15 55 0.9599 9617 9634 9652 9669 9687 9704 9721 9739 9756 369 12 15 56 0.9774 9791 9809 9826 9844 9861 9879 9896 9913 9931 369 12 15 57 0.9948 9966 9983 0001 0018 0036 0053 0071 0088 0105 369 12 15 58 1.0123 0140 0158 0175 0193 0210 0228 0245 0263 0280 369 12 15 59 1.0297 0315 0332 0350 0367 0385 0402 0420 0437 0455 369 12 15 60 1.0472 0489 0507 0524 0542 0559 0577 0594 0612 0629 369 12 15 61 1.0647 0664 0681 0699 0716 0734 0751 0769 0786 0804 369 12 15 62 1.0821 0838 0856 0873 0891 0908 0926 0943 0961 0978 369 12 15 63 1.0996 1013 1030 1048 1065 1083 1100 1118 1135 1153 369 12 15 64 1.1170 1188 1205 1222 1240 1257 1275 1292 1310 1327 369 12 15 65 1.1345 1362 1380 1397 1414 1432 1449 1467 1484 1502 369 12 15 66 1.1519 1537 1554 1572 1589 1606 1624 1641 1659 1676 369 12 15 67 1.1694 1711 1729 1746 1764 1781 1798 1816 1833 1851 369 12 15 68 1.1868 1886 1903 1921 1938 1956 1973 1990 2008 2025 369 12 15 69 1.2043 2060 2078 2095 2113 2130 2147 2165 2182 2200 369. 12 15 70 1.2217 2235 2252 2270 2287 2305 2322 2339 2357 2374 369 12 15 71 1.2392 2409 2427 2444 2462 2479 2497 2514 2531 2549 369 12 15 72 1.2566 2584 2601 2619 2636 2654 2671 2689 2706 2723 369 12 15 73 1.2741 2758 2776 2793 2811 2828 2846 2863 2881 2898 369 12 15 74 1.2915 2933 2950 2968 2985 3003 3020 3038 3055 3073 369 12 15 75 1.3090 3107 3125 3142 3160 3177 3195 3212 3230 3247 369 12 15 76 1 . 3265 3282 3299 3317 3334 3352 3369 3387 $404 3422 369 12 15 77 1 3439 3456 3474 3491 3509 3526 3544 3561 3579 3596 369 12 15 78 1.3614 3631 3648 3666 3683 3701 3718 3736 3753 3771 369 12 15 79 1.3788 3806 3823 3840 385& 3875 3893 3910 3928 3945 369 12 15 80 1.3963 3980 3998 4015 4032 4050 4067 4085 4102 4120 369 12 15 81 1.4137 4155 4172 4190 4207 4224 4242 4259 4277 4294 369 12 15 82 1.4312 4329 4347 4364 4382 4399 4416 4434 4451 4469 369 12 15 83 1.4486 4504 4521 4539 4556 4573 4591 4608 4626 4643 369 12 15 84 1.4661 4678 4696 4713 4731 4748 4765 4783 4800 4818 369 12 15 85 1.4835 4853 4870 4888 4905 4923 4940 4957 4975 4992 369 12 15 86 1.5010 5027 5045 5062 5080 5097 5115 5132 5149 5167 369 12 15 87 1.5184 5202 5219 5237 5254 5272 5289 5307 5324 5341 369 12 15 88 1.5359 5376 5394 5411 5429 5446 5464 5481 5499 5516 369 12 15 89 1.5533 5551 5568 5586 5603 5621 5638 5656 5673 5691 369 12 15 INDEX. A. Absolute measurements, 5. Accidental errors, axioms of, 29. errors, criteria of, 121. errors, definition of, 26 errors, law of, 29, 35. Adjusted effects, 149. Adjustment of the angles about a point, 81. of the angles of a plane triangle, 93. of instruments, 15, 183. of measurements, 21, 42, 63, 72. Applications of the method of least squares, 203. Arithmetical mean, characteristic errors of, 51. mean, principle of, 29. mean, properties of, 42. Average error, defined, 44. Axioms of accidental errors, 29. B. Best magnitudes for components, fundamental principles, 165. general solutions, 167. practical examples, 173. special cases, 170. C. Characteristic errors, defined, 44. errors, computation of, 53, 57, 66, 71, 99, 101, 112, 114. errors of the arithmetical mean, 51. errors, relations between, 49. Chauvenet's criterion, 127. Computation checks for normal equa- tions, 83. Conditioned measurements, 17. quantities, determination of, 92. Constant errors, elimination of, 117. errors, defined, 23. Conversion factor, defined, 3. factor, determination of, 8. Correction factors, defined, 131. Criteria of accidental errors, 121. Criticism of published results, proper basis for, 117. Curves, use of, in reducing observa- tions, 198. D. Dependent measurements, 17. Derived measurements, defined, 12. measurements, precision of, 135. quantities, defined, 95. quantities, errors of, 99. units, 4. Dimensions of units, 5. Direct measurements, defined, 11. measurements, precision of, 130. Discussion of completed observa- tions, 117. of proposed measurements, general problem, 145. of proposed measurements, prelim- inary considerations, 144. of proposed measurements, primary condition, 146. E. Effective sensitiveness of instru- ments, 183. Equal effects, principle of, 147. Equations, observation, 74. normal, 75. Error, average, 44. fractional, 101. mean, 46. probable, 47. 245 246 INDEX Error, Continued. unit, 31. weighted, 67. Errors, accidental, 26. characteristic, 44. constant, 23. definition of, 18. of adjusted measurements, 105. of derived quantities, 99. of multiples of a measured quan- tity, 98. of the algebraic sum of a number of terms, 95. of the product of a number of factors, 102. percentage, 104. personal, 25. propagation of, 95. systematic, 118. systems of, 33. Examples, see Numerical examples. F. Fractional error, defined, 101. error of the product of a number of factors, 102. Free components, 169. Functional relations, determination of, 15, 195, 198, 203. Fundamental units, 4. G. Gauss's method for the solution of normal equations, 84. General mean, 63. principles, 1. Graphical methods of reduction, 198. I. Independent measurements, 17. Indirect measurements, 11. Intrinsic sensitiveness of instru- ments, 183. Law of accidental errors, 29, 35. Laws of science, 2. Least squares, method of, 72. M. Mathematical constants, use of, in computations, 153. Mean error, defined, 46. Measurement, defined, 2. Measurements, absolute, 5. adjustment of, 21, 42, 63, 72. derived, 12. direct, 11. discussion of, 117, 144. independent, dependent, and con- ditioned, 17. indirect, 11. precision of, 19, 130, 135. weights of, 61. Method of least squares, applica- tions of, 203. of least squares, fundamental prin- ciples of, 72. Mistakes, 26. N. Negligible components, 154. effects, 151. Normal equations, computation checks for, 83. equations, derivation of, 75. equations, solution by determi- nants, 114. equations, solution by Gauss's method, 84. equations, solutions by indetermi- nate multipliers, 105. equations, solution with two in- dependent variables, 78. Numeric, defined, 2. Numerical examples: Adjustment of angles about a point, 81. Adjustment of angles of a plane triangle, 93. Application of Chauvenet's crite- rion, 129. Best magnitudes for components, 173, 175, 180. Characteristic errors of direct measurements, 56, 70. INDEX 247 Numerical examples Continued. Coefficient of linear expansion, 78. Discussion of proposed measure- ment, 157. Effective sensitiveness of potenti- ometer, 190. Errors of a derived quantity, 101. Fractional errors, 101. Precision of completed measure- ment, 140. Probable errors of adjusted meas- urements, 113, 115. Probable error of general mean, 69. Propagation of errors, 101. Solution of normal equations by Gauss's method, 88. Weighted direct measurement, 69. O. Observation, denned, 15. equations, 74. standard, 62. Observations, record of, 16. report of, 211. representation of, by curves, 198. P. Percentage errors, 104. Personal equation, 26. errors, 25. Physical tables, use of, 138. Precision constant, 35. Precision of derived measurements, 135. of direct measurements, 130. of measurement, denned, 19. Precision measure, denned, 132. Preliminary considerations for select- ing methods of measurement, 144. Primary condition, 146. Principle of the arithmetical mean, 29. of equal effects, 147. Probability curve, 32. function, 34. Probability curve Continued. function, comparison with experi- ence, 40. integral, 37. of large residuals, 124. of residuals, 30. principles of, 28. Probable error, denned, 47. error of adjusted measurements, 111, 112, 116. error of the arithmetical mean, 53. error of direct measurements, com- putation of, 54, 55, 57. error of the general mean, 66, 68. error of a single observation, 54, 68, 108. error of a standard observation, 62. Propagation of errors, 95. Publication, 209. R. Research, fundamental principles, 192. general methods, 193. Residuals, defined, 27. distribution of, 29. probability of, 30, 124. S. Sensitiveness of methods and instru- ments, 183. Separate effects of errors, 133, 135. Setting of instruments, 15. Sign-changes, defined, 123. Sign-follows, defined, 123. Significant figures, use of, 19, 58. Slugg, defined, 9. Special functions, treatment of, 155. Standard observation, defined, 62. Systematic errors, defined, 118. Systems of errors, 33. of units, 7. T. Tables, list of, ix. Transformation of units, 8. Treatment of special functions, 155. 248 INDEX U. W. Unit error, 31. Weighted errors, 67. Units, c.g.s. system, 7. mean, 63. dimensions of, 5. Weights of adjusted measurements, engineer's system, 7. 105, 112, 114. fundamental and derived, 4. of direct measurements, 61. systems in general use, 7. transformation of, 8. Use of physical tables, 138. significant figures, 19, 58. RETURN CIRCULATION DEPARTMENT TO ^ 202 Main Library LOAN PERIOD 1 HOME USE 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS Renewals and Recharges may be made 4 days prior to the due date. Books may be Renewed by calling 642-3405. SEPoPVf AS STAMPED BELOW mmt \f 5 J3J % o AUG041989! i| Rec'd UC6 A/M/S AUG 1 7 i968 FORM NO. DD6 7 UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 $ GENERAL LIBRARY - U.C. BERKELEY , THE UNIVERSITY OF CALIFORNIA LIBRARY ,..-* ,