■ ' (ttmntll %tiivemtg ff itag BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Hcuru 1*1. Sage 1891 ENGINEERING LIBRARX ~" 9755-2 Cornell University Library QA 275.M57 1909 A textbook on the method of least square 3 1924 003 967 514 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924003967514 Date Due ^1/ wnv* . K^ if - - ". 'i' ' 1"; ' JUL 1 I957 fc/ift ? o .JUS?-—. AW —s-Wrf- ijtUM ^m LflCEfr M98T ^: 1 I90J "» III fa TOO *- ZimTO WORKS OF MANSFIELD MERRIMAN PUBLISHED BY JOHN WILEY & SONS Treatise on Hydraulics. 8vo, 593 pages, $5.00 Mechanics of Materials. 8vo, 518 pages, $5.00 Precise Surveying and Geodesy. 8vo, 261 pages, $2.50 Method of Least Squares. 8vo, 238 pages, $2.00 Elements of Sanitary Engtneeking. 8vo, 252 pages, net % S2.00 Elements of Mechanics. i2mo, 172 pages, net, $1 00 Strength of Materials. i2mo, 156 pages, net, $ 00 By MERRIMAN and JACOBT TEXT-BOOK OIn ROOFS AND BRIDGES; Part I. Stresses. Svo, 326 pages, $2.50 Part II. Graphic Statics. Svo, 242 pages, $2.50 Part III. Bridge Design. 8vo, 382 pages, $2.50 Part IV. Higher Structures. 8vo, 385 pages, $2.50 By MERRIMAN and BROOKS Handbook for Surveyors, nmo, 246 pages. $2.00 Edited by MERRIMAN and WOODWARD Series of Mathematical Monographs. Eleven Volumes, 8vo. Each, $1.00 A TEXT-BOOK ON THE Method of Least Squares. BY MANSFIELD MERRIMAN, MEMBER OF AMERICAN MATHEMATICAL SOCIETY. EIGHTH EDITION, REVISED TOTAL ISSUE, SIXTH THOUSAND. NEW YORK : JOHN WILEY & SONS. London: CHAPMAN & HALL, Limited. 1909. Copyright, 1884, By MANSFIELD MERRIMAN. &V ®rlrnf Ifit JJrraa finbrrt Drummonii anil Olontpang Kan $nrk PREFACE TO THE FIRST EDITION. The "Elements of the Method of Least Squares," published in 1877, was written with two objects in view : first, to present the fundamental principles and processes of the subject in so plain a manner, and to illustrate their application by such simple and practical examples, as to render it accessible to civil engineers who have not had the benefit of 'extended mathematical training; and, secondly, to give an elementary exposition of the theory which would be adapted to the needs of a large and constantly increasing class of students. In preparing the following pages the author has likewise kept these two objects continually in mind. While the for- mer work has been used as a basis, the alterations and additions have been so numerous and radical as to render this a new and distinct book rather than a second edition. The arrangement of the theoretical and practical parts is entirely different. In Chapters I to IV is presented the mathematical development of the principles, methods, and formulas ; while in Chapters V to IX the application of these IV PREFACE. to the different classes of observations is made, and illus- trated by numerous practical examples. For the use of both students and engineers, it is believed that this plan will prove more advantageous than the one previously followed. Hagen's deduction of the law of probability of error is given, as well as that of Gauss. More attention is paid to the laws of the propagation of error, the solution of normal equa- tions, and the deduction of empirical formulas. Many new illustrative examples of the adjustment and comparison of observations have been selected from actual practice, and are discussed in detail. At the end of each chapter are given a few problems or queries ; and in the Appendix are eight tables for abridging computations. MANSFIELD MERRIMAN. Lehigh University, South Bethlehem. Pa. NOTE TO THE EIGHTH EDITION. The seventh edition was the result of a thorough revision and was enlarged by the addition of new matter on the solu- tion of normal equations, on the uncertainty of the propable error, and on the median. In this edition all known errors have been corrected and an alphabetical index has been added. M. M. 32 West Fortieth Street, New York. CONTENTS. CHAPTER I. introduction. Classification of Observations Errors of Observations ..... Principles of Probability . Problems CHAPTER II. LAW OF PROBABILITY OF ERROR. Axioms derived from Experience . The Probability Curve .... First Deduction of the Law of Error . Second Deduction of the Law of Error Discussion of the Probability Curve . The Probability Integral Comparison of Theory and Experience . Remarks on the Fundamental Formulas Problems and Queries l 3 •S 17 22 25 27 31 33 35 CHAPTER III. the adjustment of observations. "Weights of Observations ...... The Principle of Least Squares ..... Direct Observations on a Stngle Quantity Independent Observations of Equal Weight Independent Observations of Unequal Weight ^ulution of Normal Equations Conditioned Observations Pkoblems .... 3S 3S 41 43 51 56 57 65 VI CONTENTS. CHAPTER IV. THE PRECISION OF OBSERVATIONS. The Probable Error 68 Provable Error of the Arithmetical Mean t 70 Probable Error of the General Mean ...... 72 Laws of Propagation of Error • • 75 Probable Errors for Independent Observations .... 79 Probable Errors for Conditioned Observations .... 86 Problems ............. 87 CHAPTER V. DIRECT OBSERVATIONS ON A SINGLE QUANTI7Y. Observations of Equal Weight Shorter Formulas for Probable Error Problems 92 Observations of Unequal Weight ..,....< 95 99 CHAPTER VI. FUNCTIONS OF OBSERVED QUANTITIES. Linear Measurements ........... ioi Angle Measurements .......... J04 Precision of Areas ........... 106 Remarks and Problems „ 107 CHAPTER VII. INDEPENDENT OBSERVATIONS ON SEVERAL QUANTITIES. Method of Procedure , „ Discussion of Level Lines ......... Angles at a Station Empirical Constants Empirical Formulas Problems 109 1 10 117 124 130 139 CONTENTS. VII CHAPTER VIII. conditioned observations. The Two Methods of Procedure 141 Angles of a Triangle 142 Angles at a Station 145 Angles of a Quadrilateral 147 Simple Triangulation 152 Levelling 154 Problems 160 CHAPTER IX. the discussion of observations. Probability of Errors 162 The Rejection of Doubtful Observations 166 Constant Errors 169 Social Statistics 172 Problems 174 CHAPTER X. solution of normal equations. Three Normal Equations 175 Formation of Normal Equations 177 Gauss's Method of Solution ........ 181 Weighted Observations 187 Logarithmic Computations 190 Probable Errors of Adjusted Values ...... 195 Problems .. igS CHAPTER XI. appendix and tables. Observations Involving Non-Linear Equations . • . 200 Mean and Probable Error . •. 204 Uncertainty of the Probable Error .,..., 206 The Median 208 Vlll CONTENTS. History and Literature 211 Constant Numbers „ 214 Answers to Problems; and Notes 215 Description of the Tables , 219 TABLES. I. Values of the Probability Integral for Argument kx . 220 II. Values of the Probability Integral for Argument — . 221 r III. For Computing Probable Errors by Formulas (20) and (21) 222 IV. For Computing Probable Errors by Formulas (35) and (36) 223 V. Common Logarithms of Numbers 224 VI. Squares of Numbers 226 VII. For Applying Chauvenet's Criterion 228 VIII. Squares of Reciprocals 228 229 A TEXT-BOOK METHOD OF LEAST SQUARES. CHAPTER I. INTRODUCTION. i. The Method of Least Squares has for its object the adjustment and comparison of observations. The adjustment of observations is rendered necessary by the fact, that when several precise measurements are made, even upon the same quantity under apparently similar conditions, the results do not agree. The absolutely true values of the observed quantities cannot in general be found, but instead must be accepted and used values, derived from the combination and adjustment of the measurements, which are the most probable, and hence the best. The comparison of observations is necessary in order to determine the relative degrees of precision of different sets of measurements made under different circumstances, either for the purpose of properly combining and adjusting them, or to ascertain the best methods of observation. 2 INTRODUCTION. 1. Classification of Observations. 2. Direct observations are those which are made directly upon the quantity whose magnitude is to be determined. Such are measurements of a line by direct chaining, or measurements of an angle by direct reading with a transit. They occur in the daily practice of every engineer. Indirect observations are not made upon the quantity whose size is to be measured, but upon some other quantity or quanti- ties related to it. Such are measurements o' a line through a triangulation by means of a base and observed angles, meas- urements of an angle by regarding it as the sum or difference of other angles, the determination of the difference of level of points by readings upon graduated rods set up at different places, the determination of latitude by observing the altitude of stars, etc. In fact, the majority of observations in engineer- ing and physical science generally belong to this class. 3. Conditioned observations may be either direct or indirect, but are subject to some rigorous requirement or condition im- posed in advance from theoretical considerations. As such may be mentioned : the three measured angles in a plane tri- angle must be so adjusted that their sum shall be exactly 180 ; the sum of all the percentages in a chemical analysis must equal 100 ; and the sum of the northings must equal the sum of the southings in any traverse which begins, and ends at the same point. Independent observations may be either direct or indirect, but are subject to no rigorous conditions. Measurements on two of the angles of a triangle, for instance, are independent \ for the observed quantities can have no necessary geometrical dependence one upon the other. 4. As an illustration of these classes, consider the angles §5- ERRORS OF OBSERVATIONS. 3 AOB and BOC,- having their vertices at the same point, O (Fig. i). If a transit or theodolite be set at 0, and the angle AOB or BOC be measured, each of these measurements is a direct observation. If, however, an auxiliary station M be established, and the angles MOA, MOB, and MOC be read, the observations on AOB and BOC are indirect. Moreover, whether observed di- rectly or indirectly, the values obtained for AOB and BOC are M independent of each \ other. But if the three angles A OB, BOC, and AOC be measured, these observations are conditioned, or subject to the rigorous geomet- rical requirement, that, when finally adjusted, AOB plus BOC must equal AOC ; and no system of values can be adopted for these three angles which does not exactly satisfy this condition. Again : if the sides and angles of a field are measured, each observation taken alone is direct. If its area is found from the sides and angles, the measurement of that area is indirect. Further : any two sides considered are independent of each other ; but, if all the sides and angles be regarded, they must fulfil the condition, that, when plotted, they shall form a closed figure. Fig.1. Errors of Observations. 5. Constant errors are those produced by well understood causes, and which may be removed from the observations by the application of computed corrections. As such may be 4 INTRODUCTION. I- mentioned : theoretical corrections, like the effect of tempera- ture upon the length of rods used in measuring a base-line ; instrumental corrections, like those arising from a known dis- crepancy between the length of the rods and the standard of measure ; and personal corrections, like those due to the habits of the observer, who, in making a contact of the rods, might err each time by the same constant quantity. Strictly speak- ing, then, constant errors are not errors ; since they can always be eliminated from the observations, when the causes that pro- duce them are understood. The first duty of an observer, after taking his measurements, is to discuss them, and apply as far as possible the computed corrections, to remove the constant errors. 6. Mistakes are errors committed by inexperienced and occa- sionally by the most skilled observers, arising from mental confusion. As such may be mentioned : mistakes in reading a compass-needle by noting 5 8° instead of 42 ; or mistakes in measuring an angle by sighting at the wrong signal. ' Such errors often admit of correction by comparison with other sets of observations. 7. Accidental errors are those that still remain after all con- stant errors and all evident mistakes have been carefully inves- tigated, and eliminated from the numerical results. Such, for example, are the errors in levelling arising from sudden expan- sions and contractions of the instrument, or from effects of the wind, or from the anomalous and changing refraction of the atmosphere. More than all, however, such errors arise from the imperfections of the touch and sight of the observer ; which render it impossible for him to handle his instruments deli- cately, estimate accurately bisections of signals and small divis- ions of graduation, or keep them continually in adjustment. These are the errors that appear in all numerical observations, however carefully the measurements be made, and whose elimi- § 8. ERRORS OF OBSERVATIONS. 5 nation is the object of the Method of Least Squares. Al- though at first sight it might seem that such irregular errors could not come within the province of mathematical investiga- tion, it will be seen in the sequel that they are governed by a wonderful and very precise law, namely, the law of proba- bility. 8. The word " error," as used in the following pages, means an accidental error produced by causes which are numerous, and whose effects cannot be brought within the scope of physical investigation. This error is the difference between the true value of the observed quantity and the result of the measure- ment upon it. Thus, if Z be the true value of an angle, and M u J/j, and M 3 be the results of measurements made upon it,, the differences Z — M„ Z — M 2 , and Z — M^ are the errors. An error is denoted by the letter x, and subscripts are applied to it for particular errors ; thus, in the above case, Z — M l = x lt Z — M 2 = x 2 , and Z — M s = x v or, in general, x is the error of the observation M. A residual is the difference between the most probable value- of the observed quantity and the measurement upon it. This most probable value is that deduced by the application of the Method of Least Squares to the observations ; for instance, in the simple case of direct measurements on a single quantity, the arithmetical mean is the most probable value. The residual is denoted in general by the letter v. Thus, if z be the most probable value of an angle derived from the measurements M„ M„ and M v the residuals are z — M t = v„ z — M 2 = v 2 , and z — M 3 = v y Evidently the most probable value, z, will ap- proach more nearly to the true value Z, the greater the number of observations, as likewise the residuals v to the errors x. With an infinite number of precise observations, z should coin- cide with Z, and each v with the corresponding x. With a. large number of observations, the differences between the resid- t> INTRODUCTION. 1. uals and the errors will be small, so that the laws governing the two will be essentially the same. On this account residuals are often called residual errors, or sometimes even errors. Principles of Probability. 9. The word "probability," as used in mathematical reasoning, means a number less than unity, which is the ratio of the num- ber of ways in which an event may happen or fail, to the total number of possible ways ; each of the ways being supposed equally likely to occur. Thus, in throwing a coin, there are two possible cases : either head or tail may turn up, and one is as likely to occur as the other ; hence the probability of throw- ing a head is expressed by the fraction \, and the probability of throwing a tail also by A. So, in throwing a die, there are six cases equally likely to occur, one of which may be the ace : hence the probability of throwing the ace in one trial is \, and the probability of not throwing it is f . In general, if an event may happen in a ways, and fail in b ways, and each of these ways is equally likely to occur, the probability of its happening is -, and the probability of its failing is . Thus probability is always expressed by an a -\- b abstract fraction, and is a numerical measure of the degree of confidence which one has in the happening or failing of an event. As this measure may range from o to 1, so mental con- fidence may range from impossibility to certainty. If the frac- tion is o, it denotes impossibility ; if \, it denotes that the chances are equal for and against the happening of the event ; and if I, the event is certain to occur. 10. Unity is hence the mathematical symbol for certainty. And, since an event must either happen or not happen, the sum S I2 - PRINCIPLES OF PROBABILITY. 7 of the probabilities of happening and failing is unity. Thus, if P be the probability that an event will happen, i — P is the probability of its failing. For example, if the probability of drawing a prize in a lottery is ^VoTF" tne probability of not draw- ing a prize is \\\\, a large fraction. ii. When an event may happen in different independent ways, the probability of its happening is the sum of the separate probabilities. For if it may happen in a ways, and also in a 1 ways, and there are c total ways, the probability of its occur- rence (by Art. 9) is —it — ; and this is equal to the sum of the c probabilities — and — , of happening in the separate independent c c ways. For example, if there be in a bag twenty red, sixteen white, and fourteen black balls, and one is to be drawn out, the proba- bility that it will be red is -, that it will be white is -, and j 50' 50' that it will be black is ^. If, however, there be asked the probability of drawing either a red or black ball, the answer is 20 1 14 _ 34 50 "^ 50 50' 12. A compound event is one produced by the concurrence of several primary or simple events, each being independent of the other. For instance, throwing three aces with three dice in one trial is a compound event produced by the concurrence of three simple events. An error of observation may be re- garded as a compound event produced by the combination of all the small independent errors of the numerous accidental influences. The probability of the happening of a compound event is the product of the probabilities of the several primary inde- pendent events. To show this, consider two bags, one of which contains seven black and nine white balls, and the other four INTRODUCTION. I. black and eleven white balls. The probability of drawing a black ball from the first bag is ~ b , and that of drawing one from the second ~. What, now, is the probability of the compound event -of securing two black balls when drawing from both bags at once ? Since each ball in the first bag may form a pair with each one in the second, there are 16 X 15 possible ways of drawing two balls ; and, since each of the seven black balls may be associated with each of the four black balls to form a pair, there are 7X4 cases favorable to drawing two black balls. The required probability is hence ^') ; and this is equal to ^ X j-, or the product of the probabilities of the two primary independent events. To discuss the principle more generally, consider two primary events, the first of which may happen in «, ways, and fail in b z ways, and the second happen in a 2 , and fail in b 2 ways. Then there are for the first event a, -f- b, possible cases, and for the second a 2 -\- b 2 ; and each case out of the a, -\- <5, cases may be associated with each case out of the a 2 -f- b 2 cases ; and hence there are for the two events (a z -f- 6 S ) (a 2 -f- b 2 ) total cases, each of which is equally likely to occur. In a^a 2 of these cases both events happen ; in bjb 2 both fail ; in aj> 2 the first happens, and the second fails ; and in a 2 b^ the first fails, and the second hap- pens. Hence (by Art. 9) the probabilities of the compound events are — Probability that both happen Probability that both fail Prob. that first happens, and second fails . Prob. that first fails, and second happens . a z a 2 {a , + b I )(a 2 bj> 2 + b 2 ) (a, ■ + bi) (a, aj? 2 + b 2 ) (a, : + bi) (a, a 2 b T + b 2 ) § J 4- PRINCIPLES OF PROBABILITY. 9 As each of these probabilities is the product of the proba- bilities of the primary events, the principle is established for the case of two primary events. And evidently its extension to three or more is easy. Thus, if there be four events, and P„ P 2 , P v and P t be the respective probabilities of happening, the probability that all the events will happen is P 1 P 2 P 3 P^ ; and the probability that all will fail is (i — P,) (i — P 2 ) (i — P 3 ) (i - P A ). The prob- ability that the first happens and the other three fail is P, (i — P 2 ) (i - P 3 ) (i — P 4 ) ; and so on. 13. The most probable event among several is that which has the greatest mathematical probability. Thus, if two coins be thrown at the same time, there may arise the three follow- ing compound cases, having the respective probabilities as annexed : Both may be heads \ One head, and the other tail \ Both tails \ Here the case of one head and the other tail has the greatest probability, and is hence the most probable of the three com- pound events. The sum of the three probabilities, J, ^, and \, is unity ; as should be the case, since one of these events is certain to occur. If four measurements of the length of a line give the values 720.2, 720.3, 720.4, and 720.5 feet, the arithmetical mean, 720.35 feet, is universally recognized as the most probable value of the length of the line. It will be shown in the sequel that the mathematical probability of this result is greater than of any other. 14. A compound event, composed of any number of simple events, will now be considered. Let P be the probability of IO INTRODUCTION. I. the happening of an event in one trial, and Q the probability of its failing, so that P + Q = i : and let there be n such events. Then (by Art. 12) the probability that all will happen is P" ; the probability that one assigned event will fail, and ft — 1 happen, is P n ~ l Q ; and, since this may occur in n ways, the probability that one will fail, and n — 1 happen, is nP n ~ 1 Q. Similarly, the probability of two assigned events failing, and 71 — 2 happening, is P"~ 2 Q 2 ; and, since this maybe done in , —^- ways,* the probability that two out of the whole 2 number will fail, and n — 2 happen, is i 3 " -2 ^. If, 2 then, (P + Q)" be expanded by the binomial formula, thus, (P+ Qy = P« + nP» -' 2 ) + . . . + log/(*„). Now, the most probable values of the unknown quantities z t and z 2 are those which render P a maximum (Art. 13), and hence the derivative of P with respect to each of these variables must be equal to zero. Indicating the differentiation, the follow- ing equations result : dP = #(*,) + 4f(x 2 ) + . . . + d A x «) „ Q> Pdz, /(j,)^, /(*i)^«i ' ' A x *) dz i dP _ df{ Xl ) + d/(x 2 ) + _ _ _ + d/(x„) = q Pdz t A x ')dz* A x *)dz* '"" /(*«)<&> 24 LAW OF PROBABILITY OF ERROR. II. Since in general df(x) = ^(x)f(x)dx, these may be written *(*,)$-' + *(*,)^ + • • • + *(*.)§= = o, air, «z, az t <£(*.)$-' + 4>(x 2 )p + ... + *(*„)§= = o, az 2 az 2 az 2 and, being as many in number as there are unknown quantities, they will determine the values of those unknown quantities as soon as the form of the function 4> is known. Since these equations are general, and applicable to any num- ber of unknown quantities, the form of the function <£ may be determined from any special but known case. Such is that in which there is but one unknown quantity, and the observations are taken directly upon that quantity. Thus, if there be only the quantity z, and the measurements give for it the values M lt M 2 . . . M n , the errors are, x, = z — Mi, x 2 = z — M 2 , from which dz dz and the first equation above becomes 4>(x t ) + 4>(x 2 ) + 4>( Xi ) + . In this case, also, the arithmetical mean is the most probable value, and the algebraic sum of the residuals will be zero, or, if v denote any residual in general, Vi + v 2 + v 3 -+- . . . + v„ — o. Now, if the number of observations, n, is very large, the resid- uals v will coincide with the errors x (Art. 8), and Xi + x 2 + x, + . . . + x„ = o. . . x n — z — ■M m dx„ _ dz ■ + signifies multiplication by a constant, or when 4>(x L ) + (x 2 ) + . . . -|- (x„) = cx K + cx 2 + . . . + cx„. Replacing in this the values of <£(#,), 4>{x z ), etc., it becomes df(x.) d/(x 2 ) + / \ ' + etc. = cx 1 + cx 2 + etc. ; f{x l )dx 1 f(x 2 )dx 2 and, since this is true whatever be the number of observations, the corresponding terms in the two members are equal. Hence, if x be any error, and y =f(x), d A x ) = jfy_ = cx _ f(x)dx ydx « Multiplying both members by dx, and integrating, . log 7 = — + k', 2 Passing from logarithms to numbers, Here the constant c must be essentially negative, since the probability y should decrease as x increases numerically ; repla- cing it, then, by — 2/i 2 , and also putting e k> = k, there results (i) y = ke- hi *, which is the equation of the probability curve, or the equation expressing the law of probability of errors of observation. Discussion of the Curve y = ke "^ x *. 29. Since positive and negative values of x numerically equal give equal values of y, the curve is symmetrical with respect to the axis of Y. The maximum value of y is for x = o, when 26 LAW OF PROBABILITY OF ERROR. II. y = k; k is, hence, the probability of the error o. As x in- creases numerically, y decreases ; and when x = oo, y becomes o. The value of the first derivative is dy_ dx = —2kh*e- h2 * 2 x, which becomes zero when x = o and when x = ± oo, indicating that the curve is horizontal over the origin, and that the axis of x is an asymptote. The value of the second derivative is d z y dx 2 ■2kk 2 e- A2 * 2 (— 2k 2 x* + r), which becomes o when — 2/t 2 x 2 -f- i = o, indicating that the curve has an inflection-point when x = ± To show further the form of the curve, the following values have been computed, taking k and h each as unity : 1 I X y X y .0 1. 0000 <. ±1.8 0.0392 ±0.2 0.9608 ±2.0 0.0183 ±0.4 0.8521 ± 2.2 0.0079 ±0.6 0.6977 ±2.4 0.0032 ±0.8 °-5 2 73 L ±2.6 0.0012 ±1.0 0.3679 ±2.8 0.0004 ±1.2 0.2370 ±3.0 0.000 1 ±1.4 0.1409 ±1.6 0.0773 ±00 0.0000 §31- THE PROBABILITY INTEGRAL. 2J The curve in Fig. 4 is constructed from these values, the ver- tical scale being double the horizontal. C is the inflection- point, whose abscissa OM is 0.707. 30. The constant h is a quantity of the same kind as -, since the exponent fax 2 must be an abstract number. Methods will be hereafter explained by which its value may be determined for given observations. The probability of an assigned error x 1 decreases as h increases ; and hence, the more precise the ob- servations, the greater is h. For this reason h may be called "the measure of precision." The constant k is an abstract number; and, since it is the probability of the error o, it is larger for good observations than for poor ones. The more precise the measurements, the larger is k. The Probability Integral. 31. To determine the value of the constant k, and also to investigate the probability of an error falling between assigned limits, the following reasoning may be employed : Let x 1 , x u x z ... x be a series of errors, x? being the smallest, *, the next following, and x the last ; the differences between the successive values being equal, and x being any error. Then, by Art. 11, the probability of committing one of these errors, that is, the probability of committing an error lying between x 1 and x, is the sum of the separate probabilities k e -h*** t ke~ hI *?, etc. ; or, if P denote this sum, P=k(e~ v* + e~ v*? + e - *W + . . . + e-* 2 **), which may be written P=kt^e- h ^, the notation 2§> denoting summation from x* to x inclusive. 28 LAW OF PROBABILITY OF ERROR. II. To replace the sign of summation by that of integration, dx must be the interval between the successive values of the errors, and then the probability that an error will lie between any two limits x 1 and x is k k r r dxJx' e -k*x*dx. Now, it is certain that the error will lie between — oo and + °° > and, as unity is the symbol for certainty, k /> + ' dxJ-'a e~ h ^dx. The value of the definite integral in this expression is Hence k\fir ;„ V^* hdx' * The following method of determining this integral is nearly that presented by Sturm in his Cours d'Analyse, Paris, 1857, vol. ii. p. 16. The integral fe ~ * Zjr V;c expresses the area between the probability curve and the axis of X; and, since the curve is symmetrical to the axis of Y, that integral between the limits — 00 and + 00 will be equal to double the integral between the limits o and + 00 . Placing also hx — t, e -h*x* dx= l f e -P dtt and the integral in the second member is to be determined. Take three co-ordinate rectangular axes OT, OU, and OV, and change t into a, then A = J e t% dt = area between curve Vt T and axes, A = J e ~ " 2 du — area between curve Vull ^and axes, and jr=££e-*-*dtdu. § 32. THE PROBABILITY INTEGRAL. 20, from which the value of k is hdx The equation of the probability curve now becomes (2) y = hdxTr~*e~ h - x2 , and the probability that an error lies between any two given limits xf and x becomes (3) P^jJ\-'^dx. Equations (1), (2), and (3) are the fundamental ones in the theory of accidental errors of observation. 32. /The probability that an error l'es between the limits — x and'-f- x is double the probability that it lies between the limits O and -f- x, on account of the symmetry of the curve. ' Hence 2// (4) P=£[ e h ~*~dx Now v = e~ P is the equation of the curve VtT, and v = e~ " 2 is the equation of VuU, and, if either of these curves revolves about the axis of V, it generates from which P = 1 — ^^ 1 — — + — ^2 '-3-5 ■ t 1 lift I *? (2?)* ( 2 ^)^ etc -j From these two series the values of P can be found to any required degree of accuracy for all values of t or Ax. 32 LAW OF PROBABILITY OF ERROR. II. Then from the table the following values of P are taken : for x = i".o with to = 0.447 the area P = 0.473 for x = 2.0 with to = 0.894 the area P = 0.794 for x = 3.0 with hx = 1. 341 the area P = 0.942 for x = 4.0 with to = 1.788 the area P = 0.989 for x = s .0 with to = 2.235 the area ^P = 0.998 for x = 00 with to = 00 the area P = 1. 000 Now, these probabilities or areas P are proportional to the number of errors less than the corresponding values of x. Hence multiplying them by 100, the total number of errors, and subtracting each from that following, the number of theo- retical errors between the successive values of x is found. The following is a comparison of the number of actual and theoretical errors : Limits Actual Errors. Theoretical Errors. Differences. o".o and i".o 5 2 47 +5 1.0 and 2.0 3° 32 — 2 2.0 and 3.0 11 15 -4 3.0 and 4.0 4 5 — 1 4.0 and 5.0 2 1 + 1 5.0 and 6.0 1 + 1 6.0 and co The agreement between theory and experience, though not exact, is very satisfactory when the small number of observa- tions is considered. 34. Numerous comparisons like the above have been made by different authors, and substantial agreement has always been found between the actual distribution of errors and the § 35- REMARKS ON THE FUNDAMENTAL FORMULAS. 33 theoretical distribution required by equations (2) and (4). The following is a comparison by Bessel of the errors of three hun- dred observations of the right ascensions of stars : Limits. Actual Errors. Theoretical Errors. Differences. o J .o and o*.i 114 107 + 7 0.1 and 0.2 84 87 -3 0.2 and 0.3 53 57 -4 0.3 and 0.4 24 3° -6 0.4 and 0.5 14 J 3 + 1 0.5 and 0.6 6 5 + 1 0.6 and 0.7 3 1 + 2 0.7 and 0.8 1 + 1 0.8 and 0.9 1 + 1 0.9 and co The differences are here relatively smaller than in the previous case. And in general it is observed that the agreement be- tween theory and experience is closer, the greater the number of errors or residuals considered in the comparison. Whatever may be thought of the theoretical deductions of the law of probability of error, there can be no doubt but that its practical demonstration by experience is entirely satisfac- tory. Remarks on the Fundamental Formulas. 35. The two equations of the probability curve, (1) y = te-»* (2) y = hJx.v-le- ><***, are identical, and the former has already been discussed at 34 LAW OF PROBABILITY OF ERROR. II. length. In the latter, dx for any special case is the interval between successive values of x. For instance, if observations of an angle be carried to tenths of seconds, dx is o". i ; if to hundredths of seconds, dx is o".oi ; and if a continuous curve is considered, dx is the differential of x. As y is an abstract number, Ji.dx must likewise be abstract, and hence h must be a quantity of the same kind as — . The probability of the error dx Itdx O is — — ; thus in measuring angles to hundredths of seconds, the probability that an error is o".oo is — '— — . As this in- creases with h, the value of h may be regarded as a measure of the precision of the observations. Methods of determining k are given in Chap. IV. 36. The two probability integrals, h (3 ) P = T jy^ dx , 2 phx ( 4 ) P= r J o *-»***, are identical, except in their limits. The first gives the proba- bility that an error will lie between any two limits xf and x ; and the second, the probability that it lies between the limits — x and -\- x, or that it is numerically less than x. The second is then a particular case of the first. Table I refers only to (4) ; and from it by simple addition or subtraction the probability can be found for any two assigned limits. For example, the probability that an error lies between — 2".o and -f" 4'-° is tne sum of the probabilities for the limits o".o to 2".o and o".0 to 4".o ; and the probability that an error is between + 2".o and -f- 4".o is the difference of the probabilities of those limits. The integral P is simply the summation of the values of y § 37- PROBLEMS AND QUERIES. 3$ between the assigned limits, or P = 1y, as required by the principle of Art. 1 1 to express the probability of an error lying between those limits. 37. Problems and Queries. 1. Can cases be imagined where positive and negative errors are not equally probable ? 2. An angle is measured to tenths of seconds by two observers, and the value of h for the first observer is double that for the second. Draw the two curves of probability of error. 3. Show that the arithmetical mean of two measurements is the only value that can be logically chosen to represent the quantity. 4. The reciprocal of h for the bullet-marks in Art. 18 is 2.33 feet. Compare the actual distribution of errors with the theoretical. 5. Draw a curve for each of the equations y = ke~ x * and_y = ke~^, assuming a convenient value for k. Show that the value of k should have been taken different in the two equations. 6. Explain how the value of ir might be determined by experiments with the help of equation (2). }6 THE ADJUSTMENT OF OBSERVATIONS. ' IIL CHAPTER IIL THE ADJUSTMENT OF OBSERVATIONS. 38. The Method of Least Squares comprises two tolerably distinct divisions. The first is the adjustment of observations, or the determination of the most probable values of observed quantities. The second is the investigation of the precision of the observations and of the adjusted results. This chapter contains the development of the rules and methods relating to the first division. Weights of Observations. 39. Weights are numbers expressing the relative practical worth or value of observations. Thus, suppose a line to be measured twenty times with the same chain, ten measurements giving 934.2 feet, eight giving 934.0 feet, and two giving 934.6 feet ; then the numbers 10, 8, and 2 are the weights of the respective observations 934.2, 934.0, and 934.6 feet. Or, since weights express only relative worth, the numbers 5, 4, and 1, or any other numbers proportional to 10, 8, and 2, may be taken as the weights. The observation 934.2 has cost five times as much as the observation 934.6, and for combination with other measurements it should be worth five times as much. The weight of an observation expresses the number of stand- ard observations of which it is the equivalent. Thus the aver- age of n equally good direct measurements has a weight of n. §40. WEIGHTS OF OBSERVATIONS. 3/ the weight of each single measurement being unity. And any observation having a weight of p * may be regarded as the equivalent of p observations of the weight unity, and as having a practical worth or value/ times that of a single one. Hence the use of weights may be considered as a convenient method of abbreviation. Thus "934.2 with a weight of 10" expresses the same as the number 934.2 written down ten times, and regarded each time as a single observation. 40. A weighted observation is an observation multiplied by its weight. Thus if M u M z . . . M„ represent observations, and pi, A • • • P't their respective weights, the products p,M lt p 2 M 2 . . . p„M„ represent weighted observations. If x„ x 2 . . . x„ are the errors corresponding to M lt M 2 . . . M„, the products p^c lt p?x 2 . . . p^x n may be called weighted errors. As an error x is the difference between the true and measured value of the quantity observed, the product px cannot occur without implying that the corresponding observation M has a weight of p, and the same is true for the residual error v. Thus if there be two unknown quantities ?, and z„ and a measurement M be made upon f(z„ z 2 ), the residual error is v=f(z s ,z 2 ) - M if z s and z 2 denote the most probable values of the unknown quantities. Now, if the observation M be weighted with /, the residual is pv=p.f(z l ,z 2 ) -pM. Hence a weighted observation always implies a weighted re- sidual, and vice versa. The weights should be carefully distinguished from the meas- ures of precision introduced in the last chapter. The former * p is the initial of " pondus.'' 38 THE ADJUSTMENT OF OBSERVATIONS. III. are relative abstract numbers, usually so selected as to be free from fractions, while the latter are absolute quantities. The relation between them will be shown in Art. 43. The Principle of Least Squares. 41. The principle from which the term "Least Squares" arises is the following : In measurements of equal precision the most probable values of observed quantities are those that render the sum of the squares of the residual errors a minimum. To prove this, consider the general case of indirect observa- tions, and let n equally good measurements be made upon func- tions of two unknown quantities z, and z 2 . Let M u M 2 . . . M H be the results of the measurements of the functions f 1 (z 1 , z 2 ), fz{z™ ^2) • • -fn( s i, z 2 ). These measurements will not give ex- actly the true values of the functions, and the difference between the observed and true values will be small errors, x lt x 2 . . . x m or /.(z., z 2 ) —M,. = x„ / 2 (z„ z 2 ) — M 2 = x 2 . . ./„(z„ z 2 ) — M n = x„. The respective probabilities of these errors are by the fun- damental law (1) y t = ke~***i 2 , y 2 = ke~ h x * . . . y H — ke~ h x * > h being the same in all, since the observations are of equal precision. Now, by Art. 12, the probability of the compound event of committing the system of independent errors x„ x t . . . x„ is the product of these separate probabilities, or Each of these errors is a function of the quantities z t and z„ vhich are to be determined. Different values of z, and z t will § 42. THE PRINCIPLE OF LEAST SQUARES. 39 give different values for P'. The most probable system of errors will be that for which P' is a maximum (Art. 13), and the most probable values of z t and z 2 will correspond to the most probable system of errors. The probability P' will be a maximum when the exponent of e is a maximum ; that is when x? + x 2 2 + x 2 2 + . . . + Xn = a minimum. Hence the most probable system of values for z t and z 2 is that which renders the sum x* -f- x 2 + x % + • • • + x n a minimum, and the fundamental principle of Least Squares is thus proved. The errors x It x 2 . . . x„ have been thus far regarded as the true errors of the observations. As soon, however, as they are required to satisfy the condition that the sum of their squares is a minimum, they become residual errors (Art. 8), so that the condition for the most probable values of z l and z 2 is really (5) v * + v * + v i + • • • + v n — a minimum ; that is to say, if z t and z 2 be the most probable values, the com- puted residuals /i(s« z 2 ) — M 1 = »„ / 2 (z I; z 2 ) — M 2 = v 2 . . ./„(«„ z 2 ) — M— v„ will be those that satisfy the condition for a minimum. The above reasoning evidently applies to any number of unknown quantities as well as to two. 42. The more general case of the Method of Least Squares, however, is that when the observations have different degrees of precision, or different weights. In that event the general principle is the following : — In measurements of unequal weight the most probable values of observed quantities are those that render the sum of the weighted squares of the residual errors a minimum. 40 THE ADJUSTMENT OF OBSERVATIONS. III. As before, let n observations, M„ M 2 . . . M„, be made upon functions of two unknown quantities, z, and z 2 ; and let p l} p 2 . . . p„ be the respective weights of M„ M 2 . . . M„. The differences between the observations and the true values of the functions are errors, x„ x 2 . . . x n ; and the respective probabilities of these errors are y, = k x e~ h W, y 2 = k 2 e ->>*'** . . . y n = k n e~K^n% in which k and h are different for each observation. The prob- ability of the system of independent errors, x lt x 2 . . . x n , then, is P' = kX . . . &„e-(*' 2 *i' +h *'x*'+ ■ . + V^>- and the most probable system of values is that for which P' is a maximum, or that which renders ■ hfx? + h 2 2 x 2 2 + . . . + h„ 2 x„ 2 = a minimum. The values of x l} x 2 . . . x n , derived from this condition, are the residual errors, v u v 2 . . . v„ ; so that it will be well to write at once h*vf + h 2 2 v 2 2 + . . . + h n 2 v n 2 = a minimum. This expression may be divided by h 2 ; k being a constant standard measure of precision so selected, that h 2 = pji 2 , h 2 = pji 2 . . . h„ 2 = p n h 2 , where /„ p 2 . . . p n are whole numbers, which are the weights of the observations M lt M 2 . . . M„* Then it becomes (6) p l v 1 2 + p 2 v 2 2 + . . . + p„v„ 2 = a minimum ; * To show that these numbers are the weights of M v M 2 . . . M„, consider that the condition for the minimum will be fulfilled when dv T dv 2 dv n h * v * d7 z + h2V2 d7 z + • ■ ■ + h " v « d^ = > dv z dv 2 dv n h * v * d7 2 + h * v * w, + ■ • • + A « v " 7Z = °> §44- DIRECT OBSERVATIONS. 4 1 which is the principle that was to be proved. The term "weighted square" means simply v 2 multiplied by the weight /, or the product pv 2 . The conditions expressed by (5) and (6) are the fundamental ones for the establishment of the practical rules for the adjust- ment of independent observations. If the observations are of equal weight, the general condition (6) reduces to the special one (5). 43- It is here seen that the squares of the measures of pre- cision of observations are proportional to the weights, or that (7) h 2 :h 2 : k 2 : : p, : p 2 : p. The measure of precision is never used in the practical ap- plication of the Method of Least Squares, while weights are constantly employed. The quantity h, however, is- very con- venient in the theoretical discussions, and will be needed often in the next chapter : h represents an absolute quantity, while p denotes always an abstract number. Direct Observations on a Single Quantity. 44. When the observations are of equal precision, and made directly on the quantity whose value is sought, it is universally recognized that the arithmetical mean is the most probable which, after dividing by the standard h 2 , become dv z dv^ dv„ t* v - ~dJ x + t* v * ^ + ■ • ■ + P» v * 1^ = °' dv r dv 2 dv n Ma, + A", ^ + • • • + *»*n-& t = o. Here the residual v I is repeated p 1 times, v 2 is repeated p? times, and v„ is repeated f H times, and hence p v p? . . . p„ are the weights of the corresponding observations M v M t . . .M„ (Art. 40). 42 THE ADJUSTMENT OF OBSERVATIONS. 111. value of the quantity. This may be also shown from the funda- mental principle of Least Squares in the following manner : Let M [t M 2 . . M„ denote the direct observations which are all of equal weight or precision. Let z be the most probable value which is to be determined. Then the residual errors are z — M„ z — M 2 . . . z — M n , and from the fundamental principle (5) (z — M t Y + (z — M 2 y + . . . + (z — M„Y = a minimum. To apply the usual method for maxima and minima, place the first derivative of this expression equal to zero, thus 2(2 — M,) + 2(z — M 2 ) + . . . + 2(2 — M„) = o. Dividing this by 2, and solving for 2, gives (8) M T + M 2 + M l + . . . + M n that is, the most probable value z is the arithmetical mean of the 11 observations. The adjustment of direct observations of equal weight on the same quantity is hence effected by taking the arithmetical mean of the observations. 45. When the measurements of a quantity are of unequal weight or precision, the arithmetical mean does not apply. Here the more general principle (6) will furnish the proper rule to employ. Let the measurements be M„ M 2 . . . M m having the weights p„ p 2 . p n . Then, if z be the most probable value of the observed quantity, the expression (6) becomes /,(« — M x y +p 2 (z — M 2 y + . . . +p n (z — M„y = a minimum. §46. OBSERVATIONS OF EQUAL WEIGHT. 43 Placing the first derivative of this equal to zero gives p x {s - M^ + AO -M 2 ) +... + p n (z-M n ) = o, the solution of which is ( x „ _ pjM, +P*M 2 + ... + p„M„ Pi+P*+ ■■ ■ +pn that is, the most probable value of the unknown quantity g is obtained by multiplying each observation by its weight, and dividing the sum of the products by the sum of the weights. In order to distinguish this process from that of the arithmeti- cal mean, it is sometimes called the general mean, or the weighted mean. s Granting that the arithmetical mean gives the most probable value for observations of equal weight, the general mean (9) for observations of unequal weight may be readily deduced from the definitions of the word "weight " in Art. 39. The adjustment of direct observations of unequal weight on the same quantity is hence effected by taking the general mean of the observations. Independent Observations of Equal Weight. 46. The general case of independent observations comprises several unknown quantities whose values are to be determined from either direct or indirect measurements made upon them. An "observation equation" is an equation connecting the observation with the quantities sought. Thus, if M be a meas- urement of f{z s , z 2 ), the equation M=f(g„ z 2 ) is an observation equation. The number of these equations is the same as the number of observations, and generally greater than the number of unknown quantities to be determined. Hence, in general, 44 THE ADJUSTMENT OF OBSERVATIONS. III. no system of values can be found which will exactly satisfy the observation equations. They may, however, be approximately satisfied by many systems of values ; and the problem is to deter- mine that system which is the most probable, or which has the maximum probability (Art. 13). Fig.5. To illustrate, consider the following practi- " " z, cal case. Let O represent a given bench- ? ..,---'''; mark, and Z u Z z , Z v three points whose 1 _..-" s "' f elevations above are to be determined. z"—i- / Let five lines of levels be run between » — -6 z 3 these points, giving the following results : Observation 1. Z, above O = 10 feet. Observation 2. Z 2 above Z, = 7 feet. Observation 3. Z 2 above O = 18 feet. Observation 4. Z 2 above Z l = 9 feet. Observation 5. Z 3 below Z t = 2 feet. If the elevations of the points Z x , Z 2 , and Z v be designated by z 1 z„ and z v the following observation equations may be written : 2, = IO, z 2 — z L = 7, z 2 = 18, z 2 — z 3 = 9, Z L S3 — 2, each one of which is an approximation to the truth, but all of which cannot be correct. The number of these equations is five, the number of the unknown quantities is three ; and hence an exact solution cannot be made. The problem is to find the most probable values of z lt z 2 , and z 3 . The observation equations may be algebraic expressions of the first, second, or higher degrees ; or they may contain circular or logarithmic functions. Usually, however, they are of the first degree, or linear, and these alone will be considered in the §47- OBSERVATIONS OF EQUAL WEIGHT. 45 body of this work. In Art. 140 is given a method by which non-linear equations, should they occur, may always be reduced to linear ones. 47. Consider first the case of observations of equal precision or of equal weight. Let there be q unknown quantities z„ z 2 . . . Zg, and let the equations between them and the measured quantities be of the form az I + bz 2 + . . . + k q = M, in which a, b ... I are constants given by theory and absolutely known, and M the measured quantity. For each observation, there will be a similar equation, and, in all, the following n approximate observation equations : a 2 z, + b 2 z 2 + . . . + l 2 z q = M 2 , tfjZ, + b 3 z 2 + . . . 4- / 3 z ? = M 3 , a n z 1 + b^ 2 + . . . + l n z q = M„, the first of which arises from the first observation, the second from the second, and the last from the » th . Now, as the number of these observation equations is greater than that of the unknown quantities, they will not be exactly satisfied for any system of values that may be deduced. The best that can be done is to find, from the fundamental principle of Least Squares, the most probable system. Let z lt z 2 . . . z q denote the most probable values, then, if these be substituted in the observation equations, they will not reduce exactly to zero, but leave small residuals, v„ v % . . . v n ; thus strictly a l z 1 + b,.z z + . . . + l t Zq — Mi = z>„ 02«I + 1> 2 Z 2 + . . . + l 2 Zg — M 2 = V 2 , «»Zi + b„Z 2 + . . .+ Irfiq — M n = V n . 46 THE ADJUSTMENT OF OBSERVATIONS. III. The fundamental principle established in Art. 41 is, that the most probable values, z„ z 2 . . . a q , are those that render z'i 2 + v 2 + v z 2 + . . . + Vn = a minimum. Consider first what is the most probable value of the un- known quantity z u and denote the terms in the above equa- tions independent of z x by the letters N lt N 2 , N v etc. Then they become a l z I + N t = v lt a 2 z l + N 2 = v 2 , a„z x + N n = v„. Squaring both terms of each of these equations, and adding the results, gives (a.z, + JV i y+(a z z l +JV 2 y+. . . + (a n s t + N„y=v? + v 2 * + ... + v n \ In order to make this sum a minimum, its first derivative must be put equal to zero, giving c,(«A + N,) + a 2 (a 2 z, + JV 2 ) + . . . + a K {a„z 1 + N„) = o ; and this is the condition for the most probable value of #,. In like manner a similar condition may be found for each of the other unknown quantities. The number of these conditions, or "normal equations" as they are called, will be the same as that of the unknown quantities, and their solution will furnish the most probable values of z it z 2 . . . z q . 48. The following is, hence, the method for the adjustment of independent indirect observations of equal weight : For each observation write an observation equation. Form a normal equation for z, by multiplying each of the observation equations by the co-efficient of z, in that equation, and adding §48. OBSERVATIONS OF EQUAL WEIGHT. A7 the results. And, for each unknown quantity, form a normal equation by multiplying each observation equation by the co- efficient of that unknown quantity in that equation, and adding the results. The solution of these normal equations will fur- nish the most probable values of the unknown quantities. For example, let the five observation equations derived from the five observations of Art. 46 be considered, namely, z i = 10, — z t +z 2 =7, z 2 = 18, j 2 — z 3 = 9, Zl — Z 3 = 2. To form the normal equation for ^, the first observation equation is multiplied by -\- I, the second by — 1, the third by O, the fourth by o, and the fifth by -\- 1 ; the addition of the products then gives 3Z1 — z 2 — z 3 = 5. The normal equation for z 2 is formed by multiplying the first observation equation by o, the second by -f- 1, the third by -f- 1, the fourth by -f- I, and the fifth by o ; the sum of the products being — z, + 3«, — H = 34- The normal equation for z 3 is formed by multiplying the first, second, and third observation equations by o, and the fourth and fifth by — 1, the addition of which gives — Z, — Z 2 + 22 3 = — II. These three normal equations contain three unknown quanti- ties, and their solution gives z t = + iof, z 2 - + i 7 |, z 3 = + &i, 48 THE ADJUSTMENT OF OBSERVATIONS. III. which are the most probable values that can be obtained from the five observations. If now these values be substituted in the observation equations there will be found the five residuals, Vl = + 4, V2-+h Z< 3 = - I, V 4 =+h » 5 = - h and the sum of the squares of these is ■§. Of all the possible values that might be assigned to z„ z 2 , z v those above found give the minimum sum of squares of residual errors. As a second example, let three observations on the two quantities z s and z 2 give the observation equations 32, — 5z 2 = + 12.4, — 2Z, + 4Z 2 = — 10.2, Z l — 2Z 2 = -f- 8.0. To form the normal equation for 0, the observation equations are multiplied by + 3, — 2, and -\- 1, respectively, and the re- sults added. To form the normal equation for z 2 the multipliers are — 5, +4, and —2, respectively. The two normal equa- tions thus are 14^ — 2$z 2 — + 65.6, — 252, +45 2 * = — ll8 - 8 » and the solution of these gives the most probable values z l = — 3.60 and z 2 = — 4.64. 49. In order to put the above method for the formation of normal equations into algebraic language, let there be n obser- vations upon q unknown quantities which lead to the following observation equations : tfiZr + bi% 2 -)- ^,z 3 -f- . . . 4- hz q = M-l, a 2 z t 4- b 2 z 2 -\- c 2 z 3 -\- . . . 4- te q = M 2 , a„z, + b n z 2 4- s + aJ> 2 -\- a$h + • ■ ■ + a n b„, [at] = a A + a 2 l 2 -f- a,/ 3 -)-...+ a„I„, [3b] = b? +b. 2 + b 3 2 +... + *,«, [a M] = a 1 M 1 + a 2 M 2 -j- a 3 M 2 + . . . -f a n M„, and then the normal equations may be thus written: [aa]z, + [ab]z 2 -f [ac]z 3 + . . . + [_al]z q = [aM], [ba] Zl + [bb]z 2 + [bc]z 3 +... + [bl]z q = [bM\ (11) [ca\z, + [cb]z 2 + [cc\z 3 + • • • + \cl\z q = [cM\, [Ia\ Zl + [lb\z 2 + [k]z 3 + . . . + [//]% = [IM\ The co-efficients of the unknown quantities in these normal equations present a curious symmetry ; those of the first hori- zontal row being the same as those of the first vertical column, those of the second row the same as those of the second col- umn, and so on. This is due to the fact that [ba] is the same as [ab], [ca] the same as [ac], . . . and [la] the same as [al]. The notation for summation here indicated is that first used by Gauss and since generally employed in works on the Method of Least Squares in writing normal equations. The notation 2a 2 , 2ab, used by a few writers, and in former editions of this 50 THE ADJUSTMENT OF OBSERVATIONS. III. book, has the same meaning as [ad], [ab~\. The sum of the squares of the residual errors may be written either 2if or \yv~\, and in this book the former will be employed as it more readily calls to mind its name. 50. Hence the method of adjustment of indirect observa- tions of equal weight is to write for the n observations the n observation equations (10), then to form the q normal equations (11), and their solution will furnish the most probable values of the unknown quantities. Numerous examples of the appli- cation of this method will be found in Chap. VII. As a simple illustration let three observation equations be 4S1 — 2Z2 = + 6.i, 52, + 2Z 2 = + 3.8, 3Zi — $z 2 = — 0.9. Here a, = + 4, a 2 = + 5, a 3 = + 3, b 1 — — 2, b 2 = + 2, b, = - 3, M T = + 6.1, M 2 = + 3.8, M 3 = - 0.9. The forma- tion of the sums is now made, carefully regarding the signs of the co-efficients ; thus, [aa\ =+ 4 2 +5 2 + 3 2 = + 5°-°. \ab\ = — 8 + 10 — 9 =— 7.0, [aM] = + 24.4 + 19.0 — 2.7 = -f 40.7, \bb\ = 2 2 + 2 2 + 3 z = + i7-°, \bM~\ = — 12.2 -|- 7.6 + 2.7 = — 1.9. Here [ba] need not be computed, as its value is the same as [ab~\; thus the two normal equations are + 5 02; i — iz 2 = + 40.7, — 72, + 172, = — 1.9, the solution of which gives «, = + 0.8472 and z 2 = -\- 0.2371 «s the most probable values correct to the fourth decimal place. § 5 2 - OBSERVATIONS OF UNEQUAL WEIGHT. 51 Independent Observations of Unequal Weight. 51. The more usual case in practice is where the observa- tions have unequal weights. As weights are merely numbers denoting repetition, it is plain that if each observation equa- tion be written as many times as indicated by its weight, the reasoning of Art. 47 and the rule of Art. 48 applies directly to the determination of the probable values of the unknown quantities. Instead, however, of writing an observation equa- tion as many times as indicated by its weight, it will be sufficient to multiply it by its weight when forming the other products. 52. The following rule may hence be stated for the adjust- ment of independent observations of unequal weight upon several related quantities : For each weighted observation write an observation equa- tion, noting its weight. Form a normal equation for z t by multiplying each equation by the co-efficient of z 1 in that equa- tion, and also by its weight, and adding the products. In like manner form a normal equation for each of the other unknown quantities by multiplying each observation equation by the co- efficient of that unknown quantity in that equation, and also by its weight, and adding the results. The solution of these normal equations will furnish the most probable values of the unknown quantities. For example, let three observations upon two unknown quantities give the three observation equations, — 221 + 3Z 2 = -f 6, weight 3, + 22, = + 3. weight 7, — 3 Z 2 = + 5. weight 2. To form the normal equation for z I the first equation is multi- plied by the co-efficient — 2 and by the weight 3, that is, by 52 THE ADJUSTMENT OF OBSERVATIONS. III. — 6; the second is multiplied by -\- 2 and 7, that is, by -j- 14; the third is multiplied by o and 2, that is, by o ; the addition of the products gives + 402, — i8z 2 = + 6. To form the normal equation for z 2 , the first equation is mul- tiplied by -f- 3 and by 3, that is, -f- 9 ; the second by o, and the third by — 6; the sum of products being — 182, + 4522 = + 24. The solution of these two normal equations gives z l = -f- O.475 and z 2 = -f- 0.724 as the most probable values of the two quantities which were indirectly observed. 53. In order to put this method into an algebraic algorithm and at the same time review the general reasoning, let M„ M 2 , . . . M„ be the results of the n observations which have been made to determine the values of the q quantities z lt z 2 , . . . z q . As before, let each observation be represented by an observa- tion equation, thus : a 1 z l -\- 6iZ 2 + . . . + hz q = M, with weighty, , x «2Zi + b 2 z 2 + . . . + hz q = M 2 with weight p 2 , K 12 ) a«?i + b n z 2 -\- . . . + l n Zq — M n with weighty. Now, if z 1 ,z 2 ...z q denote the most probable values of the quantities sought, and these values be substituted in (12), these equations will not reduce to zero, but leave small residuals, v u v 2 . . . v„. Thus strictly, a s z, + biz 2 + . . , + hz q — Mj — v t with weight/,, a 2 zi + b 2 z 2 + . . . + l 2 z q — M2 = v 2 with weight p 2 , a„zi+ b H z 2 + . . . + lnz q — M„ = v H with weight/,, which may be called residual equations. § 53- OBSERVATIONS OF UNEQUAL WEIGHT. $$ Now, according to the general principle (6), the most probable values of the q unknown quantities z„ z 2 . . . z g are those that render the expression piVi* + p 2 v 2 2 + • • • + p n v„* = a minimum. To abbreviate, designate this quantity by 2pv 2 . Remembering that v*, v 2 . . . v 2 are functions of z„ z 2 . . . z q , it is plain that the derivative of ~2,pv 2 with reference to each variable must be zero, and that hence there are the following q conditions for the minimum : . dv 1 dv 2 , dv n A», — + p 2 v 2 — + ...+ p„v„ — = o, az t aZj aZj . dv 1 dv 2 , , , dv n plVi — - +p 2 V 2 -—+...+ p n Vn -J- = 0, dz 2 dz 2 dz 2 dv, , . dv 2 , , , dv n P1V1- 1 +p 2 v 2 --± + . . . +p„v n 7- = o. dz q dz q dz q The values of the differential co-efficients in these conditions are readily found by taking the derivatives of the residual equations with reference to each variable, thus : dv x dv 2 dv t , -=«., -j 1 = a 2 , — = by, etc. ; dz y dZt az 2 and the conditions then become /,«,#, + p 2 a 2 v 2 4- . . . + pna n v n = o, p 1 b l v l + p 2 b 2 v 2 + . . . + p„b„v„ = o, PJ,V* + PJ 2 V 2 + • ■ ■ + pJnVn = O, which are as many in number as there are unknown quantities z u z 2 . . . z g . If in these the values of v l} v 2 . . . v n be replaced 54 THE ADJUSTMENT OF OBSERVATIONS. HI. from the residual equations, the final normal equations will result. As before, the expression of the normal equations may be abbreviated by using the square bracketed notation for summation, namely, [paa] = p 1 a I ° + p 2 a 2 ^ -\- . . . -\- p„a n 2 , \_pab] = pidibt + p 2 a 2 b 2 -\- . . . -\- p n a n bn , [paM] = p^Mi -\-p 2 a 2 M 2 + . . . +p»a„M„, etc., and thus the normal equations are [paa]z t + [pab]z 2 -f- , . , -f [pal]z q = [paM ], [pba\ Zl + [pbb\z 2 +...+ [pbr[z q = [pbM], U3) [p?a]sz + [pZb]z 2 +...+ [/>//>, = lplM\ by whose solution the most probable values of z L , z 2 . . . z q may be found. The co-efficients in these equations show the same symmetry as those in Art. 48, since, as before, [pba], \_plb~], etc., are the same as \_pab~], [pbl], etc. 54. Thus, if there be 11 observations for determining q un- known quantities, the most probable values of the unknown quantities are obtained by writing n observation equations as in (12), and forming the q normal equations as in (13); then the solution of these normal equations will furnish the most probable values of the q unknown quantities. In the most common cases the co-efficients in the observation equations (12) are -+- I, — 1, or o, and, in the formation of the co-effi- cients of the normal equations, the signs must be carefully regarded. Many examples of adjustment by this method are given in Chap. VII. As a simple illustration let there be given tht following four weighted observation equations upon the two quantities z, and z, : § 54- SOLUTION OF NORMAL EQUATIONS. 55 No. i. + z t = o, weight/, = 8, 2. + z t = o, / 2 = 10, 3. + ft + 2Z 2 = + O.25, p 3 = I, 4- + z i — 3 Z * = — °-9 2 > A = 5> These co-efficients and weights, arranged in tabular form, are No. a b M P 1. + 1 8 2. + 1 10 3- + 1 + 2 + 0.25 1 4- + 1 -3 — O.92 5 The products paa, pab, etc., are now formed as below, and their summation furnishes the co-efficients for the two normal equations ; thus, No. paa pab paM pbb pbM 1. + 8 O O 2. O O + IO 3- + 1 + 2 + O.25 + 4 + 0.5& 4- + 5 - 15 — 4.60 + 45 + 13-8° + I 4 — 13 - 4-35 +59 + 14-3° Here [pba] is the same as [pab] and need not be computed. The normal equations therefore are + I4 21 ! — 13^ = — 4-35. — i3 2 i + 59^ = + i4-3°> the solution of which gives #, = — o. 102 and z 2 = -^0.225 a? the most probable values that can be derived from the four observations. If these be substituted in the observation equa. tions the adjusted values of the four observations are found to be — 0.102, 4-0.225, -fo.348, and — 0.777. 56 THE ADJUSTMENT OF OBSERVATIONS. HI. In the formation of the co-efficients of normal equations tables of squares, multiplication tables, and calculating ma- chines will often be found very useful. The method of using the table of squares at the end of this volume for the forma- tion of the products ab, ac, etc., is explained in Art. 172, and a method for checking the correctness of the co-efficients \ab\, [ac], etc., is given in Art. 142. Solution of Normal Equations. 55- The normal equations which arise in the adjustment of observations may be solved by any algebraic process. When the co-efficients consist of several digits, or when the number of unknowns is greater than three, it is desirable to follow a method by which checks may be constantly obtained upon the accuracy of the numerical work. Such a method, devised by Gauss, is presented in Chap. X. When the number of unknown quantities is two, the obser- vation equations furnish the two normal equations \aci\zv -\- [ab]z 2 = [aM], [a6] Zl + [bb]z 2 = \bM\ the solution of which may be directly effected by the formulas _ [bb]\aM] - [ab][bM] Zl ~~ [aa] [bb\ - [abf ' _ [aa\[bM] - [ab][aM] Z2 ~ [aa\[l>bY-[abf ' while checks upon the numerical work may be obtained by substituting the computed values in the given normal equa- tions which should be exactly or closely satisfied. When logarithms are used it will generally be advantageous to write the formulas thus §54- CONDITIONED OBSERVATIONS. $7 [i>o][aM]/[ab] - [i>M] Zl ~ [aa][i6]/[ao] - [ab] ' v _ [aa][oM]/[al>] - [aM] Z2 ~ [aa][diq/[ai] - [ad] ' as then the table need be entered only three times in finding the numbers corresponding to two terms in the numerator and one in the denominator, whereas by the former formulas six entries are required. As an example let the two normal equations be 9o.o7z x + 404.56^ = 295.99, 404.562! -\- 1934.10Z2 = 1306.90. Here, by the use of either numbers or logarithms, the solution gives the values / zi = + 4.1527, z 2 =— 0.1929, which, substituted in the normal equations, reduce the first to + 0.004 = o and the second to + 0.028 = O. The first is satisfied as closely as the data admit, while the error in the second can be reduced, if deemed necessary, by carrying the values of z t and z 2 to five decimal places. When the number of unknown quantities is three, general formulas for solution are best derived in the determinant form given in Art. 140. This determinant method is easily remem- bered and may be advantageously used for the case of two unknown quantities. Conditioned Observations. 56. Thus far it has been considered that the quantities to be determined by observation were independent of each other. Although they have been related to each other through the 58 THE ADJUSTMENT OF OBSERVATIONS. III. observation equations, and have been required to satisfy ap- proximately those equations, they have been so far independent, that any one unknown quantity might be supposed to vary without affecting the values of the others. All systems of values of the unknown quantities have been regarded equally possible, and the methods above developed show how to determine the most probable system. In the class of observations now to be discussed, all systems of values are not equally possible, owing to the existence of conditions which must be exactly satisfied. Thus, having measured two angles of a triangle, the adjusted value of one is entirely independent of that of the other ; but, if the third angle be measured, the three angles are subject to the rigor- ous geometrical condition that their sum must be exactly 180 . In conditioned observations there are, hence, two classes of equations, observation equations and conditional equations ; the number of the first being generally greater than the number of unknown quantities, and that of the latter always less.* 57. Designate the number of observation equations by n, the number of unknown quantities by q, and the number of condi- tional equations by «'. If no conditional equations existed, the principle of Least Squares (6) would require that the adjusted system of values should be the most probable for the n inde- pendent observations. The ;/ conditional equations, being less in number than the q unknown quantities, may be satisfied in various ways ; and, further, the final adjusted system of values must exactly satisfy them. Hence it must be concluded, that, of all the systems of values which exactly satisfy the n' condi- tional equations, that one is to be chosen, which in the n * In most books upon this subject, the term " equations of condition " is applied indiscriminately to both of these very distinct classes, and is a cause of some per- plexity to the student. The excellent distinction of the Germans, " Beobachtungs- gleichung " and " Bedingungsgleichung," ought certainly to come into use. § $8. CONDITIONED OBSERVATIONS. 59 observation equations makes the sum of the weighted squares of the residuals a minimum. The problem of conditioned observations may be, then, reduced to that of independent ones by finding from the ri conditional equations the values of ri unknown quantities in terms of the remaining q — ri quantities, and substituting them in the n observation equations. There will thus result n inde- pendent observations upon q — ri quantities. From these the normal equations are to be formed, and the most probable values of the q — ri quantities deduced. Substituting these values in the ri conditional equations, the remaining ri quan- tities become known. Thus the q quantities exactly satisfy the conditional equations, and at the same time are the most probable values for the observation equations. This, therefore, is a general solution of the problem. For example, consider the measurement of the three angles of a plane triangle. Let s„ z t , and z 3 be the most probable values of the angles, and let the observation equations be z 1 = M„ z 2 = M 2 , z 3 = M 3 , which are subject to the rigorous condition Zi + 2 2 + 2 3 = 1 8o°. From the conditional equation take the value of z 3 , and substi- tute it in the observation equations, giving z, = M„ z 2 = M 2 , z, + z 2 = i8o° — M r The most probable values of z, and z 2 may be now obtained by the method of Art. 47, since the three observation equa- tions are independent. Then the most probable value of z 2 is 1 8o° — z 1 — z 2 . 58. Although the above method is perfectly general, and very simple in theory, it gives rise in practice to tedious computa- 60 THE ADJUSTMENT OF OBSERVATIONS. III. tions whenever the number of conditional equations is large. The process generally employed is the " method of correlatives," due to Gauss, which will now be explained ; the conditional equations being considered as of the first degree, or linear, and the number of observations being the same as that of the quantities to be determined, or n = q. Consider q unknown quantities connected by the n' rigorous conditions, A, + /3,Zi + &z 2 + . . . 4- fi q z q = o, (14) K + K%i + Kz 2 + ■ • ■ + Vv Let M It M 2 . . . M q be the values found by the observations for z lt z 2 . . . z q . If these be inserted in the conditional equations, they will not reduce to zero, but leave small discrepancies, d„ d 2 . . . d n >, thus : Uo + U..M, 4- a z M 2 4- . . . 4- a q M q = d„ A, 4- p,Af, + p 2 M 2 + ...+ $ q M q = d 2 , K + KM 1 + KM 2 + . . . 4- \ q M q = d„>. Let v lt v 2 . . . v q be corrections, which when applied to M lt M 2 . . . M q , will render them the most probable values, and cause the discrepancies to disappear ; thus, if z It z 2 . . . z q be the most probable values, z, = M, + v„ z 2 = M 2 4- v 2 . . . z q = M q + v q . Then the insertion of these in (14) gives the reduced condi- tional equations a,Z\ + a 2 V 2 4- ... 4- a q V q 4- -[f>' + -4T>- + 2 =+o.2o, v 3 = — 0.55, » 4 =+o.is, ^=-0.15, and the final adjusted values are 2, = 10.65, s 2 = 6.8o, z 3 - 17.45, z„ = 9.35, z 5 = 2.55, which exactly satisfy the two conditional equations. This problem may be reduced to one of independent obser- vations by eliminating two of the unknowns by means of the conditional equations. Thus, if z 3 and z s be eliminated the observation equations are z t = 10.1, z l -\-z 1 = 18.0, ^ = 9.2, _ 2 . -}- z = 2.7, all with weight 1, and l 2 — 6.6 with weight 2. 64 THE ADJUSTMENT OF OBSERVATIONS. III. 58^. A valuable check upon the solution of the correlative normal equations is given by the necessary relation, 2pv 2 + [AV] = o, which may be proved in the following manner: Let the first equation in (14)' be- multiplied by K u the second by K 2 , the last by K n ,, and let the results be added, giving (o-.A-, + /3,JST, + ... + KK n ,) Vl + A'A + (a 2 K x + fi 2 K 2 + ...+ \ 2 K n )v 2 + K 2 d 2 + {a q K, + I3„K 2 + ...+ \K n ,)v q + K„d n , = o, Now, as shown on page 61, the coefficients of v t , v„ . . . v q , in this equation are pj>„ p 2 v 2 , . . . p q v q . Hence {p) 4- {KA +KA + ... + K n ,d n .) = o, which may be abbreviated as is done above. Thus, if the residuals be computed from the observation equations, thi? relation furnishes a check on the numerical work. For instance, in the example of the preceding article, the values of the residuals v have been found. Then 2pv 3 = 0.3025 + 0.0800 + 0.3025 + 0.0225 + 0.0225 = + °-73 Further, the values of K and d are Ki = + 0.55, K 2 = - 0.15, d, ~ — 1.3, d 2 — -\- 0.1. and accordingly [AV] = + o.ss(- 1.3) - 0.15(4- 0.1) = - 0.73. The necessary relation 2pv 2 + [Kd] = o is hence exactly sat- isfied, and the numerical work may be regarded as correct. § 59- PROBLEMS. 65 59- Problems. 1. Six indirect observations upon two quantities furnish the fol- lowing observation equations: •f*i = + 301, + 202 = — I.20, - 0, + 3s 2 = - 4.65, + 2;;,- S3 = + 6.51, 2l+ «2 = +2.35, %— S2 = + 3-70. Form the two normal equations and find the most probable values of s, and z 2 . 2. The bearing of a line is taken five times with a solar compass, giving the values A. 12' £., A*. 7' £., N. 10' W., jV. 2' IV.. N. 2'.(jE. What is the adjusted bearing of the line if the weight of the last observation is five times that of each of the others ? 3. Solve the following normal equations: 20! — z 2 +0.52 = 0, — ai + 402— 3 — 4 —0.26 = 0, — s 2 + 223 — 04 + 0.47 = o, 2 03 + 3S 4 — 05 — 1 .08 = O, - H + 325 + °-34 = 0. 4. A plane triangle has the angle A measured ten times, B meas- ured five times, and C measured once. The sum of the three ob- served values is found to differ d seconds from 180 degrees. How shall this d be divided among the three angles ? 66 THE PRECISION OF OBSERVATIONS. IV. CHAPTER IV. THE PRECISION OF OBSERVATIONS. 60. In the adjustment of observations, it is often necessary to combine measurements of different degrees of precision ; and for that purpose the determination of their weights is neces- sary. When the most probable or adjusted values have been obtained, it is also well to know what degree of confidence may be placed in them, so that comparisons may be made with values obtained under other circumstances. The comparison of observations is a very important part of the Method of Least Squares, since the knowledge of the value and precision of measurements is required for their most advantageous use. Moreover, the study of the precision of measurements is always necessary to improve and perfect the methods of observation. The Probable Error. 61 The quantity usually selected to compare the precision of observations is the probable error, of which the following is a definition : In any series of errors the probable error has such a value that the number of errors greater than it is the same as the number less than it. Or, it is an even wager that an error taken at random will be greater or less than the probable error. §62. THE PROBABLE ERROR. 67 The probable error is, then, the value of x in the probability- integral (4) when P = \, or it is the value of x given by the equation 1 2 c hx 2 \j v Jo By interpolation from Table I, Chap. X, it is found that P= 0.5 when hx = 0.4769. Hence, denoting this value of x by r, the equation (17) hr= 0.4769 gives the relation between the measure of precision h and the probable error r, and shows that h varies inversely as r. 62. To render more definite the conception of the measure of precision h and the probable error r, consider the case of two sets of observations made with different degrees of accu- racy. Let the measure of precision of the first be /i lt and of the second /z 2 ; then, from equation (2), the probability of errors in the first set will be represented by a curve whose equation is y = h l .dx.-K-*e- h ?* z , and for the second set by a curve y = hz.dx.Tr—be—t'i'**, in which dx is the constant difference between two consecutive errors. Now, suppose that the second set is twice as precise as the first, so that h^ = h, and /z 2 = 2k ; then the equations will be y = ahc- h * x * and y = 2ahe-* h * x *, in which a represents the constant ir~^dx. The curves corre- 68 THE PRECISION OF OBSERVATIONS. IV. sponding to these equations are given in Fig. 6; XB^A^^X being the one for the set of observations whose measure of precision is h l or h, and XB 2 A 2 B 2 X the one for the set whose measure of precision is h 2 , or 2/1. These curves show at a glance the relative probabilities of corresponding errors in the two sets : thus the probability of the error o is twice as much in the second as in the first set ; the probability of the error OP j is nearly the same in each ; while the probability of an error twice as large as OP, is much smaller in the second than in the first set. Now, if the lines P t B lt P 2 B 2 be drawn so that the areas P l B I A 1 B I P I and P 2 B 2 A 2 B 2 P 2 are respectively one- half of the total areas of their corresponding curves, the line OP t will be the probable error of an observation in the first set, and 0P 2 the probable error of one in the second set. Repre- senting these by the letters r, and r 2 , there must be in each case the constant relation h,r, = 0.4769, h 2 r 2 = 0.4769 ; § 63. THE PROBABLE ERROR. 69 and, since k 2 is twice h x , it follows that r 2 must be one-half of r,. The probable error, then, serves to compare the precision of observations equally as well as measures of precision. The smaller the probable error, the more precise are the measure- ments. For instance, if two sets of observations give for the length of a line in centimeters Z t = 427.32 ± 0.04 and Z 2 = 427.30 ± 0.16, in which 0.04 and 0.16 are the respective probable errors, the meaning is, that it is an even wager that the first is within 0.04 of the truth, and also an even wager that the second is within 0.16 of the true value; and the precision of the first result is to be regarded four times that of the second. The probable error thus serves as a means of comparison, and also gives an absolute idea of the uncertainty of the result. 63. In Art. 43 it was shown that the squares of measures of precision are directly proportional to weights ; and in Art. 61 it is established that measures of precision are in- versely proportional to probable errors. Hence the important relation : Weights of observations are inversely proportional to the squares of their probable errors ; or, in algebraic language, (18) /i:/2:/:: ^ : * . * Weights and probable errors are constantly employed in the practical applications of the Method of Least Squares, while h is only needed in theoretic discussions. By means of the relation just established, the weights of observed results of different degrees of precision may be found from their computed yo THE PRECISION OF OBSERVATIONS. IV. probable errors, and the observations be thus prepared for adjustment. For instance, in the two results Zi = 427.32 ± 0.04, Z 2 = 427.30 ± 0.16, it is seen that the weight of 427.32 is sixteen times that of 427.30. Probable Error of the Arithmetical Mean. 64. Let M lt M 2 . . M„ be 11 direct observations on the same quantity. The weight of each is 1, and the weight of their arithmetical mean is n. Let r be the probable error of a single observation, and r the probable error of the arithmetical mean. The principle (18) of the last article gives 1 1 n: 1 :: — :—, r a 2 r* from which (19) r °=l=-> or, the probable error of the arithmetical mean is equal to the probable error of a single observation divided by the square root of the number of observations. The probable error of the mean, hence, decreases as \fn increases. If ten observations give a certain probable error for the mean, forty observations will be necessary in order to reduce it to one-half that value. 65. To find r, the probable error of a single observation, consider the fundamental law of the probability of error (2), or §65. PROBABLE ERROR OF THE ARITHMETICAL MEAN. "Jl By Art. 12 the probability of the occurrence of the independent errors x u x^ . . . x„ is the product of the separate probabilities, or Now, for a given system of errors, it must be considered that the observations have been as precise as possible, or that h has such a value as to render P' a maximum. Putting the first derivative — • equal to zero, and reducing, gives ah n — 2h 2 ~S,x 2 = 0, or h = . / n . V 2%X 2 Since, by Art. 61, hr equals the constant 0.4769, 0.4769 /2x 2 ,= -J-p = o.6 74SV /-. Here %x 2 is the sum of the squares of the true errors, which are unknown. In a large number of observations the errors closely agree with the residuals, and 2-t 2 may be taken as equal to %v 2 ; but, for a limited number of errors, ~S,v 2 is less than S^ 2 , since, by the principle of Least Squares, the first is the mini- mum value of the second ; so that 2.v 2 = 2w 2 + « 2 , where u 2 is a quantity as yet undetermined. The absolute value of n 2 cannot be found ; but it is known to decrease as n increases, and for a given number of residuals to increase when 2.x 2 increases : as the best approximation, u 2 may be taken as equal to — . Then n 2» 2 Sjc 2 2x* = %v 2 + — , or Xx- n n 72 THE PRECISION OF OBSERVATIONS. IV. and, inserting this in the above value of r, it becomes (20) r= 0.6745V/ — ^-. » « — 1 This is the formula for the probable error of a single direct observation, or of an observation of the weight unity. To use it, the residuals are to be found by subtracting each measure- ment from the arithmetical mean, and the sum of their squares then formed. When r is known, the probable error r of the arithmetical mean is found by the formula (19), or it may be written at once (21) r = 0.67451/ Sy2 . V n(n — 1) which is the usual form for computation. Probable Error of the General Mean. 66. Let M„ M 2 . . . M„ be n direct observations having the weights p„ pi . . . p„. The weight of the general mean is P\ + Pi + • • • + Pm or 2/. Let r be the probable error of an observation of the weight unity, and r the probable error of the mean. Then, from the fundamental relation between weights and probable errors, ,. 1 1 1 : %p : : - : — , r* rj from which the probable error of the mean is (22) r ° = m'' and, in general, the probable error of any observation is equal to r divided by the square root of its weight. To find r, an § 6"]. PROBABLE ERROR OF THE GENERAL MEAN. 73 investigation like that in the preceding article could be em- ployed ; but it may be well to give one of a different character. 67. Let h be the measure of precision of an observation of the weight unity, and h„ k 2 . . . A„ those of the observations whose weights are p„ p z . p„. By formula (7) the quantities h lt h z . . . h n may be expressed in terms of the weights, thus : K* = M*> h * = P* h * ■ ■ ■ h >? = Pnh 2 ; and, in general, if x be any error, p the weight of the corre- sponding observation, and h the measure of precision of an observation of the weight unity, the probability y is, from {2), y = hpW~^dx.e~ h2 ^ 2 . Now, the quantity ~%px 2 y is the same as -^— , since each term, 11 such as P2X2, occurs ny x times in n observations ; and, for a con- tinuous series of errors, 2px 2 _ li^p_ r+ , n \j- f +a px 2 e-^^dx. J CO Taking in this hxsjp = t as the unit variable, it may be written %px* __ feJ-J The value of the integral in this expression is — ,* and hence ~Zpx* _ 1 n 2k 2 * From the footnote to Art. 31, J Coo „ |/T 74 THE PRECISION OF OBSERVATIONS. IV. From Art. 61 the value of — is ( ) : hence h 2 \0.476gJ r = 0.6745 y/^- is the probable error of an observation of the weight unity. Now, "S,fix 2 is in terms of the true unknown errors, and is greater than %pv 2 . Place, then, 2/>x 2 = 2/w 2 + u 2 , in which if is a quantity to be determined. The probability, P', of the system of errors, is P = Ke-!'^* 1 = Xe-'' 2 ^ 2 -*-" 2 ) = K'e- k2 " z . Here it is seen that the law of probability of u 2 is similar to that of an error x ; and, as in Art. 31, it may be shown that the constant K' is h.ir~*du. The mean of all the possible values of u 2 is, then, h r +to 1 W _u*e~ k2 " 2 du = — , VV 2^ Placing ? = t\s, this becomes /"» \n I e-tHdt - —p- ° 2\s Differentiating .his equation with reference to s, and regarding t as constant Dividing this by — ds, and making s = 1, it becomes /» y 71" i -t*t 2 dt — — - = one-half of the integral above, o 4 § 68. LA WS OF PROPA GA TION OF ERROR. J$ and this must be taken as the best attainable value of «*. But it was shown that — is equal to -*—. Hence 2A 2 n from which %px* ~%px* = tpt 2 H ■> 2,px* ~%pv 2 and therefore the probable error r becomes (23) r = 0.6745 \J-~TT The probable error of the general mean is now, from (22), ( 24 ) r =o.6 USS J y^ ■ (n-i)1p If the observations be all of the weight unity, ~%p becomes n, and the formulas (23) and (24) agree with (20) and (21). The probable error of any observation whose weight is p is found by dividing r by the square root of /. Laws of Propagation of Error. 68. Let z t and z^ be two independently measured quantities whose probable errors are r s and r 2 . It is required to find the probable error R of the sum z I -f- z 2 , or of the difference z, — z 2 . Let Z = z 1 ± -s^, and let the errors arising in the two cases be, x,', x", Xi", etc., for 2,, ^2 , *^"2 , *^2 , etc., tor 2f a . j6 THE PRECISION OF OBSERVATIONS. IV. Then the corresponding errors of Z are X' = xi ± */, X" = xi' ± x 2 ", X'" = *,'" ± Xl '", etc. Squaring each X, and adding the results, gives %X* = S*! 2 ± 2~%X Z X* + %xs. The products x,x x will be both positive and negative, and, on the average, %x,x 2 = o : hence 2X* = %xf + S,x^ ; and, if n be the number of errors, %X* _ 2^ Sar 2 2 n n n ~%x* Now, by Art. 65, it is known that -»— varies with y* : hence, n for the case in hand, (25) .ff* =?;-,* + r* 2 ; from which the probable error of Z is known. In like manner, if Z be the sum or difference of several inde- pendent quantities, namely, if Z = Z, ± 2 2 ± Z z ± . . . ± Z m , then the probable error of Z is given by the relation, (26) R* = r* + rj + r* + . . . + r m \ This formula is very important in the discussion of linear measurements. §69. LAWS OF PROPAGATION- OF ERROR. T] 69. Secondly, consider Z to be a multiple of an observed quantity z, so that Z ' = Az, where A is a known number. Then an error x in z produces an error Ax in Z, and .X = Ax, X 2 = A*x 2 , and %X* = ^S* 2 . Hence, as before, it is to be concluded that (27) R 2 = A 2 r*, or R = Ar. By combining the principle of the last article with that just deduced, it is seen, if Z = Az-[ ± Bz 2 ± Cz 3 ± etc., and if the probable errors of z lt z 2 , z } are r„ r 2 , r v that the prob- able error of Z is given by (28) R* = A*r* + B*rj + C~V 2 2 + etc., which is a more general formula than (26). It is interesting to note that formula (19) can be deduced from (26), and also (22) from (28). Thus, if z„ z 2 . . . z„ are n observed values of the same quantity, the probable error of their sum is, by (26), R = s/rs + rj + . . . + r„* = \jnr*, and by (27) the probable error of ^th of this sum is ]nr z _ _£_ . f — — .— j n \n which is the probable error of the arithmetical mean, as in (19). 78 THE PRECISION OF OBSERVATIONS. IV. 70. Next, consider Z to be the product of two independently observed quantities z\ and z„ whose probable errors are r x and r 2 . Let X be an error in Z corresponding to the errors x t -«id x 2 in z t and z 2 : then Z + X = (z, + Xi) (z 2 + * 2 ) = z,z 2 + z x x 2 + z 2 x l + x t x 2 . Here if = z,z 2 , and ^ r r 2 vanishes in comparison with z^c 2 and ■syr, ; so that Squaring each error X, and taking the sums, gives %X 2 = ~%Z 2 X 2 2 + %Z 2 X 2 + 222ATA, the last term of which vanishes, since the product xjc t is as likely to be positive as negative : hence %X 2 = z 2 %x 2 2 + z 2 2 5V, and accordingly, as in Art. 68, (29) R 2 = zfr 2 2 + z 2 2 rf, from which the probable error of Z may be computed. 71. Lastly, let Z be any function of the independently ob- served quantities z lt z 2 , z } . . ., or Z =f(z„ z z , z i . . .), and let it be required to find the probable error R of Z from the proba- ble errors r s , r 2 , r 3 . . . of the observed quantities. Take x lt x 2 , x % as any errors in z u z 2 , z v and X as the corresponding error in Z : then Z+X = /l(z t + x I ), (z 2 +x 2 ), (Z3+X3)...]. Now, if these errors are so small, that their second and higher §72. ERRORS FOR INDEPENDENT OBSERVATIONS. 79 powers may be neglected, the development of the function by Taylor's theorem gives „ dZ dZ dZ t az I az 2 az 3 Accordingly, by the same reasoning as in the previous articles, ldZ\ [dZ\* <*•> - = (f) v -©"- + ( *,) -'' + which is a general formula appplicable to all functions. The laws of propagation of error, given by formulas (25) to (30), are very important in forming proper rules for taking observations, as well as in discussing and comparing results. The law R = V^, 2 -f- r 2 2 , which gives the probable error of Z when i? = .c, -f- z z , or when Z = z l — z 2 , has been likened by Jordan to the celebrated geometrical theorem of Pythagoras. Probable Errors for Independent Observations. 72. In Arts. 46-50 are given methods of finding the most probable values of independent quantities which are indirectly observed. To determine the probable errors of any adjusted value, z, let p z denote its weight, and r z its probable error. Then, if rbe the probable error of an observation whose weight is unity, the relation (18) gives r? r* from which (3i) r. = ' Hence, in order to find the probable errors of z„ z 1 . . . z q , it if SO THE PRECISION OF OBSERVATIONS. IV. necessary to find r and their weights. And, in general, the probable error of any observation is equal to r divided by the square root of its weight. 73. To find the probable error of an observation whose weight is unity, the following reasoning may be employed : Suppose that the normal equations (13) have been solved, and the most probable values z„ z 2 . . . z g deduced. Let the corre- sponding true values be represented by z, -f- Sz lt z 2 -{- Sz 2 . . . z q -f- lz q , in which 8z lt Bz 2 . . . 8z q are small unknown correc- tions. Now if, in the observation equations (12), the most probable values be substituted, they will not reduce to zero, but leave small residuals v t , v 2 . . . v„ ; thus : a i z l + £,z 2 + . . . + / r z ? + m t = v 1 with weight/,, a 2 Zi + b 2 z 2 + . . . + l 2 z q + m l = v 2 , with weighty, a„z, + b n z 2 + . . . + l„z q + m„ = v„, with weight /„, while, if the corresponding true values be substituted, they will give the true errors ; thus : a, (z, + &,) + b l (z 2 + Sz 2 ) + . . . + m, = x„ a 2 (z, + &z,) + b 2 (z 2 + Sz 2 ) + . . . + m 2 = x 2 , a„(z t + Sz,) + b n (z 2 + Sz 2 ) + . . , + m i — x n . Subtracting each one of the former equations from the latter gives the following residual equations : v l + a I 8z I + b t Bz 2 4- . . . + /,8z ? = x lt v 2 + a 2 8z, + i 2 8z 2 + . . . + / 2 Sz<, = x 2 , v„ + a„8z L + b„Sz 2 + . . . + l„lz q = x„. Now, the principle of Least Squares (6) requires that 2,px* shall be made a minimum to give the most probable values of s u §73- ERRORS FOR INDEPENDENT OBSERVATIONS. 8 1 g 2 . . . z q ; and, by the solution of the normal equations, its mini- mum value is the sum %pv*. From the residual equations a relation connecting the two sums %pv 2 and %px 2 may be found by squaring both members of each of those equations, multi- plying each by its corresponding weight, and then adding the products. Without actually performing these operations, it is evident, that if the squares and products of 8z„ 8z 2 . . . 8z q be neglected as small in comparison with the first powers, the result will be of the form %pv* + kfa + k 2 8z 2 + ...-)- k q 8z q = 2px 2 , in which k„ k 2 . . . k q are co-efficients of the unknown correc- tions, and dependent only upon the known co-efficients and weights. If the number of unknown quantities is q, there will be q of these terms. Placing k^z L = u*, k z lz 2 = u 2 2 . . . k q lz q = u q 2 , it becomes tpv* + ztf + u 2 * + . . . 4- uf = 2/jc 2 . Now, the probability of the occurrence of the error x z , whose measure of precision is h lt and whose weight is /„ is, by (2) and (7), y t = hpji.dx.ir—^e — ^ti*-?, in which h is the measure of precision of an observation of the weight 1. And hence, by exactly the same reasoning as in Art. 67, it may be shown, that, when n is a large number, 2p x * = — 2/2 2 Further : if there be but one unknown quantity, there is but one tf, whose value, as shown in Art. 67, is — — . And, since 2« 2 82 THE PRECISION OF OBSERVATIONS. IV. this is true whichever unknown quantity be considered, the value of each if must be — ; and, as there are a of these Values, the above result becomes 2/» 2 + -\ = — -, from which -^ 2~2,pV 2 Therefore, from the constant relation (17) between h and r, the probable error of an observation of the weight unity is (32) ^r= 0.6745 y-' %pv 2 1 \ 74. The probable errors of the values z It z z be found from (31) as soon as their weights are known. These will now be determined. The observations M It M 2 . . . M n furnish the observation equations (12) and the normal equations (13). The solution of the latter gives the values of z„ z 2 . . . z g in terms of M lt M 2 . . . M n , and co-efficients independent of those quantities. Suppose the general solution to give z, = and the weight of z 2 is -5-; which was to be proved. 75. The following is hence a method of finding the weights of the values of the unknown quantities. Preserve the abso- lute terms of the normal equations in literal form during the solution. Then the weight of any value, as z v is equal to the reciprocal of the co-efficient of the absolute term which belonged to the normal equation for z v For example, take the normal equations 32. — z 2 — Z 3 = A v — 2i + 3 Z 2 — h = A 2) — «i — z 2 + 2Z l = A* §?6. ERRORS OF INDEPENDENT OBSERVATIONS. 8$ The solution of these by any method gives *. = 4M. + \A, + \A V z* = %A I + %A X + \A Z , z 3 = jA, + 2A2 + A s , and hence the weight of z x is f, the weight of z 2 is ■§-, and the weight of ^ 3 is i. It is evident, if it be only desired to find the weight of z„ that A 2 and A 3 need not be retained in the computation, but may be made zero. So, in finding the weight of z 2 , only A 2 need be retained in the work. 76. As an illustration of the preceding principles, let there be three observation equations of weight unity, z t = 0, z 2 = o, z x — z 2 = + 0.51. The normal equations are 2«i — z 2 = + °-5i» — z t + 22 2 = — 0.51. Writing A r and A 2 for the absolute terms the solution of these equations gives z 1 = — A 1 -\ A a Z 3 = —A 1 -f- ~A 2 > 3 3 3 3 from which the adjusted probable values are z l = -f-O.17 and z 2 = — 0.17, while the weight of each of these values is seen to be i£. The sum of the squares of the residuals is 2v 2 = 0.0578, and from (32) the probable error of an observation of weight unity is ± 0.16. This divided by V1.5 gives ± 0.13 as the probable error of the adjusted values of z t and z 2 . The adjusted value of the third observation is z 1 — z 2 = -(- O.34, and by (25) the probable error of this value is ± 0.18. It is seen that the corrections to the three observed values are here numerically equal. 86 THE PRECISION OF OBSERVATIONS. IV. Probable Errors for Conditioned Observations. 77. When conditioned observations are adjusted by the gen- eral method of Art. 57, where the q unknown quantities in the n observation equations are reduced to q — ;/ independent quantities by means of the n' conditional equations, the proba- ble error of an observation of the weight unity is evidently given by the formula (32), if q be replaced by q — n', or (33) :r = o.674sy/— — £ 7 > q + n' and the probable errors of observations or values whose weights are/,,/*, etc., are, by (31), r r r, = — , r % = — , etc. va va The weights of #„ z 2 . . . s t are to be found exactly as in Art. 75. For the case of direct observations on several quantities adjusted by the method of Art. 58, the number of observation •equations is the same as that of the unknown quantities, or n = q ; and, if n' be the number of conditional equations, the probable error of an observation of the weight unity is (34) r=o.6 145 ^r, from which the probable error of any observation of given weight can at once be deduced. In this case the residuals v are merely the differences between the observed and the adjusted values. § 78- PROBLEMS. 87 78. Problems. 1. There are two series of observations of an angle, each taken to hundredths of a second. The probable error of a single observation in the first series is o".65, and in the second i".45. Compute the proba- bilities of the error o".oo and of the error 2".oo in the two cases. 2. It is required to determine the value of an angle with a proba- ble error of o'' '.25. Twenty measurements give a mean whose probable error is o".38. How many additional measurements are necessary? 3. Find the probable error of the mean of two observations which differ by the amount a. 4. Let z„ z 2 , and z 3 be independently observed quantities whose probable errors are r u r 2 , and r v If Z= z, 2 -\-z 2 2 -j- z 3 2 find the proba- ble error of Z. 5. Let r be the probable error in log a. What is the probable error in the number a ? 6. Given the following observation equations : — z i = 4-S> w i'h weight 10, 2 2 = 1.6, with weight 5, «, — z 2 = 2.7, with weight 3. What are the most probable values of z, and z 2 with their probable errors? 7. Given the observation equations (all of equal weight) 20, — 2 4- z 3 = 3, 3*i + ¥* ~ 2 3 = '4. 40, + 2 + 403 = 21, — 50, 4- 20 2 4- 323 = s, to find the best values of e lt z 2 , and z 3 , with their probable errors. 88 DIRECT OBSERVATIONS ON A SINGLE QUANTITY. CHAPTER V. DIRECT OBSERVATIONS ON A SINGLE QUANTITY. 79. In the preceding pages the fundamental methods and formulas for the adjustment and comparison of observations have been deduced. In this and the three following chapters the application of these methods to practical examples will be presented. The most common case of observation is that of direct measurements on a single quantity, and this will form the subject of the present chapter. Observations of Equal Weight. 80. When a quantity is measured several times with equal care, so that there is no reason for preferring one observation to another, the observations are of equal weight. From re- mote antiquity the arithmetical mean of the measurements has always been regarded as the best or most probable value of the quantity sought ; and, as shown in Art. 44, this is con- firmed by the fundamental principle of the Method of Least Squares. Let z be the most probable value of the measured quantity, n the number of observations, and M any observation. Let r be the probable error of a single observation, and r Q the proba- §8 1. OBSERVATIONS OF EQUAL WEIGHT. 89 ble error of the adjusted value z. Let also v be any residual obtained by subtracting M from z. The most probable value of the quantity is the arithmetical mean, expressed, as in Art. 44, by formula (8), _ %M n The probable error of a single observation, as shown in Art. 65, is, by formula (20), r= 0.6745 Lastly, as shown in Art. 64, the probable error of the mean is, by (19), r r ° = "r v« Formula (8) indicates the method of adjustment, while (20) and (19) determine the precision of observation and of the mean. After finding a, each observation is subtracted from it, giving n values of v. The squares of these are taken, and their sum is 2» 2 ; then r is computed, and lastly, r a . If desired, r can be also found directly from formula (21), r = 0.6745 V n(n — 1) which is the same as (19). 81. As an example, consider the following twenty-four meas- urements of an angle of the primary triangulation of the United-States Coast-Survey, made at the station Pocasset in Massachusetts, and recorded in the Report for 1854: 90 DIRECT OBSERVATIONS ON A SINGLE QUANTITY. V. Observations. u. V 1 . Ii6°43'44".45 5-19 26.94 50.55 —0.91 •83 5°-95 -i-3i 1.72 48.90 0.74 •55 49.20 0.44 .19 48.85 0.79 •63 47.40 2.24 5.02 47-75 1.89 3-57 S 1 -^ — 1. 41 2.00 47-85 1.79 3.20 50.60 —0.96 .92 48.45 1. 19 1.42 5M5 — 2. 11 445 49.00 0.64 .41 5 2 -35 — 2.71 7-34 5 J -30 -1.66 2-75 5i-°5 -1.41 2.00 5!-7° — 2.06 4.24 49-05 0.59 •35 5°-55 — 0.91 •83 49- 2 5 °-39 •*5 46.75 2.89 8-35 49- 2 5 °-39 •!5 53-4o -3-76 14.14 z = n6°43'49".64 2 v 2 = 92.15 The most probable value of the angle is found by adding the observations, and dividing the sum by twenty-four. This is 1 1 6° 43' 49". 64. Subtracting from this the first reading gives §82. OBSERVATIONS OF EQUAL WEIGHT. 9 1 5.19 for the first residual, which is placed in the column headed v. The square of this is 26.94, which is placed in the column headed v 2 . The sum of all these squares is 92.15. Then from (20) the probable error of a single observation is r = 0.6745 J ^5 = I ". 3S ; V 23 and the probable error of the mean is, from (19), r a = ^H = o". 2 8 : V24 hence the final value may be written 116 43' 49".64 ± o".28. The precision of the mean of these twenty-four observations is such that o".28 is to be regarded as the error to which it is liable ; that is, it is an even wager that the mean differs from the true value of the angle by less than o".28, and of course also an even wager that it differs by more than o".28. The pre- cision of a single observation is such that i''. 35 is the error to which it is liable ; that is, half the errors should be less, and half greater, than i"-35 in a large number of observations. It will be noticed that twelve of the above residuals are less, and twelve greater, than 1 ". 3 5 . In Art. 27 it was shown that the algebraic sum of the residu- als must always equal zero. This principle may be used to furnish a check on the accuracy of the numerical work. 82. The tables in Chap. X will be found useful in abbreviat- ing computations. By the help of Table VI the squares of the residuals can be readily found. By Table III the compu- tation of r and r can be much abridged ; for instance, in the case of the last article, n =. 24, and r = 0.1406 V92.15 = i".35, r a = 0.0287 ^92. 15 = o .28. 92' DIRECT OBSERVATIONS ON A SINGLE QUANTITY. V. The table of four-figure logarithms will also prove useful in extracting roots and performing multiplications. When the tables are used, it will be found more convenient to compute r from (21) than from (19). Formula (19), however, is very important in indicating that the probable error of the mean decreases, and hence that its precision increases, with the square root of the number of observations. It should be borne in mind, that the method of the arithmeti- cal mean only applies to equally good observations on a single quantity, and that it cannot be used for the adjustment of ob- servations on several related quantities. For instance, let an angle be measured, and found to be 6oJ degrees, and again let it be measured in two parts, one being found to be 40 degrees, and the other 20 degrees. The proper adjusted value of the angle is not, as might at first be supposed, the mean of 6oi and 60, which is 6oi degrees, but, as will be seen in the next chap- ter, it is 6oi degrees, — a result which requires the correction of each observation by the same amount. Shorter Formulas for Probable Error. 83. The method of computing probable errors by formula (20) is that considered the best by all writers. Nevertheless, on account of the labor of forming the squares of the residuals, a simpler and less accurate formula is often employed, in which only the residuals themselves are used. To deduce it, let n be the number of observations, and ~%v the sum of the residuals, all taken with the positive sign, and %x the sum of all the errors ~%x taken positively. Then — is the mean of the errors ; and, by n the same reasoning as in Art. 6j, this mean is 2* 2/1 1"° „ . , — = — / xe h ***dx = n \Jn J o h\fri §84. SHORTER FORMULAS FOR PROBABLE ERROR. 93 Now since, by Art. 61, the product hr is equal to the constant 0.4769, the value of r in terms of ~%x is o 2x r= 0.8453 — n The sum of the errors 2,r is in general different from the sum of the residuals %v. Both in Art. 65 and Art. 67 it was shown that 2.x 2 _ 2w 2 « 11 — \ and it may hence be concluded, that, on the average, x* is greater than v 2 in the ratio of n to n — 1, and that, on the average, x is greater than v in the ratio of \Ju to ^ — 1, or that \Jn \Jn — 1 Accordingly the above value of r becomes 0.845327/ (35) \jn(n — 1) which gives the probable error of a single observation. By substituting this in (19), the value of r a becomes (36) ro = °J*2& n\n — 1 which is the probable error of the arithmetical mean. 84. Formulas (35) and (36) will be found much easier to use than (20) and (21). In Table IV the co-efficients of %v are tabulated for values of n from 2 to 100, and by its use the computations are much abridged. 94 DIRECT OBSERVATIONS ON A SINGLE QUANTITY. V. As an example, consider the following eight measurements of a line made with a tape twenty meters long, graduated to centimeters : Observations. V. 188.97 .88 .91 •99 •83 .80 .81 .81 0.095 .005 •°3S •i'5 •045 •075 .065 .065 188.875 0.500 Here the arithmetical mean, or most probable value of the line, is found to be 188.875 meters. The difference between this and the single observations gives the residuals v, whose sum %v = 0.5. Then, by the use of Table IV, for n =: 8, r = 0.1130 x 0.5 = 0.0565, r a = 0.0399 *- °-5 = o- 0200 - By the more accurate formulas (20) and (21) these values are r= 0.051 and r = 0.018 meters. With a larger number of observations, a closer agreement between the probable errors found by the two methods might be expected. 85. The probable error r of a single observation should always be computed, since it furnishes the means of comparing §86. OBSERVATIONS OF UNEQUAL WEIGHT. 95 the accuracy of work done with different instruments, or by different observers. Under similar conditions, r should be prac- tically a constant for a given class of measurements ; while for different classes the different values of r indicate the relative precision of the methods. For instance, suppose the same observer to measure the same angle with two different transits, and to find the probable error of a single observation with the first to be 4", and with the second 6". The relative precision of the instruments is, then, inversely as these probable errors, or as 3 to 2 ; and the weights of a single observation in the two cases are as 3 2 to 2% or as 2\ to 1 ; so that one measurement made with the first instrument is worth 2\ made with the second. These results, in order to be satisfactory, must be deduced from a large number of observations ; since the formu- las for probable error suppose that enough observations are made to exhibit the several residuals according to the law of probability of error as given by equations (1) and (2). Observations of Unequal Weight. 86. When the observations on a single quantity have differ- ent weights, the most probable value of the quantity is to be found by the use of the general arithmetical mean ; namely, by multiplying each observation by its weight, and dividing the sum of the products by the sum of the weights. Or if s be that most probable value, M any observation, and / its weight, then, as shown in Art. 45, formula (9) gives tpM z = — — The probable error of an observation of the weight unity, as shown by formula (24), Art. 67, is r= 0.6745 V (« - 1) 96 DIRECT OBSERVATIONS ON A SINGLE QUANTITY. V. in which n denotes the number of observations, and v any residual obtained by subtracting M from s. Lastly the proba- ble error of z, as shown in Art. 66, is found by (22), y/lp Formula (9) indicates the method of adjustment. Having found the most probable value z, each observation is subtracted from it, giving 11 residuals v. These are squared, and each v 2 multiplied by the corresponding weight/. The sum of these products is ~%pv 2 . Then formula (24) gives the probable error of an observation of the weight unity. Lastly, formula (22) gives the probable error of z. And in general the probable error of an observation of given weight may be found by divid- ing r by the square root of that weight. 87- As an example let the observations in the second column of the following table be the results of the repetition of an angle at different times, 18". 26 arising from five repetitions, 16". 30 from four, and so on, the weights of the observations being taken the same as the number of repetitions. Then the general mean z has the weight 21, the sum of the several weights or A M. V. V*. pv>. 5 87° 5:' i8". 2 6 — O.IO O.OIO 0.05 4 16.30 + 1.86 3.460 13.84 1 21.06 — 2.90 8.410 8.41 4 J 7-95 + 0.21 0.044 0.18 3 16.20 + 1.96 3.842 n-53 4 20.85 — 2.69 7.236 28.94 2p = 21 3 = 87° 51' i8".i6 ~2.pi) = 62 -95 §88. OBSERVATIONS OF UNEQUAL WEIGHT. 97 the number of single measures. Subtracting each M from z gives the residuals in the column v\ next from Table VI the numbers in the column z<* are found, and multiplying each of these by the corresponding weight produces the quantities pv*, whose sum is 62.95. Then, since n is 6, formula (23) gives V^ = r = 0.6745,4 / — ^ = 2 ". 39 or, by the help of Table III, r = 0.3016 t 7 62.95 = 2".39. This is the probable error of an observation of the weight unity. From (22) the probable error of the general mean is, r a — —= — o .98, T2I and the probable error of any given observation is found by dividing 2".39 by the square root of its weight. 88. The important relation (18) of Art. 63, that the weights of observations are inversely as the squares of their probable errors, furnishes, as already indicated in Art. 85, a ready means of determining weights, if the probable errors can be obtained with sufficient precision. When the weights are known, the observations can be combined by (9), and the most probable value determined. As an example, consider the two following series of meas- urements of an angle ; the first taken with a transit reading to twenty seconds, and the second with a transit reading to minutes. The angle was observed in each case ten times ; the 98 DIRECT OBSERVATIONS ON A SINGLE QUANTITY. V. circle being used in eleven different positions to eliminate errors of graduation, while each time the two verniers were read to eliminate errors of eccentricity. With First Transit. With Second Transit. M. v. V 2 . M. v. V*. 34° 55' 35" 35 20 05 75 40 10 3° 5° 30 2 2 13 28 42 7 13 3 17 3 4 4 169 784 1764 49 169 9 289 9 34° 56' 15" 55 3° 54 3° 55 '5 56 00 55 45 55 3° 55 30 56 00 55 45 39 6 66 21 24 9 6 6 24 9 1521 36 4356 441 576 81 36 36 576 81 34° 55' 33" 325 34° 55' 36" 7740 By the method of Art. 80 it is easy to find For first transit 34° 55' 33" ± 4"i For second transit 34 55 36 ± 6 .3 Hence by (18) the weights of these means are in the ratio 1 1 4i 2 '63 2 or as 12 to 5 nearly. The final adjusted value of the angle is, then, 34° 55' 33"-9> = 34 o 5S , + 33 X ,2 + 36X5 = 34 . § 89. PROBLEMS. 99 and by (18) the probable error of that value is V 17 3-4- As the probable errors of a single observation in the two cases are 13" and 20", the corresponding weights are as 400 to 169; so that one observation with the first instrument is worth about 2^ with the second. When observations upon the same quantity are known to be of different precision, and there is no way of finding the proba- ble errors, as in the example just discussed, weights should be assigned corresponding to the confidence that is placed in them, and then the general mean can be deduced. Of course, the assignment of weights in such cases is a matter requiring experience and judgment. Problems. 89. The solution of the following problems will serve to exemplify the preceding principles. 1. The latitude of station Bully Spring, on the United States northern boundary, was found by sixty-four observations to be 49 01' 09". n ± o".05i. What was the probable error of a single observation? 2. A line is measured five times, and the probable error of the mean is 0.016 feet. How many additional measurements of the same pre- cision are necessary in order that the probable error of the mean shall be only 0.004 feet? 3. An angle is measured by a theodolite and by a transit with the following results : By theodolite 24 13' 36" ± 3".! By transit 24 13 24 ± 13 .8 Find the most probable value of the angle and its probable error. IOO DIRECT OBSERVATIONS ON A SINGLE QUANTITY. V. 4. A base-line is measured five times with a steel tape reading to hundredths of a foot, and also five times with a chain reading to tenths of a foot, with the following results : — By the tape : 741.17 feet. By the chain : 741.2 feet. 741.09 feet. 74M feet. 741.22 feet. 741.0 feet. 741.12 feet. 741-3 feet. 741.10 feet. 74H feet. Find the probable errors and weights for a single observation in the two cases, and also the adjusted length of the line. Ans. 741.146 ± 0.012. 5. Eight observations of a quantity give the results 769, 768, 767, 766, 765, 764, 763, and 762, whose relative weights are 1, 3, 5, 7, 8, 6, 4, and 2. What is the probable error of the general mean, and the probable error of each observation ? 6. The length of a line is stated by one party as 683.4 ± 0.3, and by a second party as 684.9 ± °-3- What is to be inferred from the two results ? § : 9I- LINEAR MEASUREMENTS. IOI CHAPTER VI. FUNCTIONS OF OBSERVED QUANTITIES. 90. In this chapter will be discussed the determination of the precision of quantities which are computed from other measured quantities. For instance, the area of a field is a func- tion of its sides and angles : when the most probable values of these have been found by measurement, the most probable value of the area is computed by the rules of geometry, and the precision of that area will depend upon the precision of the measured quantities. Linear measurements will first receive attention ; for, although they are direct observations when the result alone is considered, yet really the length of a line is a function of its several parts, namely the sum. So, too, an observed value of an angle is a function (the difference) of two readings. All the following reasoning is based upon the laws of propagation of error deduced in Arts. 68-71. Linear Measurements. 91. As a line is measured by the continued application of a unit of measure, its probable error should increase with its length. The law of this increase is given by formula (26). If the parts are all equal, and each be taken as the unit of length, the number of parts is the same as the length of the line. Let r denote the probable error of a measurement a unit in length, R the probable error of the total observed length, and / that observed length. Then (26) reduces to (37) R=rsfl; 102 FUNCTIONS OF OBSERVED QUANTITIES. VI. that is, the probable error of a measurement of a line increases with the square root of its length. For example, the value of r for measurements with an engineer's tape on smooth ground is about 0.005 : hence, for a line 100 feet long, R is 0.05 feet, and for a line 1,000 feet long, R is 0.16 feet. Since, by (18), weights are inversely as the squares of probable errors, and, by (37), the squares of probable errors are directly as the lengths of lines, it follows that the weights of linear measurements are inversely as their lengths, or (38) *''*>'- *'---vv\ Hence, if the weight of a measurement of a unit's length be 1, the weight of a measurement of the length / will be -. This principle is to be used in combining linear measurements for which the value of r is the same. 92. The value of r may be found by measuring a line of the length / many times, and computing R by the methods of the last chapter. Then, by (37), the value of r is known. For instance, take the eight measurements of a line about 189 meters long, which are discussed in Art. 84, for which the proba- ble error of a single observation was found to be about 0.05 meters. Here R = 0.05, and then r = ' = 0.004 meters, V189 which is the probable error of a measurement of a line one meter in length. The most convenient way, however, of finding r, is to make duplicate measurements of several lines of different lengths. Let the lengths of the lines be /„ l 2 . . . /„, the differences of the duplicate measurements be d„ d 2 . . . d„, and the num- §92. LINEAR MEASUREMENTS. IO3 ber of lines be n. These differences are the true errors of a quantity whose true value is zero, and by Art. 67 the probable error of an observed difference is r = 0.6745 V^— " V n Now, from Art. 68, this probable error is also r' = \Jr 2 + r 2 = ry/I, and, by equating these two values of /, it is easy to find (39) r= 04769 V— ' V n which is the probable error of a measurement a unit long. The weight p is to be taken as — in accordance with (38). For example, the following duplicate measurements of the sides of a mountain field, made with a Gunter's chain, may be considered. No. of Side. By First Party. By Second Party. I 17.21 chains. 17.18 chains. 2 348 " 3-5 2 " 3 I5-J4 iS-!9 " 4 1.27 1.25 " 5 20.06 " 20.12 " 6 8.85 " 8.92 " 7 0.70 " 0.70 " 8 6-75 " 6.78 " 104 FUNCTIONS OF OBSERVED QUANTITIES. VL Here for the first line -3 = f V7. v» The precision of n repetitions is, hence, \Jn times greater than the mean of n independent observations. However, the errors in pointing, and other causes, render it doubtful if it is ever advantageous to make n exceed six or eight. Precision of Areas. 96. Let z t and z z be the measured sides of a rectangle, and r t and r 2 their probable errors. Then by (29) the probable error of the computed area s,s x is R = \fz*r 2 2 + z/rf. If r be the probable error of a measurement a unit in length, the law of (37) gives r, 2 = r a z x and r x * = /*Sj ; § 98. REMARKS AND PROBLEMS. \OfJ and hence the probable error in the area is R = ry/z&fa +z 2 ). For instance, let a lot 60 X 150 feet be laid out by an en- gineer's chain, for which r = 0.0 1. Then, by the formula, R = 13.75 square feet, which is the probable error of 9,000 square feet, the computed area. 97. By the application of formula (30) the probable error of any computed area can be found from the known probable errors of its sides and angles. As one of the simplest cases, take a triangle ABC, whose area is found from the angle A and the two adjacent sides AB and AC. The observed values are AB = 252.52 ± 0.06, AC = 300.01 ± 0.06, A = 42 13' 00" ± 30". The area of this triangle is \AB.AC.sinA = 25,453 square feet. To compare with (30) let AB = z„ AC = z 2 , and sin A = z i ; also r, = r 2 = 0.06, and r 3 = 0.00011 = tabular difference corre- sponding to 30". Then — = \AC.wiA, dz l — = \AB. sin A, dz 2 — = \AB.AC. By inserting all values in (30) it is easy to find R = 8.9 square feet for the probable error of the area. Remarks and Problems. 98. By the application of formulas (25) to (30) the precision of many other functions of observed quantities than those 108 FUNCTIONS OF OBSERVED QUANTITIES. VI. above noticed may be investigated. A few of the simplest cases are included among the following problems. i. The radius of a circle is observed as iooo ±0.2. Find the probable errors of its circumference and area. 2. Find the maximum probable error of sin A + cos A when the probable error of A is 20". 3. In order to determine the difference of level between two points A and B, an instrument was set up halfway between them, and twenty readings taken on rods held at each point, with the following results : Rod at A. Rod at B. 7 readings gave 7.229 feet. 3 readings gave 9.806 feet. 8 readings gave 7.230 feet. 12 readings gave 9.807 feet. 5 readings gave 7.231 feet. 5 readings gave 9,808 feet. What is the most probable difference of level between the two points and the probable error of the determination? Ans. 2.5772 ± 0.00048. 4. A block of cast-iron weighing 100 pounds rests upon a horizontal table, also of cast-iron. A horizontal force is applied to the block, and it is observed that it begins to move when the force is 15.5 pounds. If the probable error in the determination of this force is 0.5 pound, what is the probable error of the co-efficient of friction ? 5. A chronometer is rated at a certain date, and found to be g m 12^.3 fast, with a probable error of 0^.3. Ten days afterwards it is again rated, and found to be 9"' 2 1*4 fast, with the same probable error. What is the probable error of the mean daily rate? 6. A line of levels is run in the following manner : the back and fore sights are taken at distances of about 200 feet, so that there are thirteen stations per mile, and at each sight the rod is read three times. If the probable error of a single reading is 0.001 feet, what is the probable error of the difference of level of two points which are ten miles apart? § 99- METHOD OF PROCEDURE. IOO CHAPTER VII. INDEPENDENT OBSERVATIONS ON SEVERAL QUANTITIES. 99. Independent observations on several related quantities are to be adjusted by the methods of Arts. 46-50, and their precision determined by the methods of Arts. 72-76. The following are the steps of the process : 1st, Let z L , z 2 , z v etc., represent the quantities to be deter- mined, and for each observation write an observation equation ; or, if more convenient, let z u z z , z 3 , etc., be corrections to assumed approximate values of the unknown quantities. 2d, From the observation equations form the normal equa- tions, which will be as many as there are unknown quantities. 3d, Solve the normal equations : the resulting values of the unknown quantities will be their most probable values, that is, the best values that can be deduced from the given observations. 4th, Find the residuals, and the probable error of an obser- vation of the weight unity from formula (32). 5th, Find, if desired, the weights and probable errors of the adjusted values of the unknown quantities. When the number of unknown quantities exceeds four or five, it will usually be found most convenient to use the algo- rithm of formulas (10) and (1 1) for observations of equal weight, and of (12) and (13) for those of unequal weight, and to solve the normal equations by the method of Arts. 51-55. It will, however, probably be best for a beginner to form the normal IIO INDEPENDENT OBSERVATIOAS. VII. equations by the rules in Art. 48 and Art. 50, and to solve them by his own algebraic method. It will often be convenient to take the unknown quantities as corrections, rather than as the real quantities themselves ; since thus the numbers entering the computation are much smaller. The following practical examples will illustrate the whole method of procedure. Discussion of Level Lines. 100. The following observations are recorded in the Report of the United States Geological and Geographical Survey for 1873, and are here supposed to be of equal reliability or weight : 1. Z x above O, 573.08 feet, by Coast Survey and canal levels, via New York and Albany. 2. Z 2 above Z„ 2.60 feet, by observations on surface of Lake Erie. 3. Z 2 above O, 575.27 feet, by Coast Survey and railroad levels, via New York and Albany. 4. Z 3 above Z 2 , 167.33 f eet > by railroad levels. 5. Z 4 above Z 3 , 3.80 feet, by railroad levels. 6. Z 4 above Z 2 , 170.28 feet, by railroad levels, via Alliance. 7. Z 4 above Z 5 , 425.00 feet, by railroad levels. 8. Z s above O, 319.91 feet, by railroad and Coast Survey levels, via Philadelphia. 9. Z 5 above O, 319.75 feet, by railroad levels, via Baltimore. The letters here have the following meanings : O is the mean surface of the Atlantic Ocean. Z, is the mean surface of Lake Erie at Buffalo. Z, is Cleveland city datum plane. Z 3 is Depot track at Columbus, O. Z 4 is Union Depot track at Pittsburg. Z 5 is Depot track at Harrisburg. § IOO. DISCUSSION- OF LEVEL LINES. Ill It is required to adjust these observations, and to find the probable error of a single observation. ist, Represent the unknown heights of Z„ Z 2 , Z v Z ¥ and Z $ by z„ z M z v z v and z s . Then the observations give the obser- vation equations Z,= 573.08, 2 2 — z t = 2.60, 22 = 575- 2 7, 2 3 - z 2 = 167.33, «4 - z 3 = 3.80, 2 4 — z 2 = 170.28, 24 — z 5 = 425.00, z 5 = 319.91, 2 5 = 3I9-7S- 2d, Form a normal equation for z, by multiplying each equa tion in which z l occurs by its co-efficient in that equation, and adding the products ; and in the same way form a normal equa- tion for each of the other unknown quantities. This gives 23, — 2 2 = 570.48, -Si + 4 2 2 — Z 3 — 24 = 240.26, ~ 2 2 + 2Z 3 — h = 16353, — 2 2 — h 4- 324- z s = 599.08, — 24 + 3^5 = 214.66. 3d, The solution of these normal equations furnishes the following values : — z, = 572.81, z 2 = 575.14, 23=742.05, 2 4 = 745-43, 2 S = 320.03, which are the adjusted elevations of the five points above the datum O. 112 INDEPENDENT OBSERVATIONS. VII. 4th, Substitute these values in the observation equations, and find the residuals and their squares ; thus : No. V. z> 2 . I 0.27 0.073 2 .27 •073 3 •!3 .017 4 .42 .176 S .42 .176 6 .01 .000 7 .40 .160 8 .12 .014 9 .28 .078 2» 2 = 0.767 Here the number u of observations is 9, and the number q oi unknown quantities is 5. The weights / are all unity. Then, from (32), = 0.6745^/° .767 0.295 feet, which is the probable error of each observation. 5th, To determine the probable errors of the above adjusted values, it is necessary to find their weights by the method of Art. 75. For instance, to find the weight of z„ represent the absolute term in the normal equation for z 2 by B, and put all the other absolute terms equal to zero. Then the solution gives z 2 = ~B, and accordingly the weight of z t is %. Hence the probable error of the value of z 2 is 0.295 V1.96 = 0.21 1 feet; § IOI. DISCUSSION OF LEVEL LINES. 113 so that the final elevation of Z z may be written z 2 = 575- I 4 ± °-2i, and it is an even wager that the actual error in the value 575.14 is less (or greater) than the amount 0.21 feet. 101. For level lines of unequal precision the process of ad- justment is the same, except, that, before forming the normal equations, each observation equation should be multiplied by the square root of its weight. To illustrate, regard the above nine observations as of unequal weight. The least trustworthy is No. 9 ; because it is not known that mean tide at Baltimore is the same as the mean surface of the ocean, and its weight may be taken as 1. Nos. 3 to 8 inclusive are ordinary railroad levels, and may, with reference to No. 9, be given a weight of 4. Nos. 1 and 2, being the result of carefully conducted govern- ment and canal levels extending over many years, are the most reliable of all ; and a weight of 25 may be assigned them. The observation equations are the same as before ; multiplying each by the square root of its weight gives 52, = 2865.40, 5** — 5 z i = 13-°°, 22 2 = 1150.54, 22 3 — 22 2 = 334-66, 22 4 — 22 3 = 7.60, ZZ 4 — 22 2 = 34O.56, 2Z 4 — 22 5 = 85O.OO, 22 5 = 639.82, Z 5 = 3I9-75- :quations now are 502, — 252 2 = 14262.00, 252, + 372 2 — 423 — 42, = 1015.64, — 42 2 + 8z 3 — 42 4 = 654.12, 4^2 — 4*3 + I22 4 — 42 s = 2396.32, — 42 4 + 925 = —100.61, H4 INDEPENDENT OBSERVATIONS. VIL and their solution gives 2, = 572.9S, H = 745-72, ^ = 575- z 3 = 742-3 6 » 320.25. Inserting these in the observation equations, the remainders or residuals v lt v„ etc., are found, and placed in the third column below, their squares in the fourth, and the product of each square by its corresponding weight in the fifth. No. P- v. v*. pv 2 . 1 2 5 0.10 O.OIO 0.250 2 25 .11 .012 .300 3 4 .20 .040 .160 4 4 •44 .194 .776 5 4 •43 .185 .720 6 4 .02 .000 .002 7 4 .48 .210 .840 8 4 ■34 .116 .464 9 1 •5° .250 .250 2/z/ 2 = = 3-762 Then by (32) the probable error of an observation of weight unity, that is of No. 9, is 0.6 745 y : 3.762 0.635 feet » and the probable error of observations 1 and 2 is by (31) — ^ = 0.13 feet, and of those from 3 to 8 inclusive is ' = 0.32 feet. § 102. DISCUSSION OF LEVEL LINES. 1 15 In order, lastly, to find the probable errors of the above adjusted values, their weights must be determined. For in- stance, to find the weight of z v place the absolute term in the fourth normal equation equal to A, and those in the other normal equations equal to zero. Then the solution gives z 4 = ^A, and accordingly the weight of z 4 is 6.62. Hence the probable error of the value of z 4 is 0-635 V/6T62 r *4 = TrFrz — °- 2 5 feet - And in a similar way the probable error of the value of z 2 may be found to be o. 1 5 feet. 102. For such simple cases as those just presented, the abso- lute terms in the normal equations might be represented by letters, A„ A 2 , etc., and a general solution easily effected, which would give at once all the weights and unknown quantities. For instance, if the normal equations of Art. 100 are thus written 2Z, — «2 = A V -*i + 4 Z 2 — Z i - Z 4 = A 2 , — z 2 + 2Z l ~ Z 4 = A 3 , — «2 h + 3^4- Z 5 3 3 5 the solution gives Siz, = 32,4, + i$A 2 + nA 3 + 9^4 + 3^5, 5iz 2 = 13^4, + 26A 2 + 22^3 + i8A 4 + 6A 5 , 5iz 3 = nA L + 22^ + 50^3 4- 27^4 4 + gA s , i7z 4 = 3 A,+ 6A 2 + 9 A i + i2A 4 + 4A 5 , I7z 5 = A, + 2A 2 + 3^3 +- 4A 4 + 7^ 5 , where all the weights are at once seen, and from which the values of the unknown quantities can easily be found. Il6 INDEPENDENT OBSERVATIONS. VII. As indicated in Art. 99, the numerical operations may be somewhat simplified by taking the unknown quantities as cor- rections to be applied to assumed elevations of Z it Z„ etc. Thus it is seen from the observations that 573 and 575 feet are approximate elevations for Z y and Z 2 . By writing, then, elevation of Z, = 573 -f- z„ elevation of Z 2 = 575 + 2 2 , elevation of Z 3 = 742 + z 3 , elevation of Z 4 : elevation of Z s -■ = 745 + * 4 » = 3 2 ° + z s> the following simpler observation equations are obtained from the given data : 2, = 0.08, Z 2 — Z, = 0.60, 2 2 = 0.27, 2 3 — 2 2 = 2 4 - 2 3 = o-33, 0.80, 2 4 — 2 2 = 0.28, 2 4 - 2 5 = 0.00, Z 5 = - - 0.09, 2 S = " - 0.25. From these the normal equations are formed, whose first mem- bers are the same as written above, and whose second mem- bers have the values A, = — 0.52, A 2 = -\-o.26, A 2 = — 0.47, A 4 = -\- i-o8, A s = — 0.34. The solution of the normal equa- tions gives 2, = — O.I9, 2 2 = O.14, 2 3 = O.O5, 2 4 = O.43, 2 5 = O.O3 J and the final elevations are Z = 573-°° - °-i9 = 57 2 - 8l > Z 3 = 575.00 + 0.14 = 575.14, etc., which are the same as found in Art. 100. § I03- ANGLES AT A STATION. 11/ Angles at a Station. 103. When two angles and also their sum are observed at a station, the observed sum usually differs from the sum of the two measured single angles. Let the observation of the first angle give the result M„ of the second M 2 , and that of their sum M r Then M z + M 2 is greater or less than M 3 by a cer- tain discrepancy d. It is required to adjust the observations,, regarding the weights as equal, and to find the probable errors of the adjusted values. 1st, Let z, and z 2 be the most probable corrections to the observed values M, and M 2 , so that M l -f- s, and M 2 -f- z 2 are the most probable values of the first and second angles. The observation equations then are M,+z L = M„ M 2 + z 2 =M 2 , {Mi + 2.) + (M 2 + z 2 ) = M v which reduce to 2, = o, z 2 = o, 2, + z 2 = M, - (Mi + M 2 ) = d. 2d, From these, the normal equations are 22, + Z 2 = d, 2, + 2Z 2 = d. 3d, The solution of the normal equations gives 2, = \d, and z 2 = \d, for the most probable values of the corrections : hence the adjusted values are M x + id, M 2 + id, M 3 - id. n8 INDEPENDENT OBSERVATIONS. VII. 4th, The residuals are evidently the three corrections, the sum of whose squares is \d z ; then, from (32), /-=o.6745vV 2 = 0.389(7', which is the probable error of a single observed value. 5th, By the method of Art. 75 it is easy to find that the weights of the adjusted values of z, and z 2 are 1.5 : hence their probable errors are o. 389(7' = 0.318^, and evidently the probable error of the adjusted value of Zi + "2 is a l so 0.318*/. 104. When several angles are observed at a station, several sums and differences of simple angles are often taken. For example, the following are the angles observed at the Station Hillsdale, on the United States Lake Survey ; each being the mean of nearly the same number of readings, and hence re- garded as of the same weight. (See Report of United States Lake Survey, p. 449.) No. Between Stations. Observation. 1 Bunday and Wheatland 44° *5' 4 o".6i 3 2 Bunday and Pittsford 80 47 32.819 3 Wheatland and Pittsford 36 21 51.996 4 Pittsford and Reading 9 1 34 24.758 5 Pittsford and Bunday 279 12 27.619 6 Reading and Quincy 62 37 43405 7 Quincy and Bunday ™5 00 18.808 §104. ANGLES AT A STATION: II 9 The annexed figure shows the relative positions of the sta- tions and of the seven observed angles. It is required to adjust the observed results, and to find their probable errors. 1 st, Let Z"„ Z v Z 4 , and Z b be the required most probable values of four of the simple angles as indicated in Fig. 7 ; then the observation equations are Z, = 44 25' 4o".6i3, Z, + Z z = 80 47 32.819, Z 3 = 36 21 51.996, Z 4 = 91 34 24.758, 360 - (Z.+Z,) =279 12 27.619, Z, = 62 37 43405. .°-ez I + z 3 + z, + z b ) = 125 00 18.808. Assume the measured values of Z„ Z v Z v and Z b as approxi- mate, and let z n z 3 , z v and z b be the most probable corrections, thus Z, = 44 25' 4o".6i3 + z„ Z 3 = 36 21 51.996+23, Z 4 = 9 1 34 24.75 8 + z 4. Zb = 62 37 43.405 + z 6 . 120 INDEPENDENT OBSERVATIONS. VII. Then, by inserting these values in the observation equations, the following simpler observation equations are found : 2, = O, Z, + 2 3 = + 0.2IO, h = o, Z 4 = O, z, 4- z 3 = — 0.228, z b = o, z, + z 3 4- z 4 + 2 6 = 4- 0.420, in which the right-hand members denote seconds only. 2d, The normal equations are now easily written, either by the rule of Art. 48, or by the help of the algorithm of formulas (10) and (11). They are 4Z> -t- 3 Z 3 + Z 4+ z 6= + O.4O2, 3 2 . + 4 2 3 + 2 4 + «6 = 4- 0.402, 2, 4- z 3 4- 2Z 4 4- 25 = 4- 0.420, z, 4- z 3 4- z 4 4- 2Z 6 = 4- 0.420. 3d, The solution of these equations gives Z, = Z 3 = 4- o".022, Z 4 = Z 6 = o".I26. The addition of these corrections to the approximate values gives the most probable values of the angles Nos. 1, 3, 4, and 6; and from these, by simple addition, the most probable values of Nos. 2, 5, and 7, are obtained. Thus, the adjusted values are No. 1 = 44 25' 40". 635 = Z„ No. 3 = 36 21 52.018 =Z 3 , No. 4 = 91 34 24.884 = Z 4 , No. 6 = 62 37 43.531 = Z b , No. 2 = 80 47 32.653 = Z, 4- Z 3 , No. 5 = 279 12 27.347 = 360 - (Z, 4- Z 3 ), No. 7=125 00 18.932 = 360 - (Z, 4-Z 3 4-Z 4 + Z 6 ). JI04. ANGLES AT A STATION. 121 4th, The differences between the observed and the adjusted values are the residuals, which, with their squares, are thus arranged : No. Observed. Adjusted. V. V 1 . 1 4o".6i3 40".635 + 0022 0.0005 2 32.819 3 2 -6S3 — 0.166 1 .0276 3 51.996 52.018 + 0.022 .0005 4 24.758 24.884 + 0.126 .0159 5 27.619 27-347 — 0.272 .0740 6 43-405 43-53 1 + 0.126 .0159 7 18.808 18.932 + 0.124 .0154 The sum 2v 2 is here 0.1498; and hence, by formula (32), the probable error of a single observation is , /0.1498 „ r= o.6 7 45y/ — ^- = o .151. 5th, By writing A for the absolute term in the first normal equation, and zero for the absolute terms in the other nor- mal equations, the solution gives the value of z t as jA ; and hence the weight of z t is 1.7. In a similar way the weight of z A is found to be 1.4. The probable errors of the adjusted values of z x and z z are now Q-I5 1 \jT7-] o".n6; and those of the adjusted values of z A and z b aic 0£5_I o".ia8. 122 INDEPENDENT OBSERVATIONS. VII. In order to find the probable errors of angles Nos. 2, 5, and 7, it would be necessary to represent them by single letters, and to form and solve another set of normal equations. 105. As an example of angles with unequal weights, the fol- lowing observations at North Base, Keweenaw Point, on the United States Lake Survey, will next be considered : No. Between Stations. Observed Angle. Weight. I Crebassa and Middle 55° 57' 58".68 3 2 Middle and Quaquaming 48 49 x 3- 6 4 J 9 3 Crebassa and Quaquaming 104 47 12.66 17 4 Quaquaming and South Base 54 38 15-53 13 5 Middle and South Base 103 27 28.99 6 Let Z„ Z 2 , and Z^ represent the angles Nos. 1, 2, and 4; then the observation equations are £= 55° 57' 5&"-68, with weight 3, £ = 48 49 i3- 6 4, with weight 19, z t + z 2 = 104 47 12.66, with weight 17, z< = 54 38 15-53, with weight 13, z 2 + z A = 103 27 28.99, with weight 6. Let z It b 2 , and z 4 be corrections to the measured values of Z„ Z„ and Z^\ then the simpler observation equations are z, = o, with weight 3, z 2 = o, with weight 19, 2 i + z 2 = +■ 0.34, with weight 17, z 4 = o, with weight 13, z* + 2 4 = — 0.18, with weight 6. §«o 5 . ANGLES AT A STATION. 123 From these, the normal equations are formed, either by the rule of Art. 50, or by the help of the algorithm of formulas (12) and (13). They are 20z, + i7z 2 = + 5.78, i7z I + 42z 2 + 6z 4 =+4-70, 6z 2 + 1924 = — 1.08. The solution of these equations gives z, = + o".28s, z 2 = + o".oo5, z 4 = — o .059. Hence the following are the adjusted angles No. 1= 55 57'S8"965, No. 2 = 48 49 13.645, No. 3 = 104 47 12.610, No. 4 = 54 3 8 iS-47 1 . No. 5 = 103 27 29.116. To find the probable errors, the residuals are next obtained. No. Observed. Adjusted. w. v z . A pv z . 1 5 8".68 S8"-965 + 0.285 0.0812 3 0.244 2 13.64 13-645 + 0.005 .0000 19 .000 3 12.66 12.610 - 0.050 .0025 17 .042 4 15-53 I5-47 1 - 0.059 •0035 J 3 •045 5 I 28.99 29.116 + 0.126 .0159 6 •°95 The sum tpv 2 is here 0.426 ; then, by (32), ./0.426 „ r = 0.6745!/ — — = o". 3 i, 124 INDEPENDENT OBSERVATIONS. VII. which is the probable error of an observation of the weight unity. The probable error of the observed angle, No. 2, is, then, o. -ji J rr r 2 = —=- = o .07. V/19 The probable error of the final value of No. 2 must be less than o".07, since its weight is increased by the adjustment. Empirical Constants. 106. One of the most important applications of the Method of Least Squares is the deduction, from observations, of the values of physical constants or co-efficients. In all such cases a theoretical formula or law is first established, which contains the co-efficients in a literal form ; and this law serves to state as many observation equations as there are observations. The method of procedure is then exactly the same as that outlined in the first article of this chapter. The precision of the values deduced for the constants depends, of course, upon the precision and number of the observations which enter the discussion. As an example, take the determination of the ellipticity of the earth by means of experiments on the length of the seconds' pendulum. In 1743 Clairaut deduced the following remarkable law : s = S + S(§A -/)sinV, in which .S is the length of the seconds' pendulum at the equator, and s its length at any latitude /, while k is the ratio of the centrifugal force at the equator to gravity, and f is the fraction expressing the ellipticity of the earth. This may be written s = S + TiAa't. § io6. EMPIRICAL CONSTANTS. 12$ Now, by measuring s at two different latitudes, two equations would result, from which values of S and T could be found ; and, by measuring j- at many different latitudes, many equa- tions would result, from which the most probable values of 5 and T may be found. The following, for instance, are thirteen observations, taken by Sabine in the years 1822-24: Place. Latitude. Length of Seconds' Pendulum. English Inches. Spitzbergen + 79° 49' 58" 39.21469 Greenland 74 3 2 1 9 39- 2 °335 Hammerfest 70 40 5 39- I 95 I 9 Drontheim 63 2 5 54 39-!745 6 London Si 31 s 39- J 39 2 9 New York 40 42 43 39.10168 Jamaica !7 5 6 7 39.03510 Trinidad 10 38 56 39.01884 Sierra Leone 8 29 28 39.01997 St. Thomas 24 41 39.02074 Maranham — 2 31 43 39.01214 Ascension 7 55 48 39.02410 Bahia 12 59 21 39.02425 For each of these an observation equation is now to be written. Thus, for the first, s = 39.21469. /= 79° 49' 58"- sin/ = 0.9842965. sin 2 / = 0.9688402. 39.21469 = S + 0.96884027! 126 INDEPENDENT OBSERVATIONS. VII, And in like manner the following thirteen observation equations are stated : 39.21469 = S + 0.96884027I 39- 2 °335 = $ + 0.92893047: 39.19519 = S + 0.89041207". 39.17456 = S + 0.79995447. 39.13929 = S + 0.61279667. 39.10168 = S + 0.42543857. 39.03510 = 5 + 0.09482867. 39.01884 = S + 0.03414737. 39.01997 = S + 0.02180237. 39.02074 = S + 0.00005157. 39.01214 = ,S + 0.00194647. 39.02410 = S + 0.01903387. 39.02425 = S + 0.05052017. The normal equations formed from these are 508.18390 = 13.000000.S + 4.8487027J 189.94447 = 4.843702S + 3.7043947, whose solution gives S = 39.01568 inches, 7= 0.20213 inches, as the most probable values that can be deduced from the thir- teen observations. Hence the length of the seconds' pendulum at any latitude, /, may be written j- = 39.01568 + 0.20213 sin 2 /. The values thus deduced for S and T are empirical constants. To find from them the ellipticity/, it is easily seen that 7 J * S § lOy. EMPIRICAL CONSTANTS. 127 and, as the value of k is known to be — -, that of_/is /= o. 0086501; — 0.0051807 = — . J J 288.2 If desired, the precision of the constants 5 and T may be investigated by determining their weights and probable errors, and from these the precision of the value of f may also be inferred. 107. When two quantities x and y are connected by the relation y = Sx-\-T the method of the last article can, in strictness, only be applied to find the most probable values of 5 and T when the observed values of x are free from error. If x is liable to error as well as y, the following method may be used * First let the value of S be found, supposing that x is without error, and let this be called 5, . Secondly, let the value of 5 be found regarding y as without error, and let this be called S, . Let each observed value of x have the weight p, and each observed value of y have the weight unity. Then the most probable value of S is found by solving the quadratic equation and, if n be the number of pairs of observations, the formula 1 n T=-[2y- S.2x gives the most probable value of T. The following numerical example will illustrate the method. In order to determine the most probable equation of a cer- * Report U. S. Coast and Geodetic Survey, 1890, p. 687. 128 INDEPENDENT OBSERVATIONS. VII. tain straight line the abscissas and ordinates of four of its points were measured with equal precision, giving y = 0.5, 0.8, 1.0, and 1.2. x = 0.4, 0.6, 0.8, and 0.9. First, supposing that the values of x are without error, the fqur observation equations are written : — 0.5 = 0.4S + T, 0.8 = o.65 + T, 1.0 = o.ZS + T, 1.2 = o.qS + T. And then, forming and solving the normal equations, there is found 5, = 1.339. Secondly, supposing that the values of y are without error, the equation of the line must be written in the form x = l-T=Uy+V, and the observation equations are 0.4 = 0.5^ + V, 0.6 = 0.8 £7+ V, 0.8 = 1.0 17+ V, 0.9 = 1.2 U + V; from which the normal equations are derived, and by their solution U— 0.7385, or 5 2 = 1.354. These values of S, and S, give the quadratic equation S' — 0.6075 — 1 = o, whence S= 1-348, and then T is found to be — 0.035, an ^ accordingly y = 1.348* — 0.035 is the most probable equation of the line as derived from the four pairs of observations. 107'. The determination of the elements of the orbit of a comet or planet is another instance of the deduction of em- pirical constants. Here the observed quantities are the right § lOf. EMPIRICAL CONSTANTS. 1 29 ascension and declination of the body at various points in its orbit. Through any three of these points a curve may be passed, and an orbit computed by the formulas of theoretical astronomy. The problem, however, is to determine the most probable orbit by the use of all the observations. The first step, after collecting and reducing the observations, is to select a few favorably situated, and from them to compute the approximate elements of an elliptical or parabolic orbit, as the case may require. With these approximate elements, the places of the body are computed for as many dates as there are obser- vations, and the differences between the computed and observed places found. A theoretic differential formula is next estab- lished for a difference in right ascension, and another for a difference in declination, in terms of unknown corrections to the assumed elements, and of co-efficients that may be com- puted from the observations. Each observation thus furnishes a difference, and each difference an observation equation, whose unknown quantities are the corrections to the approximate ele- ments of the orbit. From the observation equations the normal equations are derived and solved, and the most probable set of corrections found. Lastly, the application of these corrections to the approximate elements furnishes the most probable ele- ments that can be deduced from the given observations. The process thus briefly described is very lengthy in its actual application. For instance, in Hall's determination of the elements of the orbit of the outer satellite of Mars * there are forty-nine observation equations, each containing seven unknown corrections, and forty-nine others, each containing six. From these the seven normal equations were formed, and by their solution the most probable values found for the correc- tions. The precision of the elements of the orbit was also deduced by computing the probable errors of the corrections. * Hall's Observations and Orbits of the Satellites of Mars ; Washington, 1878. 130 INDEPENDENT OBSERVATIONS. VII. Empirical Formulas. 108. The case of the last article is that of a rational formula with empirical constants. An empirical formula, on the other hand, is one assumed to represent certain observations, and which is not known to express the law governing them. The constants in such formulas are also best determined by the application of the Method of Least Squares. The first step in the establishment of an empirical formula is to plot the given observations, taking one observed quantity as abscissas, and the other as ordinates. Let y and x be the two quantities between which an empirical formula is to be established. The plot shows to the eye how y varies with x. If y is a continually increasing function of x, or if the curve resembles a parabola, the general equation (40) y = S + Tx+ C7x 2 +Vxi + etc., maybe written to represent the relation between y and x. This equation applies to a large class of physical phenomena, such as relations between space and velocity, volume and tempera- ture, stress and strain, and other similar related quantities. The letters S, T, U, etc., represent constants whose values are to be determined from the observations. Another large class of phenomena may be represented by the general equation t \ c , t ■ 3 6o ° , -r> 3 6o ° (41) y=S+Tsm- — x + T cos - — x m m •?6o° 360 + (7 sin ~ 2x + cV'cos 2x + etc., m m in which, as x increases, y passes through repeating cycles. As such may be mentioned the variation of temperature through- out the year, the changes of the barometer, the ebb and flow of the tides, the distribution of heat on the surface of the earth depending on latitude, and, in fact, all phenomena which repeat § 109- EMPIRICAL FORMULAS. 131 themselves like the oscillations of a pendulum. The letters S, T, U, etc., represent constants which are to be found from the observations ; while ;« is the number of equal parts into which the whole cycle is divided, and must be taken in terms of the same unit as x. If the several cycles are similar and regular, only the first three terms are required to represent the variation. Other general empirical formulas than (40) and (41) are also employed in discussing physical phenomena. Exactly what formula will apply to a given set of observations, so as to agree well with them, and at the same time be of use in other similar cases, can only be determined by trial. The investigator must, from his knowledge of physical laws, assume such an expression as seems most plausible, and then deduce the most probable values of the constants. The comparison of the observed and calculated results furnishes the residuals, from which, if desired, the probable errors may be deduced. When several empirical formulas have been determined for the same observations, that one is the best which furnishes the smallest value for the sum of the squares of the residuals. 109. Consider as a first practical example the deduction of the equation of the vertical velocity curve for the observations giver, on p. 244 of the second edition of the " Report on the Physics and Hydraulics of the.Mississippi River," by Humphreys and Abbot. The grand means of the measurements give the following results for the velocities at different depths below the surface : At surface, 3.1950 feet per second. At 0.1 depth, 3.2299 feet per second. At 0.2 depth, 3-2532 feet per second. At 0.3 depth, 3.261 1 feet per second. At 0.4 depth, 3.2516 feet per second. At 0.5 depth, 3.2282 feet per second. At 0.6 depth, 3.1807 feet per second. At 0.7 depth, 3.1266 feet per second. At 0.8 depth, 3.0594 feet per second. At 0.9 depth, 2.9759 feet per second. 132 INDEPENDENT OBSERVATIONS. VII. 2.9 S.0 3 1 r S 5. 3 3 0.1 ®s v 0.2 \_ 0.3 -X- 0.4 ~^T_ 0.5 ~lL- 0.6 rf" f~ 0.7 y 0.8 0.9 These observations may be plotted by dividing a vertical line representing the depth of the river into ten equal parts, through the points of division drawing '"••"'" horizontal lines, and laying off upon these the observed velocities. On the annexed figure the points enclosed within small circles represent the observations. Each hori- zontal division of the diagram is o. i feet per second, and each vertical division is one- tenth of the depth. Let y be the velocity at any point whose depth below the surface is x, . the total depth of the river being unity, and assume that three terms of formula (40) will give the relation between y and x, or that y = S + Tx + Ux\ This is equivalent to assuming that the curve of vertical veloci- ties is a parabola, with its axis horizontal. The observations furnish the values of y for ten values of x ; and thus, for determining .S, T, and U, there are the following ten observation equations : — 3.1950 = S + 0.0T+ o.oocV. 3.2299 = S + 0.1 7*+ 0.01U. 3.2532 — S + 0.2T + 0.04K 3.2611 = S + 0.37"+ 0.09K 3.2516 = S -f- 0.47"+ 0.16U. 3.2282 = S + 0.5 T -f- 0.25 ££ 3.1807 = S + 0.67+ 0.36K 3.1266 = S + 0.7.7 4- 0.49C. 3.0594 = S + 0.8T+ 0.64K 2.9759 = S + o.gT+ 0.81C/. § 109. EMPIRICAL FORMULAS. I3J From these the following three normal equations are found : 31.761600 = 10.00.S + 4.5007"+ 2.8500K 14.089570 = 4.50,5'+ 2.8507'+ 2.0250K 8.828813 = 2.8 5 5 + 2.0257-+ 1.5333a And their solution gives S= + 3.i95*3> T = + 0-44253, U= - 0.7653. Accordingly, the empirical formula of vertical velocities is y = 3-195 J 3 + 0.44253^ - 0.7653* 2 , where y is the velocity in feet per second at any decimal depth x. The curve corresponding to this formula is drawn on the above diagram. The following is a comparison of the observed velocities with those computed from this empirical formula : X. Observed/. Computed/. V. vf. 0.0 3-!9S° 3-J95 1 — 0.0001 0.000000 0.1 3.2299 3- 2 3!7 — 0.0018 3 0.2 3-2532 3- 2 S3° + 0.0002 °-3 3.2611 3-2590 + 0.0021 4 0.4 3-2516 3-2497 + 0.0019 4 °-5 3.2282 3-2251 + 0.0031 10 0.6 3.1807 3-1851 — 0.0044 19 0.7 3.1266 3.1299 — 0.0033 II 0.8 3-0594 3-°594 0.0000 0.9 2 -9759 2 -9735 + 0.0024 6 1.0 2.8724 134 INDEPENDENT OBSERVATIONS. VII The sum of the squares of the residuals is here 0.000057, and hence /0.0c = 0-6745V * 10 000057 0.0019 is the probable value of a residual, or the probable difference between an observed and computed velocity. The agreement between the parabola and the observed points is very close.* no. As a second example, consider the deduction of a formula to express the magnetic declination at Hartford, Conn., for which place the following observations are given on p. 225 of the United States Coast and Geodetic Survey Report for 1882 : Date. Dlu nation. 1786 5° 25' w. 1810 4 46 1824 5 45 1828-29 6 03 27 July, 1859 7 17 16 Aug., 1867 7 49-3 25 July, 1879 8 34-o From numerous records at various places, it is known that the declination oscillates slowly to and fro, passing through a cycle in a period varying, at different places, from two hundred and fifty to four hundred years. The variation in New England * See further, concerning this curve, in Journal Franklin Institute, 1877, vol. civ, p. 233; also Van Nostrand's Magazine, 1877, vol. xvii, p. 443, and 1878, vol. xviii, p. 1. The reasoning of Hagen concerning the probable errors on p. 447 of the second article is thought to be incorrect. § no. EMPIRICAL FORMULAS. 135 may be roughly represented by the annexed figure, where the ordinates to the curve show the relative values of the declina- tion at the respective years. Formula (41) is hence applicable to the discussion of the above observations. Let y be the magnetic declination at the time x, and assume the empirical relation y = S + T sin ^-x + T cos ^S-x. m m Here there are four constants, S, T, T', and m, to be found by the Method of Least Squares from the given observations. The only practical way of procedure is to assume a plausible value of m, and then to state the observation equations and normal equations, from which values of S, T, and T' may be deduced. Again : assume another value of m, and repeat the computation, thus finding other values for S, T, and T' . If necessary, the computation is to be repeated for several values of m ; and for each formula thus deduced the residuals, or differences between the observed and computed values of y, are to be found. Then that value of m and that formula is the best which makes the sum of the squares of the residuals a minimum. 360° Take for m the value 288 years ; then formula is VI is 1.25, and the S + Tsin 1.25* + T' cos 1.25* = y. Let x be the number of years counted from the epoch, Jan. 1, 136 INDEPENDENT OBSER VA TIONS. VIL 1850, and let all angles be expressed in degrees and decimals; then, for the first observation, x = 1786.5 — 1850.0 = —63.5 years, 1. 25.x = —79.4 degrees, sin 1. 25.* = —0.983, cos 1. 25.x = +0.184, y = 5.42 degrees, and hence the first observation equation is S - o, 9 8t,T + 0.1847"= 5.42. In like manner the following tabulation is made No. Date. X. 1.25*. Sin 1.25.*. COS l.2^X. y- 1 1786.5 -63-5 -79°4 -O.983 + O.184 +5°-42 2 1810.5 -39-5 -49.4 -0-759 + O.65I +4-77 3 1824.5 - 2 5-S -31-9 -0.528 + O.849 + 5-75 4 1829.0 — 21.0 -26.25 — 0.442 + O.897 + 6.05 S 1859.6 + 9-6 + 12.0 +0.208 + O.978 + 7.29 6 1867.6 + 17-6 + 22.0 +0-375 + O.927 + 7.82 7 1879.6 + 29.6 + 37-0 + 0.602 + O.799 + 8.57 From the last three columns the seven observation equations are written ; and from these the three normal equations are easily formed, either by the rule of Art. 48, or by the help of the algorithm of formulas (10) and (11). They are + 7.006-- i.5 3 r+ 5.287"- 45.67 = o, — 1 -53$ + 2.56T — 0.517"+ 5.03 = 0, + 5.286 - 0.517 + 4.537" - 35.64 = o, and their solution gives £=+8°.o6, T= +2°.6o, T'= -i°.2 9 . § IIO. EMPIRICAL FORMULAS. 1 37 Hence the empirical formula is y = 8°.o6 + 2". 60 sin 1.25X — i°.29Cos 1.25X. This may also be written y = + 8°.o6 + 2 . 90 sin {^"^x — 26°.4), which is a more convenient form for discussion.* The following is a comparison of the observed declinations with those computed from this formula : Date. A. Observed y. Computed j. V. 1786.5 -63-5 + 5°-42 5°. 2 8 +0.14 1810.5 -39-5 4-77 5- 2 5 —0.48 1824.5 -25-5 5-75 5.60 +0.15 1829.O — 21.0 6.05 5-76 +0.29 1859.6 + 9-6 7.29 7-34 -0.05 1867.6 + 17-6 7.82 7.84 —0.02 1879.6 + 29.6 8-57 8-59 — 0.02 The sum of the squares of these residuals is 0.36, and hence, by (32), = 0.6745 y/^ 3 — = 0.19, T 7 — 3 which gives the probable error of a single computed value if the observations be regarded as exact, or the probable error of an observation if the law expressed in the empirical formula be regarded as exact. * See the numerous valuable papers by Schott, in the Reports of the United States Coast and Geodetic Survey, the latest of which is in the Report for 1882, pp. 211-276. The above formula for the declination is the one there adopied, as giving the best value of the period m. 138 INDEPENDENT OBSERVATIONS. VII. in. Lastly, consider the deduction of a formula to represent certain experiments, made by Darcy and Bazin, on the flow of water in a rectangular wooden trough lined with cement. The width of the trough was 1.8 12 meters, and its slope 0.0049. Water was allowed to run through it with varying depths ; and for each depth the mean velocity was measured, and the hydrau- lic radius of the water-section computed by dividing the wetted perimeter into the area of the section. The following are the results, the hydraulic radius li being given in meters, and the mean velocity m in meters per second : No. A. PI. I 1 144 J-73 1 2 1312 1-853 3 1445 1.984 4 1579 2.081 5 1701 2. 171 6 1813 2.258 7 !9 2 5 2.326 8 2026 2 -397 9 2123 2.460 Assume the expression sh* and let it be required to find from the above experiments the most probable values of s and /. First reduce the expression to a linear form by writing it thus : log m = log j- + t\ogh. Each observation furnishes an observation equation containing log s and t. For example, the first is 0.2383 = log* — 0.9416/. §112. PROBLEMS. *39 The twelve observation equations furnish the two normal equa- tions, and their solution gives t = 0.572, log J- = 0.7767, .-. x = 5.98. Therefore the empirical formula m = 5.98A " 2 is the best of the assumed form that can be derived from the nine experiments. 112. Problems. 1. The following levels were taken to determine the elevations of five points, T, U, W, X, and Y, above the datum O : T above O = 115.52. U above T = 60.12. U above O = 177.04. W above T= 234.12. W above U =.171.00. What are the adjusted elevations ? Ans. T ': X above W= 632.25. X above Y = 211.01. Y above U = 596.12. Y above W= 427.18. 115. 61, U= 176.95, etc. 2. Four angles are observed at a station, and also their sum. The observed sum differs from the sum of the four observed parts by the discrepancy d. What are the adjusted values? 3. Adjust the following angles, taken at the station Moodus, and find the probable errors of the adjusted values. No. Between Stations. Observed Angle. Weight. I 2 3 4 5 Big Rock and Small Rock Small Rock and Tokus Small Rock and Buzzard Buzzard and Tokus Tokus and Big Rock 99° 42'i5"-6l 133 39 05.07 40 12 52.43 93 26 13.14 126 38 40.69 «37 22 57 50 20 Ans. 99° 42' i5".46, etc. I4O INDEPENDENT OBSERVATIONS. VII. 4. The following observations of the temperature at different depths were taken at the boring of the deep artesian well at Grenelle in France, the mean yearly temperature at the surface being io°.6o centigrade : 1. Temperature at a depth o£ 28 meters = 11. 71 degrees. 2. Temperature at a depth of 66 meters = 12.90 degrees. 3. Temperature at a depth of 173 meters = 16.40 degrees. 4 Temperature at a depth of 248 meters = 20.00 degrees. 5. Temperature at a depth of 298 meters = 22.20 degrees. 6. Temperature at a depth of 400 meters = 23.75 degrees. 7. Temperature at a depth of 505 meters = 26.45 degrees. S. Temperature at a depth of 548 meters = 27.70 degrees. Deduce from these observations the empirical formula t = 10°. 6 + 0.041:5* — 0.0000193X 2 , where / is the temperature at a depth of x meters. 5. Gordon's formula for the ultimate strength of columns may be written in which c is the crushing-load per unit of area of cross-section, j the ratio of the length of the column to its least diameter, and S and T are constants to be found by experiment. Determine the best values of these constants for the following four experiments on wrought-iron Phcenix columns : c = 34650, 35000, 365S0, 37030. /= 42, 33> 2 4, '9-5- 6. From several census records of the population of the United States deduce an empirical formula showing the population for any year. §113- METHOD OF PROCEDURE. 141 CHAPTER VIII. CONDITIONED OBSERVATIONS. 113. The general method of adjusting conditioned observa- tions has been deduced in Arts. 56, 57, and that of investigating the precision in Arts. J j, 78. The following is the process : 1st, Having given n observations upon q quantities subject to n' rigorous conditions, the first step is to represent the quan- tities by symbols, and state n observation equations and ti' con- ditional equations. Generally it will be found most convenient to take the unknown quantities as representing corrections to assumed approximate values, and to state the observation and conditional equations in terms of these corrections. 2d, From the «' conditional equations find the values of n' unknown quantities in terms of the remaining q — 11 quanti- ties, and substitute these values in the 11 observation equations, each of which then represents an independent observation. 3d, Adjust these n observation equations by the method of Chap. VII, and find the most probable values of the q — ri quantities. Then, by substitution in the conditional equations, the most probable values of the remaining n' quantities are known. 4th, Insert the adjusted values in the n observation equa- tions, and find the residuals, and then, from (33), the probable error of an observation of the weight unity. If desired, the weights of the adjusted values may be found by Art. 75, and their probable errors by (31). 142 CONDITIONED OBSERVATIONS. VIII. 114. The special method of correlatives, which is particu- larly valuable in the adjustment of geodetic triangulations, has been explained in Art. 58. In order to apply it, the local adjustments should first be made ; so that for each quantity, s„ z 2 . . . z q , a value, M u M 2 . . . M q , called the observed value, is known. The numbers q and n are hence equal. The fol- lowing are the steps of the practical application : 1st, For the rigorous conditions write n' conditional equa- tions, as in (14). Substitute in these the observed values, M„ M 2 . . . M q , for the quantities z lt z 2 . . . z g ; and let d x , d 2 . . . d q be the differences or discrepancies that arise. 2d, Assume ri new unknown quantities, or correlatives, K lt K 2 . . . K„>, and write the normal equations (16). Solve these normal equations, and thus find the values of the correlatives. 3d, From (15) find the corrections v lt v 2 . . . v q , which, when applied to the observed values M u M 2 . . . M q , give the most probable adjusted values. 4th, Compute the sum 2/w 2 , and from (34) find the proba- ble error of an observation of the weight unity. The probable error of any observed M is then easily found from (31), and that of the corresponding adjusted value is somewhat smaller, since the weights are increased by the adjustment. Angles of a Triangle. 115. When the three observed angles of a plane triangle are of equal weight, it is easy to show that the correction to be applied to each is one-third of the discrepancy between their sum and 180 . The following is the proof by the method of correlatives : 1st, Let M lt M 2 , and M z be the observed values, and s„ .r,, §Il6. ANCLES OF A TRIANGLE. I43 and z } the required most probable values. The conditional equation is 2 t + Z 2 + 2 3 — l8o° = O. Substitute in this the observed values, and it does not reduce to zero, but leaves a small discrepancy d ; thus M, + M 2 + M l - 180 = d. By comparison with (14) it is seen that a, = a 2 = a 3 = -f- 1. 2d, Take K as the single correlative. The weights are all equal, or p = 1. From (16) the single normal equation is [aa].K + d — o, or j,K + d = o, from which K =. -¥■ 3d, From (15) the three corrections now are v, = d 3 d 3 v i = ~ d 3 and, accordingly, the most probable values of the three angles are z = M l , « 2 = A, , z, = Ik, 3 3 3 d 2 4th, The sum of the squares of the residuals is — , and hence 3 by (34) the probable error of a single observed angle is O.^gd. By working the problem according to the general method of Art. 113, it may be shown (as in Art. 103) that the probable error of an adjusted angle is 0.32^. 116. When the three observed angles of a plane triangle are of unequal weights, it is easy to show that the corrections to b 144 CONDITIONED OBSERVATIONS. VIII. applied are inversely as the weights. For instance, take the following numerical case : M, = 36° 25' 47", with weight 4 M-, = 90 36 28, with weight 2 M 3 — 5 2 57 57, with weight 3 Sum = 180 00' 12" 1st, Take z l , z 2 , and ^ 3 as the most probable values; then, as before, the conditional equation is z t + z 2 + z 3 — 180° = o. The discrepancy is 12". To compare with (14), (15), and (16), a, = a 2 = a 3 = -f- I, />, = 4, p 2 = 2, and / 3 z= 3. 2d, Only one correlative is necessary ; and from (16) the single normal equation is and hence K = — ^ = — 1 1.08. 3d, From (15) the corrections now are v x = — =- 2". 7 7, z/ 2 = - s". S4 , z> 3 = _ 3 ".69, 4 and the adjusted angles are z r = 36 25' 44".23 z 2 = 90 36 22.46 z 3= 52 57 53-31 Sum = 180 00' oo".oo 4th, The residuals are the three corrections v„ v 2 , and v v and the sum of their weighted squares is 2/v 2 = 132.92, from which, by (34), r = 7" ' .jj for the probable error of an observa- § 1 1 8. ANGLES AT A STATION. I45 tion of the weight unity. By (31) the probable errors of the observed values are found to be ri = 3"- 8 9> r 2 = s".so, r 3 = 4".49, and the probable errors of the adjusted values are somewhat less than these. The adjustment of the angles of a spherical triangle differs from that of a plane triangle only in the introduction of the spherical excess into the conditional equation ; thus s -\- 1 -\- u = 180 -\- spherical excess. Angles at a Station. 117. When n angles, and also their sum, are observed at a station, and the weights are all equal, it is easy to show, as in Art. 103, that the correction to be applied to each observed angle is th of the discrepancy between the observed sum n -\- 1 and the sum of the observed single angles. When n angles, which close the horizon, are observed at a station, and the weights are equal, it is easy to show, as in Art. 115, that the correction to be applied to each observed angle is -th of the discrepancy between 360 and the sum of n the observed angles. When angles at a station close the horizon, or are observed by sums or differences, the adjustment may be effected, either for equal or unequal weights, by the method of Art. 113, or by that of Art. 114. The former will always reduce to the method of independent observations, as exemplified in Arts. 103-105. 118. As an example of the application of the method of cor- relatives, consider the observations of Art. 104. Represent the most probable values of the seven angles by z„ s z . . . z r I46 CONDITIONED OBSERVATIONS. VIII. From Fig. 7 the following conditions are seen : z, — z 2 + z 3 = o, z 4 — z 5 +z 6 + z 7 = o, 2, + Z 3 + 2 4 ■+• 26 + Z 7 — 360 = O. By substituting in these the observed values, the following dis- crepancies are found : — d T = — 0.210, d 2 = — 0.648, d z = — 0.420. Take K u K 2 , and X z as the correlatives to be determined. By comparison with (14), it is seen that a, = + I, a, = — I, a 3 = + I, a 4 = a 5 = a« = a 7 = O, ft = ft = ft = O, ft == ft = ft = + I, ft = - I, 7- = 73 = 74 = 76 = 7z = = + *> 7* = 7s = °- All weights are unity. The three normal equations then are, from (16), 2,K 1 4- 2K 3 — 0.210 = o, + 4 /sT 2 + 3X3 - 0.648 = o, 2K1 + $K 2 + 5^3 — 0.420 = o, and their solution gives ^,.= 4-0.167, ^=4-0.271, J£ 3 = —0.145. From (15) the corrections now are V 1 = + K, + K 3 = + o".022, v 2 — — K l = — 0.167, v z = + K x + X 3 = 4- 0.022, v 4 = + K 2 + K z = + 0.126, v s = 4- K 2 = 4- 0.271, v b = 4- K 2 4- K i = 4- 0.126, v 7 = + K 2 + K % = 4- 0.126, §"9- ANGLES OF A QUADRILATERAL. H7 and if these be applied to the observed values M„ M 2 . . . M v the adjusted values are found the same, within one or two thou- sandths of a second, as in Art. 103, the slight difference being due to the neglect of the fourth decimal places. Angles of a Quadrilateral. 119. In a quadrilateral WXYZ, the two single angles at each corner are equally well measured. It is required to ad- just them, so that the sum of the three angles in each triangle shall equal 180 , and the sum of the four angles of the quadrilater- al shall equal 360 . Let the measured angles at the corner W be denoted by W L and W 2 , and similarly for each of the other corners, as shown in Fig. 10. Let w 1 and w 2 be corrections to be applied to W 1 and W 2 in order to give the most probable values, W 1 -\- w l and W 2 + w 2 . In order to avoid writing identical equations, select any corner, as W, and take the three triangles, WXZ, ZWY, and XYW which meet at. that point, as the three triangles for cor- rection. Evidently, if the angles of these triangles add up to 180 , those of the fourth triangle will also. The three con- ditional equations now are Fig.10. w, + w 2 + x, + z 2 + d, = o, w 2 + x, + x 2 +y, + d 2 = o, w,-\- y 2 + z z + z 2 + d l = o, n which d„ d 2 , and d 3 denote the differences or discrepancies 148 CONDITIONED OBSERVATIONS. VIII. between the sum of the measured angles of the triangles and the theoretic sum 180 ; thus, for example, IV, + W 2 + X,+ Z 2 - 180° = d,. From the three conditional equations the values of the eight corrections are to be found, either by the method of Art. 113 or by that of Art. 1 14. The latter will be the shorter. As- sume, then, three correlatives, K„ K 2 , and K v and for each correction write a correlative equation, thus + K, + K z = w„ + K, + K 2 = w 2 , + K,+K 2 = x„ + K, +K Z = z 2 , + K 2 = x 2 , + K 2 =y„ ' + K,=y 2 , + K % = Zl , the co-efficients of K, being the co-efficients of the corre- sponding unknown quantities in the first conditional equation, and so on. From these equations the three normal equa- tions are 4X, + zK 2 + 2 X } + d, = o, 2K, + A K 2 + d 2 = o, 2K, + 4^ + d 3 = o, whose solution gives the values of K„ K 2 , and K 3 ; and, insert- ing these in the correlative equations,' the following values of the corrections are found : »: = z 2 = i(— 2d, + d 2 — d 3 ), w 2 =x, = %{ — 2d, — d 2 + d 3 ), x 2 =y,= i( 2d, — 5 d 2 — d } ), J* = z. = i( zd, - d 2 - 34), § I2C ANGLES OF A QUADRILATERAL. I49 and the addition of these to the observed angles gives the adjusted values. For example, let the following angles be given : JF, = 4 i° 58' 47", F, = 49° 1/ 30", W 2 = 64 08 34, , K = 53 53 5 X > X, — 36 34 15, Z, = 46 49 16, ^2 = 29 59 51, Z 2 =^37 18 18. Here the discrepancies are 4 = W, 4- W 2 4- X, + Z 2 - 180 = - 6", Z L + z L = 46 49 08.75, X 2 4- x 2 = 29 59 44.25, Z 2 4- z 2 = 37 18 19.25. These angles now fulfil all the geometrical conditions required in the statement of the problem, and are, furthermore, the most probable angles. 120. If the large angles at the corners are measured as well as the single angles, the most convenient method of procedure is, first to make the station adjustment at each corner, and then, with the eight single angles, to make a further adjustment, as in the last article. The following is an example illustrating the 150 CONDITIONED OBSERVATIONS. VIII. steps of the process for the case of unequal weights. Let the twelve measured angles be W = 106° 07' 27", weight 3, Y = 103° 11' 15" weight 2, W I = 41 58 47, weight 3, F, = 49 1 7 3°. weight 2, ^ 2 = 64 08 34, weight 3, F 2 = 53 53 51, weight 2, Jf = 66 34 03, weight 1, Z = 84 07 30, weight 4, Jf, = 36 34 21, weight 1, Z, = 46 49 16, weight 1, ^T 2 == 29 59 45, weight 1, Z 2 = 37 18 18, weight 2. First, by Art. 117, make the station adjustment at each corner, and obtain the following results : W x = 41 58' 49 ".o, weight f, Y, = 49° 17' 2 8".o, weight 3, W 2 = 64 08 36.0, weight §, F 2 = 53 S3 49.0, weight 3, -X". = 36 34 20.0, weight f, Z = 46 49 J 3-7. weight |, -^ = 29 59 44-°> weight J-, Z 2 = 37 18 16.9, weight^. Next let w„ w„ etc., be corrections to these values in order to satisfy the geometrical requirements of the figure. Then, as in the preceding article, the three conditional equations are Wi + w 2 + *i + % + i"-9 = o, w>2 + x l + x 2 + y x + 8.0 ='o, w, + y 2 + z, + z 2 + 8.6 = o. From (15) the eight correlative equations are w, = #(*, + *z), w 2 = l-(^, + ^ 2 ), x L = f(^, + X 2 ), x 2 = !( + K 2 ), J. = *( + K 2 ), y* = \{ + *",), z. = H + ■*,), Z2 = &(*". + X,). § 121, ANGLES OF A QUADRILATERAL. 1 5 1 From (16) the three normal equations now are (I + 1 + 1 + A)^. + (| + \)K % + (| + , ft)JT, + 1.9 = o, (f + |)X, + (| + 1 + | + 1)^2 +8.0 = 0, (I + i) isr, + (| + i + f + J T )^r 3 + 8.6 = o. and their solution gives the values A\ = +6.99, A' 2 = -7.53, K 3 - -9.43. from which the following corrections are found : w l = — o".5, x, = —0.4, a/ 2 = — 0.1, x 2 = — 5.0, W\= 41° 58' 4 8". S , ;f 2 = 64 08 35.9, X t = 3 6 34 i9- 6 . X 2 = 2 9 59 39-°, The adjusted values of the large angles are now obtained by simple addition of the single angles, and are 3'. = -2.5, 2, = -4.1, y 2 = -3-i» 2 2 = —O.9. ngle angles now are r, = 49 i7'2 5 ". 5 , K= S3 53 4S-9, Z,= 46 49 09.6, ^2= 37 iS 16.0. W = 106° 07' 24". 4, Y = Off/ 103 11 11 .4, X = 66 33 58.6, Z = 84 07 25.6, whose sum is exactly 360 degrees. 121. In geodetic surveys where the sides of the quadrilateral are many miles in length, the spherical excess must be con- sidered in stating the conditional equations for the three tri- angles. In such work a fourth conditional equation must also be introduced in order to insure that the length of any side shall be the same through whatever set of triangles it be com- puted. The development of the calculations for such cases belongs properly to works on geodesy, and will not here be dis- cussed. Detailed examples of the method may be seen in IS 2 CONDITIONED OBSERVATIONS. VIII. Schott's article on the adjustment of the horizontal angles of a triangulation in the United States Coast Survey Report for 1854, in Clarke's Geodesy (Oxford, 1880), and in many German works on higher surveying.* Simple Triangulation. 122. In the adjustment of a simple triangulation the method of procedure is essentially the same as for a quadrilateral. First, the adjustment of the angles at each station should be made, and then the resulting values further corrected, so as to satisfy the geometrical requirements of the figure. This method is not strictly in accordance with the fundamental principle of Least Squares. By the station adjustment a cor- rection, v n is found for each angle, and by the figure adjustment another correction, v 2 ; so that the total correction is v, -f- v 2 . The fundamental principle for observations of equal weight requires that %(v t -f- v z ) 2 should be made a minimum in order to obtain the best values of the corrections, while by the method pursued St', 2 is made a minimum in the first adjustment, and %v* a minimum in the second. The reason for deviating from the strict letter of the law is, that the general method of determin- ing the total equation at once is too laborious, owing to the large number of conditional equations involved. Usually also the difference between the final results of the two methods will be small. In the next article will be given a comparison of the two methods as applied to a simple case. * See also Merriman's Elements of Precise Surveying and Geodesy. New York, 1899. §123- SIMPLE TRIANGULATION. 153 123. The following observations were made to determine the distance between the non-intervisible stations C and D by means of a measured base AB : BAC = 2f 09' °5"-5, BAD = 5 1 34 3S-5> CAD = 24 25 27.8, ABD = 70 08 3 2 -i, ABC = 128 29 o7-5. DBC = 58 20 38.4, ACB = 24 21 46.0, ADB = 5S 16 50.8. By the strict method of Art. 113 or Art. 114 the four condi tional observations are written, one for each of the points A and B, and one for each triangle, and the adjusted values found as given in the second column of the following table : Observed. Adjusted. v. 1?. °5"-5 06.2 + 0.7 0.49 35-5 35-6 + O.I O.OI 27.8 29.4 + 1.6 2.56 32.1 32.0 — 0.1 O.OI o7-S 08.6 + 1.1 1. 21 38.4 36.6 - 1.8 3-24 46.0 4S- 2 -0.8 0.64 50.8 5 2 -4 + 1.6 2.56 The sum "Zv 2 is here 10.72, and by (34) the probable error of a single observation is r= 0.6745 y/^ = ,". 154 CONDITIONED OBSER VA TIONS. VIII. By the shorter method the local adjustment at A and B is first made, giving the results BAC = 27 09' o6".2, weight 1.5, BAD = 51 34 34.8, weight 1.5, ABD — 70 08 31. 1, weight 1.5, ABC = 128 29 08.5, weight 1.5. The triangles ABC and BAD are next separately adjusted, using these four angles and those at C and D. The results are Observed. Adjusted. V. V 1 . o 5 "-5 06.0 + O.S 0.25 35-S 35-7 + 0.2 0.04 27.8 29.7 + 1-9 3.61 32.1 32.0 — O.I 0.01 07-5 08.3 + 0.8 0.64 384 36-3 — 2.1 4.41 46.0 45-7 - 0.3 0.09 50.8 5 2 -3 + i-5 2.25 The sum 2w 2 is here 1 1.3, which is but slightly greater than that of the stricter method. A comparison of the two sets of adjusted values shows also that the differences are small. Levelling. 124. A simple discussion of the precision of levelling observa- tions involving but one conditional equation will here be given as an illustration of the general method of treatment of Art. 1 13. There are three points, A, B, and C, situated at nearly equai distances apart, but upon different levels. In order to ascertain §124- LEVELLING. 155 with accuracy th;ir relative heights, a levelling instrument was set up between A and B, and readings taken upon a rod held at those points, with the results, On rod at A, 8.7342 feet, mean of 12 readings. On rod at B, 2.3671 feet, mean of 9 readings. The instrument was then moved to a point between B and C, and the observations taken. On rod at B, 5.0247 feet, mean of 7 readings, On rod at C, 11.2069 f eet > mean of 4 readings. Lastly, the level was set up between C and A, and the rods observed. On rod at C, 0.4672 feet, mean of 5 readings, On rod at A, 0.6510 feet, mean of 3 readings. It is required to find the adjusted values of these readings, the most probable differences of level between the points, and the probable error of a single reading on the rod. First arrange these measurements as they would be written in an engineer's level-book, and, assuming the elevation of A as 0.0, find the heights of the other points. Station. Back Sight. Fore Sight. Height of Instrument. Elevation above A. , 2 A 7 B g 8-7342 5-0247 0.4672 2.3671 11.2069 O.6510 8.7342 II. 3918 0.6521 0.0 6.3671 O.1849 O.OOII The number of readings or the weight of each sight is placed in the first column preceding and following the name of the 156 CONDITIONED OBSERVATIONS. VIII. station ; thus 7 B 9 denotes that the back sight on B has a weight of 7, and the fore sight one of 9. Regarding the elevation of A as o, that of B comes out 6.3671 feet, that of C, 0.1849 feet ; and, on returning to the starting-point, it is found that A is 0.0011 feet, instead of o as it ought to be. Represent the back sights upon A, B, and C by Z„ Z v and Z v and the fore sights upon B, C, and A by Z 2 , Z v and Z b , and let z„ z v z 5 , z 2 , z v and z b be corrections to be applied to those observed values. The observation equations then are z, = o, weight 12, z 2 = o, weight 9, z 3 = o, weight 7, z 4 = o, weight 4, z 5 = o, weight 5, z 6 = o, weight 3, and the conditional equation is Zi + 2 3 + z 5 — z 2 — z 4 — Z 6 = — O.OOII. From the conditional equation take the value of z v and insert it in the observation equations, which, after multiplication by the square roots of their respective weights, become s/12 z, = o, V^z 3 = 0, V/5 z 5 = o, 3 g 2 = °. V3 z 6 = o, 2Z, 4" 2Z 3 + 2Z S — 2Z 2 — 2Z 6 = — 0.0022. From these the normal equations (Art. 48) are 162, 4- 4Z3 4- 4Z 5 — 4z 2 — 4z 6 = — 0.0044, 4Z, + 1 iz 3 4- 4Z S — 4Z 2 — 4Z 6 = — 0.0044, 42, + 4z 3 4- 9Z S — 4z 2 — 4Z 6 = — 0.0044, — 4z, — 4z 3 — 4z 5 + I3Z 2 4- 4^6 = + 0.0044, — 42, — 4Z3 — 4z 5 4- 4^2 + 7 2 6 = + 0.0044, §125- LEVELLING. 157 the first being the normal equation for z u the second for z v the third for z 5 , the fourth for z 2 , and the fifth for z 6 . The solu- tion gives the following results : «i 0.00008, 0.00014, Z s = — 0.00020, 2 2 = + O.OOOII, 2 4 = + O.OOO24, z 6 = + O.OOO33. Applying these to the observed values, the adjusted results are Station. Back Sight. Fore Sight. Elevation above A. A 8.73412 O.O B 5.02456 2.36721 6.36691 C 0.46700 II. 20714 0.18433 A 0-65133 O.O The residuals are in this case the corrections z It z v etc. Squaring these, multiplying each square by its weight, and add- ing, gives 2/w 2 = 0.000001079. From formula (34) then r — 0.6745^0.000001079 = 0.0007 f eet > which is the probable error of a single reading on the rod. 125. The adjustment of a network of level lines may also be effected by the method of conditioned observations. When the levelling is of the same precision throughout, the probable errors of differences of level should be taken as varying with the square root of the lengths of lines, being governed, in short, i 5 8 CONDITIONED OBSERVATIONS. VIII. by the same law of propagation of error as linear measure- ments (see Art. 91). Each difference of level should hence be assigned a weight inversely proportional to the length of the line between the two points. For each triangle or polygon of the network, there is the rigorous condition that the sum of the differ- ences of level shall be zero. From these conditional equations, corrections to the observed differences of level are determined by the method of Art. 1 14. As an example, consider the follow- ing eight differences of level forming three closed figures, ABE, BCFE, and CDF: No. Stations. Diff. Level. Distance. Weight. Feet. Miles. 1 B above A 120.2 4.0 O.25 2 C above B 230.6 7.2 O.14 3 D above C 143.O 5-o 0.20 4 D above F 294.4 6-3 0.16 5 C above F 150.2 2.0 0.50 6 F above E 934 4.8 0.21 7 B above E H-5 3-5 O.29 8 E above A 106.7 8-3 O.I2 It is required to find the most probable corrections to the above differences of level in order to cause the discrepancies in the three polygons to vanish § 125. LEVELLING. 1 59 Let k s , h u etc., represent the most probable differences of level. Then the three conditions are for ABE, h 1 — h 1 — h% = o, for BCFE, K — h s — h b + h 1 = o, for CDF, h i — h 4 + h-s = o. Let v v v 2 , etc., be the most probable corrections to the observed differences of level, so that h t = 120.2 + v„ h 2 = 230.6 + v 2 , etc. Then the three conditional equations become v z — v 7 — v$ — 1.0 = o, v x — v 5 — z> 6 + v 7 + 1.5 = o, #3 — V 4 + V s — 1.2 = O. From these the correlative equations are written, the weight of each v being taken as the reciprocal of the corresponding distance : v t = + 4.0X, Z>2 = + 7- 2 -X, v 3 = + S-°Kv v 4 = — 6.3X,, v 5 = — 2.oK 2 -f- 2.oX 3 , v 6 = — 4.8X, v, = - 3-5^ + 3-S-K'i v 8 = - 8.3X. Next the three normal equations are 15. 8X - 3.5X - 1.0 = o, — 3.5X + 17.5X2 — 2.oX 3 + 1.5 = o, — 2.oI 2 + 13.3X3 — 1.2 = O, and the solution of these gives K^ =z -\- 0.04848, K 2 = — 0.066855, K 3 — + 0.08017 i6o CONDITIONED OBSER VA TIONS. VIII. Lastly, by substituting these in the correlative equations, the corrections are found, which are given in the third column of the following table, while in the fourth are the adjusted results. No. Observed Diff. Level. V. Adjusted Diff. Level. I 120.2 4- 0.19 120.39 2 230.6 — 0.48 230.12 3 143.0 + 0.40 143.40 4 294.4 -0.51 293.89 5 150.2 + 0.29 i5°-49 6 934 4- 0.32 93 72 7 14-5 — 0.40 14.10 8 106.7 — 0.40 106.30 In order to ascertain the precision of the work, the correc- tions are squared, and each square multiplied by its respective weight, and the sums of these products taken. This sum is about 0.246; and then by (34) the probable error of an obser- vation of the weight unity, that is, the probable error of the difference of level of the ends of a line one mile in length, is /0.246 r = 0.6745,. / = 0.19 feet, a result that indicates a low degree of precision. 126. Problems. 1. Adjust the following angles taken at the station O: AOB = 40 52' 37", weight 16, BOC = 92 25 41, weight 4, COD = 80 6 15, weight 3, DOA = 146 35 20, weight 1. § 126. PROBLEMS. l6l 2. In a spherical triangle XYZ the three measured angles are ^"=93° 48' I5".22, with weight 30, Y= 51 55 0.18, with weight 19, Z — 34 16 49.72, with weight 13. The spherical excess is 4". 05. What are the adjusted angles? 3. In a quadrilateral WXYZ, the following angles, all of equal weight, are measured, and it is required to adjust them. W = 106 07' 30", Yi = 49 17' 23", Wi = 41 58 47, y 2 = S3 53 50, W 2 - 64 08 34, Z =84 07 18, Jf = 66 34 09, Z 2 = 37 18 12. JT, = 36 34 21, 4. Adjust the level observations in Art. 100 by the method of condi- tioned observations, taking the weights as equal. 5. Discuss the method of correcting the latitudes and departures in a compass survey of a field. 6. Two bases, AB and DE, are connected by three triangles, ABC, BCD, and CDE. The bases are measured, and also the three angles of each triangle. State the four conditional equations, and explain in detail the process of adjustment. I02 THE DISCUSSION OF OBSERVATIONS. IX. CHAPTER IX. THE DISCUSSION OF OBSERVATIONS. 127. In the preceding pages it has been shown how to adjust observations, and how to ascertain their precision by means of the probable error. By thus treating series or sets of measure- ments, a comparison or discussion may be instituted concern- ing the relative degrees of precision, the presence of constant errors, and the best way to improve the methods of observa- tion. In this chapter it is proposed to present some further remarks relating to the discussion of observations by the use of the fundamental law of probability of error, and to indicate that this law is also applicable to social statistics, and that it really governs the way in which the laws of nature are executed. Probability of Errors. 128. In Chap. II a method of investigating the probability of errors, and comparing theory with experience, was given, in which it was necessary to assume the value of the measure of precision h. For instance, in Arts. 19 and 33 there are dis- cussed one hundred residual errors, for which the value of h is stated to be — ^ — . It is now easy to see that this value may be found at once from the probable error r by means of the formula (17), while r is deduced from the formula (20). T© § 128. PROBABILITY OF ERRORS. 1 63 compare, then, the theoretical and actual distribution of errors for such cases by the use of Table I it is only necessary to deduce the value of r in the usual way, and from it to find //, which enters as an argument in the table. It is evident, then, that, in undertaking such discussions, it i? more convenient to have a table of the values of the probability integral in terms of r as an argument. Such is Table II at the end of this book, which gives, for successive values of -, the r probability that a given error is less numerically than x, or that it lies between the limits — x and +.r. To illustrate the use of Table II consider an angle for which the mean value is found to be 37 4-' i3"-92 ± o".2 5 . Now. from the definition of probable error, it is known that the probability is 1 that the actual error of the result is less than o' .25. Let it be asked what are the respective probabilities that the actual error is less than the amounts o".5 and i".o. From the table for - = ^ = ,. /> = 0.8=3. r 0.2> .r 1. 00 D for - = = 4. p = °-993- 0.25 Hence the probability that the error in the result is less than o".5 is o.S^3, or it is a fair wager of S23 to 177 that such is the case. And the probability that the error is less than i".o is 0.993, or it is a fair wager of 993 to 7 that such is the case. As the number of errors is proportional to the probability, the values of the integral need only to be multiplied by the total number of errors to give the theoretical number less than 164 THE DISCUSSION OF OBSERVATIONS. IX. certain limits. For example, in one thousand errors or residu- als, there should be 264 less than \r, and 736 greater, 500 less than r, and 500 greater, 823 less than 2r, and 177 greater, 957 less than $r, and 43 greater, 993 less than 4?-, and 7 greater, 999 less than 5?-, and 1 greater. Table II gives only four decimal places, which suffice for any ordinary investigation. By the methods of calculation explained in Chap. II more decimals may be deduced, and the following, results be found for the theoretical distribution of errors when the total number of errors is one hundred thou- sand : 95698 are less than $r, and 4302 greater, 99302 are less than \r, and 698 greater, 99926 are less than 5?', and 74 greater, 99995 are less than 6r, and 5 greater. As the frequency with which an error occurs is expressed by its probability, it is evident that errors greater than five or six times the probable error should be very rare. 129. As shown in Art. 35, the probability of the error o is -^, or, introducing for h its value , it may be written IT r dx y = 0.2691 — . r Here dx is the interval between successive values of x. If there be N errors in a series, the number having the value should hence be (42) N a = 0.2691 — N, r where r is the probable error of a single observation. §129- PROBABILITY OF ERRORS. 165 Formula (42) affords a rough comparison of theory and ex- perience without the use of tables. For instance, let the target-shots described in Art. 18 be again considered, and regard those in the middle division as having the error O, those in the next division above as having the error -f- 1 > an d so on. Then the errors, without regard to sign, are as in the first column below, their squares in the second, their weights or the number of shots in the third, and the weighted squares in the fourth. X. x\ P- JiX 2 . /■ 212 261 I I 394 394 382 2 4 282 1,128 232 3 9 89 801 93 4 16 20 320 26 5 2 5 3 75 6 2/x 2 = = 2,718 1,000 Now, the probable error of a single observation is r= 0.67451/^=1.!, V 1000 and, by formula (42), the number of errors having the value o is 0.2601 x 1 x 1000 N = z = 245, which is a satisfactory agreement with the actual number 212. In the last column of the above table are given the theoretical numbers of errors as computed from Table II. 1 66 THE DISCUSSION OF OBSERVATIONS. IX. The Rejection of Doubtful Observations. 130. The theoretical distribution of errors, according to the fundamental formula (1), is shown by the values of the proba- bility integral given in Table II ; and from these it is seen, as in Art. 128, that the number of errors greater than ^r or $r is very small. It becomes, then, a question, whether the probabil- ity of an error might not be so small that it would be justifiable to reject entirely the corresponding observation. For instance, if one thousand direct observations be taken, the probability that there will be one error greater than ;ris — ; if, then, in O J IOOO ' ' ' taking a series of, say, fifty observations, one error should exceed Sr, the probability of its occurrence would be very much smaller than j^j, and the observer would be tempted to reject that observation. But undoubtedly it would be a dangerous thing to allow an observer to decide upon his own limit of rejection. It has accordingly been proposed to attempt to establish a cri- terion by which the limit may be legitimately established from the principles of the probability of error. The criter-on pro- posed by Chauvenet is the simplest of those deduced for this purpose, and is the following : Let n be the number of direct observations, and also the number of errors. Let r denote the probable error oi a single observation as found from the n residuals by forp-ula (20). x Let x be the limiting error, and let - be called t. Le r P be the r value of the integral in Table II corresponding to /. Then (43) P= , and x = tr 271 is the criterion for the rejection of the largest residual To prove this, consider that the quantities in Table II need only be multiplied by the total number of errors to ?how the actual distribution ; so that nP indicates the number of errors §!■ REJECTION OF DOUBTFUL OBSERVATIONS. 167 less than x, and « — nP indicates the number greater than x. Now, if n — nP = \, there is but half an error greater than x, and any error greater than this x would be larger than allowed by the theoretical dis- tribution. Hence the value of x corresponding to this value of P is the limiting value, which indicates whether the greatest residual in a series may be rejected or not. 131. In order to facilitate the use of this criterion, Table VII has been computed, giving the value of t directly for several values of n. For instance, if n is 5, the value of P is —, 10 or 0.9 ; and from Table VII the corresponding value of t is 2.44. The following particular example will illustrate the method of procedure. Let there be given thirteen observations of an angle, as in the first column below. 62° 1 -'5 1" 75 2.69 7.24 4S.45 0.6l o-37 50.60 !-54 2-37 47-Sf 1. 21 1.46 5i-°5 I.99 3-9 6 47-75 J-3 1 1.72 47.40 1.66 2.76 4S.S5 0.21 0.04 49.20 0.14 0.02 4S.90 0.16 0.03 5°-95 1.89 3-57 5°-55 1.49 2.22 44-45 4.61 21.25 62 3 / tr *- 12 49 .OO ■■ — 47.01 l68 THE DISCUSSION OF OBSERVATIONS. IX. Let the mean of these be found, the residuals placed in the second column, and their squares in the third. The sum Iv 2 is 47.01 ; and hence, from (20), the probable error r of a single observation is i".32. Table VII gives t = 3.07 when n = 13 : hence, by the criterion, the limiting error is x = 3.07 x 1.3 2 = 4-05, and accordingly the largest residual 4.61 should be rejected. To ascertain if the next largest residual, 2.99, should also be. rejected, the mean of the twelve good observations should be found, and a new r computed from the twelve new residuals. But evidently the new sum 2z> 2 will not differ greatly from the former sum minus the square of the rejected residual, or new Sv- = 47.01 — 21.25 — 2 5-76, from which the new r is found to be about i".03. Then the limiting error is x= 3.02 X 1.03 = 3".n, which shows that the residual 2.99 is not to be rejected. 132. Hagen's deduction of the law of probability of error, given in Chap. II, suggests another method of finding the limiting error of observation, and a new criterion for rejection. In Art. 26 the maximum error is expressed by m&.x, and the quantity m\x* is replaced by -yj. It is hence easy, by the help of (17), to find r* (44) m&x = 4.4 -=-> ax where dx is the constant interval between successive values of the errors. For the observations discussed in Art. 129 this formula gives the limiting error mb.x as 5.3, which seems entirely satisfactory. It is not possible to apply it, however, to angle measurements like those of the last article, on account of the impossibility of assigning a proper value to the interval dx. § 133- CONSTANT ERRORS. 1 69 The same difficulty prevents the practical use of formula (42), except in cases where this constant interval is definitely known. There is another criterion, due to Peirce, which may be applied to the case of indirect observations involving several' unknown quantities, as well as to that of direct measurements ; but its development cannot be given here. In general, it should be borne in mind that the rejection of measurements for the single reason of discordance with others is not usually justi- fiable unless that discordance is considerably more than indi- cated by the criterions. A mistake is to be rejected, and an observation giving a residual greater than 4;' or $r is to be regarded with suspicion, and be certainly rejected if the note- book shows any thing unfavorable in the circumstances under which it was taken. Usually, in practice, the number of large errors is greater than should be the case, according to theory ; and this seems to indicate, either that the series is not suf- ficiently extended to give a reliable value of r, or that abnormal causes of error affect certain observations. If it were possible to increase the number of measurements, it would undoubtedly be found that the abnormal errors would be as often positive as negative, and that, for a very great number, there would be few that could be rejected by the criterion. Constant Errors. 133. In all that has preceded, it has been supposed that the constant errors of observation have been eliminated from the numerical results before discussing them by the Method of Least Squares. If this is not done, and all the measurements of a set are affected by the same constant error, that error- will also appear in the adjusted result. For instance, suppose thirty shots to be fired with the intention of hitting the centre of a target, and let their actual distribution be as shown in the figure. The most probable location of the centre, according to the records, is about two spaces to the right, and about half a 170 THE DISCUSS/ON OF OBSERVATIONS. IX. • • • • : W7 ■_,_•:.: • space below the true centre. Each shot, then, has been subject to these constant errors ; the first due, perhaps, to the wind, and the second to gravity. If, now, these marks on the target represented observations for the purpose of locating the centre, the result obtained by their adjustment would be in error by the amounts just stated. Therefore, if all the observations of a series are affected by the same constant error, the Method of Least Squares can do nothing but adjust the accidental errors; and the probable errors of the adjusted results refer only to them, and give no indication whether constant causes of error affect the measurements or net. 134. The probability of the existence of a constant error in a case like that just illustrated is evidently large, and the numerical probability of its value lying between certain limits may be found by the help of Table II. The following is an example of such a discussion : Suppose that an angle is laid out with very accurate instru- ments, and tested in many ways, so that its true value may be regarded as exactly 90 . Let twenty-five observations be taken upon it with a transit whose accuracy is to be tested, and let the mean of those measurements be 89 59' 57" ± o".8. Then it is extremely probable that a constant error of about — 3" exists in the instrument. To find the numerical expression of this probability, suppose that the true value of the angle was unknown, and ask the probability that the mean is within 2" of x 2 the truth. Then, for - = — - = 2.5, the value of the integral in 0.8 § 1 35. CONSTANT ER"RORS. \"J\ Table II is 0.908 ; so that it is a wager of 908 to 92, or of almost 10 to 1, that the mean is between the limits 89 59' 55" and 89 59' 59". Hence, since the angle is known to be 90 , it must be the same probability and the same wager that there is a constant error lying between the limits — 1" and — 5". So, also, if x = 3", it may be shown that it is a wager of 39 to 1 that there is a constant error between o" and — 6". 135. In case that several sources of constant error exist, the adjustment by the Method of Least Squares tends to elimi- nate them, and to give results nearer and nearer to the actual values, as the number of observations is increased. This will be rendered evident by considering again the illustration of the target. One marksman fires thirty balls, which are subject to a constant error, as in Fig. 13. Another marksman fires thirty more, which have a different constant error, owing to the peculiarities in his aim. A third marksman has a third con- stant error, in a still different direction. The shots of each marksman are distributed around their most probable centre in accordance with the law of probability of accidental errors. And undoubtedly these constant errors will be grouped around the true centre according to the same law ; and, as the number of marksmen increases, the constant errors will thus tend to annul each other, and ultimately make the most probable centre coincide with the true one. And so it must be in angle observations, when great pre- cision is demanded. On one day certain constant errors, due to atmospheric influences, affect all results in a certain direc- tion ; while on a second day, under different influences, new constant errors act in another direction. If the measurements be continued over many days, the number and magnitude of positive constant errors will be likely to equal the negative ones ; so that the adjustment by the Method of Least Squares will balance them, and give results near to the true values. 172 THE DISCUSSION OF OBSERVATIONS. IX. Here may be seen the reason why the number of large residuals is usually greater than the theory demands, and also a reason why a criterion for rejection cannot generally be safely applied to series of observations consisting of few measurements. Social Statistics. 136. It is found that the law of probability of error applies to many phenomena of social and political science. If men be arranged in groups, according to their heights, there will be found few dwarfs and few giants ; and the numbers in the dif- ferent groups will closely agree with the theoretical distribu- tion required by the curve of probability. The following table, which is taken from Gould's Statistics (New York, ii Height. Inches. Actual Number. Proport onal Number n 10,000. Observed. Calculated. Calc — Obs. 61 197 i°S IOO - 5 62 3 l 7 169 171 + 2 63 692 369 368 — 1 64 1289 686 675 — II 65 1961 1044 1051 + 7 66 2613 I39 1 J 399 + 8 67 2974 1584 1584 68 3017 1607 l 53* -76 69 2287 1218 1260 + 4 2 70 !599 852 884 + 3 2 7i 878 467 53i + 64 72 520 277 267 — 10 73 262 r 39 118 — 21 74 174 92 61 - 3 1 § 137- SOCIAL STATISTICS. 1 73 gives a comparison of the theoretical and observed heights of 18,780 white soldiers, including men of all nativities and ages. In the second column are recorded the actual number measured of each height, and, in the third, the proportional number in 10,000. The mean height as found by formula (9) is 67.24 inches, from this the residuals are formed ; and the probable error .if a single determination, by formula (23), is 1.676 inches. The theoretical numbers between the several limits are next derived by the help of Table II, and recorded in the fourth column, while the differences between the calculated and ob- served numbers are given in the last. 137. Numerous comparisons of this kind, made by Quetelet and others, have clearly established that stature and the other proportions of the body are governed by the law of probability of error. Nature, in fact, aims to produce certain mean pro- portions ; and the various groups into which mankind may be classified deviate from the mean according to the law of the probability curve. And the same is true of intellect. By the discussion of social statistics, then, it is possible to discover the mean type of humanity, not merely in physical proportion, but in intellect, capacity, judgment, and desires. "The aver- age man," says Quetelet, "is for a nation what the centre of gravity is for a body : to the consideration of this are referred all the phenomena of equilibrium." In fact, the distribution of social phenomena seems strictly analogous to that of the rifle-shots discussed in Art/135. Each shot may represent a person, or some property of a person, to be investigated. For all the shots there is a mean, showing the most probable result ; and also, for each group, there is a secondary mean, depending on the particular race or nation to which the person belongs. There is a type for soldiers, and another for sailors ; one for Americans, and another for Euro- peans ; one for men, and another for women ; one for the period 174 THE DISCUSSION OF OBSERVATIONS. IX. of youth, and another for that of maturity. The individuals of each type are clustered around its mean, according to the law of probability; and the several types are clustered around a general mean, according to the same law. This is true for all statistical data in which equal positive and negative deviations from the mean are equally probable ; in other cases an unsym- metric distribution may occur. 138. Problems. 1. An angle is measured by an instrument graduated to quarter - minutes, the probable error of a single reading being 12 seconds. How many observations are necessary, that it may be a wager of 5 to 1 that the mean is within one second of the truth? 2. A line is measured 500 times. If the probable error of each observation is 0.6 centimeters, how many errors will be less than 1 cen- timeter, and greater than 0.4 centimeters ? 3. The capacity of a certain large vessel is unknown : 1,600 persons guess at the number of gallons of water which it will hold, and the average of their guesses is 289 gallons. The vessel is then measured by a committee, and found to hold 297 gallons. If the probable error of a single guess be 50 gallons, and it be impossible that there can be any constant source of error in guessing, what is the probability that the committee have an error in their measurement of between 3 and 13 gallons ? 4. Determine from the data in Art. 136 the number of men per million who are more than seven feet tall. 5. Two observations differ by the amount a. A third observation differs from the mean of the first two by the amount u. Find, by Chauvenet's criterion, the value of u necessary to reject the third observation. §140. THREE NORMAL EQUATIONS. 1^5 CHAPTER X. SOLUTION OF NORMAL EQUATIONS. 139. In the preceding pages the student has been left to solve normal equations by any common algebraic process. It is usual in computing offices, however, to require them to be formed and solved by a definite method for the sake of uni- formity in making comparisons. This is, indeed, absolutely necessary when the number of unknown quantities is greater than three or four, or when the co-efficients are large, in order that checks upon the numerical work may be constantly had and the accuracy of the results be ensured. The methods in most common use will now be explained. Tliree Normal Equations. 140. The method of elimination, due to Gauss, which is de- scribed below, is probably the best for this case except when the co-efficients are small numbers. In that event the determi- nant formulas for solution may be advantageously employed. These will be here written for the general case of three linear equations, A 1 x + By + Ciz - Z>„ A 2 x + B 2 y + C 2 z = D 2 , J 3 x + Biy + C 3 z = Z> 3 , 176 SOLUTION OF NORMAL EQUATIONS. the solution of which gives the formulas, X. y = D 2 B 2 C 2 D 3 B 3 C 3 A 2 D 2 C 2 A, D 3 C 3 A, B t A A 2 B 2 D 2 A 3 B, D 3 A, B, C, A, B l Q A 3 B 3 C 3 A z B x C, ^ 2 A C 2 ^3 ^3 Q These are readily kept in mind by noticing that the denom- inator is the same for each, and that in the numerator the absolute terms D replace the co-efficients of the unknown quan- tity to be found. If C T — C 2 = C 3 = o, and A 3 B, A = o, this solution reduces to that given in Art. 55. 141. As an illustration of this method let the three normal equations be $x — y + 2Z = s, — x + 4J> + z = 6, 2x + y + 52 = 3. Then the determinant denominator, being developed, gives 3 — 1 2 1 4 1 = 3 4 1 + 1 — 1 2 + 2 — 1 2 2 1 5 1 5 1 5 4 1 = 32- Similarly the values of the three determinant numerators are found to be no, 86, and — 42. Hence * = +«, y = +tt, *=-«, which exactly satisfy the three given normal equations. §142. FORMATION OF NORMAL EQUATIONS. *77 Checks upon the results of the solution may be also obtained by writing the normal equations in another order, making for instance the third the first one, and thus obtaining different numerical determinants for development. Formation of Normal Equations. 142. Let the n observation equations between three unknown quantities be of equal weight, and let the observed quantities M v M„ . . . M n be transposed to the first term, giving a^c -+- b,y -f- C-.Z -(- ;», = o, a^x + b 2 y + C2Z -f- m 2 = o, a»x -f- b n y + c n z -\- m„ = o, and let there be formed the sums a? + <*2 2 + • • • + «» 2 = [aa], a t 6i -+- a 2 b 2 -\- . . . + ctnbn = [ab\. Then the three normal equations are [aa]a: -\- [ab]y -f- [ac]z -f- [am] = o, [ba]x + [3b] y -f- [bc]z + [bm] = o, \ca]x -f- [cb]y -f- [cc]z -\- [an] = o. Thus the formation of the normal equations consists in com- puting the co-efficients [aal, [ab], etc. This may be done by common arithmetic, by the help of Crelle's multiplication ta'ble, a logarithmic table, a table of squares, or a calculating machine. The following method of arranging and checking the work is frequently employed. Write the co-efficients and absolute terms of the observation i 7 8 SOLUTION OF NORMAL EQUATIONS. X. equations in tabular form and add a column containing the algebraic sums of these for each equation. Thus for three X a y b z c m s I 2 n unknown quantities the table has the above form, the last column containing, for each horizontal row, the algebraic sum a -\- b -f- c -f- m = s. A second table, which need not be here shown, contains fifteen columns, headed aa, ab, . . . ss, and the summation of the products in these columns gives the fifteen co-efficients and absolute quantities which are arranged in a third table as be- low. It is to be noted that [bd], [ca], \_cb~] are the same as [ab'], \_ac~], [be], and hence need not be computed. X a] y [' ?n\ A Check. [a \P [c \m [s Here the sum [bb] is placed at the right of [b and under b\ the §143- FORMATION OF NORMAL EQUATIONS. 179 sum [cs] at the right of [c and under s], and so on. The last column is used to record the results of the five checks, namely, [aa] + [ab] + [ac] + [am] = [as], [ba] + [bb] + [be] + \bm] = [bs], [ca] + [cb] + [cc] + [cm] = [«], [/««] + [ot£] -f- [/«f] 4 [mm] = [w], [>«] + [si] + [sc] + [sm] = [ss]. If these checks are all fulfilled, the normal equations may be regarded as correctly formed. In filling out the table the coefficients [da], [mc], etc., need not be written, since they are the same as [ad], [cm], etc. 143. As a simple example let five observations upon three quantities give the five observation equations — x + z — 2 = 0, — x +j> —9 = 0, +y - 18 = o, + y - z - 7 = 0. + Z — TO = O. The arrangement of the first table is then as follows : No. X y z a b c m j 1 — I O + 1 — 2 — 2 2 — I + 1 c - 9 - 9 3 + I — 18 - 17 4 + 1 — I - 7 - 7 5 O + 1 — 10 - 9 i8o SOLUTION OF NORMAL EQUATIONS. X. The products aa, ab, etc., are next computed, and the sums \aa\ \_ab~\, etc., are found. The table of co-efficients and ab- solute quantities then is X a] y z ni\ '] Check. \a + 2 — i — I + ii + ii + ii \b + 3 — I - 34 - 33 - 33 [c + 3 - 5 — 4 — 4 [m + 5S3 + 53° + 53° [s + 5°4 + S°4 and the checks being all fulfilled the computations are satis- factory. Thus the normal equations are -{- 2X — y — z-|-ii = o, - x + 3 y— z - 34 = o, — x - y + 3 z - s = o, and it will be shown in Art. 147 how these may be solved so as to continue the above system of checks throughout the entire numerical work. The advantage of the above system is more apparent in cases where the co-efficients and absolute terms consist of several digits and where the decimals must be rounded off. In such cases the number of decimals to be retained in the work should be at least sufficient to cause the checks to be fulfilled with an error not greater than one unit in the last place. The additional labor required for these checks is fully repaid by the assurance of correctness in the numerical work. § 144- GAUSS'S METHOD OF SOLUTION. l8l Gauss's Method of Solution. 144. The method of solution due to Gauss, by which is preserved throughout the work the symmetry that exists in the coefficients of the normal equations, is extensively used by computors. To illustrate it, three normal equations of equal weight will be sufficient. From the n observation equations are derived, by the method of Art. 142, the three normal equations [a«]x -f- [ a b] y -\- [ac]z -\- [am] = o, [ba]x + [bb]y + [bc]z + [bm] = o, [ca]x -\- \_cb\y -\- [cc]z -f- [cm] = o. From the first equation take the value of x and substitute it in the second and third, giving o, o. For the sake of abbreviation the quantities within the paren- theses may be denoted by [bb . 1], [be . ij, [bm . ij for the first equation, and by [cb . 1 J, [cc . 1], [cm . 1] for the second equa- tion. Then these two equations may be written [bb. \]y + [be . i]z -\- [bm . 1] = o, [cb. i]y + [cc. i]s + [cm. 1] = o, which are similar in form to the second and third normal equa- tions, except that the terms containing x have disappeared and each co-efficient is marked with a 1. These quantities, [bb . i], [bc.i], may be called "auxiliaries," and the law cf their formation is evident. 1 82 SOLUTION OF NORMAL EQUATIONS. X. From the first of these equations take the value of y and substitute it in the second, giving r [c6.i][6c.i]\ I -. [c3.i][3m.i] which may be abbreviated into [cc . z\z -\- [cm . 2] = o, where [cc . 2] and [cm, 2] may be called "second auxiliaries." The value of the quantity # now is [cm . 2] while the values of y and x are J = [3m. 1] [fo. 1] [36. 1] [63. if' X = [am] [ac] [a6] [aa] [aa] [aa] ' and the correctness of these results may be tested by inserting the computed values of x, y, z in the second and third normal equations. Or the order of computation may be reversed and the value of x be first obtained, z being first eliminated and then/; this will be necessary only in critical cases. 145. When the normal equations have been formed by the method of Art. 142, the chqcks there explained should be con- tinued by the computation of the auxiliaries [mm. ij, [bs. 1], etc.; thus, [3a] [as] [6s. 1] = [6s]- [aa] And a second table should be formed for the two equations containing y and z, by which four numerical checks are ob- tained. § I46. GAUSS'S METHOD OF SOLUTION. 1 83 In the next step also the auxiliaries [mm. 2], [«.2], etc., are found ; for example, i"-l = i«-'i -**$&>• [bb.i] and then the third table affords three numerical checks. 146. A valuable final check is obtained by computing the third set of auxiliaries ; thus, r , r , \mc . 2\cm . 2] mm . x = mm . 2 — ? — b — =, ~, L 0J L J \cc.2\ [mc. 2] [cs. 2] \ms. 3J = \ms. 2 I — i — f S -, L 0J L J \cc . 2] r 1 r i [sc.2][cs.2] \SS. 3 )=[SS.2] [^^-> and these three values are equal. Each is also equal to the quantity 2v 2 , or to the sum of the squares of the residuals ob- tained by substituting in the observation equations the values of x, y, and g, found from the normal equations. To prove this let an observation equation be ax -j- by + cz -j- m = o. Then the most probable values, x, y, z, will not reduce it to zero, but leave a small residual v. Hence, strictly, ax -)- by + cz + m = v. By squaring each of the values of v, and adding the results, the value of 2v 2 is found ; and if from this each normal equa- tion, first multiplied by its unknown quantity, be subtracted, it reduces to \am\x -\- \bm\)> + \cm\z -j- [mm] = 2v 2 . I84 SOLUTION OF NORMAL EQUATIONS. X. If this be regarded as a fourth normal equation, it becomes, after the elimination of x, [bm . \~\y -f- [cm . i]z + [mm . 1] = 2v 2 , and after eliminating^ it is [cm . 2~\z -j- [mm . 2] = ~2if; and finally, after the elimination of z, [mm . 3] = 2v 2 . Hence the auxiliary [mm . 3] is equal to the sum of the squares of the residuals ; and that [ins . 3] and [ss . 3] have the same value is shown by the method of their formation. 147. As a simple numerical example let the following ob- servation equations, all of weight unity, be taken: — x -f- z — 2 = 0, — x+y — 9 = °> + y — 18 = o, + y - z - 7 = 0. -f- z — 10 = 0. The normal equations for this case have already been formed in Art. 143, and the values of its co-efficients and check numbers wi 11 be taken from the table there cnven. S' The computation of the auxiliaries for the two equations containing j and z is now made, thus : r,, t r 7 ,-| [/«][>£] I X I L««j 2 J §'47- GAUSS'S METHOD OF SOLUTION. I8 5 r , i r , -, [ba][am] , 1 X 11 \bm. ij = \bm\ — ^-^~- = [bs.i] = [bs] [aa] [ba][as] [aa] 34 = ~ 28-S» , 1 X 11 - 33 -\ — = - 27-5. [„ T l —\,A \ca\\ac\ _ 1 X 1 . [cc.i] -[cc] - -j^ a y-+ 3- . r 2.5, I/ot .1] = [cm] — [ca][a 5 + 2 1 X 11 [aa] ~ i ~ r ~T~ -'^°-5> r 1 r 1 |/0]|W] I X II r 1 r i \ma][am] 11 X 11 [otot . ij = [mm] = + 558 — = + 497.5, r 1 r l [>««]M . 11 X 11 [»"■ J ] = M - r a 7p = + 53° " — - + 469-5. r i r i [.«][<«] , 11 X 11 . [« . 1] = [ss] - —^r-" = + 5°4 = + 443-5. \aa\ 2 and the corresponding tabulation is as follows, the four checks being exactly fulfilled: y 2 Check. *.i] ..1] OT . l] ,.1] lb + 2.5 -28.5 - 27 5 - 27-5 V + 2-5 x °-5 + i-5 + i-5 [m + 497-5 + 4 6 9-5 + 469-5 [s + 443-5 + 443-5 The coefficient [cc.'z] and the auxiliaries for the final equa- tion in z are next found ; thus, [cc . 2] = [cc . 1] — [cb. i][bc.i] + 1.6, ' 1 86 SOLUTION OF NORMAL EQUATIONS, cm. 2] - X. cs . 2] = #2#z. 2] = ms.2] — ss . 2] = L and [ss . 3] = 0.375. Also, by substituting the values of x,y, z, in the observation equations, v t =— 0.125, ^2=4-0.125, ^=-0.375, z/ 4 =+o.25o, ^=+0.375, the sum of whose squares is 2v 2 = 0.375. Hence the correct- ness of all the numerical work is assured. When the coefficients of the normal equations contain decimals these are to be rounded off as the work progresses, so that the checks may be sufficiently satisfied. Weighted Observations. 148. The method of Gauss is also directly applicable to normal equations derived from independent weighted observa- tion equations. The process will be illustrated for three unknown quantities. Let the observation equations be + x +y = o, = o, + z =0, ■y +0.92 = 0, — y + z + 1.35 = o, — X -\- z -\- 1. 00 = o, The first table is then as follows : + x p 2 = 108, A = 49. A = 165, A= 78, Pi = 60. i, A X y z No. P a b c VI S 1 85 + 1 o t + 1 2 108 + 1 + 1 3 49 + 1 + 1 4 165 + 1 — 1 + 0.92 + 0.92 5 • 78 — 1 + 1 + i-35 + i-35 6 60 — 1 — + 1 + 1 + 1 i88 SOLUTION OF NORMAL EQUATIONS. X. Next the co-efficients [pad], [pati], etc., are computed and the table of normal equations is formed, the co-efficients below the diagonal line being omitted, since [pba] is the same as [pab], and so on. X y 3 Check. a] *] f] "'1 '] [pa + 3 IQ -i6 S — 60 + 91S + 176.8 + 176.8 [pb + 35 r - 78 -257-1 — 149.1 -149.1 [pc + 187 + 165.3 + 2I4-3 + 214-3 [pm +341-8 + 34'-8 + 341-9 [ps + 583-8 + 583-8 This shows by its checks that the computations are correct, the discrepancy between 341.8 and 341.9 being due to the rounding off of decimals. Thus, 310X — 1657 — 605:+ 91.8 = 0, — 165^ + 351^— 782 — 257.1=0, - 60*— 78^+1872+165.3 = 0, are the normal equations for determining the most probable values of x, y, and z. 149. The auxiliaries [pbb.i], \_pbc.\~\, etc., are computed by exactly the same rules as before, and the table for the two y *.x] c • I] m . 1] S.I] Check. [pb + 263.2 — 109.9 — 208.2 ~ 55° - 54-9 [pc + 175-4 + 183-1 + 248.5 + 248.6 [pm + 3I4-5 + 289.4 + 289.4 [ps + 483-0 + 482.9 §iSo. WEIGHTED OBSERVATIONS. 189 reduced normal equations containing y and s is formed, the four checks being fulfilled within one unit in the last figure. The second auxiliaries [/>cc.2~\, \_pcm. 2], etc., are computed exactly as before and the table for the final equation in z is z C.2] m . 2] ,.2] Check. [pm [ps + I29.5 + 96.1 + 149-8 + 225.5 + 245-9 + 47I-5 + 225.6 + 245-9 + 47M formed and its checks found to be satisfactory. The value of z now is 96. 1 z = z = — 0.7421. 1295 which is carried to four decimals in order that y and 2 may be found correct to two decimals. From the first equations in the two tables preceding the last, the values of y and x are now obtained, thus, 208.2 y- 9 1 x = — — io 9-9 263.2 263.2 . 6o , l6 5 2 = + o. 480, y — - 0.74, 310 310 310 and hence the final results to two decimals are x — — 0.18, jc=+ 0.48, z = — 0.74, which are the most probable values of the unknown quantities. 150. Inserting these values of x, y, z in the six observation equations, the residuals are found to be v 1 = — 0.18, v 2 — -f- 0.48, v % = — 0.74, v 4 — -\- 0.26, » 5 = + °- 1 3. n — + 0.44. 190 SOLUTION OF NORMAL EQUATIONS. X. Squaring these and multiplying each square by its correspond- ing weight there results Spf = 78.57- The computation of the third auxiliaries gives Ipmm. 3] = 78.5, [pms.z] = 78.6, [pss . 3] = J&.8, an agreement which is as close as is necessary for this case Logarithmic Computations. 151. The use of logarithms is often advantageous in forming the products required in the solution of normal equations. A systematic scheme for such solutions will now be presented in which the four-place logarithmic table given at the end of this volume will be employed. In general a five- or seven-place table will be found easier to use when the co-efficients contain more than four significant figures. The scheme to be used will be as follows for three normal equations, the space for checks being in a horizontal row at the bottom and these checks referring to the auxiliaries instead of to the normal equations themselves, which are supposed to have been first formed and checked by the method of Art. 144. The form is first to be filled out by writing the numbers [ad], [ab], . . . [ms] in the places indicated. The logarithms of [ad], [ad], . . . [as] are next taken out and recorded. Then writing log [ad] on a strip of paper, it is subtracted in turn from log [ad], log [ac], log [am], log [as], and the differences are written, thus filling out the top row of squares. Log [ad] is now written on a slip of paper and added to the logarithms at the foot of the first row, thus giving the loga- rithms for the second row. Those in the third and fourth rows are similarly found by adding log [ac] and log [am] to the same ones as before. The numbers corresponding to § i5i- LOGARITHMIC COMPUTATIONS. I 9 I X y z »» J [aa] [ad] M [aw] [as] log [aa] log [ab] log [><•] log [am] log |>.r] log M log [H [am] log r , J [aa] log R [«] M [<$/»] M [aa] log^M] ^M -gM number number number number [W.i] [fc.i] [£#Z. 1] [fc.l] [«] [cm] [«] [am] . [aa] to «^M number number number [« . i] [cm. 1] [«.l] [mm] [»«] 1 [ aOT ]r 1 lQ g T-Tit""'] >o 8 g,H number number [#zwz . 1] [#z.r . 1] Checks. [&.l] |>.l] [#25. i] [W.I] [«. l] these logarithms are then taken from the table, and each number being subtracted from that at the top of the square, the co-efficients [bb 1], [be. ij, . . . [ms . 1] result. Lastly the check [bs. 1] at the foot of the second column is found by 192 SOLUTION OF NORMAL EQUATIONS. X. adding together [bb . 1], [be . 1], and [bm . 1] ; and in a similar manner [cy.i] and [ms . 1] are found. Here [ss . 1] may be determined in two ways, by the addition of the horizontal row and also by the column above it. A second similar tabulation is also made for the next opera- tion, the auxiliaries [bb . 1], be . 1], . . . [ms . 1] being transferred from the first table to the top of the squares in the seconc one. The process will be now exemplified by a numerical example. 152. Let there be given three normal equations which have arisen from a case of conditioned observations, namely, + I7-73*— 4-807— 8.132+ 4.60 = 0, — 4.80JC + 17.60)' — 2.402 + 34.89 = 0, — 8.13a:— 2.407+13.932— 7.75 = 0. Here the check sums [as], [bs], [es], [ms] are to be formed from the given co efficients ; for example, [cs] = - 8.13 - 2.40 + 13.93 - 7-75 = - 4-35. but [mm], [ms], and [ss] cannot be obtained. For the purpose of carrying through the full system of checks, one of these, say [mm], may be assumed, and the others be computed ; assuming [mm] = o, the value of [ms] is + 31.74. The co-efficients and check numbers are then arranged in the upper right-hand corners of the squares in the following table. The four-place logarithms of those in the upper row are taken out, the letter n being affixed to the logarithm of a negative num- ber. The subtractions and additions of these logarithms as required by the scheme of the last article are then made, arid the corresponding numbers taken from the logarithmic table. These subtracted from those in the upper corners give the auxiliaries [66. 1], [be. 1], etc., which are written in the lower §152. LOGARITHMIC COMPUTATIONS. 193 right-hand corners. The checks of these are then made, and found to be verified to one unit of the last decimal. X y Z m s + 17-73 — 4.80 -8.13 + 4.60 + 9.40 1 2487 0.6812K o.gioi« 0.6628 0.9731 T.4325K 1.661472 T.4141 1-7244 4- 17.60 — 2.40 + 34-89 + 45-29 0.1137 0.3426 0.0953K 0.4056?* + 1-3° + 2.20 - 1.25 - 2.54 + 16.30 — 4.60 + 3 6 - J 4 + 47-83 + 13-93 -7-75 -4-35 0.5715 0.3242K 0.6345K + 3-73 — 2. 1 1 — 4-31 -f- 10.20 -5- 6 4 — 0.04 + 0.00 + 3t-74 0.0769 0.3872 + 1.19 + 2.44 -1. 19 + 29.30 Checks + 47.84 — 0.05 + 2931 + 77.09 + 77-IO The next operation is to write the values of the auxiliaries \bb . 1 J, \bc. 1], . . . Sjns . 1] in a second table of squares, and by a similar process obtain the second set \cc . 2], . . . [tns . 2]. 194 SOLUTION OF NORMAL EQUATIONS. X. The scheme shown in the twelve upper left-hand squares of the table in Art. 151 will apply to this case if a, b, c, m be changed to b, c, m, s, and 1 added in all brackets except the y z VI j + 16.30 — 4.60 + 36-14 + 47-83 1. 2*22 o.6628« i.558o 1.6797 i.45o6» O.3458 0.4675 -\- 10.20 - 5-64 — 0.04 0.1134 i.oo36w 1. 1303* + i-3° — 10.20 - 13- 50 + 8.90 + 4-56 + 1346 - 1. 1 9 + 2 9-3° log 4.56 = 0.6590 1.9038 2.0255 log 8. go = 0-9494 + 80.13 + 106.06 log z = 1.7096" -81.32 — 76.76 — 63.30 Check. + I3-46 — 76.76 — 63.30 lowest in each square where the 1 is changed to 2. The operations are strictly analogous to those of the preceding table. A table for the computation of the third set of auxiliaries need not be formed, these being of no use, as the sum \mni\ was assumed at the start. The value of z now is 890 s = — log" 1 1.7096 = — 0.512. § 153- PROBABLE ERRORS OF ADJUSTED VALUES. 195 From the logarithms in the upper squares of the last table, y = — log -1 (0.3458) — log" 1 (i.4S o6 « + i-7°9 6 «) = — 2 -3 6z > and similarly from the logarithms in the upper squares of the first table, according to the last formula of Art. 144, x = — log" 1 (1.4141) — log -1 (1.6614* -f- 1.709672) — log" 1 (1.4325* + o.373i») = — 1. 134, which are the values that closely satisfy the given normal equations. After becoming acquainted with this method by solving several sets of normal equations the student will find it, except when the coefficients are small integers, to be gener- ally more expeditious than methods which do not employ logarithms. Probable Errors of Adjusted Values. 153. When the sum of the weighted squares of the residuals, 2pv 2 , has been computed, the probable error of an independent observation of weight unity is given by (32), namely, r= 0.6745!/— &L, n — q in which n is the number of independent observations and q the number of unknown quantities. If p x , p y , p s be the weights of the adjusted values of x,y, z, the probable errors of these adjusted values then are r r r r x = vp~' y v}~' " vp~;' and thus these are known as soon as the weights have been determined. I9& SOLUTION OF NORMAL EQUATIONS. X. 154. To find these weights the methods of Arts. 74, 75 may be conveniently employed for three unknown quantities. Using the solution in Art. 141, replacing A,, B u C z , etc., by [aa], [ad], [ac], etc., and designating by D the determinant denominator common to the three values, there are found, D _ D _ D [bb][cc] - \bcY ry [aa][cc] - [ac] 2 ' rz ~ \aa\bS\ - [ab] 2 ' which are the weights of the adjusted values of x, y, z. Referring again to Art. 74, and to the method of Gauss given in Art. 144, it is seen that the value of z is \cm . 2 1 z = — L 1 = J . \cc.2\ The negative sign here results from the fact that the absolute terms [am'], [dm], etc., are taken positive in the first members of the normal equations, and the numerator vanishes when those terms are all zero. The quantity [cc.i] is thus the reciprocal of the co-efficient of the absolute terms which be- longed to the normal equation for z and is hence the weight of z, or p z = [cc . 2]. By equating this value of p z to that found above, D may be eliminated from the three expressions, giving . _ r„ „i j, _ [«■#■'] j, _ [cc . 2][b b . i][aa\ p z - \ec . 2J, p, - ^ _ -j , p x - y b j a i _ y £ ] 2 , which are values of the weights expressed in terms of the coefficients and auxiliaries used in finding the value of x, y, z. 155. For example, consider the six observation equations of Art. 148, and let it be required to find the probable errors of the adjusted values of x, y, z. The normal equations are § 156. PROBABLE ERRORS OF ADJUSTED VALUES. 197 solved in Art. 149, giving x = — o. 1 8, y = -f- 0.48, z = — 0.74, and the value of 2pv 2 is found to be 78.6 ; thus, « 4/ 7 8 - 6 r= 0.6745V g-— = 3.45 is the probable error of an observation of weight unity. The weights of the adjusted values of x, y, z are 120. e X 26^5.2 p, = 129.5, A = ^T" 4 = 194-4, _ 129.5 X 263.2 X 3 .o _ ^- 351 X 187 - 7 * 2 77 ' 5 ' and the probable errors of the values of x,y, z are r x =-j^L= = 0.26, r, = -^= = 0.25, r,= 3 :fJL =s 0.30. V177.5 V194.4 V129.5 Accordingly the adjusted values may be written x = — 0.18 ± 0.26, y = + 0.48 ± o 25, z = — 0.74 ± 0.30, which shows the degree of mental confidence that the ad- justed values may claim. 156. When the number of unknowns is large the expres- sions for the weights of the adjusted values become quite complex, and in order to find their values it may be some- times advisable to deduce x,y, z, w, etc., by two or more dif- ferent orders of elimination. The following are formulas for the weights for the case of four unknown quantities, where w is first determined and x last : Pw _ [dd. 3], A [dd~.l]—' _ [dd . j\\cc , 2~\{bb . \} \_dd.z\cc.i\\bb.\'\\ aa\ py ~ \dd. 2 ]*[« . 1] " ' ' px \dd. 2\\cc . i] a [66] "' 198 SOLUTION OF NORMAL EQUATIONS. X. in which the subscript quantities have the following values, r 1 r 1 W [cc . 1]. = [cc] -j^y. [^. 2]. = [ =/(*i> 2 2 . . . z ? ) = M. Now, let approximate values of the unknown quantities be found, either by trial, or by the solution of a sufficient number of equations, and let them be denoted by Z„ Z t . . . Z q . Let § 159. APPENDIX AND TABLES. 201 z,', z 2 . . . z„' be the most probable corrections to these ap- proximate values ; so that Zi = Z l + z,', 2 2 = Z 2 + z 2 . . . z g = Z q + Zg. If, now, each of the functions 4> be developed by Taylor's theorem, and the products and higher powers of the correc- tions be neglected, there will be n expressions of the form t=f(Z 1 ,Z 2 ...Zg)-M + %> + f z,' +... + %' = o. az, az 2 az g Here the terms f(Z lt Z 2 . . . Z g ) are known, and may be desig- nated by N lt N 2 . . . N g ; so that the n expressions reduce to the form — z t + — z 2 + ... + -—z q = M — N, az l az 2 ciZg where the differential co-efficients are to be found by differen- tiating each of the observation equations with reference to each variable, and then substituting the approximate values Z u Z 2 . . . Zg, for z lt z 2 . . . z q . Denoting them, then, by a, b, c, etc., the n equations are of the form az 1 + bz 2 + cz % -J- . . . + h q = M— N, in which all the letters except z lt z 2 . . . z q , denote known quantities. These n equations are exactly like the observa- tion equations (io) or (12), and from them the normal equa- tions are formed, whose solution furnishes the most probable values of the corrections. If non-linear conditional equations are given, it is also neces- sary to find approximate values for the unknown quantities, and assume a system of corrections. Then the functional con- ditional equations may be developed as above, and reduced to 202 NON-LINEAR EQUATIONS. XI. linear equations containing the corrections as unknowns, which may be treated by the method of correlatives, and the most probable system of corrections determined, which, applied to the approximate values, will give the adjusted results. If these do not satisfy the original equations with sufficient accuracy, a new system of corrections should be assumed, and the process be again repeated. Certain expressions, like that in Art. Ill, may be reduced to the linear form by the help of logarithms ; and, when this is possible, it will be found a more convenient method of treat- ment than the development by Taylor's theorem. 160. As an illustration of the method, let M be the number of millions of people under the age of m years, and let it be required to find the most probable value of z in the empirical formula — 50.i6sin#z(o.996)"'z = M, which is supposed to give the relation between M and m for the population of the United States in 1880. The data are nine values of M, from the census compendium, given in the second column of the table below. The first step is to find by trial that 1^.55 is an approximate value of the angle s. The second is to compute nine values of the expression 50.16 sin »z (0.996) "'i°. 55 = N, corresponding to the nine given values of m : these are put in the third column of the table. In the fourth column are the differences M — N between the observed and computed values. The fifth column contains the values of the derivative -j- = 5o.i6wz(o.996) m cos»z(o.996)'*i°.55, corresponding to the nine values of m. § i6o. APPENDIX AND TABLES. 203 m. M. N. M— N. tf dz IO !3-39 12.90 + 0.49 466 20 24.12 24.06 + 0.06 814 30 33- 2 9 33-07 + 0.22 1004 40 39.66 40.08 — 0.42 1032 5° 44.22 44.98 — 0.76 9*3 60 47-33 48.10 - o-77 674 70 49.16 49-74 -0.58 349 80 49.94 50.14 — 0.20 29 90 50.14 49.67 + 0.47 447 Let z 1 be the most probable correction to the assumed value of z. Then the last two columns furnish the nine observation equations 466/ = + 0.49, 814/ = + 0.06, etc. From these the single normal equation is formed, and its solu- tion gives z 7 = — o IJ .ooo25 = — o°.oi4, and hence the most probable value of z is z = i°.55 - o°.oi4 = i°.536; so that the empirical formula may be written M = 50.16 sin wz (0.996) w i".536. When z is i°.SS, the sum of the squares of the residuals is about 2.25 ; and, when z is i°.S36, that sum is about 1.67 ; so that 204 MEAN ERROR AND PROBABLE ERROR. XI. the precision of the formula has not been greatly increased by the variation in the angle z. By slightly increasing the number 0.996, the formula can be made to more closely agree with the observations. Mean Error and Probable Error. 161. The probable error is an error of such a value that any given error is as likely to exceed it as be less than it, and it hence seems to be the quantity that would most naturally be selected for indicating the precision of observations. But there is another error very commonly employed for the same purpose called the "mean error," whose definition is, the error whose square is the mean of the squares of all the errors. Hence the mean error is, for direct observations, the square root of the quantity , or, in terms of the residuals, the square root of n the quantity . In general, then, the mean error can be n — 1 determined from the formulas for probable error by changing the co-efficient 0.6745 into unity. If m be the mean, and rthe probable error, the relation between them is = 1.4826^. 0.6745 In the annexed figure, OP indicates the probable error, and OM the mean error. It is seen, by Art. 29, that M is the abscissa of the point of inflection of the probability curve. In Table II, the value of the integral for the argument 1.48267- is 0.6826. Hence 0.6826 is the probability that an error is less than the mean error, or in 1,000 errors there should be 683 less than m. It is a fair wager of 1 to 1 that an error taken at random is less than the probable error ; but it is a fair wager of 683 to 317, or about 2.15 to 1, that it is less than the mean error. § 162. APPENDIX AND TABLES. 205 The mean error is generally used in German books. In this country the probable error is commonly employed ; and, being the most natural unit of comparison, it is certainly to be desired that it alone should be used, and the mean error be discarded. Fig. 14. 162. Instead of the mean or probable error, a quantity called the " huge error " might be employed to indicate the precision of measurements. The huge error is defined to be an error of such a magnitude that 999 errors out of 1,000 are less than it, and only 1 greater ; or, in other words, that the probability of an error being less than it, is 0.999. ^ u De the huge error, the relation of u to r is found from Table II. tc For P = 0.999, the argument - is 4.9 : hence f u ■= 4.gr. Accordingly, all formulas for probable error may be changed into those for huge error by writing 3.3 in place of 0.6745. For instance, the huge error of a single direct observation is given by / 2» 2 In Fig. 14 the abscissa OU represents the huge error, and the area UD AD U is 0.999 0I the total area. 206 APPENDIX AND TABLES. XL Uncertainty of the Probable Error. 163. The value of the probable error r, deduced in Art. 67, is the best attainable value, or rather the most probable value. The inquiry is now to be made as to what are the probable limits of this value of r, or what is the probable error of the probable error. Or, if the probable error r be written in the form r(i ± «), the number u is the uncertainty of the probable error, that is, it as likely that the value formed for r lies between the limits r(i — 11) and r{\ -f- u) as that it lies outside those limits. Thus ur may properly be called thw probable error of the probable error r. 164. A series of observations having been made, all having the same measure of precision h, the sum of the squares of the errors is a constant, while the probability of any value h is, by Art. 65, P' - h n {dx)"7t - i"e - h ^*' t and the value of k which renders this a maximum is the most probable value of h. Now let h -(- uh be a value greater than this probable value ; then P" = (h + uh) n (dx)"7t ~ i*e ~ < /l + «W** Is the probability of the value h -j- uh. The ratio of these probabilities is P" and taking the logarithms of both sides of this equation, P" log -p — ft log (1 + u) - (2u + u 2 )/t 2 2x 2 ; § 164. UNCERTAINTY OF THE PROBABLE ERROR. 207 also replacing log (1 -\- u) by u — \u 2 , the terms involving higher powers of u being omitted, there results P" log -pf = (n — 2/i 2 2x 2 )u — {\n -\- h 2 2x 2 )u 2 . The value of h deduced in Art. 65 causes the co-efficient of u to become zero, whence log —, = - nu 2 and -^ = e~ m \ Thus the probability of the variation nh in the value of h is expressed by the function P" = «-«*■, which is of the same form as the law of accidental error. The probability that u is less than any assigned limit is therefore, as in Art. 32, expressed by the integral W- f U e"""du = -7= /'V-'V/, 71 vo \ TC-eing the amount of correction. _ Article 67. — The reason why ~2px*y is the same as is sometimes not clear to students. If each term such as/,^,' occurs ny x times in n observations, then p 1 x 1 \ny l +p,x 3 \ny, + etc. = 2px*, or n{piX?y x +p,xy, + etc.) = 2j>x* ; whence, dividing by n, follows the statement as given. Article 89. — Problem 1: o".4o8. Problems 3 and 4: The combination of observations differing widely in precision, as in these examples, is not always safe in practice, because of the constant errors which are liable to affect the less precise series, so that the practical weight of the more precise series is often greater than that derived from the probable errors. Problem 6: It should be inferred that a constant source of error exists. Article 98. — Problem 2: 0.000137, which occurs when A is 135 degrees. Problem 4: 0.005. Problem 6: The probable error of the mean of the three readings is ' — , and that of * 3 the difference of level of two stations is this multiplied by V2; then for the 130 stations there are 129 differences of level, and the probable error of the final result is 0.0093 feet- Article 107. — The proof of this method may be made in the following manner: Let x/ and y/ be the adjusted values cf the observations x t and y x , so that the residual errors are § 171. ANSWERS TO PROBLEMS; AND NOTES. 2I 7 ar/ — x^ and j// — _y 2 . Then the most probable values of S and T are to be found from the condition ~2p(xJ — x^Y -\- •2'(_y 1 ' — y^y = a minimum. The adjusted points all lie upon the line whose equation is y = Sx -\- T. Now let a second line be drawn through the observed point whose co-ordinates are x x and jj/, , and the ad- justed point whose co-ordinates are x( and y/ ; its equatiun is y — y, = S'(x — xj. By combining this with the equation of the required line the values of the residual errors are deduced, whence the above condition becomes P + S" -^^(Sx -\- T — yf = a minimum. {s'-sy This is to be made a minimum for S', S, and T separately. Taking the derivative with respect to S' and equating to zero there is found S'S -\- p = o, which gives the inclination S' in terms of 5. Again, differentiating with respect to J> and T there are deduced two equations in 5 and T, namely, S"-~Sxy — S-TSx — S2y*+ 2 ST2y — nST 2 +pS^x ,i —p2xy + pT2x=o, SSx + nT—2y = o, and the solution of these gives values for 5 and T which agree with the results stated in the text. Article 112. — Problem 6: Let p represent the population in millions and x the number of decades since 1800. Then using the ten censuses from 1790 to 1880, there is found p = 4.97 + 0.873* + 0.581* 2 , which gives 59890000 for 1890, while the actual enumeration was 62 870 000. Again, taking the seven censuses from 1820 to 1880, there is found p = 7.29 — 0.280.x + 0.689.x 2 , which gives 60 579000 for 1890, an accordance more satisfac- tory. The sum of the squares of the residual errors for the 2l8 APPENDIX AND TABLES. XL, latter formula is 1.35, while for the same seven census years the former gives 3.34. Article 113. — Whether observations shall be independent or conditioned depends in general upon the selection of the unknown quantities whose values are to be determined. Thus if AOB, BOC, and AOC are angles measured at a station 0, the observation equations are independent if x and y be put for two of these angles. But if x, y, and z are taken as the three quantities, these are conditioned by the necessary rela- tion that the sum of two of them is equal to the third. Article 126. — Problem 1 : Refer to problem 4 of Article 59. Problem 2: x = 93° 48' I4".99, y = 51" 54' 59".84, z = 34 16' 49".22. Problem 5 : This was the problem proposed by Patterson in 1808, and by whose discussion Adrain was led to the discovery of the principle of least squares. Article 144. — The arithmetical mean of more than two ob- servations is, in strictness, the most probable value only when the results of the measurements are unknown. If the mind knows the values of the measurements, it instinctively assigns greater reliability to some than to others, and hence the weights are not equal. For example, let M x = 40, M, = 51, M 3 = 52 be three observations of the same quantity : it is reasonable to suppose that M x is of less reliability than the others, while the method of the mean assigns it the same weight. Theory has not been able to determine what theoretical weight? should be assigned in a case like this, but probably an ap- proach to them might be secured by taking the reciprocal of (Af, - MJ' -f (M, - AT,)" as the weight of M, , the reciprocal of (M, - My + (M, - M$ as the weight of Af„ and that of (M a - M,f + (Af, - AfJ as the weight of M,. For the above numerical example this gives ^-g as the weight of 40, j^-g as the weight of 51, and ^ as the weight of 52, from which results the general mean z — 49.18, whereas the arithmetical mean is 47.67. § 172. DESCRIPTION OF THE TABLES. 2ig Description of the Tables. 172. Tables I and II give values of the probability integral (4) ; the first for the argument hx, and the second for the argu- JlX X ment , or -. In both cases the arrangement is like that 0.4769 r of logarithmic tables, and needs no explanation. The use of Table I is illustrated in Arts. 32 and 33, and that of Table II in Art. 128. These tables were first given by Encke in 1832, and were computed by him from a table of the values of fe~' 2 dt, which was published by Kramp in 1799. Tables III and IV give values of the co-efficients which occur in the formulas for probable error for values of n. Table III applies to the usual formulas (20) and (21), and its use is illustrated in Art. 82. Table IV applies to the shorter formulas (35) and (36), and its use is illustrated in Art. 84. These tables were computed by Wright, and first published in "The Ana- lyst" for 1882, vol. ix, p. 74. Table V gives four-place logarithms of numbers, and Table VI gives four-place squares of numbers. The latter will be found very useful for obtaining the squares of residuals. It may be also used in forming the co-efficients in normal equa- tions, and for other purposes. For instance, the co-efficient \ab~\ may be written [a6] = *([(« + *)'] - [a 2 ] - [*']), and the sums [a 2 ], [b 7 ], and [(a -\- b) 2 ] may be easily formed with the help of the table of squares. This method has the advan- tage that no attention need be paid to the signs of a and b, except in forming the sums a -f- b. Table VII is to be used in discussing doubtful observations by Chauvenet's criterion, and its use is explained in Art. 130. 220 APPENDIX AND TABLES. Table VIII gives the squares of reciprocals of numbers from o.o to 9.0, and may be used in the computation of weights from probable errors. TABLE I. 2 nf Values of the Probability Integral -p f e'^dt for Argument t or hx. hx. 0.0 O.I 0.2 o-3 0.4 01234 56789 Diff. 0.0000 o.or 13 0.0226 0.0338 0.0451 1 125 1236 1348 1459 1569 2227 2335 2443 2550 2657 3286 3389 3491 3593 3694 42S4 4380 4475 4569 4662 0.05640.06760.07890.0901 0.1013 1680 1790 1900 2009 211S 2763 2869 2974 3079 3183 3794 3893 3992 4090 4187 4755 4847 4937 5 02 7 5"7 "3 no 106 100 92 0.5 0.6 0.7 0.8 0.9 0.5205 0.5292 0.5379 0.5465 0.5549 6039 61 17 6194 6270 6346 6778 6S47 6914 6981 7047 7421 7480 7538 7595 7651 7969 S019 8068 81 16 8163 0-5633 o-57 16 0.5798 0.5879 0.5959 6420 6494 6566 6638 6708 7112 7175 7238 7300 7361 7707 7761 7814 7S67 7918 8209 8254 8299 8342 8385 83 74 64 55 45 1.0 1.1 1.2 ■■3 1.4 0.8427 0.8468 0.8508 0.8548 0.8586 8802 8S35 8868 8900 8931 9103 9130 9155 9181 9205 9340 9361 9381 94oo 9419 9523 9539 9554 9569 95 8 3 0.8624 0.8661 0.8698 0.8733 0-8768 8961 8991 9020 9048 9076 9229 9252 9275 9297 9319 9438 9456 9473 9490 9507 9597 96n 9624 9637 9649 37 30 23 18 14 '■5 1.6 i-7 1.8 1.9 9661 0.9673 0.9684 0.9695 0.9706 9763 9772 9780 9788 9796 9838 9844 9850 9856 9861 9891 9895 9899 9903 9907 9928 993 1 9934 9937 9939 0.97 16 0.9726 0.9736 0.9745 0.9755 9804 981 1 9S18 9S25 9832 9867 9872 9877 9882 9886 991 1 9915 9918 9922 9925 9942 9944 9947 9949 9951 10 7 5 4 3 2.0 2.1 2.2 2-3 2.4 0.9953 -9955 Q-9957 0.9959 0.9961 9970 9972 9973 9974 9975 9981 9982 9983 9984 9985 9989 9989 9990 9990 9991 9993 9993 9994 9994 9994 0.9963 0.9964 0.9966 0.9967 0.9969 9976 9977 9979 9980 9980 9985 9986 9987 9987 99S8 999' 9992 9992 9992 9993 9995 9995 9995 9995 9996 2 1 1 2. -9953 -9970 0.9981 0.9989 0.9993 0.9996 0.9998 0.9999 0-9999 0-9999 CO 1 .0000 hx. 01234 56789 Diff. VALUES OF THE PROBABILITY INTEGRAL. 221 TABLE II. 2 7n 2 /* u Values of the Probability Integral -— I e~ r 'dt for Argument or -, 0.4769 r X r 1 2 3 4 5 6 7 8 9 Diff. 0.0 0.0000 0.00 S4 0.0108 ( 3.0161 3.0215 0.02690.0^23 c 2-°377 3.0430 ( 3.0484 54 o.l °533 0591 0645 0699 0752 0S06 0859 0913 0966 1020 54 0.2 ;o 7 3 1 126 1 180 1233 1286 1339 1392 1445 1498 1551 53 o-3 .603 1656 1709 1761 1814 1866 1918 1 971 2023 2075 52 0.4 2127 2179 2230 2282 2334 2385 2436 2488 2539 2590 51 0.5 0.2641 0.2691 0.2742 0.2793 ( 3.2843 0.28930.29441 3.2994 < 3-3°43 ( 3-3°93 5° 0.6 3H3 3'92 3242 3291 334° 3389 3438 3487 3535 3583 49 0.7 3 6 3 2 36S0 3728 3775 3823 3870 39'8 396S 4012 4059 46 0.8 4105 4152 4198 4244 4290 4336 4381 4427 4472 45*7 45 0.9 4562 4606 4651 4695 4739 4783 4827 4860 4914 4957 43 1.0 0.5000 0.5043 5085 ( 3.5128 < 3.5170 0.5212 i 3.52540.52950.53370.5378 41 1.1 5419 5460 5500 5540 55S1 5620 5660 5700 5739 5778 39 1.2 5817 5856 5894 5932 5970 6008 6046 6083 6120 6i57 37 x -3 6194 6231 6267 6303 6339 6375 6410 6445 6480 6515 35 1.4 6550 6584 6618 6652 6686 6719 6753 6786 6818 6851 32 '■S 0.6883 < 3.691 5 < 3.6947 3.6979 3.701 1 0.7042 0.7073 ( 3.7104 c 5-7i34< 3.7165 3° 1.6 7195 7225 7255 7284 73 r 3 7342 737i 7400 7428 7457 28 i-7 7485 7512 7540 7567 7594 7621 7648 7675 7701 7727 26 1.8 7753 7778 7804 7829 7854 7879 7904 7928 7952 7976 24 i-9 8000 8023 8047 8070 8093 8116 8138 8161 8183 8205 22 2.0 0.8227 0.8248 ( 3.8270 < 3.8291 < 3.S312 0.8332 < 5 - 8 353 < 5-S373 < 3.8394 < 3.8414 19 2.1 8433 8453 8473 8492 8511 853° 8 549 8567 8585 8604 18 2.2 8622 8639 8657 8674 8692 8709 8726 8742 8759 8775 17 2 -3 8792 SSoS 8824 8840 8855 8870 8886 8901 8916 8930 15 2.4 8945 8960 8974 898S 9002 9016 9029 9043 9056 9069 13 2.5 0.9082 0.9095 0.9108 < 3.91 2 1 ( '•9133 0.91460.91580.91700.9182 0.9193 12 2.6 9205 9217 922S 9^39 9250 9261 9272 9283 9293 93°4 10 2.7 93 ! 4 93 2 4 9334 9344 9354 9364 9373 9383 9392 9401 9 2.8 9410 9419 9428 9437 9446 9454 9463 947i 9479 9487 8 2.9 9495 95°3 951 1 95'9 9526 9534 954i 9548 9556 9563 7 3-° 0.9570 0.9577 < >-95 8 3 3.95900.9597 0.9603 0.9610 0.9616 0.9622 0.9629 6 3-i 9635 9641 9647 9652 9658 9664 9669 9675 9680 9686 5 3-2 9691 9696 9701 9706 9711 9716 9721 9726 973 1 9735 S 3-3 9740 9744 9749 9753 9757 9761 9766 977° 9774 9778 4 3-4 9782 97S6 9789 9793 9797 9800 9804 9807 981 1 9814 4 3- 0.9570 0.9635 0.9691 < 3.9740 ( 3.9782 0.9818 0.9S48 0.9874 0.9896 0.991 5 4- 993° 9943 9954 9963 9970 9976 9981 998S 9988 9990 5- 9993 9994 9996 9997 9997 9998 9998 9999 9999 9999 CO 1 .0000 \ r ' 2 3 4 5 6 7 8 9 Diff. 222 APPENDIX AND TABLES. TABLE III. For Computing Probable Errors by Formulas (20) and (21). 0.6745 0.6745 0.6745 0.6745 n. V« — 1' S/u («-l) - n. V« — i' V» (« — 1)' 40 0.1080 0.0171 41 1066 0167 2 0.6745 O.4769 42 i°53 0163 3 4769 2754 43 1041 0159 4 3 8 94 1947 44 1029 0155 5 0.3372 O.I508 45 0.1017 0.0152 6 3016 I23I 46 1005 0148 7 2754 IO4I 47 0994 0145 8 2549 O9OI 48 0984 0142 9 2385 O795 49 0974 oi39 10 0.224S 0.07 1 1 50 0.0964 0.0136 11 2133 0643 5i 0954 0134 12 2029 O587 52 0944 0131 '3 1947 O54O 53 0935 0128 H 1871 O5OO 54 0926 0126 *5 0.1S03 O.O465 55 0.0918 0.0124 16 1742 0435 56 0909 0122 17 1686 O4O9 57 0901 0119 iS 1636 O386 58 0893 0117 19 1590 03<55 59 0886 01 1 5 20 0.1547 O.O346 60 0.0878 0.0113 21 1508 0329 61 0871 OIII 22 1472 03H 62 0864 OIIO 2 3 1438 O3OO 63 0S57 0108 24 1406 O287 64 0850 0106 25 0.1377 O.O275 65 0.0843 0.0105 26 1349 O265 66 0837 0103 27 I 3 2 3 0255 67 0830 OIOI 28 1298 0245 68 0824 0100 29 1275 0237 69 0818 0098 3° 0.1252 0.0229 70 0.0812 0.0097 3 1 1231 0221 71 0806 0096 3 2 1211 0214 72 0800 0094 33 1 192 0208 73 0795 0093 34 1174 0201 74 0789 0092 35 0-1157 O.OI96 75 0.0784 0.0091 36 1 140 OI9O 80 o759 0085 3£ 1 1 24 0185 85 0736 0080 38 1109 Ol8o 90 0713 0678 0075 0068 39 1094 OI75 100 FOR COMPUTING PROBABLE ERRORS. 22J TABLE IV. For Computing Probable Errors by Formulas (35) and (36). n. 0.8453 0.8453 n\ n — 1 n. 0.8453 \n (n — i) 0.8453 n\n — 1 SJn (« _ I)' 40 0.0214 0.0034 41 0209 0033 2 0.597s 0.4227 42 0204 0031 3 345 1 '993 43 0199 0030 4 2440 1220 44 0194 0029 5 0.1890 0.0S45 45 0.0190 0.0028 6 ■543 0630 46 0186 0027 7 i3°4 0493 47 0182 0027 8 1 130 °399 48 0178 0026 9 0996 °33 2 49 0174 0025 10 0.0891 0.0282 5° 0.0171 0.0024 11 0S06 0243 5 1 0167 0023 12 0736 0212 52 0164 0023 '3 0677 0188 53 0161 0022 14 0627 0167 54 0158 0022 T 5 0.0583 0.0151 55 0.0155 0.0021 16 0546 0136 56 0152 0020 17 °5!3 0124 57 0150 0020 18 04S3 0114 58 0147 0019 19 0457 0105 59 0145 0019 20 0.0434 0.0097 60 0.0142 0.0018 21 0412 0090 61 0140 0018 22 °393 0084 62 0137 0017 2 3 0376 0078 63 OI 35 0017 24 0360 0073 64 °[33 0017 25 0.0345 0.0069 65 0.0131 0.0016 26 °33 2 0065 66 0129 0016 27 °3!9 0061 67 0127 0016 28 0307 0058 68 0125 0015 29 0297 0055 69 0123 0015 3° 0.0287 0.0052 70 0.0122 0.0015 3' 0277 0050 7i 0120 0014 3 2 0268 0047 72 0118 0014 33 0260 0045 73 0117 0014 34 0252 0043 74 0115 0013 35 0.0245 0.0041 7 J 0.01 13 0.0013 36 0238 0040 80 0106 0012 37 0232 0038 85 0100 OOI I 38 0225 0037 90 0095 0010 39 0220 °°3£ 100 0085 0008 224 APPENDIX AND TABLES. TABLE V. — Common Logarithms. » 0000 1 0043 2 0086 3 OI2S 4 0170 5 6 7 8 9 Diff. 10 0212 0253 0294 0334 0374 42 II 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 38 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1 106 35 13 "39 "73 1206 1239 1271 1303 1335 1367 '399 M3o 32 14 1461 1492 1523 1553 1584 1614 1044 1673 1703 1732 30 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 28 16 204 c 2068 2095 2122 2I4S 2175 2201 2227 2253 2279 27 17 2304 2330 2355 23SO 2405 2430 2455 2480 2504 2529 25 pa 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 24 19 27S3 2S10 2833 2856 2878 2900 2923 2945 2967 29S9 22 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 20 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 19 23 3617 3636 3655 3674 3692 37" 3729 3747 3766 3784 18 24 3S02 3820 383S 3856 3874 3S92 3909 3927 394 5 3962 18 25 3979 3997 4014 4031 404S 4065 4082 4099 4116 4133 17 26 4150 4166 4183 4200 4216 4232 4249 4265 42S1 429S 17 27 43U 4330 4346 4362 4378 4393 4409 4425 4440 4456 16 28 4472 44S7 4502 4518 4533 4548 4564 4579 4594 4609 '5 29 4624 4639 4634 4669 4683 4698 4713 4728 4742 4757 15 30 4771 4786 4800 4814 4829 4843 4857 487 r 4SS6 4900 14 31 4914 4928 4942 4955 4969 4983 4997 501 1 5024 5038 14 32 5051 5065 5079 5092 5105 5"9 5132 5M5 5159 5172 13 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 13 34 5315 5328 5340 5353 5366 5378 5391 5403 54,6 5428 13 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 12 36 5563 5575 55S7 5599 56 1 1 = 623 5635 5647 5658 5670 12 37 56S2 5694 5705 5717 5729 574° 5752 5763 5775 5786 12 3S 579S 5809 5S2I 5832 5843 5855 5866 5877 5888 5899 II 39 59" 5922 5933 5944 5955 5966 5977 59S8 5999 6010 II 40 6021 603 r 6042 6053 6064 6075 6085 6096 6107 6ci7 II 41 6128 613S 6149 6160 6170 6180 6191 6201 6212 6222 II 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 IO 43 6335 6345 6355 6365 6375 6385 fi395 6405 6415 6425 10 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 10 45 6532 6542 6531 6561 6571 6580 6590 6599 6609 661S ro 46 662S 6637 6646 6656 6665 6675 66 S4 6693 6702 6712 9 47 6721 6730 0739 6749 6758 6767 6776 67S5 6794 6S03 9 4 S 6812 6821 6830 6839 684S 6857 6866 6875 6-84 6S93 9 49 6902 69 1 1 6920 692S 6937 6946 6955 6964 6972 6981 9 5° 6990 6998 7007 7016 7024 7033 7042 7050 70J9 7067 9 51 7076 7084 7093 7101 71 10 7118 7526 7135 7143 7152 S 52 7160 7168 7177 71S5 7193 7202 7210 7218 7226 7235 8 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 8 54 7324 7332 7340 7348 7356 7364 7372 73So 73S8 7396 8 Diff. » 1 1 2 3 4 5 6 7 8 9 COMMON LOGARITHMS. 225 TABLE V. — Common Logarithms. « 1 2 3 4 5 6 7 8 9 Diff. 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 8 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 58 7b34 7042 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 £0 7782 77»9 7796 7803 7S10 781S 7825 783= 7839 7846 7 6' 7853 7860 7868 7875 7882 7889 7896 7903 7910 79'7 62 7924 7931 7938 7945 7952 7959 7866 7973 79S0 7987 t?. 7993 8c 00 8007 8014 8021 8028 8035 8041 S048 8055 % f 8062 8069 8075 80S2 8o8g 8096 8102 8109 8116 8122 05 8129 8136 8142 8149 8156 S162 8169 8176 8182 8189 7 66 8195 8202 8209 8215 S222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 8299 8 306 S312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 S420 8426 8432 8439 8445 7° 8451 8457 8463 8470 8476 S482 84S8 8494 8500 8506 6 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 85S5 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 S785 8791 8797 8802 6 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 SS65 8871 8S76 8S82 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 5 81 9085 9090 9096 gioi 9106 9112 9117 9122 9:28 9 J 33 82 9138 9143 9149 9154 9159 9165 9170 9'75 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 5 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 943° 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 958i 9586 5 91 9590 9595 9600 9605 9609 9614 9619 9624 962S 9633 92 9638 9643 9647 9652 9 6 57 9661 9666 9671 9 6 75 96S0 93 96S5 96S9 9694 9699 9703 9 70S 9713 9717 9722 9727 94 9731 9736 9741 9745 975° 9754 9759 9763 9768 9773 95 9777 97S2 9786 9791 9795 9800 9805 9809 9814 9818 4 96 9823 9827 9832 9836 9841 9845 9850 9854 9S59 9863 97 9868 9872 9877 9881 9SS6 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 n 9956 9961 99f>5 9969 9974 9978 9983 9987 9991 9996 1 2 3 4 5 6 7 8 9 Diff. 226 APPENDIX AND TABLES. TABLE VI.— Squares of Numbers. « 1 2 3 4 5 6 7 8 9 Diff. 22 I.O 1. 000 1.020 I.O40 1. 061 1.082 1. 103 1. 124 I-M5 1.166 1.1S8 I.I 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 I.488 I-5I3 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 z.280 2.3IO 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 I. g 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 2.3 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 2-5 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 2-7 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 8.5:6 3.5S5 8.644 8.703 8.762 8.821 8.880 8.940 60 3.0 9.000 9.060 g.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 9923 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 II.9O 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 I3-9 1 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 16.4O 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 I7-56 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 I9.80 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 24.9O 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. U4 10 5-2 27.04 27.14 27.25 27.35 27.46 27-56 27.67 27-77 27.88 27.98 TI 5-3 28.09 28.20 28.30 28.41 28.52 28.62 28.73 28. S4 28.94 29.05 5-4 29.16 29.27 29 3S 29.48 29.59 29.70 29.81 29.92 30.03 30.14 '' n 1 2 3 4 5 6 7 8 9 Diff. ; SQUARES OF NUMBERS. 227 TABLE VI.— Squares of Numbers. n 01234 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 33.06 33.18 33- 29 33-41 33-52 12 5-8 33-64 33-76 33.87 33.99 34.11 34.22 34.34 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-og 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 3S.94 39.06 39.19 39-3' 39-44 39-56 13 6.3 39.69 39.82 39.94 40.07 40.20 40 32 40.45 40. 5S 40.70 40-53 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. So 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 59-9 1 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 6S.72 17 8-3 68.89 6g.o6 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 74-3° 74-48 74.65 74.82 71;. co 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 7903 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. g3 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 9 2 -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 9742 97.61 97.81 20 9-9 98.01 98.21 9S.41 98. 60 98.80 99.00 99.20 99.40 99.60 99.80 20 1 « 01234 5 6 7 8 9 Diff. 22C APPENDIX AND TABLES. TABLE VII. — For Applying Chauvenet's Criterion. „. /. n. /. n. t. 3 2.05 '3 3-°7 2 3 341 4 2.27 14 3.12 24 3-43 5 244 '5 3.16 2 5 345 6 2-57 16 3- '9 3° 3-55 7 2.67 17 3.22 40 3-70 8 2.76 18 3.26 5° 3.82 9 2.84 19 3- 2 9 75 4.02 10 2 91 20 3-3 2 TOO 4.16 ii 2.96 21 3-35 200 4.48 12 3.02 22 3-3« 500 4.90 TABLE VIII. — Squares of Reciprocals. n. I n. 1 n. 1 ;/ 2 n 2 u 2 0.0 00 25 0. 1 600 5.0 0.0400 O.I IOO.OOO 2.6 0.1479 5-i 0.0384 0.2 25.000 2.7 0.1372 5-2 0.0370 °-3 1 1. Ill 28 0.1276 5-3 0.0356 0.4 6.250 2.9 o.i 1S9 54 0.0343 0.5 4.000 3-° 0. 1 1 1 1 5-5 0.0331 0.6 2.778 3- 1 0.1041 5-6 0.0319 0.7 2.041 3-2 0.0977 5-7 0.0308 o.S I.562 3-3 0.0918 5-S 0.0297 0.9 1-2.35 34 0.0865 5-9 0.0287 1.0 1. 000 3-5 0.0SI6 6.0 0.0278 I.I 0.8264 3-6 0.0772 6.1 0.0269 1.2 0.6944 3-7 0.0730 6.2 0.0260 '■3 0.5917 3-3 0.0693 6-3 0.0252 1.4 0.5102 3-9 0.0657 6.4 0.0244 '•5 0.4444 4.0 0.0625 6-5 0.0237 1 6 0.3906 4.1 0.0595 6.6 0.0230 '•7 03460 4.2 0.0567 6.7 0.0223 1. 8 0.3086 4-3 0.0541 6.8 0.0216 1.9 0.2770 44 0.0517 6.9 0.0210 2.0 0.2500 4-5 0.0494 7.0 0.0204 2.1 0.226S 4.6 0.0473 7-5 O.OI78 2.2 0.2066 4-7 0.0453 8.0 O.OI56 2 -3 0.1890 4.S 0.0434 8-5 O.OI38 2.4 0.1736 4.9 0.0416 9.0 O.OI23 INDEX. 229 INDEX. Accidental errors, 4 Adjustment, I, 36, 51, 88, 101, 109, 141, 1S7 Angle measurements, 104, 171 repetition, 106 Angles, 3, go, 98, 122, 163 at a station, 117, 145 in a quadrilateral, 147, 150 in a triangle, 142 Areas, 3, 106 Arithmetical mean, 42, 70, 211, 218 Axioms, 13 Base lines, 100, 102 Binomial formula, 10 Borings, 140 Certainty, 6 Chaining, 103 Chauvenet's criterion, 166, 228 Coins, throwing of, 9 Comparison of observations, 1, 66 Conditioned observations, 2, 57, 86, 141, 192 Constant errors, 3, 169 Constants, 214 Correlatives, 60 Criterion for rejection, 166 Curve of probability, 15, 25, 204 Declination, magnetic, 134 Direct observations, 2, 41, 88 Doubtful observations, 166 Earth, temperature of, 140 Empirical constants, 124 formulas, 130 Equal weights, 88 Equations, non-linear, 200 normal, 46, 56, 175 observation, 58 solution of, 175 Error, definition of, 5 law of, 13, 17, 22 probability of, 13, 162 propagation of, 75 Experience, axioms from, 13 Functions of observations, 90 Gauss's discussions, 22, 175, 181,, 212 General mean, 42, 72 Geodesy! 151, 214 Guessing, problem on, 174 Hagen's proof, 17, 168 History of Least Squares, 211 Huge error, 205 Impossibility, 6 Independent observations, 2, 51, 79,, 100 Indirect observations, z, 43 Instrumental errors, 4 Level lines, 44, no, 157 Levelling, 154 230 INDEX. Linear measurements, lot Literature of Least Squares, 213 Logarithmic computation, 190 Logarithms, 219, 224 Magnetic declination, 134 Mean error, 204 Measure of precision, 34, 68 Median, 208 Mistakes, 4, 169 Most probable value, 2, 9, 38 Non-linear equations, 200 Normal equations, 46, 56, 175 Observation equations, 58 Observations, adjustment of, 1, 36, 88, 101, 109, 141 classification of, 2 discussion of, 162 errors of, 3, 5, 13 precision of, 66 rejection of, 166 weights of, 36 Orbit of a planet, 129 Peirce's criterion, 169 Pendulum, 124 Population of United States, 202, 217 Principle of Least Squares, 38, 211 Probability, 6, 9. Probability curve, 13, 25, 68 integral, 27, 220, 221 of error, 13, 162, 212 Probable error, 66, 70, 72, 79, 86, 92, 195, 204 Propagation of error, 75 Quadrilateral, 147 Quetelet's statistics, 175 Reciprocals, squares of, 228 Rejection of observations, 166 Repetition of angles, 106 Residual, 5, 39 Rivers, velocity in, 131 Shooting at target, 13, 165 Social statistics, 172 Solution of equations, 56, 175 Squares of numbers, 227 reciprocals, 228 Station adjustment, 118, I45 Statistics, 162, 172 Tables, 220-228 Target shots, 13, 165, 170 Theory and experience, 31 Triangle adjustment, 59, 142 Triangulation, 152 Uncertainty of median, 210 probable error, 206 Unequal weights, 51, 95, 122 Velocity observations, 131, 138 Weighted mean, 43 observations, 37, 51, 187 residuals, 39 Weights, 36, 69, 196 Wright's probable-error tables, 219, 222, 223 Short-title Catalogue OF THE PUBLICATIONS OF JOHN WILEY & SONS New York London: CHAPMAN & HALL, Limited ARRANGED UNDER SUBJECTS Descriptive circulars sent on application. 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