UC-NRLF *C Its IDb i U fe Wi "i iTHEORY OF OBSERYATIONS LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class wm^ THEORY OF OB SE RVATION S THEORY OF OBSERVATIONS BY T. N. THIELE DIRECTOR OF THE COPENHAGEN OBSERVATOHY -►•<>>»3i5Kc>-.-. Y o! rue '>■ f UNIVERSITY ; Of . PUBLISHED HY CHARLES & EDWIN LAYTON 56 FARRINGDON STREET LONDON 1903 t4- COPENHAGEN. — PRINTED BT BIANCO LUNO CONTENTS Numbers of I. The Law of Causality. formulae Page § 1. Belief in Causality 1 § 2. The Observations and their Circumstances 2 § 3. Errors of Observations 3 § 4. Theoretical and Empirical Science , 4 II. Laws of Errors. § 5. On Rei)etitions 6 § 6. Laws of Actual Errors and Laws of Presumptive Errors 5 § 7. The Law of Large Numbers of Repetitions 6 § 8. Four Different Forms of Laws of Errors 7 III. Tabular Arrangements. § 9. Frequency and Probability 8 § 10. Repetitions with Qualitative Differences between the Results 8 § 11. Repetitions with Quantitative Differences between the Results 9 IV. Curves of Errors. § 12. Curves of Actual Errors of Observations in Discontinued Values 10 § 13. Curves of Actual Errors for Rounded Observations 10 § 14. Curves of Presumptive Errors 11 § 1.5. Typical Curves of Errors 14 § 16. Particular Measures of Curves of Errors 14 V. Functional Laws of Errors. § 17. 1. Their Determination by Interpolation 15 § 18. 2—8. The Typical or Exponential Law of Errors 16 Problems 19 § 19. 9—13. The Binomial Functions 20 § 20. 14. Some of the more general Functional Laws of Errors. Series 21 111731 Numbers of VI. Laws of Errors expressed by Symmetrical Functions. formulae Page § 21. 15—16. Coefficients of the Equation of Errors. Suras of Powers 22 § 22. 17—24. Half-luvariants 24 § 23. 25—27. Mean Values, Mean Deviations, Mean Errors 27 Examples 29 VII. Relations between Functional Laws of Errors and Half-Invariants. § 24. 28- 29. Their Kelations 30 Examples 31 § 25. 30—31. A very general Series by Half-Invariants 83 VIII. Laws of Errors of Functions of Observations. § 26. Functions of One Single Observation 35 § 27. 32—33. Half-Invariants of Linear Functions 36 § 28. Functions of Two or More Observations. Bonds 37 § 29. 34—35. Linear Functions of Unbound Observations 38 Examples 39 § 30. 36. Non-Linear Functions 41 Examples 41 §31. 37. Laws of Errors of the Mean Value. Approximation to the Typical Form 41 § 32. 38—46. Laws of Errors of the Mean Deviation and Higher Half-Invariants 44 § 33. 47. Transition between Laws of Actual and of Presumptive Errors. Rules of Prediction . . 47 Examples 50 §34. 48—50. Determination of the Law of Presumptive Errors when the Presumptive Mean Value is known beforehand 51 §35. 51. Weights of Observations. Probable Errors and other Dangerous Notions 52 IX. Free Functions. § 36. 52 - 58. Conditions of Freedom 53 Examples 55 § 37. Possibility of regarding certain Bound Observations as free 56 §38. 59—61. Every Function of Observations can be divided into a Sum of Two, which belong to Two Mutually Free Systems of Functions .56 Example 58 § 39. Every Single Observation likewise .59 § 40. Complete Sets of Free Functions 59 § 41. 62—66. Orthogonal Transformation 59 §42. 67. Schedule of Liberation 60 Example 62 General Remarks about Computation with Observed or Inexactly Given Values 63 X. Adjustment. §43. Can Laws of Errors be Determined by Means of Observations whicli are not Repeti- tions ? 64 § 44. The Principle of Adjustment 67 § 45. 68—72. Criticism ; the Method of the Least Squares 68 § 46. Adjustment by Correlates and by Elements 70 Numbers of XI. Adjustment by Correlates. formulae Page § 47. 73—81. General Solution of the Problem 71 §48. 82—84. Summary and Special Criticism 72 § 49. Schedule for Adjustment by (Correlates 73 § .'lO. Modifications of this Method 75 Examples 75 XII. Adjustment by Elements. § 51. 85 —86. One Equation for each Observation. Normal Equations 77 § 52. 87—89. Special Case of Free Normal Equations 80 § 53. 90—102. Transformation of the General Case into the Preceeding Special Case 80 § 54. 103-108. The Minimum Sum of Squares 83 § 55. Criticism 84 § .56. Under- Adjustment and Over- Adjustment 85 Examples 86 XIII. Special Auxiliary Methods. § 57. Lawful and Illegitimate Facilities 94 §58. 109—110. Adjustment upon Differences from Preceeding Computations 94 §69. Ill — 113. How to obtain Approximate Freedom of the Normal Equations 96 § 60. The Method of Fabricated Observations. Example 98 § 61. The Method of Partial Eliminations .' 99 Example 100 §62. 114—116. Rules for the General Transformation of the System of Elements 103 Example 105 §63. 117-120. The Method of Normal Places 106 Example HO § 64. Graphical Adjustment 112 XIV. The Theory of Probability. § 65. 121—123. Relation of Probability to the Laws of Errors by Half-Invariants 116 §66. 124—125. Laws of Errors for the Frequency of Repetitions. Obliquity of these Laws of Errors 118 XV. The Formal Theory of Probability. § 67. Addition and Multiplication of Probabilities 119 Examples 121 § 68. 126. Use of the Polynomial Formula for Probabilities by Repetitions 123 Examples 123 §69. 127 — 129. Linear Equations of Differences. Oppermann's Transformation 124 Examples 126 XVI. The Determination of Probabilities a Priori and a Posteriori. § 70. Theory and Experience 129 § 71. Determination a Priori 130 §72. 130-133. Determination a Posteriori and its Mean Error 132 Example 134 § 73. 134-137. The Paradox of Unanimity. Bayes's Rule 134 Numbers of XVII. Mathematical Expectation and its Mean Error. formulae Page § 74. Mathematical Expectation 137 138—140. Examples 138 §75. 141—143. Mean Errors of Mathematical Expectation of Unbound Events 139 Examples 140 § 76. 144-146. Mean En-or of Total Mathematical Expectation of the Same Trial 141 147. Examples 142 § 77. The Complete Expression of the Mean Errors 142 I. THE LAW OF CAUSALITY. § 1. We start with the assumption that everything tJiuf exists, and everything that happens, exists or happens as a necessary consequence of a previous state of things. If a state of things is repeated in every detail, it must lead to exactly the same consequences. Any difference between the results of causes that are in part the same, must be explainable by some difference in the other part of the causes. This assumption, which may be called the law of causality, cannot be proved, but must be believed; in the same way as we believe the fundamental assumptions of religion, with which it is closely and intimately connected. The law of causality forces itself upon our belief. It may be denied in theory, but not in practice. Any person who denies it, will, if he is watchful enough, catch himself constantly asking himself, if no one else, why this has happened, and not that. But in that very question he bears witness to the law of causality. If we are consistently to deny the law of causality, we must repudiate all observation, and particularly all prediction based on past experience, as useless and misleading. If we could imagine for an instant that the same complete combination of causes could have a definite number of different consequences, however small that number might be, and that among these the occurrence of the actual consequence was, in the old sense of the word, accidental, no observation would ever be of any particular value. Scientific observations cannot be reconciled with polytheism. So long as the idea prevailed that the result of a journey depended on whether the power of Njord or that of Skade was the stronger, or that victory or defeat in battle depended on whether Jove had, or had not, listened to Juno's complaints, so long were even scientists obliged to consider it below their dignity to consult observations. But if the law of causality is acknowledged to be an assumption which always holds good, then every observation gives us a revelation which, when correctly appraised and compared with others, teaches us the laws by which God rules the world. We can judge of the far-reaching consequences it would have, if there were con- ditions in which the law of causality was not valid at all, by considering tiie cases in which the effects of the law are more or less veiled. 1 In inanimate nature the relation of cause and effect is so clear that the effects are determined by observable causes belonging to the condition immediately preceding, so that the problem, within this domain, may be solved by a tabular arrangement of the several observed results according to the causing circumstances, and the transformation of the tables into laws by means of interpolation. When, however, living beings are the object of our observations, the case immediately becomes more complicated. It is the prerogative of living beings to hide and covertly to transmit the influ- ences received, and we must therefore within this domain look for the influencing causes throughout the whole of the past history. A difference in the construction of a single cell may be the only indication present at the moment of the observation that the cell is a transmitter of the still operative cause, which may date from thousands of years back. In consequence of this the naturalist, the physiologist, the physician, can only quite ex- ceptionally attain the same simple, definite, and complete accordance between the observed causes and their effects, as can be attained by the physicist and the astronomer within their domains. Within the living world, communities, particularly human ones, form a domain where the conditions of the observations are even more complex and difficult. Living beings hide, but the community deceives. For though it is not in the power of the com- munity either to change one tittle of any really divine law, or to break the bond between cause and effect, yet every community lays down its own laws also. Every community tries to give its law fixity, and to make it operate as a cause; for instance, by passing it oft' as divine or by threats of punishment; but nevertheless the laws of the community are constantly broken and changed. Statistical Science which, in the case of communities, represents observations, has therefore a very difficult task; although the observations are so numerous, we are able from them alone to answer only a very few questions in cases where the intellectual weapons of historical and speculative criticism cannot assist in the work, by independently bringing to light the truths which the communities want to conceal, and on the other hand by re- moving the wrong opinions which these believe in and propagate. § 2. An isolated sensation teaches us nothing, for it does not amount to an ob- servation. Observation is a putting together of several results of sensation which are or are supposed to be connected with each other according to the law of causality, so that some represent causes and others their effects. By virtue of the law of causality we must believe that, in all observations, we get essentially correct and true revelations; the difficulty is, to ask searchingly enough and to understand the answer correctly. In order that an observation may be free from every other assumption or hypothesis than the law of causality, it must include a perfect description of all the circumstances in the world, at least at the instant preceding that at which the phenomenon is observed. But it is clear that this far surpasses what can be done, even in the most important cases. Real observations have a much simpler form. By giving a short statement of the time and place of observation, we refer to what is known of the state of things at the instant; and, of the infinite multiplicity of circumstances connected with the observation we, generally, not only disregard everything which may be supposed to have little or no influence, but we pay attention only to a small selection of circumstances, which we call essential, because we expect, in virtue of a special hypothesis concerning the relation of cause and effect, that the observed phenomenon will be effect of these circumstances only. Nay, we are often compelled to disregard certain circumstances as unessential, though there is no doubt as to their influencing the phenomenon; and we do this either because we cannot get a sufficient amount of trustworthy information regarding them, or because it would be impracticable to trace out their connection with the eftect. For instance in statistical observations on mortality, where the age at the time of death can be regarded as the observed phenomenon, we generally mention the sex as an essential circumstance, and often give a general statement as to residence in town or country, or as to occupation. But there are other things as to which we do not get sufficient information: whether the dead person has lived in straitened or in comfortable circumstances, whether he has been more or less exposed to infectious disease, etc. ; and we must put up with this, even if it is certain that one or other of these things was the principal cause of death. And analogous cases are frequently met with both in scientific observations and in everyday occurrences. In order to obtain a perfect observation it is necessary, moreover, that our sensations should give us accurate information regarding both the phenomenon and the attendant circumstances; but all our senses may be said to give us merely approximate descriptions of any phenomenon rather than to measure it accurately. Even the finest of our senses recognizes no difference which falls short of a certain finite magnitude. This lack of accuracy is, moreover, often greatly increased by the use of arbitrary round numbers for the sake of convenience. The man who has to measure a race-course, may take into account the odd metres, but certainly not the millimetres, not to mention the microns. § 3. Owing to all this, every actual observation is affected with errors. Even our best observations are based upon hypothesis, and often even on an hypothesis that is cer- tainly wrong, namely, that only the circumstances which are regarded as essential, influence the phenomenon; and a regard for practicability, expense, and convenience makes us give approximate estimates instead of the sharpest possible determinations. Now and then the observations are affected also by ijross errors which, although 1* not introduced into them on purpose, are yet caused by such carelessness or neglect that they could have been, and ought to have been, avoided. 1 contradistinction to these we often call the more or less unavoidable errors accidetdal. For accident (or chance) is not, what the word originally meant, and what still often lingers in our ordinary acceptation of it, a capricious power which suffers events to happen without any cause, but only a name for the unknown element, involved in some relation of cause and effect, which pre- vents us from fully comprehending the connection between them. When we say that it is accidental, whether a die turns up "six" or "three", we only mean that the circumstances connected with the throwing, the fall, and the rolling of the die are so manifold that no man, not even the cleverest juggler and arithmetician united in the same person, can suc- ceed in controlling or calculating them. In many observations we reject as unessential many circumstances about which we really know more or less. We may be justified in this; but if such a circumstance is of sufficient importance as a cause, and we arrange the observations with special regard to it, we may sometimes observe that the errors of the observations show a regularity which is not found in "accidental" errors. The same may be the case if, in computations dealing with the results of observations, we make a wrong supposition as to the operation of some circumstance. Such errors are generally called systematic. § 4. It will be found that every applied science, which is well developed, may be divided into two parts, a theoretical (speculative or mathematical) part and an empirical (observational) one. Both are absolutely necessary, and the growth of a science depends very much on their influencing one another and advancing simultaneously. No lasting divergence or subordination of one to the other can be allowed. The theoretical part of the science deals with what we suppose to be accurate determinations, and the object of its reasonings is the development of the form, connection, and consequences of the hypotheses. But it must change its hypotheses as soon as it is cle£lr that they are at variance with experience and observation. The empirical side of the science procures and arranges the observations, compares them with the theoretical propositions, and is entitled by means of them to reject, if necessary, the hypotheses of the theory. By induction it can deduce laws from the obser- vations. But it must not forget — though it may have a natural inclination to do so — that, as shown above, it is itself founded on hypotheses. The very form of the observation, and especially the selection of the circumstances which are to be considered as essential and taken into account in making the several observations, must not be determined by rule of thumb, or arbitrarily, but must always be guided by theory. Subject to this it must as a rule be considered best, that the two sides of the science should work somewhat independently of one another, each in its own particular way. In what follows the empirical side will be treated exclusively, and it will be treated on a general plan, investigating not the particular way in which statistical, chemical, phy- sical, and astronomical observations are made, but the common rules according to which they are all submitted to computation. II. LAWS OF EEROES. § b. Every observation is supposed to contain information, partly as to the phenomenon in which we are particularly interested, partly as to all the circumstances, connected with it, which are regarded as essential. In comparing several observations, it makes a very great difference, whether such essential circumstances have remained unchanged, or whether one or several of them have changed between one observation and another. The treatment of the former case, that of repetitions, is far simpler than that of the latter, and is therefore more particularly the subject of our investigations; nevertheless, we must try to master also the more difficult general case in its simplest forms, which force them- selves upon us in most of the empirical sciences. By repetitions then we understand those observations, in which all the essential circumstances remain unchanged, in which therefore the results or phenomena should agree, if all the operative causes had been included among our essential circumstances. Further- more, we can without hesitation treat as repetitions those observations, in which we assume that no essential circumstance has changed, but do not know for certain that there has been no such change. Strictly speaking, this would furnish an example of observations with systematic errors; but provided there has been no change in the care with which the essential circumstances have been determined or checked, it is permissible to employ the simpler treatment applicable to the case of repetitions. This would not however be per- missible, if, for instance, the observer during the repetitions has perceived any uncertainty in the records of a circumstance, and therefore paid greater attention to the following repetitions. ^ '^ § 6. The special features of the observations, and in particular their degree of accuracy, depend on causes which have been left out as unessential circumstances, or on some overlooked uncertainty in the statement of the essential circumstances. Consequently no speculation can indicate to us the accuracy and particularities of observations. These must be estimated by comparison of the observations with each other, but only in the case of repetitions can this estimate be undertaken directly and without some preliminary work. The phrase late of errors is used as a general name for any mathematical expres- sion representing the distribution of the varying results of repetitions. Lmvs of actual errors are such as correspond to repetitions actually carried out. But observations yet unmade may also be erroneous, and where we have to speak hypo- thetically about observations, or have to do with the prediction of results of future repe- titions, we are generally obliged to employ the idea of "laws of errors". In order to pre- vent any misunderstanding we then call this idea "Zfrns of presumptive errors". The two kinds of laws of errors cannot generally be quite the same thing. Every variation in the number of repetitions must entail some variations in the corresponding law of errors; and if we compare two laws of actual errors obtained from repetitions of the same kind in equal number, we almost always observe great differences in every detail. In passing from actual repetitions to future repetitions, such differences at least are to be expected. More- over, whilst any collection of observations, which can at all be regarded as repetitions, will on examination give us its law of actual errors, it is not every series of repetitions that can be used for predictions as to future observations. If, for instance, in repeated measure- ments of an angle, the results of our first measurements all fell within the first quadrant, while the following repetitions still more frequently, and at last exclusively, fell within the second quadrant, and even commenced to pass into the third, it would evidently be wrong to predict that the future repetitions would repeat the law of actual errors for the totality of these observations. In similar cases the observations must be rejected as bad or mis- conceived, and no law of presumptive errors can be directly based upon them. § 7. Suppose, however, that on comparing repetitions of some observation we have several times determined the law of actual errors in precisely the same way, employing at first small numbers of repetitions, then larger and still larger numbers for each law. If then, on comparing these laws of actual errors with one another, we remark that they be- come more alike in proportion as the numbers of repetitions grow greater, and that the agreements extend successively to all those details of the law which are not by necessity bound to vary with the number of repetitions, then we cannot have any hesitation in using the law of actual errors, deduced from the largest possible number of repetitions, for pre- dictions concerning future observations, made under essentially the same circumstances. This, however, is wholly legitimate only, when it is to be expected that, if we could obtain repetitions in indefinitely increasing numbers, the law of errors would then approach a single definite form, namely the laiv of presumptive errors itself, and would not oscillate between several forms, or become altogether or partly indeterminate. (Note the analogy with the difference between converging and oscillating infinite series). We must therefore distinguish between good and bad observations, and only the good ones, that is those which satisfy the above mentioned condition, the Uno of large numbers, yield laws of presumptive errors and afford a basis for prediction. As we cannot repeat a thing indefinitely often, we can never be quite certain that a given method of observation may be called good. Nevertheless, we shall always rely on laws of actual errors, deduced from very large numbers of concordant repetitions, as suffi- ciently accurate approximations to the law of presumptive errors. And, moreover, the purely hypothetical assumption of the existence of a law of presumptive errors may yield some special criteria for the right behaviour of the laws of actual errors, corresponding to the increasing number of the repetitions, and establish the conditions necessary to justify their use for purposes of prediction. We must here notice that, when a series of repetitions by such a test proves bad and inapplicable, we shall nevertheless often be able, sometimes by a theoretical criticism of the method, and sometimes by watching the peculiarities in the irregularities of the laws of errors, to find out the reason why the given method of observation is not as good as others, and to change it so that the checks will at least show that it has been improved. In the case mentioned in the preceding paragraph, for instance, the remedy is obvious. The time of observation is there to be reckoned among the essential circumstances. A4id if we do not attain our object, but should fail in many attempts at throwing light upon some phenomenon by means of good observations, it may be said even at this stage, before we have been made acquainted with the various means that may be employed, and the various forms taken by the laws of errors, that absolute abandonment of the law of large numbers, as quite inapplicable to any given refractory phenomenon, will generally be out of the question. After repeated failures we may for a time give up the whole matter in despair; but even the most thorough sceptic may catch himself speculating on what may be the cause of his failure, and, in doing so, he must acknowledge that the error is never to be looked for in the objective nature of the conditions, but in an insuffi- cient development of the methods employed. From this point of view then the law of large numbers has the character of a belief. There is in all external conditions such a harmony with human thought that we, sooner or later, by the use of due sagacity, parti- cularly with regard to the essential subordinate circumstances of the case, will be able to give the observations such a form that the laws of actual errors, with respect to repetitions in increasing numbers, will show an approach towards a definite form, wliich may be con- sidered valid as the law of presumptive errors and used for predictions. § 8. Four different means of representing the law of errors must be described, and their respective merits considered, namely: Tabular arrangements. Curves of Errors, Functional Laws of Errors, Symmetric Functions of the Repetitions. In comparing these means of representing the laws of errors, we must take into consideration which of them is the easiest to employ, and neither this nor the description of the forms of the laws of errors demands any higher qualification than an elementary knowledge of mathematics. But we must take into account also, how far the diflerent forms are calculated to emphasise the important features of the laws of errors, i. e. those which may be transferred from the laws of actual errors to the laws of presumptive errors. On this single point, certainly, a more thorough knowledge of mathematics would be desirable than that which may be expected from the majority of those students who are obliged to occupy themselves with observations. As the definition of the law of presumptive errors presupposes the determination of limiting values to infinitely numerous approximations, some propositions from the differential calculus would, strictly speaking, be necessary. III. TABULAR ARRANGEMENTS. § 9. In stating the results of all the several repetitions we give the lajv of errors in its simplest form. Identical results will of course be noted by stating the number of the observations which give them. The table of errors, when arranged, will state all the various results and the fre- quency of each of them. The table of errors is certainly improved, when we include in it the relative fre- quencies of the several results, that is, the ratio which each absolute frequency bears to the total number of repetitions. It must be the relative frequencies which, according to the law of large numbers, are, as the number of observations is increased, to approach the constant values of the law of presumptive errors. Long usage gives us a special word to denote this transition in our ideas: probability is the relative frequency in a law of pre- sumptive errors, the proportion of the number of coincident results to the total number, on the supposition of infinitely numerous repetitions. There can be no objection to con- sidering the relative frequency of the law of actual errors as an approximation to the corresponding probability of the law of presumptive errors, and the doubt whether the relative frequency itself is the best approximation that can be got from the results of the given repetitions, is rather of theoretical than practical interest. Compare § 73. It makes some difference in several other respects — as well as in the one just mentioned — if the phenomenon is such that the results of the repetitions show qualitative differences or only differences of magnitude. § 10. In the former case, in which no transition occurs, but where there are such abrupt differences that none of the results are more closely connected with one another than with the rest, the tabular form will be the only possible one, in which the law of errors can 9 be given. This case frequently occurs in statistics and in games of chance, and for this reason the theory of probabilities, which is the form of the theory of observations in which these cases are particularly taken into consideration, demands special attention. All pre- vious authors have begun with it, and made it the basis of the other parts of the science of observation. I am of opinion, however, that it is both safer and easier to keep it to the last. § 11. If, however, there is such a diiference between the results of repetitions, that there is either a continuous transition between them, or that some results are nearer each other than all the rest, there will be ample opportunity to apply mathematical methods; and when the tabular form is retained, we must take care to bring together the results that are near one another. A table of the results of firing at a target may for instance have the following form : 1 foot to the left 1 foot too high 3 Central 13 1 foot too low 4 Total ... 20 134 26 180 Central 1 foot to the right Total 17 6 2G 109 19 141 8 1 13 If here the heading "1 foot to the left" means that the shot has swerved to the left between half a foot and one foot and a half, this will remind us that we cannot give the exact measures in such tables, but are obliged to give them in round numbers. The number of results then will not correspond to such as were exactly the same, but dis- regarding small differences, we gather into each column those that approach nearest to one another, and which all fall within arbitrarily chosen limits. In the simple case, where the result of the observation can be expressed by a single real number, the arranged table not only takes the extremely simple form of a table of functions with a single argument, but, as we shall see in the following chapters, leads us to the representation of the law of errors by means of curves of errors and functional laws of errors. It is an obvious course to fix the attention on the two extreme results in the table, and not seldom these alone are given, instead of a law of error, as a sort of index of the exactness of the whole series of repetitions, and as the higher and lower limits of the observed phenomenon. This index of exactness, however, must be rejected as itself too inexact for the purpose, for the oftener the observations are repeated, the farther we must expect the extremes to move from one another: and thus the most valuable series of observations will appear to possess the greatest range of discrepancy. a 10 On the other hand, if, in a table arranged according to the magnitude of the values, we select a single middle value, preceded and followed by nearly equal numbers of values, we shall get a quantity which is very well fitted to represent the whole series of repetitions. If, while we are thus counting the results arranged according to their magnitude, we also take note of those two values with which we respectively (a) leave the first sixth part of the total number, and (b) enter upon the last sixth part (more exactly we ought to say 16 per ct.), we may consider these two as indicating the limits between great and small deviations. If we state these two values along with the middle one above referred to, we give a serviceable expression for the law of errors, in a way which is very convenient, and although rough, is not to be despised. Why we ought to select just the middle value and the two sixth-part values for this purpose, will appear from the following chapters. IV. CURVES OF ERRORS. § 12. Curves of actual errors of repeated observations, each of which we must be able to express by one real number, are generally constructed as follows. On a straight line as the axis of abscissae, we mark ofl' points corresponding to the observed numerical quantities, and at each of these points we draw an ordinate, proportional to the number of the repetitions which gave the result indicated by the abscissa. We then with a free hand draw the curve of errors through the ends of the ordinates, making it as smooth and regular as possible. For quantities and their corresponding abscissae which, from the nature of the case, might have appeared, but do not really appear, among the repetitions, the ordinate will be = 0, or the point of the curve falls on the axis of abscissae. Where this case occurs very frequently, the form of the curves of errors becomes very tortuous, almost discontinuous. If the observation is essentially bound to discontinuous numbers, for instance to integers, this cannot be helped. § 13. If the observation is either of necessity or arbitrarily, in spite of some in- evitable loss of accuracy, made in round numbers, so that it gives a lower and a higher limit for each observation, a somewhat different construction of the curve of errors ought to be applied, viz. such a one, that the area included between the curve of error, the axis of abscissae, and the ordinates of the limits, is proportional to the frequency of repetitions within these limits. But in this way the curve of errors may depend very much on the degree of accuracy involved in the use of round numbers. This construction of areas can be made by laying down rectangles between the bounding ordinates, or still better, trapezoids with their free sides approximately parallel to the tangents of the curve. If the 11 limiting round numbers are equidistant, the mean heights of the trapezoids or rectangles are directly proportional to the frequencies of repetition. In this case a preliminary con- struction of curve-points can be made as in § 12, and may often be used as sufficient. It is a very common custom, but one not to be recommended, to draw a broken line between the observed points instead of a curve. § 14. There can be no doubt that the curve of errors, as a form for the law of errors, has the advantage of perspicuity, and were not the said uncertainty in so many cases a critical drawbacic, this would perhaps be sufficient. Moreover, it is in practice quite possible, and not very difficult, to pass from the curve of actual errors to one which may hold good for presumptive errors ; though, certainly, this transition cannot be founded upon any positive theory, but depends on siiill, which may be acquired by working at good examples, but must be practised judiciously. According to the law of large numbers we must expect that, when we draw curves of actual errors according to relative frequency, for a numerous series of repetitions, first based upon small numbers, afterwards redrawn every time as we get more and more repe- titions, the curves, which at first constantly changed their forms and were plentifully furnished with peaks and valleys, will gradually become more like each other, as also simpler and more smooth, so that at last, when we have a very large but finite number of observations, we cannot distinguish the successive figures we have drawn from one an- other. We may thus directly construct curves of errors, which may be approved as pictures of curves of presumptive errors, but in order to do so millions of repetitions, rather than thousands, are certainly required. If from curves of actual errors for small numbers we are to draw conclusions as to the curve of presumptive errors, we must guess, but at the same time support our guess, partly by an estimate of how great irregularities we may expect in a curve of actual errors for the.given number, partly by developing our feeling for the form of regular curves of that sort, as we must suppose that the curves of presumptive errors will be very regular. In both respects we must get some practice, but this is easy and interesting. Without feeling tied down to the particular points that determined the curve of actual errors, we shall nevertheless try to approach them, and especially not allow many large deviations on the same side to come together. We can generally regard as large deviations (the reason why will be mentioned in the chapter on the Theory of Probabilities) those that cause greater errors, as compared with the absolute frequency of the result in question, than the square root of that number (more exactly ]//t ^~ , where /; is the frequency of the result, n the number of all repetitions). But even deviations two or three times as great as this ought not always to be avoided, and Ave may be satisfied, if only one third of the deviations of the determining points must be called large. We may use 12 the word '•adjustment" (graphical) to express the operation by which a curve of presumptive errors is determined. (Comp. §64). The adjustment is called an over- adjustment, if we have approached too near to some imaginary ideal, but if we have kept too close to the curve of actual errors, then the curve is said to be tinder-adjusted. Our second guide, the regularity of the curve of errors, is as an aesthetical notion of a somewhat vague kind. The continuity of the curve is an essential condition, but it is not sufficient. The regularity here is of a somewhat diflerent kind from that seen in the examples of simple, continuous curves with which students more especially become acquainted. The curves of errors get a peculiar stamp, because we would never select the essential circumstances of the observation so absurdly that the deviations could become indefinitely large. Nor would we without necessity retain a form of observation which might bring about discontinuity. It follows that to the abscissae which indicate very large deviations, must correspond rapidly decreasing ordinates. The curve of errors must have the axis of abscissae as an asymptote, both to the right and the left. All frequency being positive, where the curve of errors deviates from the axis of abscissae, it must exclusively keep on the positive side of the latter. It must therefore more or less get the appearance of a bow, with the axis of abscissae for the string. In order to train the eye for the apprehension of this sort of regularity, we recommend the study of figs. 2 & 3, which represent curves of errors of typical forms, exponential and binomial (comp. the next chapter, p. 16, seqq.), and a comparison of them with figures which, like Nr. 1, are drawn from actual observations without any adjustment. The best way to acquire practice in drawing curves of errors, which is so important that no student ought to neglect it, may be to select a series of observations, for which the law of presumptive errors may be considered as known, and which is before us in tabular form. We commence by drawing curves of actual errors for the whole series of observa- tions; then for tolerably large groups of the same, and lastly for small groups taken at random and each containing only a few observations. On each drawing we draw also, besides the curve of actual errors, another one of the presumptive errors, on the same scale, so that the abscissae are common, and the ordinates indicate relative frequencies in proportion to the same unit of length for the total number. The proportions ought to be chosen so that the whole part of the axis of abscissae which deviates sensibly from the curve, is between 2 and 5 times as long as the largest ordinate of the purve. Prepared by the study of the differences between the curves, we pass on at last to the construction of curves of presumptive errors immediately from the scattered points of the curve which correspond to the observed frequencies. In this construction we must not consider ourselves obliged to reproduce the curve of presumptive errors which we may 13 know beforehand; our task is to represent the observations as nearly as possible by means of a curve which is as smooth and regular as that curve. The following table of 500 results, got by a game of patience, may be treated in this way as an exercise. Actual frequency for groups of: ^ errors hod, ted ~w o ° a^ III ■3 02 3Q ^ 25 repetitions 100 repetitions ] 11 III IV V I II III IV V !^ 7 1 1 1 110 1 3 0(X)03 0-0019 7 8 1 2 '^ 110 14 2 7 0-0071 00192 8 9 1 3 ] 1 5 3 1 1 3 2 2 3 1 3 1 1 2 1 6 10 7 7 5 35 00392 0-0636 9 10 9 2 9 5 6 6 6 4 5 4 8 3 3 3 5 6 4 6 3 4 25 22 20 17 17 101 0-0859 0-1005 10 11 3 6 3 3 3 6 4 4 5 5 3 5 3 7 2 5 5 6 3 8 15 17 18 17 22 89 0.1054 0-1021 11 12 8 5 3 4 3 3 2 8 3 7 4 6 5 -4 6 5 3 3 5 7 20 16 20 20 18 94 00934 0-0823 12 13 2 4 4 3 6 3 3 1 4 113 6 4 3 6 7 3 6 1 13 13 9 18 17 70 0-0705 0-0.591 13 14 1 2 2 4 1 2 3 2 12 4 3 5 4 4 2 4 9 6 9 12 10 46 0-0485 00387 14 15 1 2 2 1 1 3 '? 3 2 2 3 1 1 3 2 1 5 7 10 5 3 30 0-0298 0-0216 15 16 1 2 1 1 1 10 1 2 3 2 4 2 2 2 5 15 0-0145 0-0088 16 17 1 2 1 10 2 1 4 00046 0-0020 17 18 1 1 110 1 112 1 5 0-0007 0-0002 18 19 1 10 1 0-0000 19 Total 25 25 25 25 26 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 laj 100 100 100 100 500 0-9999 The law of presumptive errors here given is not the direct result of free-hand con- struction; but the curve so got has been improved by interpolation of the logarithms of its statements of the relative frequencies, together with the formation of mean numbers for the deviations, a proceeding which very often will give good results, but which is not strictly necessary. By this we can also determine the functional law of errors (Comp. the 14 next chapter). The equation of the curve is: Log2/ = 2-0228 + 0-0030(a'— 11) — 0-6885(«— ll)2+0-01515(j;— 11)3— 0-001678(^— 11)« § 15. By the study of many curves of presumptive errors, and especially such as represent ideal functional laws of errors, we cannot fail to get the impression that there exists a typical form of curves of errors, which is particularly distinguished by symmetry. Familiarity with this form is useful for the construction of curves of presumptive errors. But we must not expect to get it realised in all cases. For this reason I have considered it important to give, alongside of the typical curves, an example taken from real observa- tions of a skew curve of errors, which in consequence of its marked want of symmetry deviates considerably from the typical form. Fig. 4 shows this last mentioned law of presumptive errors. Deviation from the typical form does not indicate that the observations are not good. But it may become so glaring that we are forced by it to this conclusion. If, for instance, between the extreme values of repetitions — abscissae — there are intervals which are as free from finite ordinates as the space beyond the extremes, so that the curve of errors is divided into two or several smaller curves of errors beside one another, there can scarcely be any doubt that we have not a series of repetitions proper, but a combination of several; that is to say, different methods of observation have been used and the results mixed up together. In such cases we cannot expect that the law of large numbers will remain in force, and we had better, therefore, reject such observations, if we cannot retain them by tracing out the essential circumstances which distinguish the groups of the series, but have been overlooked. § 16. When a curve of presumptive errors is drawn, we can measure the magnitude of the ordinate for any given abscissa; so far then we know the law of errors perfectly, by means of the curve of errors, but certainly in the tabular form only, with all its copious- ness. Whether we can advance further depends on, whether we succeed in interpolating in the table so found, and particularly on, whether we can, either from the table or direct from the curve of errors, by measurement obtain a comparatively small number of constants, by which to determine the special peculiarities of the curve. By interpolating, by means of Newton's formula, the logarithms of the frequencies, or by drawing the curves of errors with the logarithms of the frequencies as ordinates, we often succeed, as above mentioned, in giving the curve the form of a parabola of low (and always even) degree. Still easier is it to make use of the circumstance that fairly typical curves of errors show a single maximum ordinate, and an inflexion on each side of it, near which the curve for a short distance is almost rectilinear. By measuring the co-ordinates of the maximum point and of the points of inflexion, we shall get data sufficient to enable us to 15 draw a curve of errors which, as a rule, will deviate very little from the original. All this, however, holds good only of the curves of presumptive errors. With the actual ones we cannot operate in this way, and the transition from the latter to tlie former seems in the meantime to depend on the eye's sense of beauty. V. FUNCTIONAL LAWS OF ERROES. § 17. Laws of errors may be represented in such a way that the frequency of the results of repetitions is stated as a mathematical function of the number, or numbers, expressing the results. This method only differs from that of curves of errors in the circumstance that the curve which represents the errors has been replaced by its mathema- tical formula; the relationship is so close that it is difficult, when we speak of these two methods, to maintain a strict distinction between them. In former works on the theory of observations the functional law of errors is the principal instrument. Its source is mathematical speculation; we start from the properties which are considered essential in ideally good observations. From these the formula for the typical functional law of errors is deduced; and then it remains to determine how to make computations with observations in order to obtain the most favourable or most probable results. Such investigations have been carried through with a high degree of refinement; but it must be regretted that in this way the real state of things is constantly disregarded. The study of the curves of actual errors and the functional forms of laws of actual errors have consequently been too much neglected. The representation of functional laws of errors, whether laws of actual errors or laws of presumptive errors founded on these, must necessarily begin with a table of the results of repetitions, and be founded on interpolation of this table. We may here be content to study the cases in which the arguments (i. e. the results of the repetitions) proceed by constant differences, and the interpolated function, which gives the frequency of the argument, is considered as the functional law of errors. Here the only difficulty we en- counter is that we cannot directly employ the usual Newtonian formula of interpolation, as this supposes that the function is an integral algebraic one, and gives infinite values for infinite arguments, whether positive or negative, whereas here the frequency of these infinite arguments must be = 0. We must therefore employ some artifice, and an obvious one is to interpolate, not the frequency itself, y, but its reciprocal, — . This, however, turns out to be inapplicable; for — will often become infinite for finite arguments, and will, at any rate, increase much faster than any integral function of low degree. 16 But, as we have already said, the interpolation generally succeeds, when we apply it to the logarithm of the frequency, assuming that Log y = a -\- hx ^ cx^ -\- . . . -\- gx"", where the function on the right side begins with the lowest powers of the argument x, and ends with an even power whose coefficient 8g- 2 (^) = n- 12 (a^e -^ bn^ 3x* + 3 • 5w* • Sa;^ — 1 . 3 • 5n^) e'^i^)' The law of the numerical coefficients (products of odd numbers and binomial numbers) is obvious. The general expression of D'e 2\n) can be got from a comparison of the coefficients to ( — m)'' of the two identical series for equation (3), one being the Taylor series, the other the product of e 2Vn/ and the two exponential series with m^ and m as arguments. It can also be induced from the differential equation n^iy+i^ + a; /)••+' ^ + (»• + 1) D^

V - ^ ^V + c^f- (6) where 1 IX — my

1-18133 •1645 - -313 •43 -•2 2-0 1-19629 01353 -0-271 0-41 -•3 2-1 1-20853 •1103 - -232 •38 -•3 2-2 1-21846 ■0889 - -1% •34 -•4 2-3 1-22643 -0710 - -163 •30 -•4 dz* -2 -2 -2 -2 -2 -2 -1 — 1 — 1 — 1 -0 -0 — 2-4 25 26 £ dz 2-7 1 2-8 1 2-9 1 3-0 1 3^1 1 3-2 1 3-3 1 3-4 1 3-6 1 3-6 1 3-7 1 3-8 1 3-9 1 4-0 1 4-1 1 4-2 1 4-3 1 4-4 45 23277 23775 24163 25089 25159 25210 25247 25273 25292 25304 25313 25319 25326 25328 25329 26330 25.331 i = (? -J'' dij dz (VTj 0-0561 -0-1.35 0-27 •04.39 — 110 •0340 - 089 •23 •20 d^ dz" -0-4 - -4 — -3 24462 ^0261 - ^071 24691 -0198 - -0.56 24864 -0149 - 043 ■0082 •0060 -0043 •0031 -0022 •0015 ■025 •019 •014 •Oil •008 •006 ■07 •06 •04 •03 •02 •02 •2 •0011 - -(XH 01 ■0007 - •OOS -01 -0005 — -002 01 •0 -0 -0 25323 0-0003 —0-001 001 - -0002 - -001 •OO •0001 - •OOl 00 •0001 - 000 00 ■0001 — -000 ■axx) — ■OCX) ■0000 - 000 •00 ■(X) -00 dz* -16 — -3 •14 — -3 -11 - -2 24993 0-0111 —0-0.33 009 -02 03 19 dH d Here jy, '^\, -r^ are, each of them, the same for positive and negative values of z\ the other columns of the table change signs with z. The interpolations are easily worked out by means of Taylor's theorem : „..=, = ^ + S-C + J3-C' + 5S-C- + Ag-C- + ... (7) and The typical form for the functional law of errors (2) shows that the frequency is always positive, and that it arranges itself symmetrically about the value x = >«, for which the frequency has its maximum value ij = h. For x^m^n the frequency is z/ = A • 0-60653. The corresponding points in the curve of errors are the points of inflexion. The area between the curve of errors and the axis of abscissae, reckoned from the middle to a; = m 4^ w, will be nh • 0"85562 ; and as the whole area from one asymptote to the other is nh l/2;r = nh • 2-50663, only nh • 0-39769 of it falls outside either of the inflexions, consequently not quite that sixth part (more exactly 16 per ct.) which is the foundation of the rule, given in § 11, as to the limit between the great and small errors. The above table shows how rapidly the function of the typical law of errors de- creases toward zero. In almost all practical applications of the theory of observations e~~i'^ = 0, if only z>b. Theoretically this superior assymptotical character of the function is expressed in the important theorem that, for 2; = ^j^ oo , not only e"i'^ itself is = but also all its differential coefficients; -and that, furthermore, all products of this function by every algebraic integral function and by every exponential function, and all the differential quotients of these products, are equal to zero. In consequence of this theorem, the integral \e 2'^ dz ^ V2n can be computed as the sum of equidistant values of e 2'' multiplied by the interval of the arguments without any correction. This simple method of computation is not quite correct, the underlying series for conversion of a sum into an integral being only semiconvergent in this case; for very large intervals the error can be easily stated, but as far as intervals of one unit the numbers taken out of our table are not sufficient to show this error. If the curve of errors is to give relative frequency directly, the total area must be 1 = nhV27r; h consequently ought to be put = "^^• Problem 1. Prove that every product of typical laws of errors in the functional 1 /x— m\2 form = he''~2\'~ir) ,with the same independent variable x, is itself a typical law of errors. How do the constants h, m, and n change in such a multiplication? 20 Problem 2, How small are the frequencies of errors exceeding 2, 3, or 4 times the mean error, on the supposition of the typical law of errors? Problem 3. To find the values of the definite integrals Sr = \x''e 2\ » / dx. J— 00 Answer: sn+i = and $2, = 1 • 3 • 5 . . . (2« — 1) 9r"+'\/27c. § 19. Nearly related to the typical or exponential law of errors in functional form are the binomial functions, which are known from the coefficients of the terms of the «"■ power of a binomial, regarded as a function of the number x of the term. X == n 1 2 3 4 5 6 7 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 9 1 9 36 84 126 126 84 36 10 1 10 45 120 210 252 210 120 11 1 11 55 165 330 462 462 330 12 1 12 66 220 495 792 924 792 13 1 13 78 286 715 1287 1716 1716 14 1 14 91 364 1001 2002 3003 3432 For integral values of the argument the binomial function can be computed directly by the formula M^) 1 . 2 • 3 . . . n 1.2-3...a;.l-2-3...(«- n{n — l)...{n-x+ 1) -X) fin (n — x) (9) 1.2. ..a; When the binomial numbers for n are known, those for m + 1 are easily found by the formula t3n+,(x) ^ fi„(x) + fin{X-l). (10) By substitution according to (9) we easily demonstrate the proposition that, for 21 any integral values of w, r, and t Mi)fin-t{r) = Mr)-fin-r(t), (11) which means that, when the trinomial {a + b + c)" is developed, it is indifferent whether we consider it to be ((a + b)-\- c)" or (a + (6 + c))". For fractional values of the argument x, the binomial function j3„{x) can be taken in an infinity of different ways, for instance by . , , sin;:a; This formula results from a direct application of Lagrange's method of interpolation, and leads by (10) to the more general formula ^,,_ l'2., .n sin^ra; .^„, '^"'^^ ~ (l-x){2-x)...{n^x) ~^tor' ^ ' This species of binomial function may be considered the simplest possible, and has some importance in pure mathematics; but as an expression of frequencies of observed values, or as a law of errors, it is inadmissible because, for x > n or x negative, it gives negative values alternating with positive values periodically. This, however, may be remedied. As /9o {x) has no other values than and 1, when X is integral, we can put for instance ^o(^) = ( sin Tzx \^ by (10) then TIX ) ' sin^ nx ^.(^) = [^. l)^ 2 + (13) (x—\f ' (x— 2)V ?r^ Here the values of the binomial function are constantly positive or 0. But this form is cumbersome ; and although for x = co the function and its principal coefficients are == 0, this property is lost here, when we multiply by integral algebraic or by exponen- tial functions. These unfavourable circumstances detract greatly from the merits of the binomial functions as expressions for continuous laws of errors. When, on the contrary, the observations correspond only to integral values of the argument, the original binomial functions are most valuable means for treating them. That Pn(x) = 0, if j; > w or negative, is then of great importance. But this case must be referred to special investigations. § 20. To represent non-typical laws of errors in functional form we have now the .choice between at least three different plans: 22 1) the formula (1) or y ^ (,a+IJX+yXi+...-XX^'' , 2) the products of integral algebraic functions by a typical function or (6) y I- k k l/I— m\' 3) a sum of several typical functions y = l^lh^e-^ri^l . (14) This account of the more prominent among the functional forms, which we have at our disposal for the representation of laws of errors, may prove that we certainly possess good instruments, by means of which we can even in more than one form find general series adapted for the representation of laws of errors. We do not want forms for the series, required in theoretical speculations upon laws of errors; nor is the exact representation of the actual frequencies more than reasonably difficult. If anything, we have too many forms and too few means of estimating their value correctly. As to the important transition from laws of actual errors to those of presumptive errors, the functional form of the law leaves us quite uncertain. The convergency of the series is too irregular, and cannot in the least be foreseen. We ask in vain for a fixed rule, by which we can select the most important and trustworthy forms with limited numbers of constants, to be used in predictions. And even if we should have decided to use only the typical form by the laws of presumptive errors, we still lack a method by which we can compute its constants. The answer, that the "adjustment" of the law of errors must be made by the "method of least squares", may not be given till we have attained a satisfactory proof of that method; and the attempts that have been made to deduce it by speculations on the functional laws of errors must, I think, all be regarded as failures. VI. LAWS OF ERRORS EXPEESSED BY SYMMETRICAL FUNCTIONS. § 21. All constants in a functional law of errors, every general property of a curve of errors or, generally, of a law of numerical errors, must be symmetrical functions of the several results of the repetitions, i. e. functions which are not altered by inter- changing two or more of the results. For, as all the values found by the repetitions correspond to the same essential circumstances, no interchanging whatever can have any influence on the law of errors. Conversely, any symmetrical function of the values of the 23 observations will represent some property or other of the law of errors. And we must be able to express the whole law of errors itself by every such collection of symmetrical functions, by which every property of the law of errors can be expressed as unambiguously as by the very values found by the repetitions. We have such a collection in the coefficients of that equation of the «"■ degree, whose roots are the n observed values. For if we know these coefficients, and solve the equation, we get an unambiguous determination of all the values resulting from the repe- titions, i.e. the law of errors. But other collections also fulfil the same requirements; the essential thing is that the n symmetrical functions are rational and integral, and that one of them has each of the degrees 1, 2 . . . «, and that none of them can be deduced from the others. The collection of this sort that is easiest to compute, is the sums of the powers. With the observed values Oj, 0.^, O3, ... o„ we have So - o'^,+ol + ..^Yol =n s> = o^-\-o,^..^o„ <-\-o\+ •■ + <>« \ (15) s, = 0"; + 0^ + . . -^ oi; Q and the fractions — may also be employed as an expression for the law of errors; it is only important to reduce the observations to a suitable zero which must be an average value oi 0^ . . ,o„; for if the diiferences between the observations are small, as compared with their differences from the average, then may become practically identical, and therefore unable to express more than one property of the law of errors. From a well known theorem of the theory of symmetrical functions, the equations /-^^" 1 + UiW + ai(u'^ + • • • = (1 — Oi from which the coefficients «„ are unambiguously and very easily computed, when the s„ are directly calculated. § 22. But from the sums of powers we can easily compute also another service- able collection of symmetrical functions, which for brevity we shall call the half-invariants. Starting from the suras of powers Sr, these can be defined as ^Uj, yug, //a, by the equation M, s,e\^ ^^ + f-l _2 _1_ f-i -.3 ^0 + II 7 -t- 12 ^ i- 13 ^ + (17) jl ' I |2 ' I 13 which we suppose identical with regard to r. As Sr == IV, this can be written so^f ' + t^' + t^' + -- = e'''^ + e''^^ + ...6'"'^ (18) By developing the first term of (17) as Shrt and equating the coefficients of each power of T, we get each ~ expressed as a function of //j ...fir- S3 = So(A«:i +%!/'.+/'?) (19) Taking the logarithms of (17) we get s, T , So r So r* «f.+|.'+|^ + ... = i.g(. + i^jr + i^V+i;i + -> (20) and hence s. : s„ (s,s„ — s^):sj (S3sj-3s,s..9„ + 2s;):.9j (s, sj - 4s, s, sj - 3s', si + Us, s] s„ - 6s;) : sj (21) The general law of the relation between the fi and s is more easily understood through the equations 25 S4 = /^l «3 + %2 '^2 + 3/^3 •''l +/.«4 •''O (22) where the numerical coefficients, are those of the binomial theorem. Tliese equations can be demonstrated by differentiation of (17) with regard to r, the resulting equation •s,+fr + |r^+|r3+... = (/.,+|^-r+|r^ + ...)(.s-„+^r + |r'^ + ...)(23) being satisfied for all values of r by (22). These half-invariants possess several remarkable properties. From (18) we get SqC }?. li ==e +...-{- e (24) consequently any transformation 0' = -|- c, any change of the zero of all observations 0, ...o„, affects only ^ij in the same manner, but leaves fi^, /ig, fi,, ... unaltered; any change of the unit of all observations can be compensated by the reciprocal change of the unit of r, and becomes therefore indifferent to /igr", /jIsT^, • • Not only the ratios '0 "0 ^n '"n but also the half-invariants have the property which is so important in a law of errors, of remaining unchanged when the whole series of repetitions is repeated unchanged. We have seen that the typical character of a law of errors reveals itself in the elegant functional form Now we shall see that it is fully as easy to recognize the typical laws of errors by means of their half-invariants. Here the criterion is that /^,. =0 if r>3, while /x^ =m and ^2 = n'^. This remarkable proposition has originally led me to prefer the half-invariants to every other system of symmetrical functions; it is easily demonstrated by means of (5), if we take m for the zero of the observations. We begin by forming the sums of powers .Sr of that law of errors where the fre- quency of an observed x is proportional to (p (x) = e~TV^^/ ; as this law is continuous we get iX^(p{x)dx. 4 (•+< k 26 For every differential coefficient D"'(p(x) we have (.+ " \D'"(p(x)-dx = Z>'»-V(cc) — Z)'«-i^(— c») = 0, J — 00 consequently we learn from (5) that s^r+i = 0, but s^ = l-3-n*Sg » S„ = 1 • 3 -5 • W^Sn (compare problem 3, § 18). Now the half-invariants can be found by (22) or by (17). If we use (22) we remark that S2r = w''^ (2r — 1) 82,-2; then writing for (22) Si = f^iSo = Sj — fi^So = fiiSi =0 S3 — 2^2S, =//,,s,^+ /ijSo =0 «4 — %2S2 = /^l«3 + S/igS, + /i,Su =0 Sj — 4^^283 = /i,s, + 6/i3S2 + 4//4S1 + /is.s„ = Sg — 5//2S4 = /^i>^5 + 10/Za«3 + lO/iiSa + 5/i5Si +/i„So == we see that the solution is fi^ ^n^ and ^ij ==^3 =^^ = . . . = 0. By (17) we get Equating the coefficients of z'' we get here also /ij = == w, ii^^= n'^, //,• = if r > 3. If we wish to demonstrate this important proposition without change of the zero, and without the use of the equations (3) whose general demonstration is somewhat diffi- cult, we can commence by the lemma that, for each integral and positive value of r, and also for r ^^ 0, we have for the typical law of errors Sr-,-1 = WAV + rw^ Sr_i. The function (p(x) == n'^ xr er 'iK"^) is equal to zero both for x =0) further n n n 17 31 A'a ^"4 ' ^^4 = ^" ' /^e = ^ > /^8 ^ Tg" ' /^lo ^^ ^ ^' • Example 3. What are the half-invariants of a complete binomial law of errors (the complete terms of (p + -g )\ -Wp+q) ip + qr J Example 4. A law of presumptive errors is given by its half-invariants forming a geometrical progression, ^r = ba''. Determine the several observations and their frequencies. Here the left hand side of the equation (18) is l V- b^ \ but this is = .Spe-'M +6e'"'' + -i^e^'"'' + -|3 e^'"' + ...l and has also the form of the right side of (18). Thus the observed values are 0, a, 2a, 3a, ... and the relative frequency of b'' ra is j7- == fp(r). This law of errors is nearly related to the binomial law, which can be considered as a product of two factors of this kind, ^r rjn—r 1 T--r — = ^BnivWd"--. \r \n—r |m '^^ ' It is perhaps superior to the binomial law as a representative of some skew laws of errors. Example 5. A law of errors has the peculiarity that all half-invariants of odd order are = 0, while all even half-invariants are equal to each other, Xir = 2a. Show that all the observations must be integral numbers, and that for the relative frequencies .(0)-«-(i + (f)+(|)' + -) Example 6. Determine the half-invariants of the law of presumptive errors for the irrational values in the table of a function, in whose computation fractions under J have been rejected and those over J replaced by 1 : /2r+l = 0, /,, = r2' ■"4 = T2U' ^8 = 5'5'2' ••• § 25. As a most general functional form of a continuous law of errors we have proposed (6) 1 IX— m\^ where (p(x) = e~~i\^l . 34 Now it, is a very remarkable thing that we can express the half-invariants without any ambiguity as functions of the coefficients fc,, and vice versa. By (29) we get |2 s„e[L' + '^' '' + ■•• = Y^(k,(', ==>!, — m and }! ., = k^ — m^, the computation of one set of constants by the other can, according to (17), be made by the formulae (19) and (21). We substitute only in these the A;,- for the s,-, and )! or ), for /i. It will be seen that the constants m and m, and the special typical law of errors to which they belong, are generally superfluous. This superfluity in our transformation may be useful in special cases for reasons of convergency, but in general it must be con- sidered a source of vagueness, and the constants must be fixed arbitrarily. It is easiest and most natural to put m = X^ and n"^ = X.^. In this case we get A-, =0, k^ =0, k.^ =kf^X^, k^ ^k„X^, k. ■=- k„X^, and further k, ^ kAK + '^OXl) k, = k,{X, + 3bX,X,) k. = kAX, + b6X,X, + 3bX]) The law of the coefficients is explained by writing the right side of equation (30) 35 Expressed by half-invariants in this manner the explicit I'orm of equation (6) is 1 + — ( 2 >i. l+^((^-^)'-3^(x-/,)) + (31) VIII. LAWS OF ERROES OF FUNCTIONS OF OBSERVATIONS. § 26. There is nothing inconsistent with our definitions in speaking of laws of errors relating to any group of quantities which, though not obtained by repeated observations, have the like property, namely, that repeated estimations of a single thing give rise, owing to errors of one kind or other, to multiple and slightly differing results which are prima facie equally valid. The various forms of laws of actual errors are indeed only summary expressions for such multiplicity; and the transition to the law of presumptive errors requires, besides this, only that the multiplicity is caused by fixed but unknown circum- stances, and that the values must be mutually independent in that sense that none of the circumstances have connected some repetitions to others in a manner which cannot be common to all. Compare § 24, Example 6. It is, consequently, not difficult to define the law of errors for a function of one single observation. Provided only that the function is univocal, we can from each of the observed values Oj, o.^ ... o„ determine the corresponding value of the function, and f{o^), f{0,), ...f(On) will then be the series of repetitions in the law of errors of the function, and can be treated quite like observations. With respect, however, to those forms of laws of errors which make use of the idea of frequency (probability) we must make one little reservation. Even though o,- and Oi are different, we can have /"(o,) = f (Oi), and in this case the frequencies must evidently be added together. Here, however, we need only just mention this, and remark that the laws of errors when expressed by half-invariants or other symmetrical functions are not influenced by it. Otherwise the frequency is the same for /"(o,) as for o,-, and therefore also the probability. The ordinates of the curves of errors are not changed by observations with discontinuous values; but the abscissa o,- is replaced by /"(o,), and likewise the argument in the functional law of errors. In continuous functions, on the other hand, it is the areas between corresponding ordinates which must remain unchanged. 5* 36 In the form of symmetrical functions the law of errors of functions of observations may be computed, and not only when we know all the several observed values, and can there- fore compute, for each of them, the corresponding value of the function, and at last the symmetrical functions of the latter. In many and important cases it is sufficient if we know the symmetrical functions of the observations, as we can compute the symmetrical functions of the functions directly from these. For instance, if f{o) = o^ ; for then the sums of the powers s'„ of the squares are also sums of the powers s^ of the observations, if only constantly m = 2m ; s'g = Sy, s', = s^ , s'„ = s^, etc. § 27. The principal thing is here a proposition as to laws of errors of the linear functions by half-invariants. It is almost self-evident that if o' = ao-\-h /ii = a// (32) etc. ;j.'r = a>r (»•>!) For the linear functions can always be considered as produced by the change of both zero and unity of the observations (Compare (24)). However special the linear function ao -\-b may be, we always in practice manage to get on with the formula (32). That we can succeed in this is owing to a happy circumstance, the very same as, in numerical solutions of the problems of exact mathematics, brings it about that we are but rarely, in the neighbourhood of equal roots, compelled to employ the formulae for the solution of other equations than those of the first degree. Here we are favoured by the fact that we may suppose the errors in good observations to be small, so small — to speak more exactly — that we may generally in repetitions for each series of observations Oj, Oj, ... o„ assign a number c, so near them all that the squares and products and higher powers of the differences 0, — 0,0^ — 0, . . . o„ — c without any perceptible error may be left out of consideration in computing the function: i. e., these differences are treated like differentials. The differential calculus gives a definite method, in such circumstances, for transforming any function f(o) into a linear one m = f(c) + f'{c)-(o-c). The law of errors then becomes (^ . ifio)) = f(c] + r (C) {[I , (0) - c) = /-(/i I (0)) \ 37 But also by quite elementary means and easy artifices we may often transform functions into others of linear form. If for instance f{o] = — , then we write J_ _ 1 _ c-jo -c) _ J_ _ J_ , _ X c-{-(o — c) c^' — io — c)-' c c' ^° ''-• ' and the law of errors is then — --^if^Ao)-o) - (I) - ^-M- § 28. With respect to functions of two or more observed quantities we may also, in case of repetitions, speak of laws of errors, only we must define more closely what we are to understand by repetitions. For then another consideration comes in, which was out of the question in the simpler case. It is still necessary for the idea of the law of errors of /'(o, o') that we should have, for each of the observed quantities o and o', a series of statements which severally may be looked upon as repetitions: 0,, Oj, Om o'n o\, o'„. But here this is not sufficient. Now it makes a difference if, among the special circumstances by o and o', there are or are not such as are common to observations of the different series. We want a technical expression for this. Here it is not appropriate only to speak of observations which are, respectively, dependent on one another or independent; we are led to mistake the partial dependence of observations for the functional dependence of exact quantities. I shall propose to designate these particular interdependences of repetitions of different observations by the word "bond", which presumably cannot cause any misunderstanding. Among the repetitions of a single observation, no other bonds must be found than such as equally bind all the repetitions together, and consequently belong to the pecularities of the method. But while, for instance, several pieces cast in the same mould may be fair repetitions of one another, and likewise one dimension measured once on each piece, two or more dimensions measured on the same piece must generally be supposed to be bound together. And thus there may easily exist bonds which, by community in a cir- cumstance, as here the particularities in the several castings, bind some or all the repe- titions of a series each to its repetition of another observation; and if observations thus connected are to enter into the same calculation, we must generally take these bonds into account. This, as a rule, can only be done by proposing a theory or hypothesis as to the 38 mathematical dependence between the observed objects and their common circumstance, and whether the number which expresses this is known from observation or quite unknown, the right treatment falls under those methods of adjustment which will be mentioned later on. It is then in a few special cases only that we can determine laws of errors for functions of two or more observed quantities, in ways analogous to what holds good of a single observation and its functions. If the observations o, o', o" . . ., which are to enter into the calculation of f{o, o\ o", . . .), are repeated in such a way that, in general, o,, o't, o", ... of the j'th repetition are connected by a common circumstance, the same for each i, but otherwise without any other bonds, we can for each i compute a value of the function y, = f{Oi, Oi, Oi , . . .), and laws of errors can be determined for this, in just the same way as for separately. To do so we need no knowledge at all of the special nature of the bonds. § 29. If, on the contrary, there is no bond at all between the repetitions of the observations o, o', o", ... — and this is the principal case to which we must try to reduce the others — then we must, in order to represent all the equally valid values oi y = f{o, o\ o", . . .), herein combine every observed value for o with every one for o', for o", etc., and all such values of y must be treated analogously to the simple repetitions of one single observed quantity. But while it may here easily become too great a task to com- pute y for each of the numerous combinations, we shall in this case be able to compute y's law of errors by means of the laws of errors for o, o', o" ... Concerning this a number of propositions might be laid down; but one of them is of special importance and will be almost sufficient for us in what follows, viz., that which teaches us to determine the law of errors for the sum of the observed quantities and o'. If the law of errors is given in the form of relative frequencies or probabilities, f{o) for and l. ea. +^>: (35) When the errors of observation are sufficiently small, we shall also here generally be able to give the most different functions a linear form. In consequence of this, the propositions (34) and (35) acquire an almost universal importance, and afford nearly the whole necessary foundation for the theory of the laws of errors of functions. Example 1. Determine the square of the mean error for differences of the n^"^ order of equidistant tabular values, between which there is no bond, the square of the mean error for every value being = l^. 40 A^iJ*) = >?2(o,— 403+602— 4o,+o„) = 70^2 ;rj.) = 1.1.10.14 4n-2 /^i^; 1 2 3 4 w "*• Example 2. By the observation of a meridional transit we observe two quantities, viz. the time, t, when a star is covered behind a thread, and the distance, f, from the meridian at that instant. But as it may be assumed that the time and the distance are not connected by a bond, and as the speed of the star is constant and proportional to the known value sin p {p = polar distance), we always state the observation by the one quan- tity, the time when the very meridian is passed, which we compute by the formula = t -\- f cosec p. The mean error is Example 3. A scale is constructed by making marks on it at regular intervals, in such a way that the square of the mean error on each interval is = /j- To measure the distance between two objects, we determine the distance of each object from the nearest mark, the square of the mean error of this observation being = a',. How great is the mean error in a measurement, by which there are n intervals between the marks we use? X, (length) = «>?,-!- 2/; . Example 4. Two points are supposed to be determined by bond-free and equally good (^2 = 1) measurements of their rectangular co-ordinates. The errors being small in proportion to the distance, how great is the mean error in the distance J? Ki^) = 2. Example 5. Under the same suppositions, what is the mean error in the inclina- tion to the a;-axis? Example 6. Having three points in a plane determined in the same manner by their rectangular co-ordinates (a;,,yj, (xj,?/,), {x^,^/^), find the mean error of the angle at the point (d7,,yj) J^ + Jl + J] ^An — ji J, --. Ji, J^, J;i being the sides of the triangle; Jj opposite to (^,,2/i)- 41 Examples 7 and 8. Find the mean errors in determinations of the areas of a trianglff and a plane quadrangle. X, (triangle) = \ (d\ + 2/^ + Al); k, (quadrangle) = I ( J' + A'\. § 30. Non-linear functions of more than one argument present very great difficulties. Even for integral rational functions no general expression for the law of errors can be found. Nevertheless, even in this case it is possible to indicate a method for computing the half- invariants of the function by means of those of the arguments. To do so it seems indis- pensable to transform the laws of errors into the form of systems of suras of powers. If = /"(o, o', ...0"") be integral and rational, both it and its powers 0'' can be written as sums of terms of the standard form Eko'^ • o'^ . . . oC")'', and for every such term the sum resulting from the combination of all repetitions is ksa • .s'j . . . s^P (including the cases where a or b ox d may be = 0), s^P being the sum of all «'•> powers of the repetitions of oW. Thus if Sr indicates the sum of the /•'•> powers of the function 0, we get Sr = SkSa • s't . . . sP. Of course, this operation is only practicable in the very simplest cases. Example 1. Determine the mean value and mean deviation of the product oo' = of two observations without bonds. Here S^ = s„s\ and generally Sr = s^s'^, consequently the mean value Jlf, =n^n\ and M, already takes the cumbersome form Example 2. f]xpress exactly by the half-invariants of the co-ordinates the mean value and the mean deviation of the square of the distance r^ =x''-\-y^, if x and y are observed without bonds. Here ^oir'') = So(^)«o(«/) «2 ('•' ) = ^4 i^) «o (y) + ^s^ {x) s^ (y) + So {x) s, [y) and //2 in = /U W + 4//3 i^)f^i (^) + 2 (/i, {x)y + 4fi, (x) ((,,{x))^ + + fi, (y) + ^iMAy)l^Ay) + 2(/., (ii)r + 4/., (y) {fz, (y)Y. § 31. The most important application of proposition (35) is certainly the deter- mination of the law of errors of the mean value itself. The mean value /'I = — (o, +02+...0„) known half-invariants >ij, k^, .. . ^r • • , we get according to (35) ^^ir^^) = ~{^^+--- + ^i) = ^^ ^iifij) = i,(A, + ... + ;,) = l^, and in general = m'-^.Xr. 42 is, we know, a linear function of the observed values, and we may treat the law of errors for f^^ according to the said proposition, not only where we look upon o,, ... On as per- fectly unconnected, but also where we assume that they result from repetitions made according to the same method. For, just like such repetitions, o,, ... o„ must not have any other circumstances in common as connecting bonds than such as bind them all and characterise the method. As the law of presumptive errors of o, is just the same as for Og ...Om, with the (37) While, consequently, the presumptive mean of a mean value for m repetitions is the presumptive mean itself, the mean error on the mean value u, is reduced to -r= of Vm the mean error on the single observation. When the number m is large, the formation of mean values consequently reduces the uncertainty considerably; the reduction, however, is proportionally greater with small than with large numbers. While already 4 repetitions bring down the uncertainty to half of the original, 100 repetitions are necessary in order to add one significant figure, and a million to add 3 figures to those due to the single observation. The higher half-invariants of //, are reduced still more. If the k^, k^, etc., of the single observation are so large that the law of errors cannot be called typical, no very great numbers of m will be necessary to realise the conditions ^ai/ii) = =^4(/ii) with an approximation that is sufficient in practice. It ought to be observed that this reduction is not only absolute, but it holds good also in relation to the corresponding power of the mean error |/>ijj(;Uj)2; for (37) gives r r r >ir(/i,):(-i,(/i,))^= m "^.(^,:>?f), which, for instance when tn = 4, shows that the deviation of ^3 from the typical form which appears by means of only 4 repetitions, is halved; that of /l^ is divided by 4, that of yij is divided by 8, etc. This shows clearly the reason why we attach great importance to the typical form for the law of errors and make arrangements to abide by it in practice. For it appears now that we possess in the formation of mean values a means of. making the laws of errors typical, even where they were not so originally. Therefore the standard rule for all practical observations is this: Take care not to neglect any opportunities of 43 repeating observations and parts of observations, so that you can directly form the mean values which should be substituted for the observed results; and this is to be done espe- cially in the case of observations of a novel character, or with peculiarities which lead us to doubt whether the law of errors will be typical. This remarkable property is peculiar, however, not to the mean only, but also, though with less certainty, to any linear function of several observations, provided only the coefficient of any single term is not so great relatively to the corresponding deviation from the typical form that it throws all the other terms into the shade. From (35) it is seen that, if the laws of errors of all the observations o, o', ... o<'") are typical, the law of errors for any of their linear functions will be typical too. And if the laws of errors are not typical, then that of the linear function will deviate relatively less than any of the observations o, o', ... Om- To avoid unnecessary complication we represent two terms of the linear function simply by o and o'. The deviation from the typical zero, which appears in the r*-^ half- invariants (>• > 2), measured by the corresponding power of the mean error, will be less for = o-{-o' than for the most discrepant of the terms o .and o'. The inequation Ar = Ar says only that, if the laws of errors for and 0' deviate unequally from the typical form, it is the law of errors for that deviates most. But this involves &^ (!)' or more briefly T > B\ where T is positive, r > 2. When we introduce a positive quantity U, so that r*" = f/' > R\ it is evident that (U -\- ly > (5 + 1)'', and it is easily demonstrated that (T -\- If > {U-\-ir. ^ Remembering that a; -f j;-» > 2, if x>0, we get by the binomial formula (c/r-f U-yf > U-\-U-' + 2'-2>{U^+ C/"V. Consequently (T + ir XU -{- ir =S (K + 1)' or 6» 44 and ^ ^ (>ir + ^'r V _ (ir(0)r^ K ^ ^, + r,v (A.(O)r' but this is the proposition we have asserted, for the extension to any number of terms causes no difficulty. But if it thus becomes a general law that the law of errors of linear functions must more or less approach the typical form, the same must hold good also of all mode- rately complex observations, such as those whose errors arise from a considerable number of sources. The expression "source of errors" is employed to indicate circumstances which undeniably influence the result, but which we have been obliged to pass over as unessential. If we imagined these circumstances transferred to the class of essential circumstances, and substantiated by subordinate observations, that which is now counted an observation would occur as a function, into which the subordinate observations enter as independent variables; and as we may assume, in the case of good observations, that the influence of each single source of errors is small, this function may be regarded as linear. The approximation to typical form which its law of errors would thus show, if we knew the laws of errors of the sources of error, cannot be lost, simply because we, by passing them over as unessen- tial, must consider the sources of error in the compound observation as unknown. More- over, we may take it for granted that, in systematically arranged observations, every such source of error as might dominate the rest will be the object of special investigation and, if necessary, will be included among the essential circumstances or removed by corrective calculations. The result then is that great deviations from the typical form of the law of errors are rare in practice. § 32. It is of interest, of course, also to acquire knowledge of the laws of errors for the determinations of /j^ and the higher half-invariants as functions of a given number of repeated observations. Here the method indicated in § 30 must be applied. But though the symmetry of these functions and the identity of the laws of presumptive errors for Oj, o.^, . . . Om afford very essential simplifications, still that method is too difficult. Not even for /x^ have I discovered the general law of errors. In my "Almindelig lagttagelseslcere" , Kobenhavn 1889, I have published tables up to the eighth degree of products of the sums of powers Sp s^ . . ., expressed by suras of terms of the form o\o'\o"*; these are here directly appli- cable. In W. Fiedler: "Elemente der neueren Geometrie und der Algebra der bindren Formen", Leipzig 1862, tables up to the 10* degree will be found. Their use is more difficult, because they require the preliminary transformation of the Sp to the coefficients Up of the rational equations § 21. There are such tables also in the Algebra by Meyer Hirsch, and Cayley has given others in the Philosophical Transactions 1857 (Vol. 147, 45 p. 489). 1 have computed the four principal half-invariants of n^ : mk^in.^) = (m — ^h ■m^/l.,{fi.,) = (w — l)H, + 2m {m — 1) X] m'^l.ifx.,} = (m - l]H^ + r2m {m - 1)H,L, + 4»« (m - 1) (m - 2)^J + + 8ot2(w< — l)yi5 r (38) m';},(/i,J = {m — l)^;^ + 24m (m — l)«/l6>i., + 32m (m - l)'^ (w — 2)^5/3 + + 8m (m — 1) (4m2 - 9m + Q)r, + 144m2 (m — l^XJ] + + 96m2 (m — 1) (>w — ^Ul^-z 4' 48m« (m — 1) A^ Here m is the number of repetitions. Of /ia and pt^ only the mean values and the mean errors have been found: tn'n,{/xs) = (m— l)(m — 2)^3, mnAf^;) -■= {ni-\y'(}n-2)U,+9m{m-l){m-2y^(XJ,+r,)+ \ (39) + 6m '^ (m — 1) (m — 2)^^ ; and mUi (/i J = (m - 1) (m2 — 6m + 6)X, — 6m (m — 1) AJ m';._, (^ J = (m — 1)2 (w2 — 6m + 6)Ha + + 8m (m — 1) (m''' — 6m + 6) (2m'' — Ibin + 15)/le-<2 + + 18m(m — l)(m — 2)(m — 4)(m''-6m + 6)/i5/i3+ I (40) + 2m (m — 1) (17m* — 204m3 + 852m« — 1404m -j- 828)>i; -| + 24m2 (m — 1) (3m« — 38m« + 150m — 138)^4 1', + + 144m''' (m — 1) (m — 2) (m — 4) (m — 5) /I J>1., + 4- 24m»(m — l)(m2— 6m + 24)/i*. Further I know only that m*X,{/i,) = (m— l)(m — 2)((m2-12m+12)>lf,— 60m/i2yi3}, (41) m^/ii (//g) = (m — 1) (m« — 30^=^ + 150m'' — 240m + 120)/i6 — — 30m (m — 1) (7m2 — 36m + 36)/ii /}., — — 60m(m — 1) (m — 2) (3m — 8) ^J — — 60m''(m — l)(m — 6)/l^ (42) mn^ili-,) = (»«— l)(m — 2)(m*— 60m3+420m''— 720m + 360)^, — — 630m (m — 1) (>« — 2) (jw^ — 8m + ^)k^X.> — — 210m(m — l)(m — 2)(7m«— 48>«-|-60)/i^yi3 — — 1260m2(m — l)(m — 2)(m — 10)yi3/i^ (43) MU,{fts) = (m — l)(Hi« — 126)»6 + ]806w«— 8400w'' + 16800;«2-15120w + 5040)/( — — 56w (w — 1) (31w« -^540/w'' -f 2340w'^ — 3600w + 1800)-i6yij — — 1680w(m — 1)(ot — 2)(3ot=* — 40/»2-fl20>« — OGMs/ia — — 70m(OT — l)(49w* — 720)»'' + 3168m«— 5400>M + 3240)/i5 — — 840?w2 (m— 1) (7w« — 150^2 + 576w — 540)/i, /ij — — 10080»«''' (m — 1) (m — 2) (/«'•' — 18?w + 40)yi5;.., — — 840wi«(7« — l)(m''— 30^4-90);?;. (44) Some ^I's of products of the /x,^, /i^, and /i, present in general the same charac- teristics as the above formulae. The most prominent of these characteristics are: 1) It is easily explained that X^ is only to be found in the equation ^i(/«i) = ^i; indeed no other half-invariant than the mean value can depend on the zero of the obser- vations. In my computations this characteristic property has afforded a system of multiple checks of the correctness of the above results. 2) All mean Xi(/ir) are functions of the 0* degree with regard to m, all squares of mean errors ^^(/Jtr) are of the (—1)^' degree, and generally each ^(/ir) is a function of the (1 — s)ti> degree, in perfect accordance with the law of large numbers. 3) The factor m — 1 appears universally as a necessary factor of ^(/ir), if only r>l. If r is an odd number, even the factor m — 2 appears, and, likewise, if r is an even number, this factor is constantly found in every term that is multiplied by one or more Xs with odd indices. No obliquity of the law of errors can occur unless at least three repetitions are under consideration. 4) Many particulars indicate these functions as compounds of factorials (m — 1 ) (w — 2) ... [ni — r) and powers of m. If, supposing the presumed law of errors to be typical, we put X^ = ^4 ^... = 0, then some further inductions can be made. In this case the law of errors of fi^ may be kUh) T I ^l(M T2 I / 2>l r\l=^ 1'+°° e \1 ^'\1 +-=h--f^j2 =U(o)e»^do. (45) \ / — 00 As to the squares of mean errors of fir we get under the same supposition: ^ (/^.) = ^^. L, (/^.) = ir. X, (/««) = m ^1 ^ (//J = 24 ,. m ^1 ' indicating that generally (46) 47 This proposition is of very great interest. If we have a number ni of repetitions at our disposal for the computation of a law of actual errors, tlien it will be seen that the relative mean errors of ^/j, //j , pis ■■■fir are by no means uniform, but increase with the index r. If m is large enough to give us /i, precisely and fi^ fairly well, then /i 3 and ju, can be only approximately indicated; and the higher half-invariants are only to be guessed, if the repetitions are not counted by thousands or millions. As all numerical coefficients in yij (nr) increase with r, almost in the same degree as the coefficients 1, 2, 6, and 24 of ^^, we must presume that the law of increasing uncertainty of the half-invariants has a general character. We have hitherto been justified in speaking of the principal half-invariants as the complete collection of the fir's or ;ir's with the lowest indices, considering a complete series of the first m half-invariants to be necessary to an unambiguous determination of a law of errors for m repetitions. We now accept that principle as a system of relative rank of the half-invariants with increasing uncertainty and consequently with a decreasing importance of the half- invariants with higher indices. We need scarcely say that there are some special exceptions to this rule. For instance if /i^ = — ^' , as in alternative experiments with equal chances for and against (pitch and toss), then ).^[ii^) is reduced to = 3 X\, which is only of the (—2)'"' order. § 33. Now we can undertake to solve the main problem of the theory of obser- vations, the transition from laws of actual errors to those of presumptive errors. Indeed this problem is not a mathematical one, but it is eminently practical. To reason from the actual state of a finite number of observations to the law governing infinitely numerous presumed repetitions is an evident trespass; and it is a mere attempt at prophecy to predict, by means of a law of presumptive errors, the results of future observations. The struggle for life, however, compels us to consult the oracles. But the modern oracles must be scientific; particularly when they are asked about numbers and quantities, mathematical science does not renounce its right of criticism. We claim that confusion of ideas and every ambiguous use of words must be carefully avoided; and the necessary act of will must be restrained to the acceptation of fixed principles, which must agree with the law of large numbers. It is hardly possible to propose more satisfactory principles than the following: The mean value of all available repetitions can he taken directly, without any change, as an approximation to the presumptive mean. If only one observation without repetition is known, it must itself, consequently, be considered an approximation to the presumptive mean value. The solitary value of any symmetrical and univocal function of repeated observations 48 must in the same way, as an isolated observation, be considered tlie presumptive mean of this function, for instance /ir = ^, (fir)- Thus, from the equations 37—41, we get by m repetitions: ;, = /ii . m ,2 ^ ^ {w — l)(w — 2)^'' m" {f^^+j^^mf'') [m — 1) (w, _ 2) (»j2 — \2m + 12) (/^'^ +,^^1/^2/^3); (47) as to X^, P.7, /?8 it is preferable to use the equations 42 — 44 themselves, putting only Inversely, if the presumptive law of errors is iinown in this way, or by adoption of any theory or hypothesis, we predict the future observations, or functions of observations, principalli/ by computing their presumptive mean values. These predictions however, though univocal, are never to be considered as exact values, but only as the first and most impor- tant terms of laws of errors. If necessary, we complete our predictions with the mean errors and higher half- invariants, computed for the predicted functions of observations by the presumed law of errors, which itself belongs to the single observations. These supplements may often be useful, nay necessary, for the correct interpretation of the prediction. The ancient oracles did not release the questioner from thinking and from responsibility, nor do the modern ones; yet there is a difference in the manner. If the crossing of a desert is calculated to last 20 days, with a mean error of one day, then you would be very unwise, to be sure, if you provided for exactly 20 days; by so doing you incur as great a probability of dying as of living. Even with provisions for 21 days the journey is evidently dangerous. But if you can carry with you provisions for 23—25 days, the undertaking may be reasonable. Your life must be at stake to make you set out with provisions for only 17 daysor less. In addition to the uncertainty provided against by the presumptive law of error, the prediction may be vitiated by the uncertainty of the data of the presumptive law itself. When this law has resulted from purely theoretical speculation, it is always impossible to calculate its uncertainty. It may be quite exact, or partially or absolutely false, we are left to choose between its admission and its rejection, as long as no trial of the prediction by repeated observations has given us a corresponding law of actual errors, by which it can be improved on. 49 If the law of presumptive errors has been computed by means of a law of actual errors, we can, according to (37), employ the values ^j, /{g, ... and the number m of actual observations for the determination of ^ri^i)- In this case the complete half-invari- ants of a predicted single observation are given analogously to the law of errors of the sum of two bondless observations by Xr + Xrifli)- Though we can in the same way compute the uncertainties of k^, X.^, and X^, it is far more difficult, or rather impossible, to make use of these results for the improvement of general predictions. Of the higher half-invariants we can very seldom, if ever, get so much as a rough estimate by the method of laws of actual errors. The same reasons that cause this difficulty, render it a matter of less importance to obtain any precise determination. Therefore the general rule of the formation of good laws of presumptive errors must be: 1. In determining X^ and X,^, rely almost entirely upon the actual observed values. 2. As to the half-invariants with high indices, say from X^ upwards, rely as exclusively upon theoretical considerations. 3. Employ the indications obtainable by actual observed values for the intermediate half-invariants as far as possible when you have the choice between the theories in (2). From what is said above of the properties of the typical law of errors, it is evident that no other theory can fairly rival it in the multiplicity and importance of its applications. It is not only constantly applied when X^, X^, and ^5 are proved to be very small, but it is used almost universally as long as the deviations are not very conspicuous. In these cases also great efforts will be made to reduce the observations to the typical form by modifying the methods or by substituting means of many observed values instead of the non-typical single observations. The preference for the typical observations is intensified by the difficulty of establishing an entirely correct method of adjustment (see the following chapters) of observations which are not typical. In those particular cases where X^ or X^ or ^^5 cannot be regarded as small, the theoretical considerations (proposition 2 above) as to /^ g ^^^ ^^^ higher half-invariants ought not to result in putting the latter = 0. As shown in " Videnskabernes Selskabs Oversigter" , 1899, p. 140, such laws of errors correspond to divergent series or imply the existence of imaginary observations. The coefficients kr of the functional law of errors (equation (6)) 7 50 have this merit in preference to the half-invariants, that no term implies the existence of any other. This series * (X) = k, •' + br") o]. (54) This possibility is of some importance for the treatment of those cases in which the single observations are bound. They must be treated then just like results, and we must try to represent them as functions of the circumstances which they have in common, and which must be given instead of them as original observations. This may be difficult to do, but as a principle it must be possible, and functions of bound observations must therefore always have laws of errors as well as others; only, in general, it is not possible to compute these laws of errors correctly simply by means of the laws of errors of the 54 observations only, just as we cannot, in general, compute the law of errors for aE -\-bR" by means of the laws of errors for E and E". In example 5, § 29, we found the mean error in the determination of a direction R between two points, which were given by bond-free and equally good (/.^{x) == X^iy) == 1) 2 measurements of their rectangular co-ordinates, viz.: Xi{Ii) = -^^, and then, in example 6, we determined the angle V in a triangle whose points were determined in the same way. It seems an obvious conclusion then that, as V = R' — R'\ we must have k^iV) = X^_{R')-\-k^(R") = -^ +7p2- ^"* *'^i^ i^ "^*' correct ; the solution is ^ j( F) = ~-^,2- Tj ~ . where J, J', and J" are the sides of the triangle. The cause of this is, of course, tliat the co-ordinates of the angular point enter into both directions and bind R' and R" together. But it is remarkable then that, when F is a right angle, the solutions are identical. With equally good unbound observations, Oq, Oj, o^, and O3, we get ^2(02 — 2o, + 0o) = 6-ij(o) ^2(03—202+0,) = 6^2(0), but ''2(03-302+30,-0,) = 20Xi{o), although 03— 3o2+3o,— Ofl = (03— 2oj + o,) — (Oj — 2o, + 0o), according to which we should expect to find i?2(03— 3o2+3o, — Oo) = ;2(08— 203+0,) + /2(o2—2o, + 0o) = 12^2(0). But if, on the other hand, we combine the two functions R' = O0+60, — 402 and R" = 2oi+3o2 — O3, where ki(R') = b?>Xi{p) and ;2(-K") = 14>i2(o), and from this compute ij for any function aR'-\-bR", then, curiously enough, we get as the correct result X^iaR' -\-bR") = (53a2 + 14J2)^3(o) = aU^(R') -\- bn,{R"). Gauss's general prohibition against regarding results of computations — especially those of mean errors — from the same observations as analogous to unbound observations, has long hampered the development of the theory of observations. To Oppermann and, somewhat later, to Helmert is due the honour of having discovered that the prohibition is not absolute, but that wide exceptions enable us to simplify our calculations. . We must therefore study thoroughly the conditions on which actually existing bonds may be harmless. Let Oi,...o„ be mutually unbound observations with known laws of errors, ^1(0,), ,Jg(oj), of typical form. Let two general, linear functions of them be [po] = p,o^ +...+;>„o„ [qo] = g,o, +... + ^„o„. I (56) 55 For these then we know the laws of errors X,{qo-\ = [qk,{o)l L,[ - For a general function of these, F = a\^po]-[-b[qo], the correct computation of the law of errors by means of F = {{ap-\-bq)o\ will further give X,{F) = (ap, + hq^)XAo^)^...^(apn-\-Hn)kAon) = 1 = «'^ X.,{po\ + i'^A^L^o] + 2a«.[i;5 XM} Xr{F) = for r>2. It appears then, both that the mean values can be computed unconditionally, as if [po] and [qo\ were unbound observations, and that the law of errors remains typical. Only in the square of the mean error there is a difference, as the term containing the factor 2ab in X^iF) ought not to be found in the formula, if [po] and [50] were not bound to one another. When consequently [pqX^io)] = PlqxX.^{o^) + . . . +2V/,;.,(o„) = (57) the functions [^0] and [go] can indeed be treated in all respects like unbound observations, for the law of errors for every linear function of them is found correctly determined also upon this supposition. We call such functions mutually "free functions", and for such, consequently, the formula for the mean error XApo]a + [_qo}b) = a'lj^n.m + b-'VqH.ip)] (58) holds good. If this formula holds good for one set of finite values of a and b, it holds good for all. If two functions are mutually free, each of them is said to be "free of the other", and inversely. Example 1. The sum and difference of two equally good, unbound observations are mutually free. Example 2. When the co-ordinates of a point are observed with equal accuracy and without any bonds, any transformed rectangular co-ordinates for the same will be mutually free. Example 3. The sum or the mean value of equally good, unbound observations is free of every diff'erence between two of these, and generally also free of every (linear) function of such differences. 56 Example 4. The differences between one observation and two other arbitrary, un- bound observations cannot be mutually free. Example 5. Linear functions of unbound observations, which are all different, are always free. Example 6. Functions with a constant proportion cannot be mutually free. § 37. In accordance with what we have now seen of free functions, corresponding propositions must hold good also of observations which are influenced by the same circum- stances: it is not necessary to respect all connecting bonds; it is possible that actually bound observations may be regarded as free. The conditions on which this may be the case, must be sought, as in (57), by means of the mean errors caused by each circumstance and the coefficients by which the circumstance influences the several observations. — Note particularly : Jf two observations are supposed to be connected by one single circumstance which they have in common, such a bond must not be left out of consideration, but is to be respected. Likewise, if there are several bonds, each of which influences both observations in the same direction. If, on the other hand, some common circumstances influence the observations in the same direction, others in opposite directions, and if, moreover, one class must be supposed to work as forcibly as the other, the observations may possibly be free, and the danger of treating them as unbound is at any rate less than in the other cases. § 38. Assuming that the functions of which we shall speak in the folloving are linear, or at any rate may be regarded as linear when expanded by Taylor's formula, because the errors are so small that we may reject squares and products of the deviations of the observations from fixed values; and assuming that the observations o,, ... o„, on which all the functions depend, are unbound, and that the values of ^.^C^i) • • • ^•'(On) are given, we can now demonstrate a series of important propositions. Out of the total system of all functions [po] = p,Oi + ... -\-pnO„ of the given n observations we can arbitrarily select partial systems of functions, each partial system containing all those, which can be represented as functions of a number of m < n mutually independent functions, representative of the system, [ao] = fliO, + . . . + anO„ [do] == diOi + . . . + d„o„ , of which no one can be expressed as a function of the others. We can then demonstrate the existence of other functions which are free of every function belonging to the partial 57 system represented by [ao] .... [do]. It is sufficient to prove that such a function [go] = g ^o I -\- ...-{- c/nOn IS free of [tto] . . . [do] in consequence of the equations [ga X2]^0 ... [ffd/i^] =0. For if so, [go] must be free of every function of the partial system, [{xa -\- . . . -\- zd) o] = x[ao]-j- ... '}'2[do], because [g(xa-^...+zd)?.2] = x[cia ^^] + . . . ^ z[gdL,] = 0. Any function of the total system [po] can now in one single way be resolved into a sum of two functions of the same observations, one of which is free of the partial system represented by [ao] . . . [do], while the other belongs to this system. If we call the free addendum [p'o], this proposition may be written [po] = [p'o] + {x [ao] + ... + Z [do]}. (59) By means of the conditions of freedom, [p'aL^] = ... = [p'dX^] = 0, all that concerns the unknown function [p'o] can be eliminated. We find [pa^j] = x[aa ki] -\- . . . -\- z[da X^] (GO) [pdX^] = x[ad X2] -\- . -\- z[dd X2] , from which we determine the coefficients x . . . z unambiguously. The number m of these equations is equal to the number of the unknown quantities, and they must be sufficient for the determination of the latter, because, according to a well known proposition from the theory of determinants, the determinant of the coefficients [aaXi], [dak^] [adk^], . . [ddki] = 2 dr, . . d, X^iOr) . . . X„(0s) is positive, being a sum of squares, and cannot be = 0, unless at least one of the func- tions [ao] . . . [do] could, contrary to our supposition , be represented as a function of the others. From the values of a; ... 2 thus found, we find likewise [p'o] = [po] — x[ao] — . . . — z[do]. (61) If [po] belongs to the partial system represented by [ao] .... [do] , the de- termination of X . . . . z expresses its coefficients in that system only, and then we get identically [p'o] = 0. 8 58 But if we take [po] out of the partial system, then (61) gives us [p'o] as different from zero and free of that partial system. If [po] — [qo] belongs to the partial system of [ao] . . . [fi?o], [qo] must produce in this manner the very same free function as [po]. Let [po] . . . [ro] be n — in functions, independent of one another and of the m functions [ao] . . . [c?o] ; if we then find [p'o] out of [po] and [r'o] out of [ro] as the free functional parts in respect to [ao]...[do], the n functions [ao]... [t^] and [jP'o] • • • ['"'o] may be the representative functions of the total system of the functions of Oi...o„, because no relation a[p'o]-\- ... -\- d[r'o] = is possible; for by (61) it might result in a relation a [po] -|- . . . 4- S[ro] -\- n [ao] -\-...-^

i] = [c'd'X] [dbi]-~[abk]-[daJ\:[aaX] = [d'b'^, [dcX]-[acX]-[daX]:[aaX] = [d'c'i], [ddX]-[adX]-[daX]:[aaX] = [d'd'X\ [c'c'X]-[b'c'^ . [db'X\ : [b'b'A] = [c"c"Xl [c'd'X]-[b'd'X] • [<^b'X\ : [b'b'^ = [d'd"X] [d'c>X] - [6'c'yi] '[d'b'X] : [b'b'Ji] = [d"c"X\, [d'd'X\-[b'd'^.[d'b'X] : [b'b'X] = [d"d"X] [d"d"X] - [c"d"X] • [d"d'X] : [c'V'i] = [d"'d"'X] As will be seen, there is a check by means of double computation for each of the sums of the products properly so called. The sums of the squares are of special importance as they are the squares of the mean errors of the transformed functions, X^[ao] = [aaA], X^lb'o] = [b'b'X], i^[(f'o] = [c"c"X], and k^[d"'o] = [d"'d"'X]. Example. Five equally good, unbound observations Oj, Oj, O3, o^,and O5 represent values of a table with equidistant arguments. The function tabulated is known to be an integral algebraic one, not exceeding the 3''<' degree. The transformation into free functions is to be carried out, in such a way that the higher diiferences are selected before the lower ones. (Because JS certainly, J* etc., possibly, represent equations of condition). With symbols for the diiferences, and with /(^(oi) = 1, we have then: Function VJo 0, i-J'o, FJ303-i#O3 "3 — 36^ "3 Fj03-4J^03+fFJ»03-iJ^03 Coefficients Ooi +0oj -flOg +O0, +0oj 0-110 1-210 0-1 3-3 1 1 _4 6-4 1 3 35 3 "35 2 '35 2 T 12 35 12 35 i 5 1 10 17 35 "Y 17 3 6 12 35 3 T _1 7 12 36 1 36 _1 7 A 3 "35 3 "35 12 35 2 7 Sums of the Products Factors 1 -1 -1 2 -2 3 3 -6 - 6 -10 3 36 -2 3 6- -10 - -20 1 3 -6- -10 20 35 -* 6-10- -20 35 70 is selected 35 7 2 I "~7 T -1 -f -1 5 f is selected 1 7 36 1 I is selected are free are both selected. 63 The complete set of free observations and the squares of their mean errors are thus: (0)= o, + J^o,-^y*o, = UOi + 02+0, + 0, + 0,), /,(0) = 1 (1) = Vjo,-iJ^o,+l(Vd^o,-U'o,) = ,V(-2o^-o, + o, + 2o,), >i,(l) = Jq (2) = J^o.^^J'o, = i(2o,~o^-2o,-~o, + 2o,), ;.,(2) = f (3)= VJ^o.-U^o, = H-Oi + 2o2-2o,+05), ^,(3) = | (4)= J^03 = Oi-4o,+6o3-4o, + o, , ^.,(4) = 70 Through this and the preceding chapter we have got a basis which will generally be sufficient for computations with observations and, in a wider sense, for computations with numerical values which are not given in exact form, but only by their laws of errors. We can, in the first place, compute the law of errors for a given, linear function of reci- procally free observations whose laws of presumptive errors we know. By this we can solve all problems in which there is not given a greater number of observations, and other more or less exact data, than of the reciprocally independent unknown values of the problem. When we, in such cases, by the means of the exact mathematics, have expressed each of the unknown numbers as a function of the given observations, and when we have succeeded in bringing these functions into a linear form, then we_ can, by (35), compute the laws of errors for each of the unknown numbers. Such a solution of a problem may be looked upon as a transformation, by which n observed or in other ways given values are transformed into « functions, each corre- sponding to its particular value among the independent, unknown values of the problem. It lies often near thus to look upon the solution of a problem as a transformation, when the solution of the problem is not the end but only the means of determining other un- known quantities, perhaps many other, which are all explicit functions of the independent unknowns of the problem. Thus, for instance, we compute the 6 elements of the orbit of a planet by the rectascensions and declinations corresponding to 3 times, not precisely as our end, but in order thereby to be able to compute ephemerides of the future places of the planet. But while the validity of this view is absolute in exact mathematics, it is only limited when we want to determine the presumptive laws of errors Of sought functions by the given laws of errors for the observations. Only the mean values, sought as well as given, can be treated just as exact quantities, and with these the general linear transformation of n given into u sought numbers, with altogether n^ arbitrary constants, remains valid, as also the employment of the found mean numbers as independent variables in the mean value of the explicit functions. If we want also correctly to determine the mean errors, we may employ no other transformation than that into free functions. And if, to some extent, we may choose the 64 independent unknowns of the problem as we please, we may often succeed in carrying through the treatment of a problem by transformation into free functions; for an unknown number may be chosen quite arbitrarily in all its n coefficients, and each of the following unknowns looses, as a function of the observations, only an arbitrary coefficient in com- parison to the preceding one; even the w"" unknown can still get an arbitrary factor. Altogether are \n[n-\-l) of the n"- coefficients of these transformations arbitrary. But if the problem does not admit of any solution through a transformation into free functions, the mean errors for the several unknowns, no matter how many there may be, can be computed only in such a way that each of the sought numbers are directly expressed as a linear function of the observations. The same holds good also when the laws of errors of the observations are not typical, and we are to examine how it is with X-i and the higher half-invariants in the laws of errors of the sought functions. Still greater importance, nay a privileged position as the only legitimate proceeding, gets the transformation into a complete set of free functions in the over-determined problems, which are rejected as self-contradictory in exact mathematics. When we have a collection of observations whose number is greater than the number of the independent unknowns of the problem, then the question will be to determine laws of actual errors from the standpoint of the observations. We must mediate between the observations that contradict one another, in order to determine their mean numbers, and the discrepancies themselves must be employed to determine their mean deviations, etc. But as we have not to do with repetitions, the discrepancies conceal themselves behind the changes of the circumstances and require transformations for their detection. All the functions of the observations which, as the problem is over-determined, have theoretically necessary values , as , for instance, the sum of the angles of a plane triangle, must be selected for special use. Besides, those of the unknowns of the problem, to the determination of which the theory does not contribute, must come forth by the transformation by which the problem is to be solved. As we shall see in the following chapters on Adjustment, it becomes of essential moment here that we transform into a system of free functions. The transformation begins with mutually free observations, and must not itself introduce any bond, because the trans- formed functions in various ways must come forth as observations which determine laws of actual errors. X. ADJUSTMENT. § 4.3. Pursuing the plan indicated in § 5 we now proceed to treat the determina- tion of laws of errors in some of the cases of observations made under varying or different 65 essential circumstances. But here we must be content with very small results. The general problem will hardly ever be solved. The necessary equations must be taken from the totality of the hypotheses or theories which express all the terms of each law of error — say their half-invariants — as functions of the varying or wholly different circumstances of the observations. Without great regret, however, the multiplicity of these theoretical equations can be reduced considerably, if we suppose all the laws of errors to be exclusively of the typical form. • For each observation we need then only two theoretical equations, one representing its presumptive mean value ii(o,), the other the square of its mean error X^(Oi), as func- tions of the essential circumstances. But the theoretical equations will generally contain other unknown quantities, the arbitrary constants of the theory, and these must be elimi- nated or determined together with the laws of errors. The complexity is still great enough to require a further reduction. We must, preliminarily at all events, suppose the mean errors to be given directly by theory, or at least their mutual ratios, the weights. If not, the problems require a solution by the indirect proceeding. Hypothetical assumptions concerning the X,^ {Oi) are used in the first approximation and checked and corrected by special operations which, as far as possible, we shall try to expose beside the several solutions, using for brevity the word "criticism" for these and other operations connected with them. But even if we confine our theoretical equations to the presumptive means ^1(0,) and the arbitrary unknown quantities of the theory, the solutions will only be possible if we further suppose the theoretical equations to be linear or reducible to this form. Moreover, it will generally be necessary to regard as exactly given many quantities really found by observation, on the supposition only that the corresponding mean errors will be small enough to render such irregularity inoffensive. In the solution of such problems we must rely on the found propositions about functions of observations with exactly given coefficients. In the theoretical equations of each problem sets of such functions will present themselves, some functions appearing as given, others as required. The observations, as independent variables of these functions, are, now the given observed values 0,, now the presumptive means ^1(0,); the latter are, for instance, among the unknown quantities required for the exact satisfaction of the theoretical equations. What is said here provisionally about the problems that will be treated in the following, can be illustrated by the simplest case (discussed above) of n repetitions of the same observation, resulting in the observed values Oi, ... o„. If we here write the theo- retical equations without introducing any unnecessary unknown quantities, they will show the forms == .^i(o,) — X^iot) or, generally, = /i,[a(o, — oj)]. But these equations are 9 66 evidently not sufficient for ttie determination of any ^i(Oi), which they only give if another >i,(Oifc) is found beforehand. The sought common mean cannot be formed by the introduc- tion of the observed values into any function [a{0i — 0t)], these erroneous values of the functions being useful only to check X.^ (Oj) by our criticism. But we must remember what we ijnow about free functions: that the whole system of these functions [a(o, — oj.)] is only a partial system, with w — 1 differences Oi — ot as representatives. The only w* functions which can be free of this partial system, must evidently be proportional to the sum 0, -\- . . . -\- o„, and by this we find the sought determination by ^l{Oi) = ^(Oi + ••• + "„), the presumptive mean being equal to the actual mean of the observed values. If we thus consider a general series of unbound observations, o i , ... o„, it is of the greatest importance to notice first that two sorts of special cases may occur, in which our problem may be solved immediately. It may be that the theoretical equations concern- ing the observations leave some of the observations, for instance o^, quite untouched; it may be also that the theory fully determines certain others of the observations, for instance o„. In the former case, that is when none of all the theories in any way concern the observation o,, it is evident that the observed value o, must be approved unconditionally. Even though this observation does not represent any mean value found by repetitions, but stands quite isolated, it must be accepted as the mean A,(Oi) in its law of presumptive errors, and the corresponding square of the mean error ?..^{0i) must then be taken, unchanged, from the assumed investigations of the method of observation. If, in the latter case, o„ is an observation which directly concerns a quantity that can be determined theoretically (for instance the sum of the angles of a rectilinear triangle), then it is, as such, quite superfluous as long as the theory is maintained, and then it must in all further computations be replaced by the theoretically given value; and in the same way X„ (o„) must be replaced by zero, as the square of the mean error on the errorless theoretical value. The only possible meaning of such superfluous observations must be to test the correctness of the theory for approbation or rejection (a third result is impossible when we are dealing with any real theory or hypothesis), or to be used in the criticism. In such a test it must be assumed that the theoretical value corresponding to o„, which we will call u„, is identical with the mean value in the law of presumptive errors for o„, consequently, that M„ = /ii(o„), and the condition of an affirmative result must be obtained from the square of the deviation, (o„ — m„)2 in comparison with X^iOn). The 67 equation (o„ — m„)'^ = /ig (o„) need not be exactly satisfied, but the approximation must at any rate be so close that we may expect to find ^2(^«) coming out as the mean of numerous observed values of (o„ — u„)^. Compare § 34. § 44. If then all the observations o^ ... o„ fall under one or the other of these two cases, the matter is simple enough. But generally the observations o, will be connected by theoretical equations of condition which, separately, are insufficient for the determination of the single ones. Then the question is whether we can transform the series of observations in such a way that a clear separation between the two opposite relations to the theory can be made, so that some of the transformed functions of the observations, which must be mutually free in order to be treated as unbound observations, become quite independent of the theory, while the rest are entirely dependent on it. This can be done, and the computation with observations in consequence of these principles, is what we mean by the word "adjustment". For as every theory can be fully expressed by a certain number, n — m, of theoretical equations which give the exact values of the same number of mutually independent linear functions, and as we are able, as we have seen, from every observation or linear function of the observations, in one single way, to separate a function which is free and independent of these just named theoretically given functions, and which must thus enter into another system, represented by m functions, this system must include all those functions of the ob- servations which are independent of the theory and cannot be determined by it. flach of the thus mutually separated systems can be imagined to be represented, the theoretical system by n — m, the non -theoretical or empirical system by m mutually free functions, which together represent all observations and all linear functions of the same, and which may be looked upon as a complete, transformed system of free functions, consequently as unbound obser- vations. The two systems can be separated in a single way only, although the represen- tation of each partial system, by free functions, can occur in many ways. It is the idea of the adjustment, by means of this transformation, to give the theory its due and the observations theirs, in such a way that every function of the theo- retical system, and particularly the n — m free representatives of the same, are exchanged, each with its theoretically given value, which, pursuant to the theory, is free of error. On the other hand, every function of the empiric system and, particularly, its m free representa- tives remain unchanged as the observations determine them. Every general function of the n observations \do\ and, particularly, the observations themselves are during the adjust- ment split into two univocally determined addenda: the theoretical function \d'o\ which should have a fixed value D\ and the non-theoretical one \d"d]. The former \d'o\ is by the adjustment changed into D' and made errorless, the latter is not changed at all. The result of the adjustment, D'-\-\d"d\, is called the adjusted value of the function, and may 68 be indicated as [du], the adjusted values of the observations themselves being written M,...M„. The forms of the functions are not broken, as the distributive principle f{x-[-t/) = f{^)-\-f{y) liolds good of every homogeneous linear function. The determination of the adjusted values is analogous to the formation of the mean values of laws of errors by repetitions. For theoretically determined functions the adjusted value is the mean value on the very law of presumptive errors; for the functions that are free of the whole theory, we have the extreme opposite limiting case, mean values represented by an isolated, single observation. In general the adjusted values \du\ are ana- logous to actual mean values by a more or less numerous series of repetitions. For while X^(\do\) = X^\d'o} + X^[d"o\, we have X^idu] == X^{D')^ Li[d"u] = k.[d"o], consequently smaller than k^ {do]. The ratio -fj^ is analogous to the number of the repetitions or the weight of the mean value. § 45. By "criticism" we mean the trial of the — hypothetical or theoretical — suppositions, which have been made in the adjustment, with respect to the mean errors of the observations; new determinations of the mean errors, analogous to the determinations by the square of the mean deviations, n^, will, eventually also fall under this. The basis of the criticism must be taken from a comparison of the observed and the adjusted values, for instance the differences [do] — [du]. According to the principle of § 34 we must expect the square of such a difference, on an average, to agree with the square of the correspon- ding mean error, X^ ([do] — [du]) , but as [do] — [du] = [d'o] — D, and X,i[d'o] = X-i [do] — X^ [du], we get X^ ([do] — [du]) = X^ [do] — X, [dti] , (68) which, by way of parenthesis, shows that the observed and the adjusted values of the same function or observation cannot in general be mutually free. We ought then to have ([do]-[du] )^ _ X^[do]-X^[du] ^^^> on the average; and for a sum of terms of this form we must expect the mean to approach the number of the terms, nota bene, if there are no bonds between the functions [rfo]— [rfj<]; but in general such bonds will be present, produced by the adjustment or by the selec- tion of the functions. It is no help if we select the original and unbound observations themselves, and consequently form sums such as (o — u)^ {X,(o)-XAu)y for after the adjustment and its change of the mean errors, u^ ... u„ are not generally free functions such as Oi...o„. Only one single choice is immediately safe, viz., to stick to the system of the mutually free functions which, in the adjustment, have themselves 69 represented the observations: the n — m theoreticallly given functions and the m which the adjustment determines by the observations. Only of these we know that they are free both before and after the adjustment. And as the differences of the last-mentioned m functions identically vanish, the criticism must be based upon the n — m terms corresponding to the theoretically free functions [ao\ = A,. .. [6'o] = B' of the series {[ao]-AY ••• + (\b'o\ - B'Y ([ao\-AY + ([b'o-\-Fl_ (70) „2 [ao] — >lj [aw] I ••• ' >lj [6'o] — /ij [6'm] [aak^] i ••• ' [ft'i'yij] the sum of which must be expected to be = w — m. Of course we must not expect this equation to be strictly satisfied; according to the second equation (46) the square of the mean error on 1, as the expected value of each term of the series, ought to be put down = 2 ; for the whole series, consequently, we can put down the expected value as n — m^'\/2(n— m). But now we can make use of the proposition (66) concerning the free functions. It offers us the advantage that we can base the criticism on the deviations of the several observations from their adjusted values, the latter, we know, being such a special set of values as may be compared to the observations like v^ ...v„ loc. cit. ; u^ . . . «„ are only distinguished from v^ . . .v„ by giving the functions which are free of the theory the same values as the observations. We have consequently ([g o] -Ay {[b'o]-B')^ (O — M)'^ L ^-A") = n — m^Vn — m. (71) tioned [b'b'X^-] If we compare the sum on the right side in this expression with the above men- , which we dare not approve on account of the bonds produced by (0 — m)' , by the diminution of the denomi- ^2 (o) — k^ (M) the adjustment, then there is no decided contradiction between putting down at the smaller value n — m only, while U^ (o) — Xi (u) nators, can get the value n; only we can get no certainty for it. The ratios between the corresponding terms in these two sums of squares, conse^ quently I, (o) — X^ (M) 1 ^2 (m) k,{0) , we call "scales", viz. scales for measuring the influence of the adjustment on the single observation 1 ^2 [du] More generally we call the scale for the function [do]. (72) /?2 [do] If the scale for a function- or observation has its greatest possible value, viz. 1, A^[du] = 0. The theory has then entirely decided the result of the adjustment. But if the scale sinks to its lowest limit = 0, we get just the reverse k2[du] = /{^[do], i. e. the theory has had no influence at all; the whole determination is based on the accidental 70 value of the observation, and for observations in this case we get jo — u)^ Even though the scale has a finite, but very small value it will be inadmissible to de- pend on the value of such a term becoming = 1. We understand now, therefore, the (0 - M) superiority of the sum of the squares {o — M)2 1^2 {0) AA«) L, {0) m to the sum of the squares n as a bearer of the summary criticism. We may also very well, on principle, sharpen the demand for adjustment on the (0 — m)21 must part of the criticism, so that not only the whole sum of the squares approach the value n — m, but also partial sums, extracted from the same, or even its several terms, must approach certain values. Only, they are not to be added up as numbers of units, but must be sums of the scales of the corresponding terms, we may trust to the sum of the squares , ;■ /-— -^ ciously applied, may be considered as fully justified. (o — M)2 hip) So much that this principle, when judi- The sum of the squares h [0] possesses an interesting property which all other authors have used as the basis of the adjustment, under the name of "the method of the least squares". The above sum of the squares gets by the adjustment the least possible value that - ' X, {0) can get for values v^ . . . i?„ which satisfy the conditions of the theory. The proposition (66) concerning the free functions shows that the condition of this minimum is that [c"o^^ = [c"m], . . . \d"'o'] = \d"'u\ for all the free functions which are determined by the observations, consequently just by putting for each v the corresponding adjusted value u. § 46. The carrying out of adjustments depends of course to a high degree on the form in which the theory is given. The theoretical equations will generally include some observations and, beside these, some unknown quantities, elements, in smaller number than those of the equations, which we just want to determine through the adjustment. This general form, however, is unpractical, and may also easily be transformed through the usual mathematical processes of elimination. We always go back to one or the other of two extreme forms which it is easy to handle: either, we assume that all the elements are eliminated, so that the theory is given as above assumed by n — m linear equations of condition with theoretically given coefficients and values, adjustment by correlates; or, we manage to get an equation for each observation, consequently no equations of condition between several observations. This is easily attained by making the number of the elements as large {^m) as may be necessary: we may for instance give some values of observations the name of elements. This sort of adjustment is called adjustment by elements. We 71 shall discuss these two forms in the following chapters XI and XII, first the adjustment by correlates whose rules it is easiest to deduce. In practice we prefer adjustment by correlates when m is nearly as large as n, adjustment by elements when m is small. XL ADJUSTMENT BY COREELATES. § 47. We suppose we have ascertained that the whole theory is expressed in the equations [au] ^ A, ... [cu'] = C, where the adjusted values u of the n observations are the only unknown quantities; we prefer in doubtful cases to have too many equations rather than too few, and occasionally a supernumerary equation to check the computation. The first thing the adjustment by correlates then requires is that the functions \ao\ ... [coj, corresponding to these equations, are made free of one another by the schedule in § 42. Let [ao]., . .. \c"o\ indicate the n — m mutually free functions which we have got by this operation, and let us, beside these, imagine the system of free functions completed by m other arbitrarily selected functions , [d"'o], . . . [^"o] , representatives of the empiric functions; the adjustment is then principally made by introducing the theoretical values into this system of free functions. It is finally accomplished by transforming back from the free modified functions to the adjusted observations. For this inverse transformation, according to (62), the n equations are: and according to (35) (compare also (63)) hioi) = { ^^^ • >i, [ao] + + ^jJ^ x,[g''o^] «.^ I , c"' I dr I ^ gf \,,, ^ = { (74) As the adjustment influences only the w — w* first terms of each of these equations, we have, because [au] = A, . . . [c"u] = C", and Xil^u] = ... = l^ \c"u\ = 0, {n r" il'" Q" 1 , - ; ^ ^ + . . . + ■■„,■, , C" + Y:J^,'r~^ \d"'o\ + . . . + [g'of.X^io,) (75) and 72 Consequently and , , J [ao\-A , , „\c"o]-C"\ ^.(o.)-^.(«.) = ^:Nl[aai;] + "- + [o-?Q7j) = ^^(''•-"■)- (78) Thus for the computation of all the diiferences between the observed and adjusted values of the several observations and the squares of their mean errors, and thereby indirectly for the whole adjustment, we need but use the values and the mean errors of the several observations, the coefficients in the theoretically given functions, and the two values of each of these, namely, the theoretical value, and the value which the observations would give them. The factors in the expression for o,- — «<,■, _ [ m]~A _ [c>'o]-C" ^"~" [aaX.y ^'" - [c"c"A,\ ' which are common to all the observations, are called correlates, and have given the method its name. The adjusted, improved values of the observations are computed in the easiest way by the formula M. = 0. - X, (Oi){aiKa + . . . + c7K,n}. (79) By writing the equation (78) and summing up for all values of i from 1 to n, we demonstrate the proposition concerning the sum of the scales discussed in the preceding chapter, viz. ^ _;,(«) X,(o) § 48. It deserves to be noticed that all these equations are homogeneous with respect to the symbol /ig. Therefore it makes no change at all in the results of the adjustment or the computation of the scales, if our assumed knowledge of the mean errors in the several observations has failed by a wrong estimate of the unity of the mean errors, if only the proportionality is preserved; we can adjust correctly if we know only the relative weights of the observations. The homogeneousness is not broken till we reach the equations of the criticism : 73 {[ao]-A)^ + ([c"o]-C")^ _ [o-iif L h{o) J = \{aKa + . . . + c"K,..)HM^ = n-m ± \/2{n-m) (82) It follows that criticism in this form, the "summary criticism", can only be used to try the correctness of the hypothetical unity of the mean errors, or to determine this if it has originally been quite unknown. The special criticism, on the other hand, can, where the series of observations is divided into groups, give fuller information through the sums of squares 2^^^ =^(l-f;4), (83) taken for each group. We may, for instance, test or determine the unities of the mean errors for one group by means of observations of angles, for another by measurements of distances, etc. The criticism has also other means at its disposal. Thus the differences (0 — u) ought to be small, particularly those whose mean errors have been small, and they ought to change their signs in such a way that approximately k^ (o.) (84) for natural or accidentally selected groups, especially for such series of observations as are nearly repetitions, the essential circumstances having varied very little. If, ultimately, the observations can be arranged systematically, either according to essential circumstances or to such as are considered inessential, we must expect frequent and irregular changes of the signs of — n. If not, we are to suspect the observations of systematical errors, the theory proving to be insufficient. § 49. It will not be superfluous to present in the form of a schedule of the adjustment by correlates what has been said here, also as to the working out of the free functions. We suppose then that, among 4 unbound observations o,, 0^, O3, and o^, with the squares on their mean errors -^2(01)1 '<2(''2)i ^ii'^s)^ ^^^ ^A"^*)^ there exist relations which can be expressed by the three theoretical equations \au\ = a,M, -|- a^ii^ -\- a^u^ -\- a^u^ = A \hu\ = 6j«, + 62M2 + ^^■i'"'3 + ^4**4 "= -S \cu'\ = r-iM, +C.^M2 +C3?/a +C.^U^ == C. The schedule is then as follows: 10 74 The given A B 0, /},(«, ) a, 0, ^AOt) «4 C [ao] [60] [co] [aaX\ [abX\ [acX\ [baX] [bbJi] [bcX] [caX\ [cbX] [ccX\ /? = Con-elates K, [baX\ [caX\ [aa^] ' ' [aaX] [ao] — A [aaX] Free functions C c' c' c' ir K K K \h'o\ [c'o] [b'b'k] ib'c'X] [C'6'yi] [C'c'/i] \_c'b'X\ " {b'b'X\ [6'o]-g ib'b'k] C" [c"o] [c"c">i] Adjusted values Scales 0,-M, M, ;,(o,-M,) >!,(m,) 1-;,{m,) ^2(0,) Kc" == [c"o]-C" [cVX] = 3 as proof. Criticism {o,-uy:X,{o,) Sum for proof and summary criticism The free functions are computed by means of: C = C-rA c'i = d — yai B' = B-^A b'i = bi—^Ui \b'o] = [H-/?M [c'6'yi] = \c.bX\-p\caX] Ydo\ = [c6] — Y \a6\ [c'c'K] = {ccX\-r\ca^ C" e'{ [c"o] C'-r'B c'i-r'b'i [c'o]-r'[6'o] [c"c"X] = [c'c:x\-rVh'x\ By the adjustment properly so called we compute Oi — Ui == {aiKa^b'iKt, -\-c'iKci<)Xi{o,) I n% i/n c"^ \ and for the summary criticism KliaaX^-] + K\, [b'b'k^-\ + ifV- [c'-c'-yi^] = LM J = 3±l/6. In order to get a check we ought further to compute [r«^] = A, \hu\ = B, and [cm] = C, with the values we have found for m,, m^, m^, and u^. Moreover it is useful to add a superfluous theoretical equation, for instance [(a-{-b-{-c)u\ = A-\-B-\-C, through the 75 computation of the free functions, which is correct only if such a superfluity leads to identical results. § 50. It is a deficiency in the adjustment by correlates that it cannot well be employed as an intermediate link in a computation that goes beyond it. The method is good as far as the determination of the adjusted values of the several observations and the criticism on the same, but no farther. We are often in want of the adjusted values with determinations of the mean errors of certain functions of the observations; in order to solve such problems the adjustment by correlates must be made in a modified form. The simplest course is, I think, immediately after drawing up the theoretical equations of condition to annex the whole series of the functions that are to be examined, for instance [rfo], . . . [eo], and include them in the computation of the free functions. In doing so we must take care not to mix up the theoretically and the empirically determined functions, so that the order of the operation must unconditionally give the precedence to the theoretical functions; the others are not made free till the treatment of these is quite finished. The functions [d"'o], . . . | e^o] , which are separated from these — it is scarcely necessary to mention it — remain unchanged by the adjustment both in value and in mean error. And at last the adjusted functions [du], . . .[eu], by retrograde transformation," are determined as linear functions of A, B\ C", [d"'o], . . . [e^o]. Example 1. In a plane triangle each angle has been measured several times, all measurements being made according to the same method, bondfree and with the same (unknown) mean error: for angle A has been found 70° 0' 5" as the mean number of 6 measurements 1) » jd t) » ti 50° 3 » » » " » 10 *' C <> » .1 60° 0' 2" .. » » » » 15 The adjusted values for the angles are then 70°, 50°, and 60°, the mean error for single measurement = V/SOO == 17"3, the scales 0-5, 0-3, and 0-2. Example 2. (Comp. example § 42.) Five equidistant tabular values, 12, 19, 29, 41, 55, have been obtained by taking approximate round values from an exact table, from which reason their mean errors are all = \/-^. The adjustment is performed under the successive hypotheses that the table belongs to a function of the 3"", 2°^, and 1"' degree, and the hypothesis of the second degree is varied by the special hypothesis that the 2°'' dilierence is exactly = 2, in the following schedule marked (or). The same schedule may be used for all four modifications of the problem, so that in the sums to the right in the schedule, the first term corresponds to the first modification only, and the sum of the two first terms to the second modification: 10* 76 ^2(0) 12 19 29 41 55 1 12 1 1 2 1 12 1 12 1 1 2 J* VJ' J2 (F#)' (J7=(^T (or 2) (or 2) 1 _JL 2 1 -4 -1 1 1 -^ 6 3 —2 2 -4 -3 1 -1 1 T 1 1 1 5 2 T 1 2 1 16 7 70 35 20 12 12 12 35 12 20 12 10 12 6 2 4 20 1 2 10 1 2 6 12 1 42 /? = _ 1 - 2 ' r=-f r' = o ^K. 1 70 ^K.= = - "B' T2 i(',„=8(orl) ^( 1+ 7+160 (or +20)), i(_4-14- 80 (or -10)), J^( 6+ 0-160 (or -20)), J^(_4+14_ 80 (or -10)), T^( 1- 7+160 (or +20)), /2(0— «) _( 14_ 7+20) ^(16+28+ 5) ^(36+ 0+20) _.(16+28+ 5) _( 1+ 7+20) For the summary criticism (O-M) I ^2(0) 6^ 42 7680 (or 120) 35 + 35 + 35 The hypothesis of the third degree, J^==0, where the values of 70m, and their differences are: 839 1334 2024 2874 3849 495 690 850 975 195 160 125 -35 -35, agrees too well with the observations, and must be suspected of being underadjusted, for the sum of the squares of the summary criticism is only ^, where we might expect l + |/2. The hypothesis of the second degree, J* = 0, VJ^ ^ 0, gives for 70m, and differences: 832 1348 2024 2860 3856 516 676 836 996 160 160 160. The adjustment is here good, the sum of the squares is II , and we might expect 2+1/4. The hypothesis of the first degree, A* = 0, VJ'^ .= 0, J*" = 0, gives for the adjusted values and their differences: 9-6 20-4 31-2 42-0 52-8 10-8 10-8 10-8 10-8. 77 The deviations are evidently too large {o — u is +2'4, — 1'4, —2-2, —1-0, +2-2) to be due to the use of round numbers; the sum of the squares is also 220-8 instead of 3 ±1/6, consequently, no doubt, an over-adjustment. The special adjustment of the second degree, J^=0, T^' = 0, and J* = 2, gives for M, and its differences: 11-6 19-4 29-2 41-0 54-8 7-8 9-8 11-8 13-8 The deviations o - m = 0-4, -0-4, -0-2, 0-0, +0-2 nowhere reach a, and may consequently be due to the use of round numbers; the sum of the squares _ 4-8 instead of 3 ±1/6 also agrees very well. Indeed, a constant subtraction of 0*04 from m, would lead to (3-4)2, (4.4)2^ (5.4)2^ (6-4)2, ^nd (7-4)^, from which the example is taken. Example 3. Between 4 points on a straight line the 6 distances "si are measured with equal exactness without bonds. By adjustment we find for instance we notice that every scale = |. It is recommended actually to work the example by a millimeter scale, which is displaced after the measurement of each distance in order to avoid bonds. XII. ADJUSTMENT BY ELEMENTS. § 51. Though every problem in adjustment may be solved in both ways, by correlates as well as by elements, the difficulty in so doing is often very different. The most frequent cases, where the number of equations of condition is large, are best suited for adjustment by elements, and this is therefore employed far oftener than adjustment by correlates. The adjustment by elements requires the theory in such a form that each observa- tion is represented by one equation which expresses the mean value /i , (o) explicitely as linear functions of unknown values, the ''elements", x, y, . . . z: 78 ^liOi) = P,x-\-q^y-\- ... -{-r^z (85) where the p, q, ... /• are theoretically given. All observations are supposed to be unbound. The problem is then first to determine the adjusted values of these elements X, tj, ... z, after which each of these equations (85) , which we call '^equations for the observations", gives the adjusted value u of the observation. Constantly assuming that ^^(0) is known for each observation, we can from the system (85) deduce the following normal equations: pkx{0) . h{o) = PP U2i0)\ x + Pi 1 Mo) y+- • + pr Mo) z == po [Mo)\ qX^io) I h(o) \ = \ IP 1 [LAo)\ x + q