ON THE ALGEBRAICAL AND NUMERICAL THEOKT OF ERRORS OF OBSERVATIONS AND THE COMBINATION OF OBSEEVATIONS. BY GEORGE BIDDELL AIRY, M. A. ASTRONOMER ROYAL. MACMILLAJST AND CO. CDambtifige AND 23, HENRIETTA STBEET, COVENT GABDEN, HLonfcon. 1861. PREFACE. THE Theory of Probabilities is naturally and strongly divided into two parts. One of these relates to those chances which can be altered only by the changes of entire units or integral multiples - of units in the funda mental conditions of the problem as in the instances of the number of dots exhibited by the upper surface of a die, or the numbers of black and white balls to be extracted from a bag. The other relates to those chances which have respect to insensible gradations in the value of the element measured as in the duration of life, or in the amount of error incident to an astro nomical observation. It may be difficult to commence the investigations proper for the second division of the theory without referring to principles derived from the first. Never theless, it is certain that, when the elements of the second division of the theory are established, all refer ence to the first division is laid aside and the original connexion is, by the great majority of persons who use the second division, entirely forgotten The two divi sions branch off into totally unconnected subjects those VI PREFACE. persons who habitually use one part never have occasion for the other and practically they become two different sciences. In order to spare astronomers and observers in natural philosophy the confusion and loss of time which are produced by referring to the ordinary treatises em bracing both branches of Probabilities, I have thought it desirableto draw up this tract, relating only to Errors of Observation, and to the rules, derivable from the consideration, of these Errors, for the Combination of the Results of Observations. 1 have thus also the advantage of entering somewhat more fully into several points, of interest to the observer, than can possibly be clone in a General Theory of Probabilities. No novelty, 1 believe, of fundamental character., will be found in these pages. At the same time I may state that the work lias been written without reference to or distinct recollection of any other treatise excepting only Laplaces TJitforic dcs Probability ami the me thods of treating the different problems may therefore differ in some small degrees from those commonly em ployed. G. B. AIRY. Y, OKKEHWICII, January a a, 1861 INDEX. PART I. FALLIBLE MEASURES, AND SIMPLE ERRORS OF OBSERVATION. SECTION 1. Nature of the Errors here considered. PAGE Article 2. Instance of Errors of Integers 1 3. Instance of Graduated Errors these are the sub ject of this Treatise 2 4. Errors of an intermediate class . . . . 6. 5. Instances of Mistakes ib. 6. Characteristics of the Errors considered in this Treatise ......... 3 8. The word Error really means Uncertainty, . 4 viii INDEX. SECTION 2. Law of Probability qf Errors of any given amount. Article y. Reference to ordinary theory of Chances, 4 10. Illustrations of the nature of the law ... 5 11. Illustration of the algebraic form to be expected for the law ........ 6 12. Laplaces investigation introduced .... 7 13. Algebraical combination of many independent causes of error assumed ..... t. 15. This leads to a definite integral .... 8 16. Simplification of the integral ..... 10 rw 17.Investigation of I dt. e ..... 11 CO 18. Investigation of I dt. cos rt. e- . . .,12 20. Probability that an error will fall between x and found to be T, T c l . So . . .,14 21. Other suppositions lead to the same result ., 15 22. Plausibility of this law table of values of e - a 23. Curve representing the law of Frequency of Error . 16 3. Comeguence of the Law of Probability or Frequency qf Error a applied to One Syxtewi of Measures of one Element, 25. It is assumed that the law of Probability applies equally to positive and to negative errors .18 26...