IRLF SB 35 Sfil -3O LIBRARY UNIVERSITY OF CALIFORNIA. Received Accessions No Shelf No. Jb NOTES MANAGEMENT OF CHRONOMETERS MEASUREMENT OF MERIDIAN DISTANCES. OF 1 THE " ^ UNIVERSITY SEnterrti at Stationer*' ON THE MANAGEMENT OF CHRONOMETERS AND THE MEASUREMENT OF MERIDIAN DISTANCES. BY CHARLES E. A. SHADWELL, ESQ. C.B. CAPTAIN, ROYAL NAVY. flefo dBtittion, carefullj) rcbteett. t [UNIVERSITY; LONDON: J. D. POTTER, 31 POULTRY, & 11 KING STREET, TOWER HILL. 1861. LONDON: STRANGEWAYS & WALDEN (late G. BARCLAY), Printers, 28 Castle St. Leicester Sq. ADVEETISEMENT. IN submitting to the notice of the naval profession a new edition of "Notes on the Management of Chronometers, and the Measurement of Meridian Distances/' the Author has been desirous of rendering this work still more worthy of the favour- able attention of students of nautical science. The whole work has been carefully revised, and many small improvements introduced. The writings of recent French authors, Givry, De Cornulier, Lieussou, Mouchez, Vincendon Dumoulin, Coupvent Desbois, and Charles Ploix, have been carefully examined, and many valuable extracts from their works, taken chiefly from the pages of the " Recherches Chronometriques," a most useful publication, now in course of issue, under the auspices of the Minister of Marine at Paris, enrich the present volume both in the text and notes. The important questions of the effects of change of tem- perature, and the influence of the acceleration, have been fully entered into. For this purpose a new chapter (chapter vi.) has been interpolated in this edition, in which the systems of De Cornulier, Lieussou, and Mouchez have been amply discussed and commented on. The system proposed by Hartnup, of adopting a series of "tabulated rates," obtained by direct ex- periment, corresponding to the different degrees of the thermo- VI ADVERTISEMENT. metric scale, instead of a fixed daily rate, regardless of tem- perature, as usually employed, has also been introduced to the notice of the reader, and carefully examined. The additional matter introduced by these changes, amount- ing to seventy pages of letterpress, has had the effect of altering the titles of the final chapters, chapters vii. viii. ix. and x. of this edition, respectively corresponding to chapters vi, vii. viii. and ix. of the original one. All the alterations in the work consist of additions ; all the matter given formerly being still retained. Notwithstanding these changes, and consequent increase of ex- pense in printing, the low price at which the book was originally offered to the public remains unchanged. The Author trusts that the additions to his work, above alluded to, may prove useful to his professional brethren, and that in placing before British seamen the recent researches and analytical investigations of contemporary French writers, he may have rendered some essential service to the progressive improvement of nautical science. Researches into the laws which regulate the changes caused by variations of temperature, and the influence of the accele- ration, as a function of the time, seem to be the point towards which the improvement of chronometric science is at present tending; we trust that English navigators will take their share in contributing to the development of our knowledge on these important questions. With these hopes the Author submits his labours to the notice of the naval profession, and to the attention of students of chronometric science, trusting that they may be found worthy of their favourable consideration, and may be deemed a useful contribution to nautical knowledge, Slough, March 1861. PREFACE TO THE FIRST EDITION. THE important services rendered by chronometers in the ordinary course of navigation at sea, by facilitating the daily determination of the ship's position in longitude, are well understood and fully appreciated by all intelligent seamen; but in addition to the useful ends to which, under ordinary circumstances, they are subservient in the daily conduct of the ship's affairs, they are susceptible, when placed in in- telligent hands, of being applied to higher scientific uses, and when rightly employed, are capable of affording valuable contributions towards the gradual perfection of Maritime Geography. The ordinary treatises on navigation, in use among seamen, contain ample rules and directions for the application of chro- nometers to the common purposes of determining the longi- tude at sea; but the information they afford relative to the accurate and systematic measurement of " chronometric differ- ences of longitude," or " meridian distances," is for the most part of a very meagre and insufficient character. The points involved in the discussion of meridian distances are not in themselves difficult or abstruse, and present but few impediments to those who are expert in computation and skilled in the ordinary processes of navigation ; but at the Vlll . PREFACE. same time, in order to obtain from the use of chronometers all the precision of results of which they are susceptible, and in order to treat the measurement of meridian distances in a uniform, organised, and systematic manner, many minutiae must be attended to, and many points considered, on which the usual text-books are wholly silent, and on which the ordi- nary experience of the navigator throws but little light. It may thus happen that persons without much previous expe- rience, who might be possessed of some good chronometers, and were desirous of employing them to the best advantage, would probably encounter many doubts and difficulties in the execution of their design, from the fragmentary, traditional, and irregular character of much of the knowledge at present existing on this subject. Many scattered hints, and some valuable information relative to the application of chronometers to the accurate deduction of differences of longitude, exist in many detached works, but the subject seems to require condensation on some points, amplification on others, and systematic arrangement as a whole. The object of the Author in the following pages is to attempt to remedy this existing want, and to endeavour to supply naval officers, and others entrusted with the care of chronometers, with a manual of instruction how best to use them, and how to furnish systematic results in recording the meridian distances of the several places visited during their voyages. With this end in view, the Author has freely considered, and availed himself of, several detached ideas on this subject, interspersed among various works by previous writers; among which may be enumerated, Forster's Voyage; Appendix by Tiarks. Owen on Longitude. Voyages of Adventure and Beagle; Appendix by Fitzroy. Belcher on Nautical Surveying. PREFACE. IX Raper on Longitudes, "Nautical Magazine," 1839, & c * Raper's " Practice of Navigation." Nautical Magazine, various papers ; Fisher, Bayfield, Bed- ford, &c. Memoirs Ast. Soc., Vols. III., XII., &c. Connaissance des Terns, Yols. 1835-6. Daussy sur la Marche des Chronometres. Paris, 1840. Lieussou. Recherches sur les Variations de la Marche des Chronometres, &c. Paris, 1854. &c. &c. Blending the ideas derived from these various sources with the results of his own experience, the Author trusts that he has succeeded in producing a little work which, dealing with the questions relating to chronometers in a methodical and regular manner, may be found of some assistance to those who may hereafter be inclined to undertake the measurement of meridian distances. In the execution of this design the Author has collected and arranged many detached precepts relative to the custody and management of chronometers, scattered at present among many books, or only existing in a traditional form. The questions of errors and rates, usually dismissed in books on navigation with a few brief remarks, next engage attention, and are discussed with an amplification of detail more commensurate, it is hoped, with their presumed importance. A method of combining observations for rate by the " method of least squares" is then explained, and it is hoped that, where extreme precision is sought for, the plan developed may be found useful in practice. A systematic arrangement of the formulas for meridian distances is then undertaken, in which the method of correcting for the variation of the rate, proposed by Tiarks, and followed by Fitzroy, Bayfield, and other eminent navigators, has been amply developed and pursued to its legitimate consequences. X PREFACE. Numerous examples illustrate these and the other formulae, and the work is concluded by an Appendix containing some miscellaneous matter of a useful character. In the arrangement of the formulae for the combination of observations for rate, and for the various cases which can occur in practice in the measurement of meridian distances, the Author has not deemed it necessary to translate the algebraic expressions involved into their equivalent verbal precepts, in accordance with the usual custom followed by the writers on navigation ; to have done so would have caused a considerable addition to the bulk of the book, without, perhaps, making the subject much clearer, while the comparative infrequency of calculations of this nature, contrasted with the daily use of the formulae of navigation, scarcely seemed to require such a course. The persons into whose hands these pages are likely to fall, and to whom they may practically prove useful, will doubtless, in general, be perfectly conversant with the ordinary processes of navigation, and expert in numerical and loga- rithmic computation; while in the present state of educational acquirements among naval officers, it is reasonable to assume that they will usually possess sufficient rudimentary knowledge of algebra to be able to understand, and apply with facility, the simple algebraic expressions which occur in the course of this work ; and as, moreover, there is no doubt that, after mastering the first difficulties, working from formulae is much easier and less liable to error than the blind and tedious following of verbal precepts, the Author trusts that his judgment on this point will command the assent of his readers. At the same time, mindful of the requirements of practical persons, and conscious of the necessity of vivifying the dulness of algebraic details by copious practical illustrations, the Author has been careful to explain the application of his formulae and precepts by numerous examples, for the most part taken from actual experience, and he trusts he is justified in expecting PREFACE. XI that this part of his design may be found amply to satisfy the wants of practice. Some persons may, perhaps, be of opinion, that hydrography has already attained in most parts of the world a sufficiently exact degree of precision, and that, after the correction of a few remaining gross discrepancies, further accuracy may be considered rather as the object of scientific curiosity than of practical advantage. The number of places, however, which may be considered definitely settled to a second of time, ex- cept some of the fixed observatories, are, according to Raper, very few; and, without specifying places by name, we may remark, that much remains to be done in most parts of the world towards filling up the details connected with the positions of subordinate stations ; and that, for many years to come, the leisure and opportunities of officers serving on foreign stations cannot be more usefully employed than in the accumulation of data relating to the accurate connexion in longitude of the stations they may visit during their cruises. Before concluding these remarks, the Author would wish to express his obligations to the Rev. George Fisher, Chaplain to the Greenwich Hospital Schools, for having communicated to him the valuable formula for determining the meridian distance between two stations, by means of observations, giving the travelling rates of the chronometers employed; and also to Fred. J. Evans, Esq., R.N. (his former colleague in H.M.S. " Fly," during the Australian survey), for kindly placing his manuscripts at his disposal, and for many valuable suggestions during the progress of this work. In the preparation of a work of this nature, partly treading ' on old ground, and partly entering on new, the Author cannot but be conscious of many imperfections, which must claim the indulgence of his readers ; but he trusts that the utility of his design may in some measure compensate for its defects of execution, and that his endeavours to systematise this im- Xll PREFACE. portant subject may prove useful, not only to the scientific members of his own profession, but also to intelligent officers in the Mercantile Marine; and in this hope, he confidently commits his labours to the critical judgment of the public and to the favourable consideration of his professional readers. CONTENTS. CHAPTER I. PAGE Introductory Observations ....... i CHAPTER II. Reception and Stowage of Chronometers on board Ship Effects of Magnetism Effects of Temperature Influence of other Causes on Chronometers Winding up Chronometers Standard Chronometer Distinction of Chronometers Comparison of Chronometers Chronometer Journal, &c. Assistant Watch Miscellaneous Observations . . 10 CHAPTER III. On the Determination of Time Transit Observations Sextant Observations Artificial Horizon Sextant Methods of Observation Equal Altitudes Single Altitudes Com- parisons -with the Standard Position of the Place of Observation Reduction of the Observations Precon- certed Signals . . . . . . . -43 CHAPTER IV. On Rating Chronometers, and on the Determination of their Errors on Local Mean Time Simple Method ; by Dupli- cate Observations of a similar Character, taken at a conve- nient Interval 67 XIV CONTENTS. CHAPTER V. PAGE On Rating Chronometers, and on the Determination of their Errors on Local Mean Time, by the Combination of several Observations of a similar Character, taken within any con- venient Interval 82 CHAPTER VI. On the Chronometric Determination of Meridian Distances, and on the Method of Allowing for the Change of Rates of the Chronometers, that may have taken place in the Intervals between the Observations Methods of De Cornulier, Lieussou, Mouchez, Hartnup, Tiarks Remarks on these Methods . i oo CHAPTER VII. On the Chronometric Determination of Meridian Distances Tiarks' Formulae Combination of Mouchez's Plan of Cor- rection for Temperature with Tiarks' Method Compari- son of Flinders' and Tiarks' Methods Methods ofVincen- don Dumoulin, Coupvent Desbois, and Charles Ploix. . 150 CHAPTER VIII. Various Examples of the Computation of Meridian Distances, illustrating the Application of Tiarks' Formulae given in the preceding Chapter . . . . . . .172 CHAPTER IX. On the Determination of the Meridian Distance between two Stations, by means of Observations, giving the " Travelling Rates " of the Chronometers employed . . . 195 CHAPTER X. On the Mode of Recording the Results of Chronometric Mea- surements ........ 203 CONTENTS. XV APPENDIX. PAGE Table for converting Intervals of Time, or Longitude, into Decimals of a Day . . . . . . .213 Summary of Instructions for the Management and use of Chro- nometers . . . . . . . . 2I 5 Form No. I. Chronometer Journal for the Record of Daily Comparisons of Chronometers . . . . . .218 Form No. I. (A). Chronometer Journal for fair Copy of Com- parisons, if thought necessary . . . . . ,219 Remarks on the Forms Nos. I. and I. (A) for Chronometer Journal 220 Form No. II. Return of Observed Errors and Rates of Chro- nometers . . . . . . . . . .221 Form No. III. Record of Meridian Distances Measured . 222 Form for the popular Exhibition of the Results of Meridian Distances ......... 223 Table exhibiting the Rates of the Chronometers of H.M.S. "Fly," employed Surveying on the Coasts of Australia, during a period of four years, from March 1 842 to April 1846 224 Remarks on the preceding Table 226 NOTES MANAGEMENT OF CHEONOMETERS AND THE MEASUKEMENT OF MEEIDIAN DISTANCES. CHAPTER I. INTRODUCTORY OBSERVATIONS. >^ 0^ THE ^ 'UNIVERSITY: THE application of chronometers to the accurate determination of " Meridian Distances," or the differences of longitude of distant stations, has usually formed an important object in the scientific voyages of modern times, from those undertaken in the last century by the illustrious Cook, to those performed in our own day by more recent navigators. The happy invention of the Electric Telegraph, the successful accomplishment of its submarine connexion, and its application to astronomical purposes, would seem to have completely and successfully solved the problem of differences of longitude of stations which are either situated on the same continent, or, if occupying insular positions, only separated by narrow seas ; and there can be little doubt that, before long, the various observa- tories in the British Islands and on the Continent will, by its means, be accurately linked together, and their relative positions consequently determined to the last degree of mathematical correctness. Considered as base stations, they may then be viewed as forming salient points in a network of triangles described on the surface of our globe, to which minor places B 2 ON THE MANAGEMENT OF CHRONOMETERS. can afterwards be conveniently referred by geodetic means, so as ultimately to combine them in one comprehensive whole, hitherto unexampled for accuracy in the annals of geographical science, From the very nature of its-- invention, however, and from the probable limits to its use, interposed by the difficulties of its submarine connexion, except in narrow seas, the application of the Electric Telegraph to the question of Terrestrial Longitude will doubtless be comparatively very limited ; and it must still be to the successful appliance of the ordinary means at the disposal of the navigator that we must continue to look for the final solution of questions relating to the relative longitudes of outlying stations on the ocean, and their connexion with the fixed points on the great continents. Fortunately, at a period when the successful application of the system of galvanic signals to astronomical purposes has given a great impetus to the final solution of the questions concerning the relative longitude of stations on land, the gradual improvements effected of late years in the construction of marine chronometers, and the yearly increasing extension of the appli- cation of steam to ocean navigation, seem at the same time to afford increased facilities to the Navigator for the improvement of Maritime Geography, as he will thus be enabled also to maintain an honourable rivalry with the Astronomer, and, like him, to contribute his fair share towards the ultimate perfection of geographical science. In examining the earlier history of modern hydrography, and on inquiring into the circumstances which have hitherto impeded its progress towards final perfection, notwithstanding the zealous and useful labours of numerous scientific navigators, and the voluntary contributions of many intelligent commanders, two causes will, we apprehend, be found to have tended, although accidentally, to retard its satisfactory development; first, the practice of mixing up in one indiscriminate combination the astronomical data for the positive settlement of disputed positions, and the relative evidence afforded by chronometric measure- ments; and, secondly, a want of clearness of comprehension of the relative values of absolute and differential longitudes. For instance, nothing is more common than to find, on examining even comparatively recent works on hydrography, INTRODUCTORY OBSERVATIONS. 3 that the data quoted for the settlement of a given position are of the most miscellaneous and incommensurable character: lunar observations, eclipses of Jupiter's satellites, occultations of stars by the moon, solar eclipses, and chronometric measurements by various authorities, from adjacent and independent points, all blended together in one crude and inharmonious result ; or again, than to find chronometric determinations of a purely relative character, and often measured from two or more independent stations, confounded with absolute results.* Places are fixed absolutely by astronomical observations, or relatively by chronometers. This distinction must be clearly kept in view if ever we wish to arrive at final and conclusive results, and if we desire to avoid the perpetual oscillation of ideas which a mixture of the two principles is sure to entai on us. Much has been done of late years towards simplifying the conclusions of maritime geography, by collecting from the records of astronomical and chronometric observations satisfactory details for the final establishment of several important fundamental positions. M. Daussy in France, and Mr. Raper among our own countrymen, have specially distinguished themselves by the prominent part they have taken in this useful hydrographic labour. The valuable contributions of the former writer on this important subject will be found interspersed among the pages of the " Connaissance des Terns," especially in the volumes for 1835 and 1836; and the development of the views of the latter author in various detached papers given in the " Nautical Magazine" for 1839 and subsequent years, and also in the remarks accompanying his " Table of Maritime Positions," published in the " Practice of Navigation." Raper's views on this important subject are so clear, and his remarks so apposite, that, with the view of enforcing the system that he recommends, we shall take the liberty of quoting largely from them, feeling conscious that no independent language of our own can place the subject in a more forcible light, or more strongly illustrate the necessity of its adoption. In * Horsburgh's " East India Directory," valuable work as it undoubtedly is, contains numerous illustrations of this mode of proceeding. 4 ON THE MANAGEMENT OF CHRONOMETEES. fact, we would wish it to be understood, that one principal object which we have in view in the preparation of these pages, is to facilitate the practical adoption of the plan that he recom- mends, and to systematise the determination of chronometric differences of longitude, by arranging, for the benefit of naval officers and others, the details of the operations in a uniform, organised, and consistent manner. Raper's remarks on this subject are as follows: "Pre- viously to Cook's voyages, which may be considered as the commencement of modern hydrography, the only method (besides the rude and imperfect determination of the ship's run) of obtaining the longitude of every new land made, was the lunar observation. But as that method, from its inaccuracy,* fails altogether in exhibiting truly relative positions, chro- nometers were employed in combining together the results of observations taken at different places, of which numerous instances are recorded by Horsburgh in his e East India Direc- tory.' Since, however, the observations made at two places are not in general equally good, this method of combining observa- tions with chronometric differences has the disadvantage of impairing the better determination of the two, and in consequence throws a difficulty over the connexion of either of them with a third place better known. Succeeding navigators proceeding in the same way have obtained other results of observation, and other chronometric differences ; and, in consequence, the hydro- * The first edition of this work here contained the following note : " Notwithstanding the favour with which lunar observations have been regarded by some navigators, the important service they doubtless rendered in approximately fixing positions during the earlier periods of modern hydrography, and their still inestimable value at sea, as affording to the seaman the only available independent method of determining his absolute longitude ; yet we fear, that in the present con- dition of hydrographic science they are quite unsuited to the settlement of disputed positions. In all observations for longitude connected with the moon, the errors of observation are multiplied in their effects on the resulting longitude by a factor whose mean value is about 30 ; consequently an error of 10" in a lunar distance (and we presume that, under the most favourable circumstances, we have no right to expect less) becomes 300" or 5' in the resulting longitude deduced from it : and this, be it observed, is wholly independent of the additional errors it may be liable to, depending on the moon's place in the tables, owing to the still imperfect state of the lunar theory." (See also Raper's remarks on this subject, "Naut. Mag." 1839, p. 3I9-) The opinions expressed above may now, perhaps, require some modification, in so far as regards the accuracy of the lurar tables. From and after the year 1862, the lunar calculations in the " Nautical Almanac " will be based on Hansen's new and INTRODUCTORY OBSERVATIONS. 5 grapher who has not the means afforded him of instituting a critical examination of the several positions, or of their connexion with each other, is driven to the necessity of taking a mean between each new result and those adopted from former navi- gators, and thus the whole mass of positions is kept in a state of perpetual fluctuation, from which it is impossible that universal precision can ever be obtained. " In marine surveys again, different meridians have been assumed, and different longitudes of the same meridian. In some cases the longitude of the meridian assumed has not been given ; in others, the meridian itself has not been specified at all. " If, however, instead of thus throwing open the discussion of every place at each new voyage of discovery, or surveying expedition, and unsettling all that had previously been done, without any assurance that the new series of positions would not in its turn be unsettled again, navigators and hydrographers would agree to consider, for the time being only, certain impor- tant stations as already established in longitude, whether reallv so or not, with the view of referring all the subordinate positions to them, the indistinctness which now hangs over absolute and relative positions would be forthwith cleared up. The question would be narrowed into the determination of chronometric differences alone, until favourable opportunities occurred for the definitive determination of a fundamental position. Accu- rate chronometric measures would be no longer lost to the improved tables, in the construction of which the highest resources of analytical talent, and the profoundest mathematical ability, have been successfully employed. A very high authority has recently declared, that " the residual errors of the lunar theory are now reduced to almost insignificant limits, certainly within the errors incidental to meridional observation with first-rate instruments, and very far within the limits of accuracy of observation with the sextant ;" that " Hansen's tables are practically perfect for all the purposes of navigation, and that the great nautical problem of finding the longitude at sea is now completely solved." Under these improved and altered circumstances, it is possible that a careful series of lunar observations, for the determination of the longitude of any specific station, might repay the labour bestowed on them. By making the series sufficiently numerous, embracing an equal number of observations on each side of the moon east and west and taking them with every regard to accuracy, the instrumental errors of observation would be either eliminated or very materially reduced. Such a series of observations, however, would require very careful reduction, attention to the corrections for the earth's spheroidity, and taking into account the barometric and thermometric corrections to the mean refraction, due to the actual state of the atmosphere at the time of observation. 6 ON THE MANAGEMENT OF CHRONOMETERS. world by being merged in the uncertain results of a few astronomical observations ; and the labours of each navigator would always maintain their proper value, instead of being set aside, as they must inevitably be, on the appearance of a new survey, in which the data are exhibited in a distinct form. The works of different navigators, and of the navigators of different countries, could be brought into immediate com- parison, a task which is at present often difficult and unsatis- factory, if not impossible. The labours of the hydrographer would be materially simplified; and as the points to which inquiry should next be directed would by this system be dis- tinctly brought into view, the whole subject would advance steadily to its ultimate perfection."* In furtherance of these views, it was accordingly proposed (" Naut. Mag." 1839, p. 399) "to adopt certain points under the name of Secondary Meridians, this general term being used to distinguish them from the prime Meridians, as Greenwich, Paris, &c., from which the longitudes in the tables or on the charts must be reckoned. The number of these stations at present proposed to be employed is twenty, but this is altogether matter of convenience, and will vary with the progress of the subject. The points selected are so far distant from each other that the errors of their relative positions could not be easily discoverable by the ship's chronometers ; and they themselves must ultimately depend on astronomical observations, of which it is important to remark, the number necessary for an unim- pugnable determination appears to be very great." A list of ,the Secondary Meridians selected by Raper, with the districts for which they are intended generally to serve, and their adopted longitudes, will be found in the " Practice of Navigation," p. 380. "The method of surveying by chronometers alone, to the exclusion of astronomical observation," observes Raper in continuation, "has already been extensively adopted; as in Smyth's surveys in the Mediterranean, Owen's surveys on the coast of Africa, Fitzroy's in South America, &c. The principle advanced, therefore, is not new, but in the present state of hydrography it is important to urge the necessity for making astronomical determinations a totally separate consideration, and to suggest the advantage of a common recognition * " Practice of Navigation," pp. 379, 380. INTRODUCTORY OBSERVATIONS. 7 of fundamental points in the arrangement of chronometric differences." Many concurrent circumstances of a favourable nature, and peculiar to the present time, seem to be conducive to the more systematic application of chronometers on board ship to the objects of science in the measurement of differences of longitude. The mechanical construction of chronometers has attained a high and unexampled degree of perfection, these improvements having at the same time been accompanied by a very con- siderable reduction in their cost ; so that chronometers are now no longer rare instruments only within reach of the wealthy ; and in lieu of one solitary chronometer as formerly, it is now not unusual to find three or more good chronometers on board every ship. The more general diffusion of a good practical education among young officers, both in the royal and mercantile navies, at the same time renders them more capable of applying chronometers to the accurate purposes of science and better able to appreciate their results. The increasing application of steam machinery, moreover, to men of war, whereby the average duration of passages will be much shortened and return voyages greatly facilitated, and the extension of lines of ocean steam navigation by the great mercantile companies to all parts of the globe, seem to afford, simultaneously with the above-mentioned circumstances, increased facilities for the systematic and careful measurement of chains of meridian distances. If this view of this important question be correct, and the writer's partiality for a favourite subject has not caused him to take a too favourable view of present circumstances, it is important to consider whether the time has not arrived when it might be advisable to attempt to concentrate and condense in a practical form the necessary instructions for the guidance of those, who may be desirous of undertaking the accurate and systematic measurement of chronometric differences of longitude. The information contained on this subject in the ordinary treatises on navigation is very meagre and unsatisfactory; not that this circumstance is to be imputed to them as a fault, for in truth the scientific deduction of chronometric differences forms no part of the ordinary processes of navigation, and information relating thereto might more justly be sought in a 8 ON THE MANAGEMENT OF CHRONOMETERS. work especially devoted to that subject. No such treatise, however, at present exists; many detached hints are extant in a fragmentary form, or interspersed in isolated papers among many books. The various volumes of the " Nautical Magazine," the appendices of several scientific voyages, some papers in the ee Connaissance des Terns," and numerous pamphlets, may be quoted in illustration; and, in addition, some amount of floating knowledge exists in a traditional form as the results of expe- rience, and is handed down from time to time, and from ship to ship, as one generation of officers is succeeded by another. Speaking generally, we apprehend that, unless from pecu- liarly favourable circumstances of previous service in a surveying ship, on a scientific voyage, or under a scientific chief, officers in general have no organised knowledge of the minutiae and details requisite to be attended to in the accurate measurement of meridian distances. No single book supplies the required information, and an officer furnished with some good chro- nometers, and desirous of advantageously employing them, has probably to grope his way as best he may, and devise a system for himself. This want we propose to endeavour to remedy. With this object in view, in the following pages we shall endeavour to collect together and arrange various hints relating to the custody and management of chronometers on board ship, at present existing only in a traditional form, or to be found scattered amid many books often not easily accessible. The questions relating to errors and rates next engage attention ; and subsequently, the formulae for the determination of meridian distances are dis- cussed and arranged in an organised and systematic manner, worthy, it is hoped, of the present advanced state of hydrographic science. Amid much that cannot be new, and probably much that is familiar, the scientific reader may yet, we hope, find some fresh matter worthy of attention, and much that is old presented in a new or more instructive form. The utility of chronometric determinations, and the value of their results, as well as the possibility of their advantageous incorporation with the previous labours of others, much depends on the degree of care bestowed on the minute details of their manipulations. Many series of observations have often had their value materially impaired by INTKODUCTOKY OBSERVATIONS. 9 the accidental neglect of some small particulars ; and doubtless a few measurements executed with a due regard to accuracy of detail, are many times more valuable than much larger masses of observation reduced and recorded in a loose, uncertain, and unsystematic manner. Chronometric determinations obtained with no special regard to accuracy, reduced approximately, and recorded vaguely, although perhaps formerly valuable contributions to our then knowledge, are unsuited at present to the existing condition of maritime geography. fi We are also to bear in mind," observes Raper, "that the ultimate perfection of hydrography demands very different proceedings from those which have sufficed to collect together the first rough materials of the outline ; and," continues the same writer, " it can evidently only be effected by the chronometric measurement of small distances, finally depending on unimpugnable astronomical observations."* As a contribution towards the improvement of geographical science, and in furtherance of the development of the above views, these pages have been undertaken: and if in the hands of scientific officers or intelligent travellers they are found at all conducive to the systematic realisation of this important subject, they will not have been written in vain, or the author's labour lost. * " Naut. Mag." 1839, p. 320. UNIVERSITY 10 ON THE MANAGEMENT OF CHRONOMETERS. CHAPTER II. RECEPTION AND STOWAGE OF CHRONOMETERS ON BOARD SHIP EFFECTS OF MAGNETISM EFFECTS OF TEMPERATURE INFLUENCE OF OTHER CAUSES ON CHRONOMETERS WINDING UP CHRONOMETERS STANDARD CHRONOMETER DISTINCTION OF CHRONOMETERS COMPARISON OF CHRONOMETERS CHRONOMETER JOURNAL, ETC. ASSISTANT WATCH MISCELLANEOUS OBSERYATIONS. Reception and Stowage, When chronometers are received on board ship, it is of importance that they should be at once stowed in the place prepared for their reception, in the position which it is intended they should permanently occupy ; and when once suitably located, they should on no account be subjected to subsequent removal or displacement. The possible contingency of the ship being docked for extensive repairs, in which case their temporary removal would be unavoidable, is the sole exception to this general rule, excluding, of course, the accidental necessity for sending any particular chronometer on shore for the purposes of repair. In selecting a place for their reception, much must of course depend on the size and accpmmodation of the ship, the par- ticular nature of the service on which she is employed, and on the guidance of other circumstances; but, if possible, under favourable auspices, the following general conditions should be attended to: The chronometers should be placed low down in the ship (both because there is there less motion, and because the tempe- rature is more equable); amidships; as far from the extremities, and as near the centre of motion, as convenient : not near the chain cables or other large masses of iron, so as to ensure freedom from the possible disturbance of magnetic influence ; not in drawers, where the tremor caused by opening and shutting them acts injuriously on their balances, nor suspended from the deck RECEPTION AND STOWAGE. 1 1 in cots or swinging tables, which has been proved by experience to be objectionable.* The master's cabin in men-of-war is also objectionable, because there the chronometers are not amidships. So also the casing of the rudder-head in the captain's cabin, where we have known them to be placed, and where, doubtless, they looked very ornamental and scientific, is likewise to be deprecated, because they are there subjected to the more violent motion of the extremity of the ship, to the possible vibration of the rudder, and to the injurious influence of draughts of cold or damp air from the stern- windo ws : we believe, however, that this practice has now very properly been discontinued. In vessels fitted with screw-propellers, the chronometers should be placed sufficiently far forward, as not to be affected by the vibration of the screw when under steam. In Her Majesty's ships f a suitable place is now generally fitted for the reception of the chronometers in the after-cockpit or orlop-deck. In frigates, the after-part of the steerage, while unobjectionable in other respects, would appear to be both acces- sible and convenient. In flush-deck vessels the fore-part of the captain's cabin, amidships, would seem to be the most appro- priate place ; and in merchant-ships, the master's cabin, amid- ships, and as far forward as possible, perhaps offers the only eligible position. * In the scientific voyage of the Chanticleer, under Capt. Forster, the chrono- meters were suspended in this manner. Tiarks is of opinion that the fact of several of them not retaining their rates for any length of time, and repeatedly altering them to a considerable amount in short intervals, is attributable to this cause ; and he adds, that his own experience likewise proves, that suspension from the upper deck of a vessel is not favourable to the regular going of time-pieces. (Forster's "Voyage," vol. ii., Appendix, p. 226.) Capt. Owen remarks, that when suspended, they are liable to receive a vibratory motion that may affect their balance-wheel, besides being exposed to shocks ; and, further, that the very necessity for taking them down, while cqpaparing or winding up, is a sufficiently strong objection to the practice. (" Owen on Longitude," p. 6.) To this I may add, that the Rev. Mr. Fisher informs me that he has found, in some experiments made by him on land, the rates of chrono- meters sensibly affected by placing them on swinging tables ; and if this be the case when perfectly quiescent in a room on shore, what might we not expect on board ship ? f It may be as well to state here, once for all, that the observations and precepts throughout these pages have been chiefly written with a view to the guidance of officers of H.M. Navy, and with regard to the arrangements of men-of-war. It is hoped, however, they may not be without their use to intelligent officers in the Mercantile Marine. 12 ON THE MANAGEMENT OF CHRONOMETEES. The best mode of stowing them seems to be as follows : A box, divided into as many partitions as there are chrono- meters to be stowed in it, should be securely attached by screws to a solid block* of wood, about thirty inches in height, and firmly bolted to the beams of thg deck below. Each partition should in depth be about equal to that of the largest box of the chronometers to be placed in it, and in length and breadth about two inches longer than the sides of the box it is intended to receive ; the partitions, moreover, should be furnished either with separate lids, or there should be one general lid to close the whole box. Great care should be taken that the block and partitioned box thus prepared should be entirely detached from all contact with contiguous stanchions or bulkheads; and the block and box, moreover, should be surrounded with a strong external casing, the sides and lid of which should on no account be permitted to touch it, a clear space of at least two inches being left all round. Previous to placing the boxes containing the chronometers * During the voyage of the Beagle the chronometers were placed in box parti- tions, and packed in sawdust, as described above, " upon one of two wide shelves placed in a suitable position in the ship, low down, and as near the vessel's centre of motion as could be contrived." Admiral Fitzroy states, that " thus placed, neither the running of the men upon deck, nor the firing of guns, nor the running out of chain-cables, caused the slightest vibration in the chronometers ; " and he seems to give a preference to this mode of stowing over that of on a solid block ; but as, after all, the chronometers can only be defended, do what you will, from the effects of vibration within the ship, and cannot be secured from those arising from the concussion of the ship herself, if she strike the ground, we do not see the occasion for this preference. Capt. Owen recommends (" Owen on Longitude,'' p. 6,) as the most unobjec- tionable method, " that of having a table hung on gimbols, with a weight of from 20 Ibs. to 50 Ibs. suspended underneath, in the manner of an azimuth compass, so that the centre of gravity be as near the centre of its motion as possible, to permit it to keep its level permanently without being subject to vibrate, and the axes of the gimbols working in smooth stuffed-leather sockets bearing against springs in every direction, these springs being neither too stiff nor too sensible for the weights they are to support. The pillar or stand for the sockets to be a fixture in the deck. Such a contrivance would go far to prevent the ill effects of the ship's motion or concussion from firing guns, &c. The chronometers to be placed in small partitions on the table, and to be wedged securely in their places by soft cushions. Should the above method not be resorted to, and it might not always be practicable, he further recommends in lieu " a table well secured to the deck, and kept unconnected with the adjacent bulkheads, the chronometers being placed and secured on its top as above directed." Capt. Barnett adopted a table supported on a pivot like a compass-card ; the j arring effect being relieved by a spring inside the pivot. EFFECTS OF MAGNETISM. 13 within the partitions appropriated for them, it will generally, we apprehend, be found convenient to unscrew and detach altogether their lids, because on account of the manner in which they are usually fitted ; * if this be not done, when open they occupy much more room, and will require the partitioned spaces for their reception to be made inconveniently large. No detriment to the chronometers need be apprehended from this removal of their lids, as they will subsequently be sufficiently protected from the chance of injury by the covering lids which close the partitions. Each chronometer in its box thus prepared, and moving freely in its gimbols, is then to be placed in the space allotted to it on a bed of horsehair, cotton, or shreds of bunting, about three inches thick, the interstitial spaces around its sides being stuffed with the same material to within half an inch of the top of the inner box. Of the three substances named above, we prefer horsehair, f Generally speaking, it is not the custom to receive the chronometers on board the ship until a few days before leaving the harbour, the time of the officers who will subsequently be charged with the duty of superintending their performances being at that period much occupied with other important matters con- nected with the equipment of the ship ; but we think, that as from the influence of the new circumstances under which the chrono- meters are placed, the effects of motion in removing them, change of temperature, and possible action of magnetic causes, their rates after a while may differ from their previous rates on shore, it is very advisable, when practicable, that they should be received on board at an earlier period, so that they may become naturalised in their new position, and may have settled down to a stability of rate under their new conditions before the ship is called on to proceed to sea. Effects of Magnetism. Undoubted instances are on record of the performance of chronometers being affected by the action of * Sometimes the upper part of a chronometer-box is fitted with a glass-plate, secured from injury by an additional hinged or sliding lid, so that the dial-plate may be viewed without opening the box ; but this arrangement is not favourable to accuracy of comparison, on account of the reflexion and loss of light in viewing the dial-plate through two glasses, and from the possible error of parallax, which may deceive the eye. f Coarse, dry sawdust has also frequently been used, but as it is liable to absorb moisture, and then to cake, harden, and lose its elasticity, it is not to be recommended. 14 ON THE MANAGEMENT OF CHRONOMETERS. magnetic influences, chiefly owing to the fact of the balances having accidentally acquired polarity.* Since attention has been called to this subject and greater caution exercised in the con- struction of the balances and balance-springs so as to avoid this source of error, instances of this kind are no doubt now extremely rare, and quite exceptional, in modern chronometers of good construction.! We should, however, be taking a very limited view of the subject, were we solely to confine our attention to the consideration of permanent polarity in chronometer balances, and lose sight of a possibly more prolific source of error, which, in all probability, is caused by the induced magnetism of considerable masses of iron on ship-board, in various stages of development, varying in different positions of the ship, and change of magnetic latitude. Owing, no doubt, to the difficulty of arriving at definite con- clusions on a matter so subtle and minute, as the influence of * The earliest account we have of the effects of magnetism on chronometers seems to be that of Mr. Varley (reported in the " Philosophical Magazine," vol. i. 1798), who discovered that the balance acquired polarity at two opposite points on the rim, and thus that the going of the time-piece was affected by the position of these poles with respect to the magnetic meridian. Varley, moreover, found that every new balance which he tried was already more or less polarised. In 1821, Professor Barlow, who does not appear to have been acquainted with Varley 's previous inquiries, made a very complete series of experiments, showing that the vicinity of masses of unmagnetised iron invariably affected the rate of chronometers placed near them ; no doubt from the accidental polarity of their balances. Mr. Fisher also found (" Naut. Mag.," 1837, p. 160) that the rates of chrono- meters were sensibly affected by altering the position of the marks on the dial-plate with reference to the magnetic meridian. Mr. Wackerbarth has recently pointed out a very curious instance of faulty per- formance in a chronometer, caused by the steel screws, with which the balance-wheel was loaded, having become sensibly magnetic. These screws being removed and replaced by others, the chronometer afterwards kept its rate very steadily. (" Monthly Notices, Royal Astronomical Society," vol. xix., p. 222.) f In a letter to the Hydrographer to the Admiralty on the liability of chr. = time by watch of the P.M. observation, or f. Hence reducing f=(i2 h -0 + a(x- i). Ex. Suppose, in the last set of observations quoted above, that the approximate error of the watch was 18 slow on mean time, and the equation of time io m additive to mean time, Then, h m \ h h m t =9 i j ia-f =z 59 x = 18 *(* ) = 5 6 Hence, if the observer had been at his post, at say i h 55 P.M. by his watch, he would have been in time enough. The formula is equally applicable to the case of P.M. and A.M. observations, if t represents the first, or P.M. observation, and t' the last, or A.M. one. Ex. (Raper, " Practice of Navigation," p. 274) : h m \ h li m t = 5 ai ) 1^ t = 6 39 ;. = + i 55 / 2 (A i) = + 4 18 8 - - '4 f= 7 which agrees with the recorded observations. (See Jeans' " Navigation," &c. p. 189.) SINGLE-ALTITUDE OBSERVATIONS. 57 limbs at any convenient division whatever, an equal number of observations of each limb being made as rapidly as circum- stances permit. In the two latter cases the mean of the altitudes observed will be the mean altitude of the sun's centre, corresponding to the mean of the respective times, all reference to the sun's semi- diameter being eliminated ; and in the former case, the mean of the observed altitudes will be the mean altitude of the limb observed, corresponding to the mean of the respective times as before. Either of these three methods of observing seems preferable to the practice of making the contacts by moving the tangent- screw up to the instant of observation, because there is always a tendency to error from the spring or elasticity of the index- bar;* and, moreover, as Raper observes, moving the tangent- screw diverts a portion of the attention which should be devoted to the contacts alone. When only one limb is observed, it has been recommended that the lower limb should be observed when the observations are taken in the forenoon, and the upper when they are taken in the afternoon; because the images of the sun are then receding, and the contact of the limbs, at the moment of sepa- ration, can be more correctly observed than when they are approximating. Since, in nice observations for time, the corrections to the observed altitudes for refraction should be accurately applied, the state of the barometer and thermometer at the time of observation should be carefully noted. Even when "equal altitudes " are taken, this precaution should be adopted, as from failure in obtaining the return-sights in the afternoon from cloudy weather, or other causes, the possibility of having subse- quently to reduce the forenoon observations as single altitudes should always be contemplated. * The error from this cause is often different for the onward and for the backward motion of the index. As a remedy for this, it has been proposed that all observa- tions should be taken with the same motion of the index-bar. " The onward motion being adopted as the most natural, the tangent-screw is always employed to close the object and the reflected image, and is thus always turned in the same direction." (" Practice of Navigation," p. 164.) When, however, circumstances permit, as in the observations under present discussion, it seems preferable to avoid using the tangent-screw altogether in making the contacts. 58 ON THE MANAGEMENT OF CHRONOMETERS. Circumstances may occasionally render it more convenient to obtain the time by observations of stars at night, rather than by altitudes of the sun by day. In this case, equal-altitude observations are rarely practised, on account of the inconve- nience of having to wait so long ,"at unseasonable hours for the duplicate observations west of the meridian. When, however, they are resorted to, the observations determine the time with much precision, since the declination of the body observed being invariable, there is no equation of equal altitudes to be applied. Independent altitudes of stars of nearly equal declina- tions, and nearly equidistant from the meridian, both east and west, are more usually had recourse to ; the best mode of taking them being by clamping the index a little in advance of the true (double) altitude, when the body is east of the meridian, and a little in defect of it when west, waiting in both cases till the motion of the star in altitude produces a contact ; the time is then noted, and the instrument read off; the opera- tion being repeated any convenient number of times to neutralise the small errors of observation. The best way of appreciating the moment of contact is by slowly moving the sextant round the axis of vision, making a sweep with the reflected image of the star, until it is observed exactly to coincide with the real image as it passes over it. Comparisons with the Standard. Since, as we have before remarked, the chronometers are on no account ever to be removed from their box in the " chronometer-room," the time corresponding to all observations made on shore is always to be taken in the first instance by the assistant-watch, and the corresponding indication of the " standard " at the same moment subsequently deduced by means of the comparisons. As the standard chronometer and the assistant-watch will rarely have the same rate, the comparisons before and after may exhibit an appreciable difference ; and in this case the required comparison at the moment of observation must be obtained by interpolation, on the supposition of an equable variation in the rate of change.* Even supposing the whole or a part of the variation to have been owing to real irregularity in the going of the assistant, to which even good watches are sometimes prone, when subjected to motion and careless usage, it is best to adopt this idea, as it is, of course, * See ante, p. 37. COMPARISONS WITH THE STANDARD. 59 impossible to say at what precise moment, whether before or after the observation, that the irregularity, or jump* took place. This plan, moreover, at any rate, distributes the effect of the irregu- larity over a larger period, and, in the absence of any direct knowledge on the point, is free from the objection of an arbitrary evasion of the difficulty. When equal-altitude observations are taken, the " assistant " should, therefore, be compared with the standard, both before and after the observations A.M. and P.M. ; the corresponding times shown by the standard at the moments of observation are thus obtained : the mean of these times, when corrected by the "equation of equal altitudes," gives the time shown by the standard at the moment of apparent noon ; and from this time and the " equation of time " the error of the standard, on mean time at the place at apparent noon, is at once obtained. In order to obtain the errors of the other chronometers, it is customary to get a comparison with the standard all round, about the time of apparent noon ; and these several comparisons being then applied to the error of the standard, as pointed out before (see ante, p, 33), will give the respective errors of all the other chronometers at that instant. When the time is obtained by single altitudes, the indication of the standard at the moment of observation is in like manner obtained from its comparisons with the assistant. In many ordinary cases, including the determination of the longitude by chronometer at sea, when no very rigid accuracy is required or sought for, a single comparison with the standard, either before or after the observations as may be convenient, will suffice to give its time with sufficient precision ; and in the case of fore- noon observations, the usual daily 8 A.M. comparison at the time of winding will, no doubt, be all that is necessary for all practical purposes ; and a similar plan will answer for the several other chronometers when required.f If, however, the * There is reason to suppose that the "jumps" which sometimes take place in watches and chronometers, and which are often mysterious and perplexing, proceed from defects in the jewels on which the pivots of the movable parts work. (" Rech. Chron." p. 281.) f When a ship is furnished with several chronometers, as it would be trouble- some throughout a long voyage to deduce the longitude from each of them daily, it will be quite sufficient to keep the daily reckoning by the " standard " only. The indications of the other chronometers, from their known errors and rates, may then be obtained, for the purpose of comparison with the standard, once a-week, and their 60 ON THE MANAGEMENT OF CHRONOMETERS. time by the several chronometers is accurately wanted for some special purpose, this approximate mode of proceeding will not be sufficiently exact, and it will be advisable to get a round of comparisons of each of the chronometers with the standard, both before and after the observations, so as to obtain by interpolation the exact comparisons at the required moment. In the case of forenoon observations, the usual morning com- parison at the time of winding may always be employed as one of them. When it is proposed to determine the time by " equal alti- tudes," it may frequently happen that the observer fails to obtain duplicate sights in the afternoon, and hence the forenoon sights, and also, perhaps, any independent ones that may be obtained in the afternoon, have to be subsequently reduced as single altitudes ; in this case, the required comparisons of the several chronometers with the standard, at the moment of observ- ation, may be obtained for the 'forenoon observations, by inter- polating between the comparisons with the standard, at the time of winding and noon, and those for the afternoon sights from the noon comparison, and a special one made after the completion of the afternoon operations on return on board, say at about 4 P.M. The system of double comparisons and the consequent interpolations here recommended, may appear at first sight very troublesome, and almost unnecessary ; but where in deli- cate measurements rigid precision is sought for, we are per- suaded that it will, in the long run, be deemed much more satisfactory than being obliged boldly to assume, from time to time, that the comparisons have not altered during long inter- vals; or than being compelled to interpolate for intermediate values between two consecutive daily comparisons, which will otherwise be the only alternative. Position of the Place of Observation. In selecting an appro- priate place for observations for time, much must often depend on circumstances at the moment, over which, perhaps, the choice of the observer has little or no control. If the place is a well- individual differences from the indications of the standard at that time may after- wards be applied as corrections to its results during the ensuing week, in case their values of the longitude are required to be known. In a similar manner, the weekly value of the reduction of the standard to the " general mean " may also be obtained. POSITION OF PLACE OF OBSERVATION. 6 1 known station, often visited by other navigators, and recognised in published documents as a hydrographic position, it seems advisable, when practicable, that new visitors should make their observations at the same spot, which has already been selected by previous observers. There is no advantage whatever in the unnecessary multiplication of sites of observation; on the con- trary, the practice tends to great confusion, and interposes diffi- culties in the comparison and incorporation of the results obtained by different navigators. If the spot of observation used by former observers be not known, or for some sufficient reasons not then conveniently accessible, then each new visitor may be at liberty to choose a place at his own convenience ; and in this case it is of importance that he should be careful to describe its situation accurately, and especially to give its true bearing and distance from, or the reductions in, latitude and longitude, which may be necessary to connect it with some salient point, whether natural or artificial, used in the description of the place by former voyagers, or adopted in published plans or charts, and therefore easily recognised : such as a Peak, Point, Rock, Flagstaff, Light- house, or Church, &c. &c. If the place visited be a new station, it is desirable that the place of observation selected should be generally convenient, and easy of access, and that it should be accurately described by a reference to such natural or artificial objects as exist in the vicinity, so that subsequent visitors may have no difficulty in recognising the place again from its published description. The latitude of the place of observation should be known accurately, because it is a direct element in the computation of the observations for time. The longitude of the place of observation is required approximately, as an element in the reduction of the data required from the ee Nautical Almanac ; " such as the (S sun's declination," the " equation of time," &c. If the latitude of the place be not known, it must be deter- mined. The best and most practical way of obtaining it speedily, and with facility, is by the meridian altitudes of stars observed in the artificial horizon. If an equal number of stars be ob- served north and south of the zenith, this mode is susceptible of great accuracy, as the errors of observation, both those which arise from defects in the instrument and those which are per 62 ON THE MANAGEMENT OF CHRONOMETERS. sonal to the observer, have a tendency to neutralise one another, and a skilful observer may thus in one night often obtain his latitude with a great degree of precision. The approximate longitude, it not otherwise known, may be obtained sufficiently exact -for the purposes required, by bringing it forward by the chronometer from the last station left. If any published plan or chart of the place exist, issued by competent authority, and of established reputation, no doubt the latitude and longitude of some prominent station in the plan will be given in it, and the reduction in latitude and longitude of the spot of observation to this point may easily be obtained from the scale and by measurement.* Reduction of the Observations. Ample rules for the com- putation of the time, both by the method of "equal altitudes,"! and that of " single altitudes," are to be found in the various treatises on navigation in use among seamen; and the persons into whose hands these pages may fall, and to whom they are likely to prove useful, will doubtlessly be perfectly conversant with the usual methods of computation employed. Having, therefore, nothing new to offer in a track so well beaten, we shall, in referring our readers to the rules laid down by the * The required reductions maybe effected approximately by the u traverse table" as follows: With the true bearing of the spot of observation from the meridian of the known position as a course, and the distance (expressed in miles) as a dist., enter the table. The corresponding diff. lat. will be the difference of latitude required, and the dep. the departure required : then, with the latitude as a course, and the dep. in the diff. lat. column, the corresponding dist. will be the difference of longitude in minutes of arc. If required in seconds of time, multiply it by four. (Note. The number of feet in a " nautical mile" may be assumed as a mean value at 6075.) If wanted more accurately, the reduction may be obtained by the following for- mulae : Diff. lat. in minutes of arc = [6-216454] a cos. 6. Diff. long, in minutes of arc = [6-216454] a sin. 6 . sec. lat. Diff. long, in seconds of time = [6-818514] sin. 6 . sec. lat. Where a = true distance in feet, and 6 = true bearing of the place of observation, from the known station. The quantities inclosed in brackets are the logarithms of the numerical con- stants which enter into the formulae; in using the formulae the tens are to be rejected from the indices of the sums of the logarithms. t In many of the treatises on navigation used by seamen, the rules for the com- putation of the " equation of equal altitudes" involve the use of proportional logar- ithms. Such an arrangement is not suited to cases where all possible accuracy is KEDUCTION OF THE OBSERVATIONS. 63 established writers on this subject, simply confine ourselves to a few practical remarks and observations tending to enforce careful habits of computation, and the accurate realisation of results. In reducing data from the "Nautical Almanac"* the " Greenwich date" having previously been obtained to the nearest minute the quantities should be carefully computed, as closely as the construction of the tables admits ; the sun's decli- nation, and its daily change, to the nearest second; and the equation of time to two places of decimals of a second. If star observations have been used to determine the time, the "right ascension o. Here the interval between the actual and selected epochs is o di o44, and the changes of rate for that interval, for the three chronometers respec- tively, + o-u, o s< 1 8, and + 0*^7. Hence applying these several corrections to the above errors, we have for the new errors at the epoch, May 6 d- o, h m s D -t- o 21 31*46 M + o 15 40-97 P + o 8 3i-jz and so on in similar cases. DETERMINATION OF ERRORS AND RATES. 79 thus exemplifying one of the most important ends to which the perfection of hydrography can serve."* The above examples will probably be amply sufficient in illustration of the general mode of proceeding ; a few practical observations on the subject suggest themselves in continuation. I. When the time has been obtained by " equal altitudes," the epochs to which the observations refer are the instants of apparent noon. Hence, since the "equation of time" is ever varying, in strictness they should be reduced to their correspond- ing moments of mean time, in order that the interval between any two successive epochs, and also the mean epoch, may be correctly ex^essed. In the majority of cases in practice, this degree of refine- ment will, doubtless, scarcely be necessary ; but in cases where the chronometers had large rates, and when it is wished to proceed with every regard to precision, these minutiae should be attended to. II. For the convenience of correctly describing moments of time involving fractional portions of a day, a table is given in the Appendix for converting intervals of time (or longitude) into their equivalent fractions of a day. The table will be found useful in the above instances, and also in facilitating the accurate expression of intervals, in cases where the time has been obtained by single altitudes taken, as will frequently be the case, at about, but not exactly, the same time of day. It will be sufficient to express the intervals to three places of decimals; no inconvenience can attend this nicety, since, doubtless, the arithmetic processes involved in the manipulation of the corrections for the rates of the chronometers will usually be performed by logarithms. III. The consideration as to what may be the best interval for determining the rates of chronometers is a question of some interest. If the rate of a chronometer were deduced from the comparison of two consecutive values of its error on mean time, taken at an interval of one day, the rate so obtained would be vitiated by the whole amount of the algebraic difference of any errors which might exist in the determination of the two errors on time at the place. If the rate depended on observations made at an interval of n days, the rate would be affected in like * " Nautical Magazine " for 1839, p. 406. 80 ON THE MANAGEMENT OF CHRONOMETERS. manner by the nth part of that difference. Hence, if the stability of rate of the chronometer could be implicitly relied on, its value would be determined with more and more exactness as the interval between the observations of n days increased in amount. >'* In practice, however, this theoretic view is limited in its application, by the impossibility of depending confidently on the steadiness of the rate over long periods, and by the consequent necessity for checking the performances of chronometers by frequent determinations ,of their errors, and thus breaking up the intervals on which the rates depend into short periods. As a matter of practice, therefore, it seems advisable, when circumstances permit, that the rate of a chronometer should not depend on observations made at an interval of less than five or more than ten days. Seven days will be found a convenient average interval, and in the case of eight-day chronometers, moreover, it embraces the period affected by the whole length of the chain. With the above limitations, it may be laid down as a maxim, that " chronometers cannot be rated too often when time and opportunity permit." IV. It seems advisable, moreover, when the measurement of meridian distances is in contemplation, that, in so far as may be practicable, the two rates employed should depend on observ- ations made at equal intervals of time ; since when the intervals are very unequal, the small errors of observation do not exercise an equal influence on the final results, and their values are unduly affected by the errors of observation attendant on the rate deter- mined at the shorter of the two periods. It is of importance, moreover, that the observations for the determination of the errors and rates should, whenever practicable, be of a similar character, as previously suggested (ante, p. 68). The method of "equal altitudes" is strongly recommended for adoption, as being the most correct one at the disposal of the seaman. V. Since it is expedient that those who may be desirous of undertaking the deduction of meridian distances should be ready to avail themselves of every opportunity that circum- stances may afford, we would recommend, that in cases where a ship is lying in port, and the period of her departure uncertain, observations for the errors of the chronometers on time should be made once every week, but not necessarily with any intention DETEKMINATION OF ERRORS AND RATES. 8 1 of immediate or even subsequent reduction, unless actually needed. By this plan, whenever the departure took place, however unexpectedly, the data from the two last observations would be immediately available for the determination of a working error and rate, even if time did not permit additional observations to be taken immediately before starting for the voyage. Of course, on return into port at the termination of a cruise, observations for re-determining the errors and rates should be made at the earliest convenient opportunity. 82 ON THE MANAGEMENT OF CHRONOMETEES. CHAPTER V. ON RATING CHRONOMETEES, AND ON THE DETERMINATION OF THEIR ERROES ON LOCAL MEAN TIME, BY THE COMBINATION OF SEVERAL OBSEBYATIONS OF A SIMILAR CHARACTER TAKEN WITHIN ANY CONVENIENT INTERVAL. WE shall now proceed to show how any number of similar observations can be combined for the purpose of ascertaining the error and rate. Proceeding on the same principle which guided us before, when treating of the simple method of rating by duplicate observations, viz. that equal confidence may be placed in all the observations (if not, the doubtful ones should be rejected), the determination of the error* simply consists, in general, in taking the arithmetic mean of the several errors corresponding to the respective times of observation, which will manifestly give us the mean error corresponding to the epoch indicated by the mean of those several times. It may happen that this epoch may not be an integral or exact date, and might, therefore, not be considered altogether suited for practical manipulation ; as a remedy for this possible objection we shall take occasion to show, further on, that the original error thus found can subsequently be easily reduced to any required and convenient epoch. In order to enable us to treat the observations for rate in a systematic manner, and to deal with them in accordance with mathematical principles, it is necessary to assume two postu- lates : first, that equal confidence can be placed in all the observations (if not, those of doubtful character should be rejected); and secondly, that if any change of rate is taking place, and the rate be not constant, then that the change of rate is progressing uniformly and in proportion to the time. Of course, it is not pretended that this is ever rigorously * See remarks on Example IV. p. 97. COMBINATION OF OBSERVATIONS FOR RATE. 83 true, except by mere accident; all we contend for is, that the assumption of a progressive and uniform acceleration of rate (whether in a gaining or losing direction) is admissible during short periods, and that it is highly probable that, in the long run, the actual inequalities of the rate are compensated for, and its mean condition not unsatisfactorily represented. If the rate of a chronometer were uniform, the accumulation of its rate during any given time could be graphically repre- sented by the area of a rectangle, whose base, A B, or t, should represent the time elapsed, and altitude B C, or a, the uniform rate. AB = B C = A. B Then, Area of rectangle, or accumulated rate = t a. If it be assumed that at the commencement of the period, t, the rate was a, but that at its termination the rate had altered to a + &,* the change of rate from a to a + b having been equable and uniformly proportional to the time; then the whole accumulation during the period t could be represented by the area of the figure ABED. B C = a C E = 6 B E = a + b That is, by the area of the rectangle A B C D added to the area of the triangle DOE. * The algebraic signs of a and b being positive or negative, as the case may be ; gaining rates, and the alterations of rates in a gaining direction, being considered positive, and losing rates, and the alterations of rates in a losing direction, being held to be negative. 84 ON THE MANAGEMENT OF CHRONOMETERS. Hence, Area of whole figure, or accumulated rate = ta + t- * Now let A D or B .0 = x (fig. 3) be the rate of a chrono- meter at any given epoch, and: after the lapse of m days, let the rate be supposed to have changed to B E = x + y ; then the whole accumulation of rate during the period of m days will be represented by the expression, Whole accumulation = mx H .y (i) which represents the area of the whole figure ABED. E AB = BC = BE =x + Similarly, the accumulation for any partial interval, after the lapse of n days, if A F or D G = n, will be represented by the area A F H D, or by the sum of the areas A F G D and DGH. Now DG:GH::DC:CE or y Hence, Area DGH = -? . GH and whole accumulation of rate for any partial interval, n, The rate itself at the time being represented by F H, (3) COMBINATION OP OBSERVATIONS FOE EATE. 85 Now suppose that observations for the errors of a chrono- meter on local mean time were made on several days conse- cutively, or nearly so say on the gth, nth, I2th, I4th, ijth, and 1 8th of any given month and that we assume, that although doubtless affected by errors of observation, yet that equal confidence can be placed on all of them ; then, by taking the difference between the errors on the gih and nth, 9th and I2th, Qth and I4th, and so on, we should have a series of quantities, a, 6, c . . . &c., representing the accumulations of the rate, as given by observation, for the several partial inter- vals between the first and each subsequent day's observation in succession ; and n 19 n 2 , n 3 . . . m, representing the several partial intervals, we should have, on the principle of formula (2), a series of equations of condition of the following form : * +** mx H 11 = e im If the observations from which these equations are obtained were absolutely correct., the values of the unknown quantities deduced from some of the equations ought, when substituted, to satisfy the remainder; but as this can never be the case in actual practice, it is advisable to combine the equations so as to give the most probable values of the unknown quantities, and the Method of Least Squares, well known to astronomers, seems best adapted for that purpose. The mode of proceeding in the case before us is as follows : Taking the sum of the above partial equations, we have . +w 2 which may be put under the form A*+B3, = P (4) and which gives us our first final equation. 86 ON THE MANAGEMENT OF CHRONOMETERS. Again, multiplying each of the original partial equations by the coefficient of x in it, we obtain another set of equations, as follows : ^ m Takin their sum, x -\ -- y = m e 1 j which may be put under the form, C* + Dy = Q (5) and which gives us our second final equation. From these two equations (4) and (5) we proceed to deter- mine x and y. From (4), _ P - A j y r - 3 From (5), Q- C.r Therefore, equating the values of ?/, P - A.y _ Q - C# ____ _____ Hence, DP- AD# = BQ-BC;r and (AD- BC);r = DP-BQ DP-BQ , " X = A D - B C i? being determined, ?/ becomes known from (4), since COMBINATION OP OBSERVATIONS FOR - P-A# These formulae are more simple than they at first sight appear, and will offer but little difficulty in their numerical solution. For, on examination, it will appear that A = 7Zj + n^ + n s + . . . . + m = the sum of the several partial intervals between the observations. In practice, these numbers will usually be integers, or, at any rate, it would be easy to arrange that they should be so, by taking the observations accordingly; or by reducing them to corresponding and regular epochs by approximate corrections when necessary. Again, = the sum of the squares of the several partial intervals. Also, B = the sum of the squares divided by 2 m ; and D = the sum of their cubes divided by 2 m. Likewise, P = the sum of all the partial differences (a, b, c . . . . e) between the computed errors of the chronometers on time at the place. And, Q = the sum of the same quantities multiplied respectively by the coefficients of x (viz. n 19 n zt n B . . . . m) in the first partial equations. It is also worthy of note, that if more than one chronometer is employed, and the rates of several are being determined at the same time, by the same series of observations, which will- usually be the case in actual practice, the coefficients A, B, C, and D, and hence also the factor A D ^. B c ? quantities depending on the periods between the observations, will be constants for all the chronometers. The only variable quantities will be the numbers P and Q, which will be different for each chronometer. We shall proceed to illustrate these formulae by some practical examples. 88 ON THE MANAGEMENT OF CHRONOMETERS. Example, No. I. By equal altitudes the errors of a chronometer, on local mean time at noon, were found to be as follows : May 3d 5th 8th 9th 1 2th Here, h m s Chron. fast 317 4-55 3 17 14-00 3 17 24-90 3 17 29-25 3 17 4 I '4 3 17 49'5 = 2 n,= 9 Sum 33 Whence we have A B Squares. Cubes. 4 8 25 125 36 216 81 729 121 '33 1 267 2409 22 = 267 33 LZ = 12-136 D = 2 = 109-5 22 Now, by formula (6), Differences. a = + 9-45 b = + 20-35 c = + 24-70 rf=+ 36-85 e = + 44-95 Sum = -f 136 30 Also, s n^a = + 18-90 n z b = + 10175 n a c = + 148-20 w 4 d= + 33 1<6 5 w e = + 494'45 + 1094-95 P = + 136-30 Q = +1094-95 D P-BQ " AD-BC Therefore, substituting numerical values as above, _ 14924-85 13288-27 3613-5-3240-3 1636-58 ' "373-2" COMBINATION OF OBSERVATIONS FOR RATE. 89 P-Aa? AI Also, y 136-30 144-705 12-136 __ -8-4.05 12-136 These values of x and y thus obtained are dependent on all the observations made during the period between the 3d and i4th, and from them the state of the rate at any moment during that period can be determined. Thus, on May 3d, at the moment when the first observation was made, The rate x + 4 8 '385 And on May I4th, at the moment when the last observation was taken, The rate x + y = + 4 3 '38s + (-o s -6 9 3) = 3 s -6 9 2 And at any other time, May (3 + w) day9 , by formula (3 ), The rate = x + . y y In order to determine the error of the chronometer, since we assume that equal confidence can be placed on all the observations (or otherwise they would have been rejected), we have, by taking the arithmetic mean between all the errors, the mean error corresponding to the mean of the times as a mean epoch. Hence we find in this case, May 8 d< 5, Chron. fast 3 h 17 2^-27 To find the rate corresponding to this epoch, since May 8 d> 5 is 5 d '5 in advance of May 3d, n 5-5, and by our formula (3), Rate = x + . y = 4 S '39 90 ON THE MANAGEMENT OF CHRONOMETERS. If it should be thought 'inconvenient that the mean arithmetic epoch should involve a fractional expression, and an epoch cor- responding to an integral number of days, or an exact date should be preferred, the correction due to the mean arithmetic error can easily be determined. First, let the assumed integral epoch, to which it is proposed to reduce the error and rate, be in advance of the mean arithmetic epoch. Let n = the interval between the period of the first ob- servation and the mean arithmetic epoch ; and n' = the interval between the period of the first observation and the proposed epoch. Then, by formula (2), n x -\ y = accumulation cf rate to mean arithmetic epoch. zm y And n'x -\ y = accumulation of rate to proposed integral epoch. Then, obviously, (n' n) x -\ . y V ' t vn. " , , -. n + n . n n , *. or (n r - n) x + - 2m . y (7) will be the correction to the mean arithmetic error. Again, if n' is less than n ; that is, if the proposed integral epoch is prior to the mean arithmetic epoch, then the correction will obviously be, n + n .n n / n T n . 11 y* /o\ n-n x + y For example, in the case before us, the mean arithmetic epoch is May 8 d< 5. Let it be supposed that for this we propose to sub- stitute the exact epoch, May Q d . Here ri = 6 and n = 5-5 ; therefore, formula (7), f i s , n' + n . n' n ('- ) * + ~r- y becomes 0-5 x 4 St 385 IT 5 ^ 5 . o s> 693 or 2 St i2 o s< i8i = 2 s -oi i COMBINATION OF OBSERVATIONS FOR RATE. 91 Hence, at the epoch, May 9 d , the chronometer is fast on local mean time, 3 h 17 27-27 + 2-01, or 3 h 17 29-28 Also the corresponding rate at this time, by formula (3), = 4-007 Again, let the proposed integral epoch be May 8 d , a date in the rear of the mean arithmetic epoch. Here n' 5 and n = 5-5 ; therefore the expression from formula (8), , n + n . n n n n' x A V zm 9 becomes 0-5 x 4^385 5 0-693 or 2-192 0-165 = 2-027 Hence, at the epoch May 8 d , the chronometer is fast on local mean time 3'' 17 27^27 2 5> O2J, or 3 h 17"' 2^-24., And the corresponding rate at this time by formula (3) = 4-385 -- (0-693) = 4-070 In selecting a second example, we shall take one from M. Daussy's* paper on rating chronometers, a translation of which is given in the "Nautical Magazine," vol. iii. for 1834, p. 393. * " Methode de Calcul pour obtenir la Marche des Chronometres. Par M. Daussy. Extrait de la Connoissance des Terns de 1835." This paper is reprinted in the " Recherches Chronometriqties," p. 41. 92 ON THE MANAGEMENT OF CHRONOMETERS. Example, No. II. By single altitudes P.M. taken on the nth July, and on several subsequent days, it was found that the errors of a chrono- meter on local mean time were as follows : h m s h m s July nth at 3 i 22 Chron. slow 6 41 32 i 2th 3 i 44 6 41 32 i 4 th 2 33 31 6 41 34 1 6th 2 50 40 6 41 42 i8th 3 13 52 6 41 51 i 9th 2 47 3 6 41 50 As these observations were not all made at exactly the same time in the day, the intervals between them would not be exact integers ; to remedy this inconvenience, it would be advisable to reduce the observations by means of an approximate rate to what they would have been had they all been taken at, say 3 P.M. Comparing the first and last observations on the nth and 1 9th, we have for the approximate daily rate y = 2 S< 25. Correcting the observations by means of this rate, we have, h m h m s July nth at 3 o P.M. Chron. slow 6 41 32-00 1 2th 6 41 32-00 1 4th 6 41 34-04 i6th 6 41 42*01 1 8th 6 41 50-98 I9th 6 41 50-02 And this operation, be it observed, is an illustration of what we remarked before (p. 87), viz. that if the intervals between the observations were not integral numbers, it would be easy to arrange that they should be so ; and although, in the instance before us, the corrections to the observations are very minute, yet, if the rates were large, they could not with propriety be neglected. On examining these observations, it appears that COMBINATION OF OBSERVATIONS FOR RATE. 93 n a = o-oo n z b = 6-12 n 3 c = 50-05 nd= 132-86 m e = 1 44- 1 6 ~333*I9 P = - 49-05 Q = - 333 -I 9 S quares. Cubes. s Hi = I I i a = o-oo ra 2 = 3 9 27 5= - 2-04 n 3 = 5 2 5 125 c = 10-01 rc 4 = 7 49 343 d=- 18-98 /w = 8 64 512 e = 18-02 Sum 24 148 1008 49-05 Whence we have, A = 24 B = ^ = 9' 2 S C = 148 -P. 1008 D = - And by formula (6), _ DP-BQ - AD-BC And substituting numerical values, x = 6 3 x (-49'5) -9' a 5 (~333''9) 24 x 63 9-25 x 148 _ - 3090-15- (- 3082-00) 1512 1369 'IS a = i = o s -o57 H3 Also, by formula (4), __ -49-05 -24 (-0-057) 9-25 _ -49'5 - (- ''368) 9*25 - 47-682 9-25 = - 5-154 x and y being thus determined, the state of the error and rate at any time during the period of the observations becomes known by the application of our formulae (3), (7), and (8). 94 ON THE MANAGEMENT OF CHRONOMETERS. By taking the mean of all the errors we have, at the mean epoch, July I5 d 'i25, (or July I5th, 3 P.M.), chronometer slow 6 h 41 40^17 ; and by formula (3), the rate at this time, = - s '57-5(5 s>1 54) = - 2-634 The error and rate* thus found are as good mean values as the data employed can afford. If it be thought that the values of x and y seem inconsistent, it must be remembered that the actual observations, as will appear on inspecting them, do not offer regular results ; whence it is to be inferred, either that the observations themselves were not good, or that the chronometer was really irregular in its rate. As objections may be raised to all theories, it is not im- probable that it may be advanced against the propriety of the leading idea which distinguishes our present mode of proceeding, that when we speak of the rate's altering from its original value of x to x + y, it by no means follows that any alteration really takes place, and that us may be uniform ; to which we reply, that supposing that to be the case, the formula will prove its truth by giving a value of y equal, or nearly equal, to o, The following example will illustrate this : Example, No. III. h m s June 3d Chron. slow I 10 50 4th i 10 50-3 6th i 10 5 OO2 Also, by formula (4), From the values of ss and y thus found the state of the rate at any moment can be determined. On examining these values of x and y, it appears that at the moment of the first observation the chronometer had a gaining rate of 2 S> OO2, and at the moment of the last observation a losing rate (x + y) of 2 s -5 2 3 ; and our hypothesis being that the change of rate has taken place uniformly, it follows that the rate of the chronometer first continually decreased till it came to zero, and then continued to increase its losing rate till it attained its final amount; and it follows, also, that the chronometer must have passed through a period when its error was a maximum. An examination of the data of the observations will show that both these suppositions are perfectly consistent with the apparent facts of the case. If, in order to determine the error of the chrono- meter on local mean time, we were to proceed as usual, and find the mean error corresponding to the mean arithmetic epoch, we should have d h m s Nov. 18*33 Chron. fast i o 5-65 Such a result, however, on an inspection of the observations, could not be considered satisfactory, unless we are prepared to admit great irregularities in the results of the daily observa- tions themselves. If, on the contrary, as is more probable, the daily rate of the chronometer itself had really fluctuated, and we, having confidence in the observations, agree to assume that position as true, then it will be proper to treat the deter- mination of the error somewhat differently, and after ascer- taining its successive values at the epoch of the final observa- tion, from each day's observation taken separately, then, as a final working error, to take a mean of all the separate results so determined. The computation of these several errors can be made without difficulty from an equation analogous in form to formulae (7) and (8). The correction to the first day's observation H 98 ON THE MANAGEMENT OF CHRONOMETEES. will evidently be the whole accumulation of the rate, during the whole period between the first and last observations, which is given by formula (i), m x + ~ That for the second day will be the whole accumulation diminished by that due to the partial period, n\\ that is, + n, . m y (9) Those for the subsequent observations being obtained in a similar manner ,. by substituting in (9), in lieu of n 1} the successive values w g , n s , . . . . &c. Applying formula (9) to the example before us, we shall have for the error of the chronometer by each day's observa- tion, reduced to the period of the final observation on Nov. 23d, results as follows : By the observ- ations on Nov. 1 3th ") 1 5th f 4*40 2-62 = O 178 Chron. o 7.70 570 = o 2'OO fast on o 8-90 6-99 = o I- 9 I 1 Nov. o 6-50-5-54 = o 0-96 2 3 d o 4-302-30 = o 2-OO i 2'IO . . . . = 2'10 20th 22d 23d J Chron. fast Nov. 23d, Mean Error 179 Hence, by a combination of all the observations from the 1 3th to the 23d, the error of the chronometer at the epoch as + y - Nov. 23 d , is i o 1-79 fast. And since the rate at this time by our general theory is The rate = 2 s -oo2 + ( Notwithstanding the apparent fluctuations in the rate of the chronometer, as exhibited by the above observations (since the chronometer would appear to have had a gaining rate at COMBINATION OP OBSERVATIONS FOR RATE. 99 first and a losing rate afterwards), the final results in this case, both for error and rate, may be considered as satisfactory ; and although in most cases the mean arithmetic error cor- responding to the mean of the times of observation may be considered as sufficiently accurate and convenient for practice, yet, in cases where an examination of the observations seems to indicate considerable instability or fluctuation of rate during the period of rating, and when the computer does not object to the additional labour involved in the latter more elaborate process, there is no doubt its results will usually be more satis- factory, and certainly more correct, while it will fully repay the extra trouble employed in its manipulation.* * In a memoir published in the hydrographical appendix to the "Voyage de 1'Astrolabe, Paris, 1843," by MM. Vincendon-Dumoulin and Coupvent Desbois, and since reprinted in the " Recherches Chronometriques," p. 177, a method is pro- posed, of combining any number of observations for rate, depending on the theory of probabilities. It is sufficiently simple, and may be thus explained. Let d v d 2J d y be the differences between the observations for the error of a chronometer, on local mean time, combined together in pairs (which may be done till all the possible com- binations are exhausted) ; and n v n v n 3 , . . . . &c., the corresponding intervals between the observations. Dividing each difference by its corresponding interval, we have a value of the rate, and these rates have greater chances of probable exactness, the greater are the intervals, or the denominators of the fractions, l , &c. &c., which express the rate. Applying the rules of the calculus of probabilities to determine the general mean rate during the interval, we must multiply each of these fractions by its denominator^ take their sum and then divide it by the sum of the denominators, which is the same thing as assigning to each rate a value proportional to the denominator of the fraction which gives it ; and which again, is the same as dividing the sum of all the partial differences, d lf d v &c. &c., by the sum of all the intervals, n l , n 2 , &c. &c. Hence, calling the mean concluded rate x In practice, it would be convenient to combine the observations in pairs, thus ; taking the difference between the first and the last errors, the second and last but one, third and last but two, and so on. If the number of observations were an odd number, one must be rejected. The mean rate thus obtained corresponds to the mean of the epochs of the separate partial rates. Applying the above formula to the ist example, given ante, p. 88, 44 8 '95 + *7 8 '4o + 4 8 '35 30 ^ - - - ii + 7 + i 7 6 '7 which rate corresponds to the epoch May 8 d '5, and agrees in this instance with that obtained by the method of least squares (ante, p. 89). 100 ON THE MANAGEMENT OF CHRONOMETERS. CHAPTER VI. ON THE CHRONOMETRIC DETERMINATION OF MERIDIAN DISTANCES, AND ON THE METHOD OF ALLOWING FOR THE CHANGE OF RATES OF THE CHRONOMETERS, THAT MAT HAYE TAKEN PLACE IN THE INTERVALS BETWEEN THE OBSERVATIONS METHODS OF DE CORNULIER LIEUSSOU MOUCHEZ HARTNUP TIARKS REMARKS ON THESE METHODS. THE " Meridian Distance," or f( Difference of Longitude in Time/' between any two places, is obtained chronometrically by comparing the errors on local mean time shown by a chrono- meter at the two places in succession; the error at the first place being corrected by the known rate of the chronometer in the interval, so as to give the state of the watch at the moment of the second observation. The errors of the chronometers being thus known simultaneously at the two places, their difference represents the "meridian distance," or "difference of longitude in time " between them. If the rate of the chronometer at the first place has been accurately determined, and remains constant during the transit between the two places, the "meridian distance" between them deduced from the observations so compared, will be true within the limits of the correctness of the observations themselves; but as this will rarely be the case in actual practice, since the rate of the chronometer, on account of the mechanical imper- fections in its construction, the effects of varying temperature, and other causes, is probably in a state of perpetual fluctuation, it becomes a question of considerable importance, in connexion with the accurate and systematic measurement of meridian distances, to consider how the variations of rate are to be treated, so as to produce results which, at once uniform and consistent, shall at the same time be in accordance with rea- sonable principles, and not open to any possible objection of caprice or vagueness. USE OF TEMPERATURE CORRECTIONS. 101 From the earliest times, since the attention of horologers was first directed to the construction of marine chronometers, so as to produce instruments capable of accurately determining the longitude at sea, and of carefully measuring the differences of longitude of remote maritime stations, the influence of varying temperature has been distinctly recognised as one of the principal causes of marked changes of rate. It was soon perceived, more- over, that with whatever care the adjustment of the compensation might have been affected by the artist, and within whatever narrow limits, the fluctuations of rate, due to changes of tempe- rature, might be restricted, yet that the mechanical defects of the compensation would still necessitate the application of further corrections to the results obtained from the chronometers, when accurate measurements were in question. Measures were accord- ingly proposed to allow for the irregularities of these instruments, the adoption of an " equation of temperature" being one of the first. When Harrison's timekeeper was sent on its second trial voyage to the West Indies in 1764, Harrison certified to the Admiralty beforehand, what the rate of his chronometer might be expected to be, at every tenth degree of the thermometric scale, from 42 to 82 of Fahrenheit. Since its first voyage in 1761-2, Harrison had re-arranged the compensation, and time not permitting some further final adjustments, he stated, that " as the inequalities are so small, I will abide by the rate of its gaining on a mean, one second a-day for the voyage." The account of the voyage goes on to state, that on his return from Barbadoes in July 1764, "the chronometer was found to have gained only 54" in 156 days, allowing it to have gained one second a-day, being the rate by which Harrison declared he would abide. If, how- ever, allowance be made for the variation of the thermometer, as stated by him before his departure, it will be found to have lost only 15 s ." Here we have a plain acknowledgment of the utility of temperature corrections.* Again, in Fleurieu's voyage in 1768 9,f made for the express purpose of testing at sea the performances of the marine time- * See " Mackay on the Longitude," vol. i. p. 275. f " Voyage fait par 1'ordre du Roi, en 1768 et 1769, pour e"prouver en Mer les Horloges Marines inventees par M. F. Berthoud. Paris, 1773." The expedition was under charge of M. d'Eveux de Fleurieu, who was assisted in the observations 102 ON THE MANAGEMENT OF CHRONOMETERS. pieces of Louis Berthoud, that artist furnished MM. Fleurieu and Pingre, who were charged with the conduct of the observa- tions, with a table of corrections for each of the watches, in order to " estimate their rates for the different temperatures." In the appendix to the account' of the /voyage (vol. ii.), M. Fleurieu gives a full explanation of all the observations taken for the purpose of verifying the performances of the watches ; we find that in all cases corrections, according to the daily observed mean temperature (taken from an interpolated table deduced from the primary one furnished by Berthoud), were applied to their indications of the time. Fleurieu also strongly insists (vol. ii. p. 427) on the necessity for commanders of ships being furnished, not only with information as to the actual errors and rates of their time-keepers, but also as to " the variations their movements should experience at different temperatures." He goes on to explain how a table of corrections is to be experiment- ally determined- by observations of the actual performance of the instrument for different degrees of heat and cold ; and he further points out (p. 438) the necessity of verifying the table from time to time by fresh experiments, and gives cautions, to prevent observers attributing to the effects of temperature changes of rate, in reality due to other causes. Borda, in his account of the voyage of the Flore in 1772,* holds the same language as Fleurieu had previously done ; thus (vol. i. p. 31) he says, "Although the equation of temperature was always very inconsiderable in the watches of Leroy and Berthoud, embarked on board the Flore, we did not omit always to take account of it." Again (vol. ii. p. 404), Borda refers, for the verification and use of marine watches, to the appendix of the voyage of Fleurieu. He had already said (vol. i. p. 326), " We will not repeat here the good instructions that M. de Fleurieu has given for the use of marine time-keepers," &c. ; and, p. 358, he adds, " We will not speak of the method of determining longitudes by marine chronometers ; M. de Fleurieu has already explained it with the by M. Pingre, an astronomer of note. They visited Cadiz, the Canaries, Goree, Martinique, St. Domingo, and the Azores. The narrative of the voyage is very copious, an'd full of most interesting details, reflecting infinite credit on the pains- taking assiduity of M. Fleurieu and his colleague. * " Voyages en Europe, Afrique, et Amerique, en 1771-2 ; par leChev. de Borda. Paris, 1778." USE OF TEMPEKATUKE CORRECTIONS. 103 greatest detail, and with all the precision that we can desire ; we will refer to his work." " It is impossible," observes M. De Cornulier, " to be more explicit." " Thus Borda recommends the use of the equation of temperature, and has himself employed it. The utility of this equation made itself so much felt, in fact, on the watch No. 8 of Berthoud, that he could not avoid having regard to it. This watch, which had no sensible acceleration, experienced suddenly a change of four or five seconds, when the Flore passed from the climate of the West Indies to that of Newfoundland and Iceland ; afterwards it resumed its first rate, on the return to Brest. On this occasion, Borda takes care to make the remark (vol. i. p. 323), that this rate was conformable to the table of equations that Berth oud had furnished him with."* Notwithstanding the authority and example of these dis- tinguished navigators, the practice of paying regard to equations of temperature seems to have gradually fallen into disuse. M. De Cornulier thus complains : " After the voyage of the Flore, they no longer gave for the watches a table of the equation of temperature they did not even keep any longer a register of the daily temperature of the chronometers during the voyage ; all the second chapter of the appendix of Fleurieu's voyage is put into oblivion, as if it had become useless in consequence of the perfection to which compensated balances had been carried. This conduct resembles entirely that of a man who denies the danger, and closes his eyes, so as not to see it, for fear of disturb- ing his quietude. Yet, if we examine the tables of rates of the chosen watches which have been employed in the greater part of our grand modern voyages, we shall very soon recognise that many of them have not been exactly compensated.''! English navigators appear to have been as equally unmind- ful of the refinements of temperature corrections, as their French contemporaries. No allusion is made to the subject in the account of Cook's voyages, nor does the matter seem to have engaged the attention of the commanders of our various scientific voyages, and hydrographic expeditions, 4 during the present century, in any practical degree; although the failures and anomalies which sometimes presented themselves in chronometric measurements, * " Recherches Chronometriques," p. 113. f Ibid. p. 117. 104 ON THE MANAGEMENT OF CHRONOMETERS. have often been attributed to excessive or irregular fluctuations of temperature. The causes of this neglect have probably been twofold ; first, it may have been thought, that in consequence of the great mechanical improvements in the^onstruction of chronometers, and more especially from the great success which had been attained in the adjustments of their compensations, such minute corrections were no longer needed ; and that if variations of rate took place, they could on the whole, and in the long run, be sufficiently allowed for, by assuming the theory of uniform acceleration in proportion to the interval elapsed. This theory originally, it would seem suggested by Borda, appears to have been generally adopted by modern navigators. Secondly, it is also probable, that the practical minds of seamen were often inclined to attach but little value to the minute refinements of theory, which in presence of the anomalies often revealed by observation, would appear in many respects, to verge closely on empiricism and fanciful treatment. In so far as the ordinary requirements of navigation are concerned, no doubt modern chronometers, with proper attention paid to rating them from time to time, may be sufficiently depended on without regarding these refinements ; but even the conduct of common voyages would probably be improved if the corrections or allowances for changes of temperature could be reduced to a system simple, convenient, and short.* In the future conduct of hydrographic voyages, attention to these points will perhaps become indispensable, for as maritime geography has gradually advanced towards perfection, science has become more exacting in its demands, and requires that we should avail ourselves, not only of all the mechanical improve- ments of modern art, but also of the ingenious investigations of mathematical analysis. Stimulated by these considerations, atten- tion has been recently again directed to this interesting subject, * " If accidents arising from errors in the reckoning a re rare (remarks DeCornulier), and if they do not always entail fatal consequences, it is because prudent navigators hold themselves on their guard against the infidelities of the chronometer : they control as much as they can their indications by lunar distances ; they direct their course very much at a distance from all land : often they heave to during the night. From these exaggerated precautions result passages much longer than they ought to be. A knowledge of the equation of temperature would always permit them to navigate more freely and with as much prudence." (" Rech. Chron." p. 106.) METHOD OF DE COKNTJLTER. 105 with the view of making chronometers still more available than heretofore for the accurate measurement of differences of longitude. In France, MM. De Cornulier and Lieussou (Ingenieurs Hydrographes) have recently proposed algebraic formulae for correcting the performances of chronometers, involving a con- sideration of the effects of temperature, and also the influence of the acceleration. M. Mouchez (a naval officer) has also written a memoir on the same subject, and suggested a method for cor- recting chronometric observations for the variations of tempera- ture. In this country, Mr. Hartnup (Director of the Liverpool Observatory) has proposed to recur to the use of tabulated daily rates, corresponding to the different degrees of the thermometric scale, and obtained by observation, in lieu of a mean daily rate, regardless of temperature, as is at present generally employed. This system in fact is a revival of that formerly recommended by the celebrated horologist, P. Leroy ; and as it requires none but the simplest calculation and is very easy in practical use, it seems well worthy of consideration. As the work, on which we are now engaged purports to lay before its readers all useful information on the subject of the management of chronometers, it would mani- festly be incomplete if it omitted to introduce to the notice of the scientific student the ingenious investigations of recent writers on this interesting subject. We therefore propose briefly to explain the leading features of their methods, referring our readers for fuller information to the more copious explanations of the authors themselves. Method of De Cornulier. M. De Cornulier,* an officer in the French naval service, had charge of the watches on board the Allier, during a cruise in the South Seas in 1829-31; afterwards as Director of the Naval Observatory at Lorient, he had the opportunity of enlarging his * M. De Cornulier has published, upon the calculation of the rates of chrono- meters, four memoirs, which have been successively inserted in the " Annales Maritimes " for the years 1831, 1832, 1842, and 1844. All these memoirs have for their principal object the consideration of the influence that changes of temperature exercise upon the rates of these instruments, and the necessity of taking account of it, concurrently with the variation known under the name of the acceleration, in the determination of chronometric longitudes. In his first memoir, De Cornulier explains the principles of his theory and its application ; he devotes himself in the subsequent memoirs to the development of his 106 ON THE MANAGEMENT OF CHRONOMETERS. experience and of carefully studying the performances of these instruments. His investigations led him to the following conclu- sions : First. (< The compensation of chronometers is not generally as perfect as is supposfed'-r-too m/ach confidence is accorded to it. In the present day it is necessary for many of them, to have recourse to an equation of temperature, as was the practice in the first voyages for the trial of marine chronometers." Secondly. " In the correction of longitudes obtained by chronometers, it is wrong to hold entirely to the hypothesis pro- posed by Borda in the voyage of the Flore, and according to which every rate which has varied, would have done it by a movement uniformly accelerated or retarded; this consideration is useful, but it ought to be combined with that which precedes it." Thirdly. " The equation of temperature, totally forgotten, contrary to all reason, is of the two corrections that which deserves most to engage our attention ; it is that which represents the greatest irregularities; it may be utilised immediately in the daily practice of navigation ; whilst the other, less important in itself, less regular in its effects, cannot be determined but after an accomplished voyage. It may then be employed a posteriori for the perfectionment of hydrography," that is to correct the meridian distances of the previous cruise. Finally. "The research of the equation of temperature for each watch, is one of the most essential objects, on which a nautical observer can employ himself."* In furtherance of these views, De Cornulier proposed to treat chronometric observations in the following manner : At a place where the mean daily temperature was a, let the mean daily rate of a chronometer = m. Subsequently at another place, where the mean daily temperature was a + b, let the corresponding mean daily rate = m 4- n. method, and to justifying his opinions, by supporting them by new observations that he had made, and by evidence the most suitable for corroborating them. These memoirs have for the most part been reprinted in the " Recherches Chro- nometriques." (" Rech. Chron." p. 87.) We take the opportunity of mentioning here, that in the earlier sheets of this work, De Cornulier's name has been incorrectly printed Du Cornulier, the error not being detected till the sheets had been worked off. * " Recherches Chronometriques," p. 88. METHOD OF DE CORNULIER. 107 Let the interval between the epochs of the observations for the rates be q days, and the mean daily temperature during this period a + d. Also let x be the acceleration of the daily rate of the chrono- meter, arising from imperfect mechanism ; and y the variation due to a change of one degree in the thermometric scale, x will follow the law of movements uniformly accelerated, and y will be proportional to the differences of temperature. Also let the difference between the absolute errors of the watch on mean time at the first station,* by the observations at the two places, be M. Then assuming that the change of rate n has arisen partly from the progressive influence of the acceleration, and partly from the effects of variation of temperature, n =qx + by (i) Hence from (i), y = n ~% x * M. De Cornulier writes "on mean time of the first meridian," but as it is advisable to avoid complicating the question by a reference to Greenwich or Paris time, the above is better. Then if D be the known difference of longitude of the two stations ; A. and A', the respective errors of the chronometer on mean time at the two places D = ;i'-O + M) And M =(X'-X)-D whence M may be determined, the errors >. and *! being found from observation, and D being assumed to be known. In discussing M. De Cornulier's formula?, we have thought it best to adhere throughout to his notation, for the convenience of those who might be disposed to refer to the original work. The difference of longitude, here called D, we have elsewhere called M ; while the quantity he calls M, represents the whole accumulation of the actual rate of the chronometer, in the interval between the observations. f It will be observed that M is the whole accumulation of the rate, due to the actual performance of the chronometer, in the interval between the two observations. If the chronometer had kept true mean time M = o : If the original daily rate had remained constant, then M = q m : If the chronometer had been affected by a change of mean temperature d degrees acting on it for a period of q days, then M = qm + qdy : If in addition it were influenced by a daily acceleration x acting on it for q days, then M.=qm + qdy + q (- - - J x as above. The effect of the acceleration increases uniformly by a quantity x from day to day. The sum of an arithmetic series of q terms, in which the first term equals the common difference = q - , hence this portion of the correction = q (- 108 ON THE MANAGEMENT OF CHRONOMETERS. M qm qx ( - -- J and from (2), y = - - ^ whence, solving for as, nd / x (3) Again from (i), x = M q m q dy And from (2), x = - Hence solving for y, n g --(M-mg) /.\ To obtain a? and y with exactness, it is requisite that the difference of meridians (D) of the places where the observations have been made, should be perfectly known, for it is from it that we deduce the difference of the absolute errors M. It is equally necessary that the difference of temperature b or d should be a little great, in order to obtain y with some precision. The coefficient y relative to the temperature, ought to remain very nearly constant throughout the duration of the voyage, for the errors which exist in the system of compensation of a chrono- meter are not of a nature to modify themselves of themselves, consequently when we have once determined this coefficient, we may usefully employ it during its course. It is not thus with the acceleration #,* proceeding from the mechanism of the watch ; from the very nature of the causes which engender it, it is essentially variable ; one cannot suppose * The acceleration is always very small in watches which have recently come from the hands of the artist: when they become considerable, the watch ought to be returned to him ; it no longer merits confidence. It is to the same causes which produce the acceleration, that is attributable a fact often observed ; that a watch which has stopped, rarely resumes the same rate which it had before. (" Rech. Chron." p. 96.) The opinion given above on the magnitude of the acceleration seems at first sight at variance with a previous passage from De Cornulier's memoirs, quoted in the text (ante, p. 21). We take it, that he means that chronometers recently cleaned and adjusted, and fresh from the hands of the artist, are some little time before their acceleration becomes steady, and that it is then generally small in amount. METHOD OF DE CORNULIEK. 109 it constant for the duration of a sea-passage, and we should expose ourselves to commit grave errors in wishing to determine it in advance. This element should always be concluded from the comparison of the observations at the departure with those of arrival. It would be necessary to have acquired a very decisive experience of a watch, to dare to employ in the course of one long passage^ the acceleration determined in a preceding one. The acceleration x being very variable, or at any rate having no permanent fixity, we ought not to assume it constant, but for the smallest possible intervals, especially when it is in contempla- tion to redetermine the value of the coefficient of temperature y. The ordinary circumstances of navigation are not always as favourable as could be desired for this research. It is rarely that one departs from a point well determined, to repair immediately, and in a short time to another place also well determined, and whose temperature differs much from that of the first. It is necessary, also, that one should make in each of them a stay sufficiently prolonged to determine there exactly the daily rate and absolute error of the watch, things both essentially necessary to deduce a satisfactory solution from the two equations that we have established above. In many cases, it is more important for the requirements of navigation, to know the coefficient y in a manner approximate but prompt, than to have its exact value by waiting during a long period for the circumstances requisite for its rigorous determina- tion. The knowledge of the difference of the absolute errors of the watch (on time at the meridian of the first station) is the most difficult of all to acquire. The number of points on the globe on whose longitudes we can depend with confidence, is as yet very limited; let us endeavour then to determine the coefficient y, disregarding this element, by means of the daily rates alone, m; m + n'; m + ri + n" ; m + ri + n" + ri"; &c. If after a very short sea-passage of q f days, we should have obtained two daily rates, m and m + n', for two temperatures whose difference b' is very great, we may suppose that the ac- celeration a/ has been^, nothing in this short interval : hence the equation n'-= q x' + b' y would become n'= b' y 110 ON THE MANAGEMENT OP CHRONOMETERS. whence we have y ^ (5) That is to say, that in this first approximation all the variation of the rate might be attributed to the change of temperature alone. >* Let us suppose now, that in three successive stations, separated by intervals q f , q rf of moderate length, as say twenty to twenty-five days, we should have determined the daily rates, m, m + n r and m + n + n" corresponding to differences of temperature b' and W very decided ; one might admit that the watch has had but one and the same acceleration during the two partial passages, in the intervals q and &c., by substituting for y its value in the equa- tions n' = q r x' -f y y ; n" = &c. If the accelerations appertaining to each traverse are small, and if they differ but little one from another, it is a sign that the chronometer merits great confidence. It is not probable in practice that we should ever have occa- sion to combine more than four sets of observations for the daily rate, nor is it, indeed, desirable to do so, as our theory presupposes that the permanency of the acceleration x is only to be assumed during short periods. If the number of observed daily rates during a cruise exceeded four, it would be more convenient to take them in pairs, and obtain a solution for y of the form of equation (6) jri'-j'n' y q' I" q" V Or again, any number of rate observations might be combined, and the most probable values of x and y, prevailing during the whole period of the observations, determined by the method of least squares, after the manner already explained in dealing with the combination of observations for determining the rate (ante, P . 8 5 ).* * Assuming that during a moderate interval, within which the rates have been ascertained, the value of the coefficient x, as well as y, remained constant, the several observations for rate would give us a series of equations, n' = q' x + V y n" = q" x + b"y n'" = q'"a; + b'"y &c. &c. Taking their sum (n'+ n"+ n"' + . . .) = (tf + q" + q'" + ) x + (b f + b" + b'" + . . .) y, which may be put under the form A# + By = P (i) Again, multiplying each equation by the coefficient of x in it, we obtain the new equations, n! q' = q v x + b' q' y n"q"=q' i "x+ I" q" y n'"q'"= q*'"x + b'"q'"y &c. &c. 112 ON THE MANAGEMENT OF CHRONOMETERS* This mode of proceeding, however, assumes that the accele- ration remains uniform during the whole period, and should therefore not be employed, except during moderate intervals. We may again, in certain circumstances, determine an approximate value of the 'coefficient of temperature, by means of the simple differences of the absolute errors of the chronometers on time at the first station, when we have no other observations for rate but that of the departure m. We should employ for this research the equations Hence (n'q'+ n"q" + .. . .) = (q z '+ q*" + )# + (b'q'+ b"q" + ) y, which may be put under the form C x + D y = Q, (2) p Ay From (i) y = g From (2) y = - Hence equating the values of y and reducing, we obtain DP-BQ AD-BC (3) and then again from (i), x being known, P- A# , y = -g- (4) In these equations, which would offer no difficulty in their numerical solution, A = q'+ q" + q'"+ ..... = the sum of the several partial intervals between the epochs of the rate observ- ations. B = b'+ b" = the sum of the several partial differences of temperature, corresponding to the observed rates. C = q*' + q y + q*"+ ..... = the sum of the squares of the several partial intervals. D = b'q'+ b"q" + b'"q"' + ..... = the sum of the several partial differences of temperature, multiplied into their corresponding partial intervals of time. P = the sum of the several partial differences of rate (ri + n" + n'" + ..... ). Q, = the sum of the several partial differences of rate, multiplied into the corre- sponding partial intervals of time (n' q' + n" q" + n'" q"' + ..... ) . It is also to be observed that the coefficients, A, B, C, D, and hence also the factor - , depending on the time and temperature, will be constants for all A JD .0 O the chronometers under discussion ; the quantities P and Q, depending on the diffe- rences of the observed rates, being alone variable for each chronometer. METHOD OF DE COKNULIER. 113 M"= q" (m + O + q" d"y + q" x" which give these differences ; and assuming for the accelerations d, x" analogous hypotheses to those already adopted in the former case, that is, considering the acceleration to have been uniform during the period, or x' = x" , we have from the first equation, ,__ M' g'm- q'd'y ~ And from the second, ,.(1) 1 Equating these values of d and reducing, we obtain, / / // /?' + i\ -,, , ,,/q"+ i\ d **( )-'<*") y being known, at' can then be obtained from either of the primary equations. If the results at which we thus arrive, do not present all the guarantees for correctness that we could desire, they are at least useful indications, and very superior to a vague appreciation. A circumstance very propitious for determining with pre- cision the co-efficient ?/, is when a ship, experiencing considerable difference of temperature, returns to the same point from which it had originally departed, because there is then no longer any uncertainty, as to the difference of the absolute errors of the watch, M,* and the problem is no longer complicated by the errors that might be introduced into it, by an erroneous assump- tion of the exact difference of longitude of the terminal stations. The more we should have determined the daily rates, at the * M= (x'-x)-D And since D = o M = A' x CSee ante, p. 107.) 1 14 ON THE MANAGEMENT OF CHRONOMETERS. various places in our voyage, during this interval, and the more we should have obtained the partial accelerations x r , x", x'", &c., the better should we know our chronometer and the degree of con- fidence we could accord to it, in fixing definitively the respective longitudes of the hydrographic positions that we had recognised during our cruise. It may not be out of place to remark here, in order to fix our attention on the value of the accelerations, and on the differences that we may admit among them, that if z be the error of the acceleration, and q the number of days in the passage, the error in minutes of arc, which would supervene on the resulting longi- tude, would have for its expression, | z q ( - ji thus an error of o s *oi in the acceleration would produce one of i2 / *5 at the end of a traverse of a hundred days ; thus quantities, which would seem to be otherwise insignificant, merit great considera- tion when we are dealing with the acceleration. This element of irregularity ought not to exceed a small fraction of a second in a good chronometer. The coefficients of acceleration and temperature, x and y, having been thus determined by the application, according to cir- cumstances, of some of the methods indicated in the preceding pages, they are to be subsequently utilised and employed in the computation of the several "meridian distances," between the places called at during the voyage, in determining the corrections to the observed errors on local mean time, due to the accumulation of the rate. Thus let X be the error of the chronometer at the first station, and x' at the next. Then D = A' (A + accumulated rate in interval.)] But the accumulated rate between the observations Hence D = A' ( A + q (m + dy + x ^-^) \ which expression involving the coefficients, both of acceleration and temperature, x and ?/, is probably more true than the one generally adopted, which assumes only a uniform or mean daily rate, during the period ; provided always that the coefficients have been deter- METHOD OF LIEUSSOU. 115 mined by careful observations, and the mean daily temperatures accurately noted. Such is briefly the system of chronometric treatment proposed by M. De Cornulier, and in truth it seems well worthy of atten- tive consideration. Those who may wish to study his labours in greater detail, will find them more fully developed in his memoirs,* and to them we must refer our readers for further information. Method of Lieussou. A very instructive and interesting memoir,f on the subject of chronometers, and on the causes of the variation of their rates, has recently been published in France, and submitted to the " Bureau des Longitudes." The author, M. Aristide Lieussou, availing himself of the data relative to the performances of chronometers, obtained from the public trials, made at the observatories at Greenwich and Paris, has arrived at some highly curious and interesting results. By an ingenious system of graphic projection, exhibiting the connexion between the observed daily rates, the corresponding temperatures, and the time elapsed since a given epoch, M. Lieussou has inves- tigated the laws which govern the changes to which the rates of chronometers are liable, and has established that marine chrono- meters obey with great regularity the combined action of the temperature, and the thickening of their oils, which arises from the lapse of time. In order to comprehend the effect of this double influence, let us conceive a chronometer placed within an enclosed space, and maintained at a constant temperature : its daily rate will vary insensibly with the time, and if we designate by a, this rate at the commencement of the experiment, it will be a 4- b x, at the end of a number of days x, indicating by b the variation of the daily rate in one day. * The greater part of the above remarks and information have been freely trans- lated from M. De Cornulier' s work, originally published, as already mentioned, in the " Annales Maritimes," and for the most part reprinted in the " Recherches Chronometriques," pp. 87-175. To his valuable work we are also greatly indebted for many useful extracts which enrich these pages. f " Recherches sur les Variations de la Marche des Pendules et des Chrono- metres: par M. Aristide Lieussou, Ingenieur Hydrographe de la Marine," &c. &c. " Ex trait des Annales Hydrographiques (1853). Paris, 1854." Copious extracts from this work are also given in the " Recherches Chronometriques," pp. 216-263. 116 ON THE MANAGEMENT OF CHRONOMETERS. M. Lieussou attributes this variation proportional to the time, to the defect of isochronism of the balance-spring. Since, in general, chronometer-makers arrange their balance-springs so that the small oscillations should be more rapid than the great ones, we may say that the defect of isochronism determines in general, an acceleration in the daily rate of a chronometer. According to this explanation, the constant 6, would give the measure of the precision with which the isochronism of the oscillations has been established. In the greater number of chronometers intended for sea service, this quantity rarely attains to one-hundredth of a second, and it appears to preserve the same value so long as the balance-spring is left in the same state. If the chronometer was always submitted to the same temper- ature, we should have then its daily rate by means of the expres- sion a + & x : but when the temperature varies, this expression becomes more complicated, as we shall presently see. Let us suppose that the artist should have adjusted his chro- nometer at o and 30 (centigrade), that is to say, that he should have determined the positions of the compensating weights on the balance-wheel, so that the daily rate should be exactly the same at these two extreme temperatures. If we should place this chronometer in a box, in which we could make the temperature vary from degree to degree from o to 30, we should experience immediately an increasing acceleration in the daily rate, whilst the temperature should be comprised between o and the mean temperature 15; but at 15, the daily rate would attain a maxi- mum value, and if the temperature should continue to increase from 15 to 30, the daily rate would go on continually diminish- ing, until it attained at 30 the same value it had at o. This diminution of rate would still manifest itself for temperatures below zero, and above 30, and it would be so much the more considerable, as we swerved more and more from the mean temperature 15. M. Lieussou has determined this fact in discussing the observ- ations of sixty chronometers, followed out at the observatory at Paris. He has, moreover, remarked, that for an equal variation of temperature more or less, reckoned as a point of departure from the temperature T, the arithmetic mean between the two extreme temperatures, for which the artist has adjusted his chronometer, the rate diminishes by equal quantities. He has METHOD OF LIEUSSOU. 117 investigated by graphic constructions the law of this variation, and he has found that, t being the actual temperature to which the chronometer is exposed, it is proportional to the square of the difference of temperatures T and t. Thus designating by a, the daily rate corresponding to the temperature T, we shall have the daily rate m at any temperature whatever t, by means of the expression, m = a c (T t)* c being the constant variation that the daily rate a undergoes, when the temperature to which the chronometer is subjected changes from T to T i. This constant c varies for different chronometers, but it would appear to preserve the same value for each chronometer, whilst its balance remains in the same state. It represents the precision with which the artist has adjusted the compensation. Its value is generally below o St oi5 in good chronometers, pur- chased after trial for the public service.* We see from the preceding discussion, that the correction c (T t ) 2 only depends on the half sum T, of the extreme tempera- tures selected by the artist for the adjustment of the compensation, and that it remains the same, whatever may be the temperatures, provided that their half sum be T. It is for this reason, that without troubling ourselves with the two extreme temperatures by which the experiments relative to the adjustment have really been made, we may say that the chronometer has been adjusted for this mean temperature T. The preceding expression furnishes a very simple means of appreciating the influence that the choice, apparently arbitrary, of the mean temperature T, exercises upon the rate of a chrono- meter. Taking the case of two chronometers A and B adjusted with the same success, a fact indicated by the equality of the coefficient * In M. Lieussou's researches, thermometers with the centigrade scale are referred to. If any other kind of thermometers were used, the numerical values of the co- efficient c, of the chronometer, would of course he altered ; the principle of the formulae would remain the same. If c, T, and t, refer to the centigrade scale, and c', T', and f 7 , to Fahrenheit's, c and c' become comparative, and can be obtained the one from the other, by the expression, e= o 8> oi5 of the centigrade scale, corresponds to c'=o i> oo46 for Fahrenheit's. 118 ON THE MANAGEMENT OF CHRONOMETERS. c. For the chronometer A, supposed to be adjusted for 8 and 38 (centigrade), whose mean T is 23, the influence of the tempera- ture will be represented by c (23 t)~. In like manner, for a chronometer regulated for o and 26, we should have for this influence 0(13 if. We see, by simple inspection, that when these two watches are kept at a place whose mean annual temperature . is 13 (55'4 Fahr.), as for instance, in the chronometer-room of the observatory at Paris, they would be very differently affected by changes of temperature. The chronometer B adjusted to 13 would have variations of rate much more feeble than the other, since the change of temperature would take place round and about 1 3, the mean temperature of the place : thus at Paris B would be judged superior, but if we placed the two chronometers in a room whose mean temperature was 23 (73*4 Fahr.) the contrary would happen. A would be considered superior to B. The combined influence of the thickening of the oils* and of the temperature upon the rate of a chronometer, can therefore be represented by two terms, the one proportional to the time, the other proportional to the square of the difference of temperature * " The thickening (or gradual evaporation) of the oils (with which the pivots are lubricated), which takes place by lapse of time, by augmenting the resistance, whilst the impulse to which it ought to be in equilibrium remains constant, tends to diminish the amplitude of the vibrations of the balance. The mass of the balance being very small, and the amplitude of the vibrations very great, this effect is naturally considerable; thus the arc of vibration being at a mean, 415, when the oils are fresh, it is no more than 330, when the oils are aged three years. If this great alteration of vibration affects but little the rate of a chronometer, it is only because the vibrations of the balance-spring, which determine the alternate move- ments of the balance, have sensibly the same duration in the great arcs of 415, and in the small arcs of 3 1 5. The employment of the balance-spring, as the regulator of the movements of the balance, is only then founded on a remarkable property which it enjoys. If it is short, the great vibrations are more rapid than the small ones. If it is long, the small are more rapid than the great. There is therefore in every balance-spring a length which gives to the great and small oscillations the same duration : for each chronometer this length is determined by trial, by comparing the rates corresponding to two extreme vibrations, which are obtained by varying the moving force or the tension of the main spring. In practice this special length is never rigorously obtained, and in other respects the thickening of the oils, which diminishes the amplitude of the oscillations, by 100, in three years, alters their duration con- siderably." (Lieussou, p. 27.) In M. Lieussou's system, the term, b x, in his " equation of rate," represents the accumulation of the acceleration due to the influence of time, and although chiefly viewed as a function of the age of the oils, it in reality represents the product of all the irregularities which supervene by time, wear, dirt, friction, &c. METHOD OF LIETJSSOU. 119 reckoned from the fixed temperature T, peculiar to each chrono- meter, and the daily rate may be calculated by aid of the expression, m = a + bx c(T t) z , in which, m represents the daily rate of the chronometer at the temperature t, after x days elapsed since the epoch, for which the daily rate was a for the temperature T, which has served for the adjustment of the compensation. M. Lieussou has arrived at this equation by graphically con- structing the curves of the daily rates of a great number of chronometers, followed up for a year at the Observatory at Paris : he has taken the intervals of time for abscissae, and for ordinates the daily rates observed at each epoch. For the purpose of disentangling the movement of the watches from the influence of the variations of temperature, the author considers in the curve of a chronometer, the points whose ordinates represent the daily rates observed at the same tem- perature, and he finds that these points are situated in a straight line ; for another temperature, the points of the curve are also in a straight line, parallel to the first, so that by cutting the curve of the daily rates of the chronometer by a series of parallel straight lines, making with the line of the abscissae an angle peculiar for each chronometer, the ordinates of the points of intersection, represent the daily rates, corresponding to the same temperature, the temperature varying when we pass from one straight line to another. M. Lieussou concludes from this, that the rate of a chronometer, submitted to a constant temperature, varies as the ordinate of a straight line, and may be expressed each day by the expression a + b x. With respect to the influence of variations of temperature on the daily rates, it is represented in the curve of a chronometer, by the distances which separate the parallel straight lines of which we have spoken, these distances being measured upon the ordinates themselves. After some trials, M. Lieussou has re- cognised that these distances vary in the direct ratio of the square of the difference between the actual temperature, to which the chronometer has been exposed, and a certain temperature corre- sponding to the greatest observed daily gaining rate (avance) : we have seen from the remarks which have preceded, that this 120 ON THE MANAGEMENT OF CHRONOMETERS. temperature occupies precisely the mean place between the ex- treme temperatures which have been employed in determining the position of the compensating weights on the balance. The equation of the daily rate of a chronometer considered as a function of the time v and temperature, involves therefore the four constants a, b, c, and T, of which we now know the signi- fication. These constants can be determined for each chronometer (as we shall show further on) by means of four daily rates accu- rately observed at temperatures very different; and we could obtain their values more exactly by a greater number of precise observations, furnishing a series of equations of condition. M. Lieussou has done this for many chronometers presented for public trial at the Observatory. For the greater part of them, the daily rates calculated by the aid of the formula, agree throughout the whole period of the competition, almost within two or three tenths of a second, with the observed daily rates. Although under certain circumstances, the daily rate may have varied 1 5 s in a year, or even 12 s in three months, the differences between the calculation and the observation rarely exceeded half a second.* The researches of the author have not hitherto been extended beyond observations on chronometers followed up for a year in an observatory ; but for a watch which has been exposed to all the accidents of a long voyage, experience can alone decide (observes the report of the committee to whom the work was referred) whether the constants determined before departure would serve for the whole voyage, or whether it would not be necessary to ascertain their values afresh. Nevertheless it seems natural to suppose that the constants, T and c, which depend on the adjust- ment of the compensation, could be obtained in a few days, by means of experiments made at temperatures very different, and then two daily rates, deduced from observations made during the * " Among the chronometers whose movements were considered by the author, he found many which, not having satisfied the conditions of the public trial, had been rejected for purchase by the naval department. Notwithstanding this, the method of calculation had been applied to them with equal success. The influences of time and temperature had caused in their rates considerable variations, but these variations had taken place in a manner almost as regular as in the best instruments. Thus making allowance for the particular irregularities that a watch would undergo at sea, these chronometers would have given by means of the empirical equation, longitudes as exact as the others." (Lieussou, p. 17.; METHOD OF LIEUSSOU. 121 voyage under favourable circumstances,, and at intervals of time sufficiently great, would suffice to give the value of the constants a and b, which belong to each chronometer embarked. This calculation which is sufficiently simple, (as we shall show further on,) could be made by the officer charged with the care of the watches. Determination of the Constants which enter into the Equation of the Rate of a Chronometer. The general equation of the rate of a chronometer as a function of the time and temperature, considered as independent variables, is = a 3 in which formula m expresses the actual daily rate at any moment, at a given temperature t. a is the initial daily rate of the chronometer at the temperature T. It is the measure of the imperfection of the adjustment of the watch to mean time, at the temperature T.* The constant T is the special temperature at which the chronometer takes its maximum rate, it is, as we have before stated, the arithmetic mean of the two temperatures for which the artist has established the equality of the rate ; for a well- regulated chronometer this constant ought to be comprised be- tween 15 and 20 (59 to 68 Fahrenheit). The coefficient c is the diminution of the daily rate for a change of temperature of one degree centigrade, more or less, reckoning from T. It is the measure of the imperfection of the compensation, and remains invariable, so long as the balance- spring and the balance are not modified ; for a good chronometer this coefficient ought not to exceed o s *O2. * " This initial rate augments or diminishes by a quantity b x, proportional to the number of days elapsed x ; b being generally positive (but not always), and less than o 8- oi. Artists are in the habit of adjusting the initial rate some seconds slow on mean time, in order that at the end of three years or so (for which period the oils are supposed to last) it may vary from mean time as little as possible. Assuming for instance that a equals 5 8t oo when the chronometer leaves the hands of the artist, it will be equal to 5 8- oo+548 b = o s 'oo nearly, at the end of eighteen months (or 548 days) ; and will be 5 9> oo+ 1095 b= + 5 8< oo nearly, after three years (or 1095 days) ; in cases where the coefficient of the acceleration, b, is positive, as is usually the case." (Lieussou, p. 101.) 122 ON THE MANAGEMENT OF CHRONOMETERS. The coefficient b is the change of rate of a chronometer in a unit of time. It would appear to vary a little after a long in- terval, but it is sensibly constant during a year. It may, perhaps, hence be considered, as the measure of the defect of isochronism between the great and- small oscillations of the balance ; for a good chronometer it ought not 'to exceed o s *o i per day, or o s *3O per month. The four constants, a, b, c, and T, which enter into the general equation, and whose particular values for each chrono- meter constitute its " special characteristics" (son regime special), can be determined by any four rates whatever, observed at different temperatures and different epochs. Let m ly m 2 , m 3 , w 4 be the four observed rates, corresponding to the four temperatures t ly t 2) 1 3) and 4 , let us suppose, to facilitate the calculation of the constants, separated by equal intervals h. We have then the four equations of condition, m l = a + ob c (T t^f m 2 = a + h b c (T * 2 ) 2 m 3 = a -f 2 h b c (T 3 ) 2 mi a + 3 h b c (T # 4 ) 2 whence, adding together the first and third, and subtracting twice the second, m^ 2 m 2 + m 3 = c {t 2 * 2 2 + # 3 2 2 T (^ 2 t & + * 3 )} adding together the second and fourth, and subtracting twice the third, m z - 2 m 3 -f mi = c {t - 2 f, + t? - 2 T (t 2 - 2 * 3 + * 4 )} adding together the third and fourth, and subtracting the first and second, adding together the four equations, -c {(T- O 2 + (T- # 2 ) 2 + (T- METHOD OF LIEUSSOU. 123 which for shortness may be put under the form, * =- C (/3 -2Ty) *' =:_c(/3'-2Ty') = 4 +D/I&-C {(T-O 2 + ( T -*2) 2 + (T-* 3 ) 2 -t-(T-- From the two first of these equations, which only involve the two unknowns c and T, we obtain by reduction, nl In T - .y a y c = T and c being thus known, we have from the third equation, b> = (3) and from the fourth, -O 2 } (4) These four equations determine the four constants, as functions of the daily rates and temperatures observed at four epochs (sepa- rated for the convenience of calculation by equal intervals).* * It will be instructive to give an illustration of the application of the formulae. The records of observations of the chronometer, No. 200 Winnerl, followed un- interruptedly at the Observatory at Paris, from June ist, 1847, to Sept. ist, 1848, supply the following data : Date. Daily Rate. (Mean in Ten Days.) Daily Tempe- rature. (Centigrade.) (Mean in Ten Days.) Equations of Condition. 25th Oct. 1847 + ri8 o + i5 - o + i'i8 = a+ o4-c(T 15)2 25th Jan. 1848 -1-19 + ro ri9 = 0+ gib c(T i) 2 25th April, 1848 + i'53 + J3'o + i'53 = a+i8aft-c (T-is) 2 25th July, 1848 + 1-62 + 21*0 + 1-62 = + 273 b-c (T-2i) 2 Treating the equations of condition as pointed out above, we shall obtain, Also = + 5-09 ft of =2*63 ft'= + 104 a" = + 3'l6 y = + 26 a '"= + 3-i4 y'=- 4 /S"= + 384 and y"= + i8 124 ON THE MANAGEMENT OF CHRONOMETERS. M. Lieussou states, moreover, as the result of his experience, that for a chronometer placed in an observatory, the equation, m b x c (T ff gives without sensible error, the mpan rate, in any interval which does not exceed a month, when we put for t, the mean tem- perature observed in that interval. Otherwise we ought to put for m ly w 2 , wi 3 , and m 4 ; t l9 2 , t S9 and 4 ; the rates and mean temperatures observed in ten days, Hence T I j3'-'/S 2 a y a' y becomes T I 529+1031 I 1560 2 20-36 + 68-38 2 48 and c _ cty' tty $ yB y becomes c 48 48 O--OH- 2704+1568 4272 Also b = -^{" + cr- 2 T r ")} becomes b ~ 77TT {3' l6 + ' 011 (384-585)} And = 8 '0026 Q'949 364 - { K '"-6hb + c (T- V + T- - {3'J4 1*42 + 0*011 (267-25)} -(4-66) 16. This value of a, " the initial rate," at the temperature T, corresponds to the epoch Oct. 25th, 1847. For any other date it will be given by the formula, m = a + I x Thus for the 25th Jan. 1848 (91 days afterwards) it would be s s m = 1*16 + 0*0026 x 91 1*16 + 0*2366 ^ 1*3966 = 1*40 The equation of this chronometer therefore, deduced from four mean rates of ten METHOD OF LIEUSSOU. 125 twenty days, thirty days, for the purpose of withdrawing the constants from the inevitable errors of isolated observations. The interval which seems to embrace the greatest precision, would appear to be thirty days for a good chronometer, and twenty days for one moderately good. M. 'Lieussou gives numerous tables showing the application of his formulae to chronometers under trial at the Observatories at Paris and Greenwich, and certainly the accordance of the calcu- lated and observed rates is very striking. As the result of his days' interval, separated by intervals of three months, and referred to the z$th Jan. 1848, would be, m i g< 4o + o 8> ooa6 x o 8t on (i6'25 fj* M. Lieussou afterwards proceeds to investigate the equation of rate of this chrono- meter by other observations differently combined, and shows that within allowable limits of errors of observation, similar results are obtained. (Lieussou, p. 56, &c.) The following table, selected from the numerous examples in M. Lieussou's work, will show the application of his system, and how closely it agrees with observation. Comparative Table of Mean Daily Rates (in a month) observed and calculated. a ** { 6 8< 5O ; b o 8< oo5 ; c = o 9< c>45 ; T = i6*5. Epoch of Initial Rate April 1 5th. Chronometer No. 83, Winnerl. m =6-50 + 00-005 a; o-o45 (i6'5 O 2 Mean Tem- Date. perature. (Centigrade.) Daily Rate. 6* 50 + 0" -005 # o s 'o4S (i6'-5-<) 2 Calcu- lated. Observed. Differ- ence. 1844. i s s 8 s April 1 3 3 + 6-50 0-46 + 6-04 + 6'oi + 0-03 May *5'4 + 6-65 o'o5 + 6'6o + 6-95 -o'3S June 18-4 + 6'8o 0-16 + 6-64 + 6-62 + O'O2 July 20*2 + 6-95 0-62 + 6-33 + 6-39 0*06 Aug. J 9'3 + 7*10 -o'35 + 6-75 + 6-98 -0-23 Sept. 18-4 + 7'*5 0-16 + 7-09 + 7'34 -0-25 Oct. 15*3 + 7-40 0-06 + 7'34 + 7'44 O*IO Nov. 11-4 + 7'55 - i*i? + 6-38 + 6-61 -0-23 Dec. 4'6 + 7-70 -6-37 + J-33 + 1-00 +0-33 1845. Jan. 5'4 + 7*85 - 5'54 + 2-31 + 2-37 o'o6 Feb. 4*5 + 8-00 -6- 4 8 + 1-52 + 0-24 + 0-28 (" Lieussou," p. 62.) 126 ON THE MANAGEMENT OP CHRONOMETERS. researches the author lays down, first, " that four rates and four temperatures observed at intervals of ten days determine the four constants which constitute the special characteristics of each chronometer with a precision sufficiently remarkable ; " and, secondly, " that four mean monthly rates and temperatures determine them with a rigorous exactness." Adverting to what was before stated on the probable per- manence of the constants T and c, which depend on the system of compensation, and on the subsequent determination of the other two, a and 6, by observations made by the officer charged with the care of the chronometers, it may be advisable to enter a little more into detail. The constants T and c being known by experiments previously made by the maker, or at an observatory, before delivery of the instruments on board ship ; two observations for the rate, observed under favourable circumstances of differences of temperature, and at a convenient -interval, would establish the equations of con- dition, mi = a + ob c(T ti)* m 2 = a + hb c (T t 2 ) z Taking their difference, nit-m^ = Aft c{(T fc) 8 (T O 8 } whence b = | { m z - m^ + c {(T - t z )* - (T - tj*} J (5) Again taking their sum, m 1 + m 2 = 2a+hb c {(T ^) 2 + (T * 2 ) 2 } whence a = I m^ ig- hb + c {(T- ^) 2 + (T- # 2 ) 2 } (6) From these two equations (5) and (6), made at any time during the voyage, the constants a and b become known, T and c having been previously determined. The initial rate a thus found, is the rate at the temperature T, at the epoch corresponding to the first of the two observed daily rates, used in the calculation. METHOD OF LIEUSSOU. 127 Application of the Equation of Rate of a Chronometer to the Measurement of Meridian Distances. We will now proceed to consider the application of Lieussou's formula, giving the " equation of rate " of a chronometer, to the measurement of the meridian distance between two stations, and compare it with the common method, which assumes the invari- ability of the rate during the interval, or at any rate its variation proportional to the time elapsed. Taking in the first instance the most simple case, and assuming that the variation of temperature having on the whole been nearly uniformly progressive, it may be taken to have altered regularly by a daily quantity, p. Then t representing the initial temperature at the commence- ment of the voyage, on the day of departure, or at the epoch of the last observation for rate ; after the lapse of x days, it would be t + p x. Substituting this in the general equation, m = a + bx c(T i) z we have, m= a + bx c(T t + px) z = a + bx c (T t pxf a + b x c (T * ) 2 cp* x* + 2cpx (T t ) but the expression, a c (T ) 2 , represents the initial rate m 0) at the temperature t . Hence m m + x {b + 2 cp (T * )} cpa Deducing from this by integration, the expression for the error E at the date x, (the error at the commencement of the voyage being denoted by E ). E = E 4- m . dx = E 4- Cm . (7) 128 ON THE MANAGEMENT OF CHRONOMETERS. The error E' estimated on the hypothesis that the initial rate m has remained constant during the traverse, would be, E' = E + m x and the error therefore of this estimated state of the chronometer, after a passage of x days, would be, (8) In order that this error should be nothing, it would be requisite that the coefficients b and c of the variation of the rate of the chronometer, with the time and temperature should be so also, which is never the case, at least with the coefficient c. The preceding formula (7) would be conveniently applicable in practice, and would present no difficulty in numerical solution, in cases where the alteration of temperature having been nearly uniform, it might be assumed that it had varied from day to day by a given quantity, p. Otherwise, if the temperature had been irregular, the daily rate might be determined for each day successively from the general equation, m = a + b x c (T tj- or more conveniently substituting for a, its equivalent expression m + c (T - * ), from the equation, putting for t, the daily observed temperatures, t lt t zt t 39 &c. The sum of these computed daily rates would then be the correction, to the error of the chronometer on local mean time, at the first station, to bring it up to the time of the second observation, to be employed in the deduction of the meridian distance between the two stations. Without, however, proceeding so elaborately as this, the correction may be obtained in a manner more simple, and nearly as exact, by adding to the initial error E , n times the rate corresponding to the mean temperature - (^ + 2 + 1 3 -f . . . + t n ) ; or even better still, by dividing the passage into convenient METHOD OF LIEUSSOU. 129 intervals, five, six, or seven days for example, for each of which the rate corresponding to the mean temperature being determined, their mean is to be then finally employed for the computation of the accumulated rate, as the correction to E . In dividing the whole period of the voyage into convenient intervals, the computer would be guided by circumstances, and would make them more or less long, according as the changes of temperature had been more or less abrupt. Taking then for each of these intervals, the rate due to its mean temperature as its mean rate, we should obtain without sensible error the state of the chronometer at the end of the traverse. Let us take for example a chronometer whose equation of rate, m = a + b x c (T i) 2 is m = a + o s< oo5 x o s -oi5 (24 t) z and suppose a sea-passage of two months, during which the temperature has varied gradually from o to 21 (centigrade), or by o'35 per day on the average (=>). Then its error at the end of the passage is given immediately by formula (7), = E + 6om + 3600 {0-0025 + 0<00 5 X 0-35 (72 21)} 33 o s -i If the duration of the traverse had been divided into six intervals of 10 days each, the six corresponding mean tem- peratures (in accordance with the supposition made above) would have been, + i7S5 5'25; 8 7S5 i2'255 1 S'7S'> 1 9' 2 S 5 putting these successive values of the mean temperature, for t in formula (9), m = m + c (T t o y + bx c (T tj- and substituting the above values of b and c, we have, m = m + c T 2 * -f 0-005 x ' OI 5 ( 2 4 O 8 * Since in this case t = o ; c (T- # ) 2 = c T 2 . K 130 ON THE MANAGEMENT OF CHRONOMETERS. Whence we obtain, in, 0-005 \ ' 5 \. 15 ;25 35 45 55 0^015 (4-8)* = + + o s - 1 5 - 3 s - which is the mean rate of the traverse. Consequently E' = E which agrees with the former result obtained by integration. In the method usually followed for obtaining the longitude of a ship at sea, during a voyage, it is assumed that the initial daily rate m , determined before the departure, remains constant and invariable during the passage. Hence we see, that in the case of the chronometer before us, such a supposition would involve in a passage of sixty days, an error of 330% or i 22' 30" of longitude. Again, the theory of a uniform alteration of rate in proportion to the time elapsed, as proposed by Borda and adopted by Tiarks f and modern navigators, leads to the employment of the mean of the rates, determined before departure and after arrival, in the measurement of the meridian distance, between the two terminal stations of the voyage. Now in the case before us the initial rate is m ; and the rate at the termination of the voyage, computed by the formula, m = m o + c(T-# ) will be found to be m + 8 s -8o5. c(T * The whole duration of the voyage, 60 days, being divided into intervals of 10 days, the successive values of x corresponding to the middle of each period are 5, 15, 25, &c. t See remarks on this subject further on, p. 153. % t = o ', t 21, and x = 60 by supposition. METHOD OF LIEUSSOU. 131 Hence the mean rate for the passage, m n + (m. + 8 8 ' 80 5) ii = m -f 4 S '402 and employing this to obtain the error of the watch, E' = E + 60 m + 264.*-!. The difference between this and the error estimated by Lieussou's formula is therefore 66 s or i6^ x of longitude, which therefore represents in this particular case, the error arising from the assumption of a rate uniformly accelerated, compared with the calculation of the correction by his formulae ; premising of course that the terminal rate obtained by observation, really agreed with its computed value as above, m + 8 s *8o5, as Lieussou's ex- perience has justified him in asserting that it would do, very nearly. M. Lieussou has given in his work numerous other examples of the application of his formulae, showing the near accordance of the results, whether obtained by the method of integration or by dividing the duration of the passage into convenient intervals, and deducing for them the corresponding series of mean rates. These examples, moreover, fully illustrate the serious errors that are involved in the usual assumption of the permanence of the initial rate, for longitudes at sea, or in the employment of the mean rate between that of arrival and departure, for the deter- mination of the meridian distance between the terminal stations of the voyage. For the details of the calculations we must refer our readers to the work itself, which will be found also, in all its parts, well worthy of the attention of the scientific student. The researches of this author also suggest some instructive facts for the consideration both of the artists who construct chronometers, and of the officers who are destined to employ them. The correction c (T )* depending on the system of the compensation, remains small, so long as the chronometer is maintained at a temperature nearly equal to T; as the actual temperature t varies from T, the correction increases, and the more rapidly the greater is its amount, in an increasing ratio. Hence, in the adjustment of a chronometer, the artist should consider the probable conditions of service or climate in which his instrument is likely to be employed, and make his arrangements for the adjustment of the compensation accordingly. Those also who may have to select chronometers for service at sea, and 132 ON THE MANAGEMENT OF CHRONOMETERS. especially when the equipment of a scientific voyage is under consideration, should bear this fact in mind, and choose chrono- meters for the voyage, whose values of T nearly correspond with the mean temperatures of the latitudes, in which the ship is destined to cruise. If these considerations are disregarded it may happen that a chronometer which would have been judged ex- cellent in the tropics would be deemed detestable in high latitudes, or the reverse. The complaints, often made by commanders of ships, of the performance of their chronometers, have frequently, as Lieussou observes, no other foundation. The want of accord between the mean temperature to which they are subjected, and the value of T, selected for them by the artist, is their misfortune, and not their fault. It is clear also, that in all cases, the main- tenance of uniformity of temperature is an important element in securing stability of rate. The quality of the oils, of which the term expressing the effects of the acceleration, seems to be a material function, is also most important, and well worthy of the most careful attention of horologists.* In conclusion we shall observe, that the application of M. Lieussou's formulae may be very materially facilitated by the preliminary formation of tabular corrections, giving for each chronometer the values of the terms, b x 9 and c (T ) 2 , for the arguments x and t, the days elapsed since the commencement of * The following extract from the Report of the Astronomer Royal to the Admiralty, on the annual trial of chronometers at the Observatory at Greenwich in 1860, has an important bearing on the above remarks : "An examination of the rates of the chronometers leads me to the following conclusions : " (A.) The material workmanship of all the chronometers is very good, and amongst nearly all the chronometers there is very little difference indeed in this respect. In uniform circumstances of temperature every one of the chronometers would go almost as well as an astronomical clock. " (B.) The great cause of failure is the want of compensation, or the too great compensation, for the effects of temperature. " (C.) Another very serious cause of error is brought out clearly in this trial; namely, a fault in the oil, which is injured by heat. This is very different with the chronometers of different makers. For instance, the oil used by one chronometer- maker (named in the Report) is not at all injured by heat ; while some of that used by another chronometer-maker (also named) is so bad that, after going through the same heating as those of the first- mentioned maker, the rates of the chronometers are changed (on returning to ordinary temperature) by 80 seconds per week. " (D.) I believe that nearly all the irregularities from week to week, which generally would be interpreted as proving- bad workmanship, are in reality due to the two causes (B.) and (C.)" METHOD OF MOUCHEZ. 133 the voyage, and the actual temperature. The constants b, c, and T, having been determined by previous observations. The numerical values of these corrections, taken from the tables, being then substituted in the formula (9), m = m Q + c (T - Q* + bx - c (T - *) 8 would give the actual daily rate m, corresponding to the mean daily temperature t, and might then be immediately utilised for the determination of the longitude, in the daily navigation of the ship. Method of Mouchez. Since the appearance of M. Lieussou's work in 1853, the subject of chronometric research has engaged the attention of a French naval officer, M. E. Mouchez, and his views materially differing as they do from those of Lieussou, merit separate consideration. In the present state of chronometric science any researches, based on observation and experience, are valuable contributions for the elucidation of truth. Time and experience, which can alone settle the question, have not yet given any definite sanction to any one theory of analytical investigation in preference to another. Attention has only recently been called, or rather we should say redirected, to the importance of tem- perature corrections, and in consequence, careful observations of the temperature of the chronometers in most voyages, even in- cluding those of a scientific character, have not hitherto been sufficiently accurate or extensive, to furnish data for the corro- boration of the empirical formulae, which have been suggested by analysis, as the expression of the law which controls the move- ments of these instruments ; we proceed therefore to explain Mouchez's views.* Mouchez's experience has not rendered him favourable to the employment of algebraic formulas in dealing with the performances of chronometers and the variations of their rates. " The varia- tions so frequent and so diverse of the daily rates of almost all chronometers embarked (on board ship)," observes the author, " having demonstrated to me the complete insufficiency of all the * " Observations Chronometriques, faites pendant la Campagne de Circum- navigation de la Corvette la Capricieuse, par E. Mouchez. Paris, 1855." This work has been in great measure reprinted in the " Recherches Chronometriques," p. 264, &c. 134 ON THE MANAGEMENT OF CHRONOMETERS. formulae proposed up to the present time, to correct them, I have confined my attention to the graphic constructions themselves ; for the different empirical formulas, proposed for the correction of rates, are only translations more or less feeble of these graphic curves, and consequently* can onl^f represent them in an imperfect manner." After describing at some length his system of graphical projection, by which he sought to exhibit to the eye, the move- ments of the chronometers, under the combined influence of time and varying temperature (for the details of which we must refer our readers to the work itself*), the author proceeds to comment on the conclusions to which the study of his chronometric charts has led him. The annual sheets prove that the acceleration is very variable in sign and magnitude; that the influence of temperature is sometimes regular, sometimes irregular, in fact they seem to establish an important law, which has not hitherto been noted this is " the augmentation by time of the thermometric sensibility of the chronometers, as if the compensation lost little by little its efficacious power. f This fact is revealed in the graphic con- structions, by the progressive deviations of the isothermal curves. This influence of time (he adds) " is sufficiently singular to merit confirmation, and it would be interesting to make some researches respecting it." With reference to the influence of alterations of temperature * " Recherches Chronometriques," pp. 265-8. f On this subject, De Cornulier commenting on the " progressive weakening of the compensation," discernible in the critical examination of some chronometric observations that he quotes, remarks, " Time alone is then the true cause which modifies the action of the compensations, and it does it always in the same sense, so that we may henceforth regard it as a newly-acquired fact, that a compensation which errs by defect becomes more and more inexact, whilst that which errs in excess tends to rectify itself. " This remark ought to make us much more indulgent towards the artists whom we accuse of being negligent in the adjustment of their compensations ; it is very possible that they only deliver to us watches perfectly adjusted; but time soon deranges their combinations, so that in practice most watches are insufficiently compensated." After discussing the probable causes of these changes, and arguing that. they cannot proceed from any actual alteration of mechanical condition in the metallic apparatus of the balance ; this writer adds, "it is much more natural to seek for the cause of the effects that we have observed, in a change in the state of the oils ; their progressive reduction by evaporation, perfectly corresponds to the variations, slow and progressive also, of the coefficients of temperature. On this hypothesis, the METHOD OF MOUCHEZ. 135 on the rates, M. Mouchez thinks that " we may without great error adhere to the hypothesis generally admitted hitherto, of variations of rate, simply proportional to variations of temperature. This hypothesis is gratuitous it is true, but it is nearly conform- able to facts, and has the advantage, moreover, by its simplicity, of being very easy for employment in practice." As to the accelera- tion, the author is of opinion, that it is unnecessary to take a separate account of it ; in short passages which do not exceed a month, it may be neglected on account of its smallness, and in determining differences of meridians, where we employ the rates both of arrival and departure, its effects are mixed up with, and involved in, those arising from variations of temperature. The method* employed by Mouchez for the deduction of the meridian distance between two stations, differs but little in prin- ciple from that previously proposed by De Cornulier ; in explain- ing it, therefore, we may conveniently adopt a notation nearly similar to that used by the latter, premising, however, that the matter is somewhat simplified by not separately taking cognizance of the acceleration. The coefficient of temperature y is obtained immediately from the observations, by a comparison of two daily rates obtained during the voyage at different temperatures ; the rate observations being combined in pairs for this purpose, (including as great ranges of temperature as possible,) in the most advantageous manner, according to circumstances. Then the difference between the two rates, being divided by their metallic apparatus after a time, is no longer adjusted so as to compensate for the total effects of the temperature on the balance and balance-spring, but only for that portion which remains, after a part of it has been destroyed by the effects in a contrary sense, produced by the oils. If the latter are reduced, little by little by evaporation, their compensating action diminishes with their volume, and the metallic apparatus, which works always in the same manner, becomes more and more in- sufficient." (Rech. Chron.p. 132.) These views are in accordance with the theory advanced by Lieussou, and seem to satisfactorily explain the phenomena in question. * The author, who does not use algebraic formulae, thus explains his mode of proceeding, " I have compared the mean temperature of the sea-passage with that of the stay in port, and multiplying this difference by the coefficient of temperature of each of the chronometers, I have found the correction to be made to its rate at departure to have that of the traverse. Performing the same operation on the rate of arrival, I have obtained a second rate of the traverse, which was rarely identical with the first one, on account of accidental variations, and of the acceleration ; but the mean of these two rates ought to be as near as possible to exactitude, and it is with this that I have determined the difference of meridians of the two points." (Rech. Chron. p. 269.) 136 ON THE MANAGEMENT OP CHRONOMETERS. corresponding difference of temperature, gives the value of the coefficient y, for one degree of the thermometric scale. Let m be the daily rate at departure, at the temperature #! : m + n, that of arrival, at the temperature t z : t 3 the mean temperature of the traverse, and g the number of days elapsed. Then employing the rate obtained before departure, we have for the correction of the error of the chronometer at the first station, Corr. = q m -f q (^ * 3 ) y ( I ) employing the rate found after arrival, Corr. = q (m + n) + q (/ 2 r 8 ) y (2) and employing the mean rate (3) Formula (i) may be usefully employed for the daily determination of the longitude at sea; ( 3 in this case representing the mean daily temperature of the chronometer since the departure.) Formula (3) is adapted to the determination of the meridian distance at the termination of the voyage. The following example illustrates its application to some observations made during the voyage of the Coquille : h m s Error of the watch before departure at Anhatomorin + 232 o after arrival at Paita +4 39 15 s o m = 5^5 at temp. ^ = 20-2* m + n = 5*30 at temp. 2 = 22-5 t 3 = 13-!; q = 145 days; y = o s -$22 Hence by formula (3), = H5 {- 5'43 + ( 21 '3 - "3' 1 ) '5 22 ) = 145 x (1-13) = - i6 3 s -8 * Centigrade. METHOD OF MOUCHEZ. 137 whence we have, h m s Error at Anhatomorin +232 o Correction -H 2 29 16 Error at Paita 4- 4 39 15 Difference of longitude +2 9 59 The author claims for this method the advantage, that it is very certain to correct the rate for its actual acceleration, whilst by the employment of formulae, we are obliged to make use of a mean acceleration, based upon a great interval of time, and which consequently may not correspond with the epoch with which we are then dealing. o Mouchez adds that if the records of the daily comparisons indicated any abrupt variation of the rate of the watch during the passage, he measured its amount and took account of it according to its sign. Again, if during some time, there had been in the isothermal curves, any considerable variation, indicated by many observations, he determined for that epoch, a special co- efficient of temperature, by aid of which the cotemporaneous meridian distances were subsequently obtained.* " All this," he continues, f< is doubtless only an affair of feeling one's way (tdtonnement), of approximation, and of probability ; but this method seems much more appropriate to the study of an instrument still so imperfect, and subjected to causes of pertur- bation so various and unknown, than those which, founded on the invariability, entirely imaginary, of certain quantities termed constants, or on a mathematical regularity, in the variations pro- duced by vices of construction, wish to embrace in a general formula all the movements of chronometers during entire years, and to assimilate, so to speak, the rate of this instrument to the regular phenomena of nature. This manner of employing chrono- meters will thus be at once more exact and more instructive than * These we conceive are dangerous and reprehensible practices ; they open the door to empirical treatment, and cooking of the observations, which is fatal to their integrity, as we may thus be tempted to bend them to the support of any desired solution. Much better to accept the actual indications of the chronometers for what they are worth, and reject their results altogether if they swerve beyond certain allowable limits. 138 ON THE MANAGEMENT OF CHRONOMETERS, that which has recourse to formulae, since there are numerous small variations entirely irregular, of which it is impossible to take account in a general mathematical expression. Such a formula based upon the whole of the observations, can only be capable of correcting the error proceeding from the single cause, as a function of which it is constructed; and hence accidental variations must entirely escape correction. " This is without doubt the reason," continues this writer in conclusion, "why they have always hitherto rejected on board ship the employment of different methods, proposed even by distinguished hydrographers, but who have not sufficiently re- marked the difference that there is between a new chronometer observed on shore, and a chronometer embarked for some years, and subjected to a thousand causes of derangement. Thus a single observation, even a little defective, is supposed with reason to give more exactly the actual error of a chronometer, than a formula, whatever it may be, which must support itself on observations made at distant epochs."* The author subsequently proceeds to make some strictures on the formulae proposed by Lieussou. He contends that the conclu- sions derived from experiments made in an observatory on shore, on chronometers new, or recently cleaned, and whose oils are fresh, kept in absolute repose in a mild temperature, nearly con- stant, or varying only by slow degrees, cannot be taken as finally decisive. The conditions, under which chronometers perform their functions at sea, are frequently entirely different : there they are often subjected to changes of temperature abrupt, and extend- ing beyond the limits in which they had been previously tried ; they experience frequent petty shocks and trepidations, while their oils become aged ; hence it happens that in practice at sea, we rarely find chronometers maintaining rates so admirably regular, as are to be found in the registers kept in observatories, or by chronometer-makers, and which lend themselves so com- placently to formulas of correction, containing terms which are functions of the temperature. Had M. Lieussou's formulae been applied to the performances of chronometers at sea, this writer thinks, that the agreements between results calculated and * We shall take occasion to show further on to what extent we concur in this reasoning, and in what degree we think it carried too far. (See p. 147.) METHOD OF MOUCHEZ. 139 observed would have been found much less accordant.* It may also happen from the different modes of construction and adjust- ment adopted by different artists, that each collection of chrono- meters by the same manufacturer may have a peculiar manner of action, and that that which is true for the chronometers of one maker may be quite false for those of another. Mouchez also states as a conclusion of his experience, that when the variations of temperature are sufficiently decisive to give importance to the corrections, the two expressions, c (T t) z , and c f (t ^), give in reality results much less discordant than might be at first supposed. In conclusion, this writer states his belief that these two sup- positions, like all others of the same kind that can be imagined, are more or less erroneous, and that they have an efficacy very different, according as the chronometer is by one artist or another. He thinks that the idea admitted hitherto of variations simply proportional to differences of temperature, would destroy a great part of the error : that this hypothesis is of such simplicity and of such easy application at sea, that a very great superiority in any other method should be admitted, before we are induced to abandon it ; we ought therefore to continue to employ it, until Lieussou's theory has been established by numerous observations made at sea, and especially rendered of easy application ; for * No doubt it is highly important that Lieussou's formulae, and indeed any other, that may be proposed for the correction of the rates of chronometers, should be carefully tested by their application to observations made at sea. In order that this should be done satisfactorily, the zealous co-operation of seamen themselves is absolutely necessary. The want of this verification hitherto is no fault of the hydro- graphers, who have analytically investigated this matter on shore. Lieussou thus complains, " We would have desired to verify the equation of rate of a chronometer, as a function of the age of the oils, and of the temperature, by means of observations made at sea ; unfortunately they have neglected up to the present time, even in voyages having a scientific character, to note the successive daily temperature to which the chronometers have been submitted on board. To procure this indispen- sible datum, we are reduced to make an hypothesis, and to suppose for example, that the unknown temperature of the chronometer-case has constantly varied as the mean of the temperature of the air and sea at noon." (Lieussou, p. 95.) The reader will not fail to observe, that what we want in this matter, is the actual mean daily temperature to which the chronometers are subjected in their 'case, and which is generally very different from the ever-varying level of the external or surrounding air. This temperature is best obtained by means of a maximum and minimum thermometer placed within the chronometer-case ; the mean of its indications being taken to represent the mean temperature. It is much to be hoped that in future, more attention will be paid on board ship to the acquisition of this important element. 140 ON THE MANAGEMENT OF CHRONOMETERS. above all things, and as an absolute condition, it is necessary before any new method can be admitted, in the practice of obser- vations at sea, that it should be simple and expeditious. Hartnup's Method. The subject of " Rating Chronometers " with special regard to the variations caused by changes of temperature, and with a view to the improvement of the daily practice of navigation at sea, has for some years past been an object of constant attention at the Liverpool Observatory, established at that great commercial port, by the munificence of the Town Council. In addition to the usual astronomical work which forms part of the regular business of all observatories, the rating and testing of chronometers was one of the main objects for which the Liver- pool Observatory was established ; the increase of the security of navigation, and the safer conduct of long voyages, being rightly considered as matters of vital importance. The Observatory being amply provided with all necessary apparatus, for conveniently testing chronometers, at all ranges of temperature, Mr. Hartnup, its able director, has devoted much attention to the systematic trial of chronometers before they are sent to sea, to verify their system of compensation, and to impress on all who may be concerned in their performances, the obvious connexion of fluctuations of rate with changes of temperature, and the necessity henceforth of taking account of it. As is well known, the practice ordinarily pursued, when chro- nometers are embarked for a voyage, is for the ship to be furnished with a memorandum giving the rate of the chronometers as deter- mined at the place of deposit on shore. It is assumed that this rate may be depended on, and that it will remain constant during the voyage. The navigation of the ship is then conducted with implicit reliance on this supposition, until the discrepancy between the longitude of the ship given by the chronometer, and the known longitude of the stations arrived at on making a landfall, often reveals the existence of serious errors, and is sometimes the cause of fatal disasters. For these, the chronometer is often wrongly blamed, whereas in truth its system of compensation not being exactly perfect, or its having been subjected to the influence of a range of temperature, exterior to the limits within which it had HAETNUP'S METHOD. 141 been adjusted, it had only obeyed with faithful regularity, the natural laws of temperature which control its movements, and from which it cannot escape. Mr. Hartnup proposes in future to furnish the mariner with a table of daily rates and corresponding temperatures, previously determined by experiments, which are to be employed in the daily navigation of the ship in lieu of a uniform daily rate, irrespective of temperature, as has hitherto been generally customary. This system, although a novelty in this country, at the present day, is virtually the same as that formerly proposed by the celebrated horologist, Pierre Leroy,* and in modern times, it has also been successfully practised by French naval officers in some recent voyages, f * In 1754 Pierre Leroy sent to the Academy of Sciences at Paris the " descrip- tion of a new watch suitable for use at sea," in which occurs the following passage : " The third method of avoiding the error caused by variations of temperature, and to which Ipropose to confine myself, is to fix a thermometer in the box of our watch, and to place it successively in hot-air chambers (etuves), and in places very cold : comparing then its variations with the degrees of the thermometer, we should write upon that instrument by the side of each degree the losing or gaining rate of the watch for this degree : by the aid of this precaution, the thermometer would always indicate the variations of the watch : now, in this case an error known is no longer an error. It would suffice then that the officer on duty should note in the register the degree of the thermometer, when he wound up the watch." (Lieussou, p. n.) This is, to all intents and purposes, the course of experiment pursued at the Liverpool Observatory, and the method recommended by Hartnup, for adoption henceforth at sea. f " The previous determination of the coefficient of temperature," observes M. De Cornulier, "seems to us so important, that we would wish to see all marine observatories provided with hot-air chambers and refrigerating apparatuses, in which should be placed maximum and minimum thermometers, for the purpose of testing the chronometers before their embarcation. It is a grave omission not to submit to this trial, those which are destined for employment in a scientific voyage. The knowledge of this coefficient would be even of great assistance in ordinary naviga- tion, since all the applications that we could make of it, in the calculations relative to errors and rates, would then become of extreme simplicity, as soon as it was known. " If in a place whose temperature was a, we should have determined the daily rate of a watch m, we could form a table of the rates that it would take at different temperatures, in establishing that for the temperature a + b, it would have the rate m + n = m + by. To calculate the longitude of the ship in the next sea-passage, we should then take from this table, for each day of observation, the daily rate which corresponded to the mean temperature of the preceding day, and we should add it to the accumulated rate that we had obtained the preceding day, or subtract it according as the rate was gaining or losing, to determine the total variation that the watch had undergone since the day when its error was last ascertained. It is thus that we have operated on board the Allier, and this method was crowned there with complete success." (Recherches Chronometriques, p. 104.) 142 ON THE MANAGEMENT OF CHKONOMETEES. The application of this system at sea would be extremely simple, and presents no points of difficulty. A maximum and minimum thermometer placed within the chronometer-case, would indicate daily, at the time of winding, the actual temperature ex- perienced by the chronometer during the previous day. The tabulated record of observed daily rates according to temperature, would then give immediately by inspection, the rate for the pre- ceding day ; applying this according to its sign, to the sum of the accumulated daily rates (similarly obtained), up to the previous day, we should have the whole amount of the accumulated rate on the given day, for the days elapsed since the departure from port, to be applied as a correction to the primary error of the chronometer, on starting for the voyage. If the voyage were long, it would be advisable to verify the rates of the chronometers from time to time, by observation, at any convenient opportunity that might offer during the voyage. No doubt the .rates so obtained might frequently differ from the rates previously determined at corresponding temperatures before departure, owing to the gradual influence of the acceleration, but there is every reason to believe that the differences of the rates, observed at corresponding temperatures, would always remain the same, or nearly so, while the chronometer continued in good con- dition.* Disregarding for the moment small variations due to the efflux of time and the influence of the acceleration, the experi- ments at the Liverpool Observatory prove that notwithstanding the great change of rate produced by change of temperature, in nineteen cases out of twenty the original rates return the instant the temperature is restored, On an average of about one chrono- meter in twenty it is found, that the rate changes in the most capricious manner with every change of temperature; but an examination of the records shows that the sea-rates of such chro- nometers are equally uncertain. Mr. Hartnup has published some tables of the rates of chronometers observed under varying * On this subject, De Cornulier remarks, when speaking of the performances of the chronometers on board the Allier, "In the greater part of the watches the rates observed at each stay in port vary in consequence of the accelerations ; but the change, with reference to one degree of the thermometer, is always the same constant coefficient y. The tables of the rate, that it is necessary to arrange after each period of rating, will no longer contain the same numbers, but these numbers will have between themselves, a constant difference from one degree to another." ( Rech. Chron. p. 104.) HARTNUP'S METHOD. 143 circumstances of temperature at the Observatory. From these it appears that the average change in the daily rate caused by changing the temperature from 60 to 40 (Faht.) was 6*'C)J, while the change produced by altering the temperature from 60 to 80 was only 2 8> 85, the variations, it will be seen, being much greater in the low than the high temperatures. Commenting on these facts, Hartnup observes, " It will be necessary for me to explain, that the changes of rate shown in these tables, notwith- standing the magnitude of a large majority of them, were very consistent with the change of temperature ; so much so, that after tabulating the amount of change due to temperature, the future rate of any of them might be predicted with very great certainty. It appears, moreover, from the records, that this holds good when the chronometers are at sea. We find for instance, that the average of the rates of chronometers while on the voyage between Liver- pool and New York, agrees with the average of the shore-rates of the same chronometers, in a temperature of about sixty degrees ; and the average of the rates of chronometers belonging to ships employed in the African trade, agrees with the average of the shore-rates of the same chronometers, in a temperature of about eighty degrees. Now all this shows that the rates of chronometers vary with the temperature, both at sea and on shore ; and the navigator who does not make himself acquainted with the varia- tion of rate due to temperature, peculiar to his own chronometers, must on the average make longer voyages, and expose his ship to greater danger than he would do, were he to make himself acquainted with these facts."* The application of this system to the measurement of " meri- dian distances " would be precisely analogous in all its features to that already described in speaking of the daily determination of the longitude of the ship at sea. The accumulated rate of the chronometer would be the sum of the tabulated daily rates cor- responding to the observed daily temperature instead of the mean daily rate multiplied by the number of days elapsed as usually employed. The advantages of this system seem so obvious that it is very desirable that every publicity should be given to it, and that it should be fairly tried at sea, not only as applied to the daily * Report on the Liverpool Observatory, 1853. 144 ON THE MANAGEMENT OF CHRONOMETERS. determination of the longitude in the course of the voyage, but also, with a scientific aim in view, in the accurate measurement of meridian distances. In the latter case it would be necessary to pay particular attention to the frequent redeterraination of the rates, noting care- fully their connexion with the 'temperature, and to take great care that the tabulated forms of predicted daily rates, depending partly on the primary table, and partly on the new observations, were constructed with minute regard to accuracy. For the full development of this system, the seaman must have the hearty and truthful co-operation of the chronometer- makers themselves, or the assistance of observatories or other public establishments, furnished with the necessary apparatus, for carefully testing the actual performances of chronometers under varying circumstances of temperature. At present, in so far as we are aware, Liverpool is the only port in this kingdom * where, thanks to the enlightened liberality of its municipality, a well-organised establishment for testing chronometers, and inform- ing the mariner as to their probable actual performances at sea, at present exists. Let us hope that the day is not far distant, when similar establishments maintained in some way, at the public expense, may exist, as an ordinary part of the machinery of navigation and commerce, at all our great naval and commercial ports, and that the benefits of this system may be better appre- ciated and more widely diffused. TiaM Method. English navigators, as we have before observed, have usually been in the habit of disregarding the refinements of temperature corrections, although, at least in modern times, fully recognizing change of temperature as one of the chief causes of marked changes of rate. In ordinary navigation at sea, the uniformity and invariability * A small establishment with similar objects in view, was organised by the Board of Trade a few years back, in the port of London. We regret to add, that partly from want of appreciation of its utility, and partly from the existence of professional prejudices, it failed to command that degree of support which would have justified the Government in continuing to maintain it, it was therefore, after a time, given up. Let us hope that this reproach totthe intelligence of the shipping interests of London may soon be wiped away. 145 of the initial rate of the chronometer, determined before starting on the voyage, are generally implicitly relied on during its continuance ; Lunars, the Lead, and a good Look-out, being trusted to, to help the mariner out of any difficulties that the infidelities of his chronometer might lead him into. In the measurement of " meridian distances," when scientific accuracy is aimed at, it has generally been assumed that if any change of rate has taken place it has varied uniformly and in proportion to the time. This idea, originally it would seem advanced by Borda, has been the foundation of all the methods employed by modern navigators, with few exceptions, for correct- ing the results obtained from chronometers for changes of rate. Tiarks slightly extended its application, and showed, by a simple algebraic treatment, how it might be employed to correct, in a uniform and systematic manner, not only the whole meridian distance between the terminal stations of the voyage, but also the partial differences of longitude between the intermediate stations, where the errors might alone have been determined, but not the rates. In cases where no attempt is made to correct the results by the chronometers for the errors produced by variations of temper- ature, or when the circumstances of the voyage, such as the shortness of the run, or prevalence of a uniform temperature, render such delicacy of treatment unnecessary, recourse must be had to Tiarks' plan, or to some similar system of treatment. Hence it will be necessary to enter at some length into its practical exposition. As this, however, will involve a good deal of detail, and the necessity for illustrating the formulas by examples, its amplification must needs be reserved for a separate chapter. Summary Remarks on the preceding Methods of correcting Chronometric Observations for Changes of Rate. In the preceding pages we have placed at some length before our readers the modes of treatment for correcting the deter- mination of the longitude at sea, or the measurement of meri- dian distances, obtained chronometrically, for the errors arising from changes of rate, under the combined and varying influ- ences of time and temperature, proposed by different authorities, De Cornulier, Lieussou, Mouchez, Hartnup, and Tiarks. L 146 ON THE MANAGEMENT OF CHRONOMETERS. Independently of a natural reluctance to pass a definite judg- ment on the views and opinions of authors of so much greater experience, and with so much more knowledge of the subject than himself, the writer of these pages is also restrained by the consideration that, in a matter elf this kind, resting to a certain extent on an empirical basis, experience is wanting to afford grounds for a satisfactory and final opinion. Time and expe- rience can alone decide what system of treatment is most suited for adoption, what formula of correction will prove most feasible in practice, and how far the so-called " constants," determined by observation, really continue permanent during long voyages. Unless skilfully and carefully handled, it is possible that the adoption of algebraic formulae, involving numerical coefficients of minute value, determined by observation, cannot be success- fully undertaken without incurring the risk, through a fanciful empiricism, of falling into errors greater even than those which it is our object to correct. Some few observations to assist the general judgment of our readers will, however, probably be expected of us, and will not be out of place. I. Change of temperature being now universally recognized as the principal cause of marked changes of rate, careful daily records of the actual temperature experienced by the chrono- meters during the periods of rating, and throughout the passages, become henceforth absolutely necessary and indispensable adjuncts in all reports of meridian distances. Even if the observer should not himself have employed t-hem in his calculations, they should be furnished for the use of the hydrographers, who may subse- quently have occasion to subject their results to a critical examination. II. Great care should be taken to use all accuracy in noting and recording these mean daily temperatures. Vague or erroneous records would be worse than none; the corrections sought for are always minute quantities, and careless observations might introduce greater errors than they seek to cure. III. In comparing the methods proposed by the different writers whom we have quoted, we would observe that De Cornulier and Mouchez derive their modes of correction from the immediate sea observations themselves, a very great advan- tage, since observers will thus be enabled to correct their own REMARKS ON THE PRECEDING METHODS. 147 measurements at the time. Mouchez's plan is more simple than De Cornulier's, by blending into one correction the consideration of the effects of variation of temperature and the acceleration. To this there does not seem to be any valid objection in theory, the admissibility of empirical treatment being conceded. Lieus- sou and Hartnup's plans require the aid of a careful series of observations for the determination of their constants, or tabulated rates respectively, made either at an observatory, or by the artist who has adjusted the chronometers. It is probable that these constants, and also the law of connexion which links the variation of rate to changes in the thermometric scale, may vary insensibly with the progress of time, and their reverification might then be difficult or impracticable on foreign stations at long distances from home. Both these methods require to have their utility and facile adaptation to practice, confirmed by experience gained at sea. IV. Lieussou's formulae, based upon the analytical investiga- tion of numerous careful observations, are singularly ingenious, and in the hands of a skilful computer seem well adapted to the measurement and correction of meridian distances. They are well worthy of the attention of scientific navigators, and we trust they will receive the consideration they merit. We cannot say that we concur in Mouchez's animadversions on Lieussou's formulae, or in his hostility to algebraic formulae in general. Algebra, the golden key which unlocks the portals of mathematical truth, is often only the symbolical expression of the plainest common sense. Its employment often clearly explains that which is obscure when expressed verbally. Mouchez's own views, even, are best explained, we think, by a simple algebraic treatment, as we have shown (ante, p. 136). Again, we do not think Lieussou's formulas deserve his strictures, as to their inap- plicability to practice, from their terms depending on elements deduced from byegone experiments ; for, as we understand that gentleman's writings, he has by no means proposed that algebraic formulae, involving constants derived from prior and distant ob- servations, should supersede the use of rates derived at the time from present ones. The part they play in the measurement of meridian distances, if we have rightly apprehended him, is only meant to be auxiliary and subordinate to that performed by the 148 ON THE MANAGEMENT OF CHRONOMETERS. data derived from the immediate observations of errors and rates, obtained at the time of the passage under discussion. With reference to Mouchez's remarks on the two expressions for the change of rate, considered as a function of the temperature, c (T ) 2 , and cf (t ^), we think the former, depending on the normal temperature T, has a greater probability of correctness, because we know as an independent fact, that the artist has the power of adjusting the rate between certain limits of temperature, and that as we swerve from T, the mean of the two selected tem- peratures of adjustment, the errors of the rate rapidly increase. On the other hand the term c' (t t f ), representing the variation of rate for a given difference of temperature, is not the same in all parts of the thermometric scale, as clearly shown by Hartnup's experiments before mentioned ; consequently the variation of the rate is not always correctly given by that expression, although it may be true, or nearly so, round and about the normal tempera- ture T. V. Hartnup's method, simple, and involving none but the easiest calculations, seems admirably adapted for use by the mercantile marine, in the daily practice of navigation at sea. In the merchant service, from divers obvious causes, facilities very often do not exist, for frequently checking the rates of the chro- nometers by fresh observations. On the other hand, the frequent return of merchant ships to their port of departure, affords the opportunity for reverifying the system of "tabulated rates" of their chronometers. This plan also possesses the important advantage, that it is simple, expeditious, and short. With proper precautions to ensure accuracy, as previously suggested, it might probably also be made available for the accurate measurement of r decreased uniformly by a given quantity from day to day, in vhich case the accumulated correction could be represented by ,he sum of an arithmetic series, in which the first term equals the common difference ; or that, supposing the increment or decre- nent of rate to flow on uniformly from moment to moment, its iccumulation could be expressed by the area of a right-angled triangle, whose base represented the time elapsed, and altitude the observed change of rate. We propose to adopt the latter supposition, consistently with what we have already done in treating of the combination of observations for rate at any given place, and to consider that any change of rate which is found to have taken place during a voyage between two epochs of rating at its commencement and termination, may be dealt with on precisely similar principles to those which we have shown to be practically admissible in discussing rate observations at any par- ticular place. The first idea, that of an arithmetic series, is that adopted by Flinders, King, Owen, and others; Tiarks, Fitzroy, and Bay- field* follow the latter hypothesis, viz. the area of a right-angled triangle. It will be instructive to compare the two methods, and not difficult to show that the first idea will give the correction slightly in excess of what it ought to be ; but in order not to interrupt the thread of our present discussion, this matter had better be reserved for investigation, at the close of this chapter. Let the mean daily rate of a chronometer corresponding to a particular epoch before starting on a voyage = a. * See " Voyages of Adventure and Beagle," Appendix to vol. ii. pp. 320-330 ; Owen's " Table of Longitudes," Explanation, p. 4; Forster's "Voyage," vol. ii. Appendix, p. 226; " Nautical Magazine," April 1843, p. 220; 1854, p. 169. 152 ON THE MANAGEMENT OF CHRONOMETERS. At the termination of the voyage, let the rate be again De- termined ; the interval between the epochs to which the rates are referred being represented by t. If it were found that the rate had not changed, but was still represented by a, the whole accumulation of the rate in the interval t would be given by the equation. Whole accumulation of rate = t a. Which again, as we have before observed, could be graphically represented by the area of a rectangle, whose base A B = t, the time elapsed ; and altitude B C = a, the uniform rate. AB B C If, on the contrary, it were found that the rate had changed to a + b* on the assumption that the change from a to a + b had taken place uniformly, and in proportion to the time, then, as we have formerly shown, the whole accumulation during the period t could be represented by the area of the figure ABED; that is, by the area of the rectangle A B C D added to the area of the triangle DOE. B E a + b * a and J being either positive or negative, as the case may he : gaining rates, and the alteration of rates in a gaining direction, being considered positive ; while losing rates, and the alteration of rates in a losing direction, are to be taken as negative. TIARKS' FORMULAE OF CORRECTION. Therefore, whole accumulation of rate during period t, 153 = ta + t- and as the rate, a + |, is the mean of the former rate a, and the latter rate a + b, it at once follows, as Tiarks observes,* that the adoption of our hypothesis leads directly for the whole interval t, to the use of the mean of the rates at the commencement and termination of the voyage.f In a similar manner, for any partial interval during the voyage r, for which it is proposed to determine the correction, the whole accumulation of rate may be represented by the area of the figure A F H D, if A F = r, and if r be supposed to com- mence from the epoch when a, the first rate, was determined; that is, at the commencement of the period t. B C = B E a + b Now, the area A F H D is equal to the area of the rectangle A F G D, added to the area of the triangle D G H. And by the principle of similar triangles, their areas being as the squares of their like sides, AreaDGH : areaDCE :: D G 2 : DC 2 AreaDGH : *- : : r 2 : # 2 or .-. AreaDGH == - b Hence, Whole accumulation of rate for interval T = ra + ^ b * Forster's " Voyage," vol. ii., Appendix, p. 226. + This is also in accordance with the method formerly adopted by Fleurieu. " Voyage de Fleurieu," vol. ii. p. 541. 1 54 ON THE MANAGEMENT OF CHRONOMETERS. Suppose now that after starting on a voyage from a place A, where the rate a had been determined, the ship calls at two (or more) intermediate places, B and C, &c., and finally arrives at K, where a new rate, a 4- b, is obtained, and that observations for the errors on local meaii-time onjy were made at B and C (time or circumstances not permitting the rates to be determined anew at those places), it is required to investigate formulae for the corrections due to the several meridian distances between the places A, B, C, &c., and K, using the observations for error at any two of them respectively, and the rates determined at the terminal points A and K. Let the interval between the epochs for which the rates were determined at A and K = t, between the epoch at A and the observation at B = r, and between the epoch at A and the observ- ation at C = /, and so on. Then, for the whole correction for the accumulated rate be- tween the places A and K, where the rates were determined, we have the expression, Correction = t a + t - (#) For that between A and B, Correction = r a -f ~ b (p) Similarly between A and C, Correction = r'a + b (y) By taking the difference between (7) and (/?) we obtain that between B and C, /2 <2 Correction = (T' T) a + - b and similarly that from C to K, TIAEKS' FOEMUUE OF COEEECTION. 155 These several formulae* are sufficient for all the cases which can arise in practice in the computation of meridian distances, where the rates have been determined at both ends of the chain. They are analogous to those employed by Tiarks,| and, we think, rather more simple in expression, and more satisfactory: since, by our mode of proceeding, the errors as well as the rates being referred to the same mean epoch, the notation employed admits of some simplification, and the formulae can be expressed with more conciseness. From what has been said above it follows, that if the rate of a chronometer has been found at the first terminal station only, the meridian distances between it and the places subsequently called at can only be approximately determined in the first in- stance, and will be liable to future correction should it afterwards appear that the rate of the chronometer has changed during the voyage. But if the difference of longitude between any two stations be accurately known, and the rate at one of them only has been determined, then this known difference of longitude may be used as an element in the calculation, and be itself made subservient to the correction of the longitudes of places intermediately touched at between them ; for, as we shall presently show, we then have the means of ascertaining the variation of the rate 5, which may have taken place during the voyage, and hence can apply our formulae (/3), (y), (a), and (g), to the cases in question. This position may be made clear by the following simple algebraic treatment: Let M be the "difference of longitude," or the " meridian distance," between any two stations A and K, M being considered * The number of places intermediately touched at on the voyage may be indefinite, care being taken that in the application of the formulae generally A and K represent the points of arrival and departure where the rates are determined, and B, C, &c., any intermediate places whatever between them, where the errors on time only have been ascertained. f See Forster's " Voyage," vol. ii., Appendix, p. 228. It will be observed on ^ comparing the above with Tiarks' formula, that the expression 6 in formula (/3), 2 t while it is identical in form with his formula (2), is at the same time virtually the same as his more accurate expression given in formula (i), viz. - , ' b> 2 1 + a + d in consequence of our adoption of the idea of the variation of the rate commencing at the mean epoch (that is, at the middle of the interval for which the rate is deter- mined), and consequent alteration of notation. 156 ON THE MANAGEMENT OF CHRONOMETERS. positive or negative, according as the position of the place arrived at is west or east of that sailed from ; and let X and X' be the errors of a chronometer on local mean time at the two places, the errors being reckoned as positive when considered fast, and negative when slow. Then, if the chronometer kept true mean time, M = V - A. But if the rate of the chronometer at starting from A were a, Then M = A' (A + t a) which would be a true expression for M, if a remained constant, and an approximate one, if it were found on arrival at K that the rate had changed during the voyage, and become a + b ; and in this case, in accordance with our hypothesis, M Now in this expression, if M be assumed to be known, and b, the alteration of the rate, be taken as unknown ; then, by trans- position, we can find b from it. M For t (a + |) = (A' - X) -1 , b (>.' x) M and a + - = ^- - a + -, the mean sea rate for the interval, being thus known, and a having previously been determined by observation, b can easily be found, Since b = 2 (a + - a) The whole change of rate b being thus known, it may be used by substitution in formulae (/3), (7), (5), and (i)* to determine the respective meridian distances of the places previously touched at on the voyage. * See ante, p. 154. TIARKS' FORMULAE OF CORRECTION. 157 Again, if the error alone of the chronometer had been accu- rately ascertained at the commencement of the voyage (the pres- sure of business at the moment of departure, uncertain weather, or other adverse circumstances, preventing the corresponding de- termination of the rate), but the rate a + b, had been found at its termination ; then, since a -\ , the mean sea rate, becomes known from formula (), a, the original rate at starting, and b, its alteration during the voyage, may readily be found. Since a = 2 (a -\ j (a + 6) and b = 2 (a + 6 a + -) a and b being thus known, they can be applied as needful in formulae (/3), (7), (5), and (e), as before. Recapitulating our notation and formulas for the sake of clearness and facility of reference, we have, Places called at during the voyage, being respectively represented by A, B, C, &c., and K. Rate at A = a K == a + b (a and b being positive or negative respectively, as the case may be). Error of chronometer at mean epoch at A = A K = A' at moment of observation at B = A Meridian distance A to K = M A to B = Mj B to C = M 2 C to K = M 3 Interval between the epochs of the observations, A to K = t A to B = T A to C = *' Then, for the meridian distance from A to K, we have, 158 ON THE MANAGEMENT OP CHKONOMETEES. from A to B, (2) from B to C, M 2 = * 2 - A t + 7* + j (3) from C to K, M 3 = A' - X, + =7a + ' b (4) If the places intermediately called at during the voyage are more numerous than two, viz. B and C, as we have here supposed in the arrangement of these formulae, their connexion with the points of departure and arrival, as well as with one another, can readily be obtained from the preceding formulae by assigning their appropriate values to r and r, for the several places whose differences of meridians are sought. Again, if the difference of longitude between the points of arrival and departure be assumed to be accurately known, then the mean sea rate, By the aid of which expression, if the first rate, , at the place of departure, has alone been determined, b can be found. For b = 2. (a + b - - 0) (6) And if a have not been determined, but (a + ?), the rate at the place of arrival has been found. Then a = 2 (a + *) (> + b) (7) and b = 2 (a + b - a + |) (8) The respective values of a and b being thus known, and employed by substitution in formulas (2), (3), and (4), the meridian distances of places intermediately touched at during the voyage can readily be determined. Observation I. Although in the application of the preceding formulae, for the deduction of meridian distances, to cases in 159 actual practice, attention to the algebraic signs of a and b will of course always be necessary to ensure true results ; yet, perhaps, the best course for computers, who may not have been accustomed to work from formulae, to pursue, will be to take a plain practical view of the matter, and to apply the several corrections with reference to their obvious effects, rather than hamper themselves in their treatment by too close an attention to the apparent effects of the algebraic signs. Of course, when duly interpreted, the results by the two methods will always agree. Thus, if the error of a chronometer be fast, and the rate gaining, or sloio, and the rate losing, the error will be increased by the effects of the accu- mulated rate, and the contrary if vice versa. And again, if the rate, whether gaining or losing, is increasing, it must be re- membered that the effects on the error of the two parts of the correction depending on a and b, will be in the same direction, and they must therefore be similarly applied ; but if the rate, whether gaining or losing, is decreasing, then the contrary is the case, and the two parts of the correction must be applied differently. Observation II. The intervals, t, r, and /, will usually be either integral numbers or simple decimal quantities, obtained by taking the differences between the dates of the epochs to which the observations are referred. But when the arc of longitude traversed during the voyage is considerable, it will be proper, when extreme accuracy is aimed at, to correct the intervals t, r, and /, by the fractions of a day, corresponding to the approximate difference of longitude, by means of the table given in the Ap- pendix for converting intervals of time into decimals of a day. For example, suppose the epoch of rating at the commence- ment of a voyage at Portsmouth is June i d *875 (the observations having been made at 9 A.M. mean time, on June ad), and that on the termination of the voyage to the West Indies the rates were again obtained at Barbadoes, at the epoch July 6^875, i.e. at 9 A.M. on July yth. Here the apparent interval or value of =35 d -ooo ; but since Barbadoes is 3 h 45 west of Portsmouth, the apparent interval, t, should in strictness be corrected by o d 'i62, the fraction of a day to which 3 h 54. corresponds, and the true value of t to be substituted in the calculation will become 35 d 'i6a, the in- terval being increased because the ship loses time by sailing to the westward. 160 ON THE MANAGEMENT OF CHRONOMETERS. In a similar manner, if the place of arrival is to the eastward of the point of departure, the apparent value of t should be decreased by the amount of the correction due to the approximate difference of longitude, because the ship gains time by sailing to the eastward. i* Of course the apparent values of r and / should, under like circumstances, be increased or decreased in a similar manner. Again, if any of the errors or rates have been obtained by the method of " equal altitudes," the epochs of which refer to the moment of apparent noon or midnight at the place of observation, the intervals of time t, r, and r, obtained from them by com- parison with other epochs, should in strictness be corrected for the variation in the "equation of time" in the interval; and the best mode of doing this in practice will doubtless be by reducing the moments of apparent time to the corresponding moments of mean time, by applying the equation of time for the given day to each day's observation separately, and then taking their differences as before, which will give the apparent values of t, r, and r', in mean solar time. If the errors of the chronometers on time at the place have been obtained by transits or altitudes of the stars, it will be ad- visable to reduce the epochs to which they refer to their equiva- lent expressions in mean solar time, and then proceed as before. Observation III. It will be observed, that where more than one chronometer is employed in the measurement of a meridian distance, the factors which multiply, a -and b, in the formulae, are common to all the chronometers, a fact of considerable import- ance in the reduction of labour, and rendering far less formidable than appears at first sight the manipulation of a large number of chronometers in investigations of this nature. In fact, if it be thought desirable, the ultimate mean value of the several deter- minations of each chronometer taken separately may be obtained at once, by substituting in our formulae, in lieu of the simple values of a and b for each chronometer, their mean value for the whole (regard being paid to their algebraic signs), and then pro- ceeding as for a single chronometer, using also the corresponding mean values of X, x', X 1? &c. The result will be the same as if one fictitious chronometer had alone been used, whose errors and rates represented the mean values of those appertaining to the whole number of chronometers actually employed. TIARKS' FORMULAE OF CORRECTION. 161 This mode of proceeding is obviously less laborious than applying the corrections for, and obtaining the results of, each chronometer's performance taken separately. It has, however, the disadvantage of masking the results by, or the effects of the irregularity of, any particular chronometer in the general mean, and, therefore, of diminishing the ability of the computer to judge and discriminate between the separate performances of individual chronometers ; at the same time, as affording a check on the ac- curacy of calculation, and as facilitating the computation of results in duplicate, the method has its value, and is worthy of being pointed out for the benefit of seamen. Observation I V. Since the value of chronometric differences of longitude is entirely dependent on the assumed regularity of performance of the watches employed in the measurements, it is inexpedient to place too implicit a reliance on the theory of an equable and uniform variation of rate, which forms the basis of the preceding investigations, over longer periods than is absolutely unavoidable, and advisable, on the contrary, to limit that as- sumption in practice to periods as short as the nature of circum- stances admits of. It may, moreover, be held as a general rule, that the de- pendence which may be placed on the result of any given measurement is inversely proportional to the number of days occupied in the transits from place to place, or elapsed between the epochs of the observations. It will, of course, often be out of the power of those who undertake the measurements of meridian distances to arrange the conditions under which they are to be performed, or to do more than accept for that purpose such opportunities as chance may place in their way. In so far, however, as they may have the control over their operations, it would be well for them to arrange that the obser- vations for the determination of the errors and rates before starting should immediately precede the departure from port, and that at the termination of the voyage no unnecessary delay should elapse before those elements were again determined. The diligent observer will, moreover, of course, avail himself of every opportunity of obtaining the time at intermediate stations ; and, in order to break up the measurements into as short links as M 162 ON THE MANAGEMENT OP CHRONOMETERS. possible, will seize every occasion of obtaining new rates when the ship remains sufficiently long at any station to afford time for that process. Combination of Mouchez's Plan of Correction for Temperature with Tiarks' Method. We now proceed to explain how, in cases where the fluctu- ation of temperature during the passage has been considerable, its effects on the performances of the chronometers are to be allowed for. In the present state of chronometric science, and pending the decision of time and experience on the methods proposed by Lieussou and Hartnup, we think that we cannot do wrong in recommending Mouchez's method to the attention of navigators ; although perhaps not theoretically perfect, it at any rate possesses three essential properties, which render it suitable for practice at sea, and may cause it to find favour with seamen : the corrections for temperature are derived from the actual sea observations, while the application of them is simple, expeditious, and short. Adhering in other respects to the notation* previously em- ployed in this chapter, to explain Tiarks' mode of treatment, let d be the mean daily temperature corresponding to the period of rating, when the rate was a ; = ra -\ b accumulation of the ratej Hence their difference . = b J is the excess of the former over the latter. A graphic delineation will be useful to show that this dif- OP THR - 'UNIVERSII COMPARISON OF FLINDERS AND TIARKS METH01 ference is a real excess, and that Tiarks' idea, which we adopt, the more consistent hypothesis of the two, and affords the more elegant result. . 4 G In the annexed figure, let A B or D C = t represent the whole interval between the epochs for which the rates were determined, and let A F or D G = r represent the partial interval for which it is proposed to find the correction. Also, let AD, the original rate, = a, and C E, the change of rate, = b. Let C E, the change of rate, be divided into t parts, C d, d e, ef, fg, &c., each, therefore, equal to -. Also, let A B or D C be, in like manner, divided into t parts ; each, therefore, equal to unity, since A B = t. Draw the parallels d v, et, fs, &c., and also the perpendiculars, v D, t u, s x, &c. Now, according to Tiarks' view, the whole accumulation of the rate for the interval D G or r is represented by the areas A F G D and D G H, in which the supplemental part due to the change of rate, b, is represented by the area of the triangle D G H ; and in Flinders' plan, assuming the supplemental part to be the sum of an arithmetic series, in which the first term equals the common difference, the proportion for the first day = - x i = the small rectangle D u m v ; That for the second day = - x 2 = the sum of the small rectangles uxlm and mint; 168 ON THE MANAGEMENT OP CHRONOMETERS. And similarly for the third day, the sum of the small rectangles, xypl, Ipkn, and nkos and so on. Hence, therefore, the whole supplemental correction for the days r, evidently equals the sum of all the small rectangles to the left of the line G H, which, agaia, evidently exceeds the area of the triangle, D G H, by the sum of the small triangles D m v, mnt, nos t &c. Now the area of each of these small triangles obviously equals - X -, and, therefore, for r days their sum = b ; which excess, obtained by a geometric consideration, equals that deduced before from algebraic principles. We think that, as Tiarks remarks, his formula is more correct, and rather more simple ; and it is surely more natural and more consistent to suppose that the accumulation of the rate flows on equably, following the law of the area of a right-angled triangle, than that it proceeds by successive increments corresponding to given increments of time in accordance with the law of an arithmetic series. It is worthy of note, that the idea of the area of a right- angled triangle is what the latter supposition resolves itself into, if the increment of the time dx and the increment of the rate dy are both assumed to be indefinitely small, and the limits of x and y respectively are taken from x = o to x = t, and from y = o to y = b; in which case, by integration, we shall have, Area of rectangle, or twice area of triangle = / / dx . dy *J *J And performing the integration, Twice area of triangle = tb Hence area of triangle, or whole accumula- "1 tion due to the variation of rate for the > = time t J Whence, as before, Accumulation for the partial interval T . . . . = - b See Appendix to Forster's " Voyage," vol. ii. p. 226; "Yoyage of Adventure and Beagle," Appendix to vol. ii. p. 320 ; Owen on Longitudes, Explanation, p. 4. METHODS OF DUMOULIN, DESBOIS, AND PLOIX. 169 Methods of Vincendon Dumoulin, Coupvent Desbois, and Charles Ploix. Besides the methods of chronometric treatment, explained in this and the preceding chapter, proposed by De Cornulier, Lieussou, Mouchez, Hartnup, and Tiarks, the subject has also received further illustration from the recent writings of MM. Vincendon Dumoulin, Coupvent Desbois, and Charles Ploix. The two former gentlemen are the joint authors of a hydrographical memoir, on the voyage of the Capricieuse, and since reprinted in the " Recherches Chronometriques," pp. 177-207, which we have already had occasion to quote in these pages (ante, p. 99); M. Ploix's memoir is also given in the same work.* Although their mode of proceeding is somewhat different, the principle of their two methods is essentially the same. Instead of employing only the rates of departure and arrival, as is done by Tiarks, in measuring meridian distances, they propose to utilise for this purpose all the rates which may have been obtained dur- ing the voyage, at any of the intermediate points at which the ship may have touched. The equation of rate of a chronometer may always be written under the form y = F(*) in which x represents the time elapsed since some given epoch, and y is the ordinate of the curve of the chronometer. The curve of a chronometer being defined to be the locus of a point whose abscissa is a line representing the time elapsed since a given epoch, and ordinate a line representing the actual rate of the chronometer, at that moment. If only the rates of departure and arrival are employed, the curve of the chronometer becomes a straight line, and its equation is y = a, -f- bx If several determinations of the rate have been obtained, each one furnishes a value of the ordinate y } and defines a point in the curve, whose equation may then be written under the form y = a + b x -f c x* + d x 3 + &c. in which there will be as many coefficients a, b, c, &c., to * Sur le calcul des longitudes determines au moyen des chronometres ; par M. Charles Ploix, " Ingenieur Hydrographe." "Recherches Chronometriques," p. 209. 170 ON THE MANAGEMENT OF CHRONOMETERS. determine, as the observations have furnished data to clear up the question. The whole problem then consists in the calcu- lation of these coefficients ; the definitive equation being once obtained, by putting for #, any date whatever (reckoned from the epoch of departure) the value; of y that we deduce from it, will be the rate of the watch at that date. We should have thus, the rate for every day of the passage. The whole accumulation of the rate of a chronometer in the interval between any two observations is, in any case, the area comprised between the axis of #, the curve cf the chronometer, and the two ordinates corresponding to the epochs of the ob- servations in question. When only two observations are em- ployed, the curve of the chronometer, as we before remarked, becomes a straight line, and this area is the rectangle ABED (ante, p. 152), as we have already shown while discussing Tiarks' method. In any other case, all the observed rates during the voyage being employed for the solution of the question, the area is the definite integral ry.dx, which may be determined by the application of the principles of the integral calculus. On this matter MM. Vincendon Dumoulin and Coup vent Desbois remark, " Although we do not know the nature of the curve of a chronometer, we know that it must pass through certain known points. Now to determine the form of the function y = /(V), we can employ the means furnished by the calculus of differences, which geometers use to determine the value of a function corresponding to a certain value of the vari- able, without knowing the nature of this function, provided that it be intermediate between the given values of the function, corresponding to certain values of the variable." These gentlemen then proceed to show, how the formulae of interpolation, proposed by Lagrange, may be employed for the solution of this problem. M. Ploix, in a subsequent memoir, given in the "Recherches Chronom&riques," while expressing his general concurrence with these writers' views, points out how the treatment of the question may be somewhat simplified. The mathematical discussions in these memoirs, interesting and elegant though they be, are somewhat too abstruse for these pages, and we do not therefore consider it necessary to enter more fully into their consideration, or do more than take passing METHODS OF DUMOULIN, DESBOIS, AND PLOIX. 171 notice of them, and commend them to the attention of the scientific student. We are the more reconciled to this course, because we gravely doubt whether, as a matter of practice, the application of these principles is an improvement on the plan recommended for adoption in these pages, viz. to employ Tiarks' formulae, combined with Mouchez's for correcting for temperature, to break up the meridian distances into as short links as possible, and to obtain the rates as often as possible. Every meridian distance depend- ing on two observed rates obtained before departure, and after arrival, is then complete within itself, as one of the connecting links in the chain of " differences of meridians," found chrono- metrically during the voyage. No doubt the assumption, that the curve of the chronometer in the interval is virtually a straight line, involves some errors, for probably in all cases it is really a sinuous curve, having possibly many points of flexure; hence the calculation of the area of the figure, which represents the accumulated rate during the passage, on the supposition that it is rectilinear, whereas in truth one of its containing sides is a curve, must be deficient in accuracy. The errors thus introduced may, however, be restricted within very narrow limits, by taking the intervals as small as possible, and probably a considerable part of them may be destroyed, by the application of the correction for temperature (as previously recommended), of which these errors themselves would seem to be chiefly a function. The residual error would then probably be extremely small. In other respects, the plan pro- posed by these authors would probably, in practice, involve the employment of rate observations extending over a long period. MM. V. Dumoulin and C. Desbois themselves, give an example in illustration of their method, in which the interval between the extreme observations exceeds sixty days. No doubt, in the measurement of "meridian distances" the shorter the interval the better, and as Mouchez truly observes, " the last rate obtained is always the best."* Tiarks' formulae, moreover, are so simple and so well adapted to practice, that their use ought not to be displaced on light grounds, and hence, for the present at least, we recommend that they be adhered to. * " Recherches Chronometriques," p. 273. 172 ON THE MANAGEMENT OF CHRONOMETERS. CHAPTER VIII. VARIOUS EXAMPLES OF THE COMPUTATION OF MERIDIAN DISTANCES, ILLUSTRATING THE APPLICATION OF TIARKS* FORMULA GIVEN IN THE PRECEDING CHAPTER. WE shall now proceed to illustrate the formulae for the measure* ment of meridian distances given in the preceding chapter by some numerical examples, exhibiting the varieties of the different cases which can arise in actual practice. In the arrangement of the examples we shall suppose, in all cases, that three chronometers have been employed in determining the meridian distance : this number will be conveniently suitable to the width of the page, in so far as typographical arrangements are concerned, and at the same time, without being unnecessarily diffuse, and without distracting the attention of the student by a needless amplification of numerical details, it will probably be found to afford sufficient variety in illustration of the application of the corrections for rate, and of the general manipulation of the formulae. Each example thus exhibiting the results by three chronometers, is to be taken as a type of the computations re- quired for any number whatever, the reduction of results by extra ones being made on precisely similar principles, and only involving a slight addition for each chronometer to the numerical calculations already made. Case L Meridian distance from A to K. Errors and rates determined at each place. Example I. By single-altitude A.M. observations on Jan. ist and 7th, 1 844, at Ross Bank Observatory, Hobarton, Van Diemen's Land, the errors and rates of chronometers Z, C, and I, at the mean epoch, Jan. 3 d< 875, were, Errors. Rates. h m 8 s Chron. Z 9 53 i7'58 + 0-13 C 8 47 24-63 + 10-41 I - 10 50 55-83 - 4-29 EXAMPLES OF MERIDIAN DISTANCES. 173 On arrival at Sydney, New South Wales, by equal-altitude obser- vations at Garden Island on Jan. 26th and Feb. 2d, the errors and rates at the mean epoch, Jan. 29^5, were found to be, Errors. Rates. h m 8 s Chron. Z io 9 1-02 1-14 C - 8 58 37-94 + 10-26 I ii 8 32-29 5-07 From these data the meridian distance is required. The rates of the chronometers having been determined at each place, the meridian distance is to be obtained by formula (i). M = A'- In which expression A' refers to the errors at Sydney, and A to those at Hobarton; also #, the interval between the epochs, is 25^625, and the mean rates fa + -j for the several chronometers are, Z 0-50 C + 10-33 I - 4-68 Whence we have, Chron. slow on Hobarton 'j mean time, Jan. 3^875, j> or A j Z. C. h m s h m s 9 53 i7'5 8 -8 47 24-63 I. h m s -io 50 55-83 Accumulated rate or t (a + -\ 12*81 +4 24-70 -I 5 9 - 9 2 Chron. slow on Hobarton mean time, Jan. 2o d< 5, . / 8\ ? 9 53 3 39 4 2 59'93 io 5 2 5575 or A + Do. do. at Sydney, or A' io 9 1*02 8 58 37-94 n 8 32-29 Meridian distance .. o 15 30-63 o 15 38-01 o 15 36-54 Hence, as the meridian distance by these three chronometers, we have > h m 8 Z o 15 30-63 C 38-01 1 3^54 Mean .. o 15 35-06 174 ON THE MANAGEMENT OF CHRONOMETERS. Example 2. By equal-altitude observations at Moulmain (Go- vernment Wharf), on Jan. loth and I5th, 1853, the errors and rates of chronometers A, C, and Y, at the mean epoch, Jan. i2 d> 5, were as follows : Errors. . Rates. Chron. A 6 7 14-80 5-34 C 6 21 24/56 + 2'i6 Y o 17 55-06 + 5-96 Subsequently the errors and rates at Rangoon at the mean epoch, Jan. 22 d *5, by equal altitudes at the Commodore's Wharf on Jan. I9th and 26th, were, Errors. Rates. h m s s Chron. A 6 2 11-92 4-07 C 6 15 11*02 + 2^67 Y o 10 54-27 H- 7-89 Here, as before, the rates having been obtained at each place, the meridian distance is to be obtained by formula ( i ). M = A' j A + t (a + | where A refers to the errors at Moulmain, and A' to those at Rangoon, while t= io d -o. Also, for the respective chronometers, the accumulated correction for the rate, or t (a +-) will be, Chron. A 47-00 C + 24-20 Y + 69-20 Whence we have, A. C. Y. Error of chron. on Moulmain 1 h m s h m s h m s mean time, or A, on Jan. I 6 7 14-80 6 21 24-56 o 17 55-06 I2 dt 5 Accumulated rate, or t (a + -) 47-00 +24-20 +i 9-20 Error of chron. on Moulmain 1 mean time, on Jan. 22^5, I __ 6 g r8o _ 6 2I 6 _ o , 6 86 or A + t(a + -) Error of chron. on Rangoon ^ mean time, Jan. 22^5, 162 11-92 6 15 11-02 o 10 54-27 or A' J -j-o 5 49-88 +o 549-34 +o 5,51-59 EXAMPLES OF MERIDIAN DISTANCES. 175 Therefore, by these chronometers we have, as the result for the meridian distance, Chron. A + o 5 49-88 C 49*34 Y 51-59 Mean . . + o 5 50-27 Case II. Meridian distances between A, B, C, and K. Both errors and rates determined at the terminal stations A and K ; while errors on time only were ascertained at the intermediate stations, B and C. Example i . In a voyage between La Guayra and Carthagena in the West Indies, calling on the way at Porto Cabello and Cura9oa, the following observations having been made, the respective meridian distances are required. By observations at La Guayra on May zzd and 28th, the errors and rates of chronometers F, M, and P, at the mean epoch, May 24 d> 885, were as follows: Errors. Rates. h m s s Chron. F + 4 33 7-80 077 M + 4 o 17-40 + 4-54 p + 5 9 437 + i '47 On arrival at Porto Cabello, the errors on mean time at the place on June 5^870, were ascertained to be, h m s Chron. F + 4 37 15-80 M + 4 5 31-28 P + 5 14 13-38 Passing on to Cura9oa, the errors on June 12^890 were, h m s Chron. F -f 4 40 59-20 M + 4 9 55'53 P + 5 l8 3* 2 4 And finally, on arrival at Carthagena, observations on the 25th and 29th June gave the errors and rates at the mean epoch, June 27 d , as follows: 176 ON THE MANAGEMENT OF CHRONOMETERS. Errors. Rates. h m s s Chron. F + 5 7 23-55 ' 8 5 M + 4 37 47*9 8 + 5'9 P + 5 44 34*4 2 - '3 Here, as elements of {he calculation, we have for the intervals between the epochs of the observations, #= 33 d - 139 (between La Guayra and Carthagena) ; T= 11^988 (between La Guayra and Porto Cabello); and T'=i9 d -oio (between La Guayra and Cura9oa).* Also, for the several chronometers, the mean rate, a + -, and the variation of the rate, b, is as follows : Mean Rate ( + |) Variation of Rate, 6. s s Chron. F 0-81 0-08 M + 5-22 + 1-36 P + 0-58 177 Then, first, for the meridian distance between the terminal stations La Guayra and Carthagena, where the rates were determined, we have, by formula (i), M = V- and solving numerically, Accumulated rate t ( + ) -26-84 +2 52-98 +19-22 Error at La Guayra, June 27th,l \* +43240-96 +4 310-38 +510 2-92 44 + o 34 42-59 +o 34 37-60 +o 34 31-50 Whence we have, Meridian distance La Guayra to Carthagena. h m s Chron. F + o 34 42-59 M 37-60 P 3i- Mean .. + o 34 37-23 * Applying to the several apparent intervals the corrections for the approximate difference of longitude, viz. o d> O24, o d< oo3, and o d< oo5. EXAMPLES OP MEKIDIAN DISTANCES. 177 Secondly, for the partial measurement between La Guayra and Porto Cabello we have, by formula (2), in which A refers to the errors at La Guayra, the place left, and A, to those at Porto Cabello, the place arrived at. Also the factor -^- is [0-336124],* and for the several chrono- meters we have, Chron. F 9-23 - 0-17 - 9*4 M + 54-43 + 2 '95 + 57-38 P + 17-62 - 3-84 + 1378 And applying the corrections, F. M. P. Error of chron. on La Guayra 1 h m S h M.T., or A, on May 2 S th . . / + 4 33 7'8o +4 m s h o 17-4 +5 m s 9 43-70 Accumulated rate (r a + bj -9-4 +57*38 +1378 Error at La Guayra, June 6th, 1 or (A + cor.).... ; j + 4 32 58-40 +4 11478 +5 957^8 Error at Porto Cabello, Junel 6th, or A X j+437i5-8o +4 5 31-28 +51413-38 + o 4 17-40 +o 4 16-50 +o 4 15-90 Hence, for the meridian distance between La Guayra and Porto Cabello, we have, h m s Chron. F + o 4 17-40 M 16-50 p i'o Mean . . 4- o 4 16-60 Thirdly, for the intermediate measurement between Porto Cabello and Cura9oa, by formula (3), we have, where AJ refers to Porto Cabello and Ag to Cura9oa. Also (T' r) = 7 d -022, and the factor r +r - r '-' r = [0-516425]. * A number enclosed between brackets, as above, signifies the logarithm of the numerical factor under discussion. 178 ON THE MANAGEMENT OF CHRONOMETERS. Whence, for the several chronometers, we have, / ( f [ 1 1 _/ _ J -/ n- /T + zt 0*26 + 47 - 5-81 [*. 2# s - 5-67 + 36-35 + 4-5 ' Chron. F 5-41 M + 31-88 P + 10-32 And proceeding as before, F. M. P. Error of chron. on Porto Ca- ] h m B h m B h m s bello M.T., on June 6th, S +4 37 15-80 +4 5 31*28 +5 14 13*38 or A! J Accumulated rate (as above) 5-67 +36*35 +4'5 I . + ^^o,3 +4 6 r6 3 +5 H .7-89 Error atCura9oa, June i3th, orA 2 +4 40 59-20 +4 955 '53 +518 3-24 + o 3 49-07 +o 3 47-90 +o 3 45-35 Therefore, for the meridian distance between Porto Cabello and Curagoa, we obtain, h m B Chron. F + o 3 43*07 M 47-90 P 45-35 Mean . . + o 3 47-44 Lastly, for the final link between Cura9oa and Carthagena, formula (4) gives us, M, = V - in which A 2 refers to Cura9oa and V to Carthagena. Also * r'= I4 d -i29, and the fa'ctor t + } Chron. F 10-88 o- 11-77 M + 64-15 + 15-12 + 79-27 P + 20-77 19-68 -f 1-09 EXAMPLES OF MERIDIAN DISTANCES. 179 Whence, as before, M Accumulated rate (as above) 1177 + 1 19-27 +1*09 +^"- + * '"4-8 +5-8 4-33 * S 723 ' 55 ++3747-98 + 54434-4* + o 26 36-12 +o 26 33*18 +o 26 30-09 Therefore, for the meridian distance from Cura9oa to Carthagena, we have, h m s Chron. F -f o 26 36-12 M 33-18 P 30-09 Mean . . + o 26 33*13 Finally, collecting the several partial results for examination and comparison, we have, Meridian distance, h m s h m s I. La Guayra to Porto Cabello -fo 4 17*40 -f-o 4 16-50 -fo 4 15*90 II. Porto Cabello to Cura9oa +o 3 49*07 -j-o 3 47*90 o 3 45*35 III. Cura9oa to Carthagena .. +o 26 36-12 +o 26 33*18 -j-o 26 30-09 Therefore, meridian distance T from La Guayra to Cartha- gena, by sum of partial I + 34 4*'S9 + 34 37'58 + 34 3>'34 measurements J Do. by direct measurement .. +o 34 42-59 +o 34 37*60 -fo 34 31*50 The accordance of the final results by the partial and direct measurements shows that, whatever may be the merits of the measurements, in so far as particular chronometers are concerned, the formulae here employed deal with the observations in a uniform and systematic manner, and at any rate yield consistent results. 180 ON THE MANAGEMENT OF CHRONOMETERS. Example 2. By equal-altitude observations at Cape Upstart (north-east coast of Australia) on May 2d and 6th, 1844, the errors and rates of chronometers A, B, and C, at the mean epoch, May 4 d , were as follows : Errors. / Rates. h m & s Chron. A 9 46 16-20 079 B 9 49 47-85 0-46 C 8 28 26-55 + 10-94 On arrival at Lizard Island on May 1 2th, equal-altitude observa- tions gave the errors of the chronometers on local mean time, h m s Chron. A 9 37 12-22 B - 9 40 37-92 C-8 17 48-82 And finally, equal-altitude observations at Sir Charles Hardy's Islands, on May 2ist and 25th, gave the errors and rates at the mean epoch, May 23 d , as under, Errors. Rates. h m s s Chron. A 9 29 28-82 1-39 B 9 32 58-72 2-29 C 8 7 56-27 + 10-83 From these data the several meridian distances between the stations are to be obtained. Here t= i9 d -oo (the interval between the observations at the terminal stations), and r=8 d -oo (the interval to the intermediate observation at Lizard Island). Also ~-j = [0-226396] and - ^f^- = [0*872973] while we have for the several chronometers, Mean Rate (a + - J Variation of Rate (6). 8 S Chron. A 1-09 0-60 B - 1-37 - 1-83 C + 10-88 o-n Then, first, for the meridian distance between the terminal stations, Cape Upstart to Sir C. Hardy's Islands, we have, by formula ( i ), EXAMPLES OF MERIDIAN DISTANCES. 181 And applying the formula, Error of chron on C Upstart! > , 6 . 2O _ * - _ 8 a? ^ mean time, May 4 d , or A . . J Cor. for rate t (a+-^\ 2071 26-03 +3 2672 Error at C. Upstart, May 23 d , 1 or A 4- t (a + -) ' -9 4 6 3 6 '9 I 9 52 9 A 56-63 Mean .. o 15 52-13 EXAMPLES OF MEKIDIAN DISTANCES. 187 In a similar manner the meridian distance from Amoy to Shanghai could be determined, by substituting in formula (4), M s = V - the values of b previously ascertained ; whence we should obtain, on proceeding with the computation,* h m s Chron. Z o 13 28-56 M 2578 A 2I'I2 Mean . . o 13 25*15 Comparing this with the previous measurement from Hong Kong to Amoy, their sum equals o h 29 i7 s- 28, which (small errors ex- cepted) is the same as the assumed value of the difference of longitude of the terminal stations. The same fact would be observable if we compared together the individual results of each chronometer separately. Hence we see, that by this mode of proceeding the several partial results are all consistent with the primary assumption that the difference of longitude between the terminal stations has a particular value ; and although the partial results of any given measurements by different chronometers may differ from one another, yet that the final sum of their several parts, whether taken collectively or separately, all agree. Another modification of this problem arises when the rate has been determined at the final station only, the error alone having been ascertained at the initial station. Example. In the preceding example let it be supposed that the error alone was determined before starting, as follows : Hong Kong, Nov. 24*, 1850. Chron. Z 7 44 10-06 M - 7 59 43'36 A 8 21 4-86 * In this case t r' = 10, and the factor - = [0-893545]. 188 ON THE MANAGEMENT OF CHRONOMETERS. On arrival at Amoy, on Nov. 3o d , the errors (as before) were found to be, h m s Chron. Z 7 59 56-98 M 8 15 23-85 A '-8 37 2473 and, finally, on reaching Shanghai, by observations on Dec. loth and 1 8th, the errors and rates at the mean epoch, Dec. I4 d , were found to be, Errors. Rates. Chron. Z 8 13 21-91 + 0-34 M 8 28 22-29 + 1 '99 A 8 51 25-85 2-02 Also, as before, let the difference of longitude between the terminal stations be assumed as and, consistently with this supposition, let us proceed to determine the relative longitudes of intermediate points. The observations at the terminal station, Shanghai, afford us the respective values of the final rate (a + b). And by formula (5), ~ * " M The mean sea rate (a + |) = (X/ ~ * and solving numerically for the several chronometers, Z. M. A. h m s li m s li m s Error at Shanghai, or V 8 13 21-91 8 28 22-29 "~ 8 5 1 2 5' 8 5 Error at Hong Kong, or A 7 44 10-06 7 59 43-36 8 21 4-86 (A' A) o 29 11-85 2 ^ 38*93 o 30 20-99 M o 29 17-10 o 29 17*10 o 29 17*10 (A' A) M -t-o o 5-25 +o o 38-17 o i 3-89 and dividing by t = 2O d , a + - = -f o s< 26 +i s *9i 3 s 'i9 whence, by formula (7), a (the initial rate) = 2 (a + -) (a -f- b) EXAMPLES OF MEKIDIAN DISTANCES. 189 Also, by formula (8), b (the change of rate) = And substituting numerical values, we shall find that for the several chronometers we have, a b Chron. Z + criS + 0-16 M + 1-83 -f 0-16 A - 4-36 + 2-34 With the values of a and b* thus determined, and substituted in formulas (2) and (4), we proceed, as before, to compute. the partial measurements, first from Hong Kong to Amoy, and secondly from Amoy to Shanghai. First, by formula (2), M! = A! - Here r = 6 and -j = [1*954243]. And for the several chronometers we have, - b ra + ~ it it s s s Chron. Z + 1*08 +0-14 +1-22 M + 10-98 + 0-14 -f 11*12 A 26' 1 6 + 2' 1 1 2 4*5 Whence, reducing the formula, Z. M. A. Error of chron. at Hong j _> - - ? 6 _* ^ Kong, Nov. 24 d , or A J (T 2 \ T + ^) +1-22 +II'I2 24-05 Error on Nov. 3o d , or 1 A + (T+ b] -. [ 7 44 8 ' 84 ~~ 7 59 32 ' 24 ~" 8 2I 28 ' 91 Error at Amoy, or A! ---- ~ 7 59 5 6 '9 8 8 ! 5 2 3* 8 5 8 37 2 4'73 o 15 48-14 o 15 51-61 o 15 55*82 * As a check on the accuracy of calculation it will be observed, that the above values of a and b when combined, give values of (a + J), agreeing with the previous statement of the rates at Shanghai, as given in the preceding page. 190 ON THE MANAGEMENT OF CHRONOMETERS. Hence, as the meridian distance, Hong Kong to Amoy, we have, h m s Chron. Z o 15 48-14 M 51-61 A 55-82 Mean . . o 15 51-86 Proceeding in a similar manner with the solution of formula (4),* M we should obtain, as the meridian distance from Amoy to Shanghai, h m s Chron. Z o 13 28-91 M 25-52 A 21-37 Mean .. o 13 25-27 Collating this result with that of the previous partial measure^ ment from Hong Kong to Amoy, it appears that their sum equals o h 29 I7 s< i3; a quantity, as in the preceding example, almost exactly identical with the assumed difference of longitude of the terminal stations, adopted as the basis of the determination. It will also be observed, that the individual results by each chronometer, taken separately, exhibit in a similar manner (as in the last example) the same degree of accordance ; and also, if the several partial measurements by each chronometer, employing alter- nately the initial and terminal rates, are compared together, they will be found to have a very close agreement, f Observation. The method illustrated in the preceding ex- amples of employing the known difference of longitude of the terminal stations to correct and adjust that of intermediate positions, seems capable of affording very useful results in con- ducting the work of surveys, since from its property of arranging its constituent parts in harmony with, and in subservience to, a t + T ' . t - * In this case t T' = 14, and the factor = [0-959041]. f This example is taken from the observations made during the voyage of H.M.S. Sphinx, 1850-3. EXAMPLES OF MERIDIAN DISTANCES. 191 primary and well-considered assumption, that the difference of longitude of the base stations may be taken of a given amount, it is susceptible of establishing among a mass of partial measure- ments a degree of accordance and consistency not otherwise attainable by any other mode of proceeding. The differences of longitude of the base stations having then, in the first instance, been carefully deduced and definitively established, are subsequently admitted as data in the further prosecution of the work, and, with the aid of formulae (5), (6), (7), and (8), made subservient to the ultimate systematic arrange- ment of the relative longitudes of intermediate positions. We shall conclude these examples by giving one in illus- tration of the remarks made in Observation III. (chap. vii. p. 1 60), as to the occasional convenience of combining together in an aggregrate operation the performances of individual chro- nometers, so as to obtain the final mean result by one process. A meridian distance so determined may be looked on, as the result by & fictitious chronometer, representing the average indi- cation of the several chronometers employed, and whose errors and rates respectively exhibit the mean arithmetic values of those actually appertaining to the chronometers under discussion. Example. Suppose that the following observations were made by ten chronometers to determine the meridian distance between Bahia and Rio Janeiro : Errors and rates at Bahia (Fort San Pedro) at the mean epoch, May io d , 1836: h m s s Z + 2 40 17*40 + i'6o A H- 2 38 19*00 2-40 B H- 3 i 1 8-60 1*75 C + 2 9 14-50 + 8-33 D + i 59 18-30 + 17-20 E -H 2 37 10-00 0-55 F + 2 o 11-70 + 5-10 G- + i 37 24-20 + 6-45 H + 4 9 37'3 - 2 4'3 1+3 5 27-00 + 3-10 192 ON THE MANAGEMENT OF CHRONOMETERS. After arrival at Rio Janeiro, by observations at Fort Villagagnan, the errors and rates of the chronometers at the mean epoch, May 3i d , were, Z H h 2 59 26-30, -f 1-84 A H - 2 o-ib - 2-36 B H - 3 19 8-00 1-90 C H - 2 3 26-00 + 8-05 D H - 2 24 1-50 + 18-40 E H r 2 55 43-20 + no F - h 2 20 44*3 + 6-24 G H h I 58 ^ 23-10 + 6-10 H H ^ 4 19' 57-40 22-20 I H H 3 2 5 56-00 + 5'8o Here i, the interval between the epochs of the observations, is 2 1 days; Also the mean value of a, the initial rate at Bahia, is + i s -278; While the mean value of (a + b\ the terminal rate at Rio, is + 2 S -I07J Hence the average mean rate (a + -\ to be used in the com- putation, is + i*-692. Likewise A, the average error, or that appertaining to the fictitious chronometer at Bahia, is 2 h 35 49 8< 8o; While A', the corresponding error at Rio, is 2 h 54 58 S '59. Then, by formula (i), M = *'- and solving numerically, we have, h m s Error of chron. at Bahia, May io d , or x +2 35 49-80 Correction for rate, or t (a + -^ ____ +35'53 Error onMay3i d ,or A + * (a + |) .. +2 36 25-33 Error at Rio, or A' ............... +2 54 58-59 M = +o 18 33-26 Whence, for the meridian distance between Bahia and Rio by these ten chronometers, we have as above. EXAMPLES OF MEEIDIAN DISTANCES. 1 93 If we were to proceed separately for each individual chronometer, in accordance with the usual process, we should obtain results as follows : Chron. 2 + o 1 8 32-78 A 31-08 B 27-62 C 19-51 D 29-40 E 27-53 F 33-53 a 47'23 H 28-35 I 55-55 Mean .. -f o 18 33'26 Hence it appears, that the result obtained from the aggregate process is identical with the mean result deduced in the usual manner, but by merging the individuality of performance of each chronometer in the general issue we are deprived of the power of detecting the eccentricity of any particular chronometer, or of analysing the results with a view to the adjustment of irregu- larities. If, when a large number of chronometers is employed in any given measurement, it should appear on an inspection of the results that some of them exhibit considerable divergence from the general mean; and if it should further appear, on an ex- amination of the records of the "Chronometer Journal," that the " second differences " of their daily comparisons indicated considerable instability of rate ; then it might be proper for the computer to reject the evidence of such suspicious chronometers, and to decline to receive them into the general combination. Now, in the example at present under discussion, chrono- meters C, Gr, and I, give values of the meridian distance differ- ing considerably from those indicated by the other chronometers (their divergence from the general mean being respectively I3 S '75, + I3 S *97 5 and + 22 S '29); if, then, a critical inspection of the " Chronometer Journal" afforded just reason for establish- ing against them the charge of irregularity of performance, the o 194 ON THE MANAGEMENT OF CHRONOMETERS. computer would have been justified in disallowing their evidence, and in rejecting them from the final result. Whence, in the present instance, the new mean result by the remaining seven chronometers would be, -f o h i8 ro 30 S -04 as the finally concluded meridian distance ; but it is clear that the power of this critical correction is lost, if we only confine our- selves to an aggregate mode of solution, which mode of pro- ceeding, therefore, however useful in other respects, should be restricted in practice to the purposes of duplicate reduction, as a check on the accuracy of computation. The new mean result thus obtained is usually styled the estimated mean, in contradistinction to the arithmetical mean obtained in the first instance from the evidence of all the chrono- meters ; the ultimate term, corrected mean, being held in reserve, to be finally appropriated by hydrographers to a result altered by subsequent or independent evidence. FORMULA OF TRAVELLING RATES. 195 CHAPTER IX. ON THE DETERMINATION OF THE MERIDIAN DISTANCE BETWEEN TWO STATIONS, BY MEANS OF OBSERVATIONS, GIVING THE THE problem of determining the meridian distance between two places admits of a very neat and elegant solution, in cases where, after the observations for ascertaining the errors on time at the place, at the two stations, the chronometers are again brought back to the station from whence they originally started, and their errors again determined ; it being premised that the whole of the observations on which the errors at the two places depend have been made within a reasonable interval. The formula are thus explained, At a place, A, before starting on a voyage, let the error of a chronometer be a After arrival at a station, B, let its error be /3 Before leaving B, let its error be $ And on return to the station, A, let its error be of Also, let n = the number of days between the first obser- vation at A, and the first at B ; And let m the number of days between the second obser- vation at B, and the final one at A ; Then the quantity gained or lost by the chronometer, in the whole interval between the first and last observations at A, is obviously '- ' And during the detention at B, /3'-/3 consequently the gain or loss, during the double journey from A to B and back again, will be 196 ON THE MANAGEMENT OF CHRONOMETERS. and the mean travelling rate during the double journey is ob viously Consequently, the accumulation 6f the rate during the outward voyage from A to B will be and similarly for the return or homeward voyage from B back to A, m m + n Hence, for the meridian distance from A to B, by the outward voyage, we have, M = ft - + k) k being the correction for the accumulated rate, and substituting its value, m + n a) + Wj8 na m + n _ m fff - ) + n (ft 1 - ') ( . m + n ^ ' If only one observation for the error were made at B, then $ and (3 are identical, and ft f3 = o ; hence the formula becomes simplified, and we have M = ft - * -- 2- . (X- a) (2) m + n ^ Again, for the meridian distance by the homeward voyage from B to A, M = - (p + k') mn m$' + na' n ft' m of + ma, + m$ m + n - n (ft'- of} - m (0 - ) m + _ m (& ~ ") + n (& ~~ *} m + n FORMULAE OF TRAVELLING RATES. 197 And again, as before, if only one observation were made at B, then M = '- - -2- . ('- ) (4) m + n ^ Formulas (i) and (3), it will be observed, give similar expressions for M, only with opposite signs, since obviously the meridian distances on the outward and homeward voyages are measured in opposite directions. So also, again, if formulae (2) and (4) are reduced to their simplest form, it will appear from (2) that and from (4), that M== *(-) + n(0-') m + n expressions giving equivalent values for M, but with opposite signs, and what formulae (i) and (3) resolve themselves into, when, consistently with our hypothesis (only one observation having been made at B), j3 f = (3. For facility of computation, however, formulas (2) and (4), as they stand, are to be preferred in practice in the actual reduction of observations. These formulae will be found very useful in two cases which may arise in practice: first, when it is wished to connect a maritime station with an inland position not accessible by the ship, or two inland stations with one another, when the opera- tion can be performed by means of the available pocket chrono- meters; and secondly, when, in the .course of any service, the ship leaving a given station, and calling at another, speedily returns to the first station again. In both cases, the circumstance of the determination of the meridian distance being independent of the consideration of the previous or subsequent rates of the chronometers, and solely based on their travelling rates during the double journey, deduced from the observations for error at the two stations, gives it a great advantage, in point of elegance and conciseness, over the usual process. Moreover, although an obvious remark, it is, perhaps, important to premise, that, in order that the observations should truly give the travelling rate unalloyed by any admix- ] 98 ON THE MANAGEMENT OF CHRONOMETERS. ture of that experienced by the chronometer when stationary, the observations for the error of the chronometer should be made as nearly as possible immediately before the period of departure from, and immediately after that of arrival at, the two stations; and, also, that $ie shorter the period elapsed during the double journey the better. The necessity for these precautions, especially in the case of pocket-watches, whose rates when travelling, and thereby subjected to the influence of a land journey, may be very dif- ferent from those they have when stationary, will, doubtless, on consideration, be quite apparent. We shall now proceed to give some examples to illustrate the application of these formulae Example I. In the months of July and August, 1838, the fol- lowing observations were made with three pocket chronometers, D, F, and P, to connect the observatories of Edinburgh and Greenwich : D. F. P. Errors. h m s o i 52-30 + o 10 39-60 Errors. h m s + o o 9-80 + 012 57-40 Errors. h m s o i 8-70 + o ii 32-30 July 1 1, i P.M. Gr b . 13, 8-30 P.M. Ed h . Aug. 8, i P.M. Ed h . +o 9 44-10 +o 13 25-20 +011 16-00 10, noon, Gr h . o 3 6-00 +o o 48-20 o i 29-20 Here n = 2 dt 32 2"! allowing a correction of o dp oo9 for 13, 'the and m = i d '949j approximate difference of longitude. m + n = 4^271 Also in the formula to be employed, a and <*' refer to the errors at Greenwich, and and /3' to those at Edinburgh. By formula (i), M _ (0 - ) + (/*' - *) m + n Likewise from the above observations we have, D. F. P. s s s /3 * = + 751*90 + 767-60 + 761*00 /3' '= + 770-10 + 757' + 765-20 m (/3 ot) +1465-45 +1496-08 +1483-19 w(/3'-.*') +1788-17 +175775 Sum ---- +3253-62 +3 2 53'83 +3260-11 divided by (m + n) + 761-8 +761-8 + 763-3 FORMUL/E OF TRAVELLING RATES. 199 Hence, m s m s m s M= + 12 4I'80 + 12 41-80 +12 43'30 Consequently the mean value of M, the meridian distance of Edinburgh from Greenwich, by these observations, is h m s + O 12 42*30 a result not differing much from the received longitude, + o h 12 43 s -6o (as given in the " L Nautical Almanac ").* * The formula given above, m + n although expressed in its most symmetrical and elegant form, considered alge- braically, and although conveniently adapted for general use when the numbers m and n are integral quantities, as they usually will be, and when consequently the calculation can be made by common arithmetic, is not so convenient as it might be, when the factors m and n are fractional numbers. In this case it will be better to write M m + n m + n and in computing from it, to find the logarithmic values of the factors, m n - and - m + n m + n In the case before us, the computation of the above example will then stand thus, _^. = ^949 = fi 3=[ _ r m+n 4*271 n _ 2-322 ''5437 = [-1*73533*] D. s F. 35'3 P. 347 8 '3 18-7 411-6 416-0 61-8 761-9 7 6y, s 0* m s + 12 41-9 m s + 12 43-3 m + n 4*271 TT m . Hence, (/S a ) And, ^~+~ n O 5 '" a ') = 4 l8 '7 Sum == M Hence M = + 12 41-8 which agree with the previous calculation. Of course the same reasoning applies to the reverse formula (3), M =- which may be written M=- 200 ON THE MANAGEMENT OF CHRONOMETERS. Example 2. In July, 1845, ^ e following observations were made to determine the meridian distance between Malacca Flagstaff and the Church at Sincapore : Z. A. C. Errors. Errors. Errors. h m s v j hms hms July 6 d< o Sincapore 7 44 15*43. 9 3^ 1 '7 2 4 7 ^'9 2 1 2 d> 5 Malacca 7 38 40-94 9 29 38-54 3 59 11-14 i6 d -o Malacca 7 39 8-70 ^-9 29 37*40 3 58 22-70 2i d -o Sincapore 7 46 12-68 936 1-28 4 3 40-08 Here, n = 6-5 m = 5-0 and m + n = 11-5 The correction for the difference of longitude on the intervals being very small is disregarded. Also in the formula, a, and ! refer to the errors at Sincapore, and /3 and /3' to those at Malacca. By formula (3), m + n And for the several chronometers we have, from the data above, Z. A. C. S 8 S /3-* + 334-48 + 383-18 + 47578 /?' ! + 423-98 + 383*88 + 3 J 7'38 m(fi *) +1672-40 +1915-90 +2378-90 n(p'a f ) +2755-87 +2495-22 +2062-97 Sum +4428-27 +4411-12 +4441-87 divided by (m + n) + 385-06 + 383-57 + 386-25 Hence, hms hms hms M~ o 6 25-06 o 6 23-57 o 6 26-25 And the mean value of M, the meridian distance of Sincapore east of Malacca, by the observations with these three chronometers, is, h m s o 6 24-96 Remarks, On examining the records from whence these observa- tions are extracted, and on comparing the results obtained by our formula under discussion with those .deduced in the ordinary manner, FORMULA OF TRAVELLING RATES. 201 separately, both by the outward and return voyages, we find as follows : Meridian Distance. Going .... Returning . . Mean . . Z. h m a o 6 25-24 o 6 24*93 A. h m s o 6 22-99 o 6 24.03 C. h m s o 6 26*15 o 6 26-33 o 6 25-08 o 6 23*51 o 6 26-24 Whence it appears that, in all cases, the above formula gives values of the meridian distance intermediate between those found by the ordi- nary process, and in every instance closely agreeing with their mean value. The same fact is also observable with reference to the results by several other chronometers employed on the same occasion. We shall conclude with an example in which only one observation for the error of the chronometers was made at the second station. Example 3. In June and July, 1851, observations for the meri- dian distance between Trincomalee (the Dockyard Flagstaff) and Madras Observatory* were made as follows, with three chronometers, Y, M, and ft: Y. M. R. Errors. Errors. Errors. d hms hms h m s June 23*00 Trincom. 5 39 52-82 5 45 28*82 o 24 11*32 29*33 Madras 5 36 39*30 5 41 40*30 o 24 39*30 July 1 4- oo Trincom. 5 42 15*91 5 45 54*93 o 38 58-93 Here, n n = m = 6*33 6-33 14-67 _ r 1-479185] m + n 21 ~ L The correction for the difference of longitude on the intervals being very minute may be disregarded. Also in the formula, a and #' refer to the errors at Trincomalee, while those at Madras are represented by /3. And by formula (2), M = ft - * -- (' ) m + n ^ * The errors on mean time at the Observatory at Madras were obtained by noting the time of the flash of the evening gun at 8 P.M. ; the exact Observatory mean time of which occurrence is duly published at Madras, in the Government Gazette : hence affording to ships in the roads a ready means for ascertaining the errors of their chronometers. 202 ON THE MANAGEMENT OF CHRONOMETERS. Now, for the several chronometers employed, we have from the above data, Y. M. R. hms h m B hms * + o 3 13-52 + 03 48-52 o o 27-98 '- -143-09 26-II 887-61 Y. M. R. hms hms hms ft * + o 3 13-52 + o 3 48*52 o o 27-98 n m + n ('-)- o o 43*13 o o 7-87 o 4 27*55 M +03 56-65 +03 56-39 +03 59-57 Hence, by these observations, the mean value of M, the meridian distance of Madras Observatory, west of the Dockyard Flagstaff at Trincomalee, is, h m s + 3 57-54 Again, reducing the meridian distance by formula (4), Now in this case the factor = - = [~ 1-84421 1]. 7W 4* W 21 Also for the several chronometers we have from the observed errors, Y. M. R. hms hms hms *' o 5 36-61 o 4 14*63 o 14 19-63 m (a! ct] o i 30*06 o o 18*24 o 10 20-06 m + n ^ oy y M -03 56-65 - o 3 56-39 - o 3 59-57 Hence the mean value of M, as before, is, h m a - o 3 57*54 Representing the meridian distance of Trincomalee, east of Madras. From which it appears, as we before remarked, that formulas (2) and (4) give the same results, but with opposite signs ; the one giving the meridian distance by the outward, and the other by the homeward, voyage. MODE OF RECORDINGS MERIDIAN DISTANCES. 203 CHAPTER X. ON THE MODE OF RECORDING THE RESULTS OF CHRONOMETRIC MEASUREMENTS. HAVING in the preceding- chapters explained at some length all the points necessary to be attended to relating to the gene- ral management of chronometers, and having carefully led our readers onwards through all the processes, subordinate, in the first instance, to the determination of errors and rates, and subsequently conducive to the systematic deduction of " meri- dian distances," little now remains for us to add, beyond pointing out the details to be attended to in recording the results, a matter of more importance than may at first sight appear, since the absence of minute details, or a loose habit of recording them, materially impairs the utility of chronometric measurements, and diminishes the ability of hydrographers to assign to them their just value, as compared with the previous determinations of others. Unnecessary difficulties are thus interposed to their comparison and incorporation with the labours of previous navigators, and hydrographic science in consequence derives but scanty benefit from, it may be, well- intentioned and often laborious exertions. This part of our subject has already been so fully treated by Raper,* that we can scarcely do better than follow the foot- steps of that able writer, having but little to add to the precepts enjoined by him. I. In all reports of the results of chronometric measurements, the details should be preceded by a preliminary notation, descrip- tive of the several chronometers employed, embracing the follow- ing details : the maker's name and number f the chronometer's * "Naut. Mag.," vol. vi. for 1839, P- 4 02 J and "Practice of Navigation," p. 281. f The makers' names and numbers should be recorded, in justice to the reputa- tion of the makers themselves, by facilitating the publication of the performances of first-rate chronometers ; and also, on the other hand, the failure of indifferent ones : facts which, without this precaution, are masked by the uncertainty of the 204 ON THE MANAGEMENT OF CHKONOMETERS. distinctive letter* whether box or pocket whether one, two, or eight-day where stowed in the ship, and how position of marks on dial-plate as to fore-and-aft line time of winding and comparing, &c. &c. As the greater part of these details are perma- nent facts, incapable of, or not liable to, alteration, they should be fully recorded once for all.' If any subsequent alterations take place, if accident befall any of the chronometers, or if new ones are received, &c., special notation of the fact should subse- quently be inserted in the reports from time to time, in chrono- logical order. It may also be proper to place on record, and explain any special symbols or abbreviations! which may be adopted in the subsequent pages ; and a brief reference to the modes of obser- vation and instruments employed, and to the formulae of compu- tation adopted in the reduction of the observations, if characterised by anything special, may also be given with propriety. II. It is advisable that the records of the performances of the chronometers should be specified in chronological order, and that the several measurements be numbered consecutively. " This succession," as Raper observes, " admits of specifying in its proper place any events which may relate to the chrono- meters collectively or individually as, for example, running down, accidents, exchange, or receiving new ones, &c. The chronological order, also, is evidently the only one in which we may expect to find that connexion which is absolutely necessary in considering a series of chronometric determinations, or to search with success for the origin or first appearance of an observed discrepancy among various results." literal symbols, which, however useful to the persons charged with the superin- tendence of their performances, are still only to be regarded as private distinguishing marks, and therefore unsuited for ultimate official publication, without accompanying explanation. Since, moreover, the same chronometer may be employed at different times, on different voyages, and as a knowledge of its history, as Raper remarks, influences the value of its testimony, its description by its permanent characteristics seems desirable, since the distinctive letters assigned to it on different voyages may, of course, be different, and do not necessarily convey any idea of connexion with former periods. * A, B, or C, &c. See ante, p. 26. f Raper suggests ch for chronometer; d, for days; D L, for difference of longi- tude ; and that the extreme difference of results should be denoted by the number of seconds enclosed in brackets, implying limit or boundary; thus, [7*]. As these symbols are simple and self-suggestive, they may with propriety be recommended for adoption. MODE OF RECORDING MERIDIAN DISTANCES. 205 In cases where, in any series of measurements, the ship returns to the same ground, and repeats the determination of any given chronometric link, a special notation may be made, and attention called to the fact, so that a comparison of the values of the successive results may be instituted with facility and convenience. III. It is absolutely necessary to specify or to describe the exact spot of observation at the places visited. For instance, it would be no use to report that the meridian distance by so many chronometers between Plymouth and Lisbon was "so-and-so," because Plymouth and Lisbon, being both large places, covering some miles of ground, such information respecting them is defective in precision, and however valuable they might be in other respects, such loosely-recorded facts must needs be dis- regarded. It is advisable, therefore, that the assumed latitude and longitude of the place of observation (with the authority for them when necessary) be specified, and its topographical position accurately described, in accordance with the precepts laid down on this matter in a previous chapter (see ante, p. 60) ; and this should invariably be done in recording observations made at new stations. IV. The modes by which the observations for time, on which the errors and rates employed in the computations depend, should also be stated, whether equal altitudes, single altitudes, or other- wise. Also the number of days at each place occupied in determining the rates,* and likewise the intervals between the several epochs of the observations. If any of the observations have been made with the sea horizon, the circumstance should be specially noted. V. When several chronometers are employed, the results by each chronometer should be exhibited. The general arith- metic mean should be given, and also the estimated mean, obtained * Specific particulars respecting the rates are, it would appear, of only secondary consequence to the hydrographer, as he can never be in a situation to employ this information to such advantage as the observer himself. Proper details relative to the character of the rate, such as how determined, and whether steady or unsteady, should, however, accompany every chronometric determination; and as we have before observed (p. 35), the original books containing all the details on these matters should be neatly and methodically kept, so as to be available for inspection and reference when needed. 206 ON THE MANAGEMENT OF CHRONOMETERS. by rejecting the results of those chronometers which are discordant or irregular, and merely retaining those whose performances during the voyage seem unexceptionable ; or even, in some cases, in accordance with the practice of some computers, by giving more or less weight to 'the several results, according to the per- formance of each chronometer, and of which the observer alone can be a judge. There is, perhaps, no question relating to the deduction of meridian distances, which involves considerations of greater delicacy and importance, or which requires to be approached with more caution than that relating to the establishment of an estimated mean. The computer is so liable to have his judgment warped by predilections in favour of some preconceived and desired solu- tion, and even, possibly, by his partiality for particular chrono- meters, that there is a constant temptation to be either too hasty in rejecting particular results because they do not accord either with the general mean or with previous expectations, or to be unduly desirous of retaining them because they happen to agree with those of others, or to harmonise with his previous ideas. We are by no means in favour of the practice, which has been sometimes adopted, of assigning weights to the values of particular results, dependent on the assumed character of the chronometers employed; such a practice affords an opening to the possibility of all sorts of "cooking" and " trimming," since sets of observations may thus be twisted to support any required determination, although certainly at the expense of their integrity. Even the practice of altogether rejecting the results by par- ticular chronometers, because they exhibit a divergence from those of others, is not unfraught with danger, and should not be adopted without discreet and critical consideration. The mere divergence of a given result from the general mean, unless excessive in amount, would not in itself seem to justify its rejection, unless at the same time an examination of the records of the "daily comparisons" exhibited also unmistakable indica- tions of instability of rate; and this consideration is of the greater weight when measurements occupying long periods of time are under discussion, since a degree of apparent irregularity, which might justly authorise the rejection of a particular result in a short run occupying but a few days, would be insufficient to MODE OF RECORDING MERIDIAN DISTANCES. 207 sanction that proceeding if a considerable interval elapsed during the voyage. It is, perhaps, impossible to lay down any specific rules on this subject for the guidance of computers ; all that we can do is to enjoin caution and discreetness, and to indicate, as above, some general considerations to aid the judgment.* The final results, both of the arithmetic and estimated means, should be given in arc as well as in time. In time, because it is in that form that they first develop themselves under the hand of the computer, and are most convenient for inter- comparison; and in arc, for the convenience of geographical purposes, because the unit of measure in navigation being a mile or minute, and charts having a scale divided accordingly, the diff. long, in arc is absolutely necessary in comparing two charts. VI. The extreme difference of the greatest and least results * It might, perhaps, assist to aid our judgment in estimating the values of chronometric determinations, and possibly to check the tendency to a too hasty rejection of particular results which swerve from the general mean, or do not tally with preconceived ideas, if we endeavour to institute a comparison between the accordance of the results obtained from individual chronometers, in any given measurements, and those which are found to exist between the single results of various kinds of astronomical observations. Observations of solar eclipses, occultations of stars by the moon, moon-cul- minating stars; eclipses of Jupiter's satellites and lunar distances, have all been employed, at various times, by different observers, as astronomical data for the definitive settlement of the absolute longitude of many fundamental stations. It may be instructive to make a cursory examination of the degree of accordance which is found to prevail among observations of those kinds, and it seems not unfair to contrast results by individual chronometers with the issue of single astronomical observations ; and if we mistake not, the tendency of such a comparison would be to reconcile observers to a much greater amount of discrepancy than they are usually inclined to tolerate. Accurate observations of solar eclipses and occultations of stars by the moon, especially when central, and observed under favourable circumstances, have been considered to yield very reliable results for the longitudes of the places of observa- tion. Yet among the results recorded by M. Daussy, in the " Connaissance des Terns," and collected by Raper in his paper on Maritime Positions, in the " Nautical Magazine," we find the differences between individual results ranging as high as -ji 9 for eclipses, and 56 s for occultations. The results by the eclipses of Jupiter's satellites are greatly influenced by the power of the telescope employed, and the discrepancies between single observations consequently range much higher. Raper records an instance of four eclipses of the satellites observed by the late Sir E. Home, at Port Royal, with an excellent tele- scope, in which the range between the extreme observations amounted to 214*. This, 208 ON THE MANAGEMENT OF CHRONOMETERS. by the different chronometers employed^ or the range, as it is sometimes called, should be stated, to facilitate the estimation of the general dependence to be placed on each determination, by showing whether the chronometers went well together or not; for though their going together does not prove that all or any of them are right, their not going together proves that some of them are wrong. VII. Every result should be given, without any regard to whether it agrees or not with received determinations. Many received positions, as Raper observes, are very erroneous ; and the only means by which they can be decisively rectified are the comparisons of independent and impartial evidence. VIII. Since it is known that changes of temperature in- fluence the performances of chronometers, the range of the temperature of the chronometer-room should be stated, with accompanying remarks, if its fluctuations had been excessive. doubtless, was a very extreme case, but discrepancies of as much as 30" or 40* seem not uncommon. Observations of moon-culminating stars have been greatly commended as a means of determining differences of longitude, and much has been expected from them. In practice, however, they have scarcely maintained their reputation. In a large mass of observations recorded in the " Memoirs of the Astronomical Society," connecting the Observatory at Madras with Greenwich, Cambridge, and Edinburgh, the discrepancies between extreme individual results range as high as 32*" 5 ; and in a similar series of corresponding observations, connecting the observatories at Madras and the Cape, the range of difference amounts to 3i 8> 8 ; and in those connecting Bombay with Greenwich and Bushey, to 3o s> 4 and 33 8> 9 respectively. The results by Lunar Distances are, it is well known, still more uncertain and unsatisfactory, not only as regards isolated determinations, but often even as regards the mean of masses of observations ; so much so, that the judgment of modern hydrographers seems inclined to reject the lunar method altogether as a means of definitively establishing fundamental positions. (See note, ante, p. 4.) If, then, we find such serious discrepancies to prevail between the individual results of astronomical observations, often obtained by skilful observers in established observatories, and with first-rate instruments, surely we ought to view with patience, and be inclined to reject with caution, discrepancies existing among the results of individual chronometers, which, however perfectly constructed, are but machines after all. It is, of course, impossible, in the brief limits of a note, to attempt to do justice to this interesting inquiry, or to do more than briefly suggest a few heads for con- sideration, and indicate to the studious reader the sources from which further information can be obtained ; and with these remarks we commend the subject to the critical consideration of the scientific student. (See ''Memoirs Ast. Soc.," vol. iii. p. 369 ; vol. xii. pp. 119 and 133 ; " Connaissance des Terns," vol. 1835-6, additions ; "Nautical Magazine," Raper on Longitudes, vol. 1839, &c.) MODE OF RECORDING MERIDIAN DISTANCES. 209 Uniformity of temperature would justify the inference of uni- formity of rate, while violent changes would give rise to a suspicion of irregularity. Also, since there is reason to suspect that the influence of the ship's magnetism may in some cases produce an appreciable effect on the chronometers, it is proper to indicate the general direction of the ship's head during the voyage. Precision in the indication would, indeed, be useless; but a knowledge of this element might, perhaps, be of service on some occasions. In steam-ships, where the introduction of the boilers and machi- nery concentrates large masses of iron in the interior of the ship, notations on this subject would seem to be important, and may be highly useful, since it is only by the careful record of observed facts that we can ever hope to penetrate the obscurity which at present attends the subject of magnetic influence as affecting chronometers. In accordance with the principles laid down in the preceding remarks, the records of a meridian distance would be exhibited in the following manner : Example i. No. ( ) H.M.S._ Capt. N. May, 1836. Bahia (Fort San Pedro) to Rio Janeiro (Fort Villagagnan). h Chron. Z + o 18 3278 A 31-08 B 27-62 Temp. 80 to 74. C 19-51 D 29-40 Ship's head S.S.W. E 27-53 F 33*53 Rates at both places by G- 47 -23 Eq. Alt. Intervals 7 H 2 8'35 an ^ 5 days. 1 55*55 Arithmetic mean, + o h i8 m 33 8 '26, or 4 38' 19" r ; Estimated mean, o h io m 30 8 -04, or 4 37' 3o"-6 (rejecting C, Gr, and I, whose rates were unsteady); 7 ch, 2i d , [6 s ]. P 210 ON THE MANAGEMENT OF CHRONOMETERS. Example z. No. ( ) H.M.S. Capt. N. May, 1844. Cape Upstart to Sir Charles Hardy's^Islands (N.E. Coast of Australia). h m s Chron. Z -f o 17 9*12 Spot of observation, Cape A g. Upstart (a small valley, near a white rock off the west B 15-16 point of the Cape). 6 Lat. 19 43' o" S. Long. 147 48 o E. D 16-05 Sp t of observation, Sir C. E Q'43 Hardy's Islands (a sandy beach west side of the centre F 6- 1 8 island) . G 7.63 Lat. 11 55' 15" S. Long. 143 31 o E. Temp. 76 to 82. 1 10-23 Ship's head N.N.W. K c-ci Errors and rates at both places by Eq. Alt. In- L 3*83 tervals, 4 days. Arithmetic mean, +o h I7 m 9 s -o7; Estimated mean, +o h i7 m 9 s - 10, or 4 17' 1 6"- 5 (rejecting H and L, whose rates were unsteady), 10 ch, I9 d , [i2 s -49]. No. ( ) Cape Upstart to Lizard Island. May, 1844. h Chron. Z +09 12-50 Spot of observation, Lizard A 11*31 Island (a small sandy bay, B 16-69 N.W. side of Island). C 10-40 Lat. 14 40' o" S. D 17-10 Long. 145 31 o E. E 11-73 Temp. 76 to 80. F 8-10 Ship's head N.N.W. G- 9-45 Errors on time only obtained H 15-00 at Lizard Island by Eq. I 13*70 Alt. Former rates em- K 7-16 ployed, as in preceding L 6-00 measurement. Arithmetic mean, + o h 9 m i i s> 59 ; Estimated mean, + o h 9 m i I s 81, or 2 17' 57^ (rejecting H and L as before), loch, 8 d , [9 S> 94]- MODE OF RECORDING MERIDIAN DISTANCES. 211 No. ( ) Lizard Island to Sir Charles Hardy's Islands. May, 1 844. h m Chron. Z -{-07 57*50 A 56-57 B 57-92 Temp. 80 to 82. C 53'3 D 59-10 Ship's head N.N.W. E 58-00 F 58-00 Error on time only found at Gr 57-20 Lizard Island by Eq. Alt. H 5 9- 50 Former rates employed as in I 56-00 preceding measurements. K 58-60 L 56*4 Arithmetic mean, + o h 7 5 7 s - 3 2 ; Estimated mean, + o h 7 5 7'- 1 9, or i 59' 1 8" (rejecting H and L as before), 10 ch, i i d , [6 8 -o7]. Example 3. No.( ) H.M.8.. CaptN. Shanghai to Hong Kong. h m s Chron. Z + o 2Q I C'42 Spot of observation, Shanghai, Mr. Beale's house. -o Lat. 31 15' i5"N. I2 * Long. ! 29 o E. Spot of observation, Hong Kong, v r Dent's Wharf. '5 6o Lat. 22 1 6' 27" N. Long. 114 10 o E. M 2J.-CO Temp. 43 to 57. Ship's head S.S.W. Errors and rates at both places by A l8 '95 Eq. Alt. Intervals, 8 and 19 days. Arithmetic mean, which we adopt, + o h 29"* i7 s< 29. Beale's house is 0^56 East of the Consular Flagstaff at Shanghai, and Dent's Wharf o s -26 West of Victoria Cathedral at Hong Kong. Hence we have for the corrected meridian distance between the Flagstaff and the Cathedral, o h 29 i6 s 47, or 7 19' 7", 5 ch, 22^5, [i2 8 '5o]. In the preceding examples, taken from actual observations made on board H. M. ships, we have not been able to follow out the precepts laid down and insisted on so strongly in these pages, viz. to record in all cases the mean daily temperature of the chronometers at the time of winding, as indicated by the read- 212 ON THE MANAGEMENT OF CHRONOMETERS. ings of the maximum and minimum thermometers. The actual records of these observations have not given the mean daily temperatures with sufficient precision to enable us to do so. In Form No. III. (Appendix, p. 222), which is an amplification of the above mode of record, in a /ruled form, provision has been made in the columns for the future careful notation of this highly important element. In cases where it may not be deemed necessary to record the results of the chronometric measurements with the same minuteness of detail as is exhibited in the above examples, a more succinct method may be adopted; and a tabular form adapted to that purpose will be found in the Appendix, p. 223, in which the results may be briefly exhibited in a more popular shape. 213 APPENDIX, TABLE For converting Intervals of Time, or Longitude, into Decimals of a Day. Long. Time. Decimals of a Day. Long. Time Decimals of a Day. Long. Time Decimals of a Day. h o / m o / m '5 I 0417 o 15 I 0007 7 45 31 0215 30 2 0833 o 30 2 0014 8 o 32 *O222 45 3 1250 o 45 3 0021 8 15 33 0229 60 4 1667 I O 4 0028 8 30 34 0236 75 5 2083 1 J 5 5 0035 8 45 35 0243 90 6 2500 i 30 6 0042 9 36 0250 105 7 2917 1 45 7 0049 9 15 37 0257 1 20 8 '3333 2 8 0056 9 3 38 0264 *35 9 '375 2 15 9 0062 9 45 39 0271 150 10 4167 2 30 10 0069 10 o 40 0278 165 ii 4583 2 45 ii 0076 10 15 4 1 0285 1 80 12 5000 3 o 12 0083 10 30 42 0292 '95 '3 5417 3 15 13 0090 10 45 43 0299 2IO H 5833 3 30 H 0097 I I O 44 0306 225 15 6250 3 45 15 0104 II 15 45 0312 240 16 6667 4 o 16 'Oil I II 30 46 0319 2 55 17 7083 4 15 17 oi 1 8 ii 45 47 0326 270 18 7500 4 3 18 0125 12 O 48 *333 285 1 9 7917 4 45 '9 0132 12 15 49 0340 300 20 8333 5 20 0139 12 30 5 *347 3'5 21 8750 5 15 21 0146 12 45 5 1 *354 330 22 9167 5 3 22 0153 13 o 5 2 0361 345 2 3 9583 5 45 23 0160 13 15 53 0368 360 24 I'OOOO 6 o 24 0167 '3 30 54 0375 6 15 2 5 0174 '3 45 55 0382 6 30 26 0181 14 o 56 0389 6 45 27 0187 14 15 57 0396 7 28 0194 H 3 58 0403 7 15 2 9 O2OI H 45 59 0410 7 30 3 O2O8 15 o 60 0417 2 14 ON THE MANAGEMENT OF CHRONOMETERS. Examples illustrating the Application of the preceding Table. Example i. Let it be required to express the interval between 9 h 17 A.M., on Nov. 9th; and 8 h 46$ A.M., on Nov. i/th, in decimals of a day. Here 2i h =0-8750 Also, zo h =0-8333 and 17 = 0-0118 and 46 =0-0319 Sum .. 0-8868 Sum .. 0-8652 Therefore the first date is Nov. 8-8868 second 16-8652 and hence the interval = 7*9784 Example 2. Required the interval between Apparent Noon on Feb. 20th, 1855, at Portsmouth, and March I9th, 3 h n m P.M., mean time at Bermuda. Apparent Noon on Feb. 2oth corresponds to o h 1 4 mean time (the " equation of time " being 1 4 to be added to apparent time), therefore the date at Portsmouth is, Feb. 20^0097. Again, 3 h n m correspond to 0^1326, therefore the date at Ber- muda is, March I9 d -i326, or Feb. 47^13 26. Hence the apparent interval between Feb. 20^0097 and Feb. 47 d -i326, is 27 d -i229; but the difference of longitude between Ports- mouth and Bermuda is 63 46', the fraction corresponding to which is o d -i77i, to be added to the apparent interval, because the ship loses time by sailing to the westward. Hence the true interval is * See Observation II. p. 159. APPENDIX. 215 SUMMARY OF INSTRUCTIONS FOR THE MANAGEMENT AND USE OF CHRONOMETERS. The following summary of instructions for the management and use of chronometers has recently been prepared at the Admiralty, and forms part of the new code of instructions for the government of the fleet: I. On Embarking. 1. Whenever chronometers are to be transported, clamp the catch of their gimbol rings and carry them by hand, or slung with a line, or in a handkerchief, taking care not to give them any shock or circular or oscillating motion. 2. When embarked, stow them in the case prepared for them on beds of horsehair, shreds of bunting, or raw cotton, padding them around and between with the same soft material, so as to prevent the possibility of motion or concussion. The line joining their XII. and VI. hour marks should be all in the same direction, and parallel to the keel. Never use sawdust or wood shavings. 3 . Release the catch of the gimbol rings ; see that the gimbols work freely, but not too easily. The chronometers being once secured in their places are on no account to be subsequently moved or displaced* until relanded at home. 4. The chronometer case should be covered over with a covering of coarse woollen cloth to guard them from sudden changes of temperature. II. -On Winding and Comparing. 1. Wind up daily at the same hour (8 A.M.), counting the turns, and winding carefully until the key is felt to butt; eight-day chrono- meters on Sundays. 2. Immediately after winding, compare each of the chronometers with the "standard" (the best chronometer, previously selected,) and note the comparisons in the chronometer journal, Form No. i. 3. Note also at the same time the thermometer (which should be kept within the chronometer box) and also the barometer. * The time for all observations on shore or on board is to be taken, in the first instance, by a pocket chronometer or an assistant watch showing seconds, and the times then shown by each of the chronometers are to be obtained from it, by corn- Daring it with them both before and after the observations. See Sec. II. 216 ON THE MANAGEMENT OP CHRONOMETEBS. III. On Errors and Rates. 1. Chronometers being virtually intended for determining the "difference of longitude" or "meridian distance" between places visited by the ship, as well as for her safe navigation, their errors and rates must be very carefully ascertained. The ERRORS are best found by " equal altitude " observations of the sun, or in default of them, by single altitudes A.M. or P.M.: all such observations should be made with an artificial horizon on shore. 2. The RATES are to be ascertained by comparing their errors on local mean time, obtained consecutively at convenient intervals of not less than five or more than ten days; seven days is a convenient average interval. The difference of the errors divided by the number of days elapsed between them gives the mean daily rate of the interval. 3. At places where time signals are established for giving daily " local or Greenwich mean time," they may be taken advantage of, and the errors and rates deduced from them : but independent astro- nomical observations are most to be preferred. The errors should be recorded, with the rates deduced from them, as shown in Form No. II. IV. On Meridian Distances. i . The chronometric longitudes of places visited are not required, but the "meridian distance," or difference of longitude in time, as shown by each chronometer between those places from the obser- vations,* is to be carefully recorded, as in Form No. ITI.f V. On Position of Place of Observation. i. It is very important that the exact site of the observations should be distinctly specified. At a well-frequented place, it is advisable that they be made on the site previously selected by former observers, as great confusion arises from the unnecessary multipli- cation of sites of observation. * The value of a meridian distance depends mainly on the care with which the errors and rates have been determined at each of the terminal stations, and the errors at intermediate ones ; on the shortness of the run between the places called at, and on the number of chronometers employed. f It is extremely desirable that a record of the mean daily temperature of the chronometer case, extracted from the chronometer journal, should always accompany the returns of meridian distances, forwarded at any time to the Admiralty. In the present state of chronometric science the investigation of the effects of temperature is a question of much interest, and can only be successfully accomplished by aid of the records of careful observations. APPENDIX. 217 2. If it be a new station, its situation should be carefully described, its latitude and approximate longitude be given, or its connexion with some adjacent known position. VI. The Chronometer Journal. i. The chronometer journal, kept in Form No. I., is to contain the record of the daily comparisons at the time of winding. z. Notice should be taken in the column of remarks of any circum- stances likely to affect the performance of the chronometers, such as gales of wind, violent motion of the ship, her striking the ground, heavy firing of guns in action or for exercise, accidents to or stoppage or removal of any of the chronometers, storms of thunder, lightning, and general direction of the ship's head, &c. 3. The daily rates of the chronometers, determined from time to time by observation, should also be noted as a record, and for the purpose of occasional comparison with the column of second dif- ferences. 4. The chronometer journal, being neatly and methodically kept in the manner above described, there will be no necessity for keeping a fair copy of it, the original being carefully preserved in readiness for transmission to the Admiralty, if required. VII. Official Returns. i. As the results arising from the good management of chrono- meters are only permanently required for the purpose of settling longitudes, Form No. III., when filled up, is to be transmitted annually to the Secretary of the Admiralty, with the ship's remark book. But as there is much probability of the others, No. I. and No. II., being also required, officers are strictly enjoined to pay especial attention to all the foregoing particulars, both as to the precepts they contain and the forms directed by them to be kept, in order that when these are applied for, on any occasion, for the investigation of results in longitude, they may be immediately forthcoming, in a satisfactory condition, for reference in the Hydrographic Office. On the ship being paid off the chronometer journal is to be for- warded, with the other returns, to the Secretary of the Admiralty. 218 ON THE MANAGEMENT OF CHRONOMETERS. Form No. 1. Chronometer Journal for the record of Daily Comparisons of Chronometers. H.M.S. Date. Bar. and Ther. Max. and Min. Ther. Chron. A. d Diff. Chron. B. zd Diff. Chron. C. ad Diff. Initials of Comparers. Remarks. 1860. Sun. Jan. i. in. 29-91 ?6 80 78 h m s Z 2 15 10 50 7 h m s 2 I 5 3 7 20 i U m s 2 l6 i ii 54 c.s. W.B. Heavy Gale. Much Motion. Confused Cross Sea. Less Motion, Exercised firing at Night Quarters. 3 *4 53 6 55 29 146 Mon. 2. in. 30-32 69 78 71 Z 2 39 o ii H 5*3 8 i'7 2 39 30 7 43 48-8 s I2'2 2 40 i 35 55*5 i'5 C.S. G.C. 3 2 4 54' 7 6 55 41-2 i 4 4'5 Tues. 3- in.. 30'41 7i 74 66 Z 2 24 10 59 5 0-3 2 24 30 7 28 58 10-8 2 25 I 20 56-3 0-8 G.C. A.M. 3 2 4 55 6 55 32 i 4 3'7 Wed. 4- 3*34 72 76 70 Z 2 27 II 2 4'2 0-8 2 2 7 3 7 3i 28 JO'O 2 28 I 23 57-3 I'O W.B. A.M. 3 24 55'8 6 56 2 i 4 2-7 Thur. 5- Fri. 6. Sat. 7- Sun. 8. APPENDIX. 219 Form No. 1 (A)*. Chronometer Journal for fair copy of Compa- risons, if thought necessary. H.M.S. Date. Z A zd Diff. Z B 2d Diff. Z C ad Diff. Bar. and Mean Temp. Remarks. 1860. Jan. h m s h m s h m s in. Sun. i 3 24 53 6 55 2 9 i 4 6 29-91 Heavy Gale. 79 Much Motion. 9 s s in. Mon. 2 3 24 54'7 1-7 6 55-41-2 I2'2 i 4 4-5 'S 30-32 Confused Cross 74 Sea. in. Tues. 3 3 H 55 0*3 6 55 52 ID'S i 4 3'7 0-8 30-41 Less Motion. 70 in. Wed. 4 3 24 55' 8 0-8 6 56 2 jo-o i 4 2-7 ro 30*34 Exercised firing at 73 Night Quarters. in. Thur. 5 3 H 5 6 '5 0-7 6 56 13 iro i 4 1-5 I'2 30-19 4th. 8 A.M. arrived 72 at Sincapore. in. Fri. 6 3 24 57'3 0-8 6 56 23-5 10-5 i 4 0-5 I'O 30-24 Errors and Rates 72 at the Mean in. Epoch : Sat. 7 3 24 58 0-7 6 5 6 34 10-5 i 3 597 0-8 30*16 Jan. 7 d 'i39,by 7 single -alt. P.M. in. obs. on the 4th Sun. 8 3 H 59 1*0 6 5 6 45 I I'O i 3 58'5 1*2 30-16 and loth. 72 h m s S Mon. 9 3 24 59-5 '5 6 5 6 55'5 10-5 1 3 57'5 I'O in. 30-IO _ _o Z 5 35 io-o +z'io A 9 o 8-3 +1-35 7 1 B o 31 47-0 8-50 Tues. 10 3 25 0-5 I'D 6 57 6-7 11*2 i 3 56-7 0-8 in. 30-14 C 6 39 95 +3-15 73 Wed. 1 1 3 25 i'5 ro 6 57 18-0 II'3 i 3 5 6-0 0-7 in. 30-16 72 Thur. 1 2 Fri. 13 Sat. 14 * This is the same form as Form No. 2, ante, p. 29; we here call it No. i (A), so as not to interfere with the numbering of the forms adopted by the Admiralty. 220 ON THE MANAGEMENT OF CHRONOMETERS. Remarks on the Forms Nos. I. and No. I. (A.) for Chronometer Journal. The Chronometer Journal, Form? No. /., should be kept ruled up for use in a book of the size of foolscap paper; each page may be ruled to hold ten days' comparisons, three pages thus sufficing for a month. The column for remarks should be of ample width, and if necessary, the form should be carried across the page, so as to accom- modate any required number of chronometers. Form No. I. (A.) may be ruled so as to exhibit a month's com- parisons in each page ; and when the number of chronometers renders it necessary, the ruling may be carried across into the opposite page, as before. It will be useful to note from time to time, in the column of remarks, the errors and rates of the chronometers as determined by observation. In Form No. I. (A), if a fair copy is kept; if not, in Form No. I. If preferred for neatness, the months may be described by the Roman numerals I. to XII., and the days of the week by their symbols, as follows : Sunday. D Monday. % Thursday. $ Tuesday. $ Friday. Wednesday. 1? Saturday. In Form No. I. (A.) the mean temperature is the mean of the readings of the maximum and minimum thermometer as noted at the time of winding; also the column containing the initials of the comparers is omitted, as a point of no permanent importance. As a general rule it is not recommended to keep any copy of the comparisons. The original book in Form No. I., neatly and metho- dically kept, will be sufficient. If, however, on any specific occasion, copy of the comparisons should be wanted, for the period embracing any particular measurement, Form No. I. (A.) is conveniently avail- able for that purpose. APPENDIX. 221 No. 18 Form No. II. Return of observed Errors and Rates of Chronometers. Returns to be numbered consecutively; month and year to be specified. H.M.S. A. B., Esq., Captain, Station. j Errors of Chronometer Errors of Chronometer At the Mean Epoch (3). Mean Daily 1 on Local Mean on Local Mean Temperature Place and Spot Time. Time. Concluded Concluded during of Remarks. ! By (i) Date and By (i) Date and Mean Mean Period of Rating. Observation. Time (z) Time (z) Error (4). Rate (5). z. (6.) (70 (8.) Here insert Here dis- Here insert A. the mean of tinctly specify anything the two the place and noteworthy, B. thermometer spot of likely to readings observation. affect the C. at time of If a new chronometers winding for station, give or the value D. every day its latitude of the between the and observations; periods of approximate weather on Z. ('0 By (i.) By observation longitude, days of (*.) Date for errors. or its observation, A. and Time. (2.) Date connexion &c. &c. N.B. If and Time. with B. errors alone some known have been point. C. ascertained, they are to D. be recorded in this co- lumn. Approved, A. B., Captain. C. D., Master. (i.) Here insert nature of observation : Equal altitudes, single altitudes, A.M. or P.M., &c. &c. &c. (z.) Here insert date and time. It will be convenient to express the latter decimally: thus, June 9th, 8-40 A.M. would be written June 8 d< 86i ; May zist, 3 P.M., May zi d -iz5. See Appendix, "Notes on Chronometers," Table for converting Intervals of Time into Decimals of a Day, p. 21 J. (3.) Here insert the mean date and time on which the two errors depend : thus, if the first error belonged to Jan. 5 d -88o, and the second to Jan. 12^872, the mean epoch would be Jan. o/ 1 '^. (4.) Here insert the mean of the two observed errors. (5.) Here insert the difference of the two erroi-s, divided by the interval in days elapsed. 222 ON THE MANAGEMENT OP CHRONOMETEES. No._ Form No. III. Meridian distance to Measured in H.M.S. between the Returns to be numbered consecutively, and month and year specified. _ A. B., Esq., Captain, 1 8 ,an 21 jo IV. I-0 9 O' CO 5-66 5"i'i 4-91 1-18 vj yu 0-82 0*78 0*70 O*7C 0-24 + O'42 2-77 2*21 i '47 1*20 4'49 4*84 4*38 1'OQ I*0 9 n-66 0-55 0-78 49 CO Madeira 2 1 O"2O 4"i'i 1-78 2'O4 + O'47 O'7C 2'77 i-88 2'6o 1*IO I'4Q + O*7Q 60 Teneriffe * V. o-c8 4"2Q O'OO 2*1 1 0*78 0*27 2*60 ?-68 1*71 1*17 I'OI 2*17 67 Simon's Town(C.G.H.) Ditto ... 26 VI. 14 VII. 0-15 + I- 1 r 3-05 T- 7 8 2*11 o*c6 2*33 v6? + 0-08 I" C7 1-37 ?-6S 3*35 4*^7 3*21 4*4Q 1-75 4*87 4'55 7* 14 0-79 n-8^ 2-67 7*08 62 67, Hobarton (V.D. Land) Ditto . six. AX 1 "-I) 0*62 2*16 I* C I 3-11 1*4.7 2*54 7'7-J 1*01 I'2O 2-64 I'QA 3'9 6 4-08 3*87 7*78 8*21 q-88 4*29 4*IO 0*26 0*71 1*62 1*96 61 60 Port Arthur JO ,, I"3 r I"22 O'4O 4*OO 2'OC 3*10 4" CO 4'35 IO'7C 7*7O 0*70 7,* 60 18 Sydney (N.S.Wales)... Ditto 26 22 XI. '35 O'OO r6o 1*81 0- 4 8 1*71 4'l8 4*2C I-7I I*7O 2*63 J'OC 4-30 3-78 4-48 4'55 8*55 V17 3-89 ^1*87 0*05 0-60 3*35 3*60 62 71 Port Stephen... JO O'OO I' CO I"2Q 4' co 1*60 2'CO 4*0 c 4'oc ^'2O 4'4O o - 6c 7. *6 5 7 Ditto 16 XII. O' I C i ^w T-rg 1*1 1 5*08 1*71 1*67 1*i8 4-71 7*72 4*7O o'6o 3*7,8 73 Sandy Cape ... 1843. i I 0*4.6 2'06 2*66 C*O7 1*74 0*81 7*78 4*74 O'll 1*27 1*79 3' 2 7 76 Port Bowen 27 II I'Q2 2'QC 6*26 <:*6o I'O7 1*77 rSo 4- jc 3*68 1*08 2*42 3'93 79 West Hill II III 1-67 2'77 5o6 6-28 I* c c I*7O 4"IO 4'47 2-78 6'T7 2*27 3*47 81 Cape Upstart 7 IV I7C 2"? 4. 2-8c } 1*72 1*11 AMO 4'7O 2*46 6">7 2*29 3*^8 80 Ditto 16 V. O*Q7 1*1 C i*c8 1 ) 7*07 ?,-8? c-6c 5-30 1-83 4*90 1-48 3 '47 75 Goold Island 5 I O*6o o'o6 + 0-08 ' ^ 6-71 7*2Q 7'IC c-Xc 5*27 '57 c'47 1*46 3-68 74 Sir C. Hardy's Islands Port Essington 17 VII. 25VIII. r8o 2*21 i'74 I7c 0-88 0*3 c 7*78 9-26 2* 9 ?,*6i 2-8 7 ?,*56 5-64 6'oo 5'3 C*29 1*21 3'O2 6*05 6*79 2*23 2*11 3-26 3*8i 77 79 Coepang, Timor Fremantle (Swan Riv.) Hobarton six. 15 X. 19 XI 2*20 O'lO + 0-99 stop. + 0*87 9-50 6'oi 9*21 9*18 1 0*20 3*01 4*1 8 <'I7 4*07 5*5 1*20 6*31 7*89 8*47 5*64 5-95 6*T7 3-20 + 0'2 I 2'41 6*69 4*81 4*77 2*36 '55 0*36 3*9 4*78 6*30 82 63 62 Ditto 1844. 7 I + 4*78 10*41 4-88 1*1O 8*20 fi*n I % 72 4'2Q '34 6-31 66 Sydney 2 II yu stop 10*26 4*QQ 4*70 7*Q2 6-6-7 I*89 1*O7 0*92 4*93 73 Ditto 2Q 1-26 A q.1 1*74. 10*28 4*8c 4'4Q 7'QI 6-71 2*16 1*19 0*16 5*44 72 Ditto 24. Ill 0-78 I'4-Q lO'CI C'OC 1'*1 8*57 6*79 1*38 4-69 0-68 6-24 69 APPENDIX. 225 Table exhibiting the Rates of the Chronometers of H.M.S. "Fly" employed Surveying on the Coasts of Australia, during a period of Four Years, from March 1842 to April 1846. Place. Date. Z. A. B. C. D. E. F. G. H. I. E. L. M.T. Fahr. Port Stephen 1844. 5 IV. 21 6V. *5 7 VI. 4 VII. i? 3 oVIII. aX. 3i 22 XL 13 XII. 1845. 10 I. I II. 20 ,, 22 III. 3 IV. 7 V. a5 17 VI. 16 VII. 21 iVIII iX. 5X11. 17 1846. 5 I. 22 16 II. 10 IV. + o 70 73 76 80 Si 80 80 79 84 87 86 86 83 87 86 85 84 83 81 81 84 84 83 66 72 75 69 69 74 65 0-68 1-64 2-54 379 373 3'9 3-92 476 4-85 5*95 6-15 6 '34 6-68 7-82 8-00 8-27 773 6-80 7*34 6-69 7.90 7*81 7-46 4-04 5-8 6-0 8-6 8-5 8-9 6-8 1-61 2"?8 0-60 1-03 *94 0-83 0-81 1* - VI g 4 1 3 11-70 11-36 11-65 11-98 12-52 12-68 I2*IO I2'79 12-95 T -J" CQ 4-5i 3'4 3-76 3*97 4-06 4'57 474 4-90 4'59 4-71 4-72 5*49 5-00 4-96 4-87 5-61 5'94 5-42 6-0^ 6-T3 5-07 5**3 3-96 3'4J 3-67 3'55 4*35 3'99 4^3 4*16 4'45 4*35 4'85 3'93 4-36 4'3 4-87 5-67 5'33 5'39 5*10 S'3 8 5'37 6-20 3 13 1 S 1 g- ^5 8-38 7-96 7 -6o 7*11 7*13 7'3 7'88 8-14 8*25 7'8o 7'95 8-27 8'97 8-33 8-52 8-83 9-36 10-27 9-76 10*51 10*33 IO'I2 10*38 13*20 12-96 12-76 12-99 I2-85 I3*02 I3-S5 6*60 6-83 6-49 6-08 6*20 6-12 6-32 6-86 7-03 6-76 6-84 7*00 7*27 6-80 6-62 6-87 8-06 7*86 7*14 8-54 7 '47 7*47 7-80 7-68 1 13 I S B o 1 & >5 *'33 O'O2 0-30 I*0 4 r 45 -55 0-91 + 0*30 0-13 |J 0-06 *55 i'3a 0-81 0*16 3'70 4*65 4-46 4*80 If ) F 4*38 3-60 J'53 2-97 4-89 4*64 6*16 6-44 6-56 6*25 6-55 6-44 6-87 7'39 7-39 7*18 6-71 7'45 7-78 7-81 7*54 6-84 7*38 7*06 7-16 7*24 6*73 4*80 1 B f> g; 1*08 0*87 0-91 1*97 1*67 1*30 1-27 '54 2*08 2*29 2-71 a'57 3-18 3-67 4-08 4'S 1 4-38 3-i5 3-o 3'3 4-22 6-53 6-29 5-74 5-46 6-13 5-81 6-15 6-36 7-30 6-75 6-72 7-36 6-95 7-62 6-66 5-89 673 7'55 7-60 8-91 8-30 8-27 8-24 10-86 9-71 8-77 9-6^ Sandy Cape '43 079 i'39 1-47 O'OO + i*55 1-90 3-40 3-20 1 L CO 0-92 0-84 1-31 0-90 O*II 0-50 *59 0-04 0*03 3-16 2*17 2*O 0-50 I'04 r8 2-4 * 3 I-6 9 0*46 2'29 2-92 4-68 5*18 5-90 5-10 4-30 4*45 4-62 5-01 6-25 6-64 7-50 6-89 5'54 5-87 4'93 S'53 5'S* 5-6 1-64 3'5 37 8-7 6-4 5'3 3*i Cape Upstart Sir C. Hardy's Islands Raine's Island Ditto Sir C. Hardy's Islands Ditto Port Essington Sourabaya (I. of Java) Ditto Ditto Ditto Port Essington Cape York Oomaga I. (Torres Str.) Darnley Island . ... Bramble Key Darnley Island Port Essington *j jy 13-01 14-26 13-84 1379 13-58 13-46 15*16 14*20 14-43 14-27 15-06 14-39 Malacca Sincapore r'Ri 4*2 4-38 1*28 2*84 *'S i'3 17 0-9 Ditto frit. Sydney .. Supplied to another ship Ditto Ditto Port Phillip Glenelg (S. Australia) Fremantle (SwanRiver" Simon'sTown(C.G.H.; 10-16 10-00 10-51 226 ON THE MANAGEMENT OF CHKONOMETEES. Remarks on the preceding Table. 652 Murray [2 days] 3327 French [2 days] 4300 French [i day] 201 Johnson [2 days] 814 2148 Cotterell [2 days] 9'5 18279 Litherland [2 days] 1571 Dent [2 days] 2382 Parkinson & Frodsham [2 days] 6279 Porthouse [Pocket] 640 Murray [i day] 9l8 Murray [i day] 2313 Parkinson & Frodsham [Pocket] Numbers and Makers of the foregoing Chronometers : Z No. A B C D E F H I K L All these chronometers were the property of Her Majesty's Go- vernment, except chron. C, which belonged to Captain Blackwood. Z was employed as the "standard;" H was always employed as an "assistant," and L frequently so; C was also frequently taken on shore for the purposes of astronomical observation. The other chronometers (excepting A and B) were never moved from the " chronometer room," from the time they were received on board at Devonport, in March 1 842, till the ship's return there in June 1846, except when the "Fly" was hove down for repairs at Sydney in October 1845. Chronometers A and B having stopped on various occasions were repaired at Hobarton and Sydney. The continuity of their perform- ances was thus interrupted. The rates were always determined by observations, made on shore, with the artificial horizon ; and usually by the method of equal altitudes. A critical examination of the preceding table, which exhibits the actual performances of the chronometers during a period of four years, affords some instructive results. APPENDIX. 227 Chronometers C, D, E, F, G, and L, exhibit with a surprising degree of regularity the tendency which, as we have elsewhere remarked (ante, p. 41), many chronometers have to a gradual accele- ration of rate, as time advances, since they were last adjusted. The changes from their initial rates at the commencement of the voyage to their final rates at its close were, Chron. C from +0-90 to 4-14*39 D 0-67 4- 8-02 E o'io 4- 6*20 F 4-3*20 +i3'55 G 4-1*06 4- 7*68 L 1-51 4-10-51 The variation of their rates, notwithstanding minute fluctuations, being always constantly progressive; and in a gaining direction. Chronometers A and B, until the periods when they stopped, exhibited the same tendency; respectively altering from 5 St 2i to o s> 99, and from 4*76 to 4-4 5t 38. After they were repaired, their rates, never very steady, yielded anomalous results, which it is difficult to reconcile with any regular law. In chronometers Z, H, I, and K, this tendency is not observable. The table also affords a marked illustration on two occasions of the tendency to an increase of gaining rate, caused by a decrease of temperature. First, in September and October 1843, when the ship, proceeding from Timor to the Swan River, altered the temperature from 82 to 63; a change accompanied by a marked acceleration of rate in all the chronometers except A, C, and G. Secondly, in Aug. and Oct. 1845, when the ship passed from Sincapore to Sydney, experiencing a fluctuation of temperature from 83 to 66; in which, as before, all the chronometers, except A, C, G, and H, exhibited a decided acceleration of rate. It is also worthy of note, that although at different periods the rates of the several chronometers underwent considerable changes, yet that these changes were for the most part gradual and progressive, and amid all their fluctuations in value at different periods, yet that they, on the whole, exhibit a sufficient approximation towards stability of condition, even at the termination of the voyage, to justify, during short periods of time, the general assumption of the theory of uniform 228 ON THE MANAGEMENT OF CHRONOMETERS. and equable variation of rate in proportion to the time, which we have adopted as the basis of our investigations in the preceding pages. A table similar to the above, exhibiting in chronological order an abstract of the rates of the chronometers as determined from time to time by observation, should always be kept at the commencement of the "Chronometer Journal;" since, by affording an historic record of the performances of the several chronometers, it is eminently calculated to aid the judgment of the observer, and to assist him in rightly estimating the values of his results. THE END. LONDON STRANGEWAYS AND WALDEN (late G. BARCLAY), Printers, Castle St. Leicester Sq. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. LD 21-207n-5, '39 (9269s) YC 22253