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 LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
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 Accessions No 
 
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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<mo- 
 meters to be disturbed by the earth's magnetism (Nov. 1857), Mr. Airy has recently 
 stated that "of the hundreds of chronometers which have passed before me, and have 
 been regularly put under magnetic trial, only one, viz. Buckland's, No. 42 5, has been 
 sensibly affected by the earth's magnetism." An interesting paper by Mr. Airy relative 
 to the performance of the chronometer alluded to, caused by the effects of the earth's 
 magnetism on its balance, owing to its having accidentally become polarised, will be 
 found in the " Nautical Magazine" for April 1840, p. 231, together with an explana- 
 tion of the successful measures adopted by Mr. Airy to neutralise its effects. 
 
 Some interesting experiments " on the influence of the magnetic condition of 
 ships on the rates of chronometers," have recently been made at the " Depot de la 
 Marine," at Paris, by MM. Delamarche and Ploix, "Ingenieurs hydrographes." 
 
EFFECTS OF MAGNETISM. 15 
 
 magnetic causes, in which the phenomena produced are so liable 
 to be masked by extraneous circumstances, considerable variety 
 of opinion prevails, even among highly competent authorities ; * 
 however, inasmuch as, although probably in very rare and iso- 
 lated cases, magnetic causes may possibly exert an injurious 
 effect on the performance of chronometers, prudence, we think, 
 requires us to recommend precautionary measures to obviate 
 them. The following precautions are therefore suggested: 
 
 The marks on the dial-plates of the chronometers should all 
 occupy the same relative positions; that is, the line joining 
 the twelfth and sixth hour-marks should all be parallel to one 
 another, in order that, retaining the same invariable position with 
 reference to the " fore-and-aft " line of the ship, they may be 
 similarly affected by the possible local magnetic attraction of the 
 ship's iron, and that in cases where the balances of the chrono- 
 meters have accidentally acquired any degree of polarity, their 
 mutual influences on each other may be reciprocal. Proximity 
 
 These experiments were made on nine chronometers, between June 1858 and 
 January 1859, and were of the following nature. The chronometers were subjected 
 to the influence of magnetized bars at different distances, and in different positions ; 
 the rates before, during, and after the trial, being very carefully ascertained, these 
 magnets having previously, when similarly placed, with reference to a compass-needle, 
 produced deflections from the magnetic meridian, varying from 15 to 40 degrees. 
 
 The variations of rate observed under these circumstances fluctuated from o"'O2 
 to o'5o, the mean being only o s- i4. MM. Delamarche and Ploix state, as their 
 final conclusion, " The result of our experience is, then, contrary to the opinion en- 
 tertained by some physicists, and seems to indicate that the magnetic condition of 
 ships has no sensible influence on the rates of chronometers." (" Comptes Rendus 
 des Seances de 1' Academic des Sciences," tome xlviii. seance du 7 Mars, 1859.) 
 
 * Admiral Fitzroy suggested, as the possible cause of the discrepancy existing in 
 the Beagle's chain of meridian distances, measured round the globe, which amounted 
 to as much as 33 seconds, that chronometers may be affected by magnetic action in 
 consequence of a ship's head being for a considerable time in any given direction. 
 
 Mr. Fisher thinks it probable that many discrepancies that have been observed by 
 intelligent naval officers (which have generally been attributed to changes of tem- 
 perature), have arisen from this cause. 
 
 On the other hand, Mr. Airy, in the communication to the Hydrographer before 
 referred to (November, 1857), " On the magnitude of the magnetic forces in an iron 
 ship," observes, 
 
 "I have never been in any part of a ship in which a chronometer would be 
 placed, in which the compass -needle was reversed, or in which it followed rudely the 
 movements of the ship ; there are places possessing this character, as very near the 
 funnel, or very near some iron stanchions or knees, but nowhere where a chrono- 
 meter would probably be placed. That is to say: The ship's magnetic force 
 (practically) is never equal to the earth's magnetic force, and the earth's magnetic 
 force will never be doubled by the addition of the ship's force. Therefore if the 
 earth alone will not sensibly affect a chronometer, the earth and ship together will 
 scarcely disturb it sensibly." 
 
16 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 to vertical iron stanchions or knees, arm-stands of muskets or 
 rifles, any large masses of unmagnetised iron, and in iron- built 
 ships vertical iron bulk-heads, should also, in all cases, be very 
 carefully avoided. 
 
 Effects of Temperature. Ever- since, in scientific voyages of 
 modern times, the value of the chronometer has been recognised, 
 as an instrument capable of being employed in the accurate 
 determination of differences of longitude, changes of temperature 
 have been universally admitted to be one of the principal causes 
 of marked changes of rate. Great attention has in consequence 
 been paid by the makers of these instruments to the mechanical 
 details of their construction, to the improvement of their balances, 
 and to the adjustment of their compensations. Considerable 
 ingenuity has been employed, and with much success, in en- 
 deavours to eliminate or control the fluctuations of rate produced 
 by variations of temperature. 
 
 The important effects which alterations of rate, caused by 
 changes of temperature, would exercise on the measurement of 
 meridian distances, seem to have engaged the attention of 
 French navigators soon after chronometers were first employed 
 for that purpose. Fleurieu, in his Voyage de I'Isis, in 1768, 
 enters at length into this question, and proposes the employment 
 of an " equation of temperature," to correct the irregularities of 
 their rates. This matter is also alluded to in the accounts of the 
 Voyages of M. Bordainthe Flore, in 1772, and by M. Rossel in 
 that of D' Entrecasleaux , in 1791. 
 
 This matter has also seriously engaged the attention of 
 scientific officers in our own day. Admiral Fitzroy thus remarks : 
 " During eight years' observation of the movements of chronometers, 
 I have become gradually convinced that the ordinary motions of 
 a ship, such as pitching and rolling moderately, do not affect 
 tolerably good timekeepers, which are fixed in one place, and 
 defended from vibration as well as concussion. Frequently 
 employing chronometers in boats, and in very small vessels, has 
 strengthened my conviction, that temperature is the chief, if not 
 the only cause (usually speaking) of marked changes of rate. 
 The balances of but few watches are so well compensated, as to 
 be proof against a long continuance of higher or lower tempera- 
 ture. It often happens that the air in port, or near the land, is 
 at a temperature very different from that over the open sea in 
 the vicinity, and hence the difference sometimes found between 
 
EFFECTS OF TEMPERATURE. 17 
 
 harbour and sea-rates. The changes so frequently noticed to take 
 place in the rate of chronometers moved from the shore to the 
 ship and the reverse, are well known to be caused partly by 
 change of temperature,, and partly by change of situation."* 
 
 "Some chronometrical measurements," continues the same 
 authority, " have erred and caused much perplexity in the fol- 
 lowing manner. The chronometers were rated in air whose 
 average temperature was, let us suppose for example, 70. They 
 were then carried through air, either considerably hotter, or 
 considerably colder, and again rated in a temperature nearly 
 equal to that specified. The rates were not found to differ 
 much, and it was supposed the chronometers had been going 
 extremely well, though, in truth, the rates of most of the watches 
 had differed extremely (from those found in port) during the 
 voyage ; but they had returned nearly to their old rates upon 
 reaching nearly equal temperature." f 
 
 The general tendency of alteration of temperature in an 
 uncompensated\ chronometer is, that an increase of temperature 
 causes the rate to be retarded, or to alter in a losing direction ; 
 
 * Notwithstanding their suspension in gimbols, chronometers are liable to a 
 certain degree of oscillation, and alteration of horizontal position, when the ship is iu 
 motion. It has been suggested that this may affect their rates, by disturbing the 
 regularity of the balance. 
 
 f "Voyages of Adventure and Beagle," Appendix, vol. ii., p. 326. 
 
 % MM. Delamarche and Ploix recently made some very interesting experiments 
 on the effects of temperature on an uncompensated chronometer, placed at their dis- 
 posal for that purpose by M . Breguet, a French maker of celebrity. 
 
 The experiments were divided into two series, from May to September, and again 
 during November and December, 1858, during which the chronometer was exposed 
 to temperatures ranging from o to 35 Centigrade (^ to 95 Fahr.), while the cor- 
 responding rates were carefully observed. 
 
 The results showed a striking connexion between the rates and corresponding 
 temperatures. At o the observed rate was + 3"' 33 S '5, and at 35 it was 2,"' 5 2 "* 
 Between these two limits the rate constantly diminished as the temperature increased. 
 The difference between these two extreme rates is 6 m 25"- 5, which divided by 35 gives 
 i I s as the average change due to one degree of temperature centigrade (which corre- 
 sponds to 6 s - 1 1 for a degree of Fahrenheit). Assuming, then, that the rate varied 
 mathematically in proportion to the temperature, these gentlemen then formed a 
 table giving the probable temperature corresponding to the observed rates, which 
 was then compared with the observed temperatures. The column of differences in 
 no case shows a greater variation than i-6. In 115 differences 77 are below half 
 a degree, and in 7 only some tenths above i. When we consider that these results 
 may be affected by slight errors in the mode of determining the mean daily tem- 
 perature, viz. by taking a mean of the indications of a maximum and minimum 
 thermometer, they are certainly very remarkable ; in fact, as MM. Delamarche and 
 
 C 
 
18 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 a decrease of temperature, on the contrary, accelerates the rates, 
 or produces an alteration in a gaining direction. The object of 
 the compensation is to correct this defect, and to produce unifor- 
 mity of rate in spite of alteration of temperature. The artist has 
 the power, by the adjustment of the compensation weights, to 
 cause the chronometer to maintain the same rate at the two 
 selected temperatures for which he purposes to regulate the com- 
 pensation. At intermediate temperatures, and at temperatures 
 outside the chosen limits on either side, the rates may be expected 
 to vary ; for the further correction of which " secondary or 
 auxiliary compensations"* have been proposed. The smallness of 
 
 Ploix observe, "we are then justified in considering this uncompensated chrono- 
 meter as a thermometer giving the mean temperature (n degrees) during the 24 
 hours by a daily rate equal to 
 
 (" Comptes Rendus des Seances de 1' Academic des Sciences," tome xlviii., seance du 
 24 Jan. 1859). 
 
 The results of these experiments suggest some very important considerations. 
 It is clear that in an uncompensated chronometer the corresponding variations of 
 rate and temperature have a much more simple relation to one another than when 
 the effects of the latter are modified by the compensation ; no difficulty, therefore, 
 would be experienced in arranging a series of tabulated rates corresponding to 
 the different degrees of the thermometric scale. If, then, a chronometer with an 
 uncompensated balance could be constructed with its variation of rate for tem- 
 perature confined within very moderate limits, while at the same time careful 
 means could be devised for estimating the mean daily temperature to which it 
 was subjected, such an instrument would be as available for scientific uses, and 
 probably as efficient, as the costly chronometers now in use. The time and trouble 
 bestowed on the compensation is a large element in the cost of chronometers ; 
 uncompensated chronometers, being so much simplified in construction, might be 
 sold at a greatly reduced price. By constructing the balance of gold or platinum, 
 and no inner rim of steel being required, such a chronometer moreover would be 
 entirely free from the possibility of magnetic influences. 
 
 * "When chronometers had been brought to great perfection, so as to go with 
 scarcely any sensible variation of rate, while they were kept within moderate limits 
 of temperature, it was observed that they always lost if the temperature either rose 
 or fell beyond those limits ; and on the other hand, if the compensation was adjusted 
 for two extreme temperatures, then the watch always gained at mean ones. The 
 cause of this was pointed out by the Rev. Geo. Fisher to the late Mr. E. J. Dent, and 
 embodied by the latter in a Pamphlet on the " Errors of Chronometers," &c., which 
 gave the following illustration of it. The diminution of force in the spring proceeds 
 uniformly in proportion to the increase of heat, and may therefore be represented 
 by a straight line inclined at some angle to another straight line which is divided 
 into degrees of temperature. But the inertia of a compound balance cannot be 
 made to decrease quite as fast as the heat increases ; and therefore its rate of varia- 
 tion can only be represented by a curve, and can therefore only coincide with 
 the straight line representing the variation of force in the spring, in two points, 
 either the two extremes or two means, or one mean and one extreme point ; or, 
 
EFFECTS OF TEMPERATURE. 19 
 
 the variation, and the stability of the rate under varying tem- 
 perature, is the test of the skill and success of the maker in 
 the adjustment of his instrument. It sometimes happens that 
 chronometers are " over-compensated," so that the effects alluded 
 to above are reversed, the rate increasing with increase of tem- 
 perature, and vice versa. In consequence, however, of the diffi- 
 culty of exactly regulating the compensation, it is to be under- 
 stood, that these effects in practice are extremely uncertain, and 
 hence it is impossible to predicate with any certainty what the 
 effects on the rate caused by fluctuation of temperature in any 
 particular chronometer may be. 
 
 If the size and arrangements of the ship permit, a separate 
 cabin may be advantageously appropriated for the reception of 
 the chronometers, a solid block and fittings, as already described, 
 being placed within it. Such an arrangement affords the means, 
 moreover, of adopting measures for maintaining in the chrono- 
 meter-room "uniformity of temperature," which, under the circum- 
 stances already discussed, is a point of great importance. In the 
 surveying voyage of Baron Roussin on the coast of Africa and 
 Brazil in 1816-20 the temperature of the chronometer -box on 
 board the Bayadere was maintained at a uniform standard (30 
 Centigrade, or 86 Fahr.) by means of a lamp; the admission 
 of air being regulated by an aperture, the size of which could be 
 adjusted at pleasure by means of a slide, the temperature of the 
 interior air being shown by a thermometer. These precautions 
 were rewarded by the watches performing their functions with 
 extreme regularity.* In the surveying voyage, on the coast of 
 Australia, of H.M.S. Fly, under the late Captain Black wood, 
 the temperature of the chronometer - room was in a similar 
 manner maintained by a lamp at an equable standard, at times 
 
 in other words, the compensation can only be exact for some two temperatures for 
 which you may choose to adjust it. The selected temperatures are generally 45 and 
 75, 60 being the mean ; for a greater range than 30, the adjustment cannot be 
 depended on. 
 
 t( The correction of this error is called the ' secondary or auxiliary compensation/ 
 and it is the point towards which nearly all recent inventions for the improvement 
 of chronometers have been chiefly directed." (See " Denison on Clocks and 
 Watches ; " Weale's Series, p. 292 ; also " Dent on the Errors of Chronometers ; " 
 London, 1842.) 
 
 * " Recherche s Chronometriques," p. 77. It has also been further remarked 
 that artificially keeping the chronometers at a uniform temperature, likewise insures 
 their being kept in dry air, and the maintenance of their oils in a state of fluidity. 
 
20 ON THE MANAGEMENT OF CHKONOMETEKS. 
 
 when, from natural causes, it would otherwise have been liable to 
 excessive variation.* 
 
 Another precaution was also adopted in the Fly to protect 
 the chronometers from sudden draughts of cold air and to prevent 
 the radiation of heat. 'Their box^was covered over with a cover- 
 ing of coarse woollen (fearnought)', over the partition containing 
 the "standard"! chronometer, there was a flap in the cover, 
 so that at any moment a comparison with the " standard " could 
 be obtained without uncovering the rest. By this arrangement, 
 under ordinary circumstances, it was only necessary to uncover 
 the chronometer-box once a-day, for the purpose of winding and 
 comparing ; except in harbour, on those days when observations 
 for the rates were made, when it was required to be done more 
 frequently. As this device is certainly simple and easy of execu- 
 tion, we think it may safely be recommended for imitation. 
 
 Influence of oilier Causes on Chronometers. Besides the possible 
 effects of magnetism, and the undoubted effects of variations of 
 temperature, experience shows that there are other causes in 
 existence which exercise a very sensible influence on the perform- 
 ances of chronometers. 
 
 It has frequently been observed that many chronometers have 
 a tendency towards a constant and progressive increase of rate, 
 sometimes in a gaining, sometimes in a losing direction ; but more 
 frequently the former, altogether independent of the changes 
 caused by variation of temperature. To this tendency has been 
 given the name of the acceleration ; great attention has been paid 
 to its investigation by recent French writers, MM. du Cornulier 
 and LieussouJ to whose valuable labours nautical science is much 
 indebted. 
 
 * It may be necessary to add a word of caution to those who may be disposed to 
 adopt this device. The lamp and ventilating apparatus should be very carefully 
 attended to, for if the temperature be allowed to become excessive, or the lamp 
 incautiously permitted to go out for any time, the extreme variations of temperature 
 to which the chronometers would thereby become liable, might probably be more 
 injurious than if they were simply exposed to the natural changes of temperature of 
 the atmosphere. 
 
 t See p. 26. 
 
 J The memoirs of M. du Cornulier, an officer in the French naval service, were 
 originally published in the " Annales Maritimes " for 1831, 1832, 1842, and 1844; 
 copious extracts from them have since been republished in the "Recherches Chrono- 
 metriques," a most useful work, now in course of publication by the Depot de la 
 Marine at Paris, under the auspices of the Minister of Marine. 
 
 M. Aristide Lieussou, Ingenieur Hydrographe de la Marine, &c., is the author 
 of a valuable work, "Recherches sur les Variations de la Marche des Pendules 
 
INFLUENCE OF THE ACCELEEATION. 21 
 
 Each of these gentlemen has independently* proposed alge- 
 braic formulae for correcting the differences of longitude obtained 
 by chronometers^ which embrace not only the fluctuation produced 
 by varying temperature, but also the effects of the progressive 
 influence of the acceleration. In a subsequent part of this work 
 we shall have occasion to refer more at length to the ingenious 
 speculations and investigations of these authors ; our only 
 object here, is to introduce the matter generally to the notice 
 of our readers, and to accompany it by a brief and popular ex- 
 planation. 
 
 The " acceleration" may be defined to be the product of the 
 mechanical imperfections which unavoidably exist even in the very 
 best constructed chronometers, and is attributable to the effects of 
 " wear, dirt, and the thickening of the oils (with which the pivots 
 are lubricated), and which, in consequence, hinders or facilitates 
 more and more the play of the moveable parts. Their effect is 
 at first insensible ; but as they are continually in action, the errors 
 that they incessantly produce accumulate, and it becomes indis- 
 pensable to take account of them after a certain lapse of time." f 
 In most watches the acceleration is a variable quantity, and 
 presents no guarantee of permanent fixity ;J we ought not, 
 therefore, to assume it constant, but for small intervals ; but, 
 inasmuch as the performance of the chronometers can be checked, 
 and their errors ascertained from time to time, by the necessary 
 observations, " the hypothesis of the uniformity of the acceleration 
 during moderate intervals," observes M. du Cornulier, " seems 
 generally admissible." This uniformity is one of the principal 
 conditions of the goodness of a chronometer. " In general," con- 
 tinues the same authority, "the acceleration seems to be so much 
 
 et des Chronometres," Paris, 1854; which abounds with matter useful and in- 
 structive to those who take an interest in chronometric science. 
 
 * M. Lieussou states (note p. 30) that he was not aware of the previous labours 
 of Du Cornulier, until he saw the report of the Consultative Committee of the Depot 
 de la Marine on his own memoir, in which the work of his predecessor was 
 alluded to. 
 
 f " Recherches Chronometriques," p. 94. 
 
 " An abrupt shock may alter the acceleration, because it may produce in an 
 imperfect mechanism a new obstacle to movement, or destroy one which already 
 exists. It is for this reason, that one ought to avoid putting a watch in a place where it 
 would be exposed to any shock acting upon it directly or by transmission ; a sharp 
 pitching motion, a salvo of artillery, the proximity of a pump in movement, the fall 
 of a heavy body near a watch, may suffice to develope the germ of a new accelera- 
 tion." (" Rech. Chron." p. 96.) 
 
22 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 the greater as the watch has been in movement for a less long- 
 time. It seems that these instruments require some months' 
 exercise to settle down to a rate a little stable. The irregu- 
 larities reappear after a long service, whether it be in consequence 
 of wearing out, dirt, or the thickening of the oils ; whatever may 
 be the cause of the acceleration,* it is wise to bestow no con- 
 fidence on watches which are affected by it to a considerable 
 degree ; it is always the sign of an essential vice, and often the 
 prelude of a total derangement. "f 
 
 Besides the progressive deterioration caused by wear, dirt, 
 and the thickening or gradual evaporation of the oils, chrono- 
 meters are sometimes injuriously affected by damp, which rusts 
 the balance-spring and other parts composed of steel ; when this 
 happens, the effects are very destructive. To guard against it, 
 some artists have lately adopted the precaution of hermetically 
 closing the box which contains the works. 
 
 There is reason to believe that chronometers have sometimes 
 been much affected by violent storms of thunder and lightning, 
 not only when the ship has been struck by lightning, but even 
 when the ship has only been in the midst of the storm. It is 
 stated on good authority that such was the case in the voyage of 
 the Coquille, under M. Duperrey in 1823. In the voyage from 
 Amboyna to Port Jackson the frigate experienced some severe 
 thunder-storms near the islands of Savu and Timor ; when the 
 vessel arrived at her destination the chronometers were found to 
 have changed their rates enormously, from 5*3, 26-2, and 
 + 10*1 respectively, to + 7-0, 187, and +27-6. The interval 
 between the epochs of rating was 84 days, a long period doubtless, 
 and possibly some of the change was due to the effects of tem- 
 
 * " Recherches Chronometriques," p. 129. 
 
 f " We conceive," says M. du Cornulier, " that the tendency to advance (i. e. to 
 go faster) results from a play more and more easy in all the parts of the mechanism, 
 and especially of the balance upon its pivot-; whatever may be the degree of exact- 
 ness and of polish that the artist may have succeeded in giving to these pieces, the 
 natural friction might still bring them to a more perfect degree of agreement, render 
 their forms more regular, and annihilate the small asperities which had escaped him. 
 Whilst this tendency is feeble we willingly recognise in it a perfection in the instru- 
 ment itself." 
 
 " The tendency to retard (i. e. to go slower) seems to us to result from the pro- 
 gressive increase of rust, or from wear which injures the polish and becomes destruc- 
 tive to it ; this is why we always suspect it. A degree of wear, which attains to a 
 destructive limit, may equally give rise, according to circumstances, to an exaggerated 
 tendency to advance." (" Rech. Chron." p. 127.) 
 
WINDING-UP CHRONOMETERS. 23 
 
 perature the chronometers being taken into colder latitudes and 
 a lower temperature ; but still the facts are very remarkable, and 
 it would be unwise to deny the possible influence of the unwonted 
 electric conditions of the atmosphere during these storms on their 
 performances.* 
 
 Winding-up Chronometers. The chronometers having been 
 received on board, and stowed in the manner previously described 
 in the place appropriated for their reception, it is of importance 
 at once to commence and adopt a uniform and systematic manner 
 of winding them up, and comparing them daily with one another. 
 
 Methodical arrangements in these particulars favour the 
 stability of rate of the chronometers, assist in the detection of 
 irregularities, and diminish the probabilities of their being 
 accidentally allowed to run down;f while in the reduction of 
 chronometric observations for the determination of meridian 
 distances, systematic plans of comparison are indispensable to 
 accuracy of computation, and very influential in diminishing 
 the amount of labour required, if the number of chronometers 
 employed is at all large. 
 
 Jt is of course a matter of purely arbitrary convenience 
 what time is selected for the daily winding and comparison of 
 the chronometers; the only point of importance is, that what- 
 ever arrangement be adopted, it be uniformly and systematically 
 adhered to. 
 
 The practice of different scientific officers, of acknowledged 
 repute, engaged on surveying duties, has varied considerably. 
 
 Capt. Owen states, that their chronometers were always 
 wound and compared at noon ; Capt. Fitzroy's plan was to wind 
 up at 9 A.M., and compare at noon ; while Sir Edward Belcher 
 
 * These facts are stated by M. Givry, " Ingenieur hydrographe de la marine," 
 in a memoir " Sur 1'emploi des Chronometres a la mer," published in 1840, and which 
 has been in great part reprinted in the u Recherches Chronometriques ; " see p. 81. 
 
 f It is advisable to adopt, in the daily routine of the ship's duties, some pre- 
 caution to prevent the possibility of the chronometers being accidentally omitted 
 to be wound up, and to devise some plan by which this important duty should not 
 be altogether dependent on the memory of any one single person. In many ships 
 the cabin-door or steerage sentry reports both to the master and to the officer of 
 the watch when the time has arrived for the performance of this duty ; and he is 
 subsequently not allowed to be relieved from his post until the serjeant or corporal 
 of the guard has ascertained from the officer in charge of the chronometers that the 
 operation has been performed, and has duly reported the same to the officer of the 
 watch and to the commanding officer. Some arrangement of this nature is extremely 
 necessary, and should always be adopted. 
 
24 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 prefers 6 A.M. for both, operations, thus leaving the whole of the 
 day available for more active duties. 
 
 On the whole, we think 8 A.M. to be a time conveniently 
 adapted to this duty, and consistent with the general arrange- 
 ments, of a man-of-war. .-. The officers generally charged with this 
 important function are not, under ordinary circumstances, likely 
 to be then absent on detached duty, nor are they usually engaged 
 in the customary morning evolutions. In cases, moreover, where 
 it is proposed to take sights for time, during the forenoon, the 
 usual morning comparison may be made subservient to that end, 
 and taking the place of the first comparison before the observa- 
 tions, may save the necessity for an extra special comparison. 
 
 The chronometers should be wound first and compared after- 
 wards. In winding them they should habitually be attended to, 
 in the same order, from day to day, one by one, as they lie in 
 their places ; so that the mechanical habits of regularity in this 
 particular may be a safeguard against the caprice of memory or 
 accidental distraction from any disturbing cause. From want 
 of system in this particular, we have known instances where the 
 attention of the officer engaged in this duty being accidentally 
 distracted during its performance, the chronometers have been 
 compared, but some or all of them not wound up, that operation 
 being forgotten, and the omission not detected till the chrono- 
 meters ran down. 
 
 In winding up chronometers the turns of the key should 
 always be counted, and the last turn made gently and carefully, 
 until it is felt to butt. It is said, that it has sometimes happened 
 to persons over careful, that they have let their chronometers 
 run down by having counted the number of turns, and never 
 winding up close for fear of injury to the chain or works, by 
 which they have always lost a little of the chain each day ; so 
 that, after a time, the chronometer is found to stop just at the 
 time it should be wound up. 
 
 In winding up box-chronometers, the chronometer should be 
 inverted carefully in its gimbols, held firmly in the left hand, and 
 the key pressed close home with the right ; after the operation is 
 performed, and the key withdrawn, care should be taken that the 
 keyhole is again covered with the slider to secure it from dust or 
 damp, and then the chronometer should be gently eased down into 
 its natural position without violence or jerk. 
 
WINDING-UP CHRONOMETERS. 25 
 
 Many chronometers are fitted with permanent keys* at their 
 backs, which in some respects seem preferable to moveable ones, 
 especially if they are fitted with rack-work to prevent the ill 
 effects of attempting to turn them the wrong way. 
 
 Sir E. Belcher suggests the advantage of furnishing box- 
 chronometers with the means of performing the operation of 
 winding without disturbing their position, by adapting a spindle 
 to their sides ; such an arrangement could easily be effected, and 
 would, possibly, be advantageous. Probably, also, the opera- 
 tion of winding in bad weather would be attended with less diffi- 
 culty and risk.f 
 
 In winding up pocket-chronometers, the watch should be held 
 firmly in the left hand with the wrist pressed gently against the 
 breast, the key should be turned equably and steadily with the 
 right, care being taken to avoid giving the w r atch any circular 
 motion upon the key. The common but vicious practice of turn- 
 ing the left as well as the right hand is injurious, for two reasons: 
 first, because the circular motion affects the regularity of the 
 balance ; and, secondly, because the compound motion of the two 
 hands doubles the velocity of winding, and increases the chances 
 of straining or snapping the spring from the jerk at the conclusion 
 of the operation. 
 
 Chronometers and watches which are wound at the back 
 usually require the key to be turned from right to left. Those 
 having their keyholes on the dial-plate, on the contrary, from left 
 to right. 
 
 The chronometers should be wound daily, whether con- 
 structed to go for one or two days ; eight-day chronometers once 
 a-week, say on Sunday, a day easily remembered. If the 
 number of chronometers is large, the precaution should be taken 
 
 * The chronometer-makers do not generally seem to be in favour of this arrange- 
 ment. It is thought that dust and damp are more likely to penetrate through the 
 fittings of the permanent key, than when the key-hole is closed in the usual manner 
 by a spring slider. 
 
 f M. Givry states that " MM. Motel and Breguet, two French watchmakers of 
 celebrity, constructed chronometers which were wound up from the top of the box 
 in which they were enclosed (how is not exactly stated), and that this ingenious in- 
 novation dispensed with the necessity of inverting their suspension." 
 
 On this, M. Delamarche, the editor of the " Recherches Chronometriques," 
 remarks, " These artists have returned to the plan of winding up watches from 
 the bottom, and consequently by reversing them; they say, that this movement is 
 salutary on account of its distributing the oils over the friction points." (" Rech. 
 Chron." p. 74.) 
 
26 ON THE MANAGEMENT OF CHRONOMETEES. 
 
 of examining them after winding, to see that none have been 
 accidentally forgotten, either by looking at the winding index, if 
 furnished with that apparatus, or by trying the key. 
 
 Standard Chronometer. The chronometers having been all 
 wound up, they should next be,;compared. To facilitate their 
 systematic comparison, and to organise more easily the reduction 
 of chronometric observations, it is customary to select one chrono- 
 meter as a " standard" to which all observations for the determi- 
 nation of the time are in the first instance referred ; the indication 
 of the other chronometers at the same moment being subsequently 
 obtained by means of the comparisons. The chronometer selected 
 to perform the duty of the " standard" should be one of first-rate 
 character, and by a maker of established repute : it will be con- 
 venient that it should have a clear and distinct beat to half- 
 seconds ; also, that its dial-plate be well marked ; and it is advis- 
 able, although not indispensable, that its rate (at any rate at 
 starting on the voyage) should be small. Stability of rate, how- 
 ever, is at all times much more important than the smallness of 
 its amount. For the facility of comparison with the other chrono- 
 meters it will be found convenient to arrange the chronometers in 
 their box in such a manner that the standard shall occupy a 
 central position among them. The XII and VI hour marks 
 should all be parallel to one another, and to the " fore and aft " 
 line of the ship. 
 
 Distinction of Chronometers. As the description of chrono- 
 meters by means of their makers' names and numbers is tedious 
 and troublesome, it is often customary to denote each chronometer 
 emphatically by a single letter, A, B, or C, &c. &c., an arrange- 
 ment sufficiently distinctive to the persons who have to use them, 
 and at the same time brief and concise when a series of results 
 obtained from them has to be recorded ; care being taken that in 
 the commencement of the chronometric journal, and also in the 
 records of their performances transmitted at any time to head- 
 quarters, a preliminary notation is made of the actual maker's 
 name and number belonging to each chronometer. 
 
 In accordance with this arrangement, it will be found con- 
 venient emphatically to denote the "standard chronometer ; * by 
 the letter Z, the final letter of the alphabet, the other chrono- 
 meters being represented by the initial letters, A, B, C, D, &c., 
 in order of alphabetical sequence, or by the initial letter of their 
 
COMPARISON OF CHRONOMETERS. 27 
 
 makers' names, if preferred : such as A, for Arnold; D, for Dent; 
 F, for Frodsham; M, for Murray, &c. &c. ;* these little arrange- 
 ments being, of course, matters of mere fancy and convenience. 
 They are sanctioned, however, by the practice of high authori- 
 ties; and having found them very convenient in our own ex- 
 perience, we think them worthy of being recommended for adop- 
 tion, especially since, in addition to other advantages, as will be 
 seen further on, this mode of description facilitates the algebraic 
 discussion of the questions of errors and rates. 
 
 Comparison of Chronometers. It is very desirable, when practi- 
 cable, that the comparisons of the chronometers with the standard 
 should all be made by one person. The effects of personal equa- 
 tion are thereby much simplified, and the chances of small contin- 
 gent errors materially diminished. 
 
 The mode of proceeding is as follows : the observer, with 
 book in hand, containing a ruled form for the entry of the time 
 shown by the several chronometers takes a beat from the standard 
 five seconds before the arrival of the second hand at any five or 
 ten seconds mark, and then quickly casting his eye on the chrono- 
 meter to be compared, counts with his ear the ten f beats which 
 elapse before the second-hand arrives at the mark selected; at the 
 completion of the interval he reads the other chronometer, and 
 the comparison is effected. The operation being repeated a second 
 time to correct the first judgment, the final observation is recorded. 
 It will be found convenient to make the first comparison at 50% 
 or at 2O S , and again at the minute or half minute for the final 
 result. 
 
 This is done in succession for each chronometer. Half-a- 
 minute elapsing between each comparison affords ample time to 
 record the last one and to prepare for the next, while the fact of 
 the standard being only noted at the minute and half-minute, 
 makes the notation of its time a mere matter of form, and allows 
 the writer's attention to be concentrated on the " second " of the 
 other chronometer. When the round is completed, he takes the 
 
 * It will be found convenient to have the distinctive letter of each chronometer 
 legibly printed on pieces of card-board, and conspicuously attached to their respective 
 partitions in the chronometer-box. 
 
 f We are here supposing the "standard" to beat to half-seconds; if the beats 
 are otherwise arranged, the observer must regulate his counting accordingly. Box- 
 chronometers generally beat to half-seconds ; pocket ones frequently beat five to 
 two seconds. 
 
28 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 precaution of seeing that the " minute " of the standard has been 
 duly noted, so as to ensure that no constant error affects all the 
 comparisons. 
 
 To compare chronometers, however, satisfactorily in this 
 manner, requires much practice ajid self-confidence, and demands 
 also a more delicate organisation of ear and eye than is possessed 
 by all persons. The deficiency of natural or artificial light -in the 
 chronometer-room, moreover, and the difficulty of self- abstraction 
 amid the babel of noises which sometimes prevails aboard ship, 
 may render the assistance of a second person indispensable. The 
 mode of comparison by two persons is as follows : 
 
 The principal, who notes the time by the standard, also re- 
 cords the result in the ruled book previously prepared. The 
 comparisons are made at the minute and half-minute. As the 
 second-hand approaches 50* or 2O S , the principal says, Look out; 
 at 5O S or 2O S , Stop ; sharply and audibly : but this is only a 
 caution. At the minute or half-minute he repeats the word 
 Stop. The assistant corrects his first impression by his second 
 observation, and reports his reading to the principal, who writes 
 it down. If necessary, an intermediate comparison, to assist the 
 accurate observation of the subdivision of the second into tenths, 
 may be made at the 55 s or 25% but this is rarely necessary to two 
 persons accustomed to work together. 
 
 In stating the indication it is customary and convenient for 
 the assistant to mention the seconds and tenths first, as most 
 important; then the minutes; and, lastly, the hour, as of least 
 consequence, since gross errors are always easy of detection: 
 thus, the record of J h 54 23^3 is thus given: "twenty-three, 
 three," " fifty-four," " seven ; " the principal writing it down (as 
 spoken) in reverse order. 
 
 If any mistake occur or confusion arise, whether the com- 
 parisons be made singly or with the aid of an assistant, it is 
 advisable to pass that chronometer by for the moment, complete 
 the comparisons of the others in regular order, and then return 
 to the faulty one again. 
 
 The comparisons when completed will stand recorded in the 
 chronometer-journal as in the first form in the following page. 
 
 If it be thought advisable to keep a fair copy of the com- 
 parisons, the second form will be found convenient. 
 
FOEMS FOR CHRONOMETER JOURNAL. 
 
 29 
 
 Form No. 1.* Chronometer Journal for record of daily com- 
 parisons of chronometers. 
 
 H.M.S. 
 
 Date. 
 
 Bar. 
 
 Max. 
 and 
 Min. 
 Ther. 
 
 Chron. 
 A. 
 
 2d 
 Diff. 
 
 Chron. 
 B. 
 
 2d 
 
 Diff. 
 
 Chron. 
 C. 
 
 2d 
 Diff. 
 
 '5 o 
 flg 
 
 Remarks. 
 
 1860. 
 Sun. 
 Jan. 
 
 ISt. 
 
 in. 
 29-91 
 
 80 
 78 
 
 h m s 
 Z 2 15 
 
 10 50 7 
 
 
 h m s 
 2 15 30 
 
 7 20 i 
 
 
 h m s 
 2l6o 
 
 i ii 54 
 
 
 C.S. 
 W.B. 
 
 Heavy Gale 
 Much Motion 
 
 3 24 53 
 
 6 55 29 
 
 i 4 6 
 
 Mon. 
 ad. 
 
 in. 
 30-32 
 
 78 
 71 
 
 Z 2 39 o 
 ii 14 5'3 
 
 s 
 
 2 39 30 
 7 43 48-8 
 
 H 
 I2'2 
 
 2 40 
 
 i 35 55'5 
 
 s 
 
 C.S. 
 G.C. 
 
 Confused Cross 
 Sea 
 
 3 24 54-7 
 
 6 55 41-2 
 
 i 4 4'5 
 
 Form No. 2.* Chronometer Journal for fair copy of compa- 
 risons, if thought necessary. 
 
 H.M.S. 
 
 Date. 
 
 Z A 
 
 2d 
 Diff. 
 
 Z-B 
 
 zd 
 Diff. 
 
 Z-C 
 
 2d 
 Diff. 
 
 Bar. 
 and 
 Mean 
 Temp. 
 
 Remarks. 
 
 1860. 
 
 
 
 
 
 
 
 
 
 Jan. 
 
 h m s 
 
 
 h m s 
 
 
 h m s 
 
 
 in. 
 
 
 Sun. i 
 
 3 M 53 
 
 
 6 55 29 
 
 
 146 
 
 
 29-91 
 
 Heavy Gale 
 
 
 
 
 
 
 
 
 79 
 
 Much Motion 
 
 
 
 s 
 
 
 8 
 
 
 s 
 
 in. 
 
 
 Mon. 2 
 
 3 24 54'7 
 
 1*7 
 
 6 55 41-2 
 
 12-2 
 
 i 4 4'5 
 
 i'5 
 
 30-32 
 
 Confused Cross 
 
 
 
 
 
 
 
 
 74 
 
 Sea 
 
 
 
 
 
 
 
 
 in. 
 
 
 Tues. 3 
 
 3 24 55 
 
 '3 
 
 6 55 52 
 
 I0'8 
 
 i 4 3*7 
 
 0-8 
 
 3'4i 
 
 Less Motion 
 
 
 
 
 
 
 
 
 70 
 
 
 
 
 
 
 
 
 
 in. 
 
 
 Wed. 4 
 
 3 2 4 55'8 
 
 0-8 
 
 6 56 2 
 
 I0'0 
 
 i 4 2-7 
 
 I'D 
 
 30-34 
 
 73 
 
 Exercised Firing at 
 Night Quarters 
 
 It is advisable that the operation of winding and compar- 
 ing should, as far as possible, always be performed by the same 
 
 * See also Forms, as amplified in the Appendix. 
 
30 ON THE MANAGEMENT OF CHRONOMETEES. 
 
 persons,* that these duties should be confided to as few people 
 as possible ; and in order that the errors arising from personal 
 equation f should always lie in the same direction, that in making 
 the comparison the parts performed by the principal and his 
 assistants should not be interchanged. In order to guard, how- 
 ever, against the possibility of the persons usually charged with 
 this duty being unable to do so from sickness or temporary 
 absence, it is advisable that one or more persons should be 
 instructed as occasional assistants ; and it seems convenient that 
 officers of the civil branch of the service should be chosen in 
 preference, as less liable to be absent on detached duty. 
 
 Since, as has been previously observed, change of temperature 
 is supposed generally to be the principal cause of marked changes 
 of rate, it is desirable that the temperature of the chronometer- 
 box be duly recorded, and accordingly means should be devised 
 for placing within it a maximum and minimum thermometer, 
 the indications of which, at the time of winding, J should be duly 
 noted daily in the comparison book, so that if decided fluctuations 
 of rate are observable in any of the chronometers, its cause may 
 be sought among the records of the changes of temperature. The 
 mean of the readings of the maximum and minimum thermometer 
 is to be taken as the mean temperature for the preceding 24 hrs . 
 The height of the barometer may also with propriety be noted at 
 the same time. 
 
 As a matter of technical convenience, and as favouring uni- 
 formity of treatment, it is advisable that, for the purposes of com- 
 parison, the " standard" be always considered fast on the other 
 chronometers, whatever their actual indications may be, 12 hrs 
 being added to the time shown by the standard in taking their 
 difference when necessary. 
 
 The utility of systematic comparison may be easily exempli- 
 
 * The initials of the persons who compare should be inserted in the comparison- 
 book, in case a reference to them should be subsequently required to explain any 
 seeming discrepancy. 
 
 f If all the observations were affected by a constant personal error, the errors of 
 the chronometers on time only would be vitiated thereby, the rates which depend 
 on the difference of errors not being influenced by them. So also, in measuring 
 a meridian distance which depends on the comparison of two errors, its value would 
 not be altered by a correction equally affecting both the times. 
 
 J The thermometers should be read off as soon as the box is opened, and before 
 the admission of a fresh volume of air has had time to influence the indications 
 of those instruments. 
 
COMPARISON OF CHRONOMETERS. 31 
 
 fied as follows: If Z and A represent the standard and any 
 other chronometer, we have from the records of the comparison- 
 book the daily consecutive values of Z A. 
 
 If the daily rates of the two chronometers happened to be 
 the same, Z A would remain a constant quantity; if, as is 
 more probable, the two chronometers had different rates, the 
 consecutive values of Z A, from day to day, would vary by the 
 algebraic difference of their rates ; if the rates remained uniform, 
 this second difference would remain constant (disregarding, of 
 course, small discrepancies arising from errors of comparison); 
 if, on the contrary, the second difference were not uniform, it 
 would show that the rate of one or both the chronometers was 
 altering, and without further evidence it would be impossible 
 to decide which was going astray. 
 
 If, however, we have more than two chronometers, a third 
 one may be brought in to assist our judgment. 
 
 In this case, having three chronometers, Z, A, and B, our 
 comparisons give us not only Z A, but also Z B, whence, 
 without the trouble of direct comparison with each other, the 
 differences of B and A may be obtained. 
 
 For (Z - A) - (Z - B) = B - A ; 
 
 or (Z - B) - (Z - A) = A- B ; 
 
 similarly for any other chronometer, C, 
 
 (Z-B)- (Z-C) = C-B; 
 or (Z-C)- (Z-B) = B-C; 
 
 and so on for any number of chronometers: in which processes 
 all trace of the presence of Z vanishes.* 
 
 If, then, on examination of the records of the daily dif- 
 ferences, Z A and A B, it be observed that the second 
 difference of the daily value of A B remained constant, 
 while that of Z A is irregular, we may safely infer that 
 
 * This may appear, at first sight, a very circuitous and clumsy method of finding 
 the differences of A and B, B and C, &c. ; and so it would be if the number of 
 chronometers were small, and these differences always required: but as the number 
 of chronometers employed may be large, and as these differences are only required 
 when there is a necessity for instituting a critical examination of the performances of 
 the chronometers, perhaps at some subsequent period, this plan, we apprehend, will 
 be found very convenient, and not liable to any just objection. 
 
32 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 chronometers A and B are going steadily, and that Z is alter- 
 ing; if, on the other hand, we perceived that the second 
 differences of Z A and A B were both irregular, while 
 that of Z B remained steady, it would indicate that A was 
 the culprit, and Z and B retaining their rates, and so on.* 
 
 The system of the intercomparison of each chronometer 
 separately with the standard, which we are here recommending, 
 gives also great facilities for the manipulation of the results 
 when a large number of chronometers is employed. 
 
 It is only necessary, in the first instance, to take into 
 account the indications of the standard, as if it alone were con- 
 cerned; the indications of the other chronometers are after- 
 wards separately obtained by means of their respective com- 
 parisons. 
 
 Thus, Z being the time shown by the standard at the 
 moment of any phenomenon, or at any given instant, the 
 corresponding times indicated by the other chronometers are 
 obtained at once by applying their comparisons, since 
 
 Z (Z - A) = A 
 Z _ (Z - B) = B 
 
 Z _ (Z - C) = C 
 
 &c. &c. 
 
 That is, by subtracting the comparison for each chronometer 
 from the time shown by the standard, we obtain the indications 
 of the other chronometers in succession. 
 
 Thus, again, the primary result of any set of observations 
 for the errors of the standard on local mean time, L, gives us 
 
 * In Her Majesty's Navy it was formerly customary to furnish every ship with 
 one chronometer. If the captain supplied a private one in addition, then the 
 Government gave another, so as to make three chronometers in all. It was argued 
 that if a ship had but one chronometer, it would be unwise implicitly to trust it ; 
 and, therefore, great caution was necessary in navigation. If the ship had two, 
 and they happened to differ, it would be impossible to tell which was right. If, 
 however, she had three, the coincidence of any two of them would throw a strong 
 probability on the truth of their results ; while the mean of the three could probably 
 be more safely relied on than any one of them taken singly : added to which, the 
 examination of their intercomparisons, as pointed out above, gave the means of 
 detecting which of the three was irregular. 
 
 By a recent more liberal regulation, the Admiralty now furnish all sea-going 
 ships with three chronometers. 
 
COMPARISON OF CHRONOMETERS. 33 
 
 Z L if the error be considered fast, and L Z if, on the 
 contrary, it be estimated as slow. 
 
 In the first case, for the errors of the other chronometers, 
 we then have 
 
 (Z-L) - (Z A) = A-L 
 
 (Z L) - (Z-B) = B-L 
 
 (Z-L) - (Z-C) = C-L, 
 
 &c. &c. 
 
 That is, by subtracting the comparison for each chronometer from 
 the error of the standard (adding 12 hours to the latter, when 
 necessary), we obtain the error of that particular chronometer, 
 and so on for the others in succession. 
 
 Again, if the chronometers are to be considered as slow, we 
 have 
 
 (L Z) + (Z A) = L-A 
 
 (L Z) + (Z B) = L B 
 
 (L-Z) + (Z-C) = L-C, 
 
 &c. &c. 
 
 That is, by adding the comparison for each chronometer to the 
 error of the standard, we obtain the error of that particular 
 chronometer (rejecting 12 hours when it exceeds 12 hours), and 
 so on for the others in succession. 
 
 In the practical application of this system two points are 
 essential : first, that, for the purposes of comparison, we agree to 
 consider the standard fast of all the other chronometers, whatever 
 their actual indications may be; and secondly, that whatever the 
 times shown by the several chronometers actually may be (and, 
 of course, during a long voyage they may exhibit considerable 
 divergence, from the diversity of their rates), they may be treated 
 in a uniform manner ; that is, considered either to be all fast on 
 local mean time, or else all stow. If this be done, we apprehend 
 the management of any number of chronometers whatever will 
 be a matter of comparative facility, and not nearly so formidable 
 as might at first sight appear. It may also be useful to remark, 
 that in all cases of uncertainty as to the proper application of the 
 comparisons, the matter will generally be greatly simplified by 
 expressing it algebraically, as in the preceding illustrations. 
 
 D 
 
34 ON THE MANAGEMENT OF CHRONOMETEBS. 
 
 Chronometer Journal, fyc. A book (styled the " Chronometer 
 Journal ") should be kept for the daily record of the comparisons 
 of the chronometers. The form No L, given in the Appendix 
 (which is the same as the first one given at page 29), seems con- 
 veniently adapted for all practical purposes.* The book should 
 be the size of foolscap or large letter-paper; it should be kept 
 ruled up ; the entries, in the first instance, should be in pencil ; 
 the daily differences of the several chronometers from the 
 standard should be taken at the time of comparison, so that any 
 error or discrepancy might be immediately detected, and a fresh 
 comparison made, if necessary ; the several entries should after- 
 wards be inked in while the matter was still fresh on the memory; 
 a wide column should be left at the right-hand margin for 
 remarks, and notice should be taken therein of any circumstance 
 likely to affect the performance of the chronometers, such as 
 violent motion of the ship, her striking the ground, heavy firing 
 of guns for salute or exercise, stoppage or removal of any of the 
 chronometers, storms of thunder and lightning, &c. &c. The 
 daily rates of the chronometers, determined from time to time by 
 observation, should also be noted in the book, as a record, and 
 for the purpose of occasional comparison with the column of 
 second differences. 
 
 The chronometer journal, kept in the form and manner above 
 described, should be solely appropriated to the daily record of the 
 intercomparisons of the several chronometers with the standard at 
 the time of winding. All other comparisons made at any other 
 time for the purposes of observation for time, or for any other 
 
 * In the Admiralty instructions (for the government of Her Majesty's naval 
 service), chap. v. p. 174, the captain is instructed to keep a chronometer journal, 
 and a formof journal adapted for this purpose is given in the Appendix, p. 54. 
 With every respect for Admiralty forms, we think those given in the Appendix to 
 this work much to be preferred for all practical purposes. In the Admiralty form 
 four columns are given for the comparison of each chronometer with the standard : 
 the first two containing the times shown by the two chronometers at the moment 
 of comparison ; the third, their difference ; and the fourth, their second difference. 
 If this form be intended for daily practical use, it is very inconvenient to have 
 two quantities, whose difference is to be taken, placed in parallel columns ; and if 
 intended for a permanent record, the first two columns are superfluous, as the actual 
 times shown by the chronometers are a matter of no moment, their differences and 
 subsequent second differences being alone of any importance. Whatever view be 
 adopted as to the propriety of keeping a fair copy of the comparisons with a view to 
 a permanent record, one thing is quite clear, that in everything connected with these 
 duties condensation should be studied, and all superfluities carefully avoided. We 
 believe it is intended, shortly, to alter the form of these returns. 
 
CHRONOMETER JOURNAL, ETC. 35 
 
 special object, should be made, on the principles previously recom- 
 mended, in a separate memorandum-book set apart for that use. 
 
 If the chronometer journal be kept neatly and carefully in the 
 manner above recommended, there does not seem to be any 
 necessity for keeping a fair copy of it.* Copies involve a large 
 additional amount of clerical labour. With every care, it is 
 almost impossible wholly to avoid errors of transcription, espe- 
 cially in a matter involving a large mass of figures, in which the 
 copyist can in most cases only ta,ke a mechanical interest ; and, 
 moreover, in case of doubt or dispute, the best-written copy has 
 no value compared with the original. 
 
 The sole use of the chronometer journal is to enable the 
 persons engaged in the management of the chronometers to detect 
 irregularities, and to estimate the value of their results ; and also 
 to obtain, when necessary, by interpolation, their comparisons 
 with the standard for any required moment when special compa- 
 risons have been omitted to be made. 
 
 Besides the " Chronometer Journal," it will also be found 
 advisable to devote specially to the service of the chronometers, and 
 to the computation of the meridian distances, a " Sight-Book," and 
 a " Chronometer Work-Book." In the former are to be entered 
 all observations of " single or equal altitudes " for the determina- 
 tion of time, errors, and rates, together with all occasional special 
 comparisons and memoranda connected therewith. The latter 
 should be exclusively appropriated to the calculations and reduc- 
 tions of the observations, for the determination of the errors and 
 rates, and the computation of the meridian distances. 
 
 If fair copies be kept of the one, why not of the others ? they 
 are all equally essential and necessary; but, in truth, as Capt. 
 Fitzroy observes, in recording observations " it would be of little 
 use to give computations without comparisons, or comparisons 
 without rates, or rates without the calculations on which they 
 depend, or any part of these without the whole, which in any 
 extended voyage would constitute a mass of figures filling several 
 thick books. "f The original books, therefore, containing the records 
 of the comparisons, observations, and computations relating to the 
 
 * If, however, it be deemed necessary to keep a fair copy of the comparisons, the 
 Journal form No. II. in the Appendix (which is the same as the second form at 
 p. 29), is recommended for that purpose. 
 
 f Voyages of Adventure and Beagle, Appendix, vol. ii. p. 330. 
 
36 ON THE MANAGEMENT OF CHEONOMETEES. 
 
 chronometers, should be neatly and methodically kept, so that 
 when their results are finally reported they may be available for 
 inspection and reference if needed. 
 
 The " Chronometer Sight-Book," and " Work-Book," being as 
 above recommended wholly and" exclusively reserved for their 
 own specific uses, all miscellaneous memoranda of common obser- 
 vations connected with the keeping of the ship's reckoning, or the 
 calculations relative thereto, as well as generally all extraneous 
 matter of any kind whatsoever, are to be rigidly excluded from 
 them, and made in other books destined particularly for those 
 uses. It may often happen, amid the multiplicity of affairs which 
 engage an officer's attention on board ship, that the reduction of 
 the observations relating to the chronometers cannot be made till 
 a leisure opportunity offers, some time after they have been 
 taken. It is then a great satisfaction to find all the necessary 
 data clearly arranged, and recorded in such a manner that they 
 may be confidently relied on. 
 
 Some of these remarks and recommendations may appear at 
 first sight rather dogmatical and unnecessary ; and, perhaps, only 
 those who have been much engaged in operations of this nature 
 can fully appreciate the advantages of methodical and systematic 
 habits of comparison, observation, and computation, or can ade- 
 quately realise the saving of time and labour involved in a 
 careful attention to these minutias. 
 
 Assistant -Watch. We have already established, as a funda- 
 mental maxim, that when the chronometers have been received 
 on board and once stowed in their places, they should on no 
 account be subsequently removed. The practice, therefore, of 
 taking a chronometer on deck or on shore, for the purpose of 
 observation, should never be resorted to ; a pocket-chronometer, 
 or a good pocket-watch with a second-hand, should on these 
 occasions be used as an assistant.* The actual time of any 
 observed phenomena being then, in the first instance, taken by 
 the assistant-watch ; the corresponding times shown by the chrono- 
 meters at the moment are subsequently obtained by applying the 
 comparison. 
 
 * Tn the French naval service every vessel carrying a chronometer is, by a recent 
 regulation, furnished also with a " compteur," for the purpose of the comparisons. 
 (" Rech. Chron." p. 75.) A " compteur " is a small chronometer, fitted in a wooden 
 box, without any gimbol suspension. They are used for " taking sights '' with, and 
 are supposed to go sufficiently well for that purpose. Their price is about 25^. 
 
ASSIST ANT- WATCH. 37 
 
 In comparing the assistant with the standard it will be found 
 convenient, as in other cases, to consider the standard always 
 fast : hence the comparison will give us Z W, and W being 
 the time shown by the watch, 
 
 W + (Z-W) = Z; 
 
 that is, the comparison being added to the time shown by the 
 watch, gives the corresponding time shown by the standard. 
 
 As the assistant will rarely have exactly the same rate as the 
 standard, the value of the comparison, Z W, will be constantly 
 varying ; and hence, if the interval between the time of compa- 
 rison and observation is at all large, the comparisons should be 
 made both before and after, and their exact value at the moment 
 of observation obtained by interpolation, and this should be done 
 in all cases where observations are made on shore. 
 
 Thus, supposing the following comparisons to have been made : 
 
 Before. After. 
 
 Z 5 h 14 30 s 6 h 24 o s 
 
 W 3 ii 16 4 20 48*7 
 
 Diff. 2314 2 3 11-3 
 
 Also let the time by watch, corresponding to the observation on shore, 
 be 3 h 56 I9 S '5; here, in i h io m , the comparison Z W has varied 
 by 2 S 7, and the interval between the time by watch at the first com- 
 parison and the observation is 45, whence we have 
 
 70 : 45 : : 2 S 7 : x 
 
 . x = 45 X 2-7 
 70 
 
 = I S 74 
 
 Consequently the comparison corresponding to the moment of obser- 
 vation will be 2 h 3 m 14 s i s 74 = 2 h 3 i2 s -26, and the time of 
 observation by the watch 3 h 56 i9 s< 5 + 2 h 3 1 2^-26 = $ h 59 3i s 76, 
 the corresponding time by the standard. 
 
 The interpolation is made on the only admissible assumption, 
 viz. that the variation of the comparison has taken place uniformly 
 and in proportion to the time ; and it may be computed either by 
 common arithmetic or by the aid of proportional logarithms, as 
 may be thought convenient. 
 
 It will be advisable, if possible^ to appropriate a pocket-chro- 
 
38 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 nometer to the purpose of performing the duty of assistant-watch; 
 if a pocket-chronometer is not available, or cannot be spared, a 
 good pocket-watch with a second-hand must be substituted: in 
 either case, in order to give the watch so employed a fair chance 
 of maintaining the stability of its' rate, it should be kept in the 
 chronometer-box, except when actually in use, and subjected to 
 the same careful usage and delicacy of treatment as the other 
 chronometers. Pocket-chronometers, or good watches employed 
 as such, notwithstanding their name, should never be worn in the 
 pocket. When carried in the pocket, they are placed in a vertical 
 position, and subjected to the warmth arising from contact with 
 the body. When removed from the pocket for use, they are 
 held in a horizontal position, and probably subjected to a con- 
 siderable alteration of temperature. The alteration of position 
 is liable to disturb the regularity of the balance, and the varia- 
 tion of temperature to produce fluctuations of rate. A chro- 
 nometer is merely a machine, after all; and a perfect chrono- 
 meter, like a perfect circle or a perfect straight line, only a 
 happy mathematical accident ; and it is therefore unreasonable to 
 suppose that it can be expected to perform its functions accu- 
 rately, unless delicately treated and subjected to uniformity of 
 physical conditions. 
 
 It will be advisable, therefore, to keep the assistant- watch in 
 a small box, which, when carried on shore or on deck for the 
 purpose of observation, should be maintained in a horizontal 
 position, and, in its passage to and fro, should always be carried 
 by hand. Care should also be taken that, when the box is open 
 for the purpose of noting time, the watch should be kept in 
 the shade, and not exposed to the direct action of the sun's 
 rays, which, especially in a tropical climate, is likely to be very 
 injurious. 
 
 Miscellaneous Observations. I . When it may be necessary 
 to move box-chronometers from- the shore to the ship, or on 
 other occasions,* they should be locked in their gimbols, so as to 
 prevent an unusual oscillatory motion and to retain their hori- 
 
 * The Admiralty have recently issued the following instructions for sending 
 home defective chronometers from abroad, or from the outports, to the Royal 
 Observatory at Greenwich : 
 
 " Defective chronometers being very liable to injury on their passage from abroad, 
 or when travelling on shore, the following precautions in sending them are to be 
 adopted : 
 
MISCELLANEOUS OBSERVATIONS. 39 
 
 zontal position, under which condition alone their balances act 
 with perfect fairness. In carrying them by hand, the best way 
 is to sling them in a kerchief, passing the bight under the 
 bottom of the box and the ends through the handles, tying 
 them at the top in a reef-knot; care should be taken not to 
 give the chronometers a vibrating or oscillating motion, and to 
 carry them as steadily as possible. When conveyed in a boat, 
 they should be held suspended by the hand.* 
 
 It has been sometimes customary, when salutes have been 
 
 " I. Wedge the balance with cork, so as to prevent any movement in it what- 
 ever. 
 
 " IT. Turn the screw-catch into the gimbol-ring and the metal-box of the chro- 
 nometer, and screw it there fast, so as to prevent either of them from moving. 
 If there appears to be the slightest probability that the gimbolling can come loose, 
 or the gimbol-block can fail, secure them by tying the screw-catch fast, putting a 
 wood screw into the block, or adopting any other suitable measures. 
 
 " III. Fill the whole space between the metal-box and its wooden case with any 
 stuffing, such as dry oakum free from dust, or dry paper shavings free from dust 
 (wood shavings are sometimes objectionable, as bringing turpentine and dirt), or any 
 other clean and soft material, so as to prevent the possibility of movement of the 
 chronometer. 
 
 " IV. Having closed the wooden chronometer-box, it is to be placed in a wicker 
 basket or hamper, or in a box of partially yielding character, and is to be packed in 
 it with abundance of soft stuffing. If there is no wicker basket at hand, or nothing 
 except a wooden box, this box must be surrounded with stuffing, and inclosed in 
 canvas, so that it can never receive a jarring blow. 
 
 " V. Two or more chronometers (secured from injury in their boxes by the fore- 
 going measures I. to III.) may be packed in a yielding case or basket, but all contact 
 between them must be prevented by precautions similar to the above with straw or 
 some stuffing material. 
 
 "VI. Address the package to the Astronomer Royal, in characters distinctly 
 legible, adding the words 'With care,' the date, and name of the ship sent from, 
 and, when by land, that of the rail by which it goes to Greenwich ; a mail or 
 passenger train being adopted, but never a luggage train. 
 
 " Admiralty Hydrographic Office, 
 December 1859." 
 
 * The difference of longitude between Greenwich and Valentia was recently mea- 
 sured with great accuracy by means of chronometers, under the immediate superin- 
 tendence of Mr. Airy, the Astronomer Royal. Thirty pocket-chronometers were 
 employed on the occasion, it being considered that they were less susceptible of 
 injury by travelling than box-chronometers. The mode of carrying them was as fol- 
 lows : they were carefully packed in two cases divided each into fifteen compartments, 
 with springs under each chronometer, pressing it upwards firmly, but gently, against 
 a padded lid. The sides and tops of each case were well wadded outside to protect 
 them from any violence or jar. A number of boxes was then made, each of which 
 would just hold the two cases, placed one above the other ; and to every railway- 
 carriage, steam-boat, or mail-coach to be used in the course of their transmission to 
 and fro between the several stations, one of these boxes was screwed down. 
 
40 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 fired on board ship, or the guns fired for exercise, to remove the 
 chronometers from their places, and hold them by hand ; but if 
 the chronometers are stowed in horsehair on a solid block, as 
 previously recommended in these pages, we believe this precau- 
 tion to be entirely unnecessary a#d superfluous, and likely to be 
 rather detrimental than otherwise. 
 
 2. It would be advantageous, when chronometers are delivered 
 from the depot to any ship, if information were furnished not only 
 of the errors and rates, by the latest observations, as is at present 
 customary, but also of the age of the oils (that is, the date when 
 they were last renewed by the maker, when the chronometer was 
 cleaned or repaired), and the manner in which the chronometers are 
 affected by temperature, as is the custom in the French naval ser- 
 vice.* The same remarks are equally applicable to chronometers 
 supplied for the use of merchant shipping. 
 
 The age of the oils is very important,! an( ^ a knowledge of it 
 will often account for a deterioration in the performance of a 
 chronometer. The manner in which changes of temperature 
 affect the performance of these instruments is, perhaps, even 
 more so, and an acquaintance with it would often reconcile 
 apparent anomalies otherwise inexplicable ; as, for instance, when 
 two chronometers which happen to be differently or even very 
 unequally acted on by change of temperature, diverge in their 
 indications. 
 
 3. The acoustic properties of tubes are well known, and the 
 convenience of gutta-percha tubing has already caused it to be 
 extensively introduced on board ship, for the purpose of con- 
 veying with facility messages from the deck to the engine-room 
 and other places. We think it is well worthy of consideration, 
 whether gutta-percha tubes might not also be applied with great 
 
 * In the French naval service, when chronometers are returned to the depot, a 
 return is also forwarded giving a statement, ist, of all the observed errors and rates 
 referred to noon, mean time at the place of 'observation; ad, the mean of the tem- 
 peratures at 9 A.M. (the time of winding) of the chronometer chest, in the intervals 
 of the observations; 3d, a summary indication of the circumstances of the voyage 
 which have seemed to alter the rates of the watches. " Rech. Chron." p. 14. 
 
 t To show the importance attached to the age of the oils by the French hydro- 
 graphers, we may mention that by the regulations of the Ministry of Marine, chrono- 
 meters are permitted to be returned to the naval observatories, when the age of the 
 oils exceeds three years, or when the vessel is about to proceed to a destination such, 
 that there is reason to presume, that the age of the oils will exceed three years, before 
 the end of the voyage. "Rech. Chron." p. 15. 
 
MISCELLANEOUS OBSERVATIONS. 41 
 
 advantage for the purpose of conveying signals to the chrono- 
 meter-room. In taking observations at sea, the assistant-watch 
 could thus be altogether dispensed with, and the chances of 
 contingent errors arising from its use would thus disappear ; no 
 difficulty would probably be experienced in attaching the mouth- 
 piece to the mouth of the observer (both his hands being engaged 
 with his sextant), as the attachment could be made by an elastic 
 India-rubber band passing round the back of his neck ; and the 
 tubing, being perfectly flexible, might be accommodated to his 
 position without inconvenience. In noting the times of extra- 
 neous phenomena, signals from other ships, flashes of guns, 
 bursting of rockets, &c. &c., its use would be equally applicable; 
 and, doubtless, in bad weather at sea it would be of great advan- 
 tage in common observations. The transmission of sound is 
 practically instantaneous for short distances, and the advantage 
 of dispensing, if possible, with the assistance of an intermediate 
 watch very great. 
 
 4. If a chronometer has run down, it may be set a-going, 
 after being again wound up, by giving it a small but moderately 
 quick horizontal circular motion. In the absence of any direct 
 evidence to the contrary, it may be assumed that the chronometer 
 has resumed its former rate; but this is by no means certain, as 
 it is frequently found to differ. 
 
 5. It has been found that chronometers generally perform best 
 at the commencement of a voyage ; as time advances since they 
 left their makers' hands, their performances fall off, their results 
 are less regular, arid their rates generally accelerate.* These 
 
 * A table will be found in the Appendix, exhibiting the performances of the chrono- 
 meters of H. M.S. Fly, employed surveying on the coasts of Australia from 1842 to 
 1846. The table exhibits the results of the experience of four years, during which the 
 chronometers were subjected to great variety of climate, and to all the vicissitudes of 
 a long voyage. The conclusions to be deduced from it are highly instructive, and 
 tend to show that the general, though not invariable, tendency of all chronometers is 
 to accelerate their rates as time advances since they were last adjusted by their 
 makers. It also shows that, although after the lapse of a long period since the com- 
 mencement of the voyage, the rates may exhibit considerable divergence from the 
 values they had at starting, yet that these changes for the most part are gradual and 
 progressive ; and amid all their minute oscillations of amount, there is a much greater 
 approximation to stability of condition, at any given period, than on a casual inspec- 
 tion might at first sight appear. This consideration tends to increase the confidence 
 which may be placed in the performances of good chronometers during long voyages, 
 and seems to show that they may be more fully relied on than is generally supposed. 
 
 A similar table will be found in the " Nautical Magazine " for April 1843, 
 
42 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 changes, no doubt, are due to the general mechanical imperfection 
 to which their construction is liable, to the thickening of the oil 
 with which their pivots are lubricated, and to the effects of 
 friction, dust, and damp ; for these reasons it seems advisable, 
 in regulating the arrangement of the work of surveys, to measure 
 the principal chronometric runs, if possible, at the commencement 
 of the expedition, while the chronometers are in good order, their 
 rates regular, and performances unexceptionable: advantage has 
 been taken of these considerations in some recent survevs. 
 
 * 
 
 Stability of rate, as we have elsewhere remarked, is of much 
 more importance than the smallness of its amount: chronometers 
 with small rates are, no doubt, more convenient to work with 
 than those whose rates are very large ; but we have known 
 chronometers with very large rates, say 40* or 5O S a-day, give 
 most unexceptionable results in the measurement of meridian 
 distances : they are, therefore, by no means to be despised or 
 rejected. 5 * 
 
 giving the rates of some chronometers employed by Captain Barnett, during a period 
 of three years in the West Indies. The general results are very remarkable, and show 
 that, in this case, those which had originally a gaining rate, increased their rate of 
 gaining ; while those with an original losing rate decreased it, and gradually acquired 
 a gaining one : that is, the tendency of the acceleration in all cases seems to have 
 been towards a gradual increase of gaining rate. 
 
 The chronometers which have passed under our notice in other ships have usually 
 exhibited the same tendency ; and even in the best-constructed chronometers a gra- 
 dual acceleration of rate may, we believe, be generally anticipated. 
 
 * We once had experience of an eight-day chronometer, whose rate was large and 
 irregular. Its performance was much improved by adopting the practice of winding 
 it daily, instead of once a- week. 
 
ON THE DETERMINATION OF TIME. 43 
 
 CHAPTER III. 
 
 ON THE DETERMINATION OF TIME TRANSIT OBSERVATIONS SEXTANT 
 OBSERVATIONS ARTIFICIAL HORIZON SEXTANT METHODS OF 
 OBSERVATION EQUAL ALTITUDES SINGLE ALTITUDES COMPA- 
 RISONS WITH THE STANDARD POSITION OF THE PLACE OF OB- 
 SERVATION REDUCTION OF THE OBSERVATIONS PRECONCERTED 
 SIGNALS. 
 
 BEFORE we proceed to discuss the important questions relating to 
 the determination of the errors and rates of chronometers, it will 
 be proper to make some brief and practical observations on the 
 modes of determining the time correctly, which are available to 
 officers serving on board ship, and within compass of the means 
 ordinarily at their disposal. 
 
 The observations which may be employed are of three 
 kinds : 
 
 I. Transit observations. 
 II. Those requiring the use of a sextant. 
 III. Preconcerted signals. 
 
 1. Transit Observations. The first class of observations may 
 be briefly dismissed with but few remarks. Ships employed on 
 scientific voyages and on surveying duties have occasionally been 
 furnished with portable transit instruments ; of course, they will 
 very rarely be found among the apparatus employed on board 
 ships engaged in the ordinary duties of the service. Even among 
 scientific officers of acknowledged ability and experience, the use 
 of the portable transit does not seem to have met with much 
 favour or approbation. 
 
 " The correct adjustment of a transit instrument," says Capt. 
 Owen,* "for nice astronomical observations, is a delicate and 
 tedious operation ; and not by any means so simple as is generally 
 
 * Owen on Longitude, p. 11. 
 
44 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 supposed : it is, therefore, not an instrument likely to be much in 
 use among navigators, as it requires not only very strict attention 
 to preserve it in adjustment, but the adjustment itself requires 
 more time than can generally be given to it, either in men-of-war 
 or merchant-vessels. It will likewise frequently be difficult to 
 find a convenient place to fix it in for the period necessary to make 
 it useful." And again, Capt. Fitzroy* observes, that " a transit 
 instrument requires what is not always at command, viz. time ; to 
 be well placed ; skill in its use ; and habits of observing : which 
 are neither readily nor easily acquired." And in the measure- 
 ments of his extensive series of Meridian Distances, the time was 
 always obtained by equal altitudes observed with a sextant. 
 
 Our own experience in this matter tends in the same 
 direction. Satisfactory observations with a transit instrument 
 require much time and many conveniencies not always easily 
 attainable under the ordinary conditions of service at sea. Several 
 requisites are indispensable ; a convenient situation, command- 
 ing a clear view of the heavens, north and south ; a firm and 
 stable support for the instrument, so as to ensure freedom from 
 accidental tremor ; a tent, or portable observatory of some kind, 
 to afford shelter to the instrument; time, opportunity, and a 
 favourable turn of fine weather. Unless these requisites are 
 likely to be obtained, it is scarcely worth while to set up a transit 
 instrument, or to take the trouble of forming a temporary 
 establishment on shore, which it necessarily involves. 
 
 Small transit instruments, moreover, are very difficult to 
 maintain in strict adjustment; the standards, or piers, being made 
 of metal, are constantly altering their state of expansion, from the 
 varying effects of temperature which, in a small and temporary 
 observatory (more especially in a tropical climate), it is impossible 
 to maintain in a state at all approaching to uniformity ; hence the 
 level error is constantly varying; and, let the adjustments 
 be effected with what care they may, it is impossible to depend on 
 their stability even for a short period. For these, and other 
 obvious reasons, officers in general give the preference to sextant 
 observations, which, when undertaken by a skilful observer, give 
 the time with an accuracy but little, if at all, inferior to that 
 attainable by transit observations : persons, however, who may be 
 
 * Voyages of Adventure and Beagle, Appendix, vol. ii. p. 328. 
 
AKTIFICIAL HORIZON. 45 
 
 furnished with portable transit instruments, and disposed to use 
 them, will find ample directions for the necessary adjustments 
 and manipulations in many works devoted to such subjects.* 
 
 II. Sextant Observations. The simplest and most convenient 
 kind of astronomical observations available for the determination 
 of time are those which can be made with a sextant and artificial 
 horizon. In any observations having any pretensions to scientific 
 accuracy, it is always to be assumed, as a matter of course, that 
 they have been made on shore with the artificial horizon ; observ- 
 ations made with the sea horizon can only be considered as 
 approximations, and would only be admissible as data in the dis- 
 cussion of meridian distances, in cases where, from any peculiar 
 circumstances, it had been impossible to obtain observations on 
 shore. Altitudes observed with the sea horizon, even under the 
 most favourable circumstances, are liable to be vitiated by three 
 causes : first, the inaccuracy of the contact ; secondly, the uncer- 
 tain effects of refraction on the appearance of the horizon ; and 
 thirdly, the errors arising from an inaccurate estimation of the 
 correction for dip. 
 
 In observations made with a mercurial horizon on shore, the 
 contact of the limbs of the sun or of the images of a star is 
 susceptible of being made with great precision, while the second 
 and third sources of error vanish altogether ; and, moreover, the 
 effects of any error in the estimation of the index correction of the 
 instrument, as well as those of observation, are halved in the 
 resulting altitudes : while those proceeding from small errors in 
 the assumed value of the sun's semi-diameter may be eliminated 
 by observing both limbs. 
 
 Artificial Horizon. Of the various kinds of artificial horizons 
 that have been proposed, the common mercurial horizon is, 
 without doubt, the best. It consists of a shallow trough filled 
 with mercury, and screened from the wind, when necessary, by 
 a glass roof. In using it, care should be taken to select a con- 
 venient place, where it will not be subjected to accidental 
 tremors ; rocky soil is the best, if attainable ; sandy soil, or 
 made ground of any kind, being very susceptible of tremulous 
 motion. When the mercury is poured out, the forefinger being 
 placed on the mouth of the bottle, and the bottle having been 
 
 * See Galbraith's Tables : Simms on Mathematical Instruments ; "Penny Cyclo- 
 paedia," article "Transit;" Pearson's " Practical Astronomy," &c. &c. 
 
46 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 well shaken, it should be inverted, so that all the scum and 
 dirt may rise to the surface, and the pure mercury alone pass 
 into the trough. Care should be taken completely to fill the 
 trough, so that the capillary attraction of the mercury to the 
 sides may not interfere' With its Assuming a perfectly horizontal 
 surface ; and at the same time not to drain the bottle, so that no 
 accidental impurities may be poured into the trough. If this 
 happens, it is often very difficult to clear the surface ; the best 
 way of doing it, is with the edge of a silk handkerchief passed 
 along it, but it is often easier to pour the mercury back into the 
 bottle again, and pour it out again de now. 
 
 If from the admixture of foreign dirt, or from the accumu- 
 lation of its oxydation, the mercury become very foul, it should 
 be cleaned from time to time by straining it through a piece 
 of wash-leather. Care should be taken that the leather so used 
 should not afterwards be employed to clean a sextant with, as 
 the minute pariicles of mercury remaining in it, will form an 
 amalgam with the silver of the arc and vernier, and may do 
 them great injury. 
 
 The glass roof should be placed fairly over the trough, so 
 that the rays of light from the object observed may pass 
 through it in a vertical plane, and not obliquely, so as to avoid 
 as much as possible the errors arising from any imperfection 
 in, or non-parallelism of, the surface of the glass plates. The 
 ends of the roof should be marked, and it should always 
 be placed over the trough in the same invariable position, so 
 that any errors arising from its use may be constants equally 
 affecting all the observations. Many careful observers, how- 
 ever, are in the habit of reversing the roof during their obser- 
 vations; half being made with the roof in one position, and 
 half in the other, the errors become neutralised. When the 
 air is sufficiently calm, it is advisable to dispense with the roof 
 altogether. 
 
 Sextant. It seems desirable, when otherwise convenient, 
 that the sextant devoted to observations for time on - shore 
 should not be indifferently hacked about for other purposes ; it 
 is of great importance that an observer should be able to place 
 reliance on the trustworthiness of his instrument, and on the 
 stability of its index error. If, therefore, he has the command 
 of more than one instrument, the best may be suitably reserved 
 
USE OF THE SEXTANT. VV *7^' 
 
 for the finer kinds of observation, such as lunars 3fe^^j EiQ H^jb^ 
 determining the latitude and time on shore ; and the othe"r" 
 to the common observations required in the ordinary navigation 
 of the ship at sea. 
 
 The index error of the instrument should be frequently 
 examined, and its value determined from time to time; ample 
 instructions on this head will be found in the several treatises 
 on Navigation. The actual amount of the index correction is a 
 point of no importance, provided its stability can be depended 
 on. The constancy of the value of this error is a test of a 
 good instrument. In an instrument supposed to be good, 
 no attempt should ever be made to eliminate the error by 
 adjusting the screws ; nothing tends to injure sextants more 
 than tampering with the instrumental adjustments which have 
 been made by the maker. From the alternate expansion and 
 contraction of the metal from the effects of temperature, the 
 values of the instrumental errors are probably in a state of 
 constant minute fluctuation. These may be allowed for by a 
 careful determination of the index error from time to time; 
 but it is impossible ever wholly to remove them by mechanical 
 adjustments.* 
 
 Observations with a sextant are liable to minute sources of 
 error, from the imperfections of the shades or dark glasses. 
 The sides of the coloured glasses are not always parallel, and, 
 consequently, the divergence of the rays of light in passing 
 through them vitiates the value of the angles measured. Again, 
 glasses of different shades give different values of angles 
 measured, and apparently different results for the value of the 
 index errors. This chiefly arises from optical causes affecting 
 the appreciation of the contacts. On this account it has been 
 recommended to employ a dark glass at the eye end of the 
 telescope instead. If this glass is not perfect, the rays from the 
 object and the image are affected alike, and the angle between 
 them remains unchanged. In cloudy weather, however, when 
 
 * "It is better that error should exist (observes Mr. Raper), provided that it is 
 allowed for nearly, than that mischief should ensue to the instrument from ignorant 
 attempts at a perfect adjustment ; and the skilful observer, instead of implicitly 
 depending upon the supposed perfection of his instrument, will endeavour to avail 
 himself of those cases in which errors, if they exist, will destroy each other." 
 (" Practice of Navigation," p. 165 ; which see, also, for other important remarks on 
 the sextant.) 
 
48 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 the brilliancy of the real and reflected images in the horizon is 
 constantly varying, the use of the eye-piece is not practicable, 
 and the coloured shades must be used instead. In this case 
 neutral-tint shades seem far preferable to the common red, 
 yellow, and green ones : they ar^'more natural, less fatiguing to 
 the eye, and, by permitting the two images to be adjusted to the 
 same shade of colour, more favourable to freedom from optical 
 defects of observation. 
 
 Time and trouble are often saved by having the tubes 
 marked to the observer's focus.* Previous to observing, the 
 precaution should be taken of seeing that there is plenty of drift 
 for the working of the tangent-screw ; after a set of observations 
 has commenced, it is very unsatisfactory to have it interrupted 
 by finding that the tangent-screw is choked, an annoyance that 
 a little previous foresight would have avoided. 
 
 Care should be taken, especially in tropical climates, that 
 the sextant is -not unnecessarily exposed to the action of the 
 sun's rays, and, when not actually in use during the pauses 
 between the observations, it should be kept in the shade, or 
 screened with a handkerchief: nothing tends to ruin sextants 
 more, than unnecessarily roasting them in the sun. 
 
 The degree of accuracy and satisfaction with which obser- 
 vations can be made with the artificial horizon depends much 
 on the bodily health and condition of the observer. A clear 
 head, a steady hand, a sharp eye, and a mind at once calm in 
 its general disposition and acute in its perceptions, are elements 
 in the success of the perfect observer. Hence, previous to 
 observing, everything should be avoided which tends to agitate 
 the nervous system, or to increase the action of the heart ; such 
 as violent exercise, or running : so, also, carrying the instru- 
 ments to the spot of observation should be avoided, and per- 
 formed by deputy, as carrying heavy weights with the arm in 
 a constrained position tends to produce muscular tremor very 
 unfavourable to the duty to be performed. The eye also should 
 
 * It would be extremely convenient if the fittings of sextant-boxes permitted the 
 tubes to be put away in them, adjusted to focus ; time is often lost in doing so ; at 
 night it is often very troublesome ; moreover, the constant habit of pulling the 
 eye-tube in and out frequently makes it work slack. If sextant-boxes were made 
 square, as subsequently recommended, there would be plenty of room in them to 
 permit this. 
 
METHODS OF OBSERVATION FOR TIME. 49 
 
 not be unduly fatigued previous to observing, and should not be 
 unnecessarily exposed to glare. 
 
 Methods of Observation. The modes of observation for the 
 determination of time usual in practice, are either " equal or cor- 
 responding altitudes," or " single or independent altitudes." The 
 objects observed may either be the sun or stars. The former is 
 the most convenient, and generally preferred ; the latter have 
 some advantages, in a mathematical point of view, which, how- 
 ever, are perhaps neutralised in practice by the greater demand 
 on the skill of the observer, and by the material inconveniences 
 which attend night observations ; we shall, therefore, here suppose 
 generally the object observed to be the sun. 
 
 When the time is determined by single altitudes, the result 
 is directly dependent on the exact correctness of the elements 
 employed the latitude of the place the declination of the 
 object observed, and its altitude. The former may not be cor- 
 rectly known ; the second depends on the assumed perfection 
 of the tables employed, and on the right estimation of the 
 approximate longitude ; the last on the skill of the observer, the 
 correctness of his instruments, and on the true allowance for 
 the value of the refraction. From the combination of these 
 small causes discrepancies often arise in the results for time, 
 especially when the observations are made on opposite sides of 
 the meridian, which are very difficult to reconcile. Equal 
 altitude observations, on the other hand, are in a great measure 
 independent of these sources of error. If the object observed 
 is a star, a knowledge of the true time flows at once from the 
 observation without subsequent reduction; arid if the object 
 observed be the sun, as is more generally the case, the " equation 
 of equal altitudes " required on the occasion is but slightly 
 dependent on an exact knowledge of the latitude of the observer, 
 or of the sun's declination. The value of the method of equal 
 altitudes consists in the same altitude being noted on the opposite 
 side of the meridian, without regard to the precise measure 
 of it. Hence, as Raper observes, "the moving of the index 
 destroys the integrity of the method, since the second altitude 
 is no longer identical with the first, but is merely inferred to 
 be equal to it from the reading.* The errors, however, are 
 
 * It would be a great advantage if the mathematical-instrument makers would 
 abandon the absurd shape in which sextant-boxes are usually constructed, and make 
 
 E 
 
50 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 greatly diminished by taking numerous altitudes ; or a number 
 of instruments may be employed, set to different altitudes," So, 
 also, the theory of the method supposes that the value of the 
 refraction remains the same at the period of the two observa- 
 tions ; and thus that' equal apparent altitudes actually corre- 
 spond to the same true altitudes on each side of the meridian. 
 No doubt, in ordinary practice, this assumption does not involve 
 any appreciable error. In cases, however, where there is reason 
 to suppose a considerable variation in the amount of the refraction, 
 it has been proposed to allow a correction for it, and a formula 
 for that purpose is given in the " Memoirs of the Astronomical 
 Society," in a paper by Mr. Riddle, vol. iii. p. 216. 
 
 Equal Altitudes. A common mode of taking equal-altitude 
 observations is to set the instrument at any even division, a little 
 in advance of the sun's double altitude at the moment, when 
 viewed in the artificial horizon ; the images of the sun are then 
 overlapping, and gradually uncovering. At the moment of 
 separation of the limbs the time by watch is noted by an assistant 
 on a signal from the principal* The index of the sextant is 
 then advanced through ic/ (or one division of a degree), and 
 the operation repeated. This is done in succession through 
 from five to ten observations, f so that in taking a mean, small 
 errors may be eliminated. In the afternoon the observations 
 are repeated in reverse order, the sextant being set to the same 
 divisions as in the forenoon, beginning with the last observed. 
 If any of the observations are lost, the corresponding one in the 
 forenoon must be rejected. 
 
 The corresponding times of the forenoon and afternoon observ- 
 ations are then to be added together, and their sums meaned; 
 
 them rectangular, and so arranged that the sextant might be put away in them, with 
 the index clamped at any part of the arc : for the convenience of night observations, 
 when it may not always be practicable to read off at the moment, and in taking 
 equal altitudes, when the last observation in the forenoon is the first in the afternoon, 
 this facility would be found very useful. 
 
 * Skilful observers are sometimes in the habit of taking the time for themselves ; 
 no doubt there is great advantage in dispensing with the assistance of a second person 
 when it can be done, but it requires much practice and habits of self possession ; and, 
 moreover, the watch employed must have a very clear and distinct beat. 
 
 f Our own practice has frequently been to observe through two degrees : this 
 gives thirteen observations (the sextant being cut to 10"), and, after retaining 
 ten, allows an overplus of three for faulty contacts, or for the loss of duplicates in 
 the afternoon. 
 
EQUAL-ALTITUDE OBSERVATIONS, 51 
 
 the half of which mean result will be the mean approximate time 
 by watch at the moment of apparent noon. 
 
 The observations will stand recorded in the following form : 
 
 Altitudes. Times, A.M. Times, P.M. Sums, 
 
 o / h m s h in s h m s 
 
 65 o 3 25 29 8 36 13-5 12 i 42-5 
 10 25 50 35 52 42 
 20 26 11-5 35 31-5 43 
 30 26 327 35 10 427 
 40 26 53-0 34 49-5 42-5 
 50 27 13-5 34 28-3 41-8 
 
 66 o 27 35-3 34 7-5 42-8 
 10 27 56-5 Lost. .... 
 20 Lost. 33 25-0 .... 
 
 30 28 38 33 3-5 41-5 
 
 Mean . . 121 42-28 
 
 Approximate time by watch at apparent noon . . 6051-14 
 
 The above mode of proceeding is conveniently applicable when 
 the regularity of performance of the assistant-watch can be 
 depended on throughout the period embraced by the observations, 
 for in this case the " equation of equal altitudes.," and the " noon 
 comparison with the standard," being applied to the above 
 " approximate noon by watch," will give the time shown at noon 
 by the standard. 
 
 If, however, the going of the assistant-watch cannot be 
 implicitly relied on, it will be better to take the mean of the 
 corresponding A.M. and P.M. observations separately; apply to 
 each mean the comparison with the standard,* and thus obtain 
 its times corresponding to the A.M. and P.M. observations : the 
 half sum of these two times will be the approximate noon by the 
 standard, to which the " equation of equal altitudes " is subse- 
 quently to be applied as before. 
 
 In noting the observations it is advisable, as a check at the 
 time on their accuracy, to take the sum of the corresponding 
 times, A.M. and P.M., in the margin of the memorandum-book in 
 which the sights are recorded; these several sums should all 
 
 * See the remarks on the subject of the comparisons, in the section under 
 that head, p. 58. 
 
52 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 agree (each one representing twice the time of approximate 
 noon by the watch). Their accordance, or otherwise, therefore, 
 affords a criterion of the accuracy with which the observations 
 have been made, both of the degree of precision with which the 
 observer has appreciated the contacts, and the assistant noted 
 the times. 3 * This practice will be found very useful, and should 
 always be adopted by beginners ; since, by affording an unerring 
 test of their skill, it enables them gradually to acquire a proper 
 but not undue confidence in the value of their observations. 
 
 It may be proper to remark here, that it seems advisable, as 
 far as may be practicable, that an observer should always em- 
 ploy the same assistant to note the time for him : two persons 
 accustomed to work together are less liable to fall into any 
 error from the want of a mutual intelligence between them, 
 and, after a time, it is reasonable to suppose that the effects of 
 personal equation will approximate to a condition of constancy, 
 both in quantity and direction; whereas, with a frequent change 
 of persons, this element of error is in a state of perpetual change. 
 Certainly the same assistant who commences a set of observa- 
 tions should be employed to complete them, and the times A.M. 
 and P.M. should always be noted by the same party. 
 
 Many careful observers are in the habit of noting the con- 
 tact of both limbs of the sun in taking " equal altitudes ; " the 
 advantage of this method being, that it has a tendency to neu- 
 tralise the errors of observation arising from any personal 
 habit in the observer, which may induce him, in estimating the 
 moment of contact, to be either always a little too soon, or always 
 a little too late : the chances being, that these errors tell in 
 opposite directions, according as the observed limbs of the sun 
 are approaching or receding. 
 
 The mode of observation is as follows: The sextant is 
 set to any convenient division, a little in advance, in point of 
 time, of the true (double) altitude (that is, to a greater altitude 
 in the forenoon and a less in the afternoon), so that the reflected 
 
 * It may also be useful to point out, that after the first two observations, the 
 assistant who is taking the time becomes aware of the interval between the contacts 
 (in the above example about 2i s ), and is thus prepared to note the time of the 
 succeeding contacts, looking out sharply as the moment approaches, without the 
 necessity of any premonitory signal fro^n the observer. The attention of the 
 assistant thus concentrated more readily seizes the subdivision of the " second," and 
 the attention of the principal is not distracted by superfluous speaking. 
 
EQUAL- ALTITUDE OBSERVATIONS. 53 
 
 suns may be seen in the field of the telescope, separate from 
 each other, and approaching. Then the stationary image seen 
 directly in the quicksilver will always be the upper in the 
 forenoon, and the lower in the afternoon ; (when the inverting 
 telescope is used, these appearances will be reversed). At the 
 moment of contact of the limbs the time is noted, the reflected 
 image of the sun then passes over the true image ; and at the 
 moment of separation the time is again noted : by this means 
 two sights are obtained with only once setting the instrument. 
 
 The index is then moved through two degrees, which will 
 allow ample time to prepare for the next set ; after a little prac- 
 tice one degree and a half will be found sufficient. This mat- 
 ter being also in some degree regulated by the quickness of the 
 sun's motion in altitude at the time ; the observations are then 
 repeated, and this is done in succession through five sets (or any 
 convenient number). In the afternoon the observations are 
 repeated in reverse order (the last set in the forenoon being the 
 first set in the afternoon) ; the time by the assistant- watch 
 being noted as before. 
 
 The observations, when completed, will stand recorded in the 
 following manner: 
 
 Altitudes. Times, A.M Times, P.M. Sums, 
 
 o / hms hms hms 
 
 5 '9 '7 
 Lost. 
 
 i 
 
 10 31 20 
 
 15 5 37 
 
 5 2 5 3i 
 28 51-5 
 
 f 
 
 10 25 5-5 
 21 45 
 
 36-5 
 36-5 
 
 5 3 J 9'3 
 33 4 2 '5 
 
 1 
 
 Lost. 
 
 10 16 54^ 
 
 36-8 
 
 5 35 ll 
 3 8 39'5 
 
 { 
 
 10 15 26 
 
 ii 57-5 
 
 37 
 37 
 
 5 4 97 
 43 4 2 
 
 { 
 
 IO IO 26 
 
 6 54*3 
 
 357 
 36-3 
 
 70 o 
 
 72 o 
 
 73 3 
 
 75 
 
 76 30 
 
 k T J T~ V 
 
 Mean. . 15 50 
 
 Approximate time by watch at apparent noon . . 7 55 18*30 
 
 The corresponding times of the forenoon and afternoon observ- 
 ations are then to be added together, and their sums meaned ; 
 
54 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 the half of which mean result will be the approximate time by 
 the watch at the moment of apparent noon; from which, by 
 applying the " equation of time," and the noon comparison, the 
 time by the standard at apparent noon is determined: or, as 
 before pointed out, the comparisons of the assistant with the 
 standard are to be applied separately to the A.M. and P.M. times 
 of observation, and the time of the standard thus directly ascer- 
 tained. 
 
 In high latitudes, when the sun's motion in altitude is very 
 slow, it may not be convenient to wait for the sun to pass over 
 its own diameter ; in which case, five altitudes of each limb (or 
 any convenient number), with their corresponding times, may 
 be observed on each side of the meridian. The only way in 
 which this method is inferior to the last is in having to shift 
 the index twice as often to the same number of observations. 
 
 If cloudy weather should interfere to prevent observations 
 being taken through a regular series of sets, as recommended 
 in the foregoing methods, several independent sights should, if 
 possible, be obtained, clamping the index at any convenient 
 division, and carefully noting the times corresponding to each 
 altitude : several observations may thus be noted of either limb. 
 In the afternoon the observations are to be repeated in reverse 
 order, noting the corresponding contacts of the limbs, care being 
 taken that the same observations are made of each limb, and 
 remembering that in the forenoon lower limbs will be found to 
 separate, and upper limbs to close, the reverse taking place in 
 the afternoon.^ 
 
 As some of the corresponding times in the afternoon may be 
 lost, it is advisable, in taking the mean of the sums of the A.M. 
 and P.M. times, that an equal number of observations of each 
 limb should be retained ; superfluous ones should therefore be 
 rejected, so as to equalize the numbers. 
 
 The observations, when completed, will stand recorded as 
 follows : 
 
 * In order to avoid any confusion between the two limbs, beginners will find 
 it convenient to make their approximate contacts, and satisfy their minds on the 
 matter in the first instance, without using any tube ; the inverting tube being 
 afterwards screwed in, when they are all ready to commence observing. 
 
EQUAL-ALTITUDE OBSERVATIONS. 
 
 55 
 
 Altitudes. 
 
 Times, A.M. 
 
 Times, P.M. 
 
 L. L. 
 
 
 
 / a 
 
 h m s 
 
 h m s 
 
 97 45 40 
 
 8 50 44 
 
 2 13 39'5 
 
 98 17 30 
 
 5 1 54'5 
 
 12 29-5 
 
 98 53 10 
 
 53 13 
 
 11 II 
 
 99 28 20 
 
 54 29*5 
 
 Lost. 
 
 100 3 30 
 
 55 47 
 
 8 36-3 
 
 IT. L. 
 
 
 
 ioi 40 20 
 
 8 57 i 
 
 2 7 21-5 
 
 102 9 20 
 
 5* 5 
 
 6 19-5 
 
 102 38 40 
 
 59 8'3. 
 
 5 H 
 
 103 7 40 
 
 9 o 11-5 
 
 4 I3-5 
 
 103 37 50 
 
 i 17 
 
 3 67 
 
 Sums. 
 
 23 4 2 3'5 
 24 
 
 24 
 
 22-5 
 
 24*5 
 22-3 
 Rejected. 
 
 23*3 
 Mean . . 23 4 23-42 
 
 Approximate time by watch at apparent noon . . 1 1 32 11-71 
 
 In the foregoing remarks we have supposed that, generally 
 speaking, equal-altitude observations of the sun are taken by 
 observing A.M. and P.M., and thus deducing the error of the 
 watch at the moment of apparent noon : it may, however, be 
 occasionally necessary or convenient to reverse the process, and 
 take the observations, first in the afternoon, and then again on 
 the forenoon of the following day. The mode of observation is 
 exactly the same, except that they are taken in reverse order, the 
 last of the afternoon being the first in the following forenoon. 
 The reductions required are made in the same manner, and the 
 final result is the time by the watch at the moment of apparent 
 midnight. 
 
 In taking equal altitudes, in order that the observer may be 
 prepared to get the return sights in the afternoon, without wasting 
 unnecessary time beforehand in waiting for them, or running the 
 risk of losing them altogether, if too late, it is advisable that he 
 should compute approximately beforehand the time by watch 
 when the first return sight is due. 
 
 The following convenient formula for this purpose has been 
 proposed by Mr. Jeans : 
 
 If t = time by watch when the forenoon observation was taken, 
 '= time of corresponding sight in the afternoon, 
 A = approximate error of watch or local mean time, +fast and 
 slow, 
 
56 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 i = approximate equation of time, + when additive to mean 
 time, when subtr active; 
 
 Then t'= (i2 h i) + 2 (x g). 
 
 On which formula it is only necessary to remark, that attention 
 must be paid to the algebraic sigrfg of X and ?.* 
 
 Single Altitudes. When want of time, the state of the weather, 
 or other circumstances, do not permit " equal altitudes " to be 
 taken, the method of " single or independent altitudes " must be 
 resorted to. 
 
 The observations may be made in either of the three ways 
 mentioned above, in treating of " equal altitudes ; " that is, by 
 setting the index to any given division, noting the contact, and 
 then advancing the index successively through the following 
 consecutive divisions, and repeating the operation ; or by noting 
 the contact of both limbs in succession, allowing the image of 
 the sun to pass over itself; or by observing the contact of the 
 
 * The demonstration of this formula is as follows : 
 
 Since t = time by watch, when the A.M. observation was taken, 
 
 t X = mean time of A.M. observation, 
 and t A 4- s = apparent time of A.M. observation. 
 
 Hence iz h (t x + i) = the interval to apparent noon, or, the apparent time 
 
 of the P.M. observation, 
 
 .'. I2 h (t X + s = the mean time of P.M. observation, 
 and (iz h (t X + ) ) + >. = 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 <jf the mean sun" should in like manner be accurately 
 reduced, and the "stars' right ascension" taken for the given 
 day from the table of their apparent places in the ephemeris, 
 and not from the table of their mean places, which only gives 
 their approximate values. 
 
 The latitude, true altitude, and declination (or their com- 
 plements, as the case may be), should be expressed to the nearest 
 second ; the interval of elapsed time (required as the argument 
 for the tables of A and B, for the reduction of " equal altitudes") 
 will be sufficiently exact to the nearest minute. The logarithms 
 of the various quantities employed should be taken out accu- 
 
 required, and it seems then better to make the calculation by the aid of tables of 
 common logarithms : rules adapted to their use are given in Riddle's and Norie's 
 Navigation, and also in Raper's (in a note, p. 273). 
 
 Sometimes the rules for assigning their proper signs to the two parts of the 
 equation are so expressed as apparently to involve an unnecessary number of distinc- 
 tion of cases. It will appear on examination that the following rule holds good : 
 
 The sign of the first part of the equation is positive from the summer to the win- 
 ter solstice, and negative from the winter to the summer solstice : (according to the 
 hemisphere in which the observer is : that is, considering the summer solstice to be in 
 December, and the winter solstice to be in June, when the observer is in south 
 latitude). That of the second part positive between the equinoxes and the solstices, 
 and negative between the solstices and the equinoxes : (but if the elapsed time is 
 greater than twelve hours, the sign of the second part must be reversed); the advan- 
 tage of which rule is, that it is expressed as a simple function of the season of the 
 year, and is apparently independent of mathematical conditions. 
 
 * It is desirable that the quantities required should always be taken from the 
 " Nautical Almanac," and not from the rough tables given for the purpose of super- 
 seding its use, which are to be found in many of the treatises on navigation : such 
 tables, although, perhaps, useful enough for rude purposes in ordinary navigation at 
 sea, are wholly unsuited to the delicate determination of time from observations on 
 shore. 
 
64 ON THE MANAGEMENT OP CHRONOMETERS. 
 
 rately, and proportional parts obtained for odd seconds, &c., 
 when necessary, by interpolation.* 
 
 All results in time, the equation of equal altitudes, hour 
 angles, errors of chronometers on time, comparisons with one 
 another, and rates, should be earned to two places of decimals ; 
 not that it is for one moment eitlier supposed or pretended that 
 in practice such a degree of minute accuracy is either really 
 attainable or necessary, even if it were possible, but theoretic 
 considerations seem strongly in favour of such refinements as 
 applied to computation. In the first place, there can be no 
 doubt that precision of reduction, and closeness of work, tend 
 to encourage in the computer accurate habits of calculation; 
 secondly, it should be remembered, that the results sought for 
 are often minute quantities, and it is vain to expect that they 
 should be obtained satisfactorily if the means employed are 
 only rude and approximate; and lastly, the practice may be 
 fairly defended by a consideration which may be aptly alluded 
 to by parodying a well-known aphorism ; and we may remind 
 our readers, that if we look after the hundredths and tenths, 
 the seconds and minutes will take care of themselves. In thus 
 recommending closeness of reduction and calculation, one caution, 
 however, is necessary, the computer should ever bear in 
 mind that no degree of accurate computation can ever com- 
 pensate for any inherent defects, or erroneous assumptions, 
 which may exist in the data employed. The influence of such 
 errors will make itself felt, whether the subsequent computations 
 are performed closely or rudely ; but it is not unreasonable to 
 expect that the practice of precise calculation may reciprocally 
 engender in the observer the habit of obtaining the elements of 
 his operations as accurately and as free from errors as circum- 
 stances permit. 
 
 Preconcerted Signals. Sometimes, when it is not convenient 
 to obtain the errors of the chronometers on time by means of 
 
 * Many logarithmic tables contain either proportional parts or columns of differ- 
 ences, to enable the student to compute to the nearest second. If such tables are 
 not at hand, or their use not approved of, the computer may easily make the neces- 
 sary reduction himself, by taking the difference between the consecutive logarithms 
 as they stand in the tables, and obtain the proportional part required for the seconds 
 in excess by a short calculation by practice, or the rule of three. 
 
DETERMINING TIME BY SIGNALS. 65 
 
 independent astronomical observations, advantage may be taken 
 of the arrangements existing for that purpose at various ports, 
 for obtaining them by preconcerted signals ; or special arrange- 
 ments for that object may be devised for the occasion. 
 
 Under the general term, {f preconcerted signals," we mean 
 to include the various arrangements which exist at several 
 ports and places for notifying at precise moments the mean 
 time at the place, by the fall of time-balls, the display of signal- 
 flags, the flashes of guns, &c. &c. ; and also those special measures 
 which may occasionally be devised between different ships for 
 the intercomparison of their chronometers. 
 
 The arrangements for the former class of signals will, of 
 course, vary very much at different places ; and it is of import- 
 ance, in order that the seaman may be able to avail himself 
 fully of the advantages of this plan, that he should make him- 
 self fully conversant by inquiry with the nature of the local 
 regulations on the matter. Sometimes the final signal is pre- 
 ceded a short time before by some distinctive announcement, 
 to call the attention of observers ; sometimes a premonitory 
 signal, at a given interval, precedes the final one ; sometimes 
 the signals, of whatever nature they be, are repeated two or three 
 times, for the convenience of observers. 
 
 When the fall of time-balls is regulated by galvanic appa- 
 ratus in connexion with the Mean Time Clock of an established 
 observatory,* the fall of the ball at the precise pre-arranged 
 and expected moment may be implicitly depended on. When 
 this, however, is not the case, when the ball is liberated by 
 hand, or by some ordinary mechanical appliance, or when the 
 time is indicated by the flash of a gun fired at (about) some 
 pre-established instant,! the exact moment of the occurrence 
 of the phenomenon is usually subsequently published for general 
 information. All these matters, depending on local arrange- 
 ments, should be duly inquired into by those desirous of profiting 
 by them. 
 
 In noting these phenomena on board the ship, it is desirable 
 that, when circumstances permit, the time should be at once taken 
 by the " standard," and that the intervention of an intermediate 
 
 * As at Greenwich and Edinburgh, for instance. 
 
 f As at Madras, where the evening gun is fired from the Fort at 8 P.M., and the 
 true mean time of the flash, by the Observatory clock, subsequently published in the 
 Government Gazette. 
 
 F 
 
66 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 watch should, if possible, be dispensed with. Notwithstanding the 
 apparently inconvenient position of the chronometers, low down in 
 the interior of the ship, while the signals have to be observed 
 from the deck, there will generally be no great difficulty in 
 attaining this end, if the ' arrangements for the observation are 
 properly organised, and discipline interpose to secure the needful 
 silence during the brief period occupied by the operation. The 
 signals may be transmitted from the observer on deck to the one 
 stationed at the chronometer, either sharply by word of mouth, or 
 by a smart blow on the deck by a stick or by the foot. 
 
 Sometimes it may be necessary or convenient to compare the 
 chronometers with those of other ships. In this case, naval 
 ingenuity will have no difficulty in arranging means by which 
 flags giving the required signals should be smartly dipped; or, 
 better still, in devising for the occasion an impromptu time-ball. 
 Again, by a proper system of leading blocks, it is easy to arrange, 
 that the observer at the chronometer on board the ship giving the 
 signals, should himself give the instant of observation, by letting 
 go from his hand the rope suspending the ball; thus making the 
 signal more instantaneous, and dispensing with the intervention of 
 a second party. 
 
 Sometimes, when observations for time are made on shore, 
 the place of observation may not be conveniently accessible from 
 the ship ; or from the state of the weather, or other causes, com- 
 munication with the ship may be difficult or impracticable : in this 
 case, the comparisons of the assistant-watch with the standard 
 must be made by signals in the manner alluded to above. 
 
 In all cases of comparisons between ship and ship, or from the 
 shore to the ship, it is important that the system intended to be 
 acted on be thoroughly understood by all parties concerned ; and 
 when the operations are completed, the parties should compare 
 notes as soon as possible, so that discrepancies should be at once 
 detected, and suitable measures adopted for rectifying them when 
 necessary. 
 
 In all cases of the intercomparison of the chronometers of 
 different ships, the comparisons are supposed to be made in the 
 first instance between their " standard " chronometers. A round 
 of comparisons of each of the other chronometers with their 
 respective "standards " should then be taken immediately after- 
 wards, if their indications are also required. 
 
DETERMINATION OF ERRORS AND RATES. 67 
 
 CHAPTER IV. 
 
 ON RATING CHRONOMETERS, AND ON THE DETERMINATION OF THEIR 
 
 ERRORS ON LOCAL MEAN TIME SIMPLE METHOD; BY DUPLICATE 
 
 OBSERVATIONS OF A SIMILAR CHARACTER, TAKEN AT A CON- 
 VENIENT INTERVAL. 
 
 THE chronometers having been received on board, and carefully 
 stowed, in the manner previously recommended, in the places 
 allotted for their reception, it next becomes of importance to 
 ascertain their rates, and also their errors on local mean time. 
 
 It is frequently customary to adopt, at the commencement of a 
 voyage, the errors and rates received with the chronometers from 
 their makers, or from their place of deposit, or rating on shore ; 
 it being assumed that these data, which, no doubt, have been 
 determined with a proper regard to accuracy, may be sufficiently 
 depended on for all practical purposes, until more ample leisure 
 and a favourable opportunity afford ready facilities for the re- 
 determination of these elements. For the rough and approximate 
 requirements of the ordinary processes of navigation there may, 
 perhaps, be no great harm in thus confidently assuming the per- 
 manency of the data thus received; but if it is proposed to employ 
 the chronometers for higher uses than their application to the daily 
 navigation of the ship, and if an ambition exists to make them 
 subservient to the more exacting requirements of science, and to 
 obtain from them satisfactory contributions towards the improve- 
 ment of hydrography, it will not be proper to rely on the stability 
 of their rates previously obtained on shore; while at the same 
 time the greater importance of the results sought for, demands, 
 and should receive, a more careful attention to the conditions 
 necessary to ensure accuracy. 
 
 It is found in practice, that the rates of chronometers, after 
 they have been received on board ship, rarely agree with those 
 they previously had when on shore ; the causes of this variation 
 
68 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 are probably to be sought for in the disturbing effects of acci- 
 dental circular motion, or slight shocks during the process of 
 removal, in the alteration of temperature to which they have been 
 subjected, and in the possible effects of magnetic influences to 
 which they may become liable in $ieir new position : be this as it 
 may, it seems always advisable, if possible, when accuracy is 
 aimed at, to rate the chronometers anew before starting on the 
 voyage, after they have been some days on board, and after 
 they have become, if we may so express it, naturalised in their 
 new position, and have settled down to a stability of rate under 
 the new conditions to which they are submitted. It is with 
 this object in view that, in a preceding chapter, we have re- 
 commended their being received on board, during the equipment 
 of the ship, at a much earlier period than is generally customary. 
 
 By the " rate of a chronometer," we mean the difference of 
 its error on local mean time from day to day. It is called gaining, 
 when it goes too fast, and losing, when it goes too slow. 
 
 The usual method of obtaining the rate consists in deter- 
 mining by duplicate observations of a similar character,* taken 
 at a convenient interval, the error of the chronometer on mean 
 solar time at the place. If at the period of the second observa- 
 tion it be found that the error of the chronometer remains 
 unchanged, it shows that the chronometer has no rate, and is, 
 therefore, keeping true mean solar time ; if, on the other hand, 
 it appear that the error on local mean time is no longer the 
 same, it shows that the chronometer has a sensible rate, and 
 
 * By " observations of a similar character," we mean observations similar, as far 
 as may be, in their mathematical conditions, and therefore liable to be similarly affected 
 by the possible errors in the data on which they depend. 
 
 For instance, in the comparison of observations for the deduction of errors and 
 rates, it is of importance that equal altitudes should be used in combination with 
 equal altitudes, because any possible errors in the assumed latitude of the place of 
 observation would be neutralised in its effects, in so far as the rate is concerned, while 
 it would equally affect the absolute errors on, time; and, at the same time, it is not 
 unreasonable to suppose that the errors arising from any personal defect of the observer 
 would equally vitiate the two observations. 
 
 In a similar manner, it is fair to suppose, that since they are taken at nearly the 
 same time, and under similar conditions, the errors of refraction and observation 
 would probably equally affect single-altitude observations taken A.M. at any two 
 periods conveniently consecutive; and that they might, therefore, with propriety, be 
 used together for the determination of errors and rates. Similar reasoning applies to 
 the combination of two P.M. observations taken consecutively. 
 
 If, in default of equal altitudes, the mean of the times obtained from A.M. and 
 P.M. sights, at nearly equal intervals from noon, be used, they should, in like manner, 
 
DETERMINATION OF ERRORS AND RATES. 69 
 
 the difference* of the two errors divided by the interval elapsed 
 between the observations expressed in days will give the " mean 
 daily rate" of the chronometer during that interval: which rate, 
 moreover, it is important to remark, in strictness belongs to 
 the epoch corresponding to the middle of the interval between 
 the observations. / i. ;. 
 
 If the chronometer were slow on local mean time at the first 
 observation, and less slow at the second, or fast and more fast, 
 it shows that the rate of the chronometer is +, or gaining. If, 
 on the other hand, it were fast at the first observation, and less 
 fast at the second, or slow and more slow, then the rate of the 
 chronometer is , or losing. 
 
 In indicating the condition of the error of a chronometer, it 
 will be convenient, as tending to facilitate subsequent algebraic 
 discussion, to consider errors fast on local mean time as positive, 
 or +, and errors slow, as negative, or . Of course, mathe- 
 matically considered, it is immaterial and altogether arbitrary, 
 whether we choose to consider a chronometer fast on local 
 mean time by a given quantity, or its complement to 12 hours 
 slow, or vice versa, slow, or its complement fast: thus, if at noon 
 at a given place a chronometer show 9 h 54 n s , it may either be 
 said to be that quantity fast, or its complement to 12 hours, 
 viz. 2 h 5 m 49* slow. In practice we apprehend it will be found 
 more convenient to consider all the chronometers slow on local 
 mean time when voyaging in east longitude, and fast when in 
 west. When the measurement of meridian distances is pro- 
 be combined with similar data obtained on a subsequent occasion : but inasmuch 
 as the small errors of refraction and observation are probably neutralised in the final 
 results, there does not seem to be any particular objection to comparing the mean of 
 results by A.M. and P.M. observations with those obtained from equal altitudes on a 
 previous or subsequent occasion, provided the value of the latitude used in the calcu- 
 lations can be accurately depended on. 
 
 In like manner transit observations should be compared with transit observa- 
 tions; and generally, of whatever nature the observations for the determination of 
 the time may be, they should be referred to corresponding observations of a similar 
 character. 
 
 It should be borne in mind, that the general tendency of this mode of proceeding 
 will be to neutralise the effects of the errors of observation on the rate, and to make 
 it a constant on the error on local mean time, since the former is obtained by com- 
 paring the difference between given observations, and the latter from the observations 
 themselves, or by taking their mean. 
 
 * This difference is their algebraic difference ; therefore, when the error is found 
 to have changed from fast to slow, or from slow to fast, the rate is the sum of the 
 errors divided by the number of days elapsed. 
 
70 ON THE MANAGEMENT OP CHRONOMETERS. 
 
 posed to be undertaken, some uniform arrangement of this 
 nature will be found very convenient ; and as, moreover, it is 
 reasonable to suppose, that before the commencement of the 
 voyage the chronometers were originally adjusted approxi- 
 mately to Greenwich mean time ,^or that of the first meridian 
 employed), the plan here recommended will maintain that 
 analogy, whatever apparent discrepancies may ultimately ex- 
 hibit themselves through the gradual divergence due to the 
 diversities of their rates, whichever way the matter may be 
 viewed, it is of importance that all the chronometers be treated 
 alike, and that the plan originally adopted be steadily adhered 
 to throughout. 
 
 For the determination of the error on local mean time, it has 
 formerly generally been the practice in the measurement of 
 meridian distances, or on taking a departure from a port, to 
 adopt as a starting-point the error shown by the last observ- 
 ation used in the determination of the rate; but inasmuch as, 
 in the deduction of the latter, we necessarily place equal confi- 
 dence in both the observations, there seems to be no good 
 reason why we should not do the same in assuming the error, 
 and adopt as our working error the mean of the two errors on 
 which we have already agreed to make the rate depend. 
 
 The mean error thus introduced may probably be assumed 
 to be more accurate than either of the single elements on which 
 it depends ; while this mode of proceeding, moreover, will have 
 the advantage of referring both the error and the rate to the 
 same epoch : a circumstance which, when we come by and by, 
 in the discussion of the formulas for meridian distances, to treat 
 of the corrections to be applied for the changes of rate which 
 may have taken place in the transit from port to port, will be 
 found to introduce considerable simplicity into the equations 
 employed.* 
 
 The immediate result of any observation, at any place, for 
 the state of the chronometer at any given moment, will be its 
 error on local mean time. This error, depending on observa- 
 
 * On this subject M. du Conulier observes, " Conformably to a remark already 
 made by M. Barral (' Annales Maritimes,' Jan. 1829). It is the mean day of 
 the observations and not the last, that we ought to take, for that on which the daily 
 rate of the watch has been determined ; and it is to this mean day, that one ought to 
 refer its absolute error." ("Rech. Chron." p. 108.) We may add that this practice 
 also had the approval of the late Mr. Raper. 
 
DETERMINATION OF ERRORS AND RATES. 71 
 
 tions having reference to astronomical or other data, will be true 
 within the limits of the correctness of the elements employed ; 
 and in any particular case, by multiplying the observations, or 
 by adopting specific practical precautions suitable to the occa- 
 sion, a great part of the probable errors may generally be almost 
 entirely eliminated, and the residual error reduced within very 
 small limits : and it is important to remark, that the error on 
 local mean time thus obtained is wholly independent of any 
 previous arbitrary assumption of the actual longitude of the 
 place of observation.* 
 
 In the measurement of meridian distances, which is simply 
 a careful comparison of the difference of time under two meri- 
 dians, it is of importance that the question should not be com- 
 plicated by any reference to the relation which either place 
 bears to Greenwich or any other " prime meridian/' 
 
 In the development of the science of hydrography, the 
 questions of the relative longitudes of outlying stations must be 
 settled by the navigator, that of absolute longitude belongs 
 exclusively to the hydrographer. The former carefully measures 
 differences of longitude, and reports their results; the latter 
 compares results, collates authorities, settles from the evidence 
 of astronomical data the positions of the " secondary meridians," 
 and unites the whole in one harmonious combination. It is, 
 therefore, much to be desired, that navigators would solely con- 
 fine their attention to the accurate measurement of differences 
 of meridians by comparing the errors of their chronometers on 
 local mean time at the two places, without any reference what- 
 ever to what the actual longitudes of the places of observation 
 may be. 
 
 It is true that by carefully carrying on a chain of meridian 
 distances from the commencement of a voyage, a consistent 
 series of the values of the longitude of each place visited may 
 be obtained. The values of all the longitudes thus deduced are 
 all equally affected by any primary error which may exist in 
 the assumed longitude of the place originally started from ; 
 while each particular longitude is, in addition, affected by the 
 algebraic sum of the small errors existing in the individual links 
 
 * Except in so far as the reduction of the quantities required from the astronomical 
 ephemeris is concerned, on which the value of the assumed longitude of the place of 
 observation exercises a certain influence. 
 
72 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 in the chain of which it is composed. It is reasonable to sup- 
 pose that the algebraic sum of such small errors may, in well- 
 executed measurements, frequently be very small, and that the 
 minute errors of the several parts may often nearly eliminate 
 one another from the contrariety ;6f their signs; but there can 
 be no doubt that the value of such a series, as a contribution to 
 accurate hydrography, is greatly enhanced by viewing its con- 
 stituent parts separately, link by link, and comparing their 
 values with those of other authorities, rather than when, taking 
 the chain as a whole, it is attempted during the successive stages 
 of the voyage to settle the values of the absolute longitudes of 
 the places visited. Pursuing the analogy suggested by likening 
 the series to a chain, it should be remembered that as one 
 faulty link renders a chain valueless, while the several other 
 links taken separately may be perfectly good; so also one 
 unsatisfactory measurement vitiates the value of a series of 
 meridian distances, in settling chronometrically the actual longi- 
 tude of any given station, while, at the same time, the other 
 several parts of which it is composed may be unexceptionable in 
 character, and of great and acknowledged value,* 
 
 The mode of proceeding, moreover, here recommended, has 
 another great and decided advantage; each measurement of 
 the meridian distance between two places visited during the 
 
 J. O 
 
 * The extensive series of meridian distances measured by Capt. Fitzroy, in H.M.S. 
 Beagle (1831-6), may be quoted in illustration of the above remarks. The Beagle 
 circumnavigated the globe, sailing from east to west, during a voyage occupying 
 nearly five years. She was furnished at starting with twenty chronometers, which 
 number, from various causes, was reduced to eleven at the termination of the voyage. 
 The whole sum of the meridian distances exceeded 24 h by about 33 s . It is clear, 
 therefore, that errors exist somewhere ; the separate measurements, comprising the 
 links of the chain, when examined one by one, appear unexceptionable in point of 
 character, and remarkably accordant with those of other authorities. And what 
 makes the ultimate error more perplexing is, that it is in the highest degree impro- 
 bable that the ordinary errors of observation and the discrepancies to which chrono- 
 metric measurements are liable, should all lie in the same direction, and that no 
 compensation of error should take place from the contrariety of their algebraic signs. 
 
 Capt. Fitzroy suggested, that some unknown effects of magnetism, on account of 
 the ship's head being a long time in a given direction, may possibly have affected the 
 performance of the chronometers. This point is well worthy of future investigation, 
 and its consideration is suggestive of the interesting results to science that might 
 arise if a ship were sent on a scientific voyage charged with measuring a series of 
 meridian distances round the world in both directions, from east to west; and, again, 
 from west to east. Rio Janeiro, or Bahia, would be a convenient starting-point; and 
 with proper arrangements, the double voyage might easily be performed in the course 
 of three years. Other objects of scientific interest might be united with it. 
 
DETERMINATION OF EREORS AND RATES. 73 
 
 voyage, (made in the manner to be hereafter explained), is 
 complete within itself, and has no necessary connexion with 
 those that may precede or follow it. This consideration permits 
 the operations connected with the chronometers to be carried 
 on with greater freedom, and relieves the persons engaged 
 in them from any feeling of irksomeness or restraint. When 
 circumstances are unfavourable, and pressure of business, adverse 
 weather, or other cause, prevent the necessary observations, 
 the work may be dropped for the time ; on the other hand, when 
 fortune is again propitious, and ample leisure, fine weather, and 
 other favouring contingencies invite the undertaking, advantage 
 may again be taken of the opportunity offered, and the measure- 
 ments resumed. 
 
 For the ordinary purposes of navigation, a reference to the 
 absolute longitude of the place of observation from Greenwich, 
 or other prime meridian for which the charts employed are 
 constructed, is, of course, indispensable ; and the error of the 
 chronometer on local mean time having been obtained as an 
 immediate result from the observations in the first instance, its 
 error on Greenwich mean time is subsequently deduced from 
 it, by applying thereto the longitude in time of the place of 
 observation ; * but it is important that the navigator should 
 bear in mind that all longitudes at any time obtained, and all 
 
 We cannot refrain, in closing these observations, from remarking that Capt. (now 
 Admiral) Fitzroy is deserving of the highest praise for the thorough honesty with 
 which he reported the results of his chronometric measurements, although apparently 
 reflecting on his own success. How easily, by a little " cooking " and " trimming," 
 judiciously paring off a second or two here, and a second or two there, and thus 
 altering the estimated means, the results might have been " squared in," so that the 
 final sum should have been exactly 24 hours, neither more nor less ; or, perhaps, two 
 or three seconds over or under, for conscience sake, or to give the appearance of 
 extreme honesty ! A sciolist would have yielded to this temptation: happily, for the 
 credit of science, this officer was superior to such influences. (See "Voyages of 
 Adventure and Beagle," Appendix, vol. ii. p. 331.) 
 
 * The general expression for the chronometric difference of longitude, or the 
 meridian distance between any two stations, is 
 
 M = x'-x 
 
 (See remarks on this subject in Chap. VII.}, in which expression M represents the 
 meridian distance, while x' and X represent the errors of the chronometer at the same 
 moment, on local mean time, at the two places, the former referring to the place 
 arrived at, and the latter to the place sailed from, and being considered positive when 
 fast, and negative when slow ; hence, M will be positive when measured towards the 
 west, and negative when towards the east. 
 
 If the place to which x refers be Greenwich, then M represents the longitude of 
 
74 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 positions in longitudes which it may be his duty to report during 
 the conduct of his voyage, depend on, and will be vitiated by, 
 any error in the assumed longitude of the place where his last 
 error was obtained ; hence, in making any report of this nature, 
 in which he may mention his chropbmetric value of the absolute 
 longitude of any place, the assumed longitude of his starting- 
 point should be specified ; or better still, for hydrographic uses 
 (as we have before remarked), if he should content himself with 
 giving the difference of longitude from the place where his error 
 was last obtained. 
 
 The observations usually employed by navigators for the 
 determination of errors and rates may be comprised under the 
 following heads: 
 
 I. Transit observations. 
 II. Equal altitudes. 
 III. Single altitudes, A.M. or P.M. 
 
 IY. Single altitude A.M. and P.M. observations combined. 
 Y. By comparison with a time-ball, flash of a signal gun, 
 or rocket fired at a given moment; with an obser- 
 vatory clock or with another ship, by preconcerted 
 signals. 
 
 Of whatever nature the observations may have been from 
 which the time at the place has been determined, the subse- 
 quent deduction of the errors and rates from them involves the 
 same processes, and is similar for any number of chronometers 
 employed. 
 
 We shall proceed to give some numerical examples : 
 
 Example i. By equal-altitude observations at Trincomalee Dock- 
 yard, on June 2d, 1851. The errors of chronometers Z, R, and Y, on 
 local mean time, were as follows : 
 
 the place to which x' refers, and x' and M being known, X the error of the chrono- 
 meter on Greenwich mean time, becomes known by transposition, since 
 
 X = X'-M. 
 
 In applying which formula to actual practice, attention must be paid to the algebraic 
 signs, west longitudes and errors fast being considered positive, and east longitudes 
 and errors slow being taken as negative. 
 
 It will be observed, moreover, that x and x' represent respectively, for the several 
 chronometers employed, the quantities elsewhere denoted by Z -L, A L, B L, 
 &c. ; or by L-Z, L A, L-B, &c. (See ante, p. 33.) 
 
DETERMINATION OF ERRORS AND RATES. 75 
 
 h m s 
 
 z -5 29 54-23 
 R_o 9 33-23 
 Y-5 37 43-23 
 
 On June 9th, by similar observations, the errors were, 
 
 h m s 
 
 Z-5 2 9 35'3 8 
 R o 14 28-68 
 
 Y-5 38 23-58 
 
 Subtracting the respective errors of the chronometers on June 2d 
 from their corresponding errors on June 9th, the differences are as 
 follows : 
 
 Z + 18-85 
 R- 295-45 
 Y- 40-35 
 
 The interval between the epochs of the observations is 7 days ; hence 
 the above differences, divided by 7, will give the mean daily rate 
 during the interval, while the mean of the respective errors gives the 
 mean error corresponding to the mean epoch for which the rate has 
 been determined. 
 
 Hence, in this case, we have at the mean epoch, June 5^5 : 
 
 Errors. Rates. 
 
 h m s g 
 
 Chron. Z 5 29 44-80 + 2-69 
 
 R o 12 0-95 42-21 
 
 Y-5 38 3'4 ~ 5-76 
 
 If, for the convenience of the ordinary processes of navigation, the 
 errors of the chronometers on Greenwich mean time were required, 
 they would be obtained from the preceding errors on local mean time, 
 by applying the longitude in time to them, by the formula 
 
 A = / - M. 
 
 According to the most approved authorities, M, the longitude 
 of the Dockyard flagstaff at Trincomalee may be assumed as 
 5 h 24 5 3 s * 8. Applying this to the above errors of the chrono- 
 meters on local mean time, we shall have for their corresponding 
 errors on Greenwich mean time as follows : 
 
76 
 
 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 h m s 
 
 Chron. Z o 4 51-00 
 R-6 47 7-15* 
 Y o 13 9-60 
 
 Example 2. By single-altitude observations, taken at 9 A.M. on 
 Feb. 6th, 1 844, the errors of chronometers A, C, and I, on mean time 
 at Garden Island, Sydney, New South Wales, were as follows : 
 
 A 10 
 C - 8 
 I - ii 
 
 i 
 
 57 
 9 
 
 28-10 
 21-55 
 10-75 
 
 Similar observations on Feb. 1 7th gave the respective errors as fol- 
 lows : 
 
 h m s 
 
 A 10 i 13*87 
 C - 8 55 32-27 
 I ii 10 7-17 
 
 Hence, in this case, the differences of the errors for the several chrono- 
 meters are as follows : 
 
 A + 14-23 
 C + 109-28 
 
 i - 56-42 
 
 The interval between the epochs is 
 the preceding example, we have, 
 
 1 days. Hence, proceeding as in 
 
 h 
 
 Chron. A 10 
 C 8 
 I ii 
 
 Errors. 
 
 m 
 
 i 
 
 56 
 9 
 
 Rates. 
 
 20-98 
 26-91 
 38-96 
 
 9'93 
 
 Which errors and rates correspond to the mean epoch between Feb. 
 5^875, and Feb. i6 d -875 (the observations having been made at 9 
 A.M.), that is, to the period Feb. i i d '375. 
 
 If the errors of the chronometers on Greenwich mean time were 
 
 * It might, at first sight, seem more consistent to consider chronometer R fast 
 on Greenwich + 5 h ii ro S^'^S but si nce ? as we have before observed, chronometers 
 may be indifferently considered fast, or their complements to 12 hours slow, and vice 
 versa, and as we recommend all the chronometers to be subjected to uniformity of 
 treatment, it is preferable in this case to consider its error slow, like that of the two 
 other chronometers under discussion with it. 
 
DETERMINATION OF ERRORS AND RATES. 77 
 
 then required, they could be obtained, as in the preceding example, by 
 applying the longitude in time of the spot of observation to the above 
 errors on local mean time. 
 
 The preceding examples suppose (as will generally be the 
 case) that both the observations for the time at the place have 
 been made under the same meridian. It may, however, occa- 
 sionally be necessary to employ observations made under two 
 different meridians, since it may sometimes happen that a 
 vessel may not remain sufficiently long at a given station to be 
 able to obtain two observations at a suitable interval for the 
 deduction of the rate ; but, after obtaining one observation at the 
 one station, may proceed to another, and obtain a second inde- 
 pendent determination of the error on local mean time at that 
 place. In this case, if the difference of longitude of the two 
 stations be accurately known, the observations may be com- 
 bined as before for the determination of the mean error and 
 rate. 
 
 Example 3. By observations of the fall of the time-ball at Ports- 
 mouth Dockyard, on May 3d, 1854, the errors of the chronometers on 
 Greenwich mean time at i P.M. of a vessel lying at Spithead were as 
 under, 
 
 h m s 
 
 D + o i 13*50 
 
 M o 4 17-90 
 P o ii 58*00 
 
 On the vessel's arrival at Falmouth, on May pth, single-altitude 
 observations at 9 h 1 3 A.M., gave the errors on local mean time as 
 follows : 
 
 h m s 
 
 D + o 21 38*40 
 M + o 15 29-40 
 P + o 8 48-90 
 
 From these observations the respective errors and rates are required. 
 The difference of longitude between Greenwich and Falmouth 
 (Pendennis Castle) is + o h 20 io s '8o. Hence, by the observations 
 at Portsmouth on May 3d, the errors of the chronometers on Fal- 
 mouth mean time were, 
 
 h m s 
 
 D + o 21 24-30 
 M + o 15 52-90 
 P -f o 8 12-80 
 
78 ON THE MANAGEMENT OF CHRONOMETEES. 
 
 Comparing these with the Falmouth observations on May 9th, it 
 appears that the differences were, 
 
 D + 14-10 
 M 23-50 
 P -f 36-10 
 
 Also the interval between May 3^042 Greenwich mean time, and 
 May 8 d< 884 Falmouth mean time, is 5^856 (allowing a correction of 
 o d -oi4 for the difference of longitude). 
 
 Hence the errors of the chronometers on Falmouth mean time at 
 the mean epoch, May 5^956,* and the corresponding rates, were, 
 
 Errors. Rates. 
 
 h ra s s 
 
 D + o 21 31-35 + 2-41 
 
 M + o 15 41*15 4'oi 
 
 P + o 8 30-85 + 6-16 
 
 This example affords an instance in which, in default of 
 duplicate observations of a similar character, the errors and 
 rates are of necessity deduced from independent observations 
 of different kinds : it also affords an illustration of an import- 
 ant remark of Raper's, "that as the longitudes of the several 
 places approach to precision, ships will employ the difference 
 of longitude as a means of obtaining directly the sea rates of 
 their chronometers, instead of waiting to obtain harbour rates ; 
 
 * In this and the preceding examples, as will often happen in practice, the date, 
 corresponding to the mean epoch of the errors and rates, is not an exact or integral 
 date. This will be of little consequence in the measurement of meridian distances, 
 because when the intervals which enter into the equations are accurately expressed, 
 they are almost certain to involve fractional quantities, even when the observations 
 are referred to mean noon, owing to the correction to the intervals, for difference of 
 longitude. 
 
 If, however, this be thought troublesome in ordinary practice, the errors may be 
 brought up to any convenient exact integral date, by applying to them a propor- 
 tional part of the mean daily rate, due to the interval between the actual and 
 selected epochs. 
 
 Thus, in the last example, suppose it was proposed to express the errors for the 
 exact date, May 6 d> 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<D'7 
 
 7th . i 10 50-8 
 
 9th i 10 51-1 
 
 nth i 10 51-7 
 
 I3th i 10 52*2 
 
 * Comparing the above result with that given in M. Daussy's paper, it will be 
 found that, employing the results there given, and reducing the errors and rates to 
 the same epoch as above, July ifth, we shall obtain identical results. (See " Naut. 
 Mag." 1834, p. 395, or " Rech. Chron." p. 48.) 
 
COMBINATION OF OBSERVATIONS FOR RATE. 
 
 95 
 
 Hence we have, 
 
 
 
 Squares. 
 
 Cubes. 
 
 
 s 
 
 g 
 
 % = 
 
 i 
 
 I 
 
 I 
 
 a = 
 
 - 0-3 
 
 HI a 0*3 
 
 n z = 
 
 3 
 
 9 
 
 27 
 
 b = 
 
 -07 
 
 n z b 2-1 
 
 n s = 
 
 4 
 
 16 
 
 64 
 
 C '^ = - 
 
 0-8 
 
 n 3 c 3-2 
 
 ^^4 = 
 
 6 
 
 36 
 
 216 
 
 d = 
 
 ri 
 
 n d 6'6 
 
 n 5 = 
 
 8 
 
 64 
 
 512 
 
 e = 
 
 - 17 
 
 n 5 e 13-6 
 
 m = 
 
 10 
 
 IOO 
 
 IOOO 
 
 /f 
 
 2'2 
 
 mf 22-0 
 
 Sum 32 226 
 
 Whence we obtain, 
 
 1820 
 
 -6-8 
 
 -47-8 
 
 A = 32 
 
 T-. 226 
 
 B = = 11*3 
 
 20 
 
 C=226 
 T. 1820 
 
 Dz= = 91 
 
 20 y 
 
 And by formula (6), 
 
 DP-BQ 
 AD-BC 
 
 And substituting numerical values, it will be found that 
 
 X = O S '2I9 
 
 Again, 
 
 = o s -oi8 
 
 Which results, on examination, will appear very consistent with 
 the data employed. 
 
 Also, at the mean arithmetic epoch, June 7 d *57, the chro- 
 nometer was slow i h io m 50 St 97 ; and by formula (3), the rate at 
 this time, 
 
 S '2IQ + ^- . O s '0l8 
 
96 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 And reducing these results to the mean integral epoch, June 
 8 d , we shall have, by formulas (7) and (3), 
 
 h m s 
 
 June 8 d Chron. slow i 1051 '06 
 Daily rate* 0-21 
 
 We shall conclude these examples with one purposely se- 
 lected, on account of the apparent fluctuations in the rate, in 
 order to show that this circumstance will not prevent our 
 formula from doing itself justice, and from exhibiting probably 
 true values of x and y. 
 
 Example, No. IV. 
 
 h 
 
 Nov. 1 3th Chron. fast i o 4-4 
 
 I5th i o 77 
 
 1 7th i o 8-9 
 
 20th i o 6-5 
 
 22d i o 4*3 
 
 2 3 d i o 2-1 
 
 Here, 
 
 
 Squares. 
 
 Cubes. 
 
 
 
 
 
 
 8 
 
 B 
 
 W 1= = 2 
 
 4 
 
 8 
 
 = +3'3 
 
 w 1 a = -j- 6'6 
 
 w 2 = 4 
 
 16 
 
 64 
 
 b = + 4-5 
 
 ra 3 & = + 1 8-0 
 
 w 3 = 7 
 
 49 
 
 343 
 
 C = + 2'I 
 
 n s c = + 147 
 
 w 4 = 9 
 
 81 
 
 729 
 
 rf=-0'I 
 
 w 4 d = 0-9 
 
 m = 10 
 
 100 
 
 1000 
 
 e= 2-3 
 
 m e = 23*0 
 
 Sum 32 
 
 250 
 
 2144 
 
 + 9'9 
 
 + 39*3 
 
 
 
 
 -2-4 
 
 -23-9 
 
 + 7'5 +IS-4 
 
 Whence we have, 
 
 7o- = I2 '5 P = + 7'5 
 
 250 Q= 4-15-4 
 
 = = 107-2 
 
 20 ' 
 
COMBINATION OF OBSERVATIONS FOR RATE. 97 
 
 And by formula (6), 
 
 _DP-BQ 
 "AD-BC 
 
 And substituting numerical valuer, 
 
 X = 2 8> 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 </' r ; that is to say, one might admit that 
 a/= x". 
 
 We should then have the two equations, 
 
 n 1 qx' + b'y. And ra"= q" x' + b" y, 
 from which equating the values of x 1 , we should obtain, 
 
 which value would be rigorously exact, so long as we can assume 
 that the acceleration x' shall have been regular. 
 
 Again, if we should have observed four successive daily rates, 
 m ; m + n' ; m + n' + n ff ; and m + n 1 + n" + n'". 
 
 We might suppose that the acceleration of the chronometer 
 during the middle passage has been a mean between the accelera- 
 tions which have prevailed during the two extreme ones ; that is 
 to say, that we might make, 
 
 x " tfjfaf " 
 
 2 
 
 We should then have the three following equations, 
 
 n q' x' + b' y 
 
 /x' + aT"\ , 
 n =q - +b y 
 
 Taking in the first and third equations, the values of x' and a/", 
 and substituting them in the second, there would then remain but 
 the one equation, 
 
 '- b'y . n'" b" 
 
METHOD OF DE CORNULIER. Ill 
 
 
 
 containing but one unknown quantity y, and from which by 
 reduction, we obtain, 
 
 When we have obtained the coefficient of temperature ?/, we 
 can immediately determine the successive accelerations of the 
 watch a/ 9 x fr , x'" > &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 
 <f meridian distances.'' 
 
 VI. Mouchez's plan not differing essentially in principle, 
 although simplified in form and expression, from that proposed by 
 De Cornulier, seems well adapted for general practice ; and, when 
 skilfully combined with the formulae of Tiarks' method, is that 
 which, for the present, we recommend for adoption by English 
 navigators. 
 
 Possibly experience may by and by decide more in favour of 
 Lieussou's or Hartnup's methods, but at present they require the 
 confirmation of practical trials at sea. 
 
REMARKS ON THE PRECEDING METHODS. 149 
 
 Very many persons, we doubt not, while admitting the patent 
 irregularities produced by changes of temperature, will object 
 altogether to the use of formulae of correction, not so much on ac- 
 count of the additional trouble that they devolve on the computer, 
 as because they seem to a great extent fanciful and empirical. 
 
 We would remind such objectors, that if they disapprove of 
 formulae for correcting for temperature, they may have the matter 
 pretty much in their own hands, if they will be at the trouble to 
 adopt suitable and obvious precautions. 
 
 If care is taken to maintain the chronometers at a uniform and 
 equable standard (using artificial means if necessary), and if the 
 errors and rates of the chronometers are redetermined by frequent 
 observations, and the measurement of meridian distances, broken 
 into as short links as possible and executed as rapidly as possible, 
 the necessity for correcting for temperature and acceleration will 
 well-nigh vanish altogether. The elimination of error is better 
 than its correction, while the application of Tiarks' formulae will 
 satisfy all the wants of practice. 
 
 
150 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 - 
 
 CHAPTER VII. 
 
 ON THE CHRONOMETRIC DETERMINATION OF MERIDIAN DISTANCES 
 TIARKS' FORMULA COMBINATION OF MOUCHEZ's PLAN OF CORREC- 
 TION FOR TEMPERATURE, WITH TIARKS' METHOD COMPARISON OF 
 FLINDERS' AND TIARKS' METHODS METHODS OF YINCENDON BU- 
 
 MOIJLIN, COUPVENT DESBOIS, AND CHARLES PLOIX. 
 
 IN the preceding chapter we have placed before the student the 
 methods proposed by various writers for correcting the immediate 
 results obtained by chronometric observations, for changes of 
 temperature, and the influence of the acceleration. 
 
 English navigators generally, have not been in the habit of 
 attending to these refinements. When the chronometers are 
 kept in a uniform temperature, and when the runs are short, 
 the changes which are produced by variations of temperature, 
 and the influence of the acceleration, are minute quantities, which 
 become mixed up with and masked by the petty irregularities of 
 the observation, and even to a certain extent allowed for, when the 
 mean of the rates at departure and on arrival is employed in the 
 measurement of the meridian distances. 
 
 We proceed, therefore, to explain the mode of proceeding 
 under general circumstances, no special account being taken of 
 the influence of the temperature or the acceleration. 
 
 In the correction of meridian distances for the alterations 
 which may have taken place in the rates of the chronometers 
 employed, we propose to adopt the same hypothesis as we have 
 already assumed* in the combination of observations for the 
 determination of the rates, and to consider that any variation, 
 which the rates may be found to have undergone between the 
 two epochs for which they have been determined, has taken 
 place uniformly and in proportion to the time. Of course, this 
 
 * See ante, p. 82. 
 
TIARKS' FORMULAE OF CORRECTION. 151 
 
 idea is to be accepted under the same conditions as we have 
 already laid down when treating of the combination of observa- 
 tions for the deduction of rates, and is to be taken with the same 
 reservations, as to its ultimate and approximate truth, as we noted 
 3n that occasion. 
 
 The idea of an equable variation of rate, in proportion to the 
 ime elapsed, has usually been viewed by computers under one of 
 ;wo aspects. 
 
 It has either been considered that the change of rate increased 
 >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 ; <f the corresponding temperature 
 during the period when the rate was subsequently found to be 
 a + b ; #! the mean daily temperature during the period r in the 
 partial passage from A to B; 2 the mean daily temperature 
 during the period / r in the partial passage from B to C ; & n the 
 mean temperature for the whole period of the passage from 
 A to K during the interval t. 
 
 Then the value of the coefficient of temperature y can be 
 immediately ascertained from the comparison of any two ob- 
 servations for the rate made under favourable circumstances of 
 
 * We must claim the indulgence of our readers for some apparent confusion of 
 notation. In discussing Mouchez's method, ante, p. 136, we have called the tem- 
 perature t, because it was necessary subsequently to compare his plan with Lieussou's. 
 In Tiarks' formulae t is already appropriated to signify time, hence we propose here 
 to use 6 (theta), the initial letter of the word thermometer. In discussing the 
 system of previous writers it is always advisable to adhere as nearly as possible to 
 their notation, introducing as few changes as possible, which principle has guided us 
 on the present occasion. 
 
CORRECTIONS FOR TEMPERATURE. 163 
 
 considerable differences of temperature, and is given by the 
 formula, 
 
 6-f 
 
 There are good grounds for supposing that, in a good chrono- 
 meter, whose compensation for temperature has been accurately 
 adjusted, the value of ?/, once carefully determined, remains 
 sensibly constant, while the instrument is in good condition. 
 The diligent observer, who studies his chronometer with the view 
 to obtaining a complete confidence in its performances, will not 
 fail, however, to verify the value of this coefficient, whenever the 
 circumstances of his observation for rate have been propitious for 
 it. When the chronometer is aged, its oils thickened, or partially 
 evaporated, and its movements affected by the petty causes of 
 derangement supervening by time, it may be expected that this 
 coefficient will undergo a sensible alteration as time advances. 
 In the system under discussion the influence of the acceleration 
 is mixed up with, and involved in, that due to the change of 
 temperature ; hence, as the performance of the chronometer gradu- 
 ally deteriorates by the lapse of time, the value of the coefficient 
 y may be expected slowly to change. 
 
 Again, the researches of Lieussou, and Hartnup's experiments, 
 all tend to show that the correction for, or coefficient of, tem- 
 perature is not exactly the same in all parts of the thermometric 
 scale (see ante, p. 143) ; the value of y, in fact, determined from 
 time to time by any given observations, in strictness actually 
 corresponds at the time to the mean temperature . 
 
 Bearing these facts in mind, the diligent navigator will of 
 course avail himself of every opportunity for verifying the value 
 of the coefficient y, as frequently as possible, and under as varying 
 circumstances of temperature as possible, for each of his chrono- 
 meters ; he himself will then be the best judge what degree of 
 confidence he can bestow on its permanence. 
 
 For the correction of the longitude of the ship at sea, the 
 value of the coefficient y previously determined will be no doubt, 
 in general, sufficiently accurate, in cases where it is considered 
 necessary to apply it. 
 
 In this case, instead of assuming, as is usually done, the in- 
 
164 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 variability of the initial rate a, and obtaining the longitude of the 
 ship from the formula, 
 
 M = A' 
 
 we should use the equation, 
 M = A'- 
 
 on which we shall only further observe, that attention must be 
 paid to the algebraic sign of the coefficient y, or, which comes 
 to the same thing, the correction for change of temperature 
 t (d^), y must be applied according to its actual observed effects 
 in altering the rate. 
 
 Again, in a similar manner, the four formulae given ante, 
 pp. 157-8, for the measurement of meridian distances, become 
 by the introduction of the temperature corrections as follows : 
 
 For the whole meridian distance from A to K : 
 
 (9) 
 
 From A to B: 
 M, = *,-{* + (ra + -fj- 6) +T ('--,)y 
 
 From B to C : 
 
 From CtoK: 
 
 M, = A, - x s + f=77a + * + *^ ~ * 3 (1 2) 
 
 The example already given at p. 136, is an illustration of the 
 application to practice of formula (9) for the measurement of a 
 
 * Recapitulating the notation for the sake of clearness : M is the longitude of 
 the ship sought for ; X the original error of the chronometer on Greenwich mean time 
 at starting on the voyage ; x' the error of the chronometer on time at the ship, deter- 
 mined by observation on the day in question ; a the initial daily rate, ascertained before 
 departure, at the mean temperature 6 ; 6 V the mean of the mean daily temperatures of 
 the chronometers, observed since leaving port for the time t, elapsed since the 
 epoch, when the rate was a; y the coefficient of temperature previously determined. 
 
COMPARISON OF FLINDERS' AND TIARKS' METHODS. 165 
 
 meridian distance between two terminal stations. The formulae 
 are themselves so simple, and their application to numerical 
 calculations so easy, merely involving the use of one additional 
 term, in the correction of the error of the chronometer on local 
 mean time, at the first station, that their further amplification 
 seems superfluous.* 
 
 We have only further to observe that it will always be 
 advisable, if possible, to obtain the coefficient y from the actual 
 rate observations of the passage under discussion, combining them 
 together for this purpose in pairs in the most advantageous 
 manner, according to circumstances. If this cannot be done the 
 value of y, derived from previous observations, must needs be 
 employed in lieu. 
 
 The actual coefficient of temperature at the period is, however, 
 so much more preferable, that officers should not refuse the trouble 
 of obtaining it if possible. It involves not only the influence of the 
 acceleration then existing, but necessarily depends on that parti- 
 cular range of temperature within which the observations have 
 been made. We have no certainty that this is the case, if a 
 coefficient determined by prior and distant observations is em- 
 ployed. The permanence, or small variation of the coefficient y, 
 affords a test of the goodness of the chronometer. 
 
 Comparison of Flinders' and Tiarks' Methods. 
 
 We shall now proceed to institute a comparison between 
 the methods of correcting for changes of rate usually adopted 
 by different English navigators. 
 
 If we assume that at the epoch of the first observation the rate 
 were a, and that being subsequently found to have changed to 
 a + b, we consider the change to have taken place uniformly and 
 in proportion to the time, following the law of an "arithmetic 
 series" (which is the hypothesis adopted by Flinders, King, Owen, 
 and others); then the interval between the epochs being t, the 
 
 * We honestly confess that this is rather slurring over a difficulty ; we should 
 much have preferred, in accordance with our usual custom, to have illustrated these 
 formulae by applying them to examples of actual observations. Unfortunately the 
 paucity of sea observations, in which the records of temperature of the chronometers 
 have been carefully made hitherto is such, that data are not accessible, even if existing 
 at all, for this purpose. 
 
166 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 daily variation would be - 9 and the accumulation of the rate for 
 any partial interval, r, would be 
 
 in which expression it is to be noted that the first term in the 
 series equals the common difference. 
 
 Now in any arithmetic series, in which the first term equals 
 the common difference, 
 
 S being the sum of the terms, n their number, and q the first 
 term, or common difference. 
 
 Hence our expression above, in a similar manner, becomes, 
 
 Accumulated rate = ra + (r + i) -~ b 
 
 in which expression the two last terms denote the correction for 
 interpolation for the change of rate, or, as it has been sometimes 
 called, the supplemental error. 
 
 Capt. Owen's formula (see Owen on Chronometers, p. 4) is, in 
 principle, identically the same with the above, only with a dif- 
 ference of notation, the factor J being also introduced to reduce 
 the correction from seconds of time to minutes of arc. 
 
 Hence we see that, according to the view adopted by Flinders, 
 King, Owen, and others, 
 
 The accumulated rate for any*! _ ** , . *_ , 
 
 partial interval, r j it it 
 
 And according to Tiarks, BayO 
 
 field, &c. (ante, p. 153) the > = 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 + <r '^* *' = [1*045988]. 
 Likewise for the several chronometers we have, 
 
 t + r' . t r' ( , t + r . t - 
 
 t T a ; b ; \ t r'a + 
 
 ->} 
 
 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 5<? i3' 88 ~ 8 2 4 59' 8 3 
 
 Error at Sir C. Hardy's Isles, j 2g . g2 g _ g 6 
 
 or A , on May 23 d J y y 
 
 4-0 17 8-09 4-0 17 15-16 4-0 17 3-56 
 Whence we have, 
 
 Meridian distance Cape Upstart to Sir C. Hardy's Islands, 
 
 km s 
 
 Chron. A 4- o 17 8-09 
 B 15-16 
 
 c 3-56 
 
 Mean .. 4-0 17 8-93 
 
 Again, for the partial measurement from Cape Upstart to Lizard 
 Island, by formula (2), 
 
 And solving numerically, 
 
 A. B. C. 
 
 Error of chron on C Upstart j ' fi - -. * J ^ 
 
 mean time, May 4 d , or A . . J 
 
 Cor. for rate (T + 77 *) 7*33 6-76 4- 1 27-33 
 
 Error on May 1 2 d 9 46 23-53 9 49 54*61 8 26 59-22 
 
 Error at Lizard Island, or AJ 9 37 12-22 9 40 37*92 8 17 48-82 
 
 -j-o 9 11-31 +o 9 16-69 + 9 I0 '4 
 
 Therefore, for the meridian distance, Cape Upstart to Lizard Island, 
 we have, 
 
 h m s 
 
 Chron. A + o 9 11*31 
 B 16-69 
 
 C 10-40 
 
 Mean . . + o 91 2-60 
 
182 ON THE MANAGEMENT OF CHRONOMETEBS. 
 
 Lastly, from Lizard Island to Sir C. Hardy's Islands, by formula (4),' 
 
 M = x' - A + *-=? a 
 
 A.- B. C. 
 
 Errorofchron-onLizardlsland 
 
 _ _ , 8 
 
 mean time, May 1 2 d , or A 2 J 
 
 -13-17 -1872 -fi 59 
 
 Cor. for rate, or 1 
 
 ' 
 
 _ 
 
 Error on May 23 d , or(x 2 + cor.) 9 37 25-39 9 40 56-64 8 15 49*30 
 
 Error at Sir C. Hardy's Isles, j 28 . 82 _ 9 32 ^.^ __ g ? ^ 
 
 or A ............ ......J 
 
 4-0 7 56-57 +o 7 57-92 +o 7 53-03 
 
 Hence, for the meridian distance, Lizard Island to Sir Charles 
 Hardy's Islands, 
 
 h m s 
 
 Chron. A + o 7 56-57 
 B 57-92 
 
 C 53'3 
 
 Mean . . + o 7 55*84 
 
 Collating the results as before, for the purpose of comparison and 
 verification, we have, 
 
 A T ri 
 
 Meridian distance, 
 
 ra m 
 
 756-57 +o 757 . 92 + o 753-03 
 
 Sum .. +o 17 7-88 -fo 17 14-61 +o 17 
 
 t measure- 1 
 ment J 
 
 The same by direct measure- \ +Q iy g . 09 +Q 1? 6 +o I? 
 
 Which results, like those in the preceding example, exhibit a 
 satisfactory degree of accordance. 
 
 * We employ formula (4) in preference to (3), (see ante, p. 158), because it is 
 the one connecting the last intermediate with the final station (C with K). Formula 
 (3), connecting the intermediate stations (B and C), would serve equally well, if we 
 consider t to become r, and treat the final as a second intermediate station. In fact, 
 as will be observed, the two formulae are identical in form. 
 
EXAMPLES OF MERIDIAN DISTANCES. 183 
 
 The preceding examples, illustrating Cases I. and II., afford 
 instances of all the varieties which can arise in practice in what 
 we may term perfect cases of the measurement of meridian dis- 
 tances ; that is, cases in which, observations for the determination 
 of the rates having been made both at the commencement and 
 termination of the voyage, we have, in point of theory, the means 
 of ascertaining the indication of the chronometers at any moment, 
 and hence of obtaining results for differences of longitude, theo- 
 retically perfect, in so far as the formulae on which they depend 
 are concerned; although, of course, liable practically to many 
 imperfections, from defects of observation, and the mechanical 
 irregularities to which chronometers are constitutionally prone. 
 
 We now pass on to the consideration of the imperfect cases, in 
 which the incompleteness of the data relative to our knowledge of 
 the rates, and consequent ignorance of the state of the chrono- 
 meter throughout the period embraced by the observations, pre- 
 vent our arriving at more than an approximate solution, liable 
 subsequently to be rejected, modified, or amended, when circum- 
 stances arise more favourable to a perfect determination. 
 
 Case III. Approximate determination of meridian distance 
 between A and K, the rate having been found at one of the 
 stations only. 
 
 Example. By observations at Hong Kong (at Dent's Wharf), on 
 Nov. loth and 24th, 1850, the errors and rates of chronometers Z, M, 
 and A, at the mean epoch, Nov. 1 7 d , were, 
 
 Errors. Rates. 
 
 h m s s 
 
 Chron. Z 7 44 12-89 + '4 
 
 M 7 59 52-29 + 1-28 
 
 A 8 20 22-29 ~ 6-08 
 
 On arrival at Shanghai, on Dec. io d , the errors of the chrono- 
 meters were again determined (at Beale's House) as follows : 
 
 h m s 
 
 Chron. Z 8 13 23-26 
 M 8 28 30*26 
 A 8 51 17-76 
 
 From these data we propose to determine the approximate 
 meridian distance. 
 
184 ON THE MANAGEMENT OF CHRONOMETERS. 
 
 For this purpose our formula is, 
 
 M = V (A + to)* 
 
 in which A refers to the errors at Hong Kong, and A' to those at 
 Shanghai; also , the interval between the epochs to which the errors 
 refer, is 23 d . 
 
 Therefore we have, 
 
 Z. M. A. 
 
 Cor. for rate (t a) ............ +9*20 +29-44 2 19*84 
 
 Error on Dec. i o d , or (A -f t a) 7 44 3-69 7 59 22-85 8 22 42-13 
 Error at Shanghai, or A' 8 13 23-26 8 28 30-26 8 51 17-76 
 
 o 29 19-57 o 29 7-41 o 28 35-63 
 
 Whence, for the meridian distance from Hong Kong to Shanghai, 
 we have, by 
 
 h m s 
 
 Chron. Z o 29 19*57 
 M 29 7-41 
 
 A 28 35-63 
 
 which results exhibit a considerable divergence from one another 
 (indicating that at least some of the chronometers have altered their 
 rates), and therefore could not be considered at all satisfactory. 
 
 Results obtained in this manner, the rate being only deter- 
 mined at the commencement of the voyage, can only be con- 
 sidered as first approximations, to be subsequently confirmed 
 or amended, when further observations show whether the rate, 
 a, on which they depend, has remained constant or undergone a 
 change during the voyage. 
 
 A similar remark applies to cases in which the measurement 
 depended on a rate obtained at the termination of the voyage 
 (circumstances not having permitted that element to be obtained 
 before its commencement), since there would be no evidence to 
 show that the subsequent rate had not changed during the voyage, 
 and become materially different from its value before starting. 
 
 If, however, the difference of longitude between the two 
 extreme stations should be accurately known, it may be made 
 subservient to the correction of the relative longitudes of inter- 
 mediate points. 
 
 * See ante, p. 156. 
 
EXAMPLES OF MERIDIAN DISTANCES. 185 
 
 Case IV. Rate determined at one station only. Difference 
 of longitude between extreme stations assumed to be known, and 
 applied to correct the relative longitudes of intermediate positions. 
 
 Example. Suppose that, on the occasion quoted in the last 
 example, observations for time only had been made at Amoy, as 
 follows : 
 
 Nov. 3o d , 1850. By observations at the Cornwallis Rocks, Amoy, 
 the errors of the chronometers were, 
 
 hms 
 
 Chron. Z 7 59 56-98 
 M 8 15 23-85 
 A - 8 37 24-73 
 
 And also that we assume that the true difference of longitude between 
 the places of observation at Hong Kong and Shanghai is, 
 
 o h 29 i7 8 -io 
 
 we proceed to apply this to the correction of the position of the 
 intermediate station at Amoy. 
 
 By formula (5), the errors of the chronometers A and A' at the two 
 stations, and M, their difference of meridians, being known, the mean 
 ' sea rate, 
 
 * *' ~ ~ M 
 
 2 t 
 
 Hence, for the several chronometers, we have, 
 
 Z. M. A. 
 
 hms h m a hms 
 
 Error at Shanghai, or A' 8 13 23-26 8 28 30-26 8 51 17-76 
 
 Error at Hong Kong, or A 7 44 12-89 7 59 5 2 ' 2 9 "-8 20 22*29 
 
 '_ A o 29 10-37 o 28 37-97 o 30 55-47 
 M o 29 17-10 o 29 17-10 o 29 17-10 
 
 (A' A) M +o o 6-73 -fo o 39-13 o i 38-37 
 Dividing by t = 23, 
 
 + 1*70 4 s -28 
 
186 ON THE MANAGEMENT OF CHRONOMETEKS. 
 
 And by formula (6), 
 
 = 2 ( + ^ ) 
 
 Whence, for the respective chronometers, the values of b, the variation 
 of the rate during the vo'ya'ge, are,-^ 
 
 g 
 
 Chron. Z 0-22 
 M + 0-84 
 A + 3-60 
 
 Then, for the meridian distance between Hong Kong and Amoy, 
 by formula (2), 
 
 + ra + -- 
 
 Here, r = 13 d and = [0-565128]. 
 
 Also, for the several chronometers, we obtain, 
 
 ** , ** , 
 
 ra - b r a + - o 
 
 ^t ^t 
 
 Chron. Z + 5-20 0-8 1 -f 4-39 
 
 M -f 16-64 + 3-09 + 1973 
 
 A 79-04 + 13-23 65-81 
 
 And solving the formula numerically, 
 
 Z. M. A. 
 
 Error of chron. at Hong Kong " - - * - 
 
 Nov. i; d , or A 
 
 , | _ 
 J 
 
 
 Cor. for rate ^a-\- b +4'39 + I 973 i 5' Sl 
 
 d 
 
 10 
 
 Error on November 3o d , orl 
 
 Jx + f a4- b\l C ~~ 7 44 8 ' 5 -7 59 3 2 '5 6 -82128- 
 
 ( h \ 2^/j J 
 
 Error at Amoy, or AJ 7 59 5 6 '9 8 8 15 2 3' 8 5 8 37 2 4'73 
 
 o 15 48^48 o 15 51*29 o 15 56-63 
 Hence, for the meridian distance, Hong Kong to Amoy, we have, 
 
 Chron. Z o 15 48-48 
 
 M 5 I>2 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<f 1 8 
 
 
 
 Observed Mean Daily 
 Temperature, 
 
 
 
 
 in Interval between the 
 
 
 Chron. 
 
 Meridian Distance. 
 
 Observations. 
 
 Remarks. 
 
 
 
 Date. 
 
 Mean 
 Temp. 
 
 
 
 h m 
 
 
 
 
 Z. 
 
 
 
 
 
 A. 
 
 
 
 
 
 B. 
 
 
 
 
 
 C. 
 
 
 
 
 
 &c. 
 
 
 
 
 
 &c. 
 
 
 - 
 
 
 
 h 
 
 Arithmetic mean 
 
 Estimated mean or 
 
 rejecting [any particular chronometers to be mentioned, for reasons specified]. 
 
 Chronometers (i) days 0) range (3) 
 
 Errors and rates at determined by 
 
 at by 
 
 General direction of ship's head during the voyage 
 Approved, A. B., Captain. C. I)., Officer in charge of Chronometers. 
 
 Notes. 
 
 (i.) Number of chronometers employed in estimated mean to be specified. 
 (2.) Number of days between the epochs of the observation. 
 
 (3.) The range is the greatest difference of the resulting meridian distance by any two chronom 
 N.B. Every meridian distance measured is to be reported separately in this form. The column of Remarks 
 is to contain anything noteworthy, likely to have affected the chronometers or the value of the meridian distance. 
 
APPENDIX. 
 
 223 
 
 fcD 
 
 o 
 
 w 
 
 TJ1 . 
 
 ri 
 
 
 s 
 
 s 
 
 
 *(2 
 
 S's 
 
 -aduiaj, 
 
 to which th 
 t. refers, or 
 n reduced. 
 
 O 
 
 H 
 
 W 
 
 vo oc 
 
 3 ^n ^ 
 
 W 
 
 ^o 
 
 .eo 
 OO 
 
 2 % 
 .2 o 
 
 f 
 
 
 
 8 
 
 
224 
 
 ON THE MANAGEMENT OF CHKONOMETERS. 
 
 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. 
 
 K. 
 
 L. 
 
 M.T. 
 Fahr. 
 
 
 
 + 
 
 ' 
 
 '- 
 
 j 
 
 * 
 
 
 
 -h 
 
 - 
 
 - 
 
 + 
 
 
 
 
 Greenwich Observatory 
 Devonport (Mr Cox's) 
 
 7 III. 
 
 2'77 
 
 S'21 
 
 4-76 
 
 + 
 
 O'QO 
 
 0*67 
 
 O' JO 
 
 + 
 7'2O 
 
 i'o6 
 
 5-16 
 
 3-00 
 
 i'93 
 
 I* 5 I 
 
 
 Devonport (on bd. ship) 
 Falmouth 
 
 * 3 > 
 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. 
 
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