C 5 B 201 B6 EART JbfeKKfiLlS I LL&KA& J DEPARTMENT OF (COMMERCE U. S. COAST AND GEODETIC SURVEY O. 1 I. rri MUPKUIXTKSDKNT ASTRONOMY DETERMINATION OF TIME, LONGITUDE LATITUDE, AND AZIMUTH FIFTH EDITION BY BOWIE Inspector of G-eodetio WorU and Cliief of tlie Computing Division TJ. S. Coast and Geodetic Sui^vey SPECIAL PUBLICATION No. 14 WASHINGTON GOVERNMENT PRINTING OFFICB 1917 DEPARTMENT OF COMMERCE U. S. COAST AND GEODETIC SURVEY O. H. TI SUPERINTENDENT ASTRONOMY DETERMINATION OF TIME, LONGITUDE LATITUDE, AND AZIMUTH FIFTH EDITION BY WILLIAM BCTWIK Inspector of Geodetic "Work and Chief of the Computing Division. TJ. S. Coast and G-eodetic Survey SPECIAL PUBLICATION No. 14 PRICE, 65 CENTS Sold only by the Superintendent of Documents, Government Printing Office, Washington, t>. C. WASHINGTON GOVERNMENT PRINTING OFFICE 1917 CONTENTS. Page. Introduction 5 PART I. DETERMINATION OF TIME. General remarks 7 Transit instrument 7 Transit micrometer '. 8 Chronograph 11 Theory of the transit instrument 13 Adjustments of the transit instrument 14 Transit observations 17 Computation of transit observations: Usual method of computing time set 20 Second method of computing time set 28 Least square method of computing time set when azimuth stars are observed 39 Complete least square method of computing time set 41 Determination of instrumental constants 43 Discussion of errors 48 Other methods of determining time 51 The vertical circle 52 Star factors 60 PART II. THE DETERMINATION OF THE DIFFERENCE OF LONGITUDE OF TWO STATIONS. Introductory 78 Program and apparatus of the telegraphic method 79 Computation of difference of longitude when transit micrometer is used 84 Discussion of errors, transit micrometer method 85 Program where no transit micrometer is used 87 Computation of difference of longitude when no transit micrometer is used 87 Personal equation 90 Discussion of errors, key method 93 Statement of costs 94 Longitude by the chronometric method 95 Computation of longitude, chronometric method 97 Discussion of errors, chronometric method 100 PART III. THE DETERMINATION OF LATITUDE BY MEANS OF THE ZENITH TELESCOPE. Introductory 103 Instructions for latitude work 103 Instruments 104 Adjustment of instruments 106 Latitude observations 107 Computation of latitude Ill Apparent places 116 Corrections 117 Combination of results 119 Instrumental constants .-. 124 Computation of micrometer value 126 Reductions for elevation and pole variation 130 Discussion of errors 132 Economics of latitude observations 135 PART IV. THE DETERMINATION OF THE ASTRONOMIC AZIMUTH OF A DIRECTION. General remarks 138 Primary azimuth 138 Instruments 139 General considerations 142 General formula 143 3 4 CONTENTS. Page. PART IV. THE DETERMINATION OF THE ASTRONOMIC AZIMUTH OF A DIRECTION Contd. Direction method 145 Method of repetitions 153 Micrometric method 155 Discussion of errors 158 Statement of costs 160 Azimuth from time observations 160 Correction for elevation of mark and variation of the pole 164 Table of log --L_ 165 1 a Index 175 TABLES. Diurnal aberration () 24 For use in computation of incomplete transits 32 Intervals of lines of transit No. 18 from mean line 33 Weights for incomplete transits, eye and ear observations 36 Weights for incomplete transits, chronographic observations 38 Relative weights to transits depending on the star's declination 39 Refraction 58 Sun's parallax 60 Star factors 62 Relative personal equation 92 Correction to latitude for differential refraction 118 Correction to latitude for reduction to meridian 119 Correction for curvature of apparent path of star in computation of micrometer value 127 Reduction of latitude to sea level 131 Curvature correction 150 2 ^ * T . . 151 sin \" Logj-L.. 165 ILLUSTRATIONS. 1. Large portable transit (equipped with transit micrometer) 8 2. Broken telescope transit 8 3. Meridian telescope 8 4. Transit micrometer 10 5. Transit micrometer 11 6. Chronograph 12 7. Portion of chronograph record 13 8. Vertical circle - 52 9. Nomogram for obtaining star factors 60 10. Arrangement of electrical connections, telegraphic longitude transit-micrometer method 80 11. Arrangement of electrical connections, telegraphic longitude key method 81 12. Switchboard telegraphic longitude 82 13. Zenith telescope 104 14. Observatory 106 15. Observatory 107 16. Observiag tent 108 17. Observiag tent 108 18. Twelve-inch direction theodolite 138 19. Seven-inch repeating theodolite 138 20. Four-inch theodolite 138 21. Small acetylene signal lamp 140 22. Large acetylene signal lamp 141 23. Eighty-foot signal 142 24. Wooden pier used for theodolite and zenith telescope 142 25. Structure for elevating signal lamp over triangulation station used as mark 144 26. Structure for elevating signal lamp over triangulation station used as mark 144 27. Azimuth mark 145 28 . Circum polar stars 146 29. Diagram showing directions to triangulation stations and Polaris 147 DETERMINATION OF TIME, LONGITUDE, LATITUDE, AND AZIMUTH. By WILLIAM BOWIE, Inspector of Geodetic Work and Chief of the Computing Division, U. S. Coast and Geodetic Survey. INTRODUCTION. From time to tune during many years publications have been issued describing the instruments and methods used by the Coast and Geodetic Survey in the determination of time, longitude, latitude, and azimuth. The general aim has been to provide a working manual which would serve as a guide to the observer in the field and the computer in the office in carrying on the astronomic work of the Survey in a systematic manner. The exhaustion of previous editions and the introduction of new instruments and methods have made necessary the suc- cessive editions, in each of which much has been repeated from the preceding one. The edition of the last publication is now exhausted, which gave in one volume descriptions of the instruments and methods, and was entitled "Determination of Time, Longitude, Latitude, and Azimuth." It was published as Appendix No. 7, Report for 1898. The needs of the members of this Survey for a similar manual, and requests for it by others, make it desirable to issue the present and fifth edition. The subject matter includes most of that in the fourth edition, with a number of changes, however. Some of the most important additions to the previous edition arc : The determination of time and longitude, using the transit micrometer; the description of the transit micrometer; determination of time with the vertical circle for use in connection with azimuth observations; a description of the method of observing azimuth coincidently with horizontal directions in primary triangulation ; an example of the determination of an azimuth in Alaska with a transit equipped with a transit micrometer; examples of the records and computations in the different classes of work, as actually made at present by the Survey; and statements of the field cost of the different classes of work. A number of new illustrations have been added. The writer takes pleasure in acknowledging here his indebtedness to Mr. H. C. Mitchell, Mr. C. R. Duvall, and several other members of the Computing Division who assisted in preparing this edition. The material is principally the work of former Assistant C. A. Schott, who prepared the first three editions, and of former Assistant John F. Hayford, who prepared the fourth edition. It has not been deemed necessary to insert the derivation of formulae, except in the few rare cases in which such derivation can not be found readily in textbooks on astronomy. For general developments the reader is therefore referred to Chauvenet's Astronomy, to Doolittle's Practical Astronomy, and to Hayford's Geodetic Astronomy. The last-mentioned book and the fourth edition of this publication appeared about the same time, and as they were by the same author it is natural that some of the text is identical in the two. Much of this publication was copied from the fourth edition without change, and some portions are necessarily identical with the corresponding parts of Prof. Hayford's textbook. In addition to this manual on geodetic astronomy, the American Ephemeras and Nautical Almanac for the year of observation will be required in time and azimuth work, and the Boss Preliminary General Catalogue of 6188 stars, together with the Cape Tables, by Finlay, in latitude determinations. WILLIAM BOWIE, Inspector of Geodetic Work, Chit f of the Computing Division. 5 PART I. DETERMINATION OF TIME. GENERAL REMARKS. This part deals almost exclusively with the portable transit instrument in its several forms as used in the Coast and Geodetic Survey, and when mounted in the plane of the meridian for the purpose of determining local sidereal time from observations of transits of stars, in connection with an astronomic clock or chronometer regulated to sidereal time. The use of this instrument when mounted in the vertical plane of a close circumpolar star out of the meridian is not recom- mended on account of the greater complexity both in field and office work, as compared with the usual method herein discussed, especially when one considers the ease with which a transit may be placed approximately in the meridian. (See p. 16.) The observations are made either by the method of "eye and ear," or by chronographic registration. The latter method is used exclu- sively for all telegraphic longitude work and in making time observations for determining the periods of the pendulums in gravity determinations. In using the first method the observer will, of course, mark his own time; that is, he will pick up the beats of the chronometer and carry them forward mentally up to the time of transit of the star, which he will estimate to the nearest tenth of a second. In using the second method the chronograph record will be produced in one of two ways: First, when the observer sees the star bisected by a line of the diaphragm he will press an observing key (break-circuit) held in his hand and cause a record of that instant to appear on the chronograph sheet; or, second, he will follow the star across the field of the telescope with the movable wire of the transit micrometer, the star being continuously bisected as nearly as possible by the wire, and the record on the chronograph sheet will be made automatically by the make-circuit device of the micrometer. DESCRIPTION OF LARGE PORTABLE TRANSIT. Several sizes of portable transits are used in this Survey. The largest and oldest ones, made by Troughton & Simms, of London, were intended for use exclusively on the telegraphic determinations of longitude, but in 1888 a slightly smaller t} r pe of transit (described below) was made at the Survey office, and has been used very extensively since that time on the same class of work as the largest type. The smallest type of transit, known as the meridian telescope (described on p. 8), is used in the determination of the local time needed while observing astronomic azimuths and latitudes, and for other purposes. In the hands of skillful observers the instruments used for longitude determinations give results which compare favorably with the results obtained with the much larger transits usually employed at astronomic observatories, where special difficulties are encountered in consequence of strains or temporary instability of the instrument due to reversal of axis, and the more serious effect of flexure. In case of necessity, and when an approximate degree of accuracy suffices, any theodolite or altazimuth instrument may be converted temporarily into and used as an astronomic transit. Illustration No. 1 shows Transit No. 18, 1 one of the second-sized portable transits made in the Survey office in 1888. It has a focal length of 94 cm. and a clear aperture of 76 mm. The magnifying power with the diagonal eyepiece ordinarly used is 104 diameters. It is provided with a convenient reversing apparatus, by means of which it can be reversed without lifting the 1 For a full description of this instrument, see Appendix 9, Report for 1889, by Edwin Smith, Assistant. 8 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. telescope by hand. The value of one division ( = 2 mm.) of the striding level is 1".35. The setting circles are 4 inches in diameter, are graduated to 20' spaces, and arc read by verniers to single minutes. Until about 1905 this, as well as the other transits of the Coast and Geodetic Survey, was supplied with a glass diaphragm, but, with the adoption of the transit-micrometer, the glass diaphragms were discarded. The glass diaphragm carries two horizontal lines which are simply to define the limits within which all observations should be made, and 13 vertical lines, 11 of which are used in making time observations with the chronograph and observing key and 5 of which (longer than the others) are used in making eye and ear observations. The shortest time interval between lines for chronographic observations is about 2 seconds and for eye and ear observa- tions about 10 seconds. The transit micrometer and its use are described below. Transit No. 18 is provided with a sub-base which is firmly secured to the supporting pier. The transit proper is supported on this sub-base by three foot screws. At the left of the base in the illustration is shown a pair of opposing screws which serve to adjust the instrument in azimuth. One of these screws carries a graduated head which enables one to set the instrument very nearly in the meridian as soon as the azimuth error is known. This instrument may serve as a typical illustration of the class of large portable transits. The broken telescope transit, like that shown in illustration NQ. 2, has been used with marked success by other countries. This instrument may also be used in the determination of latitude by the Talcott method. This manual can be used with either type of instrument (broken or straight telescope) . DESCRIPTION OF MERIDIAN TELESCOPE. Certain instruments are known in this Survey as meridian telescopes. 1 They are fitted both for time observations and for latitude observations by the Horrebow-Talcott method (see p. 103) and are provided with a frame which may be folded up for convenience in transpor- tation. Illustration No. 3 shows Meridian Telescope No. 13, which may serve as an illustration of the type of smaller instruments used for time observations in this Survev. This telescope has a focal length of 66 cm., a clear aperture of 5 cm., and a magnifying power of 72 diameters. The value of one division ( = 2 mm.) of the striding level is about 2J". During time observations the telescope is reversed by hand; during latitude observations it may be reversed by turning the upper half of the double base on the lower half. One of the two setting circles carries a delicate level for use in making latitude observations, and the eyepiece is fitted with a micrometer for measuring differences of zenith distance, in addition to the diaphragm carrying fixed vertical lines for use in making time observations. On one side of the base (the left-hand side in the illustration) is a slow-motion screw for accurate adjustment in azimuth. THE TRANSIT MICROMETER. The transit micrometer is a form of registering micrometer placed with its movable wire in the focal plane of an astronomic transit and at right angles to the direction of motion of the image of the star which is being observed at and near meridian transit. Certain contact points on the micrometer head serve to make an electric circuit as they pass a fixed contact spring, thus causing to be recorded upon the chronograph sheet each separate instant at which the microm- eter wire reaches a position corresponding to a contact. The transit micrometer in use on the transits of this Survey is hand driven and was designed by Mr. E. G. Fischer, Chief of the Instrument Division of the Survey, and made in that division. Much of the following description is copied from pages 458-460 of Appendix No. 8, Report for 1904, entitled "A test of the transit micrometer." The pages referred to were written by Mr. Fischer. 1 See Appendix No. 7, Report for 1879, for a " Description of the Davidson Meridian Instrument. " No. 1. LARGE PORTABLE TRANSIT (EQUIPPED WITH TRANSIT MICROMETER). No. 2. BROKEN TELESCOPE TRANSIT. No. 3. -#-* MERIDIAN TELESCOPE. DETERMINATION OF TIME. 9 DESCRIPTION OF THE HAND-DRIVEN TRANSIT MICROMETER, MADE FOR COAST AND GEODETIC SURVEY TRANSIT NO. 2. Before considering the details of this micrometer, three points were determined upon as being essential to insure accurate and decisive action, durability, and convenience in reading the chronograph record made by it. First, it was decided that the mechanism of the slide carrying the wire should be of the form in which the screw is mounted in bearings at the extreme ends of the box or case holding the slide, the micrometer head being fast upon the end of the screw projecting from the box, because this insures greater stability under the side stress of the gears connecting the screw with the handwheel shaft than the form usually employed in theodolite and ocular micrometers, in which the screw is fastened to the slide and therefore takes part of whatever play there may be in the latter. Second, it was decided that the electric recording device of the micrometer should be of the make-circuit form, transmitting its records to the chronograph, which is in the break-circuit of the chronometer, through a relay. This permits the use of a strong current through the contact points of the micrometer head, and therefore a minimum of pressure upon the latter by the contact spring. Third, in order that the micrometer transmit no records except those made within an accepted space on either side of the line of collimation and forming the observations of the star transits proper, an automatic cut-out must be provided. Illustrations 4 and 5 show the micrometer with draw tube and eye end of the telescope. The telescope has a focal length of 115 cm. and an aperture of 77 mm. It is of the straight type of the same general form as that shown in illustration No. 1 of Appendix 7 of the Report for 1898. (Illustration No. 1 of this publication.) The micrometer box or case is 46 mm. in length and 31 mm. wide. Within it and near to one side is mounted the micrometer screw. Upon the latter fits, by a thread and cylindrical bearing, a rectangular frame forming the slide, which is 31 mm. long and 23 mm. wide. All play or lost motion, both of the slide upon the screw and the screw in its bearings, is taken up by means of a helical spring within the box, which, pressing from the inner end of the box against the slide and through it against the screw, holds the latter firmly against the point of an adjustable abutting screw, without impeding its free rotary motion. Upon the slide, at right angles to its line of motion, is mounted the single spider thread, which is used for bisecting the star during its passage across the field. Two threads, parallel to the line of motion, about four time seconds apart, and mounted against the inner surface of the box, define the space within which the observations should be made. A short comb of five teeth, with distances equal to one turn of the screw between them, is also provided and indicates the four whole turns of the screw within which the observations are to be made. The diameter of the field of view through the Airy diagonal eyepiece, which has an equivalent focal length of 12 mm., is something over 24 turns of the screw, thus giving a space of fully 10 turns of the screw on each side of the 4 turns in the center of the field. That portion of the micrometer screw which projects through the box has the micrometer head fitted upon it and secured in position by a clamp nut. The cylindrical surface of this head, graduated at the edge nearest the box to 100 parts (g, illustration No. 4), also carries near its opposite edge a screw thread, t, of three turns with a pitch of 1 mm. and a diameter of 32 mm. Sunk into the outer face of the head and fitted concentrically with it is a thin metallic shell, which has fitted upon it a hollow cylinder, e, made of ebonite, 6 mm. long and 26 mm. in diameter. Five strips of platinum, each 0.4 mm. thick, and corresponding to the 12.5, 25.0, 50.0, 75.0, and 87.5 division points of the graduation, g, are slotted into the edge of the ebonite cylinder and secured in such manner as to make metallic contact with the micrometer head proper, and through it with the screw, micrometer box, telescope and telescope pivots, and the iron uprights of the transit. By releasing the clamp nut within the ebonite ring the graduated 10 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. head, with its thread, t, can be adjusted, in a rotary sense, in relation to the thread of the screw, and therefore also to the spider thread upon the slide. At the same time the position of the platinum contact strips can be set to correspond to the zero of the graduation, g, which latter is read by the index, i, illustration No. 5. A small ebonite plate, p, illustration No. 4, secured to the micrometer box, carries upon its outer end, mounted in a suitable metal block, the contact spring, s, which ends in a piece of platinum turned over so as to rest radially upon the ebonite cylinder. The width of this piece of platinum is 4 mm., and its thickness that of the contact strips, i. e., 0.4 mm. A small screw, c, illustration No. 5, serves to adjust the pressure of the spring upon the cylinder. Against one end of the micrometer box is fastened a small bracket, upon which is centered a small worm wheel, w, illustration No. 4, gearing into the screw thread, t, of the micrometer head. It has 40 teeth, and moves 1 tooth for each turn of the micrometer head. To this worm wheel is fastened a cup-shaped cylinder, r, wliich has cut into its rim a notch or depression with sloping ends not visible in the illustrations. A small steel pin in the end of the lever, I, rests upon the edge of this cup-shaped cylinder. The other end of the lever, I, fitted with a small ivory tip, presses upon the end of the contact spring, &, which is mounted upon an ebonite plate, and is therefore insulated electrically from the instrument. When the small steel pin rests upon the edge of the cup-shaped cylinder, the ivory tip presses the contact spring away from the platinum-tipped screw, a. When, however, the notch or depression comes below the steel pin, the contact spring, 6, is free to press against the platinum-tipped screw, thus allowing the flow of an electric current through the coiled wires, m and n, and the contact spring, s. The length of the notch is chosen so as to allow the circuit to be closed during four revolutions of the micrometer head. As the ends of the notch are sloping, it will be seen that by raising or lowering the platinum-tipped screw, and consequently lowering or raising respectively the steel pin in the lever I, the time during which the current can flow can be made to correspond exactly to that of four revolutions of the micrometer head. But it is also important that the four revolutions during which the current can flow and record the contacts made on the ebonite cylinder, e, are those disposed symmetrically about the zero position of the micrometer, wliich indicates the meridian. This is accomplished for adjustments requiring corrections greater than one tooth of the worm wheel w, by removing the latter from its axis, turning and replacing it with the proper tooth engaging the screw thread, t. The adjustment for amounts less than that of one tooth, as the micrometer is now arranged, is made by loosening a capstan-headed screw (hidden in the illustration by the lever 1), and turning to right or left the two screws z, thus moving the plate carrying the lever I, until the small steel pin at the end of lever I is in proper relation to the notch or depression in the cup-shaped cylinder r. It will be seen, therefore, that tlu's arrangement permits of the motion of the spider thread across the entire field without transnu'tting records to the chronograph, except during the four revolutions symmetrically disposed about the line of collimation. Against the inner face of the micrometer head is fastened a spur wheel, k, illustration No. 5, with 36 teeth of 48 diametral (inch) pitch, into which gears the wheel/, with 72 teeth, mounted on the handwheel shaft, d. This shaft is supported by arms from the micrometer box, as can readily be seen from illustration No. 5. The handwheels have a diameter of 33 mm., are 1 16 mm. apart, and equidistant from the middle of the telescope, allowing ample space for manipulating in either position of the eyepiece. The pitch of the micrometer screw is about 48.4 threads per centimeter, or 123 per inch. In the telescope of Transit No. 2 the angular value of one revolution of the screw is 2.5 equatorial time seconds, nearly. As the gearing of the handwheel shaft to the micrometer screw is as 2 to 1 it follows that the hands must produce rotary motion of one revolution in about 5 s for an equatorial star. The adjustment for collimation is made by means of two nuts, x, illustration No. 4, upon a small screw fastened to the micrometer box, which in turn is mounted by dovetail slides upon a short flanged cylinder, y. The latter is fixed in position by the screws, h, which, when loosened, also permit of a rotary motion for adjusting the transit wire into the vertical. Neither No 4. TRANSIT MICROMETER. No. 5. TRANSIT MICROMETER. DETERMINATION OF TIME. 11 of these adjustments will disturb the rather delicate relations between the zero of the transit wire, the contact breaks upon the micrometer head, and the worm wheel with its electric cut-out attachment. As indicated in the description of the ebonite head with its five platinum contact strips, the instrument itself is used as part of the electric conductor forming the transit circuit. The relay of 20 ohms resistance converts the makes of the transit circuit into breaks in the chrono- graph circuit. From the contact spring, 6, through wire, m, connection is made with an insu- lated binding post at the eye end of the telescope tube, from which a wire leads along the tele- scope to and into the telescope axis and within the latter to an insulated metal cylinder pro- jecting from the transit pivot. Each of the wye bearings of the transit has fastened to it an insulated contact spring, which, being connected with an insulated binding post at the foot of the instrument, establishes the circuit whether the telescope lies in either an east or west posi- tion. Another binding post, screwed directly into the iron foot of the transit, affords a ready means for making the necessary connection to begin observations. It is necessary to use both hands in order to impart to the wire a steady motion. As explained above, the cut-out device allows only a limited portion of the field of observation to be registered, by automatically breaking the transit circuit while the wire is outside the limits. It requires four complete revolutions of the micrometer head to carry the wire across the field of record and as there are five contact strips on the micrometer head, the complete record of the observation of the transit of a given star consists of 20 breaks registered on the chrono- graph sheet. As the five contact strips are not equally spaced around the head of the microm- eter wheel, it follows that the record is in four groups of five observations each. This facilitates the reading of the chronograph sheet. The transit of an equatorial star across the field of record occupies only about 10 seconds of time, a fact which makes it possible to observe stars which are quite close together in right ascension. Adjustments of the transit micrometer. Before using the transit micrometer it should be carefully examined to see that there is no loose play in any of its parts, that its contact strips and contact spring are clean and bright, and that the cut-out attachment permits the recording of 20 breaks which are symmetrical about the mean position of the micrometer wire. If a symmetrical record is not obtained, the adjustment must be made, as described on page 10. The adjustment of the micrometer wire for collimation and verticality are described on page 15, under the heading "Adjustment of the transit instrument." THE CHRONOGRAPH. Illustration No. 6 shows the form of chronograph now in use in the Survey. The train of gears seen at the right is driven by a falling weight. It drives the speed governor (seen above the case containing the gears), the cylinder iipon which the record sheet is wound, and the screw which gives the pen carriage a slow motion parallel to the axis of the record cylinder. When the speed governor is first released, the speed continually increases until the governor balls have moved far enough away from the axis of revolution to cause a small projection upon one of them to strike a small hook. This impact and the effect of the friction at the base of the weight attached to the hook causes the speed to decrease continually until the hook is released. The speed then increases again until the hook is engaged, decreases until it is released, and so on. The total range of variation in the speed is, however, surprisingly small, so small that in interpreting the record of the chronograph the speed is assumed. to be uniform during the intervals between chronometer breaks. The speed may be regulated by screwing or unscrewing the movable weights which are above the governor balls and attached to the same arm. This moves them nearer to or farther from the axis, and thus decreases or increases the critical speed at which the hook is engaged. To get a convenient record it is desirable to adjust the speed so that the record cylinder makes just one revolution per minute with the ordinary arrangement of the train of gears. The gears may also be changed quickly to another combination which will run the record cylinder at double speed. This will require additional driving weights. 12 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. The chronograph circuit, passing through the coils of the pen magnet, is operated by a battery of two dry cells in series, so that a relatively strong spring may be used to draw the pen armature away from the pen magnet when the circuit is broken. This insures a sharp lateral movement of the recording pen, which is attached to the pen armature, on the breaking of the circuit, and a correspondingly sharp offset or break is secured in the helix which the pen traces on the drum. When observations are made on the lines of a reticle, an observing key is placed in the chronograph circuit, which normally keeps the circuit closed, and breaks it only when the key is pressed by the observer as the star is bisected by each of the lines of the reticle. When the transit micrometer is used, the transit circuit, passing through the transit, the micrometer head and the coils of the transit relay, and operated by two dry cells in series, is connected with the chronograph circuit through the points of the transit relay. The observing key and the transit circuit with its relay may be regarded as interchangeable, as either one may be joined into the chronograph circuit in the place of the other. The chronometer circuit is operated by a single dry cell, and passes through the coils of a relay, through the points of which it is connected with the chronograph circuit. Breaks in the chronometer circuit are transmitted into breaks in the chronograph circuit by means of the chronometer relay. A condenser should be placed in the circuit across the terminals of the chronometer to prevent sparking and consequent injury to the contact points of the break circuit wheel in the chronometer. The strength of the current, the tightness of the spring which draws back the pen armature, the distance of that armature from the magnet core, and the range of movement of the armature must all be adjusted relatively to each other so that the pen will furnish a neat and complete record of all the breaks in the circuit. The driving weight must be heavy enough to overcome all friction and cause the governor hook to be engaged frequently, but it must not be so heavy as to cause the hook to be carried forward continuously after it is once engaged. Where a transit micrometer is used and the chronograph circuit is broken by means of a relay placed in the transit circuit, this relay also must be adjusted to produce a short neat break of the chrono- graph circuit. In operation the chronometer breaks the circuit automatically every second (or every two seconds) and the pen records the breaks upon the moving record sheet at equal or very nearly equal linear intervals. The chronometer is usually arranged to indicate the beginning of each minute by failing to make a break for the fifty-ninth second, or if it is a two-second chronometer, by making a break for the fifty-ninth second. The hours and minutes may be identified by writing upon some point of the record sheet the corresponding reading of the face of the chronometer. In longitude work it is not essential to have the hours and minutes on the chronograph sheet correspond to those shown on the face of the chronometer. It is customary to mark on the chronograph sheet such hours and minutes as will give the clock a correction of less than one minute, which is equivalent to setting the chronometer to produce that reading. The record of the exact time of the transit of a star is obtained in the following manner : Where a transit micrometer is used the star is bisected with the wire of the micrometer soon after it enters the field of view of the telescope (see p. 18), and the observer endeavors to keep the star bisected as it crosses the field. As the wire passes the various positions corresponding to contacts on the micrometer head the transit circuit is automatically made, and through the action of a relay it automatically breaks the chronograph circuit and produces a record on the chronograph sheet. Where an observing key is used the observer breaks the chronograph circuit directly by pressing the key wliich he holds in his hand ; this is done as the star transits each line of the reticle. In each case the position of the additional break or record on the chro- nograph sheet, with reference to the record made by the chronometer, indicates accurately the chronometer time at wliich it was made, the chronograph being assumed to run uniformly between adjacent chronometer breaks. (See illustration No. 7.) To read the fractions of seconds from the chronograph sheet one may use either a glass scale on wliich converging lines make it possible to divide varying lengths of seconds into 10 equal spaces, or a small linear -i. i ij f v ^tj JJs 111111 tJTJJJJtn *<< ti-d-rf^tJ J 10 M to n V) 1 Ml 1 Vw en O) Tl (ft H 1 1 lo 1 M O z J o H 1m 30 i , , j c 111! (A t 1 t* J ^^TIT J J J IT 1 ' 1 1 1 , M T| 3" O I] " 11 S " ^ P" W J t 1 o g 1 11V v> X I III II III J o j r J 'IJ J J, o ; i Ijll' T l{JJJ|^ T^ S T T i to n ir \ - w -i, 51 1 T i i S \ i 111 1 n i^ C i a ** ! 1 J I ^ x iT.trr tO 11 iii ' IB -1 ill ! J 0' J J J C 1 T 1* -J 1 , J J J J < Oi 33 m 1 T T S "S >1 i o '" i o (A J J J J J Jen 1 1 1 ] fi 51 33 m r- 30 J JJiUJ t JJ, iniS 1 !^ 1 tJJJJJ MM ii I D i n 11 1 1 v V) w t cn 1 1 ITT 1 ! DETERMINATION OF TIME. 13 rule, so divided that 10 of its spaces fit closely a second's interval of the chronograph, when the chronograph is making exactly one revolution per minute. Some of the chronographs now in use in the Survey are so constructed that when in perfect adjustment one second on the record will be exactly 1 cm. in length. Such a record may be easily read by using a meter scale. When the linear scale does not fit the chronograph record exactly a satisfactory reading is obtained by a slight shifting of the scale to fit the adjacent seconds marks as the transit records are successively read. This linear scale is much preferred to the glass scale, as it enables one to read the complete record for a star with one setting of the scale. Also by placing the mark of the scale on an even 10-second mark (0, 10, 20, etc.) immediately preceding the stai's record, not only the fractional part of the second may be read at once, but also the number of the second. The beginning of each break made by the observer and by the chronometer is the exact point to be used in reading the chronograph record, the break of the circuit being sharp and definite, while the make is indefinite. When an observing key is used and 11 breaks constitute a full record for a star, the star transits are usually read from the record sheet to the nearest half-tenths (0.05) of a second; when a transit micrometer is used and 20 obser- vations constitute the full record of a transit, the readings are made to the nearest tenth (0.1) of a second only. In longitude work it is customary to read the time signals to the nearest hundredth (0.01) of a second, the chronograph then being run at double speed. There will occasionally be a slight interference between the chronometer and the star transit record caused by overlapping, but the time of the observation can usually be identified and closely estimated by comparing the distances between the successive breaks. A correction, called the contact correction, is sometimes applied to the chronograph record of transits observed with a micrometer to account for the time required for the contact spring to cross the contact strip on the head of the micrometer. In order to insure a satisfactory record the contact strips on the micrometer are given material width, since if they were reduced too much there would be an occasional skipping of a record. The micrometer wire travels from a different side of the instrument for upper and lower culminating stars, and also before and after reversal of the telescope in its wyes, so that the contact spring produces a record sometimes from one edge of the contact strip and sometimes from the other. Theoretically, the proper reduction would be to correct all observations for one-half the movement of the micrometer wire from the beginning of the contact to its end. This may be measured on the micrometer head. The micrometer is turned very slowly until the armature of a relay, in the transit circuit is heard to make the circuit; the micrometer head is then read. The motion is continued until the armature sounds the breaking of the circuit, and the micrometer is read again. The difference between the two readings is the movement of the wire in terms of divisions on the micrometer head. This may be reduced to time when the equatorial value of the micrometer division is known. This correction is always plus, since the middle of the strip must always come under the contact spring later than does its near edge. But being very small and having nearly the same effect on all time determinations with similar instruments it is without appre- ciable effect on the observed differences of longitude. Nor is this correction necessary in time determinations for gravity observations with pendulums. If we designate the contact correction on an equatorial star for any transit micrometer as n, then the contact correction for any star is n sec dorn C, where C, the collimation factor, is obtained directly from the table on pages 62-77, or graphically as shown in illustration No. 9. The equatorial contact correction on transit No. 18 is 0.008 second. THEORY OF THE TRANSIT INSTRUMENT. The meaning of the phrase line of collimation used in the preceding edition of this publication vAppendix No. 7, of 1898) is adhered to in the present publication. The line of collimation may be defined as the line through the optical center of the objective and the middle point of the mean vertical line of the diaphragm or the micrometer wire in its mean position. It may be considered synonymous with the pointing line, sight line, or line of sight. The term collimation axis as used in this publication may be defined as the line through the optical center of the 14 U. S. COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 14. objective, and perpendicular to the horizontal axis (axis of rotation) of the telescope. The line of collimation and collimation axis of a telescope coincide only when there is 110 error of collimation hi the instrument. If a transit instrument were in perfect adjustment the line of collimation of the telescope would be at right angles to the transverse axis upon which the telescope rotates, and that transverse axis would be horizontal and in the prime vertical. Under these circum- stances the line of collimation would always lie in the meridian plane, and local sidereal time at the instant when a given star crossed the line of collimation would necessarily be the same as the right ascension of that star. The difference then between the chronometer time of transit of a given star across the line of collimation and the right ascension of that star would be the error of the chronometer on local sidereal time. Before observing meridian transits for the deter- mination of time, the conditions stated in the first sentence of this paragraph are fulfilled as nearly as possible by careful adjustment of the instrument. The time observations them- selves and certain, auxiliary observations are then made in such a manner that the small remain- ing errors of adjustment may be determined, and the observed times of transit are corrected as nearly as may be to what they would have been had the observations been made with a perfectly adjusted instrument. The observed chronometer time of transit of any star across the line of collimation as thus corrected being subtracted from the right ascension of that star gives the correction (on local sidereal time) of the chronometer used during the observations. ADJUSTMENTS OF THE TRANSIT INSTRUMENT. Let it be supposed that observations are about to bo commenced at a new station at which the pier and shelter for the transit have been prepared. (See p. 105.) By daylight make the preparations described below for the work' of the night. By whatever .means are available determine the approximate direction of the meridian and mark it on the top of the pier or by an outside natural or artificial signal. Place the sub-base or footplates of the instrument in such position that the telescope will swing closely in the meridian. It is well to fix the sub-base or footplates firmly in place by cementing them to the pier with plaster of Paris when a stone, concrete, or brick pier is used, and by screws or bolts when a wooden pier is used. The meridian may be determined with sufficient accuracy for this purpose by means of a compass needle, the magnetic declination being known and allowed for. A known direction from triangulation or from previous azimuth observations may be utilized. All that is required is that the telescope shall be so nearly in the meridian that the final adjustment will come within the scope of the screws provided upon the instru- ment for the azimuth adjustment. Set up the instrument and inspect it. The pivots and wyes of both instrument and level should be cleaned with watch oil, which must be wiped off to prevent its accumulating dust. They should be carefully inspected to insure that there is 110 dirt gummed to them. The lens should be examined occasionally to see that it is tight in its cell. It mav be dusted off witli a camel's-hair brush, and when necessary may be cleaned by rubbing gently with soft, clean tissue paper, first moistening the glass slightly by breathing on it. Focus the eyepiece by turning the telescope up to the sky and moving the eyepiece in and out until that position is found in which the most distinct vision is obtained of the micrometer wire. If any external objects are visible through the eyepiece in addition to the micrometer wire seen projected against a uniform background (the sky, for example) the eye will attempt, in spite of its owner, to focus upon those objects as well as upon the micrometer wire and the object of the adjustment, namely, to secure a focus corresponding to a minimum strain upon the eye, will be defeated to a certain extent. Focus the objective by directing the teloscope to some well-defined object, not less than a mile away, and changing the distance of the objective from the plane in which the micrometer wire moves until there is no apparent change of relative position (or parallax) of the micrometer wire and the image of the object when the eye is shifted about the front of the eyepiece. The DETERMINATION OF TIME. 15 object of the adjustment, namely, to bring the image formed by the objective into coincidence with the micrometer wire is then accomplished. If the eyepiece has been properly focused this position of the objective will also be ths position of most distinct vision. The focus of the objective will need to be inspected at night, using a star as the object, and corrected if necessary. Unless the focus is made nearly right by daylight none but the brightest stars will be seen at all at night and the observer may lose time trying to learn the cause of the trouble. If the objective is focused at night a preliminary adjustment should be made on a bright star and the final adjustment on a faint star, as it is almost impossible to get a very sharp image of a large star. A planet or the moon is an ideal object on which to focus the objective. A scratch upon the draw- tube to indicate its approximate position for sidereal focus will be found a convenience. After a satisfactory focus has been found the drawtube is clamped in position with screws provided for that purpose. Methods exactly similar to those described in the two preceding paragraplis are employed in focusing the eyepiece and objective when a diaphragm is used instead of the micrometer. If unusual difficulty is had with the illumination at night, it is advisable to remove the eyepiece and look directly at the reflecting mirror in the telescope tube. The whole surface of the mirror should be uniformly illuminated. If tliis is not the case, the mirror should be rotated until a satisfactory illumination is obtained. Occasionally the mirror must be removed from the telescope and its supporting arm bent in order to make the reflected rays of light approximately parallel with the tube of the telescope. Adjust the striding level in the ordinary manner, placing it on the pivots direct and reversed. If the level is already in perfect adjustment the difference of the two east (or west) end read- ings will be zero for a level numbered in both directions from the middle, or the sum of the two east (or west) end readings will be double the reading of the middle of the tube for a level num- bered continuously from one end to the other. The level must also be adjusted for wind. In other words, if the axis of the level tube is not parallel to the line joining the wyes, the bubble will move longitudinally when the level is rocked back and forth on the pivots. The adjustment for wind is made by means of the side adjusting screws at one end of the level. To adjust for wind, move the level forward and then back and note the total movement of the bubble. The wind will be eliminated by moving the bubble back one-half of the total displacement by means of the side adjusting screws. Then test again for wind, and repeat adjustment if necessary. In placing the level upon the pivots it should always be rocked slightly to insure its being in a central position and in good contact. Level the horizontal axis of the telescope. This adjustment may, of course, be combined with that of the striding level. Test the verticality of the micrometer wire (or of the lines of the diaphragm) by pointing on some well-defined distant object, using the apparent upper part of the wire (or of the middle line of the diaphragm). Rotate the telescope slightly about its horizontal axis until the object is seen upon the apparent lower part of the line. If the pointing is no longer perfect, the micrometer box (or reticle) must be rotated about the axis of figure of the telescope until the wire (or line) is in such a position that this test fails to discover any error. To adjust the collimation proceed in the following manner: If a transit micrometer is used, place the micrometer wire in its mean position, as indicated by the middle point of the rack or comb in the apparent upper (or lower) edge of the field, the graduated head reading zero. Point on some well-defined distant object by means of the azimuth screws, keeping the wire in the position indicated above. Reverse the telescope in its wyes and again observe the distant object. If the wire again bisects the object, the instrument has no error of collimation. If upon reversal the wire does not again bisect the object, then the adjustment is made by bringing the wire halfway back to the object with the screw x, illustration No. 5. Set on the object again, using the azimuth screws, and test the adjustment by a second reversal of the telescope, If the transit has a diaphragm instead of a transit micrometer, the process is very similar to that described above, though simpler. Point on some well-defined distant object, using the 16 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. middle vertical line of the diaphragm. Reverse the instrument in its wyes and again obseive the same distant object. If after reversal the wire covers the object no adjustment is needed. If an adjustment is necessary it is made by moving the diaphragm halfway back to the object by means of the adjusting screws which hold it in place. A second test should be made to show whether the desired condition has been obtained. Wherever practicable, the adjustment for collimation should be made at sidereal focus on a terrestrial object at least 1 mile distant, or on the cross wires of a theodolite or collimator which has previously been adjusted to sidereal focus, set up just in front of the telescope of the transit. If necessary- the lines of the theodolite are artificially illuminated. Occasionally, if neither a distant object nor a theodolite is available for making the collimation adjustment, a near object may be used for the purpose. In this case, however, collimation error may exist when the telescope is in sidereal focus. If such error is not large, the method of computations of the observations will eliminate its effect from the results. A rapid and careful observer may sometimes be able to make this collimation adjustment on a slow-moving close circumpolar star. In so doing he will have to estimate the amount the star moves while he is reversing his instrument and securing the second pointing. No attempt should be made to adjust the collimation error to zero. If it is already less than say 0.2 second of time it should not be changed, for experience has shown that frequent adjustment of an instrument causes looseness in the screws and the movable parts. To test a finder circle which is supposed to read zenith distances, point upon some object, placing the image of the object midway between the two horizontal lines (guide lines) ; bring the bubble of the finder circle level to the center and read the circle. Next reverse the telescope and point again on the same object; bring the bubble to the center and read the same finder circle as before. The mean of the two readings is the true zenith distance of the object, and their half difference is the index error of the circle. The index error may be made zero by set- ting the circle to read the true zenith distance, pointing on the object, and bringing the vernier bubble to the center with the level adjusting screw. At night this adjustment may be made by keeping a known star between the horizontal lines as it transits the meridian. While the telescope remains clamped in this position set the finder circle to read the known zenith dis- tance of the star and bring the bubble to the middle position of the tube as before. A quick test when there are two finder circles is to set them at the same angle and see if the bubbles come to the center for the same position of the telescope. Adjust the transit micrometer so that it will give 20 records which are symmetrical about the mean position of the micrometer wire. For a description of this adjustment see page 10. The preceding adjustments can not always be made in the order named, as, for instance, when a distant mark cannot be seen in the meridian, nor need they all be made at every station. The observer must examine and correct them often enough to make certain that the errors are always within allowable limits. The azimuth adjustment. In the evening, before the regular observations are commenced, it will be necessary to put the telescope more accurately in the meridian. If the chronometer correction is only known approximately, say within one or two minutes, set the telescope for some bright star which is about to transit within 10, say, of the zenith. Observe the chro- nometer time of transit of the star. This star being nearly in the zenith, its time of transit will be but little affected by the azimuth error of the instrument. 1 The collimation and level errors having previously been made small by adjustment, the right ascension of this star minus its chronometer time of transit will be a close approximation to the chronometer correction. Now set the telescope for some star of large dech'nation (slow-moving) which is about to transit well to the northward of the zenith. Compute its chronometer time of transit, using the chro- nometer correction just found. As that time approaches bisect the star with the micrometer 1 To avoid waiting for stars close to the zenith the chronometer correction may also be estimated closely by comparing observations of two stars not very distant from the zenith, one north and one south, and these at tte same time will give some idea of the amount and direction of the azimuth error. DETERMINATION OF TIME. 17 wire in its mean position or with the middle vertical line of the diaphragm and keep it bisected, following the motion of the star in azimuth by the slow-motion screws provided for that pur- pose, until the chronometer indicates that the star is on the meridian. The adjustment may be tested by repeating the process; that is, by obtaining a closer approximation to the chronometer error by observing another star near the zenith and then comparing the computed chronometer time of transit of a slow-moving northern star with the observed chronometer time of transit. If the star transits apparently too late, the objective is too far west (if the star is above the pole), and vice versa. The slow-motion azimuth screw may then be used to reduce the azimuth error. This process of reducing the azimuth error will be much more rapid and certain if, instead of simply guessing at the movement which must be given the azimuth screw, one computes rouglily what fraction of a turn must be given to it. This may be done by computing the azimuth error of the instrument rouglily by the method indicated on page 35, having previously determined the value of one turn of the screw. 1 If from previous observations the chronometer correction is known within, say, five seconds, the above process of approximation may be commenced by using a northern star at once, instead of first observing a zenith star as indicated above. Or, the clironometer correction being known approximately, and the instrument being fur- nished with a screw or graduated arc with which a small horizontal angle may be measured, the first approximation to the meridian may be made by observing upon Polaris, computing the azimuth approximately by use of tables of azimuth of Polaris at different hour angles then by means of the screw or graduated arc swinging the instrument into the meridian. The tables referred to are given in Appendix No. 10 of the Report for 1895, in "Principal Facts of the Earth's Magnetism, etc.," (a publication of the Coast and Geodetic Survey), or in the Ameri- can Ephemeris and Nautical Almanac. Where saving of time is an important consideration, the latter method has the advantage that Polaris may be found in daylight, when the sun is not too high, by setting the telescope at the computed altitude and moving it slowly in azi- muth near the meridian. It is advisable to use a hack chronometer and the eye and ear method in making the azimuth adjustments, the chronograph being unnecessary for this pur- pose, even when available. OBSERVING LIST. The following is an example of the list of stars selected for time observations at stations of a lower latitude than 50. The second time set shown in this list is computed on page 26, and enters into the longitude determination shown on page 84. Each set consists of two half sets of six stars each, selected hi accordance with the instructions shown on page 80. Such a list prepared in easily legible figures, should be posted in the observatory. 1 Some, of the meridian telescopes carry a small graduated arc on the double base of the frame, which may be used for measuring the small angle here required. 813C 13 2 18 Form 250.* XI. S. COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 14. Star list for Key West, Fla. 5 Aurigae 5.5 40 02 +43 40 N 19 07 -0.45 1.38 1.31 -0.03 B 18 Monocer. 4.7 43 01 + 2 31 S 22 02 +0.37 1.01 0.93 -0.02 B 6 Geminor. 3.4 46 40 +34 04 N 9 31 -0.20 1.21 1.19 -0.02 B Geminor. 3.8 58 36 +20 42 S 3 51 +0.07 1.07 1.07 -0.02 B 63 Aurigae 5.0 7 05 16 +39 28 N 14 55 -0.34 1.30 1.25 -0.02 B t Geminor. 3.8 7 19 57 +27 59 N 3 26 -0.07 1.13 1.13 -0.02 B /? Canis Min. 2.9 22 06 + 8 29 S 16 04 4-0. 28 1.02 0.97 -0.02 B a Canis Min. 0.5 34 26 + 5 28 S 19 05 +0.33 1.01 0.95 -0.02 B /? Geminor. 1.1 39 38 +28 15 N 3 42 -0.08 1.13 1.13 -0. 02 B JT Geminor. 5.5 41 31 +33 39 N 9 06 -0.19 1.21 1.18 -0.02 A Geminor. 5.0 47 48 +27 00 N 2 27 -0.05 1.12 1.12 -0.02 * Form 25fi, known as "Coast and Geodetic Survey, Longitude Record and Computation," is a book containing all the different forms used in observing and computing, time and longitude, except form 34 shown on p. 20. fBerliner Astronomisches Jahrbuch. t American Ephemeris and Nautical Almanac. DIRECTIONS FOR OBSERVING. Everything being in readiness and the instrument completely adjusted set the tele- scope for the first star. It is not advisable to use the horizontal axis clamp during obser- vations, for its action may have a slight tendency to raise one end of the axis. See to it, loading one end if necessary, that the center of gravity of the telescope is at its horizontal axis, and then depend upon the friction at the pivots to keep the telescope in whatever position it is placed. Watch the chronometer 1 so as to know when to expect the star to appear in the field of view of the telescope. When the star enters the field, bring it between the horizontal lines of the diaphragm, if it is not already there, by tapping the telescope lightly. If a transit micrometer is used the process of observing consists simply in bisecting the star's image with the micrometer wire soon after it appears and in keeping it bisected as it moves across the field of the telescope. The record is made automatically by the contact of a spring with certain metal strips on the micrometer head. A cut-out device allows only 10 such con- tacts on either side of the moan position of the micrometer wire to register on the chronograph. The observer learns by experience at what part of the field the wire begins to register and he should endeavor to keep the star bisected several seconds before it reaches that point. Similarly, he knows when the record is complete and he can cease observing a particular star, and set for the next one on his observing list. If an instrument with a diaphragm is being used in connection with a chronograph, the process of observing the transit of a star across a line of the diaphragm consists in waiting, observing key in hand, until the instant when the star is apparently bisected by the line and then pressing the key as soon as possible thereafter. The time record thus made on the chrono- i When achronograph is being used, it is customary to keep the chronometer which is connected with the chronograph protected as carefully as possible from rapid changes of temperature and from jars. During the observations it is not usually removed from its protecting box, but instead an extra chronometer (sometimes called a hack chronometer) is used at the instrument. DETERMINATION OF TIME. 19 graph will always follow the event by a time interval, known as personal equation, which depends mainly on the rapidity of the action of the nerves and brain of the observer. It may occur to a new observer to attempt to make this time interval zero by anticipating the bisection of the star's image, and this he may succeed in doing. He may even make the personal equation negative. The accumulated experience of many observers, however, is that it is better to observe in the manner first indicated and have a large and constant personal equation, rather than to reduce this personal equation to a small but at the same tune rather variable quantity. The method of observing with a transit micrometer practically eliminates the personal equation from the tune observations. In other methods it may be eliminated from the results by special observations, or by programs of observing especially devised for that purpose. (See p. 91.) At about the middle of the observations which are to constitute a set the telescope should be reversed, so that the effects of the error of collimation and inequality of pivots upon the apparent times of transit may be reversed in sign. Three or four readings of the striding level, in each of its positions (direct and reversed) should be taken during each half set. To eliminate, in part at least, the effects of irregularities in the figure of the pivots upon the determination of the inclination of the axis, it is desirable to take the level readings with the telescope inclined at the various practicable angles at which stars are observed, and to make half of them with the objective to the northward and half with the objective southward. Great care should be taken to avoid unequal heating of the two ends of the striding level. The level readings may be checked and possible errors often detected by the fact that the bubble length should be constant except for the effect of change of temperature (the bubble shortens with rise of tem- perature) and in observing and computing this should be kept in mind. A very short length of bubble should not be used on account of increased tendency to stick, and extreme length should be avoided because of danger of running off the graduation. In using the striding level it is important that the bubble be given tune to come to rest before reading. The only difference between the eye and ear method of observing time and the chronograph and key method just described is in the process of observing and recording the times of transit of the star image across the separate lines of the diaphragm. Before using the eye and ear method the observer must first learn to pick up the beat of a chronometer and to carry it even while paying attention to other matters. To pick up the beat of a chronometer, first look at some second's mark two or more seconds ahead of the second hand. Fix the number of that second in mind as the second hand approaches it. Name it exactly with the tick at which the second hand reaches it. Then, keeping the rhythm of the chronometer beat, count the seconds and half seconds (aloud, in a whisper, or mentally), always keeping the count exactly with the tick of the chronometer. In counting it will be found easier to keep the rhythm if the names of the numerals are elided in such a way as to leave but a single staccato syllable in each. The half -second beat should be marked by the word "half," thus one, half, two, half, three . . . twenty, half, twenty-one, half, twenty-too . . . and so on. 1 With practice, an observer can carry the count of the beat for an indefinite period without looking at the chronometer face if he can hear the tick. If he becomes expert, he will even be able to carry the count for a half minute or more during which he has not even heard the tick. The chronometer should, of course, be placed where it can be seen and heard by the observer with as little effort as possible. To observe the time of transit of a star across a given line the observer first picks up the beat of the chronometer as the star approaches the line. At the last tick of the chronometer occurring before the transit he notes mentally the number of the tick, and also carefully observes the apparent distance of the star from the line. At the next tick the star is on the other side of the line and the observer notes again the apparent distance of the star from the line. By a mental comparison of these two distances he estimates fifths of the time interval between the two ticks of the chronometer and obtains his estimate of the time of transit to the nearest tenth of a second. Though the mental processes involved may seem difficult at first, practice soon makes them easy. An experienced observer using this process is able to estimate the tune of transit i Another method often used is to count only to 10 (thus using only words of one syllable) and to glance at the chronometer alter the obser- vation to show the position in the minute. 20 U. S. COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 14. of a star's image across a line of the diaphragm with a probable error of about s .l. It is conducive to accuracy for the observer to acquire the habit of deciding definitely, without hesitation, upon the second and tenth as soon as the event is complete. Hesitation in this matter is likely to cause inaccuracy. EXAMPLE OF RECORD AND PART OF THE COMPUTATIONS. There are shown on pages 18, 20-22 examples of the list of stars and the original transit level readings made in the observatory at the time of the observations, a set of time observations as read from the chronograph sheet, and the computation of a t (right ascension minus the chronometer time of transit) for each star. The computation of AT (the mean correction to the chronometer) is shown on page 26. These computations are for the second set of stars given on page 18. These observations were made under the General Instructions for Longitude Determina- tions with the Transit-Micrometer, which are given on page 79 of this publication. Form 34. Longitude record. [Station, Key West. Date, Feb. 14, 1907. Instrument, Transit No. 2. Observer, J. S. Hill.) Set I Set II Stars Levels W E Stars Levels W E d N d d N d Clamp or band, W 17. 7 58. 8 Clamp or band, W 62. 20. ft Tauri 60. 1 19. S Monocer. 17. 7 59. 5 % Aurigae 5 Aurigae i Orionis S 18 Monocer. S o Aurigae 17. 7 58. 8 6 Geminor. 61. 2 19. 4 v Aurigae 61. 2 20. Geminor. 17. 7 59. 6 63 Aurigae N N 17. 5 58. 9 61.5 19.5 60.7 19.3 17. 7 59. 7 S 17.6 59.0 61. 7 20. 2 N N Clamp or band, E 17. 58. 7 Clamp or band, E 16. 8 58. 9 S Aurigae 61.3 19.7 i Geminor. 61. 6 19. 5 6 Aurigae ft Canis Min. j] Geminor. S a Canis Min. S 8 Monocer. 17.2 59.0 ft Geminor. 17.4 59.7 10 Monocer. 61. 9 20. n Geminor. 62. 1 19. 7 Geminor. N N 16. 8 58. 7 17. 59. 4 61.3 19.4 62. 19. 5 S 16. 9 59. 4 62. 3 19. 9 1 div. of level scale 2". 322. Chronometer 1824. Pivot inequality = 0.000. Remarks: Cable was used direct, without repeaters, between Miama and Key West. DETERMINATION OF TIME. 21 While the following method of computing was devised for observations with the transit micrometer, it is not limited in its use to such observations. The star list for which observa- tions and computations are shown on the following pages could have been observed with a key and the computation made in the same manner as the one which foUows. The only differ- ence is that had the observations been made with a key not so many records would have been obtained and the observations would have been subject to a large observation error, called personal equation. (See p. 90.) Explanation of the formulae and methods used hi this computation follows the examples ol the record and computation. Form 256.* [Station, Key West. Date, Feb. 14, 1907. Instrument, transit No. 2, with transit micrometer. Observer, J. S. Hill. Recorder, J. S. Hill. Cnro- nometer, Sidereal 1824.] Star: S. Monoccr. ifi' Aurigae 18 Monocer. Geminor. C Geminor. 63 Aurigae Clamp: W W W W VV W Lev el: W E W E W E d d d d d d N62.0 20.0 S61.2 19.4 N61.5 19.5 17.7 59.5 17. 7 59. 6 17.7 59.7 +44.3 -39.5 +43.5 -40.2 +43.8 -40.2 +4.S +3.3 +3.f i Computatior of level constant: Me anN+4.20 S+3.30 s + 3. 75X0.039- +0.140= b w h m h m h m h m h m K m 6 35 6 39 6 42 6 46 6 58 7 04 s s Sums s s Sums s s Sums s s Sums s s Sums s s Sums 32.0 41.4 73.4 41.3 54.0 95.3 41.5 50.5 92.0 19. 5 30. 4 49.9 16.2 26.0 42.2 55.3 67.0 122.3 32.4 41.1 .5 41.8 53.5 .3 41.9 50.2 .1 20. 30. 1 50.1 16.5 25.5 2.0 55. 6 66. 5 .1 33.1 40.4 .5 42.8 52.6 .4 42. 5 49. 7 .2 20. 6 29. 4 .0 17. 2 24. 8 2.0 56. 4 65. 8 .2 33.6 39.8 .4 43.5 51.9 .4 43. 1 49. 1 .2 21.3 28.7 .0 17.7 24.3 2.0 57. 1 65. 1 .2 33.9 39.5 .4 43.9 51.4 .3 43. 3 48. 8 .1 21. 7 28. 3 .0 18. 23. 9 1.9 57. 5 64. 6 .1 34.6 38.8 .4 44.7 50.6 .3 44. 48. 1 .1 22. 3 27. 6 49.9 18. 8 23. 1 1.9 58. 4 63. 9 .3 35.0 38.5 .5 45.3 50.3 .6 44. 3 47. 9 .2 22. 8 27. 1 9.9 19.1 22.9 2.0 58. 8 63. 4 .2 35.6 37.9 .5 46.0 49.3 .3 44.8 47.3 .1 23.6 26.4 50.0 19. 8 22. 3 2.1 59. 5 62. 6 .1 36.1 37.4 .5 46.9 48.5 .4 45. 4 46. 6 .0 24. 3 25. 7 .0 20. 5 21. 6 2.1 60.3 61.9 .2 36.4 37.1 .5 47. 2 48. 1 .3 45. 7 46. 3 .0 24. 6 25. 4 .0 20.7 21.4 2.1 60.7 61.5 .2 Sum 734. 6 Sum 953.6 Sum 921. Sum 499. 8 Sum 420. 3 Sum 1221.9 Mean 36.73 47.68 46.05 24.99 21.02 01.10 Rt K - .02 - .03 - .02 - .02 - .02 - .02 Bb + .14 + .19 + .14 + .17 + .16 + .18 t 6 35 36.85 6 39 47.84 6 42 46.17 6 46 25.14 6 58 21.16 7 05 01.26 a. 6 35 51.85 6 40 02.92 6 43 01.21 6 46 40.17 6 58 36.16 7 05 15.28 (a-) + 15.00 + 15.08 + 15.04 + 15.03 + 15.00 +15.02 * See note below table on p. 18. t K, correction for rate, is negligible in this time set. 22 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Form 256.* [Station, Key West. Date, Feb. 14, 1907. Instrument, transit No. 2, with transit micrometer. Observer, J. S. Hill. Recorder, J. S. Hill. Chro- nometer, Sidereal 1824.] Star : t Geminor. p Canis Min. tt Canis Min. f Geminor. ?r Geminor. $ Geminor Clamp: E E E E E E Level: \V E W E W E W E d d d d d d d d N 16.8 58.9 S 17.4 59.7 N 17. 59.4 S 16. 9 59. 4 61.6 19.5 62. 1 19. 7 62.0 19.5 62. 3 19. 9 +44.8 -39.4 +44.7 -40.0 +45.0 -39.9 +4.5.4 39.5 +5.4 +4.7 +5.1 +5.9 d Computation of level constant: Mean N +5. 25 S+5.30 +5. 28X0.039= +0.206= & E ft 771 h m h m h m h m h in 7 19 7 21 1 34 7 3!) 7 41 7 47 s s Sums s 5 Sums a s Sums s s Sums s s Sums s s Sums 37.8 48.3 86.1 47.9 57.1 105.0 07.5 16.7 24.2 18. 5 28. 8 47.3 11.3 22.3 33.6 29. 5 39. 6 69. 1 38.3 47.9 .2 48. 2 56. 8 5. 07. 8 16. 4 .2 18. 8 28. 5 .3 11.6 21.9 .5 29.8 39.4 .2 38.9 47.3 .2 48. 7 56. 1 4. 8 08. 4 15. 7 .1 19.5 27.7 .2 12.5 21.1 .6 30.3 38.5 68.8 39.6 46.5 .1 49.3 55.5 4.8 09. 15. 1 .1 20.1 27.0 .1 13. 2 20. 4 .6 31. 37. 8 .8 39.9 46.3 .2 49. 7 55. 2 4. 9 09. 2 14. 8 .0 20. 5 26. 8 .3 13. 6 20. 1 .7 31.3 37.5 .8 40.7 45.6 .3 50. 2 54. 6 4. 8 09.9 14.2 .1 21. 2 26. 1 .3 14. 3 19. 4 .7 32. 36. 8 .8 41.0 45.1 .1 50. 6 54. 4 5. 10. 2 13. 9 .1 21. 6 25. 7 .3 14. 7 19. .7 32. 3 36. 5 .8 41.7 44.6 .3 51. 1 53. 7 4. 8 10.8 13.3 .1 22.3 25.0 .3 15.4 18.3 .7 33. 1 35. 9 69. 42.5 43.8 .3 51. 8 53. 4. 8 11. 4 12. 6 .0 23.1 24.3 .4 16.1 17.5 .6 33. 8 35. 1 68. 9 42.8 43.4 .2 52. 1 52. 7 4. 8 11.7 12.3 .0 23.3 24.1 .4 16.3 17.2 .5 34. 1 34. 8 .9 Sum 862. Sum 1048.7 Sum 240. 9 Sum 472.9 Sum 336. 2 Sum 689. 1 Mean 43.10 52.44 12.04 23.64 16.81 34.46 Rt X - .02 - .02 - .02 - .02 - .02 - .02 Bb + .23 + .20 + .20 + .23 + .24 + .23 t 7 19 43.31 7 21 52.63 7 34 12.22 7 39 23.85 7 41 17.03 7 47 34.67 a 7 19 57.74 7 22 07. 08 7 34 26.67 7 39 38.26 7 41 31.45 7 47 49.14 (a t ) + 14.43 + 14.45 + 14.45 + 14.41 + 14.42 + 14.47 * Eee note below table on p. 18. t R, correction lor rate, is negligible in this time set. CORRECTION FOR INCLINATION OF AXIS. If the horizontal axis of the telescope is slightly inclined to the horizon and the telescope is otherwise in perfect adjustment, the line of collimation will, when the telescope is rotated about its horizontal axis, describe a plane which passes through the north and south points of the horizon and makes an angle with the meridian plane equal to the inclination of the axis to the horizon. If the eastern end of the axis is too high, the transits of all the stars above the pole (apparently moving westward) will be observed too late, and the transits of all subpolars will be observed too early, and it is therefore necessary to correct the observed times of transit by means of the readings of the striding level, taking into account the inequality of the pivots, if appreciable. Let w and e be the readings of the west and east ends, respectively, of the bubble of the striding level for a given position of the telescope axis. Let w' and e,' be the corresponding west and east readings after the level is reversed, the telescope axis remaining as it was. Let d be the value of a division of the level in seconds of arc. Then for /3, the apparent inclination of the DETERMINATION OF TIME. 23 telescope axis expressed in seconds of time, we may write, if the level divisions are numbered in both directions from the middle : f ) - (e + e 1 ) } ~ = [ (w + w f ) -( + ') 1 4 ) 1O I J DU in whicli ^ is a constant for the level, -r-= being the value of one division of the level in seconds ou lo of time. If the level divisions are numbered continuously from one end of the level to the other the above formula takes the form /?= (w-w f ) + (-') L in whicli the primed letters refer to that position of the level in which the zero end of the tube is to the west. 1 Inequality of pivots. The level readings give a determination of the inclination of the line joining the points of the two pivots, which are midway between the lines of contact of the pivots and the wyes of the level, but do not give the required inclination of the axis of rotation of the telescope (which is the line joining the centers of the two pivots) unless the pivots are of the same size. Let p, the pivot inequality, be the angle, expressed in seconds of time, between the line joining the centers of the pivots and the line whose inclination is determined by the level readings, and let this angle be called positive if the pivot nearest the designating mark (band, clamp, or illumination) is the smaller. Then and b E = 3 e - 2 in which b is the required inclination of the axis of rotation of the telescope. The subscripts indicate the position, to the westward or to the eastward, of the bright band, the clamp, or the illumination, or whatever mark is used to distinguish between the two positions of the telescope axis. The pivot inequality, p, is ordinarily derived from a special series of observations taken for that purpose. For an example of such a series, with the corresponding formula and com- putation, see page 44. The correction to the observed time of transit of any star for inclination is b cos sec d = bB, in which d is the declination of the star and is its zenith distance ( = S for all stars above the pole, and = + d 180 for subpolar stars) . The factor B = cos sec 3 is tabulated on pages 62-77, but is much more easily obtained with the graphical device shown in illustration No. 9 and explained on page 61. It is positive for stars above the pole and negative for subpolars. It is the present practice in this Survey to assume that b, the inclination, is constant for each half set, and it is computed in the following manner: Within each half set the mean of the observed values of j) with objective northward is first derived, then the corresponding mean with objective southward, and finally the mean of these two means is taken as the /? for the half set. The value of B for each star, as taken from either the table on pages 62-77 or the graphical device shown in illustration No. 9, is given in the observing list on page 18. i As w is always greater than w' and is always less than t', the sign of the west difference is always + and of the east difference is always , so that when the differences are taken vertically, the resulting sign of the level correction will at once be apparent, as shown in the following example: West East d d 62. 20. 17.7 S9.S +44.3 -39.5 +4.8 s These formulae are exact only in case the angle of the level wyes is the same as the angle of the supporting wyes. 24 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. INCOMPLETE TRANSITS WITH TRANSIT-MICROMETER. If the transit of a star observed with the transit-micrometer is incomplete, only the obser- vations which are symmetrical with regard to the mean position of the micrometer wire are used and those for wliich the symmetrical observations are lacking are rejected. (See General Instructions for Longitude Determinations, p. 79.) Incomplete transits by other methods of observing are utilized by a method of reduction shown on page 32. CORRECTION FOR RATE. If the chronometer rate is not zero, the chronometer correction changes during the progress of the time set. To reduce each observed time of transit across the mean line to what it would have been had the rate been zero (and the correction equal to that which actually existed at the mean epoch of the set) apply the following correction : R=(t-T )r h in which t is the chronometer time of transit of a star, T is the mean epoch of the time set, that is, the mean of ah 1 the chronometer times of transit, and r h is the hourly rate of the chronometer on sidereal time, + when losing and -- when gaining. The quantity (t T ) is expressed in hours. The above is the correction as applied to the observed time of transit of the star; applied to a t, the sign is reversed. The correction for rate may be looked upon as a refinement which is not always essential. If a time set has perfect symmetry of arrangement, the effect of introducing a rate correction into the computation will be shown only in the residuals, as it will have no effect on the com- puted clock correction. If the daily rate of the chronometer is less than five seconds, it can be ignored in the computation of all time sets except those in which one of the half sets contains many more or less stars than the other, or in which one of the half sets extends over a very much longer period of time than the other. In all cases where the rate is greater than five seconds per day it should be considered, and it should be omitted only after a preliminary test shows its effect on the chronometer correction to be negligible. CORRECTION FOR DIURNAL ABERRATION. The effect of the annual aberration due to the motion of the earth in its orbit is taken into account in computing apparent star places and need not be considered here. The correction for diurnal aberration to be applied to an observed tune of transit across the meridian is K=0 8 .021 cos < sec This correction may be obtained easily by the graphical device shown in illustration No. 9 and described on page 61, but it is also given in the following table. It is minus for all stars observed at upper culmination and plus for stars observed at lower culmination. Table of diurnal aberration (K). Latitude Declination-,? -* 0" 10 20 30 40 50 60 70 75 80 85 S s S S S S S S S * * 0.02 0.02 0.02 0.02 0.03 0.03 0.04 0.06 0.08 0.12 0.24 10 .02 .02 .02 .02 .03 .03 .04 .06 .08 .12 .24 20 .02 .02 .02 .02 .03 03 .04 .06 .08 .11 .23 30 .02 .02 .02 .02 .02 .03 .04 .05 .07 .10 .21 40 .02 .02 .02 .02 .02 .03 .03 .05 .06 .09 .18 50 .01 .01 .01 .02 .02 .02 .03 .04 .05 .08 .15 60 .01 .01 .01 .01 .01 .02 .02 .03 .04 .06 .12 70 .01 .01 .01 .01 .01 .01 .01 .02 .03 .04 .08 80 .00 .00 .00 .00 .00 .01 .01 .01 .01 .02 .04 DETERMINATION OF TIME. 25 DERIVATION OF (-<) The correction for diurnal aberration, inclination of axis, and rate (if considered) being applied to the observed time of transit across the mean position of the micrometer wire (or mean line of the diaphragm) as shown in the computation on pages 21-22, the result ist, an approxi- mate time of transit across the meridian. The apparent right ascension at the time of observa- tion is taken from some star catalogue, giving apparent places, such as the American Ephemeris and Nautical Almanac or the Berliner Astronomisches Jahrbuch (pieferably the former) The difference between t and the right ascension, a, of the star at the time of observation, is (ac t). an approximate correction to the chronometer time. In taking right ascensions from the star catalogue it is necessary to interpolate for the longitude of the observer, and to consider second differences when they affect the result by as much as a hundred tli of a second. THE COLLIMATION CORRECTION. If the instrument is otherwise in perfect adjustment, but has a small error in collimation, the micrometer wire in its mean position (or the mean line of the diaphragm) will describe a small circle parallel to the meridian and at an angular distance, the error of collimation, from it, when the telescope is rotated about its horizontal axis. The collimation correction = c sec o = Cc, in which c is the angle, expressed in seconds of time, between the line of sight defined by the micrometer wire when in its mean position (or by the mean line of the diaphragm) and a plane perpendicular to the horizontal axis of the telescope. In other words, c is the angle between the line of collimation and the collimation axis. (See p. 13.) It is considered positive for a given telescope if the line of sight is too far east (and stars at upper culmination are therefore observed too soon) when the illumination (or bright band) is to the westward. This convention of sign is purely arbitrary, however, c is derived from the time computations by one of the processes shown on pages 26, 34, and 42. The factor C is written for sec d and is tabulated on pages 62-77. It is more easily obtained from the graphical device shown in illustration No. 9 and described on page 61. For observa- tions made with illumination (or band) to the westward C is to be considered positive for stars at upper culmination and negative for stars at lower culmination. The signs are reversed with illumination (or band) east. THE AZIMUTH CORRECTION. If the instrument is otherwise in adjustment, but has a small error in azimuth, the microme- ter wire in its mean position (or the mean line of the diaphragm) will describe a vertical circle on the celestial sphere at an angle with the meridian. The correction in seconds to an observed time of transit for this azimuth error is, Azimuth correction = a sin sec d = Aa, in which a is the angle expressed in seconds of time between the meridian and the vertical circle described by the mean position of the micrometer wire. 1 It is considered positive when the collimation axis is too far to the east with the telescope pointed south. For convenience A is written for sin sec 3 and will be found tabulated on pages 62-77. It can be more easily obtained with the graphical device shown in illustration No. 9 and described on page 61. The factor A is considered positive for all stars except those between the zenith and the pole. ' In practice there always exists an error of collimation, so in general a is tha angle between the meridian and the axis of collimation. 26 TJ. S. COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 14. a is derived from the observations by one of the processes shown on pages 26, 34, 39, and 42, attention being paid to sign as indicated above. COMPUTATION OF AT, c, AND a WITHOUT LEAST SQUARES. The following method of computation was devised shortly after the tune (1905) the transit- micrometer was adopted by this survey for use on longitude work and it is used both in the field and in the office for the final computation of ah 1 tune observations made with the transit microme- ter at stations in latitude less than 50. In all latitudes greater than 50 the least-square solution is used in obtaining the final results. There is also a somewhat different method of computation (shown on p. 34) used when the stars of a time set consist of four time stars and one azimuth star. This method was used in the field for a number of years. Form 256.* Computation of time set. [Station, Key West, Florida. Date, Feb. 14, 1907. Set,2. Observer, 3. S. Hill. Computer, J. S. Hill.] Star 1. S Monocer. 2. 5 Aurigae 3. 18 Monocer. 4. 6 Geminor. 5. Geminor. 6. 63 Aurigae 7. t Geminor. 8. j9 Can. Min. 9. a Can. Min. 10. /? Geminor. 11. ic Geminor. 12. Geminor. Clamp W w W w w w E E E E E E s +15.00 +15. 08 +15. 04 + 15.03 +15.00 +15. 02 + 14.43 +14. 45 + 14.45 +14.41 +14.42 +14.47 0.00 +0.08 +0.04 +0.03 0.00 +0.02 -0.57 -0.55 -0.55 -0.59 -0.58 -0.53 + 1.02 +1.38 + 1.01 + 1.21 +1.07 +1.30 -1.13 -1.02 -1.01 -1.13 -1.21 -1.12 +0.26 -0.45 +0.37 -0.20 +0.07 -0.34 -0.07 +0.28 +0.33 -0.08 -0.19 -0.05 Cc s +0.27 +0.36 +0.26 +0.32 +0.28 +0.34 -0.30 -0.27 -0.26 -0.30 -0.32 -0.29 Aa +0.02 -0.03 +0.03 -0.01 0.00 -0.02 0.00 +0.01 +0.01 0.00 -0.01 0.00 (a-0- Cc-Aa Mean AT= +14. 71 + 14.75 + 14.75 + 14.72 + 14.72 +14. 70 +14. 73 +14.71 +14. 70 +14.71 + 14.75 + 14.76 .727 1. 3.00 (M+3. 10 c+0. 70 o w -0.04=0 2. 3.00 O U< . UU 18.00 I 1_. BO 3.30 57.30 X 1.11 12.30 3.20 48.00 6.60 14 22 03. 60 4.20 38.70 2.95 - - 1.41 15.20 3.40 51.80 6.60 07.80 4.20 49.50 2.65 14 06 02. 90 17.70 3.20 55.25 6.40 11.95 4.25 14 29 00. 05 3.20 46.17 01.63 6.95 33.29 3.20 47.14 1.55 56.55 72.10 .00 - .01 - .01 - .02 - .03 - .02 - .02 - .02 - .03 - .06 + .11 + .11 + .15 + .17 + .35 14 05 46.26 14 11 01.71 14 12 33. 41 14 21 47.26 14 27 56. 81 14 05 42.32 14 10 57. 90 14 12 29. 18 14 21 42. 97 14 27 51.37 -3.94 -3.81 -4.23 -4.29 -5.44 32 U. S. COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 14. REDUCTION OF INCOMPLETE TRANSITS. If the transit of a star across every line of the diaphragm is observed, the mean of the times is the required time of transit across the mean line. In obtaining the sum of the several observed times any gross error in any one of the times may be detected by using the auxiliary sums, shown in the example on pages 30-31, in the little column just after the observed times, namely, the sum of the first and last times, of the second and last but one, third and last but two, etc. These auxiliary sums should be nearly the same and nearly equal to double the time on the middle line. This is also a convenient method of taking means, as it is in general only necessary to sum the decimal columns. When the star was observed on some of the lines but missed upon the others, the time of transit over the mean of all the lines may be found as follows: t m = mean of observed times (sum of equatorial intervals of observed lines) (sec number of observed lines. or (sum of equatorial intervals of missed lines) (sec S) = mean of observed times + - number ofobserved line^T The first of these formulae is the more convenient if but few lines were observed and the second the more convenient if but few lines were missed. The two incomplete transits shown in the example on pages 30-31 were reduced by the second formula. t m is the time of transit across the mean of all the lines of the diaphragm. The equatorial interval of a given line is the time which would elapse between the transit of an equatorial star over the mean line of the diaphragm and the transit over the line in question. It is, in seconds of time, ^ the angular interval between the lines expressed in seconds of arc. An equatorial interval is called positive when the transit across the line in question occurs later than the transit across the mean line. The signs of all the equatorial intervals are therefore reversed when the horizontal axis of the telescope is reversed. For an example of the method of computing the equatorial intervals see page 44. The above formulae for reduction to the mean line are approximate, and the maximum possible error of the approximation increases with an increase in the declination of the star and with an increase in the equatorial intervals of the extreme lines. If the extreme equatorial interval is 60 s , the maximum error is less than 8 .01 for a star of which hic observations. Number o'lines For large portable transits For small portable transits JV-25 JV=17 N- 13 ff~n 2V- 15 W=13 jy-ii A T =9 P VP P VP" P VP p VP~ P VP" P VP p VP~ P VP 1 .41 .64 .42 .65 .43 .66 .44 .66 .38 .62 .38 .62 .39 .63 .41 .64 2 .59 .77 .61 .78 .62 .79 .64 .80 .56 .75 .58 .76 .59 .77 .61 .78 3 .69 .83 .71 .84 .73 .86 .75 .86 .68 .83 .69 .83 .71 .84 .73 .86 4 .76 .87 .78 .88 .80 .90 .82 .90 .75 .87 .77 .88 .79 .89 .82 .90 5 .81 .90 .83 .91 .85 .92 .87 .93 .81 .90 .82 .91 .84 .92 .87 .93 6 .84 .91 .86 .93 .89 .94 .90 .95 .85 .92 .87 .93 .89 .94 .92 .96 7 .87 .93 .89 .94 .91 .96 .93 .97 .88 .94 .90 .95 .92 .96 .95 .97 8 .89 .94 .91 .95 .94 .97 .96 .98 .91 .95 .92 .96 .95 .97 .98 .99 9 .90 .95 .93 .96 .95 .98 .97 .99 .92 .96 .94 .97 .97 .98 1.00 1.00 10 .92 .96 .94 .97 .97 .98 .99 .99 .94 .97 .96 .98 .99 .99 11 .93 .96 .95 .98 .98 .99 1.00 1.00 .96 .98 .98 .99 1.00 1.00 12 .94 .97 .96 .98 .99 1.00 .97 .99 .99 1.00 13 .95 .97 .97 .99 1.00 1.00 .98 .99 1.00 1.00 14 .96 .97 .98 .99 .99 1.00 15 .96 .98 .99 1.00 1.00 1.00 16 .97 .98 1.00 1.00 17 .97 .98 1.00 1.00 18 .98 .99 19 .98 .99 20 .98 .99 21 .99 .99 22 .99 1.00 23 .99 1.00 24 1.00 1.00 25 1.00 1.00 RELATIVE WEIGHTS TO TRANSITS DEPENDING ON THE STAR'S DECLINATION. The following tables of the probable error (e) of an observation of a transit of a star over a single line have been derived from a discussion of 1047 transits taken in February and March, 1869, at San Francisco, by Assistant G. Davidson, with the large transit C. S. No. 3 (aperture 2f inches, magnifying power 85); and 875 transits taken about the same time at Cambridge by Assistant A. T. Mosman, including some observations by Subassistant F. Blake, with the large transit C. S. No. 5 (aperture 2| inches, magnifying power 100). For the discussion of obser- vations with a smaller instrument, 330 transits were used, taken in September, October, and November, 1871, at Cleveland, Ohio; and 585 transits, taken in December and January, 1871-72, at Falmouth, Ky., by Assistant E. Goodfellow, with a meridian telescope C. S. No. 13 (aperture If inches, magnifying power about 70). Transit No. 3 Transit No. 5 Meridian telescope No. 13 Meridian telescope No. 13 W S () 3 () a () s o s o s s 87.2 0.74 86.9 0.66 81.9 0.62 76.3 0.20 86.6 0.49 80.0 0.20 76.9 0.18 68.2 0.16 83.0 0.38 76.3 0.19 67.4 0.11 55.8 0.13 81.0 0.31 72.6 0.12 62.0 0.14 48.4 0.15 68.4 0.12 68.8 0.11 55.8 0.09 23.2 0.102 62.9 0.088 3.2 0.066 44.8 0.088 20.4 0.089 48 6 075 29 7 067 170 01 1 n 28.5 0.058 7 071 6 1 OAQA 7.8 0.060 DETERMINATION OF TIME. These tabular values are fairly represented by the expressions Transit, No. 3 0) = V(0.060) 2 +(0.036) 2 tan 2 d 39 Transit, No. 5 ( )=V(0-066) 2 +(0.036) 2 tan 2 d Meridian telescope, No. 13 ( )=V(0.069) 2 +(0.078) 2 tan 2 3 Meridian telescope, No. 13 (s)=V(0.087) 2 +(0.055) 2 tan 2 8 Combining these expressions for the larger and smaller instruments, we obtain (e) = V(0.063) 2 +(0.036) 2 tan 2 J and (e) = V(0.080) 3 + (0.063) 2 tan 2 d respectively, 1 from which the following tables of probable errors (s), of relative weights p, and of the multipliers -^Jp for the conditional equations, have been computed: Table of weights to transits depending on the star's declination. For large portable transits For small portable transits () P Jp w p VP / s s " 0.06 1 1 0.08 1 1 10 .06 1 1 .08 0.98 1 20 .06 0.98 1 .08 .92 0.96 30 .07 .91 0.95 .09 .83 .91 40 .07 .82 .90 .10 .70 .83 45 .07 .76 .87 .10 .62 .79 50 .08 .69 .83 .11 .53 .73 55 .08 .61 .78 .12 .44 .66 60 .09 .51 .71 .14 .34 .59 65 .10 .40 .63 .16 .26 .51 70 .12 .29 .54 .19 .18 .42 75 .15 .18 .43 .25 .10 .32 80 .21 .09 .30 .37 .05 .22 85 .42 .02 .15 .72 .01 .11 d Ursse Minoris 86 37 0.61 0.011 0.103 1.1 0.006 0.075 51 Cephei 87 12 0.74 0.007 0.085 1.3 0.004 0.062 ft Ursse Minoris 88 46 1.7 0.001 0.037 2.9 0.001 0.027 A Ursse Minoris 88 59 2.0 0.001 0.031 3.5 0.001 0.023 COMPUTATION OF AT AND a BY LEAST SQUARES. A field computation made by the approximate method indicated on page 34 gives values for d T, a, and c, which are of a high degree of accuracy. It should be noted that the derived values of a and c depend upon all the observations and not simply upon observations on a few stars only of the set, as is frequently the case with other approximate methods. Experience shows that the value of c especially, as thus derived in the field computation, is so accurate that a value derived from a subsequent rigid least square adjustment will in general be sub- stantially identical with it, provided the stars of the set are chosen as indicated on pages 34 and 43. Accordingly, in the final computations by this method, only the unknowns a w , a E , and A T are to be determined by least squares, while c is taken from the field computations, revised and corrected if necessary. This method of computation is shown below. Let Ate = (a t} Cc in which t is the chronometer time of transit across the mean line of the diaphragm corrected for rate, diurnal aberration and inclination and ct t is therefore the 1 The following formula has been published by Dr. Albrecht on p. 23 of his Formeln und Hiilfstafeln, etc., Leipzig, 1894, viz: d)=y (0.05)+ sec* ) Putting v= 85 for the magnifying power and changing sec into tan, this expression is equivalent to (e) V(0-062)+(0.037) s tan" ) 40 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. quantity on the last line of the field record and computation as shown on pages 30-31. Let At be an assumed value of the chronometer correction and dt a correction to At to be derived from the computation. The final value of the chronometer correction will then be AT=At + dt. Let d, for each star=Ji c At. Then for each star observed an observation equation of the form Vp St + -JpAa = V? d, may be written, in which the weights p are assigned according to the tables on pages 38-39. In forming the normal equations each half set, made with the horizontal axis in one posi- tion, is treated independently of the other half set. The normal equations corresponding to the half set made with illumination (or bright band) to the westward are Ipdt + IpAa w = Ipd IpAdt + IpA^ = Ip Ad and similarly for the other half set. The most convenient arrangement of this computation is shown below, this example being a computation of the time set treated on pages 29-31 and 34. WASHINGTON, D. C., May 17, 1896. c=+.032 J(=-4 S .01 Star Band tt-( C Cc J(c d A P* pA pA* pd pAd Aa AT J pA pffi 17 H.Can.Ven. W -4.07 +1.26 +.04 -4.11 - .10 + .02 .83 +.02. . 00 -.08 .00 + .01 -4.12 + .10 +.08 .0083 13 Urs. Maj. W -4.09 + 1.55 + .05 -4.14 - .13 - .30 .69 -.21 .06 -.09 +.03 - .18 -3.96 -.06 -.04 25 jj Bootis W -3.69 +1.06 + .03 -3.72 + .29 + .36 .98 +.35 .13 + .28 +.10 + .22 -3.94 -.08 -.08 63 II Bootis W -3.89 +1.13 +.04 -3.93 + .08 + .22 .93 +.20 .04 + .07 +.02 + .13 -4.06 + .04 + .04 15 a Draconis W -4.52 +2.36 + .08 -4.60 - .59 -1.03 .40 -.41 .42 -.24 +.24 - .62 -3.98 -.04 -.02 06 3.83 -.05 .65 -.06 + .39 04 d Bootis E -3.94 -1.11 -.04 -3.90 + .11 + .25 .93 + .23 .06 + .10 +.03 + .14 -4.04 +.02 + .02 a Bootis E -3.81 -1.06 -.03 -3.78 + .23 + .35 .98 + .34 .12 + .23 + .08 + .19 -3.97 -.05 -.05 25 A Bootis E -4.23 -1.46 -.05 -4.18 - .17 - .19 .74 -.14 .03 -.13 + .02 - .10 -4.08 + .06 + .04 27 e Bootis E -4.29 -1.64 -.05 -4.24 - .23 - .38 .65 -.25 .09 -.15 + .06 - .21 -4.03 + .01 + .01 01 5 Urs. Min. E -5.44 -4.18 -.13 -5.31 -1.30 -2.53 .16 -.40 1.02 -.21 + .53 -1.37 -3.94 -.08 -.01 10 3.46 -.22 1.32 -.16 +.72 .0259 * These weights are taken from the column headed " For large portable transits " in the table on p. 39. Normal equations: +3.83 d t-.Oo a w =- .06 - 05d*+.65a w = + .39 a w =+.601 +3.46 St- .22a E =- .16 - .22 St+1.32 a E = + .72 a E =+.543 3t=-'.012 +7.29 Q- .27 7=1 - .27 Q+1.97 q=0 At 14 h 02 m JT=-4 3 .020 Q=0.138 [='.044 =-!-s.016 In the above computation a check on the correctness of the assumed value of c is furnished by the nearness of agreement of the two values of dt resulting from the two groups of stars. The normal equations are solved most conveniently by successive approximations, as, for DETERMINATION OF TIME. 41 instance, in the second equation the value of a w can be closely derived at once on the assumption that dt is small. The residuals (J) are taken for each group separately, using its own dt 1 to derive a A T for this purpose, and the sums of the pJ's should of course nearly equal zero for each set. The probable error of a single observation of unit weight is ., = 0.674^1 ^^ \ n - where 2pJ 2 is the sum of the weighted squares of the residuals (last column in form), n is the number of stars and n e is the number of unknown quantities or number of normal equations, remembering in this example that there are four unknowns, dt, a w , a E , and c, the latter being taken from the field computation. To obtain the probable error of the computed AT, add the corresponding normal equations of the two sets, put Q in place of dt, g in place of a, 1 in place of 2pd, and in place of 2pAd, as shown. Then = e^Q. THE COMPLETE LEAST SQUARE COMPUTATION. When time observations are taken in Alaska unusual conditions are encountered, arising from the high latitude of the station from 55 to 65 for the regions in which the Survey observers are called upon to observe most frequently. Zenith stars are there slow-moving stars (and consequently have small weights) ; for stars between the zenith and the pole pA is com- paratively small; the rapidly moving stars are far to the southward of the zenith, and it is easy to observe subpolars, as the northern horizon is far below the pole. Moreover the very prevalent cloudy weather is apt to break in . upon any previously arranged program. The combined result of these conditions is in general that the sets of stars actually observed are poorly balanced; that is, the algebraic sum of the A factors for each half set and of the C factors for the whole set will differ considerably from zero. In extreme cases it is sometimes desirable to resort to the complete least square computation in which c, a w , a E , and AT are all derived by the principle of least squares. We here start with a t (as shown on pp. 30-31), and the remaining notation stands as on page 40, except that we must here distinguish by the subscripts w and E between A factors belong- ing to the two half sets. An observation equation of one of the following forms may be written for each star observed: Jpdt + -JpA a E -Jpdt The normal equations will be IpCdt + IpA E Ca E The following will serve as a concrete illustration of this method of computation. The only preliminary assumption in this computation is an approximate value of the chronometer correc- tion, At. Owing to the high latitude of St. Michael, 63 29', the time stars are all south of the zenith, and the average value of A is far from zero. 1 Tile two 3t's here happen to be so nearly equal that J's are the same as if taken by using the J T for the whole group. 42 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. ST. MICHAEL, ALASKA, March 19, 1891. J<= -20.10. Star Clamp a-t d A C P pA pC pA* pAC pC' pi pAd pCd Aa Cc AT J pA T>f s s 1 E -21.27 -1.17 + .66 -1.13 0.9 + .59 -1.02 .39 - .67 1.15 -1.05 - .69 + 1.-19 .89 - .21 -20.17 + .05 +.04 .0022 2 E -21.22 1.12 + .72 -1.08 0.9 + .65 - .97 .47 - .70 1.05 -1.01 - .73 + 1.09 - .97 - .20 -20. 05 -.07 -.06 44 3 E -21.40 -1.30 + .76 -1.05 0.9 + .68 - .94 .52 - .72 .99 -1.17 - .89 + 1.23 -1.02 - .19 -20. 19 + .07 + .06 44 4 E -23.09 -2.99 +2.89 +4.58 0.08 + .23 + .37 .66 +1.06 1.68 - .24 - .69 -1.10 -3.88 + .84 -20.05 -.07 -.01 04 5 E -21.23 -1.13 + .73 -1.07 0.9 + .66 - .96 .48 - .70 1.03 -1.02 - .74 + 1.09 - .98 - .20 -20.05 -.07 -.06 44 +2.81 2.52 -1.73 -3.74 01 6 W j-20.98 -0.88 + .85 +1.01 1.0 + .85 + 1.01 .72 + .86 1.02- .88 - .75 - .89 -1.05 + .18 -20.11 -.01 -.01 7 W -20.86 -0.76 + .72 + 1.08 0.9 + .65+ .97 .47 + .70 1.08 - . 68 - .49 - .73 - .89 + .20 20.17 +.05 +.04 22 8 W -20.70 -0.60 + .64 + 1.14 0.9 + .58+1.03 .37 + .66 1.17 - .54 - .35 - .62 - .79 + .21 -20.12 .00 .00 00 9 W -20.95 -0.85 + .85 + 1.01 1.0 + .85 + 1.01 .72 + .86 1.02 -.85 - .72 - .86 -1.05 + .18 -20.08 -.04 -.04 16 10 W -25.39 -5.29 +3.46 -5.83 0.05 + .17 - .29 .60 -1.01 1.70 - .26 - .92 + 1.54 -4.27 -1.07 -20.05 -.07 .00 02 7.53 +3.10 +0.21 2.88 +2.07 11.86 -7.70 -3.23 + 1.94 0199 Normal equations: +7. 53 St +2. 81 a E +3. 10 a w + 0. 21 c =-7. 70 +2. 81 St +2. 52 at upper culmination For stars witlu'n 10 of the pole (as for d Urs. Min., 51 Cephei, Polaris, and A Urs. Min.) use the formulae: ij = (<, t m ) cos d -/ cos TJ etc. ^___ ^n = ( 4 Level 4 = $W Level 4 = Pc = P W. end E.end W.end E.end h m Of div div div div div div div 38 9 43 a. m. 33 33.5 22.0 33.4 21.7 20.8 34.8 - .625 21.0 34.0 -.325 +.075 43 20.4 33.9 21.0 34.0 32.4 21.9 - .750 33.1 21.8 -.425 +.081 48 20.2 33.9 20.3 33.4 32.2 21.9 - .850 32. 1 21.8 -.700 +.038 43 31.8 21.9 32.7 21.1 19.7 33.9 -1.075 20.1 33.3 -.400 +. 169 38 10 03 a. m. 35 19.7 33.8 20.1 33.1 31.9 21.3 - .875 32.0 21.1 -.525 +.088 Mean, band west, object glass south, and band east, object +.090 glass north Band west Rand east Object glass north Object glass south Zenith distance Time Temper- ature sw-le Sw-Se Pl-Pw 4 Level 4 =ffu> Level 4 t = P W.end E.end W.end E.end o h m op div div div div div div div 38 10 07 a. m. 35 19.7 33.1 19.4 33.6 31.9 20.9 -.600 31.9 20.9 -.800 -.050 43 31.9 20 9 31.7 20.9 19.1 33.3 -.800 19.1 33.2 -.825 -.006 48 19.3 33.0 19.1 33.3 31.5 20.9 -.775 31.7 20.9 -.850 -.019 43 31.3 20.9 31.1 21.0 19.0 33.2 -.950 18.9 33.2 -1.050 -.025 33 10 27 a. m. 36 19.0 33.1 18.8 33.7 31.7 20.5 -.725 31.2 20.9 -1. 150 -.106 Mean, band west, object glass -.041 north, and band east, object glass south Mean, band west, object glass +.090 south, and band east, object glass north Mean +.024 1 division of striding level=l // .850=OM23 p= + .024 div.=OM23X.024=+0.003 sec- ond of time 46 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. In determining the pivot inequality the level readings are made as in observing time, reversing the telescope between the readings. Observations should be made in two groups, reversing the relation between the positions of the band and object glass as shown in the example. This is done to partially eliminate the effect of the pivots not being truly circular in cross section. In the example shown there is a systematic though unimportant difference in p for the two positions A complete investigation of the pivots would involve level readings at all angles from the zenith, from to 90, but the ordinary form of level will not permit readings closer than 30 or 40, and stars are not often observed more than 50 from the zenith. In the example; given the observations were from 38 to 48 zenith distance, less weight being given to the latter angle at which few star observations are made. A less satisfactory value for the pivot inequality may be obtained from the level readings made in connection with the time observations. Since the correction for pivot inequality has opposite signs for the two halves of a time set, its effect on the determined clock correction is very small for a set which has the same number of stars in each half. The question of when the pivot inequality correction is to be applied and when not, should be decided after a consideration of the absolute value of the correction but the difference in the sums of the B factors for the two half sets should also be considered. Most of the instruments used at present in this Survey have had their pivots refinished and their pivot inequality made practically zero. With these instruments it is not usually necessary to consider this correction when making the computations for time. DETERMINATION OF LEVEL VALUE. The most accurate way of determining the value of one division of a level is by means of a level-trier, wliich consists of a bar the support of which at one end is a micrometer screw. The level tube to be tested is placed on this bar. The method of observing and computing is shown in the following example. In the level-trier used one division of the micrometer head equals one second of arc; that is, a movement of one division changes the angular position of the bar by one second. The first part of these observations was simply for the purpose of test- ing the uniformity of the tube, changing the angle by 5" intervals. In determining the level value about the same length of bubble is employed that is used in the field observations. DETERMINATION OF TIME. 47 Determination of value of one division of stride level of meridian telescope No. 9. Chamber vial 175 mm. by 15 mm., marked 7526, 2" .02 K. and E., mounted by springs. Length of bubble used, 35 div. = 70 mm. E. G. F., observer. Mean temperature, 12. 3 C. Chamber left Chamber right Bubble reading Movement Bubble reading Movement Level- trier reading Value of one divi- sion of level Level- trier reading Value of one divi- sion of level Left end TUsht end Level- trier Bubble. Mean of two ends Left end Right end Level- trier Bubble. Mean of two ends // div div // div // // div div // div // 25 -0.1 35.2 75 GO. 4 25.8 30 35 40 45 50 2.4 4.9 7.4 10.1 12.7 37.7 40.2 42.7 45.4 48.0 5 5 5 5 2.5 2.5 2.7 2.6 80 85 90 95 100 57.7 55.3 52.9 50.2 47.5 23. 1 20.7 18.3 15.6 12.9 5 5 5 5 2.4 2.4 2.7 2.7 55 15.3 50.6 * O ft 105 44.9 10.3 K o 7 60 65 70 75 17.9 20.3 22.9 25.5 53.2 55. 6 58.2 60.8 5 5 5 2.4 2.6 2.6 110 115 120 125 42. 2 39^6 37.0 34.5 7.6 5.0 2.4 -0.1 5 5 5 2.6 2.6 2.5 25 75 -0.2 25.5 35.0 60.7 50 25.7 1.945 75 125 60.9 34.6 26.3 0.0 26.3 1.901 35 65 4.7 20.5 39.9- 55.7 30 15.8 1.899 85 115 55.9 39.8 21. 2 5.1 16.1 1.863 40 CO 7.4 17.9 42 6 53.1 20 10.5 1.905 90 110 53.2 42.4 18.5 7. 7 20 10.8 1. 852 45 55 10.1 15.4 45.3 50.6 10 5.3 1.887 95 105 50.4 44.9 15.8 10.3 10 5.5 1.818 Mean , chamber left 1 909 ! Mean chamb ?r ric;ht 1. 859 Final mean 1 div =2 mm . = 1 /X .SJ !4 at 12 .30 If the level vial is so held in its metallic mounting that there is any possibility that it may be put under stress by a change of temperature, it is advisable to determine the value of a division with the tube in its mounting at two or more widely different temperatures. Level vials are now usually mounted witli springs, so as to avoid such stresses. If an observer is forced to determine the value of a level division in the field, remote from a level-trier -after some accident, for example he must devise some method of utilizing what- ever apparatus is at Ids disposal for that purpose. If a telescope having an eyepiece micrometer fitted for measuring altitudes or zenith dis- tances is available, the unknown angular value of a level division may be found by comparison with the known angular value of a division of the micrometer. Place the level in an extempo- rized mounting fixed to the telescope so that the level vial is parallel to the plane in winch the telescope rotates (about its horizontal axis). Point with the micrometer upon some distant well-defined fixed object and read the micrometer and level. Change the micrometer reading by an integral number of divisions, point to the same object again by a movement of the tele- scope as a whole, and note the new reading of the level. Every repetition of tin's process gives a determination of the level value in terms of the micrometer value. If another level of sufficient sensibility and of which the value is well known is available, it may be used as a standard with which to compare the unknown level. Put the unknown level in an extemporized mounting, fastened to that of the known level in such a way that the two level vials are parallel or nearly so. Adjust so that both bubbles are near the middle at once. Compare corresponding movements of the two bubbles for small changes of inclination common to the two levels. 48 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. DISCUSSION OF ERRORS. The various errors which affect the final result of any astronomic observation may be grouped into three separate classes with respect to their sources, and consequently the pre- cautions which must be taken against them fall under the same general heads. They are: (1) External errors, or errors arising from conditions outside the observer; (2) instrumental errors, due to the instrument, and arising from imperfect construction 1 or imperfect condition of the instrument, from instability of the relative positions of the different parts, etc.; (3) observer's errors, due directly to the observer, arising from liis unavoidable errors of judgment as to what he sees and hears and from the fact that nerves and brain do not act instantaneously. By the phrase "Errors of observation" is meant the combined errors arising from all these sources. The principal external errors in transit observations for time arise from errors in the assumed right ascensions of the stars and from lateral refraction of the light from the stars. If the right ascensions of all stars observed are taken from the American Ephemeris and Nautical Almanac or the Berliner Astronomisches Jahrbuch, the probable error of a right ascension will be upon an average about 0. S 03, except for stars of large declination, for which this estimate must be increased. The right ascensions are subject also to small constant errors with which the geodesist is hardly concerned, because of their smallness and because they are almost completely eliminated from Ms final results. When the same stars are used at both stations in determining a difference of longitude the errors of the right ascensions are com- pletely eliminated from the determined difference of longitude. If one considers how small are the lateral refractions which affect measurements of hori- zontal angles and azimuth observations, in which lines of sight are close to the ground, it seems certain that the effects of lateral refraction upon transit time observations in which all lines of sight are elevated high above the horizon must be almost or quite inappreciable. Tin's is probably the case whenever proper precautions are taken to avoid local refraction within a few feet of the instrument. If, however, the temperature within the observatory is much above that outside, or if active chimneys or other powerful sources of heat are near the observatory, warm columns of air rising from or passing over the observatory may produce a sensible lateral refraction. The lateral refraction is included, with many other errors from wliich it can not be separated, in the culmination error, (s,), estimated on pages 38-39. In addition to the lateral refraction referred to in the preceding paragraph and tacitly assumed to be constant during the interval of a few seconds in wliich a star is being observed upon, there are usually momentary lateral refractions which serve merely to make the apparent rate of progress of the star variable and to make the observer's errors greater than they other- wise would be. Among the instrumental errors in transit observations for time may be mentioned those arising from the chronograph and the reading of the chronograph sheet, from poor focusing, from nonverticality of the micrometer wire or of the lines of the diaphragm, from changes in azimuth and colhmation, from errors in the measured collimation, from errors in the measured inclination, from irregularity of pivots, and from changes in the rate of the chronometer. All of these except the first two are included in the culmination error, (s^, as estimated on pages 38 and 39. As already noted the chronographs of the form now used operate so well that no appreci- able error is introduced by the assumption that the speed of the chronograph is constant between successive breaks of the chronometer. The chronograph sheet is read to hundredths of seconds for the exchange of arbitrary signals between stations in telegraphic longitude work. In observations made with an observing key, marking the times of transit across the lines of a diaphragm, the chronograph record of the observations is read for each line to the nearest 0. 8 05. ' By imperfect construction is here meant the failure to satisfy fully the rigid geometric conditions imposed by theory, but necessarily attained out imperfectly by the instrument maker, as, for example, the condition that the cross section of a pivot should be a perfect circle and remain so. Imperfect construction is therefore not meant to imply poor construction, that is, construction much below the attainable degree of excellence. DETERMINATION OF TIME. 49 By so doing, a probable error of about 0. S 01 on each single line is introduced into the readings; but this is too small in comparison with the other errors concerned in transit work to warrant a closer reading. In observations made with a transit equipped with a transit micrometer, where 20 observations on each star are recorded, the chronograph record of these observations is read to the nearest 0. 8 1. The probable error of a single record (position of micrometer wire) from this source is about 0. S 02, but the number of such records obtained on a star makes the probable error of the mean of these observations less than 0. 8 01, showing that a closer reading of the chronograph sheet is not justifiable. Poor focusing of either the objective or the eyepiece leads to increased accidental errors because of poor definition. But poor focusing of the objective is especially objectionable, because it puts the diaphragm (or plane of the micrometer wire) and the star image in different planes, and so produces parallax. The parallax errors may be avoided to a large extent by keep- ing the eyepiece centered carefully over the part of the diaphragm wliich is being observed upon, if proper longitudinal motion of the eyepiece is provided for that purpose. If the lines of the diaphragm do not make an angle of exactly 90 with the horizontal axis of the telescope a star observed above or below the middle of the diaphragm will be observed too late or too early. A similar error will be caused in the case of the transit micrometer if the movable wire does not, in each of its positions, make an angle of 90 with the horizontal axis. Errors from this source may be made very small by careful adjustment and by observing within the narrow limits given by two horizontal lines or wires. The mean errors of azimuth and of collimation, being determined by the time observations themselves, are canceled out from the final result with a thoroughness which depends upon the success attained in selecting stars. The process of elimination depends upon the assumption that the error of azimuth remains constant during each half set and that the collimation error remains constant during the whole set. The changes in these errors during the intervals named, arising from changes of temperature, shocks to the instrument, or other causes, produce errors in the final result. These errors will evidently be smaller the more rapidly the observations are made, the more carefully the instrument is handled, and the more symmetrical and constant are the temperature conditions. In general, these errors are small but not inappreciable. In this connection the stability of the pier on which the instrument rests is of especial importance, and also the degree to which it is protected from shocks such as, for instance, the observer's walk- ing in its immediate vicinity, if there is no floor to the observatory or tent. It is mainly in the light of the preceding paragraph that the number of stars to be observed in a time set must be determined. If the number of stars hi a tune set and the length of tune over which it extends be increased, the errors due to accumulated changes in the azimuth and collimation are increased. On the other hand, if the number of stars is decreased below the present standard (12) the number of observations rapidly approaches equality with the number of unknowns (4), and the accuracy with which the unknowns are determined decreases very rapidly. From these considerations it would seem that 12 stars per set is about the most advantageous number when the highest degree of accuracy is desired. 1 Under normal condi- tions this number involves the necessity of depending upon the constancy of the instrument in azimuth for about 30 minutes and in collimation for about 1 hour. If greater accuracy is desired than can be obtained from a set of 12 stars, it is necessary to continue observing half sets of 6 stars each, with a reversal of the instrument in its wyes between each two half sets, but the number of stars in a half set should not be materially increased. To a considerable extent the preceding two paragraphs also apply to the inclination error. The changes in inclination during each half set produce errors in addition to those arising from uncertainty as to the mean inclination, hence again the desirability of rapid manipulation. The mean inclination is determined from the indications of the striding level, which are more or less in error. Different observers seem to differ radically as to the probable magnitude of * When only a minor degree of accuracy is desired, the number of stars may, of course, be much less than 12. 8136 13 4 50 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. errors from this source, but the best observers are, prone to use the striding level with peat care. However small this error may be under the best conditions and most skillful manipulations, there can be no doubt that careless handling of the striding level, or a little heedlessness about bringing a warm reading lamp too near it, 1 may easily make this error one of the largest affecting the result. An error of 0.0002 inch in the determination of the difference of elevation of the two pivots of a transit like that shown in illustration No. 1 produces an error of more than s . 1 in the deduced time of transit of a star near the zenith. The method of treating the level readings given on page 22 is based upon two assumptions: First, that the indications of the striding level are not sufficiently accurate to determine the small changes of inclination during the progress of a half set, and, second, that if (as is generally the case) there is any systematic difference between the inclination as defined by level readings with objective northward and with objective southward the mean of these two inclinations is the required most probable value corresponding to intermediate positions of the telescope in which it points to stars near the zenith (time stars). There may be individual cases in which the first of these assumptions should be reversed and each star transit reduced by using the level reading which is nearest to it in time, upon the supposition that the actual changes of incli- nation are so large that the level indications furnish a real measure of them. In general, however,' the method of treating the level readings shown on pages 21-23 is probably the best. The errors in the computed time arising from inequality and irregularity of pivots are prob- ably negligible for first-class instruments in good condition. Any small error in the adopted mean value of the inequality will appear in the computation with nearly its full value in the derived error of collimation, but will be almost completely eliminated from the computed chronometer correction. It is only the difference of the irregularities of the two pivots which affect the observed times, and it should be noted that corresponding points on the two pivots are always under about the same pressure at the same time, and that therefore irregularities due to wear tend to be the same for the two pivots. Changes in the rate of the chronometer during the progress of a set of observations evidently produce errors in the computed chronometer correction at the mean epoch of the set. Under ordinary circumstances such errors must be exceedingly small. If, however, an observer is forced to use a poor timepiece, or if clouds interfere so as to extend the time required to make a set of observations over several hours, this error may become appreciable. The observer's errors are by far the most serious of any class of errors in transit observations for time. The observer is subject to both accidental and constant 2 errors in his observations of the times of transit and in his readings of the striding level. The level reading errors (such as errors in estimating tenths) are inappreciable in their effect upon the computed time, but the errors in observations of time of transit enter into the computed time with full value. The observer's accidental errors are estimated under the heading ''Relative Weights to Transits Depending on the Star's Declination" (pp. 38 and 39). His constant error in estimating the 1 The longitudinal section of the upper inner surface of a level vial is made as nearly a perfect circle as possible. If an observer will consider how great this radius of curvature is in asensitivestridinglevel he will understand why very small deformations of the level vial by unequal changes of temperature have a marked effect upon the position of the bubble. The radius of curvature for a level of which each division is 2mm long and equivalent to 1} seconds of arc is more than 300 m (about 1000 feet). * In discussing errors, and especially when discussing them with reference to their ultimate effects, it is quite important to keep clearly in mind the distinctions between accidental errors, constant errors, and systematic errors. A constant error is one which has the same effect upon all the observations of the series or portion of a series under consideration. Accidental errors are not constant from observation to observation; they are as apt to be minus as plus, and they presumably follow the law of error which is the basis of the theory ofleast squares. A systematic error is one of which the algebraic sign, and, to a certain extent, the magnitude, bears a fixed relation to some condition or set of conditions. Thus, for example, the phase error in observations of horizontal directions is systematic with respect to the azimuth of the sun and of the line of sight. The expression "constant error" is often used loosely in contradistinction to "accidental error," in such a way as to include both strictly constant errors and sys- tematic errors. The effect of accidental errors upon the final result may be diminished by continued repetition of the observations and by the least square method of computation. The effects of constant errors and of systematic errors must be eliminated by other processes; for example, by changing the method or program of observations, by special investigations or special observations designed to evaluate a constant error or to determine the exact law of a systematic error. The above discussion applies with full force, in so far as the observer is directly concerned, to errors arising from imperfect perception or judgment rather than to blunders or mistakes, such as reading a level five divisions wrong or estimating a Urn? one second wrong. If a mistake is so large that it is caught by the checks which are used for that purpose it is usually without effect upon the computed result, since it is either corrected or the observation concerned is rejected. A mistake which is not caught is, in its effect upon the com- puted result, an accidental error and, if proper checks have been used to detect mistakes, will lie within the limits of magnitude of the accidental errors. A similar distinction between instrumental errors and instrumental blunders may be drawn; for example, a blunder rather than error is caused by the movement of an objective which is loose in its cell. DETERMINATION OF TIME. 51 time of transit when observing with a key, or by the eye and ear method, is known as personal equation and may amount to half a second or even a whole second in an extreme case. In observations with a transit micrometer this error if it exists at all is very small and may te neglected. The personal equation, and the methods of measuring it and of eliminating it from the final results, will be treated more fully in connection with longitude determinations. In the same place will be found a discussion of the data which indicate that the personal equation in observations made with a transit micrometer is so small that it may be neglected in longitude work. To sum up, it may be stated that the accidental error in the determination of a chronometer correction from observations with a portable transit instrument upon twelve stars may be reduced within limits indicated by a probable error of from s .01 to MO. However, in observations made without the transit micrometer the chronometer correction may be subject to u large constant error, the observer's absolute personal equation, which may be many times as great as the probable (accidental) error. If the observations have been made with the transit micrometer, there is practically no personal equation, and the results may be considered free from constant errors due to that source. OTHER METHODS OF DETERMINING TIME. In the field it is sometimes necessary to use other instruments as transits for the determi- nation of time. A theodolite, when so used, is apt to give results of a higher degree of accuracy than would be expected from an instrument of its size, unless one has in mind that the princi- pal errors in transit time observations are those due directly to the observer. On the other hand, zenith telescopes of the form in which the telescope does not swing in a plane passing through the vertical axis of the instrument have been found to give disappointing results when iised in the meridian for time, perhaps because of the asymmetry of the instrument and of the fact that there can be no reversal of the horizontal axis in its bearings, but only of the instrument as a whole. The time may, however, be thus determined with sufficient accuracy for use in connection with determinations of latitude with the zenith telescope. 1 The determination of time by the use of the transit in any position out of the meridan has been advocated, but has not seemed advisable. The additional difficulty of making the com- putation, over that for a transit nearly in the meridian, and other incidental inconveniences, much more than offset the fact that the adjustment for putting the transit in the meridian is then unnecessary. The use of the transit in the vertical plane passing through Polaris at the time of observa- tion has been advocated, and has been used to a considerable extent in Europe and in Canada. It is not used by this Survey. The advantage of this method over the meridian method is that the stability of the instrument is depended upon for only about 5 minutes instead of 30 minutes or more. This method is open, though to a less extent, to the objections stated in the preceding paragraph against the method of observing in any position out of the meridian. If a mark nearly in the meridian has been established and its azimuth determined the chronometer correction may be determined at noon within a half second by observing the transit of the sun as follows: Point on the meridian mark just before apparent noon; observe the transit of the preceding limb of the sun across the lines of the diaphragm; reverse the horizontal axis of the telescope and observe the transit of the following limb across the lines of the diaphragm. If the transit micrometer is used, the west limb of the sun is followed across the center of the field by the micrometer wire, and then the telescope is reversed and the east limb is followed by the wire. The record of observations on each limb is recorded automatically on the chronograph. The striding level should be read just before the transit of the preceding limb and just after the transit of the following limb. The mean of all the observed times is the chronometer time of transit of the sun's center across the plane of the instrument. This 1 For methods of determining time witli a zenith telescope by using it as an equal-altitude instrument, see Coast Survey Report for 1869, Appen- dix No. 12, pp. 226-232. 52 U. S. COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 14. time corrected for azimuth error, as determined by the pointing on the meridian mark, and for inclination, is the chronometer time of the sun's transit across the meridian. During the observations the instrument should be sheltered from the direct rays of the sun. This may be done by hanging in front of it a cloth with a hole cut in it opposite the objective. This method of determining time may sometimes be found desirable in connection with chronometric determi- nations of longitude in Alaska when continuous cloudy weather prevents star observations. When setting up a transit at a new station it is sometimes difficult to get a close approxi- mation to the local time with which to make the first setting of the transit in the meridian. The following method has been used to furnish a rough value of the local time, and makes it possible to put the instrument so closely in the meridian on the initial trial that there is almost no time lost from the regular observations. At a Little before local noon commence observing the sun, following it by moving the telescope both in azimuth and altitude. While the sun is still rising appreciably, clamp the telescope in altitude, and mark the time of the transit of the sun's limbs across the horizontal wire of the telescope; then keeping the telescope fixed in altitude swing it slightly in azimuth to meet the descending sun and mark the transit of the sun's limbs across the same wire as before. The mean of the times will be approximately the chronom- eter time of the sun's passage across the local meridian, and the chronometer correction on apparent solar time can be determined, and finally its correction on local sidereal time. With this correction, using an azimuth star first in the final placing of the instrument in azimuth, it will be found that two approximations will usually be all that are required to set the instrument close enough for actual observations. With the meridian telescope form of instrument this method may be easily and accurately followed. Sextant observations for time by measuring the altitude of the sun give sufficiently accurate results for many purposes. 1 For example, the chronometer correction may thus be determined with sufficient accuracy for use in zenith telescope determinations of latitude or in observations for azimuth made upon a circumpolar star within an hour of elongation. If a specially constructed vertical circle 2 is used, illustration No. 8, the time may be determined from observed altitudes of a star or the sun with sufficient accuracy for all purposes in observations for latitude and azimuth. The sun or star should be observed near the prime vertical if possible. This is the method used at present by nearly all the parties of this Survey engaged in latitude and azimuth observations. With time obtained in this way azimuth observations may be made on Polaris at any hour angle. This method is also used by the field parties engaged in making magnetic observations. 3 As this method is so frequently used a sample record of observations and of the computations is given below with such explanations as are necessary. DESCRIPTION OF THE VERTICAL CIRCLE AND ITS ADJUSTMENTS. The vertical circles in use in the Coast and Geodetic Survey are, in general form, like that shown in illustration No. 8. The instrument is practically a theodolite with the graduated circle in a vertical position and the axis horizontal, with the telescope fastened rigidly to the alidade. The circle and alidade are fastened to a horizontal support which rests upon the top of a vertical axis, the latter fitting into a stand. There is a counterpoise to the circle and alidade on the opposite side of the vertical axis. The stand has three leveling screws, and there may be a graduated circle near its base for measuring horizontal angles approximately. 1 For convenient instructions, formulae, and tables for sextant observations for time and other approximate astronomic methods, sec Bowditch's American Practical Navigator, published by the U. S. Navy Department. ' Such an instrument is used in observing vertical angles or zenith distances in primary triangulation. The circles of these instruments are from 8 to 10 inches in diameter and are graduated very accurately. 1 See p. 45, Directions for Magnetic Measurements, Coast and Geodetic Survey. No. 8. VERTICAL CIRCLE. DETERMINATION OF TIME. 53 Before starting observations the usual adjustments of the eyepiece and object glass should be made and the crosswires should be brought approximately into the center of the field. There is no adjustment for collimation in either the vertical or horizontal plane. A coarse stride level is used to make the horizontal axis of the circle truly horizontal and, consequently, the circle vertical, and a sensitive level is placed parallel with and fastened to the circle to define a hori- zontal line through the instrument. If, after leveling by the two levels, the instrument is rotated on its vertical axis through 180 and the bubbles remain on the graduated scales of the level vials then the adjustments for level are satisfactory. TIME FROM OBSERVATIONS ON A STAR WITH A VERTICAL CIRCLE. When making the observations the star's image is brought into the field of the telescope and the telescope clamped with the horizontal wire slightly ahead of the star. As the star crosses the horizontal wire the observer notes the time of the chronometer by the eye-and-ear method, or, at the instant of crossing, he calls "Mark" to the recorder, who notes the chronome- ter time. Readings are made of the bubble of the fixed level and of the verniers of the vertical circle. The telescope is then rotated on its horizontal axis and revolved 180 about the vertical axis of the instrument. A second observation is made on the star and the level and vertical circle are read again. These observations constitute one complete determination of the time. It is advisable to take at least four such sets of observations for the determination of the chro- nometer correction if the results are used for primary azimuth work where Polaris or some other close circumpolar star is observed at any hour angle. If, upon revolving the instrument through 180 in azimuth for the second reading on the star for any one set, it is found that one end of the bubble extends beyond the graduations of the level vial, it may be brought back by the foot screws of the instrument. It should never be brought back to the graduations by moving the tangent screw which controls the relation between the bubble and the graduations of the circle. In other words, the relation between the fixed level and the vertical circle qf the instrument should remain undisturbed during a set. If the level is badly out of adjustment, it should be adjusted between sets. Whenever practicable one. half of the sets of observations should be made on a star in the east and the other half on a west star, both stars being nearly in the prime vertical and at about the same elevation, in order to eliminate instrumental errors and errors due to refraction. The above two paragraphs apply also to observations on the sun, except, of course, the last sentence of the second paragraph. The instrumental and refraction errors may be minimized by observing the sun in the morning and again in the afternoon at about the same angular distance from the meridian. RECORD OF OBSERVATIONS ON STARS. The following record shows four sets of observations with the vertical circle, all on an eastern star. These observations were made in connection with primary azimuth observations at Sears triangulation station in Texas. The azimuth observations and computations are shown on pages 147 to 149 of this publication. It will be noticed that the zenith distances of the star cor- rected for level are computed in the record. 54 Forir. 252. U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Double zenith distances* IStation: Sears triangulation station. Observer: \V. Bowie. State: Texas. County: Jones. Instrument: Vertical eircle No. 46. Date: Dec. 22, 1908.] Object observed Time Level Circle, right orleft Circle read- ing Vernitrs Zenith dis- tance Remarks O E A // 40 50 00 00 30 20 20 20 B ft 60 50 40 20 60 40 60 40 C D* Mean a Tauri a Tauri a Tauri crTaun h m s 1 03 49 1 06 02 5 d 14 1 14 4 d 12 11 8 23.8 11.8 14.3 R L L K R L L R 49 57 50 01 49 49 48 59 48 36 48 47 48 31 47 34 20 60 in 30 10 50 40 20 40.0 53.3 06.7 56. 7 33.3 16.7 40.0 26.7 49 59 46. 6 - 3.0 43.6 49 24 01. 7 0.0 01.7 48 41 55.0 - 5.0 50.0 48 03 03. 4 - 1.5 01.9 Sidereal chronometer No. 1769 was used. Temperature, 5 C. Ba- rometer, 716 ram Value of one division of level bub- ble=2".58 1 04 55.8 1 07 05.0 1 08 28.5 2S.5 -4.7 14.3 11 8 1 07 46.8 1 10 06.5 1 12 00.5 26.1 0.0 16.8 13.4 26.1 09.5 12.9 1 11 03.5 1 13 14.5 1 15 13.0 30.2 -7.8 13.2 14.1 22.4 12.8 12.2 1 14 13. 8 27.3 -2.3 25.0 * Vertical circle No. 46 differs from the usual type of this instrument in use by the Survey in the number of verniers and in the numbering of the graduations of the circle. There are four verniers as a rule, and the circle graduations are generally numbered continuously, so that the differ- ence of the two circle readings, Circle R and Circle L, gives the double zenith distance. No. 46has only three verniers and the verticalcircle gradu- ations arc numbered from to 180 both ways from the zenith. In the column of remarks is given such information as is necessary for the proper inter- pretation of the record by the computer. In this column should also be given notes on any unusual occurrence, such as the jarring of the instrument or the adjustment of the instrument during the period of observations. The above form is bound in books of octavo size, which are furnished to field parties upon request. The level correction, which is shown in the column headed "Level" and is applied to the observed zenith distance in the next to the last column, is computed by the formula: When the level graduations are numbered continuously, the formula is: in which O and E are the readings of the level when the larger numbers are at the object end of the le*vel vial, and d is the value in seconds of arc of one division of the vial. The formula used in computing time from observations with a vertical circle on a star or on the sun is sn sin _r t = cos $ cos d in which t is the hour angle, d the declination, the zenith distance of the object observed, and is the latitude of the station. In the following form (No. 381a) the usual method of computation is shown. This form is designed especially for the computation of time from the observed altitudes of a star. DETERMINATION OF TIME. Computation of time, observations on a star with vertical circle. Form 381 a. (State, Texas. Station, Sears triangulation station. Chronometer, 1769 Sidereal. Date, Dec. 22, 1908. Barometer, 716 rr.m. Temperature, 5 C.] 55 Star: a Tauri Star: a Tauri Ti m s , h m s , Chron. reading, Zenith dist. 1 04 55. 8 49 59 44 1 07 46.8 49 24 02 Refraction + 1 06 + 1 05 Corrected Z. D.=c 50 00 50 49 25 07 log cos , $, 9. 9257458 32 33 31 9. 9257458 32 33 31 log cos S, 3 9. 9821234 16 19 37 9. 9821234 16 19 37 log cos ^+log cos 3 log D, (j> i 9. 9078692 16 13 54 9. 9078692 16 13 54 log sin J [C+(#-], i IC+tt-a)) 9. 7375385 33 07 22 9. 7340593 32 49 30 log sin j [C-W-,5)], j [:--! 9. 4632265 16 53 28 9.4557230 16 35 36 Sum two !.->g sines=log X, 9. 2007650 9. 1897823 log N-log D=log sin 2 j t, 9. 2928958 9. 2819131 log sin J (, J < (arc) 9. 6464479 26 17 54 9.6409566 25 56 35 h m S h m s < (time), I (arc) 3 30 23.2 52 35 48 3 27 32.7 51 53 10 Right ascension of star, 4 30 41. 9 4 30 41.9 Sidereal time, 1 00 18.7 1 03 09.2 Chronometer reading, 1 04 55.8 1 07 46. 8 Chronometer correction, -04 37.1 -04 37.6 The correction is plus if the chronometer is slow and minus if fast. Carry all angles to seconds only, all times to tenths of seconds, and all logarithms to seven decimal places. In space below, compute rate of chronometer, etc. Mean Epoch h m 1 10 4 58 Star ft Tauri 3 Geminor. Chronometer correction m s -4 37.7 -4 36.7 Clock rate=0'.263 per hour losing. Tn the above computation the correction for refraction was obtained from the tables on pages 58-59 of this publication. The apparent declination and right ascension of the star were obtained from the American Ephemeris and Nautical Almanac for 1908 (the year of observation). TIME FROM OBSERVATIONS ON THE SUN WITH THE VERTICAL CIRCLE. When the sun is the object observed upon a slightly different program of observations is required. The telescope is pointed on the sun's upper limb (the horizontal wire of the telescope made tangent to the disk of the sun) with the circle right and immediately afterward with the circle left. At each pointing the time of contact, the level reading, and the reading of the vertical circle are noted. The letters R and L (right and left) are used to designate the posi- tion of the circle with reference to the vertical axis of the instrument. Two quarter sets similar to the above are then made in quick succession on the sun's lower limb, and finally another quarter set on the upper limb. These are recorded on the form shown below, on which are also computed the zenith distances of the sun's limbs corrected for level. 56 Form 252. U. S. COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 14. Double zenith distances. [Station Tilden. Observer, W. Bowie. State, Minnesota. County, Poik. Instrument, Vertical circle No. 63. Date, Sept. 6 1906.] Object observed Time Level Circle right or left Circle reading Verniers Zenith distance Remarks E A B C D Mean Sun's upper limb h m s 8 47 39. 5 d 32.3 d 11.0 R Value of one division of 49 02 24 M 45 30 38.2 8 48 47.0 07.6 29.2 L 147 30 36 08 15 06 15.8 49 13 48. 8 the level vial =4" .00 24.7 18.2 -6.5 -6.5 49 13 42. 3 Sun's lower limb Q 8 50 12.5 32.5 11.2 R 8 51 17.5 07. 8 29. 3 L 246 26 24 21 00 36 20.2 49 28 02.2 Chronometer, Sidereal 24.7 18.1 -6.6 No. 102 -6.6 49 27 55.6 Temperature, 27 C Sun's lower limb Q 8 52 57.0 31.2 09.8 R Barometer not read 8 53 45. 2 08.0 29.8 L 344 46 15 05 30 00 12.8 49 09 56.3 23.2 20.0 -3.2 -3.2 49 09 53.1 Sun's upper limb (3 8 55 08.2 31.5 10.0 R 8 55 52.0 09.3 31.0 L 81 32 54 81 48 45 57.0 48 23 22. 1 22.2 21.0 -1.2 -1.2 48 23 20. 9 The observations on the upper limb are computed separately from those on the lower limb in order that one may make more exact corrections for refraction. Computation of time, observations on sun with vertical circle. Form 381. [Station, Tilden. Date, Sept. 6, 1906. Chronometer, Sidereal 102. Temperature, 27 C. Barometer (not read).] Sun's upper limb Sun's lower limb h m s . , h m s , Chron. reading, Zenith dist. 8 48 13. 2 49 13 42 8 50 45.0 49 27 56 Chron. reading, Zenith dist. 8 55 30. 1 48 23 21 8 53 21.1 49 09 53 Mean, Mean 8 51 51.6 48 48 32 8 52 03.0 49 18 54 Parallax 07 - 07 Refraction + 1 03 + 1 04 Semidiameter + 15 54 -1.5 54 Corrected Z. D.= J 49 05 22 49 03 57 log cos ^, 9. 8279861 47 42 16 9.8279861 47 42 16 log cos 3, S 9. 9970883 6 37 38 9. 9970S83 6 37 38 log cos <4+log cos i)=log D, 4>3 9. 8250744 41 04 38 9. 8250744 41 04 38 log sin J [c+(^-j)J, J (c+W _,j)j 9.8501157 45 05 00 8. 8500254 45 04 17 log sin J [c_(0-)] F J(C-W-J)] 8. 8442464 4 00 22 8. 8429S19 3 59 40 Sum two log sines=log N, 8. 6943621 8. 6930073 log N-log D=log sin ' i t, 8. 8692897 8. 8679329 log sin J4, It (arc) 9.4346438 15 47 10 9.4339664 15 45 39 h m s h m s ((time), (arc) 2 06 17.3 31 34 20 2 06 05.2 31 31 IS Local apparent time, 21 53 42.7 21 53 54.8 Equation of time, -1 31.4 -1 31.4 Local mean time, 21 52 11.3 21 52 23.4 Local sidereal time, 8 51 33.8 8 51 45.9 Chronometer reading, 8 51 51.6 8 52 03.0 Chronometer correction -17.8 -17.1 ft m Longitude from Greenwich, =6 25.3 Estimated local mean time of observation, =9 52 Greenwich mean time of observation, =4 17 Interpolation interval, from Greenwich mean noon, =4.3 hours h m =6 25.3 =9 53 =4 18 =4.3 hours. DETERMINATION OF TIME. 57 In this computation the correction for refraction was obtained from the tables on pages 58-59 of this publication. The argument used was the apparent altitude. The first table gives the mean refraction, or the refraction under an assumed standard condition of 760 mm. ( = 29. 9 in.) pressure and 10 C. ( = 50 F.) temperature. The second table gives the factor C B , by which the mean refraction as obtained from the first table must be multiplied, on account of a barometer reading different from 760 mm. In the third table is obtained the factor C T by which the mean refraction must be multiplied on account of a temperature different from the standard (10 C.). The resulting refraction is then r = r u X C B X C T in which r u is the refraction under standard conditions obtained from the first table and C B and C T are the factors obtained from the second and third tables, respectively. 1 The reduction for semidiameter, and the values for the sun's declination and for the equa- tion of time were obtained from the American Ephemeris and Nautical Almanac for 1906 (the year of observations). The parallax was obtained from the table on page 60, which was also taken from Hayford's Geodetic Astronomy. The semidiameter was obtained from page 405 of the Ephemeris. The declination and the equation of time were obtained from pages 146 and 147 of the Ephemeris. The interpolation of these quantities for the time of observation is made by the use of the interpolation interval obtained at the bottom of the computation. The mean of the observations on either limb, reduced for parallax, refraction, and semi- diameter gives the true zenith distance of the sun's center. The computation is by the same formula as is given for the reduction of the observations on a star. (See p. 54.) As the above observations were made using a sidereal chronometer, and as the correction on sidereal time was required, it was necessary to reduce the computed mean time of the observa- tion to its corresponding local sidereal time before a comparison was made with the time as read from the chronometer face. The following computation shows the various steps of this reduction for the observations on the sun's upper limb: h m s Local mean time of observation (Sept. 5, 1906) 2 21 52 11. 3 Reduction to sidereal interval (Table III, Ephemeris) 3 35. 6 Right ascension of mean sun, Greenwich mean noon September 5, 1906 10 54 43. 6 Increase in right ascension of mean sun, at Tilden mean noon September 5, 1906 (Table III, Ephemeris, 6" 25 m .3 west) 1 03. 3 Sum, local sidereal time of observation at Tilden 8 51 33. 8 For several reasons the observations on a star are more satisfactory than those on the sun. When used in connection with other astronomic observations, such as the determination of azimuth, a chronometer correction from observations on a star may be obtained close to the epoch of the observations, since any one of many available stars may be used. The computa- tion is more easily made as there is no reduction for semidiameter or for parallax, and the declination and right ascension of a star are practically constant during an entire set of observa- tions and therefore easily and quickly obtained from a star list. No equation of time is intro- duced. The observer should have a star chart 3 for use in identifying the stars observed upon. 1 These tables were copied from A Text Book of Geodetic Astronomy by John F. Hayford, formerly inspector of geodetic work and Chief of the Computing Division, U. S. Coast and Geodetic Survey. John Wiley & Sons, 1898. 3 It must be remembered that the day of the Ephermis is astronomic, and begins at noon of the civil day of the same date. Sept. 5, 21& 52w 11O, astronomic mean time is the forenoon of Sept. S, civil time. 8 Star Charts are published by the Hydrographic Office of the U. S. Navy and may be obtained from the Navy Department, Washington, D. C. Star Charts are also contained in A Field Book of the Stars, by W. T. Olcott (G. P. Putnam's Sons, publishers). 58 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Mean refraction (r M ) [Barometer, 760 millimeters (=29.9 inches). Temperature, 10 C.(=50 F).] Alti- tude Mean re- fraction Change per minute Alti- tude Mean re- fraction Change per minute Alti- tude Mean re- fraction Change per minute Alti- tude Mean re- fraction Change per minute Alti- tude Mean re- fraction Change minute 00 34 08.6 11.66 7 00 7 24.2 0.95 19 00 2 47.6 0.16 33 00 1 29.4 0.06 52 30 44. 7 0.03 10 32 15.9 10.88 10 7 14.9 0.91 20 2 44.6 0.15 20 1 28. 2 0.06 53 00 43.9 0.03 20 30 31.1 10.10 20 7 06.0 0.88 40 2 41.6 0.15 40 1 27.1 0.05 30 43.1 0.03 30 23 53.9 9.64 30 6 57.4 0.84 20 00 2 38.7 0.14 34 00 1 26.1 0.05 54 00 42.3 0.03 40 27 18. 2 9.20 40 6 49.1 81 20 2 35.9 0.14 20 1 25.0 0.05 30 41.6 0.03 50 25 49.8 8.50 50 6 41.2 0.78 40 2 33.2 0.13 40 1 24.0 0.05 55 00 40.8 0.03 1 00 24 28.3 7.82 8 00 6 33.5 0.76 21 00 2 30.6 0.13 35 00 1 23.0 0.05 30 40.0 0.03 10 23 13.5 7.17 10 6 26.0 73 20 2 28.1 0.13 20 1 22.0 0.05 56 00 39.3 0. 025 20 22 04.9 6.58 20 6 18.9 0.70 40 2 25.6 0.12 40 1 21.0 0.05 57 00 37.8 0.024 30 21 01. 8 6.06 30 6 12.0 0.68 22 00 2 23.2 0.12 36 00 1 20.0 0.05 58 00 36.4 0.023 40 20 03.7 5.60 40 6 05.3 0.66 20 2 20.9 0.12 30 1 18.5 0.05 59 00 35.0 0. 023 50 19 09.8 5.20 50 5 58.9 63 40 2 18.6 0.11 37 00 1 17.1 0.04 60 00 33. 6 0.022 2 00 18 19.7 4.84 9 00 5 52.7 0.61 23 00 2 16.4 0.11 30 1 15.7 04 61 00 32.3 0.022 10 17 33.1 4.50 20 5 40.8 0.58 20 2 14.2 0.11 38 00 1 14.4 0.04 62 00 31.0 0.022 20 16 49.7 4.18 40 5 29.7 0.54 40 2 12.1 0.10 30 1 13.1 0.04 63 00 29.7 0.022 30 16 09.5 3.88 10 00 5 19.2 0.51 24 00 2 10.1 0.10 39 00 1 11.8 0.04 64 00 28.4 0.021 40 15 32. 1 3.62 20 5 09.4 48 20 2 08.1 0.10 30 1 10.5 0.04 65 00 27.2 0.021 50 14 57.1 3.39 40 5 00.1 0.46 40 2 06. 1 0.10 40 00 1 09.3 0.04 66 00 25.9 0.021 3 00 14 24.3 3.18 11 00 4 51.2 0.43 25 00 2 04 2 0.09 30 1 08.1 0.04 67 00 24.7 0.020 10 13 53.6 2.98 20 4 42.8 0.40 20 2 02.4 0.09 41 00 1 06.9 0.04 68 00 23.6 0. 020 20 13 24.8 2.79 40 4 35.0 0.38 40 2 00.6 0.09 30 1 05. 7 0.04 69 00 22.4 0.020 30 12 57.8 2.61 12 00 4 27.5 0.37 26 00 1 58.8 0.09 42 00 1 04.6 0.04 70 00 21.2 0.019 40 12 32.5 2.46 20 4 20.3 0.35 20 1 57.1 0.09 30 1 03.5 0.04 71 00 20.1 0.019 50 12 08.7 2.33 40 4 13.5 0.33 40 1 55.4 0.08 43 00 1 02.4 0.04 72 00 18.9 0.019 4 00 11 46.0 2.20 13 00 4 07.1 0.32 27 00 1 53.8 0.08 30 1 01.3 0.04 73 00 17.8 0.018 10 11 24.6 2.09 20 4 00. 9 0.30 20 1 52.2 0.08 44 00 1 00.2 0.03 74 00 16.7 0.018 20 11 04.2 1.98 40 3 55.1 0.28 40 1 50.6 0.08 30 59.2 0.03 75 00 15.6 0.018 30 10 44.9 1.88 14 00 3 49.5 0.27 28 00 1 49.1 08 45 00 58.2 0.03 76 00 14.5 0.018 40 10 26. 5 1.79 20 3 44.2 0.26 20 1 47.6 0.07 30 57.2 0.03 77 00 13.5 0.018 50 10 09.1 1.70 40 3 39.1 0.25 40 1 46.1 0.07 46 00 56.2 0.03 78 00 12.4 0.018 5 00 9 52.6 1.61 15 00 3 34.1 0.24 29 00 1 44.6 0.07 30 55.2 0.03 79 00 11.3 0.018 10 9 36.9 1.54 20 3 29.4 0.23 20 1 43.2 0.07 47 00 54.2 0.03 80 00 10. 3 0.018 20 9 21 9 1.46 40 3 24.8 0.23 40 1 41.8 0.07 30 53.3 0.03 81 00 09.2 0.018 30 9 07.6 1.40 16 00 3 20.4 0.22 30 00 1 40.5 0.07 48 00 52.5 0.03 82 00 08.2 0.018 40 8 54.0 1.33 20 3 16.1 0.21 20 1 39.1 0.07 30 51.6 0.03 83 00 07.2 0.018 50 8 41.0 1.27 40 3 12.0 0.20 40 1 37.8 0.06 49 00 50. 7 0.03 84 00 06. 1 018 6 00 8 28.6 1.22 17 00 3 08.2 0.19 31 00 1 36.6 0.06 30 49.8 0.03 85 00 05. 1 0.018 10 8 16.7 1.16 20 3 04 5 0.19 20 1 35.3 0.06 50 00 48.9 0.03 86 00 04. 1 0.017 20 8 05.3 1.12 40 3 00.9 0.18 40 1 34.1 0.06 30 48.0 0.03 87 00 03.1 0.017 30 7 54.3 1.07 18 00 2 57.4 0.17 32 00 1 33.0 0.06 51 00 47.2 0.03 88 00 02.0 0.017 40 7 43. 9 1.02 20 2 54. 0.17 20 1 31.8 0.06 30 46.3 0.03 89 00 01. 0.017 50 7 33.9 0.98 40 2 50. 7 0.16 40 1 30.6 0.06 52 00 45.5 0.03 90 00 00.0 0.017 DETERMINATION OF TIME. 59 Correction to mean refraction as given on page 58, depending upon the reading of the 'barometer. Barometer CB Barometer CB Barometer c, Barometer CB Barometer CB Inches mm Inches mm Inches mm Inches mm Inches mm 20.0 508 0.670 22.4 569 0.749 24.8 630 0.829 27.2 691 0.909 29.6 752 0.989 20.1 511 0.673 22.5 572 0.752 24.9 632 0.832 27.3 693 0.912 29.7 754 0.992 20.2 513 0.676 22.6 574 0.755 25.0 635 0.835 27.4 696 0.916 29.8 757 0.996 20.3 516 0.679 22.7 576 0.759 25.1 637 0.838 27.5 699 0.920 29.9 759 0.999 20.4 518 0.6&2 22.8 579 0.762 25.2 640 0.842 27.6 701 0.923 30.0 762 1.003 20.5 521 0.685 22.9 582 0.766 25.3 643 0.846 27.7 704 0.926 30.1 765 1.007 20.6 523 0.688 23.0 584 0.770 25.4 645 0.849 27.8 706 0.929 30.2 767 1.010 20.7 526 0.692 23.1 587 0.773 25.5 648 0.853 27.9 709 0.933 30.3 770 1.013 20.8 528 0.696 23.2 589 0.776 25.6 650 0.856 28.0 711 0.936 30.4 772 1.016 20.9 531 0.699 23.3 592 0.779 25.7 653 0.859 28.1 714 0.939 30.5 775 1.020 21.0 533 0. 703 23.4 594 0.783 25.8 655 0.862 28.2 716 0.942 30.6 777 1.023 21.1 536 0.706 23.5 597 0.786 25.9 658 0.866 28. 3 719 0.946 30.7 780 1.026 21.2 538 0.709 23.6 599 0.789 26.0 660 0.869 28.4 721 0.949 30.8 782 1.029 21.3 541 0.712 23.7 602 0.792 26.1 663 0.872 28.5 724 0.953 30.9 785 1.033 21.4 544 0.716 23.8 605 0.796 26.2 665 0.875 28.6 726 0.956 31.0 787 1.036 21.5 546 0.719 23.9 607 0.799 26.3 668 0.879 28.7 729 0.959 21.6 549 0.722 24.0 610 0.803 26.4 671 0.882 28.8 732 0.963 21.7 551 0.725 24.1 612 0.806 26.5 673 0.885 28.9 734 0.966 21.8 554 0.729 24.2 615 0.809 26.6 676 0.889 29.0 737 0.970 21.9 556 0.732 24.3 617 0.813 26.7 678 0.892 29.1 739 0.973 22.0 559 0.735 24.4 620 0.816 26.8 681 0.896 29.2 742 0.976 22.1 561 0.739 24.5 622 0.820 26.9 683 0.899 29.3 744 0.979 22.2 564 0.742 24.6 625 0.823 27.0 686 0.902 29.4 747 0.983 22.3 566 0.746 24.7 627 0.826 27.1 688 0.905 29.5 749 0.986 Correction to mean refraction as given on page 58, depending upon the reading of the detached thermometer. Temperature C T Temperature C T Temperature C T Temperature C T Temperature CT Fahren- heit Centi- grade Fahren- heit Centi- grade Fahren- heit Centi- grade Fahren- heit Centi- grade Fahren- heit Centi- grade -25 -31.7 1.172 8 -13.3 1.089 41 5.0 1.018 74 23.3 0.955 107 41.7 0.900 24 -31.1 1.169 9 -12.8 1.087 42 5.6 1.016 75 23.9 0.953 108 42.2 0.899 -23 -30.6 1.166 10 -12.2 1.085 43 6.1 1.014 76 24.4 0.952 109 42.8 0.897 -22 30.0 1.164 11 -11.7 1.082 44 6.7 1.012 77 25.0 0.950 110 43.3 0.895 21 29.4 1.161 12 11.1 1.08C 45 7.2 1.010 78 25.6 0.948 Til 43.9 0.894 -20 -28.9 1.158 13 -10.6 1.078 46 7.8 1.008 79 26.1 0.946 112 44.4 0.892 -19 -28.3 1.156 14 -10.0 1.076 47 8.3 1.006 80 26.7 0.945 113 45.0 0.891 -18 -27.8 1.153 15 -9.4 1.073 48 8.9 1.004 81 27.2 0.943 114 46.6 890 -17 -Z7.2 1.151 16 - 8.9 1.071 49 9.4 1.002 82 27.8 0.941 115 46.1 0.888 16 -26.7 1.148 17 - 8.3 1.069 59 10.0 1.000 83 28.3 0.939 116 46.7 0.886 15 -26.1 1.145 18 - 7.8 1.067 51 10.6 0.998 84 28.9 0.938 117 47.2 0.885 -14 -25.6 1.143 19 - 7.2 1.064 52 11.1 0.996 85 29.4 0.936 118 47.8 0.884 13 25.0 1.140 20 -6.7 1.062 53 11.7 0.994 86 30.0 0.934 119 48.3 0.882 12 24.4 1.138 21 - 6.1 1.060 54 12.2 0.992 87 30.6 0.933 120 48.9 0.881 -11 -23.9 1.135 22 - 5.6 1.058 55 12.8 0.990 88 31.1 0.931 121 49.4 0.880 10 23.3 1.133 23 - 5.0 1.056 56 13.3 0.988 89 31.7 0.929 122 50.0 0.878 Q -22.8 1.130 24 - 4.4 1.054 57 13.9 0.986 90 32.2 0. 928 123 50.6 0.877 - 8 22.2 1.128 25 - 3.9 1.051 58 14.4 0.985 91 32.8 0.926 124 51.1 0.876 7 -21.7 1.125 26 -3.3 1.049 59 15.0 0.983 92 33.3 0.924 125 51.7 0.874 6 -21.1 1.123 27 - 2.8 1.047 60 15.6 0.981 93 33.9 0.923 126 52.2 0.873 - 5 -20.6 1.120 28 -2.2 1.045 61 16.1 0.979 94 34.4 0.921 127 52.8 0.871 4 -20.0 1.118 29 1.7 1.043 62 16.7 0.977 95 35.0 0.919 128 53.3 0.870 3 -19.4 1.115 30 - 1.1 1.041 63 17.2 0.975 96 35.6 0.917 129 53.9 0.868 2 -18.9 1.113 31 - 0.6 1.039 64 17.8 0.973 97 36.1 0.916 130 54.4 0.867 _ 1 -18.3 1.111 32 0.0 1.036 65 18.3 0.972 98 36.7 0.914 -17.8 1.108 33 + 0.6 1.034 66 18.9 0.970 99 37.2 0.912 + 1 -17.2 1.106 34 1.1 1.032 67 19.4 0.968 100 37.8 0.911 2 -16.7 1.103 35 1.7 1.030 68 20 0.966 101 38.3 0.909 3 -16.1 1.101 36 2.2 1.028 69 20.6 0.964 102 38.9 0.908 4 -15.6 1.099 37 2.8 1.026 70 21 1 0.962 103 39.4 0.906 5 -15.0 1.096 38 3.3 1.021 71 21.7 0.961 104 40 0.905 6 -14.4 1.094 39 3.9 1.022 72 22 2 0.959 105 40.6 0.903 7 -13.9 1.092 40 4.4 1.0?0 73 22.8 0.957 106 41.1 0.902 60 U. S. COAST AND GEODETIC SUSVEY SPECIAL PUBLICATION NO. 14. The parallax of the sun (p) for the first day of each month. Altitude Jan. 1 Feb.l Dec.l Mar. 1 Nov. 1 Apr. 1 Oct.l May 1 Sept,' 1 June 1 Aug. 1 July 1 Zenith distance 9.0 9.0 8.9 8.9 8.8 8.7 8.7 90 3 9.0 9.0 8.9 8.8 8.8 8.7 8.7 87 6 9.0 8.9 8.9 8.8 8.7 8.7 8.7 84 9 8.9 8.9 8.8 8.8 8.7 8.6 8.6 81 12 8.8 8.8 8.7 8.7 8.6 8.5 8.5 78 15 8.7 8.7 8.6 8.6 8.5 8.4 8.4 75 18 8.6 8.6 8.5 8.4 8.4 8.3 8.3 72 21 8.4 8.4 8.3 8.3 8.2 8.2 8.1 69 24 8.2 8.2 8.2 8.1 8.0 8.0 8.0 66 27 8.0 8.0 8.0 7.9 7.8 7.8 7.8 63 30 7.8 7.8 7.7 7. 7 7.6 7.6 7.6 60 33 7.6 7.5 7.5 7.4 7.4 7.3 7.3 57 36 7.3 7.3 7.2 7.2 7.1 7.1 7.0 54 39 7.0 7.0 6.9 6.9 6.8 6.8 6.8 51 42 6.7 6.7 6.6 6.6 6.5 6.5 6.5 48 44 6.5 6.5 6.4 6.4 6.3 6.3 6.3 46 46 6.3 6.2 6.2 6.2 6.1 6.1 6.0 44 48 6.0 6.0 6.0 5.9 5.9 5.8 5.8 42 50 5.8 5.8 5.7 5.7 5.6 5.6 5.6 40 52 5.6 5.5 5.5 5.4 5.4 5.4 5.4- 38 54 5.3 5.3 5.2 5.2 5.2 5.1 5.1 36 56 5.0 5.0 5.0 5.0 4.9 4.9 4.9 4 6 34 32 58 60 4. 8 4.5 4. 8 4.5 4.5 4.4 4.4 4.4 4^4 30 62 4.2 4.2 4.2 4.2 4.1 4.1 4.1 28 64 4.0 3.9 3.9 3.9 3.8 3.8 3.8 26 66 3.7 3.7 3.6 3.6 3.6 3.6 3.5 24 68 3.4 3.4 3.4 3.3 3.3 3.3 3.3 22 70 3.1 3.1 3.1 3.0 3.0 3.0 3.0 20 72 2.8 2.8 2.8 2.7 2.7 2.7 2.7 18 74 2.5 2.5 2.5 2.4 2.4 2.4 2.4 16 76 2.2 2.2 2.2 2.1 2.1 2.1 2.1 14 78 1.9 1.9 1.9 1.8 1.8 1.8 1.8 12 80 1.6 1.6 1.6 1.6 1.5 1.5 1.5 10 82 1.2 1.2 1.2 1.2 1.2 1.2 1.2 8 84 0.9 0.9 0.9 0.9 0.9 0.9 0.9 6 86 0.6 0.6 0.6 0.6 0.6 0.6 0.6 4 88 0.3 0.3 0.3 0.3 0.3 0.3 0.3 2 90 0.0 0.0 0.0 0.0 0.0 0.0 0.0 A, B, C, FACTORS. These factors are referred to in the computations of time from observations with the transit on pages 23 and 25. Their arithmetical values are as follows: Azimuth factor = A = sin sec 8 Level factor =5 = cos sec d domination f actor = C=sec d where <5 = declination and = zenith distance = < - or c- (180-T, "noT* ~fyrj' ' ' ' decimeters. 62 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (i). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (;). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on opposite page.) C 10 15 20 22 24 26 28 30 32 34 36 38 40 41 42 C 1 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 89 2 .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 .05 .05 .05 88 3 .05 .05 .05 .03 .06 .06 .06 .06 .06 .06 .06 .06 .07 .07 .07 .07 87 4 .07 .07 .07 .07 .08 .08 .08 .08 .08 .08 .08 .09 .09 .09 .09 .09 86 5 .09 .09 .09 .09 .09 .10 .10 .10 .10 .10 .10 .11 .11 .11 .11 .12 85 6 .11 .11 .11 .11 .11 .11 .12 .12 .12 .12 .13 .13 .13 .14 .14 .14 84 7 .12 .12 .13 .13 .13 .13 .14 .14 .14 .14 .15 .15 .15 .16 .16 .16 83 8 .14 .14 .14 .15 .15 .15 .16 .16 .16 .16 .17 .17 .18 .18 .18 .19 82 9 .16 .16 .16 .17 .17 .17 .17 .18 .18 .18 .19 .19 .20 .20 .21 .21 81 10 .17 .18 .18 .19 .19 .19 .19 .20 .20 .?! .21 .21 .22 .23 .23 .23 80 11 .19 .19 .20 .20 .21 .21 .21 .22 .22 .23 .23 .24 .24 .25 .25 .26 79 12 .21 .21 .22 .22 .22 .23 .23 .24 .24 .25 .25 .26 .26 .27 .27 .28 78 13 .22 .23 .23 .24 .24 .25 .25 .2ti .26 .27 .27 .28 .29 .29 .30 .30 77 14 .24 .25 .25 .28 .26 .27 .27 .27 .28 .29 .29 .30 .31 .32 .32 .33 76 15 .26 .26 .27 .28 .28 .28 .29 .29 .30 .31 .31 .32 .33 .34 .34 .35 75 16 .28 .28 .29 .29 .30 .30 .31 .31 .32 .33 .33 .34 .35 .36 .37 .37 74 17 .29 .30 .30 .31 .31 .32 .33 .33 .34 .34 .35 .36 .37 .38 .39 .39 73 18 .31 .31 .32 .33 .33 .33 .34 .35 .36 .36 .37 .38 .39 .40 .41 .42 72 19 .33 .33 .34 .35 .35 .36 .36 .37 .38 .38 .39 .40 .41 .42 .43 .44 71 20 .34 .35 .35 .36 .37 .37 .38 .39 .40 .40 .41 .42 .43 .45 .45 .46 70 21 .36 .36 .37 .38 .39 .39 .40 .41 .41 .42 .43 .44 .45 .47 .47 .48 69 22 .37 .38 .39 .40 .40 .41 .42 .42 .43 .44 .45 .46 . .48 .49 .50 .50 68 23 .39 .40 .41 .42 .42 .43 .44 .44 .45 .46 .47 1 .48 .50 .51 .52 .53 67 24 .41 .41 .42 .43 .44 .45 .45 .46 .47 .48 .49 .50 .52 .53 .54 .55 66 25 .42 .43 .44 .45 .46 .46 .47 .48 .49 .50 .51 .52 .54 .55 .56 .57 05 26 .44 .45 .45 .47 .47 .48 .49 .50 .51 .52 .53 .54 .56 .57 .58 .59 64 27 .45 .46 .47 .48 .49 .50 .51 .51 .52 .54 .55 .56 .58 .59 .CO .61 63 28 .47 .48 .49 .50 .51 .51 .52 .53 .54 .55 .57 .58 .CO .61 .62 .63 62 29 .48 .49 .50 .52 .52 .53 .54 .55 .56 .57 .58 .60 .61 .63 .64 .65 61 30 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .62 .63 .65 .66 .67 60 31 .52 .52 .53 .55 .56 .56 .57 .58 .59 .61 .62 .64 .68 .67 .68 .69 59 32 .53 .54 .55 .56 ,57 .58 .59 .60 .61 .63 .64 .65 .67 .69 .70 .71 58 33 .55 .55 .56 .58 .59 .60 .61 .62 .63 .64 .66 .67 .69 .71 .72 .73 57 34 .56 .57 .58 .59 .60 .61 .62 .63 .65 .66 .67 .69 .71 .73 .74 . 75 56 35 .57 .58 .59 .61 .62 .63 .64 .65 .66 .68 .69 .71 .73 .75 .76 .77 55 36 .59 .60 .61 .63 .63 ,64 .65 .67 .68 .66 .71 .73 .75 ,77 .78 .79 54 37 .60 .61 .62 .64 .65 .66 .67 .68 .70 .71 .73 .74 .76 .79 .80 .81 53 38 .62 .63 .64 .66 .66 .67 .69 .70 .71 .73 .74 .76 .78 .80 .82 .83 52 39 .63 .64 .65 .67 .68 .69 .70 .71 .73 .74 .76 .78 .80 .82 .83 .85 51 40 .64 .65 .67 .68 .69 .70 .72 .73 .74 .76 .77 .79 .82 .84 .85 .86 50 41 .66 .67 ,68 .70 .71 .72 .73 .74 .76 .77 .79 .81 .S3 .86 .87 .88 49 42 .67 .68 .69 .71 .72 .73 .74 .76 .77 .79 .81 .83 .85 .87 .89 .90 48 43 .68 .69 .71 .73 .74 .75 .76 .77 .79 .80 .82 .84 .86 .89 .90 .92 47 44 .69 .71 .72 .74 . .76 .77 .79 .80 .82 .84 .86 .88 .91 .92 .93 4C 45 .71 .72 .73 .75 .76 .77 .79 .80 .82 .83 j .85 .87 .90 .92 .94 .95 45 10 15 20 22 24 26 28 30 32 ! 34 36 3S 40 41 42 DETERMINATION OF TIME. 63 Table of factors for reduction of transit observations. TOP ARGUMENT=STAR'S DECLINATION (<). SIDE ARGUMENT" STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on this page.] : 10 15 20" 22 24 26 28 30 32 34 36 38 40 41 42 C 4fi 72 .73 .74 . 77 .78 .79 .80 .82 .83 .85 .87 .89 .91 .94 .95 .97 44 47 .73 .74 .76 .73 .79 .80 .81 .83 .84 .86 .88 .90 .93 .95 .97 .98 43 48 .74 .76 .77 .79 .80 .81 .83 .84 .86 .88 .90 .92 .94 .97 .98 1.00 42 49 . 75 .77 .78 .80 .81 .83 .84 .86 .87 .89 .91 .93 .96 .99 1.00 1.02 41 > .77 .78 .79 .82 .83 .84 .85 .87 .89 .90 .92 .95 .97 1.00 1.01 1.03 40 .78 .79 .80 .83 .84 .85 .87 .88 .90 .92 .94 .96 .99 1.01 1.03 1.05 39 5 .79 .80 .82 .84 .85 .86 .88 .89 .91 .93 .95 .97 1.00 1.03 1.04 1.06 38 53 .80 .81 .83 .85 .86 .87 .89 .91 .92 .94 .96 .99 1.01 1.04 1.06 1.07 37 54 .81 .82 .84 .86 .87 .89 .90 .92 .93 .96 .98 1.00 1.03 1.06 1.07 1.09 36 55 .82 .83 .85 .87 .88 .90 .91 .93 .95 .97 .99 1.01 1.04 1.07 1.08 1.10 35 56 .83 .84 .86 .88 .89 .91 .92 .94 .96 .98 1.00 1.02 1.05 1.08 1.10 1.12 34 57 .84 .85 .87 .89 .90 .92 .93 .95 .97 .99 1.01 1.04 1.06 1.09 1.11 1.13 33 58 .85 .86 .88 .90 .91 .93 .94 .96 .98 1.00 1.02 1.05 1.08 1.11 1.12 1.14 32 59 .86 .87 .89 .91 .92 .94 .95 .97 .99 1.01 1.03 1.06 1.09 1.12 1.14 1.15 31 60 .87 .88 .90 .92 .93 .95 .96 .98 1.00 1.02 1.04 1.07 1.10 1.13 1.15 1.17 SO 61 .87 .89 .91 .93 .94 .96 .97 .99 1.01 1.03 1.05 1.08 1.11 1.14 1.16 1.18 29 62 .88 .90 .91 .94 .95 .97 .98 1.00 1.02 1.04 1.06 1.09 1.12 1.15 1.17 1.19 28 63 .89 .91 .92 .95 .96 .98 .99 1.01 1.03 1.05 1.07 1.10 1.13 1.16 1.18 1.20 27 64 .90 .91 .93 .96 .97 .98 1.00 1.02 1.04 1.06 1.08 1.11 1.14 1.17 1.19 1.21 26 65 .91 .92 .94 .96 .98 .99 1.01 1.03 1 05 1.07 1.09 1.12 1.15 1.18 1.20 1.22 25 66 .91 .93 .95 .97 .99 1.00 1.02 1.04 1.06 1.08 1.10 1.13 1.16 1.19 1.21 1.23 24 67 .92 .94 .95 .98 .99 1.01 1.02 1.04 1.06 1.09 1.11 1.14 1.17 1.20 1.22 1.24 23 68 .93 .94 .96 .99 1.00 1.02 1.03 1.05 1.07 1.09 1.12 1.15 1.18 1.21 1.23 1.25 22 69 .93 .95 .97 .99 1.01 1.02 1.04 1.06 1.08 1.10 1.13 1.15 1.18 1.22 1.24 1.26 21 70 .94 .95 .97 1.00 1.01 1.03 1.05 1.06 1.09 1.11 1.13 1.16 1.19 1.23 1.25 1.26 20 71 .95 .96 .93 1.01 1.02 1.04 1.05 1.07 1.09 1.12 1.14 1.17 1.20 1.23 1.25 1.27 19 72 .95 .97 .98 1.01 1.03 1.04 1.06 1.08 1.10 1.12 1.15 1.17 1.21 1.24 1.26 1.28 18 73 .96 .97 .99 1.02 1.03 1.05 1.06 1.08 1.10 1.13 1.15 1.18 1.21 1.25 1.27 1.29 17 74 .96 .98 1.00 1.02 1.04 1.05 1.07 1.09 1.11 1.13 1.16 1.19 1.22 1.25 1.27 1.29 16 75 .97 .98 1.00 1.03 1.04 1.06 1.08 1.09 1.12 1.14 1.16 1.19 1.23 1.26 1.28 1.30 15 76 .97 .99 1.00 1.03 1.05 1.06 1.08 1.10 .12 1.14 1.17 1.20 1.23 1.27 1.29 1.31 14 77 .97 .99 1.01 1.04 1.05 1.07 1.08 1.10 .13 1.15 1.17 1.20 1.24 1.27 1.29 1.31 13 78 .98 .99 1.01 1.04 1.05 1.07 1.09 1.11 .13 1.15 1.18 1.21 1.24 1.28 1.30 1.32 12 79 .98 1.00 1.02 1.04 1.06 1.08 1.09 1.11 .13 1.16 1.18 1.21 1.25 1.28 1.30 1.32 11 80 .98 1.00 1.02 1.05 1.06 1.08 1.10 1.12 .14 1.16 1.19 1 22 1.25 1.29 1.30 1.33 10 81 .99 1 00 .02 1.05 1.07 1.08 1.10 1.12 .14 1.17 1.19 1.22 1.25 1.29 1.31 1.33 9 82 .99 1.01 .03 1.05 1.07 1.08 1.10 1.12 .14 1.17 1.19 1.22 1.26 1.29 1.31 1.33 8 83 .99 1.01 .03 1.06 1.07 1.09 1.10 1.12 .15 1.17 1.20 1.23 1.26 1.30 1.32 1.34 7 84 .99 1.01 .03 1.06 1.07 1.09 1.11 1.13 .15 1.17 1.20 1.23 1.26 1.30 1.32 1.34 6 85 1.00 1.01 .03 1 06 1.07 1.09 1.11 1.13 .15 1.17 1.20 1.23 1.26 1.30 1.32 1.34 5 86 1.00 1.01 .03 1.06 1.08 1.09 1.11 1.13 .15 1.18 1.20 1.23 1.27 1.30 1.32 1.34 4 87 1.00 1.01 .03 1.06 1.08 1.09 1.11 1.13 .15 1.18 1.20 1.23 1.27 1.30 1.32 1.34 3 88 1.00 1.01 .03 1.06 1.08 1.09 1.11 1.13 .15 1.18 1.20 1.23 1.27 1.30 1.32 1.34 2 89 1.00 1.02 .04 1.06 1.03 1.09 1.11 1.13 .15 1.18 1.21 1.24 1.27 1.31 1.32 1.35 1 90 1.00 1.02 .04 1.06 l.OS 1.09 1.11 1.13 .15 1.1S 1.21 1.24 1.27 1.31 1.32 1.35 10 15 20 22 24 26 28 30 32 34 36 38 40 41 42 64 U. S. COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 14. Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (<). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on opposi/e page.] C 42' 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 C a 1 .02 .02 .02 .02 .02 .03 .03 .03 .03 .03 .03 .03 .03 .03 .03 .03 89 2 .05 .05 .05 .05 .05 .05 .05 .05 .05 .06 .06 .06 .06 .06 .06 .06 88 3 .07 .07 .07 .07 .07 .08 .08 .08 .08 .08 .08 .09 .09 .09 .09 .10 87 4 .09 .10 .10 .10 .10 .10 .10 .11 .11 .11 .11 .12 .12 .12 .12 .13 86 5 .12 .12 .12 .12 .13 .13 .13 .13 .13 .14 .14 .14 .15 .15 .16 .16 85 6 .14 .14 .15 .15 .15 .15 .16 .16 .16 .17 .17 .17 .18 .18 .19 .19 84 7 .16 .17 .17 .17 .18 .18 .18 .19 .19 .19 .20 .20 .21 .21 .22 .22 83 8 .19 .19 .19 .20 .20 .20 .21 .21 .22 .22 .23 .23 .24 .24 .25 .26 82 9 .21 .21 .22 .22 .22 .23 .23 .24 .24 .25 .25 .26 .27 .27 .28 .29 81 10 .23 .24 .24 .25 .25 .25 .26 .26 .27 .28 .28 .29 .30 .30 .31 .32 80 11 .26 .26 .27 .27 .28 .28 .28 .29 .30 .30 .31 .32 .32 .33 .34 .35 79 12 .28 .28 .29 .29 .30 .30 .31 .32 .32 .33 .34 .35 .35 .36 .37 .38 78 13 .30 .31 .31 .32 .32 .33 .34 .34 .35 .36 .36 .37 .38 .39 .40 .41 77 14 .33 .33 .34 .34 .35 .35 .36 .37 .38 .38 .39 .40 .41 .42 .43 .44 76 15 .35 .35 .36 .37 .37 .38 .39 .39 .40 .41 .42 .43 .44 .45 .46 .48 75 16 .37 .38 .38 .39 .40 .40 .41 .42 .43 .44 .45 .46 .47 .48 .49 .51 74 17 .39 .40 .41 .41 .42 .43 .44 .45 .45 .46 .47 .49 .50 .51 .52 .54 73 18 .42 .42 .43 .44 .44 .45 .46 .47 .48 .49 .50 .51 .53 .54 .55 .57 72 19 .44 .45 .45 .46 .47 .48 .49 .50 .51 .52 .53 .54 .55 .57 .58 .60 71 20 .46 .47 .48 .48 .49 .50 .51 .52 .53 .54 .56 .57 .58 .60 .61 .63 70 21 .48 .49 .50 .51 .52 .52 .54 .55 .56 .57 .58 .59 .61 .62 .64 .66 69 22 .50 .51 .52 .53 .54 .55 .56 .57 .58 .60 .61 .62 .64 .65 .67 .69 68 23 .53 .53 .54 .55 .56 .57 .58 .60 .61 .62 .63 .65 .66 .68 .70 .72 67 24 .55 .56 .57 .58 .59 .60 .61 .62 .63 .65 .66 .68 .69 .71 .73 .75 66 25 .57 .58 .59 .60 .61 .62 .63 .64 .66 .67 .69 .70 .72 .74 .76 .78 65 26 .59 .60 .61 .62 .63 .64 .65 .67 .68 .70 .71 .73 .75 .76 .78 .80 64 27 .61 .62 .63 .64 .65 .67 .68 .69 .71 .72 .74 .75 .77 .79 .81 .83 63 28 .63 .64 .65 .66 .68 .69 .70 .72 .73 .75 .76 .78 .80 .82 .84 .86 62 29 .65 .66 .67 .69 .70 .71 .72 .74 .75 .77 .79 .81 .82 .84 .87 .89 61 30 .67 .68 .69 .71 .72 .73 .75 .76 .78 .79 .81 .83 .85 .87 .89 .92 60 31 .69 .70 .72 .73 .74 .75 .77 .78 .80 .82 .84 .86 .88 .90 .92 .95 59 32 .71 .72 .74 75 .76 .78 .79 .81 .82 .84 .86 .88 .90 .92 .95 .97 58 33 .73 .74 .76 .77 .78 .80 .81 .83 .85 .87 .88 .91 .93 .95 .97 1.00 57 34 .75 .76 .78 .79 .80 .82 .84 .85 .87 .89 .91 .93 .95 .97 1.00 1.03 56 35 .77 .78 .80 .81 .83 .84 .86 .87 .89 .91 .93 .95 .98 1.00 1.03 1.05 55 36 .79 .80 .82 .83 .85 .86 .88 .90 .91 .93 .95 .98 1.00 1.03 1.05 1.08 54 37 .81 .82 .84 .85 .87 .88 .90 .92 .94 .96 .98 .00 1.02 1.05 1.08 .10 53 38 .83 .84 .86 .87 .89 .90 .92 .94 .96 .98 1.00 .02 1.05 1.07 1.10 .13 52 39 .85 .86 .87 .89 .91 .92 .94 .96 .98 1.00 1.02 .05 1.07 1.10 1.12 .15 51 40 .86 .88 .89 .91 .93 .94 .96 .98 1.00 1.02 1.04 .07 1.09 1.12 1.15 .18 50 41 .88 .90 .91 .93 .94 .96 .98 1.00 1.02 1.04 1.07 .09 1.12 1.14 1.17 .20 49 42 .90 .91 .93 .95 .96 .98 1.00 1.02 1.04 1.06 1.09 .11 1.14 1.17 1.20 .23 48 43 .92 .93 .95 .96 .98 1.00 1.02 1.04 1.06 1.08 1.11 .13 1.16 1.19 1.22 .25 47 44 .93 .95 .97 .98 1.00 1.02 1.04 1.06 1.08 1.10 1.13 1.15 1.18 1.21 1.24 .28 46 45 .95 .97 .98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.15 1.17 1.20 1.23 1.26 .30 4.5 42 43 44 45 46 47 4S 49 50 51 52 53 54 55 56 57 DETERMINATION OF TIME. Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (<>). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on this page.l 65 C 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 C 46 .97 .98 1.00 1.02 1.04 1.05 1.07 1.10 1.12 1.14 1.17 1.19 1.22 1.25 1.29 1.32 44 47 .98 1.00 1.02 1.03 1.05 1.07 1.09 1.11 1.14 1.16 1.19 1.21 1.24 1.27 1.31 1.34 43 48 1.00 1.02 1.03 1.05 1.07 1.09 1.11 1.13 1.16 1.18 1.21 1.23 1.26 1.30 1.33 1.36 42 49 1.02 1.03 1.05 1.07 1.09 1.11 .13 1.15 1.17 1.20 1.23 1.25 1.28 1.32 1.35 1.39 41 50 1.03 1.05 1.06 1.08 1.10 1.12 .14 1.17 1.19 1.22 1.24 1.27 1.30 1.34 1.37 1.41 40 51 1.05 1.06 1.08 1.10 1.12 1.14 .16 1.18 1.21 1.23 1.26 1.29 1.32 1.35 1.39 1.43 39 52 1.06 1.08 1.10 1.11 1.13 1.15 .18 1.20 1.23 1.25 1.28 1.31 1.34 1.37 1.41 1.45 38 53 1.07 1.09 1.11 1.13 1.15 1.17 .19 1.22 1.24 1.27 1.30 1.33 1.36 1.39 1.43 1.47 37 54 1.09 1.11 1.12 1.14 1.16 1.19 .21 1.23 1.26 1.29 1.31 1.34 1.38 1.41 1.45 1.49 36 55 1.10 1 12 1 14 1 16 1.18 1.20 .22 1.25 1.27 1.30 1.33 1.36 1.39 1.43 1.46 1.50 35 56 1.12 1.13 1.15 1.17 1.19 1.22 .24 1.26 1.29 1.32 1.35 1.38 1.41 1.45 1.48 1.52 34 57 1.13 1.15 1.17 1.19 1.21 1.23 .25 1.28 1.31 1.33 1.36 .39 1.43 1.46 1.50 1.54 33 58 1.14 1.16 1.18 1.20 1.22 1.24 .27 1.29 1.32 1.35 1.38 .41 1.44 1.48 1.52 1.56 32 59 1.15 1.17 1.19 1.21 1.23 1.26 .28 1.31 1.33 1.36 1.39 .42 1.46 1.49 1.53 1.57 31 60 1.17 1.18 1.20 1.22 1.25 1.27 .29 1.32 1.35 1.38 1.41 .44 1.47 1.51 1.55 1.59 30 61 1.18 1.20 1.22 1.24 1.26 1.28 .31 1.33 1.36 1.39 1.42 .45 1.49 1.53 1.56 .61 29 63 1.19 1.21 1.23 1.25 1.27 1.29 .32 1.35 1.37 1.40 1.43 .47 1.50 1.54 1.58 .62 28 63 1.20 1.22 1.24 1.26 1.28 1.31 .33 1.36 1.39 1.42 1.45 .48 1.52 1.55 1.59 .64 27 64 1.21 1.23 1.25 1.27 1.29 1.32 1.34 1.37 1.40 1.43 1.46 .49 1.53 1.57 1.61 .65 26 65 1.22 1.24 1.26 1.28 1.30 1.33 1.35 1.38 1.41 1.44 1.47 .51 1.54 1.58 1.62 .66 25 66 1.23 1.25 1.27 1.29 1.32 1.34 1.37 1.39 1.42 .45 1.48 1.52 1.55 1.59 1.63 .68 24 67 1.24 1.26 1.28 1.30 1.33 1.35 1.38 1.40 1.43 .46 1.50 1.53 1.57 1.60 1.65 .69 23 68 1.25 1.27 1.29 1.31 1.33 1.36 1.39 1.41 1.44 .47 1.51 1.54 1.58 1.62 1.66 .70 22 69 1.26 1.28 1.30 1.32 1.34 1.37 1.40 1.42 1.45 .48 1.52 1.55 1.59 1.63 1.67 1.71 21 70 1.26 1.28 1.31 1.33 1.35 1.38 1.40 1.43 1.46 .49 1.53 1.56 1.60 1.64 1.68 1.73 20 71 1.27 1.29 1.31 1.34 1.36 1.39 1.41 1.44 1.47 .50 1.54 1.57 1.61 1.65 1.69 1.74 19 72 1.28 1.30 1.32 1.34 1.37 1.39 1.42 1.45 1.48 .51 1.54 1.58 1.62 1.66 1.70 1.75 18 73 1.29 1.31 1.33 1.35 1.38 1.40 1.43 1.46 1.49 .52 1.55 1.59 1.63 1.67 1.71 1.76 17 74 1.29 1.31 1.34 1.36 1.38 1.41 1.44 1.46 1.49 .53 1.56 1.60 1.63 1.68 1.72 1.76 16 75 1.30 1.32 1.34 1.37 1.39 1.42 1.44 1.47 1.50 .53 1.57 1.60 1.64 1.68 1.73 1.77 15 76 1.31 1.33 1.35 1.37 1.40 1.42 1.45 1.48 1.51 1.54 1.58 1.61 1.65 1.69 .73 .78 14 77 1.31 1.33 1.35 1.38 1.40 .43 1.46 .48 1.52 1.55 1.58 1.62 1.66 1.70 .74 .79 13 78 1.32 1.34 1.36 1.38 1.41 .43 1.46 .49 1.52 ' 1.55 1.59 1.62 1.66 1.70 .75 .80 12 79 1.32 1.34 1.36 1.39 1.41 .44 1.47 .50 1.53 1.56 1.59 1.63 1.67 1.71 .76 .80 11 80 1.33 1.35 1.37 1.39 1.42 .44 1.47 .50 1.53 1.56 1.60 1.64 1.87 1.72 .76 .81 10 81 1.33 1.35 .37 1.40 1.42 .45 .48 .51 1.54 1.57 1.60 1.64 1.68 1.72 .77 .81 9 82 1.33 1.35 .38 1.40 1.43 .45 .48 .51 1.54 1.57 1.61 1.64 1.68 1.73 .77 .82 8 83 1.34 1.36 .38 1.40 1.43 .46 .48 .51 1.54 1.58 1.61 1.65 1.69 1.73 .77 .82 7 84 1.34 1.36 .38 1.41 1.43 .46 .49 .52 1.55 1.58 1.62 1.65 1.69 1.73 .78 .83 6 85 1.34 1.36 .38 1.41 1.43 .46 .49 .52 1.55 1.58 1.62 1.65 1.69 1.74 .78 .83 5 86 1.34 1.36 1.39 1.41 1.44 1.46 .49 1.52 1.55 1.59 1.62 1.66 1.70 1.74 1.78 .83 4 87 1.34 1.37 1.39 1.41 1.44 1.46 .49 1.52 1.55 1.59 1.62 1.66 1.70 1.74 1.79 .83 3 88 1.34 1.37 1.39 1.41 1.44 1.46 .49 1.52 1.55 1.59 1.62 1.66 1.70 1.74 1.79 .83 2 89 1.35 1.37 1.39 1.41 1.44 1.47 1.49 1.52 1.56 1.59 1.62 1.66 1.70 1.74 1.79 .84 1 90 1.35 1.37 1.39 1.41 1.44 1.47 1.49 1.52 1.56 1.59 1.62 1.66 1.70 1.74 1.79 .84 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56" 57 8136 13 5 66 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (S). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on opposite page, j C 57 58 58J 59 595 60 60 J 61 61J 62 62j 63 63J 64 645 65 C 1 .03 .03 .03 .03 .03 .03 .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 89 2 .06 .07 .07 .07 .07 .07 .07 .07 .07 .07 .08 .08 .08 .08 .08 .08 88 3 .10 .10 .10 .10 .10 .10 .11 .11 .11 .11 .11 .12 .12 .12 .12 .12 87 4 .13 .13 .13 .14 .14 .14 .14 .14 .15 .15 .15 .15 .16 .16 .16 .17 86 5 .16 .16 .17 .17 .17 .17 .18 .18 .18 .19 .19 .19 .19 .20 .20 .21 85 6 .19 .20 .20 .20 .21 .21 .21 .22 .22 .22 .23 .23 .23 .24 .24 .25 84 7 .22 .23 .23 .24 .24 .24 .25 .25 .26 .26 .26 .27 .27 .28 .28 .29 83 8 .26 .26 .27 .27 .27 .28 .28 .29 .29 .30 .30 .31 .31 .32 .32 .33 82 9 .29 .29 .30 .30 .31 .31 .32 .32 .33 .33 .34 .35 .35 .36 .36 .37 81 10 .32 .33 .33 .34 .34 .35 .35 .36 .36 .37 .38 .38 .39 .40 .40 .41 80 11 .35 .36 .36 .37 .38 .38 .39 .39 .40 ..41 .41 .42 .43 .44 .44 .45 79 12 .38 .39 .40 .40 .41 .42 .42 .43 .44 .44 .45 .46 .47 .47 .48 .49 78 13 .41 .42 .43 .44 .44 .45 .46 .46 .47 .48 .49 .50 .50 .51 .52 .53 77 14 .44 .46 .46 .47 .48 .48 .49 .50 .51 .52 .52 .53 .54 .55 .56 ,57 76 15 .48 .49 .50 .50 .51 .52 .53 .53 .54 .55 .56 .57 .58 .59 .60 .61 75 16 .51 .52 .53 .54 .54 .55 .56 .57 .58 .59 .60 .61 .62 .63 .64 .65 74 17 .54 .55 .56 .57 .58 .58 .59 .60 .61 .62 .63 .64 .66 .67 .68 .69 73 18 .57 .58 .59 .60 .61 .62 .63 .64 .65 .66 .67 .68 .69 .70 .72 .73 72 19 .60 .61 .62 .63 .64 .65 .66 .67 .68 .69 .70 .72 .73 .74 .76 .77 71 20 .63 .64 .65 .66 .67 .68 .69 .70 .72 .73 .74 .75 .77 .78 .79 .81 70 21 .66 .68 .69 .70 .71 .72 .73 .74 .75 .76 .78 .79 .80 .82 .83 .85 69 22 .69 .71 .72 .73 .74 .75 .76 .77 .78 .80 .81 .82 .84 .85 .87 .89 68 23 .72 .74 .75 .76 .77 .78 .79 .81 .82 .83 .85 .86 .88 .89 .91 .92 67 24 .75 .77 .78 .79 .80 .81 .83 .84 .85 .87 .88 .90 .91 .93 .94 .96 66 25 .78 .80 .81 .82 .83 .85 .86 .87 .89 .90 .92 .93 .95 .96 .98 1.00 65 26 .80 .83 .84 .85 .86 .88 .89 .90 .92 .93 .95 .97 .98 1.00 1.02 1.04 64 27 .83 .86 .87 .88 .89 .91 .92 .94 .95 .97 .98 1.00 1.02 1.04 1.05 1.07 63 28 .86 .89 .90 .91 .93 .94 .95 .97 .98 1.00 1.02 1.03 1.05 1.07 1.09 1.11 62 29 .89 .91 .93 .94 .96 .97 .98 1.00 1.02 1.03 1.05 1.07 1.09 1.11 1.13 1.15 61 30 .92 .94 .96 .97 .99 1.00 1.01 1.03 1.05 1.07 1.08 1.10 1.12 1.14 1.16 1.18 60 31 .95 .97 .99 1.00 1.01 1.03 .05 1.06 1.08 1.10 1.11 .13 1.15 1.17 1.20 1.22 59 32 .97 1.00 1.01 1.03 1.04 1.06 .08 1.09 1.11 1.13 1.15 .17 1.19 1.21 1.23 1.25 58 33 1.00 .03 1.04 1.06 1.07 1.09 .11 1.12 1.14 1.16 1.18 .20 1.22 1.24 1.26 1.29 57 34 1.03 .05 1.07 1.09 1.10 1.12 .14 1.15 1.17 1.19 1.21 .23 1.25 1.27 1.30 1.32 56 35 1.05 .08 1.10 1.11 1.13 1.15 .16 1.18 1.20 1.22 1.24 .26 1.29 1.31 1.33 1.36 55 36 1.08 .11 1.12 1.14 1.16 1.18 .19 1.21 1.23 1.25 1.27 .30 1.32 1.34 1.37 1.39 54 37 1.10 .14 1.15 1.17 1.19 1.20 .22 1.24 1.26 1.28 1.30 .33 1.35 1.37 1.40 1.42 53 38 1.13 .16 1.18 1.20 1.21 1.23 1.25 1.27 1.29 1.31 1.33 1.36 1.38 1.40 1.43 1.46 52 39 1.15 .19 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36 1.39 1.41 1.43 1.46 1.49 51 40 1.18 .21' 1.23 1.25 1.27 1.29 1.31 1.33 1.35 1.37 1.39 1.42 1.44 1.47 1.49 1.52 50 41 .20 1.24 1.26 1.27 1.29 1.31 1.33 1.35 1.37 1.40 1.42 1.45 1.17 1.50 1.52 1.55 49 42 .23 1.26 1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.45 1.47 1.50 1.53 1.55 1.58 48 43 .25 1.29 1.30 1.32 1.34 1.36 1.39 1.41 1.43 1.45 1.48 1.50 1.53 1.56 1.58 1.61 47 44 .28 1.31 1.33 1.35 1.37 1.39 1.41 1.43 1.46 1.48 1.50 1.53 1.56 1.58 1.61 1.64 46 45 .30 1.33 1.35 1.37 1.39 1.41 1.44 1.46 1.48 1.51 1.53 1.56 1.58 1.61 1.64 1.67 45 57 58 58} 59 59J" 60 0j 61" 615 62 62J 63 635 64 64J 65 DETERMINATION OP TIME. Table of factors for reduction of transit observations. TOP ARGUMENT=STAR'S DECLINATION (J). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on thi* page.] 67 C 57 58 58J 59 59j 60 60i 61 61J 62 62j 63 63J 64 64{ 65 C 46 1.32 1.36 1.38 1.40 .42 1.44 1.46 1.48 .51 1.53 1.58 1.58 1.61 1.64 1.67 1.70 44 47 1.34 1.38 1.40 1.42 .44 1.46 1.49 1.51 .53 1.56 1.58 1.61 1.64 1.67 1.70 1.73 43 48 1.36 1.40 1.42 1.44 .46 1.48 1.51 1.53 .55 .58 1.60 1.53 1.66 1.69 1.72 1.76 42 49 1.39 1.42 1.44 1.47 .49 1.51 1.53 1.56 .58 .61 .K 1.66 .69 1.72 1.75 1.79 41 50 1.41 1.44 1.47 1.49 .51 1.53 1.56 1.58 .60 .63 .66 1.69 .72 1.75 1.78 1.81 40 51 1.43 1.47 1.49 1.51 .53 1.55 1.58 1.60 .63 .66 .68 1.71 .74 1.77 1.80 1.84 39 52 1.45 1.49 1.51 1.53 .55 1.58 1.60 1.63 .65 .68 .71 1.74 .77 1.80 1.83 1.86 38 53 1.47 1.51 1.53 1.55 .57 1.60 1.62 1.65 .67 .70 .73 1.76 .79 1.82 1.85 1.89 37 54 1.49 1.53 1.55 1.57 .59 1.62 1.64 1.67 1.69 .72 1.75 1.78 .81 1.85 1.88 1.91 36 55 1.50 1.55 1.57 1.59 .61 1.64 1.66 1.69 1.72 1.74 1.77 1.80 .84 1.87 1.90 1.94 35 56 1.52 1.56 1.59 1.61 .63 1.66 1.68 1.71 1.74 1.77 1.80 1.83 1.86 1.89 1.93 1.96 34 57 1.54 1.58 1.61 1.63 .65 1.68 1.70 1.73 1.76 1.79 1.82 1.85 1.88 1.91 1.95 1.98 33 58 1.56 1.60 1.62 1.65 .67 1.70 1.72 1.75 1.78 1.81 1.84 1.87 1.90 1.93 1.97 2.01 32 59 1.57 1.62 1.64 1.66 .69 1.71 .74 1.77 1.80 1.83 1.86 1.89 1.92 1.96 1 99 2 03 31 60 1.59 1.63 1.66 1.68 .71 1.73 .76 1.79 1.81 1.84 1.88 1.91 1.94 1.98 2.01 2.05 30 61 1.61 1.65 1.67 1.70 .72 1.75 .78 1.80 .83 1.86 1.89 1.93 1.96 2.00 2.03 2.07 29 62 1.62 1.67 1.69 1.71 .74 1.77 .79 1.82 .85 1.88 1.91 1.94 1.98 2.01 2.05 2.09 28 63 1.64 1.68 1.70 1.73 .76 1.78 .81 1.84 .87 1.90 1.93 1.96 2.00 2.03 2.07 2.11 27 64 1.65 1.70 1.72 1.75 .77 1.80 .83 1.85 .88 1.91 1.95 1.98 2.02 2.05 2.09 2.13 26 65 1.66 1.71 1.73 1.76 .79 1.81 .84 1.87 .90 1.93 1.96 2.00 2.03 2.07 2.11 2.14 25 66 1.68 1.72 1.75 1.77 .80 1.83 .85 1.88 .91 1.95 1.98 2.01 2.05 2.08 2.12 2.16 24 67 1.69 1.74 1.76. 1.79 .81 1.84 .87 1.90 .93 1.96 1.99 2.03 2.06 2.10 2.14 2.18 23 68 1.70 1.75 1.77 1.80 .83 1.85 .88 1.91 .94 1.97 2.01 2.04 2.08 2.11 2.15 2.19 22 69 1.71 1.76 1.79 1.81 1.84 1.87 .90 1.93 .96 1.99 2.02 2.06 2.09 2.13 2.17 2.21 21 70 1.73 1.77 1.80 1.82 1.85 1.88 .91 1.94 .97 2.00 2.03 2.07 2.11 2.14 2.18 2.22 20 71 1.74 1.78 1.81 1.84 1.86 1.89 .92 1.95 .98 2.01 2.05 2.08 2.12 2.16 2.20 2.24 19 72 1.75 1.79 1.82 1.85 1.87 1.90 .93 1.96 .99 2.03 2.06 2.09 2.13 2.17 2.21 2.25 18 73 1.76 1.80 1.83 1.86 1.88 1.91 .94 1.97 2.00 2.04 2.07 2.11 2.14 2.18 2.22 2.26 17 74 1.76 1.81 1.84 1.87 1.89 1.92 1.95 1.98 2.01 2.05 2.08 2.12 2.15 2.19 2.23 2.27 16 75 1.77 1.82 1.85 1.88 1.90 1.93 1.96 1.99 2.02 2.06 2.09 2.13 2.16 2.20 2.24 2.29 15 76 1.78 1.83 1.86 1.88 1.91 1.94 1.97 2.00 2.03 2.07 2.10 2.14 2.17 2.21 2.25 2.30 14 77 1.79 1.84 1.86 1.89 1.92 1.95 1.98 2.01 2.04 2.07 2.11 2.15 2.18 2.22 2.26 2.31 13 78 1.80 1.85 1.87 1.90 1.93 1.96 1.99 2.02 2.05 2.08 2.12 2.15 2.19 2.23 2.27 2.31 12 79 1.80 1.85 1.88 1.91 1.93 1.96 1.99 2.02 2.06 2.09 2.13 2.16 2.20 2.24 2.28 2.32 11 80 1.81 1.86 1.88 1.91 1.94 1.97 2.00 2.03 2.06 2.10 2.13 2.17 2.21 2.25 2.29 2.33 10 81 1.81 1.86 1.89 1.92 1.95 1.98 2.01 2.04 2.07 2.10 2.14 2.18 2.21 2.25 2.29 2.34 9 82 1.82 1.87 1.90 1.92 1.95 1.98 2.01 2.04 2.08 2.11 2.15 2.18 2.22 2.26 2.30 2.34 8 83 1.82 1.87 1.90 1.93 1.96 1.99 2.02 2.05 2.08 2.12 2.15 2.19 2.22 2.26 2.31 2.35 7 84 1.83 1.88 1.90 1.93 1.96 1.99 2.02 2.05 2.08 2.12 2.15 2.19 2.23 2.27 2.31 2.35 6 85 1.83 1.88 1.91 1.93 1.96 1.99 2.02 2.05 2.09 2.12 2.16 2.19 2.23 2.27 2.31 2.36 5 86 1.83 1.88 1.91 1.94 1.97 2.00 2.03 2.06 2.09 2.13 2.16 2.20 2.24 2.28 2.32 2.36 4 NT 1.83 1.88 1.91 1.94 1.97 2.00 2.03 2.06 2.09 2.13 2.16 2.20 2.24 2.28 2.32 2.36 3 \S 1.83 1.89 1.91 1.94 1.97 2.00 2.03 2.06 2.09 2.13 2.16 2.20 2.24 2.28 2.32 2.36 2 89 1.84 1.89 1.91 1.94 1.97 2.00 2.03 2.06 2.10 2.13 2.17 2.20 2.24 2.28 2.32 2.37 1 !M> 1.84 1.89 1.91 1.94 1.97 2.00 2.03 2.06 2.10 2.13 2.17 2.20 2.24 2.28 2.32 2.37 57 58 58J" 59 59J 60 60J 61 81} 62 62J 63 63} 64 64J 65" 68 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Table of factors for reduction of transit observations. TOP ARGUMENT=STAR'S DECLINATION (3). SIDE ARGUMENT=STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on opposite page.] C 65" 65} 66 66} 67 67J 68 681 69 69 10' 69 20' 69 30' 69 40' 69 50' 70 70 10' ; 1 .04 .04 .04 .04 .04 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 89 2 .08 .08 .09 .09 .09 .09 .09 .10 .10 .10 .10 .10 .10 .10 .10 .10 88 3 .12 .13 .13 .13 .13 .14 .14 .14 .15 .15 .15 .15 .15 .15 .15 .15 87 4 .17 .17 .17 .18 .18 .18 .19 .19 .20 .20 .20 .20 .20 .20 .20 .20 86 5 .21 .21 .21 .22 .22 .23 .23 .24 .24 .24 .25 .25 .25 .25 .25 .26 85 6 .25 .25 .26 .26 .27 .27 .28 .28 .29 .29 .30 .30 .30 .30 .31 .31 84 7 .29 .29 .30 .31 .31 .32 .33 .33 .34 .34 .34 .35 .35 .35 .36 .36 83 8 .33 .34 .34 .35 .36 .36 .37 .38 .39 .39 .39 .40 .40 .40 .41 .41 82 9 .37 .38 .39 .39 .40 .41 .42 .43 .44 .44 .44 .45 .45 .45 .46 .46 81 10 .41 .42 .43 .43 .44 .45 .46 .47 .48 .49 .49 .50 .50 .50 .51 .51 80 11 .45 .46 .47 .48 .49 .50 .51 .52 .53 .54 .54 .54 .55 .55 .56 .56 79 12 .49 .50 .51 .52 .53 .54 .56 .57 .58 .58 .59 .59 .60 .60 .61 .61 78 13 .53 .54 .55 .56 .58 .59 .60 .61 .63 .63 .64 .64 .65 .65 .66 .66 77 14 .57 .58 .59 .61 .62 .63 .65 .66 .67 .68 .68 .69 .70 .70 .71 .71 76 15 .61 .62 .64 .65 .66 .68 .69 .71 .72 .73 .73 .74 .74 .75 .76 .76 75 16 .65 .66 .68 .60 .71 .72 .74 .75 .77 .78 .78 .79 .79 .80 .81 .81 74 17 .69 .70 .72 .73 .75 .76 .78 .80 .81 .82 .83 .83 .84 .85 .85 .86 73 18 .73 .74 .76 .77 .79 .81 .83 .84 .86 .87 .88 .88 .89 .90 .90 .91 72 19 .77 .78 .80 .82 .83 .85 .87 .89 .91 .92 .92 .93 .94 .94 .95 .96 71 20 .81 .82 .84 .86 .88 .89 .91 .93 .95 .96 .97 .98 .98 .99 1.00 1.01 70 21 .85 .86 .88 .90 .92 .94 .96 .98 1.00 .01 1.02 .02 1.03 1.04 1.05 1.06 69 22 .89 .90 .92 .94 .96 .98 1.00 1.02 1.05 .05 1.06 .07 1.08 . 1.09 1.09 1.10 68 23 .92 .94 .96 .98 1.00 1.02 1.04 1.07 1.09 .10 1.11 .12 1.12 ' 1.13 1.14 1.15 67 24 .96 .98 1.00 1.02 1.04 1.06 1.09 1.11 1.14 .14 1.15 .16 1.17 1.18 1.19 1.20 66 25 1.00 1.02 1.04 1.06 1.08 1.10 1.13 1.15 1.18 .19 1.20 .21 1.22 1.23 1.24 1.25 65 26 1.04 1.06 1.08 1.10 1.12 1.15 1.17 1.20 1.22 1.23 .24 .25 1.26 .27 1.28 1.29 64 27 1.07 1.09 1.12 1.14 1.16 1.19 1.21 1.24 1.27 1.28 .29 .30 1.31 .32 1.33 1.34 63 28 1.11 1.13 1.15 1.18 1.20 1.23 1.25 1.28 1.31 1.32 .33 .34 1.35 .36 1.37 1.38 62 29 1.15 1.17 1.19 .22 1.24 1.27 1.29 1.32 1.35 1.36 .37 .38 1.40 .41 1.42 1.43 61 30 1.18 1.21 1.23 .25 1.28 1.31 1.33 1.36 1.39 1.41 .42 .43 1.44 .45 1.46 1.47 60 31 1.22 1.24 .27 .29 1.32 1.35 1.38 1.40 1.44 1.45 .46 .47 1.48 .49 1.51 1.52 59 32 1.25 1.28 .30 .33 1.36 1.39 1.42 .45 1.48 1.49 .50 .51 1.52 .54 1.55 1.56 58 33 1.29 1.31 .34 .37 1.39 1.42 1.45 .49 1.52 1.53 .54 .55 1.57 .58 1.59 1.60 57 34 1.32 1.35 .37 .40 1.43 1.46 1.49 .53 1.56 1.57 .58 .60 1.61 .62 1.63 1.65 56 35 1.36 1.38 .41 .44 1.47 1.50 1.53 .56 1.60 1.61 .62 1.64 1.65 .66 1.68 1.69 55 36 1.39 1.42 .45 .47 1.51 1.54 1.57 .60 1.64 1.55 .66 1.68 1.69 .70 1.72 1.73 54 37 1.42 1.45 .48 .51 1.54 1.57 1.61 .64 1.68 1.69 .70 1.72 1.73 .74 1.76 1.77 53 38 1.46 1.48 .51 .54 1.58 1.61 1.64 .68 1.72 1.73 .74 1.76 1.77 .79 1.80 1.82 52 39 1.49 1.52 .55 .58 1.61 1.65 1.68 .72 1.75 1.77 .78 1.80 1.81 .82 1.84 1.86 51 40 1.52 1.55 .58 .61 1.65 1.68 1.72 .75 1.79 1.81 .82 1.84 1.85 .86 1.88 1.89 50 41 1.55 1.58 .61 .64 1.68 .71 1.75 .79 1.83 1.84 .86 1.87 1.89 .90 1.92 1.93 49 42 1.58 1.61 .64 .68 1.71 .75 1.79 .83 1.87 1.88 .90 1.91 1.93 .94 1.96 1.97 48 43 1.61 1.64 .68 .71 1.75 .78 1.82 .86 1.90 1.92 .93 1.95 1.96 .98 1.99 2.01 47 44 1.64 1.67 .71 .74 1.78 .82 1.85 1.90 1.94 1.95 .97 1.98 2.00 2.02 2.03 2.05 46 45 1.67 1.70 .74 1.77 1.81 .85 1.89 1.93 1.97 1.99 2.00 2.02 2.04 2.05 2.07 2.08 45 65 65} 66 661 67 671 68 68} 69 69 10' 69 20' 69 30' 69 40' 69 50' 70 70 10' DETERMINATION OF TIME. 69 Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (d). SIDE ARGUMENT=STAR'S ZENITH DISTANCE (0 [ For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on this page.] C 65 65J 66 66* 67 67 j 68 68J 69 69 10' 69 20' 69 30* 69 40' 69 50' 70 70 10' C 46 1.70 1.74 1.77 1.80 1.84 1.88 1.92 1.96 2.01 2.02 2.04 2.05 2.07 2.09 2.10 2.12 44 47 1.73 1.76 1.80 1.83 1.87 1.91 1.95 2.00 2.04 2.06 2.07 2.09 2.10 2.12 2.14 2.16 43 48 1.76 1.79 1.83 1.86 1.90 1.94 1.98 2.03 2.07 2.09 2.11 2.12 2.14 2.16 2.17 2.19 42 49 1.79 1.82 1.86 1.89 1.93 1.97 2.01 2.06 2.11 2.12 2.14 2.16 2.17 2.19 2.21 2.22 41 50 1.81 1.85 1.88 1.92 1.96 2.00 2.04 2.09 2.14 2.15 2.17 2.19 2.20 2.22 2.24 2.26 40 51 1.84 1.87 1.91 1.95 1.99 2.03 i 2.07 2.12 2.17 2.18 2.20 2.22 2.24 2.25 2.27 2.29 39 52 1.86 1.90 1.94 1.98 2.02 2.06 2.10 2.15 2.20 2.22 2.23 2.25 2.27 2.29 2.30 2.32 38 53 1.89 1.93 1.96 2.00 2.04 2.09 2.13 2.18 2.23 2.25 2.26 2.28 2.30 2.32 2.33 2.35 37 54 1.91 1.95 1.99 2.03 2.07 2.11 2.16 2.21 2.26 2.28 '2.29 2.31 2.33 2.35 2.37 2.38 36 55 1.94 1.98 2.01 2.05 2.10 2.14 2.19 2.23 2.29 2.30 2.32 2.34 2.36 2.38 2.40 2.41 35 56 1.96 2.00 2.04 2.08 2.12 2.17 2.21 2.26 2.31 2.33 2.35 2.37 2.39 2.40 2.42 2.44 34 57 1.98 2.02 2.06 2.10 2.15 2.19 2.24 2.29 2.34 2.36 2.38 2.39 2.41 2.43 2.45 2.47 33 58 2.01 2.05 2.08 2.13 2.17 2.22 2.26 2.31 2.37 2.38 2.40 2.42 2.44 2.46 2.48 2.50 32 59 2.03 2.07 2.11 2.15 2.19 2.24 2.29 2.34 2.39 2.41 2.43 2.45 2.47 2.49 2.51 2.53 31 60 2.05 2.09 2.13 2.17 2.22 2.26 2.31 2.36 2.42 2.44 2.45 2.47 2.49 2.51 2.53 2.55 30 61 2.07 2.11 2.15 2.19 2.24 2.29 2.33 2.39 2.44 2.46 2.48 2.50 2.52 2.54 2.56 2.58 29 62 2.09 2.13 2.17 2.21 2.26 2.31 2.36 2.41 2.46 2.48 2.50 2.52 2.54 2.56 2.58 2.60 28 63 2.11 2.15 2.19 2.23 2.28 2.33 2.38 2.43 2.49 2.50 2.52 2.54 2.56 2.58 2.60 2.63 27 64 2.13 2.17 2.21 2.25 2.30 2.35 ! 2.40 2.45 2.51 2.53 2.55 2.57 2.59 2.61 2.63 2.65 26 65 2.14 2.19 2.23 2.27 2.32 2.37 2.42 2.47 2.53 2.55 2.57 2.59 2.61 2.63 2.65 2.67 25 66 2.16 2.20 2.25 2.29 2.34 2.39 2.44 2.49 2.55 2.57 2.59 2.61 2.63 2.65 2.67 2.69 24 67 2.18 2.22 2.26 2.31 2.36 2.41 2.46 2.51 2.57 2.59 2.61 2.63 2.65 2.67 2.69 2.71 23 68 2.19 2.24 2.28 2.32 2.37 2.42 2.47 2.53 2.59 2.61 2.63 2.65 2.67 2.69 2.71 2.73 22 69 2.21 2.25 2.30 2.34 2.39 2.44 2.49 2.55 2.61 2.62 2.64 2.67 2.69 2.71 2.73 2.75 21 JO 2.22 2.27 2.31 2.36 2.40 2.46 2.51 2.56 2.62 2.64 2.66 2.68 2.70 2.73 2.75 2.77 20 71 2.24 2.28 2.32 2.37 2.42 2.47 2.52 2.58 2.64 2.66 2.68 2.70 2.72 2.74 2.77 2.79 19 72 2.25 2.29 2.34 2.38 2.43 2.49 2.54 2.59 2.65 2.67 2.70 2.72 2.74 2.76 2.78 2.80 18 73 2.26 2.31 2.35 2.40 2.45 2.50 2.55 2.61 2.67 2.69 2.71 2.73 2.75 2.77 2.80 2.82 17 74 2.27 2.32 2.36 2.41 2.46 2.51 2.57 2.62 2.68 2.70 2.72 2.74 2.77 2.79 2.81 2.83 16 75 2.29 2.33 2.37 2.42 2.47 2.52 2.58 2.64 2.70 2.72 2.74 2.76 2.78 2.80 2.82 2.85 15 76 2.30 2.34 2.39 2.43 2.48 2.54 2.59 2.65 2.71 2.73 2.75 2.77 2.79 2.81 2.84 2.86 14 77 2.31 2.35 2.40 2.44 2.49 2.55 2.60 2.66 2.72 2.74 2.76 2.78 2.80 2.83 2.85 2.87 13 78 2.31 2.36 2.40 2.45 2.50 2.56 2.61 2.67 2.73 2.75 2.77 2.79 2.81 2.84 2.86 2.88 12 79 2.32 2.37 2.41 2.46 2.51 2.57 2.62 2.68 2.74 2.76 2.78 2.80 2.82 2.85 2.87 2.89 11 80 2.33 2.38 2.42 2.47 2.52 2.57 2.63 2.69 2.75 2.77 2.79 2.81 2.83 2.86 2.88 2.90 10 81 2.34 2.38 2.43 2.48 2.53 2.58 2.64 2.69 2.76 2.78 2.80 2.82 2.84 2.86 2.89 2.91 9 82 2.34 2.39 2.43 2.48 2.53 2.59 2.64 2.70 2.76 2.78 2.81 2.83 2.85 2.87 2.90 2.92 8 83 2.35 2.39 2.44 2.49 2.54 2.59 2.65 2.71 2.77 2.79 2.81 2.83 2.86 2.88 2.90 2.92 7 84 2.35 2.40 2.45 2.49 2.55 2.60 2.66 2.71 2.78 2.80 2.82 2.84 2.86 2.88 2.91 2.93 6 85 2.36 2.40 2.45 2.50 2.56 2.60 2.66 2.72 2.78 2.80 2.82 2.84 2.87 2.89 2.91 2.94 5 86 2.36 2.41 2.45 2.50 2.55 2.61 2.66 2.72 2.78 2.80 2.83 2.85 2.87 2.89 2.92 2.94 4 87 2.36 2.41 2.46 2.50 2.56 2.61 2.67 2.72 2.79 2.81 2.83 2.85 2.87 2.90 2.92 2.94 3 88 2.36 2.41 2.46 2.51 2.56 2.61 2.67 2.73 2.79 2.81 2.83 2.85 2.88 2.90 2.92 2.95 2 89 2.37 2.41 2.46 2.51 2.56 2.61 2.67 2.73 2.79 2.81 2.83 2.86 2.88 2.90 2.92 2.95 1 90 2.37 2.41 2.46 2.51 2 56 2.61 2.67 2.73 2.79 2.81 2.83 2.86 2.88 2.90 2.92 2.95 65 65j 66 66J 67 67j 68 esr 69 69 10 7 69 yy 69 30' 69 40' 69 50' 70 70 10* 70 TJ. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Table of factors for reduction of transit observations. TOP AROUMENT=STAR'S DECLINATION (a). SIDE ARGUMENT-STAB'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor S use right-hand argument. For factor C use bottom line on opposite page.] C 70 10' 70 20' 70 30' 70 40' 70 50' 71 71 10' 71 20' 71 30' 714CK 71 50' 72 72 10' 72 20' 72 30' 72 40' C o 1 .05 .05 .05 .05 .05 .05 .05 .05 .05 .06 .06 .06 .06 .06 .06 .06 89 2 .10 .10 .10 .10 .11 .11 .11 .11 .11 .11 .11 .11 .11 .12 .12 .12 88 3 .15 .16 .16 .16 .16 .16 .16 .18 .16 .17 .17 .17 .17 .17 .17 .18 87 4 .20 .21 .21 .21 .21 .21 .22 .22 .22 .22 .22 .23 .23 .23 .23 .23 86 5 .26 .26 .26 .26 .26 .27 .27 .27 .27 .28 .28 .28 .28 .29 .29 .29 85 6 .31 .31 .31 .32 .32 .32 .32 .33 .33 .33 .34 .34 .34 .34 .35 .35 84 7 .36 .36 .37 .37 .37 .37 .38 .38 .38 .39 .39 .39 .40 .40 .41 .41 83 8 .41 .41 .42 .42 .42 .43 .43 .44 .44 .44 .45 .45 .45 .46 .46 .47 82 9 .46 .46 .47 .47 .48 .48 .48 .49 .49 .50 .50 .51 .51 .52 .52 .52 81 10 .51 .52 .52 .52 .53 .53 .54 .54 .55 .55 .56 .56 .57 .57 .58 .58 80 11 .56 .57 .57 .58 .58 .59 .59 .60 .60 .61 .61 .62 .62 .63 .63 .64 79 12 .61 .62 .62 .63 .63 .64 .64 .65 .66 .66 .67 .67 .68 .68 .69 .70 78 13 .66 .67 .67 .68 .68 .69 .70 .70 .71 .72 .72 .73 .74 .74 .75 .76 77 14 .71 .72 .72 .73 .74 .74 .75 .76 .76 .77 .78 .78 .79 .80 .80 .81 76 15 .76 .77 .78 .78 .79 .79 .80 .81 .81 .82 .83 .84 .84 .85 .86 .87 75 16 .81 .82 .83 .83 .84 .85 .85 .86 .87 .88 .88 .89 .90 .91 .92 .92 74 17 .86 .87 .88 .88 .89 .90 .90 .91 .92 .93 .94 .95 .96 .96 .97 .98 73 18 .91 .92 .93 .93 .94 .95 .96 .96 .97 .98 .99 1.00 1.01 1.02 1.03 1.04 72 19 .96 .97 .98 .98 .99 1.00 1.01 1.02 1.03 1.04 1.04 1.05 1.06 1.07 1.08 1.09 71 20 1.01 1.02 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 70 21 1.06 .06 1.07 .08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 69 22 1.10 .11 1.12 .13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.24 1.25 1.26 68 23 1.15 .16 1.17 .18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.28 1.29 1.30 1.31 67 24 1.20 .21 1.22 .23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.32 1.33 1.34 1.35 1.36 66 25 1.25 .26 1.27 .28 1.29 1.30 1.31 1.32 1.33 1.34 1.36 1.37 1.38 1.39 1.41 1.42 65 26 1.29 .30 1.31 1.32 1.34 1.35 1.36 1.37 1.38 1.39 1.41 1.42 1.43 1.44 1.46 1.47 64 27 1.34 .35 1.36 1.37 1.38 1.39 1.41 1.42 1.43 1.44 1.46 1.47 1.48 1.50 1.51 1.52 63 28 1.38 .40 1.41 1.42 1.43 1.44 1.45 1.47 1.48 1.49 1.51 1.52 1.53 1.55 1.56 1.58 62 29 1.43 .44 1.45 1.46 1.48 1.49 1.50 1.52 1.53 1.54 1.56 1.57 1.58 1.60 1.61 1.63 61 30 1.47 .49 1 50 1 51 1.52 1.54 1.55 1.56 1.58 1.59 1.60 1.62 1.63 1.65 1.66 1.68 60 31 1.52 .53 1.54 1.56 1.57 1.S8 1.60 1.61 1.62 1.64 1.65 1.67 1.68 1.70 1.71 1.73 59 32 1.56 .57 1.59 1.60 1.61 1.63 1.64 1.66 1.67 1.68 1.70 1.71 1.73 1.75 1.76 1.78 58 33 1.60 .62 1.63 1.64 1.66 1.67 1.69 .70 1.72 1 73 1 75 1.76 1.78 1.80 1.81 1.83 57 34 1.65 .66 1.68 1.69 1.70 1.72 1.73 .75 1.76 1.78 1.79 1.81 1.83 1.84 1.86 1.88 56 35 1.69 .70 1.72 1.73 1.75 1.76 1.78 .79 1.81 1.82 1.84 1.86 1.87 1.89 1.91 1.92 55 36 1.73 .75 1.76 1.78 1.79 1.80 1.82 .84 1.85 1.87 1.88 1.90 1.92 1.94 1.95 1.97 54 37 1.77 .79 1.80 1.82 1.83 1.85 1.86 .88 1.90 1.91 1.93 1.95 1.96 1.98 2.00 2.02 53 38 1.82 .83 1.84 1.86 1.88 1.89 1.91 .92 1.94 1.96 1.98 1.99 2.01 2.03 2.05 2.07 52 39 1.86 .87 1.89 1.90 1.92 1.93 1.95 .97 1.98 2.00 2.02 2.04 2.06 2.07 2.09 2.11 51 40 1.89 .91 1.93 1.94 1.96 1.97 1.99 2.01 2.03 2.04 2.06 2.08 2.10 2.12 2.14 2.16 50 41 1.93 1.95 1.96 1.98 2.00 2.01 2.03 2.05 2.07 2.09 2.10 2.12 2.14 2.16 2.18 2.20 49 42 1 97 1 99 2.00 2.02 2 04 2.05 2.07 2.09 2.11 2.13 2.15 2.16 2.18 2.20 2.22 2.25 48 43 2.01 2.03 2.04 2.06 2.08 2.09 2.11 2.13 2.15 2.17 2.19 2.21 2.23 2.25 2.27 2.29 47 44 2.05 2.06 2.08 2.10 2.12 2.13 2.1.5 2.17 2.19 2.21 2.23 2.25 2.27 2.29 2.31 2.33 46 45 2.08 2.10 2.12 2.14 2.15 2.17 2.19 2.21 2.23 2.25 2.27 2.29 2.31 2.33 2.35 2.37 45 7010' 70 20' 70 30' 70 40' 70 50' 71 71 Itr 71-W 71*30' 71 40' 71 50' 72 72 10' 72 20' 72 30' 72 40' DETERMINATION OF TIME. 71 Table of factors for reduction of transit observations, TOP ARGUMENT- STAR'S DECLINATION (J). SIDE ARGUMENT-STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on this paee.] C 70 10' 70 20' 70 30' 70 40' 70 50' 71 71 10' 71 20' 71 30' 71 40' 71 50' 72 72 10' 72 20' 72 30' 72 40' C 46 2.12 2.14 2.15 2.17 2.19 2.21 2.23 2.25 2.27 2.29 2.31 2.33 2.35 2.37 2.39 2.41 o 44 47 2.16 2.17 2.19 2.21 2.23 2.25 2.27 2.28 2.30 2.32 2.35 2.37 2.39 2.41 2.43 2.45 43 48 2.19 2.21 2.22 2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.43 2.45 2.47 2.49 42 49 2.22 2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46 2.49 2.51 2.53 41 50 2.26 2.28 2.29 2.31 2.33 2.35 2.37 2.39 2.41 2.44 2.46 2.48 2.50 2.52 2.55 2.57 40 51 2.29 2.31 2.33 2.35 2.37 2.39 2.41 2.43 2.45 2.47 2.49 2.51 2.54 2.56 2.58 2.61 39 52 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46 2.48 2.50 2.53 2.55 2.57 2.60 2.62 2.64 38 53 2.35 2.37 2.39 2.41 2.43 2.45 2.47 2.50 2.52 2.54 2.56 2.58 2.61 2.63 2.66 2.68 37 54 2.38 2.40 2.42 2.44 2.46 2.48 2.51 2.53 2.55 2.57 2.60 2.62 2.64 2.67 2.69 2.72 36 55 2.41 2.43 2.45 2.47 2.50 2.52 2.54 2.56 2.58 2.60 2.63 2.65 2.68 2.70 2.72 2.75 35 56 2.44 2.46 2.48 2.50 2.52 2.55 2.57 2.59 2.61 2.64 2.66 2.68 2.71 2.73 2.76 2.78 34 57 2.47 2.49 2.51 2.53 2.55 2.58 2.60 2.62 2.64 2.67 2.69 2.71 2.74 2.76 2.79 2.82 33 58 2.50 2.52 2.54 2.56 2.58 2.61 2.63 2.65 2.67 2.70 2.72 2.74 2.77 2.79 2.82 2.85 32 59 2.53 2.55 2.57 i 2.59 2.61 2.63 2.66 2.68 2.70 2.72 2.75 2.77 2.80 2.82 2.85 2.88 31 60 2.55 2.57 2.59 2.62 2.64 2.66 2.68 2.71 2.73 2.75 2.78 2.80 2.83 2.85 2.88 2.91 30 61 2.5S 2.60 2.62 2.64 2.66 2.69 2.71 2.73 2.76 2.78 2.80 2.83 2.86 2.88 2.91 2.94 29 62 2.60 2.62 2.64 2.67 2.69 2.71 2.74 2.76 2.78 2.81 2.83 2.86 2.88 2.91 2.94 2.96 28 63 2.63 2.65 2.67 2.69 2.71 2.74 2.76 2.78 2.81 2.83 2.86 2.88 2.91 2.94 2.96 2.99 27 64 2.65 2.67 2.69 2.72 2.74 2.76 2.78 2.81 2.83 2.86 2.88 2.91 2.94 2.96 2.99 3.02 26 65 2.67 2.69 2.71 2.74 2.76 2.78 2.81 2.83 2.86 2.88 2.91 2.93 2.96 2.99 3.01 3.04 25 66 2.69 2.71 2.74 2.76 2.78 2.81 2.83 2.85 2.88 2.90 2.93 2.96 2.98 3.01 3.04 3.07 24 67 2.71 2.74 2.76 2.78 2.80 2.83 2.85 2.88 2.90 2.93 2.95 2.98 3.01 3.03 3.06 3.09 23 68 2.73 2.76 2.78 2.80 2.82 2.85 2.87 2.90 2.92 2.95 2.97 3.00 3.03 3.06 3.08 3.11 22 69 2.75 2.77 2.80 2.82 2.84 2.87 2.89 2.92 2.94 2.97 2.99 3.02 3.05 3.08 3.10 3.13 21 JO 2.77 2.79 2.81 2.84 2.86 2.89 2.91 2.94 2.96 2.99 3.01 3.04 3.07 3.10 3.12 3.15 20 71 2.79 2.81 2.83 2.86 2.88 2.90 2.93 2.95 2.98 3.01 3.03 3.06 3.09 3.12 3.14 3.17 19 72 2.80 2.83 2.85 2.87 2.90 2.92 2.95 2.97 3.00 3.02 3.05 3.08 3.10 3.13 3.16 3.19 18 73 2.82 2.84 2.86 2.89 2.91 2.94 2.96 2.99 3.01 3.04 3.07 3.09 3.12 3.15 3.18 3.21 17 74 2.83 2.86 2.88 2.90 2.93 2.95 2.98 3.00 3.03 3.06 3.08 3.11 3.14 3.17 3.20 3.23 16 75 2.85 2.87 2.89 2.92 2.94 2.97 2.99 3.02 3.04 3.07 3.10 3.13 3.15 3.18 3.21 3.24 15 76 2.86 2.88 2.91 2.93 2.96 2.98 3.01 3.03 3.06 3. OS 3.11 3.14 3.17 3.20 3.23 3.26 14 77 2.87 2.90 2.92 2.94 2.97 2.99 3.02 3.04 3.07 3.10 3.12 3.15 3.18 3.21 3.24 3.27 13 78 2.88 2.91 2.93 2.95 2.9S 3.00 3.03 3.06 3. OS 3.11 3.14 3.16 3.19 3.22 3.25 3.28 12 79 2.89 2.92 2.94 2.96 2.99 3.02 3.04 3.07 3.09 3.12 3.15 3.18 3.20 3.23 3.28 3.29 11 80 2.90 2.93 2.95 2.97 3.00 3.02 3.05 3.08 3.10 3.13 3.16 3.19 3.22 3.24 3.27 3.31 10 81 2.91 2.94 2.96 2.98 3.01 3.03 3.06 3.09 3.11 3.14 3.17 3.20 3.23 3.25 3.28 3.32 9 82 2.92 2.94 2.97 2.99 3.02 3.04 3.07 3.09 3.U 3.15 3.18 3.20 3.23 3.26 3.29 3.32 8 83 2.92 2.95 2.97 3.00 3.02 3.05 3.08 3.10 3.13 3.16 3.18 3.21 3.24 3.27 3.30 3.33 7 84 2.93 2.96 2.98 3.00 3.03 3.06 3.08 3.11 3.13 3.16 3.19 3.22 3.25 3.28 3.31 3.34 6 85 2.94 2.96 2.98 3.01 3.03 3.08 3.09 3.11 3.14 3.17 3.20 3.22 3.25 3.28 3.31 3.34 5 86 2.94 2.96 2.99 3.01 3.04 3.06 3.09 3.12 3.14 3.17 3.20 3.23 3.26 3.29 3.32 '3.35 4 87 2.94 2.97 2.99 3.02 3.04 3.07 3.09 3.12 3.15 3.18 3.20 3.23 3.26 3.29 3.32 3.35 3 88 2.95 2.97 2.99 3.02 3.04 3.07 3.10 3.12 3.15 3.18 3.20 3.23 3.26 3.29 3.32 3.35 2 89 2.95 2.97 3.00 3.02 3.04 3.07 3.10 3.12 3.15 3.18 3.21 3.24 3.27 3.30 3.33 3.36 1 90 2.95 2.97 3.00 3.02 3.05 3.07 3.10 3.12 3.15 3.18 3.21 3.24 3.27 3.30 3.33 3.36 70 10' 7020' 70 30' 70 40' 70 50' 71 71 10' 71 20' 71 30' 71 40" 71 5V 72 72 10 7 72 20' 72 30' 72 40' 72 U. S. COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 14. Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (3). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on opposite page.] C 72" 40' 72 50' 73 73 ICC 73 20' 73 30' 73 40' 73 50' 74 74 W 74 20' 74 30' 74 40' 74 50' 75 75 10' <. 1 .06 .06 .06 .06 .06 .06 .06 .06 .06 .06 .06 .06 .07 .07 .07 .07 89 2 .12 .12 .12 .12 .12 .12 .12 .12 .13 .13 .13 .13 .13 .13 .13 .14 88 3 .18 .18 .18 .18 .18 .18 .19 .19 .19 .19 .19 .20 .20 .20 .20 .20 87 4 .23 .24 .24 .24 .24 .24 .25 .25 .25 .26 .26 .26 .26 .27 .27 .27 86 5 .29 .30 .30 .30 .30 .31 .31 .31 .32 .32 .32 .33 .33 .33 .34 .34 85 6 .35 .35 .36 .36 .36 .37 .37 .38 .38 .38 .39 .39 .40 .40 .40 .41 84 7 .41 .41 .42 .42 .42 .43 .43 .44 .44 .45 .45 .46 .46 .47 .47 .48 83 8 .47 .47 .48 .48 .48 .49 .50 .50 .50 .51 .52 .52 .53 .53 .54 .54 82 9 .52 .53 .53 .54 .54 .55 .56 .56 .57 .57 .58 .58 .59 .60 .60 .61 81 10 .58 .59 .59 .60 .60 .61 .62 .62 .63 .64 .64 .65 .66 .66 .67 .68 80 11 .64 .65 .65 .66 .66 .67 .68 .68 .69 .70 .71 .71 .72 .73 .74 .74 79 12 .70 .70 .71 .72 .72 .73 .74 .75 .75 .76 .77 .78 .79 .79 .80 .81 78 13 .76 .76 .77 .78 .78 .79 .80 .81 .82 .82 .83 .84 .85 .86 .87 .88 77 14 .81 .82 .83 .84 .84 .85 .86 .87 .88 .89 .90 .91 .92 .93 .94 .95 76 15 .87 .88 .89 .89 .90 .91 .92 .93 .94 .95 .96 .97 .98 .99 1.00 1.01 75 16 .92 .93 .94 .95 .96 .97 .98 .99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.08 74 17 .98 .99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.11 1.12 1.13 1.14 73 18 1.04 .05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.16 1.17 1.18 1.19 1.21 72 19 1.09 .10 1.11 1.12 1.14 1.15 1.16 1.17 1.18 1.19 1.21 1.22 1.23 1.24 1.26 1.27 71 20 1.15 .16 1.17 1.18 1.19 1.20 1.22 1.23 1.24 1.25 1.27 1.28 1.29 1.31 1.32 1.34 70 21 1.20 .21 1.22 1.24 1.25 1.26 .27 1.29 1.30 1.31 1.33 1.34 1.36 1.37 1.38 1.40 69 22 1.26 .27 1.28 1.29 1.31 1.32 .33 1.34 1.36 1.37 1.39 1.40 1.42 1.43 1.45 1.46 68 23 1.31 .32 1.34 1.35 1.36 1.38 .39 1.40 1.42 1.43 1.45 1.46 1.48 1.49 1.51 1.53 67 24 1.36 .38 .39 1.40 1.42 1.43 .45 1.46 1.48 1.49 1.51 1.52 1.54 1.55 1.57 1.59 66 25 1.42 .43 .45 1.46 1.47 1.49 .50 1.52 1.53 1.55 1.56 1.58 1.60 1.62 1.63 1.65 65 26 1.47 1.48 .50 1.51 1.53 1.54 1.56 1.58 1.59 1.61 1.62 1.64 1.66 1.68 1.69 1.71 64 27 1.52 1.54 .55 1.57 1.58 1.60 1.61 1.63 1.65 1.66 1.68 1.70 1.72 1.74 1.75 1.77 63 28 1.58 1.59 .60 1.62 1.64 1.65 1.67 1.69 1.70 1.72 1.74 1.76 1.78 1.79 1.81 1.83 62 29 1.63 1.64 .66 1.67 1.69 1.71 1.72 1.74 1.76 1.78 1.80 1.81 1.83 1.85 1.87 1.89 61 30 1.68 1.69 .71 1.73 1.74 1.76 1.78 1.80 1.81 1.83 1.85 1.87 1.89 1.91 1.93 1.95 60 31 1.73 1.74 .76 1.78 1.80 1.81 1.83 1.85 1.87 1.89 1.91 1.93 1.95 1.97 1.99 2.01 59 32 1.78 1.80 .81 1.83 1.85 1.87 1.88 1.90 1.92 1.94 1.96 1.98 2.00 2.02 2.05 2.07 58 33 1.83 1.85 .86 1.88 1.90 1.92 1.94 1.96 1.98 2.00 2.02 2.04 2.06 2.08 2.10 2.13 57 34 1.88 1.89 .91 1.93 1.95 1.97 1.99 2.01 2.03 2.05 2.07 2.09 2.12 2.14 2.16 2.18 56 35 1.92 1.94 .96 1.98 2.00 2.02 2.04 2.06 2.08 2.10 2.12 2.15 2.17 2.19 2.22 2.24 55 36 1.97 1.99 2.01 2.03 2.05 2.07 2.09 2.11 2.13 2.15 2.18 2.20 2.22 2.25 2.27 2.30 54 37 2.02 2.04 2.06 2.08 2.10 2.12 2.14 2.16 2.18 2.21 2.23 2.25 2.28 2.30 2.33 2.35 53 38 2.07 2.09 2.11 2.13 2.15 2.17 2.19 2.21 2.23 2.26 2.28 2.30 2.33 2.35 2.38 2.40 52 39 2.11 2.13 2.15 2.17 2.19 2.22 2.24 2.26 2.28 2.31 2.33 2.35 2.38 2.40 2.43 2.46 51 40 2.16 2.18 2.20 2.22 2.24 2.26 2.29 2.31 2.33 2.36 2.38 2.40 2.43 2.46 2.48 2.51 50 41 2.20 2.22 2.24 2.26 2.29 2.31 2.33 2.36 2.38 2.40 2.43 2.45 2.48 2.51 2.53 2.56 49 42 2.25 2.27 2.29 2.31 2.33 2.36 2.38 2.40 2.43 2.45 2.48 2.50 2.53 2.56 2.58 2.61 48 43 2.20 2.31 2.33 2.36 2.38 2.40 2.42 2.45 2.47 2.50 2.53 2.55 2.58 2.61 2.63 2.66 47 44 2.33 2.35 2.38 2.40 2.42 2.45 2.47 2.50 2.52 2.55 2.57 2.60 2.63 2.66 2.68 2.71 46 45 2.37 2.40 2.42 2.44 2.46 2.49 2.51 2.54 2.56 2.59 2.62 2.65 2.67 2.70 2.73 2.76 45 72 40' 72 50' 73 73 10' 73 20' 73 30' 73 40' 73 50' 74" 74 10' 74 20' 74 30' 74 40' 74 50' 75 75 HC DETERMINATION OF TIME. 73 Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (3). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on this page.] C 72" 40' 72 50' 73 73 10' 7320' 73 30' 73- 40' 73-50' 74 74 10' 74-20' 74-30' 74-40' 74-50' 75 75-10' C 46 2.41 2.44 2.46 2.48 2.51 2.53 2.56 2.58 2.61 2.64 2.66 2.69 2.72 2.75 2.78 2.81 44 47 2.45 2.48 2.50 2.52 2.55 2.57 2.60 2.63 2.65 2.6'i 2.71 2.74 2.77 2.80 2.83 2.86 43 48 2.49 2.52 2.54 2.57 2.59 2.62 2.64 2.67 2.70 2.72 2.75 2.78 2.81 2.84 2.87 2.90 42 49 2.53 2.56 2.58 2.61 2.63 2.66 2.68 2.71 2.74 2.77 2.80 2.82 2.85 2.88 2.92 2.95 41 50 2.57 2.60 2.62 2.64 2.67 2.70 2.72 2.75 2.78 2.81 2.84 2.87 2.90 2.93 2.96 2.99 40 51 2.61 2.63 2.66 2.6S 2.71 2.74 2.76 2.79 2.82 2.85 2.88 2.91 2.94 2.97 3.00 3.04 39 52 2.64 2.67 2.69 2.72 2.75 2.77 2.80 2.83 2.86 2.89 2.92 2.95 2.98 3.01 3.04 3.08 33 53 2.68 2.71 2.73 2.76 2.78 2.81 2.84 2.87 2.90 2.93 2.96 2.99 3.02 3.05 3.09 3.12 37 54 2.72 2.74 2.77 2.79 2.82 2.85 2.88 2.91 2.94 2.97 3.00 3.03 3.06 3.09 3.13 3.16 36 55 2.75 2.78 2.80 2.83 2.86 2.88 2.91 2.94 2.97 3.00 3.03 3.07 3.10 3.13 3.16 3.20 35 56 2.78 2.81 2.84 2.86 2.89 2.92 2.95 2.98 3.01 3.04 3.07 3.10 3.14 3.17 3.20 3.24 ' 34 57 2.82 2.84 2.87 2.90 2.92 2.93 2.98 3.01 3.04 3.07 3.11 3.14 3.17 3.21 3.24 3.28 33 58 2.85 2.87 2.90 2.93 2.96 2.99 3.02 3.05 3.08 3.11 3.14 3.17 3.21 3.24 3.23 3.31 32 59 2.88 2.90 2.93 2.% 2.99 3.02 3.05 3.08 3.11 3.14 3.17 3.21 3.24 3.28 3.31 3.35 31 60 2.91 2.93 2.96 2.99 3.02 3.05 3.08 3.11 3.14 3.17 3.21 3.24 3.28 3.31 3.35 3.3S 30 61 2.94 2.96 2.99 3.02 3.05 3.08 3.11 3.14 3.17 3.21 3.24 3.27 3.31 3.34 3.38 3.42 29 62 2.96 2.99 3.02 3.05 3.08 3.11 3.14 3.17 3.20 3.24 3.27 3.30 3.34 3.3S 3.41 3.45 28 63 2.99 3.02 3.05 3. OS 3.11 3.14 3.17 3.20 3.23 3.27 3.30 3.33 3.37 3.41 3.44 3.43 27 64 3.02 3.04 3.07 3.10 3.13 3.16 3.20 3.23 3.26 3.29 3.33 3.36 3.40 3.44 3.47 3.51 26 65 3.04 3.07 3.10 3.13 3.16 3.19 3.22 3.26 3.29 3.32 3.36 3.39 3.43 3.46 3.50 3.54 25 66 3.07 3.10 3.13 3.16 3.18 3.22 3.25 3.28 3.31 3.35 3.38 3.42 3.46 3.49 3.53 3.57 24 67 3.09 3.12 3.15 3.18 3.21 3.24 3.27 3.31 3.34 3.37 3.41 3.44 3.48 3.52 3.56 3.60 23 68 3.11 3.14 3.17 3.20 3.23 3.26 3.30 3.33 3.36 3.40 3.43 3.47 3.51 3.54 3.58 3.62 22 69 3.13 3.16 3.19 3.22 3.26 3.29 3.32 3.35 3.39 3.42 3.46 3.49 3.53 3.57 3.61 3.65 21 70 3.15 3.18 3.21 3.24 3.28 3.31 3.34 3.38 3.41 3.44 3.48 3.52 3.55 3.59 3.63 3.67 20 71 3.17 3.20 3.23 3.26 3.30 3.33 3.36 3.40 3.43 3.47 3.50 3.54 3.58 3.61 3.65 3.69 19 72 3.19 3.22 3.25 3.28 3.32 3.35 3.38 3.42 3.45 3.49 3.52 3.56 3.60 3.63 3.67 3.72 IS 73 3.21 3.24 3.27 3.30 ?.33 3.37 3.40 3.44 3.47 3.50 3.54 3.58 3.62 3.65 3.69 3.74 17 74 3.23 3.26 3.29 3.32 3.35 3.38 3.42 3.45 3.49 3.52 3.56 3.60 3.64 3.67 3.71 3.76 16 75 3.24 3.27 3.30 3.34 3.37 3.40 3.44 3.47 3.5C 3.54 3.58 3.61 3.65 3.69 3.73 3.77 15 76 3.26 3.29 3.32 3.35 3.38 3.42 3.45 3.48 3.52 3.56 3.59 3.63 3.67 3.71 3.75 3.79 14 77 3.27 3.30 3.33 3.36 3.40 3.43 3.46 3.50 3.54 3.57 3.61 3.65 3.68 3.72 3.76 3.81 13 78 3.28 3.31 3.34 3.38 3.41 3.44 3.48 3.51 3.55 3.58 3.62 3.66 3.70 3.74 3.78 3.82 12 79 3.29 3.33 3.36 3.39 3.42 3.46 3.49 3.53 3.56 3.60 3.64 3.67 3.71 3.75 3.79 3.83 11 80 3.31 3.34 3.37 3.40 3.43 3.47 3.50 3.54 3.57 3.61 3.65 3.68 3.72 3.76 3.81 3.85 10 81 3.32 3.35 3.38 3.41 3.44 3.48 3.51 3.55 3.58 3.62 3.66 3.70 3.74 3.78 3.82 3.86 9 82 3.32 3.36 3.39 3.42 3.45 3.49 3.52 3.56 3.59 3.63 3.67 3.71 3.75 3.79 3.83 3.87 8 83 3.33 3.36 3.40 3.43 3.46 3.49 3.53 3.56 3.60 3.64 3.68 3.72 3.75 3.79 3.84 3.88 7 84 3.34 3.37 3.40 3.43 3.47 3.50 3.54 3.57 3.61 3.64 3.68 3.72 3.76 3.80 3.84 3.88 6 85 3.34 3.38 3.41 3.44 3.47 3.51 3.54 3.58 3.61 3.65 3.69 3.73 3.77 3.81 3.85 3.89 5 86 3.35 3.38 3.41 3.44 3.48 3.51 3.55 3.58 3.62 3.66 3.69 3.73 3.77 3.81 3.85 3.90 4 87 3.35 3.38 3.42 3.45 3.48 3.52 3.55 3.59 3.62 3.66 3.70 3.74 3.78 3.82 3.86 3.90 3 88 3.35 3.39 3.42 3.45 3.48 3.52 3.55 3.59 3.62 3.66 3.70 3.74 3.78 3.82 3.86 3.90 2 89 3.36 3.39 3.42 3.45 3.49 3.52 3.56 3.59 3.63 3.66 3.70 3.74 3.78 3.82 3.86 3.91 1 90 3.36 3.39 3.42 3.45 3.49 3.52 3.56 3.59 3.63 3.66 3.70 3.74 3.78 3.82 3.86 3.91 72-40' 72 50' 73 73 10* 73 20' 73 30! 73 40' 73-50' 74 7410 / 74 20' 74-30' 74 W 74-50' 75 75 W 74 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on opposite page.] C 75 W 75 20' 75 30' 75 40' 75 50' J6 76 HK 76 20' 76 30' 76 40' 76-50' 77 77 10' 77 20' 77 30' 77 40' C 1 .07 .07 .07 .07 .07 .07 .07 .07 .07 .08 .08 .08 .08 .08 .08 .08 89 2 .14 .14 .14 .14 .14 .14 .15 .15 .15 .15 .15 .16 .16 .16 .16 .16 88 3 .20 .21 .21 .21 .21 .22 .22 .22 .22 .23 .23 .23 .24 .24 .24 .24 87 4 .27 .28 .28 .28 .28 .29 .29 .30 .30 .30 .31 .31 .31 .32 .32 .33 86 5 .34 .34 .35 .35 .36 .36 .36 .37 .37 .38 .38 .39 .39 .40 .40 .41 85 6 .41 .41 .42 .42 .43 .43 .44 .44 .45 .45 .46 .46 .47 .48 .48 .49 84 7 .48 .48 .49 .49 .50 .50 .51 .52 .52 .53 .54 .54 .55 .56 .56 .57 83 g .54 .55 .56 .56 .57 .58 .58 .59 .60 .60 .61 .62 .63 .64 .64 .65 82 9 .61 .62 .62 .63 .64 .65 .65 .66 .67 .68 .69 .70 .70 .71 .72 .73 81 10 .68 .69 .69 .70 .71 .72 .73 .74 .74 .75 .76 . 77 .78 .79 .80 .81 80 11 1 .74 .75 .76 .77 .78 .79 .80 .81 .82 .83 .84 .85 .86 .87 .88 .89 79 12 .81 .82 .83 .84 .85 .86 .87 .88 .89 .90 *.91 .92 .94 .95 .96 .97 78 13 .88 .89 .90 .91 .92 .93 .94 .95 .96 .98 .99 1.00 1.01 1.03 1.04 1.05 77 14 .95 .96 .97 .98 .99 1.00 1.01 1.02 .04 1.05 1.06 1.08 1.09 1.10 1.12 1.13 76 15 1.01 1.02 1.03 1.04 1.08 1.07 1.08 1.10 .11 1.12 1.14 1.15 1.16 1.18 1.20 1.21 75 16 .08 1.09 1.10 1.11 1.13 1.14 1.15 1.17 .18 1.20 1.21 1.23 1.24 1.26 1.28 1.29 74 17 .14 1.16 1.17 1.18 1.20 1.21 1.22 1.24 .25 1.27 1.28 1.30 1.32 1.33 1.35 1.37 73 18 .21 1.22 1.23 1.25 1.26 1.28 1.29 1.31 .32 1.34 1.36 1.37 1.39 1.41 1.43 1.45 72 19 .27 1.29 1.30 1.32 1.33 1.35 1.36 1.38 .39 1.41 1.43 1.45 1.47 1.48 1.50 1.52 71 20 .34 1.35 1.37 1.38 1.40 1.41 1.43 1.45 .47 1.48 1.50 1.52 1.54 1.56 1.58 1.60 70 21 .40 1.12 1.43 1.45 1.46 1.48 .50 1.52 .54 1.55 1.57 1.59 1.61 1.63 1.65 1.68 69 22 .46 1.48 1.50 1.51 1.53 1.55 .57 1.58 .60 1.62 1.64 1.66 1.69 1.71 1.73 1.75 68 23 .53 1.54 1.56 1.58 1.60 1.62 .63 1.65 .67 1.69 1.72 1.74 1.76 1.78 1.81 1.83 67 24 .59 1.61 1.63 1.64 1.66 1.68 .70 1.72 .74 1.76 1.79 1.81 1.83 1.86 1.88 1.90 66 25 .65 1.67 1.69 1.71 1.73 1.75 .77 1.79 .81 1.83 1.86 1.88 1.90 1.93 1.95 1.98 65 26 .71 1.73 1.75 1.77 1.79 1.81 .83 1.86 .88 1.90 1.92 1.95 1.97 2.00 2.02 2.05 64 27 .77 1.79 1.81 1.83 1.86 1.88 .90 1.92 .95 1.97 1.99 2.02 2.04 2.07 2.10 2.12 63 23 .83 1.85 1.87 1.90 1.92 1.94 .96 1.99 2.01 2.04 2.06 2.09 2.11 2.14 2.17 2.20 62 29 1.89 1.92 1.94 1.96 1.98 2.00 2.03 2.05 2.08 2.10 2.13 2.15 2.18 2.21 2.24 2.27 61 30 1.95 1.98 2.00 2.02 2.04 2.07 2.09 2.12 2.14 2.17 2.20 2.22 2.25 2.28 2.31 2.34 60 31 2.01 2.03 2.06 2.08 2.10 2.13 2.15 2.18 2.21 2.23 2.26 2.29 2.32 2.35 2.38 2.41 59 32 2.07 2.09 2.12 2.14 2.16 2.19 2.22 2.24 2.27 2.30 2.33 2.36 2.39 2.42 2.45 2.48 58 33 2.13 2.15 2.18 2.28 2.22 2.25 2.28 2.30 2.33 2.36 2.39 2.42 2.45 2.48 2.52 2.55 57 34 2.18 2.21 2.23 2.26 2.28 2.31 2.34 2.37 2.40 2.42 2.46 2.49 2.52 2.55 2.58 2.62 56 35 2.24 2.26 2.29 2.32 2.34 2.37 2.40 2.43 2.46 2.49 2.52 2.55 2.58 2.62 2.65 2.68 55 36 2.30 2.32 2.35 2.37 2.40 2.43 2.46 2.49 2.52 2.55 2.58 2.61 2.65 2.68 2.72 2.75 54 37 2.35 2.38 2.40 2.43 2.46 2.49 2.52 2.55 2.58 2.61 2.64 2.67 2.71 2.74 2.78 2.82 53 38 2.40 2.43 2.46 2.49 2.52 2.55 2.58 2.61 2.64 2.67 2.70 2.74 2.77 2.81 2.85 2.88 52 39 2.46 2.49 2.51 2.54 2.57 2.60 2.63 2.66 2.70 2.73 2.76 2.80 2.83 2.87 2.91 2.95 51 40 2.51 2.54 2.57 2.60 2.63 2.66 2.69 2.72 2.75 2.79 2.82 2.86 2.89 2.93 2.97 3.01 50 41 2.56 2.59 2.62 2.65 2.68 2.71 2.74 2.78 2.81 2.84 2.88 2.92 2.95 2.99 3.03 3.07 49 42 2.61 2.64 2.67 2.70 2.73 2.77 2.80 2.83 2.87 2.90 2.94 2.97 3.01 3.05 3.09 3.13 48 43 2.66 2.69 2.72 2.76 2.79 2.82 2.85 2.89 2.92 2.96 2.99 3.03 3.07 3.11 3.15 3.19 47 44 2.71 2.74 2.77 2.81 2.84 2.87 2.90 2.94 2.98 3.01 3.05 3.09 3.13 3.17 3.21 3.25 46 45 2.76 2.79 2.82 2.86 2.89 2.92 2.96 2.99 3.03 3.07 3.10 3.14 3.18 3.22 3.27 3.31 45 75 10' 75 20' 75 30' 75 40' 75 50' 76 76 10' 76-20' 76 30' 76 40' 76 50' 77 77 10' 77 20' 77 30' 77" W DETEEMINATION OF TIME. 75 Table of factors for reduction of transit observations. TOP ARGUMENT- STAR'S DECLINATION (). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on this page.] C 75 10' 75 20' 75 30' 75" 40' 75 50' 78 76 IV 76 20' 76 30' 76 40' 76 50' 77 77 10' 77 20' 77 30' 77 40' C 46 2.81 2.84 2.87 2.91 2.94 2.97 3.01 3.04 3.08 3.12 3.16 3.20 3.24 3.28 3.32 3.37 44 47 2.86 2.89 2.92 2.95 2.99 3.02 3.06 3.10 3.13 3.17 3.21 3.25 3.29 3.34 3.38 3.42 43 48 2.90 2.94 2.97 3.00 3.04 3.07 3.11 3.15 3.18 3.22 3.26 3.30 3.35 3.39 3.43 3.48 42 49 2.95 2.98 3.01 3.05 3.08 3.12 3.16 3.19 3.23 3.27 3.31 3.36 3.40 3.44 3.49 3.53 41 50 2.99 3.02 3.06 3.09 3.13 3.17 3.20 3.24 3.28 3.32 3.36 3.41 3.45 3.49 3.54 3.59 40 51 3.04 3.07 3.10 3.14 3.18 3.21 3.25 3.29 3.33 3.37 3.41 3.45 3.50 3.54 3.59 3.64 39 52 3.08 3.11 3.15 3.18 3.22 3.26 3.30 3.34 3.38 3.42 3.46 3.50 3.55 3.59 3.64 3.69 38 53 3.12 3.15 3.19 ?.23 3.26 3.30 3.34 3.38 3.42 3.46 3.51 3.55 3.60 3.64 3.69 3.74 37 54 3.16 3.20 3.23 3.27 3.31 3.34 3.38 3.42 3.47 3.51 3.55 3.60 3.64 3.69 3.74 3.79 36 55 3.20 3.24 3.27 3.31 3.35 3.39 3.43 3.47 3.51 3.55 3.60 3.64 3.69 3.74 3.78 3.83 35 56 3.24 3.27 3.31 3.35 3.39 3.43 3.47 3.51 3.55 3.60 3.64 3.68 3.73 3.78 3.83 3.88 34 57 3.28 3.31 3.35 3.39 3.43 3.47 3.51 3.55 3.59 3.64 3.6S 3.73 3.78 3.83 3.88 3.93 33 58 3.31 3.35 3.39 3.43 3.47 3.51 3.55 3.59 3.63 3.68 3.72 3.77 3.82 3.87 3.92 3.97 32 59 3.35 3.38 3.42 3.46 3.50 3.54 3.58 3.63 3.67 3.72 3.76 3.81 3.86 3.91 3.96 4.01 31 60 3.38 3.42 3.46 3.50 3.54 3.58 3.62 3.66 3.71 3.76 3.80 3.85 3.90 3.95 4.00 4.05 30 61 3.42 3.45 3.49 3.53 3.57 3.62 3.66 3.70 3.75 3.79 3.84 3.89 3.94 3.99 .04 4.09 29 62 3.45 3.49 3.53 3.57 3.61 3.65 3.69 3.74 3.78 3.83 3.88 3.93 3.98 4.03 .08 4.13 28 63 3.48 3.52 3.56 3.60 3.64 3.68 3.73 3.77 3.82 3.86 3.91 3.96 4.01 4.06 .12 4.17 27 64 3.51 3.55 3.59 3.63 3.67 3.72 3.76 3.80 3.85 3.90 3.95 4.00 4.05 4.10 .15 4.21 26 65 3.54 3.58 3.62 3.66 3.70 3.75 3.79 3.84 3.88 3.93 3.98 4.03 4.08 4.13 .19 4.24 25 66 3.57 3.61 3.65 3.69 3.73 3.78 3.82 3.87 3.91 3.96 4.01 4.06 4.11 4.17 .22 4.28 24 67 3.60 3.64 3.68 3.72 3.76 3.81 3. &5 3.90 3.94 3.99 4.04 4.09 4.14 4.20 .25 4.31 23 68 3.62 3.66 3.70 3.74 3.79 3.83 3.88 3.92 3.97 4.02 4.07 4.12 4.17 4.23 .28 4.34 22 69 3.65 3.69 3.73 3.77 3.82 3.86 3.90 3.95 .00 4.05 4.10 4.15 4.20 4.26 .31 4.37 21 70 3.67 3.71 3.75 3.80 3.84 3.89 3.93 3.98 .03 4.08 4.12 4.18 4.23 4.28 .34 4.40 20 71 3.69 3.73 3.78 3.82 3.86 3.91 3.96 .00 .05 4.10 4.15 4.20 4.26 4.31 .37 4.43 19 72 3.72 3.76 3.80 3.84 3.89 3.93 3.98 .02 .07 4.12 4.18 4.23 4.28 4.34 .39 4.45 18 73 3.74 3.78 3.82 3.86 3.91 3.95 4.00 .05 .10 4.15 4.20 4.25 4.30 4.36 .42 4.48 17 74 3.76 3.80 3.84 3.88 3.93 3.97 4.02 .07 .12 4.17 4.22 4.27 4.33 4.38 .44 4.50 16 75 3.77 DO O. O4 3.86 3.90 3.95 3.99 4.04 .09 .14 4.19 4.24 4.29 4.35 4.40 .46 4.52 15 76 3.79 3.83 3.88 3.92 3.96 4.01 4.06 .11 .16 4.21 4.26 4.31 4.37 4.42 .48 4.54 14 77 3.81 3.85 3.89 3.94 3.98 4.03 4.08 .12 .17 4.22 4.28 4.33 4.39 4.44 .50 4.56 13 78 3.82 3.86 3.91 3.95 4.00 4.04 4.09 .14 .19 4.24 4.29 4.35 4.40 4.46 .52 4.58 12 79 3.83 3.88 3.92 3.96 4.01 4.06 4.11 4.16 4.21 4.26 4.31 4.36 4.42 4.48 .54 4.60 11 80 3.85 3.89 3.93 3.98 4.02 4.07 4.12 4.17 4.22 4.27 4.32 4.38 4.43 4.49 .55 4.61 10 81 3.86 3.90 3.94 3.99 4.04 4.08 4.13 4.18 4.23 4.28 4.34 4.39 .45 4.50 .56 4.62 9 82 3.87 3.91 3.96 4.00 4.05 4.09 4.14 4.19 4.24 4.29 4.35 4.40 .46 4.52 .58 4.64 8 83 3.88 3.92 3.96 4.01 4.06 4.10 4.15 4.20 4.25 4.30 4.36 4.41 .47 4.53 .59 4.65 7 84 3.88 3.93 3.97 4.02 4.06 4.11 4.16 4.21 4.26 4.31 4.37 4.42 .48 4.54 .60 4.66 6 85 3.89 3.93 3.98 4.02 4.07 4.12 4.17 4.22 4.27 4.32 4.37 4.43 .48 4.54 4.60 4.66 5 86 3.90 3.94 3.98 4.03 4.08 4.12 4.17 4.22 4.27 4.33 4.38 4.43 .49 4.55 4.61 4.67 4 87 3.90 3.94 3.99 4.03 4.08 4.13 4.18 4.23 4.28 4.33 4.38 4.44 .50 4.55 4.61 4.68 3 88 3.90 3.95 3.99 4.04 4.08 4.13 4.18 4.23 4.28 4.33 4.39 4.44 .50 4.56 4.62 4.68 2 89 3.91 3.95 3.99 4.04 4.08 4.13 4.18 4.23 4.28 4.34 4.39 4.44 .50 4.56 4.62 4.68 1 90 3.91 3.95 3.99 4.04 4.09 4.13 4.18 4.23 4.28 4.34 4.39 4.44 .50 4.56 4.62 4.68 75 10' 75 20' 75 30' 75 40' 75 50' 76 76 10' 76 20' 76 30' 76 40' 76 50' 77 77 10' 77 20' 77 bO' 77 40' 76 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Table of factors for reduction of transit observations. TOP ARGUMENT=STAR'S DECLINATION (J). SIDE ARGUMENT- STAR'S ZENITH DISTANCE (C). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on opposite page.] C 77 40" 77 50' 78 78 10' 78 20' 78 30' 73 40' 78 50' 79 79 10' 79 20' 79 30' 79 40' 79 50' 80 C 1 .OS .08 .08 .08 .09 .09 .09 .09 .09 .09 .09 .10 .10 .10 .10 89 2 .16 .17 .17 .17 .17 .18 .18 .18 .IS .19 .19 .19 .20 .20 .20 88 3 .24 .25 .25 .26 .26 .26 .27 .27 .27 .28 .28 .29 .29 .30 .30 87 4 .33 .33 .34 .34 .34 .35 .36 .36 .37 .37 .33 .38 .39 .40 .40 86 5 .41 .41 .42 .42 .43 .44 .44 .45 .46 .46 .47 .4S .49 .49 .50 85 6 .49 .50 .51 .51 .52 .52 .53 .54 .55 .56 .56 .57 .58 .59 .60 84 7 .57 .58 .59 .59 .60 .61 .62 .63 .64 .65 .66 .67 .68 .69 .70 83 8 .65 .66 .67 .68 .69 .70 .71 .72 .73 .74 .75 .76 .78 .79 .80 82 9 .73 .74 .75 .76 .77 .78 .80 .81 .82 .83 .84 .86 .87 .89 .90 81 10 .81 .82 .84 .85 .86 .87 .88 .90 .91 .92 .94 .95 .97 .98 1.00 80 11 .89 .90 .92 .93 .94 .96 .97 .93 1.00 1.02 1.03 1.05 1.06 1.08 1.10 79 12 .97 .99 .00 1.01 1.03 1.04 1.00 1.07 1.09 1.11 1.12 1.14 1.16 1.18 1.20 78 13 1.05 1.07 .08 1.10 1.11 1.13 1.14 1.16 1.18 1.20 1.22 1.23 1.25 1.27 1.30 77 14 1.13 1.15 .16 1.18 1.20 1.21 .23 1.25 1.27 1.29 1.31 1.33 1.35 1.37 1.39 76 15 1.21 1.23 .25 1.26 1.28 1.30 .32 1.34 1.36 1.3S 1.40 1.42 1.44 1.47 1.49 75 16 1.29 1.31 .33 1.34 1.36 1.38 .40 1.42 1.44 1.47 1.49 1.51 1.54 1.56 1.59 74 17 1.37 1.39 .41 1.43 1.45 1.47 .49 1.51 1.53 1.56 1.58 1.60 1.63 1.66 1.68 73 18 1.45 1.47 .49 1.51 1.53 1.55 .57 1.60 1.62 1.64 1.67 1.70 1.72 1.75 1.78 72 19 1.52 1.54 .57 1.59 1.61 1.63 .66 1.68 1.71 1.73 1.76 1.79 1.82 1.84 1.87 71 20 1.60 1.62 .65 1.67 1.69 1.72 .74 1.77 1.79 1.82 1.85 1.88 1.91 1.94 1.97 70 21 1.68 1.70 .72 1.75 1.77 1.80 1.82 1.85 1.88 1.91 1.94 1.97 2.00 2.03 2.06 69 22 1.75 1.78 .80 1.83 1.85 1.88 1.91 1.93 1.% 1.99 2.02 2.06 2.09 2.12 2.16 OS 23 1.83 1.85 .88 1.90 1.93 1.% 1.99 2.02 2.05 2.03 2.11 2.14 2.18 2.21 2.25 67 24 1.90 1.93 .96 1.98 2.01 2.04 2.07 2.10 2.13 2.16 2.20 2.23 2.27 2.30 2.34 66 25 1.98 2.00 2.03 2.06 2.09 2.12 2.15 2.18 2.22 2.25 2.28 2.32 2.36 2.39 2.43 65 26 2.05 2.08 2.11 2.14 2.17 2.20 2.23 2.26 2.30 2.33 2.37 2.41 2.44 2.48 2.52 04 27 2.12 2.15 2.18 2.21 2.24 2.28 2.31 2.34 2.38 2.42 2.45 2.49 2.53 2.57 2.61 63 28 2.20 2.23 2.26 2.29 2.32 2.36 2.39 2.42 2.46 2.50 2.54 2.58 2.62 2.66 2.70 62 29 2.27 2.30 2.33 2.36 2.40 2.43 2.47 2.50 2.54 2.58 2.62 2.66 2.70 2.75 2.79 61 30 2.34 2.37 2.40 2.44 2.47 2.51 2.54 2.58 2.62 2.66 2.70 2.74 2.79 2.83 2.88 60 31 2.41 2.44 2.48 2.51 2.55 2.58 2.62 2.66 2.70 2.74 2.78 2.83 2.87 2.92 2.97 59 32 2.48 2.51 2.55 2.58 2.62 2.66 2.70 2.74 2.78 2.82 2.86 2.91 2.95 3.00 3.05 58 33 2.55 2.58 2.62 2.66 2.69 2.73 2.77 2.81 2.85 2.90 2.94 2.99 3.04 3.09 3.14 57 34 2.62 2.65 2.69 2.73 2.76 2.80 2.84 2.89 2.93 2.98 3.02 3.07 3.12 3.17 3.22 56 35 2.68 2.72 2.76 2.80 2.84 2.88 2.92 2.96 3.01 3.05 3.10 3.15 3.20 3.25 3.30 55 36 2.75 2.79 2.83 2.87 2.91 2.95 2.99 3.04 3.08 3.13 3.18 3.23 3.28 3.33 3.38 54 37 2.82 2.86 2.90 2.94 2.98 3.02 3.06 3.11 3.15 3.20 3.25 3.30 3.36 3.41 3.47 53 38 2.88 2.92 2.96 3.00 3.04 3.09 3.13 3.18 3.23 3.28 3.33 3.38 3.43 3.49 3.55 52 39 40 2.95 3.01 2.99 3.05 3.03 3.09 3.07 3.14 3.11 3.18 3.16 3.22 3.20 3.27 3.25 3.32 3.30 3:37 3.35 3.42 3.40 3.47 3.45 3.53 3.51 3.58 3.56 3.64 3.62 3.70 51 SO 41 3.07 3.11 3.16 3.20 3.24 3.29 3.34 3.39 3.44 3.49 3.54 3.60 3. 66 3.72 3.78 49 42 43 3.13 3.19 3.18 3.24 3.22 3.28 3.26 3.83 3.31 3.37 3.36 3.42 3.41 3.47 3.46 3.52 3.51 3.57 3.56 3.63 3.61 3.68 3.67 3.74 3.73 3.80 3.79 3.86 3.85 3.93 48 47 44 3.25 3.30 3.34 3.39 3.43 3.48 3.54 3.59 3.64 3.70 3.75 3.81 3.87 3.94 4.00 46 45 3.31 3.36 3.40 3.45 3.50 3.55 3.60 3.65 3.71 3.73 3.82 3.83 3.94 4.01 4.07 45 77 40' 77 50' 78 78 W 78 20" 78 30' 78 40' 78 50' 79 79 10' 79 20' 79 30' 79 W 79' 50' 80 DETERMINATION OP TIME. 77 Table of factors for reduction of transit observations. TOP ARGUMENT=STAR'S DECLINATION (}). SIDE ARGUMENT- STAR'S ZENITH DISTANCE ). [For factor A use left-hand argument. For factor B use right-hand argument. For factor C use bottom line on this page.) C 77" 40" 77 SO 1 78 78 10' 78 20" 78 30' 78 40* 78 50' 79 79 W 79 20* 79 3W 79 40' 79 ay 80 C 40 3.37 3.41 3.46 3.51 3.56 3.61 3.66 3.71 3.77 3.83 3.89 3.95 4.01 4.08 4.14 44 47 3.42 3.47 3.52 3.57 3.62 3.67 3.72 3.78 3.83 3.89 3.95 .01 4.08 4.14 4.21 43 48 3.48 3.53 3.57 3.62 3.68 3.73 3.78 3.84 3.89 3.95 4.02 .08 4.14 4.21 4.28 42 49 3.53 3.58 3.63 3.68 3.73 3.79 3.84 3.90 3.96 4.02 4.08 .14 4.21 4.28 4.35 41 so 3.59 3.63 3.68 3.74 3.79 3.84 3.90 3.96 4.02 4.08 4.14 .20 4.27 4.34 4.41 40 ol 3.64 3.69 3.74 3.79 3.84 3.90 3.96 4.01 4.07 4.14 4.20 .26 4.33 4.40 4.48 39 52 3.69 3.74 3.79 3.84 3.90 3.95 4.01 4.07 4.13 4.19 4.26 .32 4.39 4.46 4.54 38 53 3.74 3.79 3.84 3.89 3.95 4.01 4.06 4.12 4.19 4.25 4.32 .38 4.45 4.52 4.60 37 54 3.79 3.84 3.89 3.94 4.00 4.00 4.12 4.18 4.24 4.30 4.37 4.44 4.51 4.58 4.66 36 55 3.83 3.89 3.94 3.99 4.05 4.11 4.17 4.23 4.29 4.36 4.43 4.50 4.57 4.64 4.72 35 56 3.88 3.93 3.99 4.04 4.10 .16 .22 4.28 4.34 4.41 4.48 4.55 4.62 4.70 4.77 34 57 3.93 3.98 4.04 4.09 4.15 .21 .27 4.33 4.39 4.46 4.53 4.60 4.68 4.75 4.83 33 58 3.97 4.02 4.08 4.14 4.19 .25 .32 4.38 4.44 4.51 4.58 4.65 4.73 4.80 4.88 32 59 4.01 4.07 4.12 4.18 4.24 .30 .3fi 4.43 4.49 4.56 4.63 4.70 4.78 4.86 4.94 31 1*1 4.05 4.11 4.17 4.22 4.28 .34 .41 4.4,7 4.54 4.61 4.68 4.75 4.83 4.91 4.99 SO 61 4.09 4.15 4.21 4.26 4.32 4.39 .45 4.52 4.58 4.65 4.72 4.80 4.88 4.96 5.04 29 62 4.13 4.19 4.25 4.31 4.37 4.43 .49 4.56 4.63 4.70 4.77 4.85 4.92 5.00 5.08 28 63 4.17 4.23 4.29 4.35 4.41 4.47 4.55 4.60 4.67 4.74 .81 4.89 4.97 5.05 5.13 27 64 4.21 4.26 4.32 4.38 4.44 4.51 4.57 4.64 4.71 4.78 .86 4.93 5.01 5.09 5.18 26 65 4.24 4.30 4.36 4.42 4.48 4.55 4.61 4.68 4.75 4.82 .90 4.97 5.05 5.14 5.22 25 66 4. 28 4.34 .40 4.46 4.52 4.58 4.65 4.72 4.79 4.86 .94 5.01 5.09 5.18 5. 26 24 67 4.31 4.37 .43 4.49 4.55 4.62 4.68 4.75 4.82 4.90 .97 5.05 5.13 5.22 5.30 23 68 4.34 4.40 .46 4.52 4.58 4.65 4.72 4.79 4.86 4.93 5.01 5.09 5.17 5.25 5.34 22 69 4.37 4.43 .49 4.55 4.62 4.68 4.75 4.82 4.89 4.97 5.04 5.12 5.20 5.29 5.38 21 70 4.40 4.46 .52 4.58 4.65 4.71 4.78 4.85 4.93 5.00 5.08 5.16 5.24 5.32 5.41 20 71 4.43 4.49 .55 4.61 4.68 4.74 4.81 4.88 4.96 5.03 5.11 5.19 5.27 5.36 5.45 19 72 4.45 4.51 .57 4.64 4.70 4.77 4.84 4.91 4.98 5.06 5.14 5.22 5.30 5.39 5.48 18 73 4.48 4.54 .60 4.66 4.73 4.80 4.87 4.94 5.01 5.09 5.17 5.25 5.33 5.42 5.51 17 74 4.50 4.56 .62 4.69 4.75 4.82 4.89 4.% 5.04 5.11 5.19 5.27 5.36 5.45 5.53 16 75 4.52 4.58 .65 4.71 4.78 4.85 4.92 4.99 5.06 5.14 5.22 5.30 5.38 5.47 5.56 15 76 4.54 4.60 4.67 4.73 4.80 4.87 4.94 5.01 5.09 5.16 5.24 5.32 5.41 5.50 5.59 14 77 4.56 4.62 4.68 4.75 4.82 4.89 4.96 5.03 5.11 5.18 5.26 5.3S 5.43 5.52 5.61 13 78 4.58 4.64 4.70 4.77 4.84 4.91 4.98 5.05 5.13 5.20 5.28 5.37 5.46 5.54 5.63 12 79 4.60 4.66 4.72 4.79 4.85 4.92 5.00 5.07 5.14 5.22 5.30 5.39 5.47 5.56 5.65 11 80 4.61 4.67 4.74 4.80 4.87 4.94 5.01 5.08 5.16 5.24 5.32 5.40 5.49 5.58 5.67 10 81 4.62 4.69 4.75 4.82 4.88 4.95 5.03 5.10 5.18 5.26 5.34 5.42 5.51 5.60 5.69 9 82 4.64 4.70 4.76 4.83 4.90 4.97 5.04 5.11 5.19 5.27 5.35 5.43 5.52 5.61 5.70 8 83 4.65 4.71 4.78 4.84 4.91 4.98 5.05 5.13 5.20 5.28 5.36 5.45 5.53 5.62 5.72 7 84 4.66 4.72 4.79 4.85 4.92 4.99 5.06 5.14 5.21 5.29 5.37 5.46 5.54 5.63 5.73 6 85 4.66 4.73 4.79 4.86 4.93 5.00 5.07 5.14 5.22 5.30 5.38 5.47 5.55 5.64 5.74 5 86 4.67 4.73 4.80 4.86 4.93 5.00 5.08 5.15 5.23 5.31 5.39 5.47 5.56 5.65 5.74 4 87 4.68 4.74 4.81 4.87 4.94 5.01 5.08 5.16 5.23 5.31 5.40 5.48 5.57 5.66 5.75 3 88 4.68 4.74 4.81 4.87 4.94 5.01 5.09 5.16 5.24 5.32 5.40 5.48 5.57 5.66 5.75 2 89 4.68 4.74 4.81 4.88 4.94 5.01 5.09 5.16 5.24 5.32 5.40 5.49 5.57 5.66 5.76 1 90 4.68 4.74 4.81 4.88 4.94 5.02 5.09 5.16 5.24 5.32 5.40 5.49 5.58 5.67 5.76 C 77" 40' 77 50' 78 78 10' 78 20' 78 30' 78 40' 78 50< 79 79 10" 79 20' 79 30' 79 40' 79 50' 80 PART TI. THE DETERMINATION OF THE DIFFERENCE OF LONGITUDE OF TWO STATIONS. INTRODUCTORY. The meridian at Greenwich having been adopted as the initial one to which all longitudes in the United States are to be referred, the determination of the longitude of a new station consists simply in the determination of the difference of longitude of the new station and of Greenwich, or some station of which the longitude reckoned from Greenwich is known. The determination of a difference of astronomic longitude is nothing more nor less than the deter- mination of the difference of the local times of the stations. 1 There are three general methods of determining longitude now in use, viz, the telegraphic, the chronometric, and the lunar. In the telegraphic method the error of the local chronometer on local sidereal time is deter- mined at each of the two stations by the methods stated in Part I of this publication, and the two chronometer times are then compared by telegraphic signals sent between the stations. In the chronometric method certain chronometers which are transported back and forth between the stations take the place of the telegraphic signals and thus serve merely to compare the station chronometers. In each of the lunar methods the observer at a station of which the longitude is required observes the position of the moon, or at least one coordinate of that position, and notes the local time at which his observation was made. He may then consult the Ephemeris and find at what instant of Greenwich time the moon was actually in the position in which he observed it. The difference between this time and the local time of his observation is his longitude reckoned from Greenwich. One coordinate fixing the position of the moon may be determined to serve as a means of deriving a longitude by measuring the right ascension of the moon at a transit across the meridian; by measuring the angular distance between the moon and the sun or one of the four larger planets, or between the moon and one of the brighter stars or by observing the times of disappearance and reappearance (immersion and emersion) of a known star behind the moon the lunar distance of the star at those instants being the angle sub- tended by the moon's radius. In each case the Greenwich time at which the moon occupied the position in which it was observed is obtained either from the Ephemeris, from observations at Greenwich at about the time in question, or from similar observations at some station of known longitude. The determination of longitude by wireless telegraph is not discussed in this publication. This method has been used to a certain extent by some countries with apparently satisfactory results. It will no doubt be used to a considerable extent in the location of islands which have no cable connections. The writer believes that it is much less expensive and more satisfactory at present to use the ordinary telegraph lines for the determination of longitude for geodetic purposes within the United States. These conditions may be reversed in the not distant future. 1 The times may he either sidereal or mean solar. Usually the sidereal times are compared because the time observations are nearly always made upon stars. 78 DETERMINATION OF LONGITUDE. 79 The telegraphic method 1 is the most accurate known method of determining differences of longitude. It is always used in this Survey for all longitude determinations in regions penetrated by telegraph lines, and is therefore set forth fully in this publication. A method suitable for use in regions not reached by the telegraph, 2 is the chronometric method. As this has been extensively used at coast stations in Alaska and will probably continue to be so used during some years to come, it is also here treated in full. To use the chronometric method one must be able to travel back and forth carrying chro- nometers between the two stations. The cost of such a longitude determination increases with increased cost of travel between stations, and its accuracy decreases as the time required to make a round trip increases. These facts cause the chronometric method to give way to lunar methods in certain comparatively rare situations. The points at which the boundary between Alaska and British America (one hundred and forty-first meridian) crosses the Yukon and Porcupine Rivers were determined by lunar methods. 3 Comparatively few such cases have occurred in late years in this Survey in which it was desirable to resort to observations upon the moon to determine important longitudes. 4 To have determined these longitudes by trans- portation of chronometers would have been exceedingly difficult and costly, and would have given results of a low order of accuracy, for there are more than a thousand miles of slow river navigation between the mouth of the Yukon and either station. As the lunar methods will probably be used less and less with the lapse of time and the increase of traveling facilities, it does not seem desirable to incorporate details in regard to them in this publication, especially as such details would greatly increase its size. The computa- tions involved are long, complex, and difficult. Those who wish to study the lunar methods are referred for details to Doolittle's Practical Astronomy, to Chauvenet's Astronomy, Volume I, and to the American Ephemeris (aside from the tables), especially to the pages in the back of each volume headed "Use of tables." PROGRAM AND APPARATUS OF THE TELEGRAPHIC METHOD. During more than 60 years of its use by the Coast and Geodetic Survey the telegraphic method was gradually modified, but with the adoption of the transit micrometer about 1904 the program of the determination of primary longitudes underwent radical changes. The pro- gram and apparatus used at present in the Survey will be described first and then the method formerly used will be briefly explained. The introduction of the transit micrometer practically eliminated from the time determina- tions, and consequently from the longitude determinations, the large error which was known as the observer's personal equation. The program of longitude observations was formerly designed to eliminate the personal equation from the results. GENERAL INSTRUCTIONS FOR LONGITUDE DETERMINATION BY THE COAST AND GEODETIC SURVEY WITH TRANSIT MICROMETERS IN LOW LATITUDES (LESS THAN 50). 1. The observations upon each star should be given unit weight, regardless of the declina- tion of the star and of whether or not the observation of the transit is complete. If an observed transit is incomplete, only those observations should be used for which the positions of the observing wire are symmetrical with reference to the middle point of the registration interval of the screw; that is, each record is to be rejected for which the symmetrical record is missing. 1 The telegraphic method of determining differences of longitude was originated by the Coast Survey in 1846, two years after the first trans- mission of telegraphic messages over wires. During the long interval since that time the method has gradually been brought to its present high state of perfection. For a historical note on this subject see Appendix No. 2, Report for 1897, pp. 202-203. 2 In certain cases in which the telegraph line is wanting, the same principles may be used with the substitution of a flash of light between sta- tions in the place of the electric wave. For example, one might so determine the longitudes of the Aleutian Islands of Alaska, the successive islands being in general intervisible. This method has not, however, been used by this Survey. The cost of determining longitudes by this method will in general bo so much greater than by the chronometric method (because of the many intermediate stations which will be required between distant stations), as to more than offset its greater accuracy. * In the final demarcation of the boundary between Alaska and British Columbia, an initial point on the one hundred and forty-first meridian was determined telegraphically, using transits equipped with transit micrometers. The telegraphic longitude came within the range of three determinations by lunar methods. The total range of the several lunar determinations of longitude in different years was 1.1 seconds of time. 4 A statement of the results of these determinations, which is especially interesting as showing what errors may be expected in such observa- tions, is given in Appendix No. 3 of the Report for 1895. 80 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. 2. The limit of rejection for an observation upon one star (whether the observed transit is complete or not) is a residual of 0.20 second. No observation corresponding to a residual smaller than this should be rejected unless the rejection is made at the time of observation. 3. Each half set of time observations should consist of observations on from 5 to 7 stars (6 preferred). In rare cases a half set may consist of only four stars. All of these are to be time stars; that is, no azimuth stars are to be observed. For the purpose of this paragraph an azimuth star is defined as one for which the azimuth factor, A, is greater than unity. The alge- braic sum of the A factors in each half set should be kept less than unity unless it is found that to secure such a half set considerable delays would be necessary. It is desirable to have the algebraic sum of the A factors as small for each half set as it is possible to make it by the use of good judgment in selecting the stars, but it is not desirable to reduce the number of stars per hour to be observed in order to improve the balancing of the A factors, if said balancing is already within the specified limit. 4. In selecting lists of stars to be observed, one should endeavor to secure the maximum number of stars per hour possible, subject to the conditions of paragraph 3 and to the necessity of securing level readings, reversing the instrument, exchanging signals, et cetera. To observe the same stars at both stations involved in a longitude difference is desirable, but it is of less importance than to secure rapid observations with well-balanced A factors in each half set. 5. The telescope should be placed in the position "illumination west" for the first half set of each night and it should be reversed before the beginning of each of the other half sets. 6. The observations on each night should consist, under normal conditions, of four such half sets as are defined in paragraph 3. In case of -interference with the normal progress of the observations by clouds or other causes, a determination on a given night may be allowed to depend upon a smaller number of stars and of half sets at each station. But the determination of the longitude difference on any night is to be rejected if, at either station, there has been no reversal of the instrument, or if less than twelve stars with two reversals are successfully observed at either station, or if the exchange of signals takes place at either station outside the interval covered by the time observations at that station. 7. There is to be no exchange of observers during the determination of any difference of longitude. 8. A determination of a difference of longitude will consist of either three or four such nights of observations as are specified in paragraph 6. If, before an opportunity occurs to take observations upon a fourth night, it becomes known that the result from each of the first three nights of observations agrees with the mean result within 0.070, no observations on a fourth night should be taken. If one or more of the first three nights give results differing by 0*.070 or more from the mean, or if observations are secured on a fourth night before the results from the first three nights are all known, then observations on four nights are to con- stitute a complete determination of a difference of longitude. 9. When referring a longitude station to a triangulation station the angle and distance measurements should be made with a check and with such accuracy that if necessary the longitude station may replace the triangulation station for future surveys. 10. The field computations are to be kept as closely up to date as practicable. 11. In making the computations of time observations in the field, the method shown on pages 21 to 27 of this publication should be followed. GENERAL INSTRUCTIONS FOR LONGITUDE DETERMINATION BY THE COAST AND GEODETIC SURVEY WITH TRANSIT MICROMETERS IN HIGH LATITUDES (GREATER THAN 50). The observing and the field computations for the work in connection with the telegraphic determination of longitude in latitudes greater than 50 should be done in accordance with the instructions for work in latitudes less than 50 except that: (a) The stars of a set are given different weights depending upon their positions. (V) No rejection limit is fixed for use by the observer; rejections are made, if necessary, in the office after the least square computations have been made, (c) It will be impossible, as a rule, to have a half set with all time stars and No. 10. (Chronometer (Condenser rConde -=p-Battery Chronometer Relay Battery -=- III Chronograph (Relay Battery Transit Micrometer Battery -SST Telegrapher's & Signal Key Mam Line During Time Observations /Chronometer (Condenser Battery Telegrapher's & Signal Key Battery ~=F During Exchange of Signals ARRANGEMENT OF ELECTRICAL CONNECTIONS, TELEGRAPHIC LONGITUDE TRANSIT-M ICROM ETER METHOD. No. 11. (Chronometer /-Condenser Vx^^x 1 - Chronometer Relay Battery -=- Chronograph I Observing Key LJ Battery + 4 Signal Relay 1 J ^Sounder Relay O Telegn __ apher's & Signal Key Main Line During Time Observations (Chronometer (Condenser yffTT^ Battery -d= During Exchange of Signals ARRANGEMENT OF ELECTRICAL CONNECTIONS, TELEGRAPHIC LONGITUDE-KEY METHOD. DETERMINATION OF LONGITUDE. 81 hence, the half sets are to be made up of time and azimuth stars. (An azimuth star is one hav- ing an A factor greater than unity.) (d) In making the computation of the time observations the observer will use his discretion as to the method to be used, provided it is one of those given in this pubb'cation. USUAL METHOD OF OPERATIONS. As the personal equation is very small, if it exists at all, it is not considered necessary in determining astronomic longitudes for geodetic or geographic purposes to have an exchange of observers, nor is it necessary that a new station should be in a closed circuit. The normal determination of longitude between two stations using transit micrometers consists of three nights' observations without exchange of observers. (Under the general instructions a fourth night is sometimes required.) Each night's observations consist of four half-sets of six stars each, the instrument being reversed in its wyes between each two half-sets. Arbitrary signals are usually exchanged between the two stations by telegraph in the interval between the second and third half-sets. This places the arbitrary signals, by which the chro- nometers at the two stations are compared, as nearly as possible in the middle of the observing period and it makes the longitude determined depend equally on each of the time sets. The two observatories must, of course, be connected by means of a telegraph line. An arrangement is made with the telegraph company for a direct connection between the stations, at the required time, on nights of observation. This is accomplished by running wires from the longitude stations to the switchboards of the local telegraph offices. If possible the line should be without repeaters. The advisability of having the station convenient to the telegraph office should have some weight in determining its location. Occasionally the station may have to be con- nected directly with a main wire instead of with the telegraph office switchboard. The general arrangement of the electrical apparatus at each station during star observa- tions and also during exchange of signals is shown in the diagrams of illustrations Nos. 10 and 11. Illustration No. 12 shows the actual switchboard and instruments used in these operations. This board carries an ordinary telegrapher's key, sounder relay, and signal relay, all of which may be included in the telegraph circuit. If desired the signal relay or the sounder relay and key may be cut out by means of plug switches. The sounder is worked by the sounder relay through a separate battery. When the operator is clearing the line or communicating with the operator at the other observatory, the signal relay is cut out, and when signals are being sent it is again cut in, and it operates the pen of the chronograph through a separate battery. Thus, at each station, when the signal relay is on the main line, every break of the telegrapher's key operates the two signal relays and makes records on the chronograph sheets at both stations. The chronometers being placed in the local circuits at both stations continue their records on the chronograph sheets, the circuits being break circuits, and so it is possible to read from the chronograph sheet at each station the chronometer time of sending and receiving the arbitrary signals. The local circuit, as explained on page 12, consists of one principal circuit, the chronograph circuit, to which the chronometer circuit and the transit circuit are joined through the points of their respective relays. The observing key, when used, replaces the transit circuit. The chronograph circuit, connected with the proper binding posts of the switchboard, includes the points of the signal relay, except when cut out by a plug switch. This plug is kept in during time observations, and taken out only during the exchange of signals. A few minutes before the time for exchange of signals the telegraph operator secures a clear line between stations, ascertains whether the observations at the other station are pro- ceeding successfully, and telegraphs the exact epoch at which signals will be exchanged. This epoch is arranged, if practicable, not to interfere with the star observations at either station. If at one of the stations floating clouds or other causes are making it difficult to get observations the observer at that station should choose the epoch, for the loss of one or more stars by him might cause the loss of a night's work. When the epoch arrives the points of the signal relay 8136 13 6 82 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. are placed in the local circuit at each station by the removal of a plug of each switchboard Any break in the main-line circuit will now cause corresponding breaks in the local circuits, and a signal made with the telegraph key 1 will be recorded on both chronographs. The observer at the western station customarily sends signals first, by releasing the telegraph key for an instant between the breaks of his chronometer at an average interval of two seconds. He times these signals so that they will not interfere with his own chronometer record, and he must also be prepared to shift them to another portion of the second, if they are conflicting with the record of the chronometer at the other station. Notice of an interference is given by the other observer by breaking into the circuit and making a succession of quick breaks with the key. After 15 to 20 signals have been sent from the western station, covering a period of over half a minute, double that number of signals are sent by the eastern observer, and then 15 to 20 more are sent by the western observer. This makes a total of 30 to 40 signals each way, with the mean epoch of the signals from the two different directions agreeing closely. The signals, as a rule, cover a total period of less than three minutes. It is well to make a succession of quick breaks at the beginning and end of each series of signals. It is also desirable to vary the position of each of several signals with reference to the chronometer breaks at the beginning of a series or to make several signals at intervals of one second in order to facilitate the identification of corresponding records at the two stations. The number of signals exchanged is arranged to cover a period greater than one minute each way, with a view of eliminating errors in the contact wheel of the chronometer. A signal sent from one station to the other will be recorded on the chronograph of the sending station slightly before it is on the distant chronograph, and this difference in time of record is called the transmission time. It depends, in fact, both on the retardation of the signal in the telegraph line between the two stations, and on the difference in the time of action of the signal relays at the two stations. 2 Signals sent from west to east will make the difference in longitude too large, and signals from east to west will make it too small by the amount of the transmission time. By taking the mean of the differences as given by the signals in both directions this source of error is eliminated, provided the transmission time is the same in both directions. 3 During exchange of signals the chronographs are run at double speed, so that the signals may be read to hundredths of seconds. The advantage in sending signals by making arbitrary breaks of the circuit is that they will come at varying parts of the seconds, thus tending to elimi- nate personal equation in the reading of the fractional parts of the second. 4 If portions of the record are missed, the corresponding signals at the two stations may still be identified by com- paring the successive differences between signals. RECORD OF AN EXCHANGE OF SIGNALS. The following is one night's record of an actual exchange of signals between two stations, written as read from the chronograph sheet on a special form used for the purpose, on which is also made the computation of the epochs of the signals at the two stations, the computation of the final difference of signals, and the transmission time. 1 It is to be noted that these signals are made by breaking the circuit, which is opposite to the ordinary correspondence use of the key. 2 The latter is probably a small quantity. Some measurements of the armature time of one of the quick-acting relays used in these longitude determinations showed it to vary from 0.005 to 0.015 second with extreme changes in adjustments and current. 3 There is always some uncertainty on this score when repeaters are used in the mam telegraph line, because of the distinct mechanical arrange- ments for repeating the signals in the two directions. Repeaters are therefore to be avoided as far as practicable. * Chronometer signals were formerly used that is, the chronometers were alternately made to send their breaks through the main-line circuit, recording on both chronographs. Some of the objections to this method were liability of damage to the points of the break circuit wheel of the chronometer when put on the main line, possibility of the record of one chronometer interfering with the record of the other, and personal equation in reading a record that always occurred at the same part of a second. D - 5 z o I A, < o: u _i u 1- Q DETERMINATION OF LONGITUDE. 83 Arbitrary signals. Form 256. [Station, Key West, Fla. Date, Feb. 14, 1907. Observer, J. S. Hill. Recorder, J. S. Hill.] From Key West to Miami From Miami to Key West* Miami record Key West record Diff. Miami record Key West record Difl. T> m s ft TO s m s Ti m s It m > m 8 6 33 35. 10 6 27 38. 28 5 56.82 6 34 54.41 6 28 57.71 5 56. 70 36.42 39.63 .79 56.32 59.63 .69 37.50 40.70 .80 58.31 29 01. 60 .71 38.50 41.71 .79 35 00.22 03.52 .70 39.45 42.67 .78 02.35 05.64 .71 41.47 44.67 .80 04.30 07.58 .72 43.43 46.63 .80 06.54 09.83 .71 45.50 48.69 .81 08.31 11.60 .71 47.50 50.70 .80 10.26 13.54 .72 49.58 52.77 .81 12.24 15.53 .71 51.60 54.78 .82 14.31 17.61 .70 53.57 56.77 .80 16.22 19.51 .71 55.65 58.85 .80 18.22 21.52 .70 58.18 28 01.37 .81 20.28 23.57 .71 34 00.51 03.71 .80 24.29 27.58 .71 02.52 05.72 .80 26.22 29.51 .71 03.67 06.88 .79 28.23 31.53 .70 04.77 07.95 .82 30.28 33.57 .71 35 42.20 29 45.40 .80 31.24 34.54 .70 43.50 46.72 .78 32.43 35.73 .70 45.08 48.29 .79 34.25 37.54 .71 47.50 50.70 .80 49.56 52.74 .82 51.50 54.70 .80 53.47 56.67 .80 55.64 58.84 .80 57.59 30 00.80 .79 59.57 02.79 .78 36 01.57 04.77 .80 03.58 06.80 .78 05.55 08.76 .79 07.55 10.76 .79 09.60 12.80 .80 11.53 14.74 .79 12. 62 15.85 .77 13.57 16.77 .80 14. 61 17.81 .80 15.54 18.73 .81 Means: 6 34.9 6 29.0 5 56.798 6 35.3 6 29.3 5 56.707 6 34.9 6 29.0 5 56.798 Means 6 35. 1 6 29.1 5 56. 752 Transmission time= . 046 * Complete set of signals from Miami to Key West not obtained. In the foregoing table the mean epochs are shown for the record of signals on each chrono- graph sheet, the mean of all the differences of the chronometer records, and the transmission time. It is usually sufficient, in obtaining the mean epochs of signals, where they are symmet- rically arranged, to take the mean of the first five and the last five signals. CHRONOMETER CORRECTIONS AND RATES. On the following form are tabulated the epochs (T ) for which chronometer corrections were determined at both stations, the corrections (AT) determined, and the rate per minute computed from the two time sets observed on each night. In each case the mean of the epochs and of the corrections is given on the third line for each date. These means furnish a correction for the chronometer very nearly at the epoch of the signals, and they thus reduce the work of computing the chronometer corrections for the epochs of the signals. 84 U. S. COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 14. Chronometer corrections and rates. Date Key West, Fla. Rate per minute Miami, Fla. Rate per minute To JT To AT 1907. Feb. 14 1 m 5 49.6 7 11.0 6 30.3 s +14.691 + 14.726 + 14.708 1 +0.00043 h m S 41.4 7 19.1 6 30.2 +45. !77 +45.493 +45.335 s +0. 00323 15 5 50.0 7 47.9 6 49.0 + 14.327 + 14.220 + 14.274 -0.00091 5 46.9 7 11.0 6 29.0 +50. 182 +50.449 +50. 316 +0.00317 16 5 50.1 7 09.0 6 29.5 + 13.479 + 13.460 + 13.470 -0.00024 5 54.7 7 22.4 6 38.6 +55.337 +55.551 +55.444 +0.00244 COMPUTATION OF DIFFERENCE OF LONGITUDE. The next step is the computation of the difference of longitude from the mean of the signals sent in each direction. Each night's observations represents a complete determination of this difference, and a separate and complete computation is accordingly made for each night. The epoch of signals and difference of chronometers are taken from the record of signals for each night, and the chronometer corrections at these epochs are computed for each station and each night, using the rates per minute given in the preceding form. To the difference in chronometers is then applied the difference in chronometer corrections (eastern minus western chronometer), which gives the difference of longitude in time as determined by the night's observations. From this determination the transmission time has already been eliminated by taking the means of eastern and western signals. The chronometer correction A T at the time of exchange T and its probable error r are expressed by - 71), and r where AT^ and r\ are the chronometer correction and its probable error derived from the first set of time observations at epoch T l} and AT 2 and r 2 are the same quantities, respec- tively, for the second set at epoch T 2 . Computation of difference of longitude. BETWEEN MIAMI AND KEY WEST, FLA. T AT Date Diff: JT Difl. Of signals Ji V Trans- mission time Miami Key West Miami Key West 1907. h m h m 3 s s m s m s s s Feb. 14 6 35.1 6 29.1 +45.351 +14.709 +30.642 5 56.752 6 27.394 -0.031 0.046 15 6 31.7 6 25.8 +50.325 + 14.295 +36.030 5 51.285 6 27. 315 + .048 .051 16 6 33.6 6 27.8 +55.432 + 13.470 +41.962 5 45.418 6 27.380 - .017 .047 Mean.. 6 27.363 Reduction to longitude pier of 1896= .97 meter Reduction to mean position of pole ' = +0.002 - 0.000 m t Miami longitude station east of Key West longitude station- 6 27.365 = 136'50".525 In the example shown above the second column gives the mean epoch of the exchange of signals as read from the chronograph sheet at the eastern station, Miami, and the fourth column gives the correction to the chronometer at Miami for the mean epoch of the signals, this correction being computed from the corrections to the chronometer and the rate deduced from the time observations. The third and the fifth columns give similar data for the western i See Astronomische Nachrichten No. 4253. DETERMINATION OF LONGITUDE. 85 station, Key West. The difference between the chronometer corrections (AT) given in the fourth and fifth columns is shown in the sixth column and equals the correction at the eastern station minus the correction at the western station. In the next column is given the difference of signals (eastern minus western). The difference of longitude, AX, is then the combination of the difference between the A T's at the two stations and the difference of signals. The trans- mission time is taken from the form on which the record of signals and their reduction is shown, and is placed in the last column, while in the column immediately preceding is placed the differ- ence between each night's determination and the mean of the determinations of all the nights. The values from the various nights are each given unit weight, and their mean is then considered to be the observed difference of longitude between the transit instruments at the two stations. In the example given this difference has a correction applied to it to reduce it to what it would have been had the transit at the base station, Key West, been placed exactly over the position occupied by the transit in 1896 (adjusted in the longitude net of the United States) 1 instead of at a position 0.97 meters east of it. The particular example given is one of a series of differences of longitude determined in 1907, commencing at Key West and closing on Atlanta. There is also at the latter place an adjusted longitude station of the longitude net of the United States. The longitudes of these two stations, at Key West and Atlanta, being held fixed, a closing discrepancy was developed which was distributed equally among the various differences, each difference being given unit weight. The following table shows the differences of longitude determined between Key West and Atlanta and the distribution of the closing error: Computation of closing error between Key West and Atlanta. Observed difference Miami west of Key West Jupiter west of Miami Sebastian west of Jupiter Daytona west of Sebastian Fernandina west of Daytona Atlanta west of Fernandina Atlanta west of Key West Atlanta west of Key West (From adjusted longitude net of United States) m - 6 - + 1 + 2 + 1 +11 27. 365 27.404 33. 654 11. 332 46. 878 42. 609 + 10 +10 19.704 19. 759 Correc- tion to close circuit + .009 +.009 +.009 +.009 +.009 +.010 +.055 Adjusted difference m - 6 - + 1 + 2 + 1 +11 27. 356 27. 395 33. 663 11. 341 46. 887 42. 619 +10 19. 759 Closing error= + .055 CORRECTION FOR VARIATION OF THE POLE. v A correction is necessary to reduce the observed astronomic longitude to the mean posi- tion of the pole. About the middle of each year the Latitude Service of the International Geodetic Association publishes in the Astronomische Nachrichten provisional values of the coordinates of the instantaneous pole for the preceding calendar year, together with tables to reduce observed latitudes, longitudes, and azimuths to the mean position of the pole. The proper correction to the longitude may be computed by means of these tables, knowing the time of observation and the latitude and longitude of the observing station. DISCUSSION OF ERRORS WHEN TRANSIT MICROMETER IS USED. Let it be supposed that the regular program for observations with a transit micrometer, three nights' observations without exchange of observers, has been carried out. The computed result, the difference of astronomic longitude of the two places, is subject to the following errors : 1 See Appendix 2 of the Report for 1897. 86 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. First. An accidental error arising from the accidental errors of observations of about T2 stars at each station. If the accidental error of observation of a single star be estimated at 8 .07, which may be considered sufficiently large to cover both the observer's errors and those instrumental errors which belong to the accidental class, then the probable error of the final result arising from this cause would be s .07-n V36= s .012. Second. An accidental error arising from the accidental errors in the adopted right ascen- sions of such stars as are observed at one station on a given night but not at the other. It is in such cases only that errors in right ascension have any effect on the computed result. If entirely different stars were observed at the two stations, 24 at each station, and if s .03 be accepted as the probable error of a right ascension, then the probable error of the result for one night arising from this source would be 8 .03^ V 12 = s -009. In ordinary cases, in which the number of stars not common to both stations is less than 10 per cent, this accidental error is reduced to less than 8 .001. Third. Errors due to the assumption that the rate of the chronometer is constant during and between the two time sets of a night. As the interval between the mean epochs of the sets is ordinarily only about one hour, these errors are probably exceedingly small. In order to make these errors inappreciable, longitude observers should use chronometers known to show but small variations in rate, and should protect them as thoroughly as is feasible while in use against jars and sudden changes of temperature. The errors from this source will be of about the same value whether the exchange of signals is made at about the mean epoch of the two sets of time observations, or is made at any other epoch within the interval covered by the two sets. Fourth. The question of the personal equation with the transit micrometer is discussed fully on pages 90 and 91. Fifth. Errors arising from lateral refraction. The probable minuteness of these errors in time observations has already been commented upon (see p. 48). It is not impossible, however, that small constant errors may arise from this source at stations established in closely built-up portions of great cities, particularly of manufacturing centers. Sixth. Errors arising from variation of transmission time. By transmission time is meant the interval that elapses from the instant at which the signal relay breaks the local circuit at the sending station to that at which the signal relay breaks the local circuit at the receiving station. This interval is made up of armature time, induction time, and the true transmission time of the electric wave passing along the wire. It is only the variation in transmission time occurring during the exchange of signals on each night that introduces error into the computed result. As this interval is not much over a minute the error is probably insensible if there is a continuous wire connection between stations. If the line between stations passes through a "repeater" the transmission time in one direction through the repeater will be different from that in the other direction unless the two magnets of the repeater are adjusted exactly alike, and half this difference will enter into the computed result as an error. The repeaters used in ordinary telegraph service are not specially designed for quick action, as are the signal relays on the Coast and Geodetic Survey switch board, nor is their adjustment in the control of the longitude observers. Hence the desirability of a continuous wire connection. Any change in transmission time within the local circuit during the exchange of signals will produce an error in the computed longitude, but such changes are probably insensible. A change at any other time in the local circuit will appear in the observations as a change in the chronometer correction and will probably have no appreciable effect on the final result for the night. Seventh. The difference of the transmission time through the two signal relays and also the difference in the transmission time through the two transit micrometer relays enter as errors in the final result. These errors are made very small in the present longitude work of the Survey by using relays which are as nearly alike as can be made, and which are specially designed to act very quickly. DETERMINATION OF LONGITUDE. 87 If the difference of longitude which is being measured is large, it becomes necessary to abandon the practice of observing the same stars at both stations in order to make the exchange of arbitrary signals come within the period of the night's observations at each station. How- ever, the errors of right ascension thus introduced will not be large. The combination of the numerical values of the above errors will not fully account for the error of the result as computed from the separate determinations, that is from the residuals, but it may be that some of the above errors for which no numerical values are estimated are much larger than supposed. The discussion of errors of time observations on pages 48-51 of this publication applies to a certain degree to longitude work. See also Discussion of Errors, when the key method is used, on page 93. PROGRAM WHERE NO TRANSIT MICROMETER IS USED. Before the adoption of the transit micrometer for longitude work, when the chronograph and key method was in use, it was necessary in all determinations of differences of longitude to arrange the program of observations so as to eliminate the personal equation of the observers making the time observations. The personal equation was eliminated either directly by exchange of observers, or indirectly by supplementary observations, themselves independent of the longitude observations, but which gave a value for the personal equation to be introduced into the computations. Further on, page 90, the question of personal equation and its deter- mination will be more fully discussed. In the determination of primary differences of longitude the personal equation was elimi- nated by the observers exchanging stations when one-half of the observations had been made. One-half the sum of the mean determinations before and after exchange of observers gave a resulting difference of longitude which was independent of the personal equations of the observers provided these personal equations remained constant. Except for this, the program of observations was the same as for observations with a transit micrometer (see p. 81). The arrangement of the telegraphic apparatus was the same as described on page 81. The observing key took the place of the relay points of the transit micrometer. Illustration No. 11 shows the arrangement of the local and main circuits while time observations were being made, and also while signals were being exchanged. The switchboard is the same as used in transit micrometer observations, and is shown in illustration No. 12. The following records and computations show the various steps in observing and computing an actual difference of longitude. Record of exchange of signals, and computation of difference of chronometers. [Station, Atlanta, Ga. Date, Mar. 7, 1896. Observer, G. R. P. Recorder, G. R. P.] ARBITRARY SIGNALS. From Atlanta to Key West From Key West to Atlanta Key West record Atlanta record Difference of chronometers Key West record Atlanta record Difference of chronometers h m s 7 35 59. 97 36 01. 90 04.03 05.96 * h m s 7 25 42. 39 44.30 46.47 48.39 * TO S 10 17.58 .60 .56 .57 # h m s 7 37 08. 76 10.82 12.78 15.28 * h m s 1 26 51. 51 53.60 55.52 58.04 # TO S 10 17. 25 .22 .26 .24 * # * * # * * 56.90 58.91 26 39. 30 41.34 .60 .57 38 38. 48 40.60 28 21. 21 23.33 .27 .27 h m Means 7 36. 5 h m 7 26.2 TO S 10 17. 570 h m 7 37.9 h m 7 27.6 TO S 10 17. 249 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. SUMMARY OF RESULTS OF TIME DETERMINATIONS AT ATLANTA. Azimuth Date Epoch (by face of chro- nometer) Chronometer correction J7V Rate per minute Collimation West East 1896. h m S s S S S Mar. 7 6 56.4 -13.546 +. 00261 +.03 -.154 + .035 7 8 12.6 -13.347 -.01 -.036 -.070 8 6 56.3 - 7. 742 +. 00310 -.01 +.115 +.089 8 8 12.5 - 7.506 -.06 +.190 +.313 # * * # # * # * * * -X- * * ft 27 8 12.6 -12. 660 +.00043 -.18 +.183 4-. 155 27 9 22.4 -12.630 -.22 +.378 +.167 SUMMARY OF RESULTS OF TIME DETERMINATIONS AT KEY WEST. Azimuth Date Epoch (by face of chro- nometer) Chronometer correction tr, Rate per minute Collimation West East 1896. h m s s S S S Mar. 7 6 56.4 -11. 157 -.00232 -.05 -1.108 -1. 236 7 8 12.6 -11.334 -.03 -1.220 -1.108 8 6 56.4 -13.994 -.00227 -.06 -1. 649 -1.447 8 8 12.6 -14. 167 -.03 -1.644 -1.580 # # * # tt # # * * * # * # * 27 8 12.4 - 4.992 -.00223 -.06 -0. 181 -0. 256 27 9 21.8 - 5.147 -.00 -0. 121 -0.144 FROM WESTERN OR ATLANTA SIGNALS.* Date Epoch of signalsf Difference of chronome- ters Chronometer corrections J V (from western signals) Key West *J Atlanta TW Key West "M Atlanta *TW Difference J T E -J T w 1896. Mar. 7 8 * h m 7 36.5 7 36.6 * h m 7 26.2 7. 26. 2 * m s 10 17.570 10 26. 199 it s -11. 250 -14.085 * s -13.468 - 7.649 * S + 2. 218 -6.436 # m s 10 19. 788 19. 763 * # * # * * # # * 27 8 51.3 8 41.1 10 12.507 - 5.079 -12.648 +7. 569 20. 076 * Unconnected for transmission time and personal equation, f By face of chronometer. FROM EASTERN OR KEY WEST SIGNALS.* Date Epoch of signals f Difference of chronome- ters Chronometer corrections (from eastern signals) Key West Atlanta TW Key West Atlanta J7V Difference 1896. Mar. 7 8 * h m 7 37.9 7 37.6 * h m 7 27.6 7 27.2 * m s 10 17. 249 10 25. 881 * S -11. 253 -14.087 * S -13.464 - 7.646 * S + 2. 211 -6.441 * m s 10 19.460 .440 # * * * # * * * 27 8 53.3 8 43.1 10 12. 136 - 5.083 -12. 647 +7. 564 .700 * Unoorrected for transmission time and personal equation. t By face of chronometer. DETERMINATION OF LONGITUDE. 89 COMBINATION OF LONGITUDE RESULTS. At one time it was the custom in the Coast and Geodetic Survey to combine the resulting differences of longitude for the various nights' observations by deducing weights and assigning them to the various values. This custom is not now practiced where transit micrometers are used, nor is it followed where an accepted program is carried out even if no micrometers are used. If a regular program is carried out the various nights' determinations are given equal weight, and direct means are taken for the final value of the difference of longitude. How- ever, the following discussion of the combination of longitude results where the different nights' observations are assigned different weights is given here as occasion might arise where the information would be of value. The following table gives the collection of the results for the different nights and their combination to develop and eliminate the transmission time and personal equation. The mean of the differences of longitude as derived from the western and eastern signals will be free from the transmission time, and their difference is double the transmission time. The rela- tive weights for the resulting differences of longitude for different nights are derived from the expression p = f_ 2 , where p l and p 3 are the weights of the determinations of the chronom- eter corrections at the epoch of exchange of signals at the two stations, respectively, or p! = and p t = - 2 in which ^ and r 2 are the probable errors of the chronometer corrections. r \~ r 2 To obtain the personal equation the weighted means are taken for each position of the observers, and half their difference is the personal equation to be applied with opposite signs to the two groups. This gives the corrected result for difference of longitude for each night, and the weighted mean of all the nights is the final difference of longitude. The probable error of the latter is 0.674-/v ^j^y v~ where n is the number of nights of observation and 2 is the number of unknowns (longitude and personal equation). In the table the means in the seventh and ninth columns are weighted means. The personal equation is one-half the difference in the weighted results for the two posi- tions of the observers, or the sign indicating that S observes later than P. The probable error 1 of the personal equa- tion may be taken as identical with that of the resulting difference of longitude. The transmission time, as stated, is one-half the difference between the results from western 338 and eastern signals, or in this example, = ~o~ = s - 169, an unusually large value, due to the marine cable, between Key West and the mainland. Table of resulting difference of longitude between Atlanta, Ga., and Key West, Flo. Date Observer at From western or Atlanta signals J From eastern or Key West signals s Double trans- mission time M w -"" Mean of W.andE. signals Personal equation Difference of longitude Combi- nation weight P Resid- uals V A KW 1896. Mar. 7 8 9 13 14 Mar. 20 21 25 26 27 P. P. P. P. P. S. S. S. s. s. S. 8. 8. S. S. P. P. P. P. P. TO S 10 19. 788 .763 .754 .802 .842 10 20.018 .075 .102 .074 .076 m s 10 19. 460 .440 .445 .495 .522 Mean 10 19.686 .705 .737 .721 .700 Mean a 328 .323 .309 .307 .320 m 10 19. 624 .602 .600 .648 .682 1 +0.120 -0.120 m s 10 19.744 .722 .720 .768 .802 .732 .770 .799 .777 .768 S 11 4 13 21 9 5 4 8 6 s -.021 -.043 -.045 +.003 +.037 -.033 +.005 + .034 +.012 + .003 .317 10 19. 645 .332 .370 .365 .353 .376 10 19. 852 .890 .919 .897 .888 0.359 10 19.884 10 19.765 0.007 1 Practically the same result is obtained by deriving separate values for the personal equation by comparing each result in the first position of the observers with the corresponding result in the second position and computing the probable error from the variations in these separate values. 90 TJ. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. The above formulae and forms are used in. the office computation. The field computation differs from that made in the office in that the time computation is made by an approximate field method shown on page 26 or page 34 instead of the least square method given on page 41, and that in the field no probable errors or weights are computed and indiscriminate means are taken instead of weighted means. In the past some of the forms used in the field have been slightly different from those shown above. The office computation will be facilitated by making the field computation as here indicated. PERSONAL EQUATION. The absolute personal equation in time observations with a transit is the interval of time from the actual instant of transit of a star image across a line of the diaphragm to the instant to which the transit is assigned by the observer. When the time is observed using a chronograph and an observing key the absolute personal equation is simply the time required for the nerves and the portions of the brain concerned in an observation to perform their functions. In the case of observations by the eye and ear method the mental process becomes more involved, and the personal equation depends on a much more complicated set of physical and psychological conditions than when the observations are made with a key and chronograph. Although the personal equation has been studied by many persons and for many years, little more can be confidently said in regard to the laws which govern its magnitude than that it is a function of the observer's personality, that probably whatever affects the observer's physical or mental condition affects its value, that it tends to become constant with experience, that it probably differs for slow moving and fast moving stars, and that it is different for very famt stars which the observer sees with difficulty from what it is for stars easily seen. A systematic error may be present which is due to the tendency of the observer to place the wire always to the right or to the left of the center of the star's image. This tendency is due to the delects in the observer's eye and the error resulting is called the bisection error. At some astronomic observatories a reversing prism is used which reverses the image of the star midway in the observations. Thus, during one half of the observations the wire would be placed too far east and during the other half too far west of the center of the star's image (or vice versa) and the mean of all the observations would be free from a bisection error. No numerical values are available for the effect of the bisection error but it is known to be so small that it may be neglected in all time and longitude work for the usual geodetic and geographic purposes. (See remarks under the Description of the Zenith Telescope on p. 105.) There are various mechanical devices for the determination of the absolute personal equation of an observer, but as these are seldom used they will not be discussed here. The relative personal equation of two observers is the difference of their absolute equations. When observing time with a transit micrometer the personal equation, if any, may be neg- lected. The observing does not consist of a series of independent consecutive operations, but rather of a continuous performance, the star's image being bisected by the micrometer wire before the record is begun and kept bisected till after the record is ended. In Appendix 8 of the Report for 1904, entitled "A Test of the Transit Micrometer," it was shown that if there is an actual personal equation in observing star transits with a transit micrometer it is so small as to be masked by the other errors of observation. Viewed in the light of several years of actual longitude observations with the transit micrometer this conclusion is fully justified. These longitude observations involved four simple or compound loop closures, and one determination with exchange of observers. In observing differences of longitude to close a loop the same observer always kept in front as the work progressed around the loop, thus introducing into the loop closure an accumulation of any relative personal equation that might exist. In 1906 four differences determined with the transit micrometer between Seattle, Wash., and the point where the one hundred and forty-first meridian boundary of Alaska intersects the Yukon River, were combined with certain Canadian results to form a loop, and the loop closure was reduced to zero by applying a correction of only 0.008 second to each observed difference of longitude. DETERMINATION OF LONGITUDE. 91 In Texas in 1906 the three differences of longitude between the three points, Austin, Alice, and Isabel, were determined, using transit micrometers and a program as indicated above. This would introduce into the closure three times any relative personal equation of the observers. The loop closure was 0.038 second, making necessary corrections on the three differences of 0. 8 013, 0. 8 013, and 0. S 012. In 1907 a series of longitude differences was determined, using transit micrometers, between Key West and Atlanta, for both of which stations adjusted values are given in the longitude net of the United States, 1 and these adjusted values were held fixed. Six longitude differences between these two stations were determined in such a way as to accumulate any relative personal equation between the two observers. The results are shown on page 85. The correction required to be applied to each observed difference to close the loop was 0. 8 009. A second loop, closing on one of the links of the first loop or forming with all but the last difference of the first loop a new loop of eight links between the fixed stations, Key West and Atlanta, obtained corrections of only 0. 8 008 per link to close. The corrections in both loops were of the same sign. Later in 1 907 a series of longitude differences was determined in Minnesota, Dakota, Nebraska, and Iowa, using the transit micrometer. The points held fixed were the stations of the longi- tude net at Bismarck and Omaha. There were four condition equations and ten unknowns involved in the adjustment of this secondary net. - The largest correction to an observed differ- ence of longitude obtained was 0. 8 038 and the smallest was 0. 8 003. Four of the corrections obtained were less than 0. S 010 and seven were less than 0. 8 015. Where possible the program of observations was arranged to produce an accumulation of any existing relative personal equation. In 1908 the difference of longitude between the observatory of the new University o'f Wasliington at Seattle and the old longitude station in Seattle was determined, using transit micrometers. Observations were made on six nights, the observers changing stations after each night's observations. The apparent relative personal equation determined by this method of observation amounted to only 0.008 second. The above evidence justifies the present method of longitude observations with transit micrometers without exchange of observers. The evidence is sufficient to justify the continua- tion of the present method of carrying on telegraphic longitude work for geographic and geodetic purposes, for the personal equation, if present, is much smaller than the probable errors of the determinations. However, where the greatest accuracy is required, as in the determination of the difference of longitude between two fixed observatories, then an exchange of observers is desirable to eliminate any possible personal equation. An exchange of instruments is also required to eliminate differences in the total relay and armature times at the two ends of the line. For a complete elimination of this error the adjustments of the relays and magnets should be the same before and after exchange. The accuracy of the telegraphic determination of the difference of longitude, where no transit micrometer is used, depends largely upon the accuracy of the determination of the relative personal equation of the two observers, and upon its constancy. The relative personal equation of two observers may be determined in various ways. The method to be selected in a given case depends upon circumstances, involving the question of cost, the difficulty of exchange of observers, and to some degree the desired accuracy of the result. In primary longitude determinations, where cost and ease of transportation are not prohibi- tive, the relative personal equation of the observers is eliminated from the result by the observers changing stations after about one-half of the observing has been done. In this way the relative personal equation will enter the resulting differences of longitude before and after exchange of observers with different signs and the mean of such determinations will be the resulting differ- ence of longitude with the effect of personal equation eliminated. The relative personal equation may be determined independently of the longitude observa- tions by the use of two transits placed in the same observatory or in separate observatories close together, and by having the two observers observe independently the same stars, which should be arranged in time sets. If the two instruments are on the same meridian, or nearly so, and use is made of only one chronometer and chronograph to record both sets of observations, 1 See Appendix 2, Report for 1897. 92 TJ. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. it may be necessary to throw one instrument out of adjustment (in collimation) more than the other in order to avoid having the observations overlap. A better arrangement would be to have two chronographs controlled by the same chronometer by means of local relays, and have the chronograph records of the two instruments independent of one another. The difference of the two chronometer corrections thus determined, corrected for the very small longitude differ- ence between the two transit instruments, is the personal equation of the two observers. Some- times different chronometers are used and compared in the same manner as in actual longitude determinations. The relative personal equation may also be observed with a single transit instrument as follows: On the first star A observes the transits over the lines of the first half of the diaphragm, then quickly gives place to B who observes the transits across the remainder of the lines, omitting the middle line. On the second star B observes on the first half of the diaphragm and A follows. After observing a series of stars thus, each leading alternately, each observer computes for each star, from the known equatorial intervals of the lines, and from h's own observations, the time of transit of the star across the mean line of the diaphragm. The difference of the two deduced times of transit across the mean line is the relative personal equation. If each has led the same number of times in observing, the result is independent of any error in the assumed equatorial intervals of the lines. No readings of the striding level need be taken, and the result is less affected by the instability of the instrument than in the other method. If the stars observed by this method are so selected as to form time sets, and the chronometer corrections are computed from each observer's observations independently, the difference of these chronometer corrections will be the relative personal equation. As the accuracy of the telegraphic determination of longitude without the use of the transit micrometer depends also upon the constancy of the relative personal equation of the two obser- vers concerned, there is shown below a table which gives some values of the relative personal equation as derived from telegraphic longitude observations (key and chronograph method). The values in this table indicate to what extent the relative personal equation may be expected to vary from month to month and year to year. The plus sign indicates that the observer first named observes later (slower) than the other. Relative personal equation (not reduced to equator). C. H. Sinclair E. Smith [14 years] C. H. Sinclair R. A. Man- [4 years] C. H. Sinclair G. R. Putnam [5 years] s s s i s s 1881 Aug. and Sept. -0.123 0.008 1886 Sept. and Oct. +0.288 008 1891 May and June +0. 184 0.011 1881 Nov. and Dec. -.085 06 1888 Sept. + .210 09 1891 June and July + . 140 08 1885 Apr. and May - . 047 08 1888 Oct. and Nov. + .144 11 1891 July + .172 06 1885 May and June - . 131 03 1888-9 Dec. and Jan. .+ .214 10 1891 Aug. + .161 10 1885 July and Aug. - . 110 10 1889 Jan. + .233 05 1891 Aug. and Sept. + . 176 11 1886 May and June - . 062 08 1889 Jan. and Feb. + .225 07 1892 Feb. and Mar. + . 160 06 1886 June and July +.010 06 1889 Feb. and Mar. + .267 07 1892 Mar. + . 192 04 1886 July and Aug. -.023 12 1889 Mar. and Apr. + .278 12 1892 Mar. and Apr. + . 140 02 1886 Aug. and Sept. +.056 04 1889 Apr. and May + .217 12 1892 Apr. + . 150 05 1887 May and June + . 038 10 1889 May and June + .282 18 1892 Apr. and May + . 126 04 1887 June, July, and 1 1+ 109 13 1889 June and July + .246 07 1892 June and July + . 109 10 Aug. JT 1889 July + .275 08 1893 Feb. and Mar. + .082 10 1887 Sept. + .111 13 1889 July + .265 05 1896 Feb. and Mar. + . 155 03 1887 Sept. and Oct. + . 160 09 1889 July and Aug. + .228 15 1896 Mar. + . 129 07 1895 Feb. and Mar. +.093 11 1889 Aug. + .284 08 1896 Apr. + . 122 05 1895 Mar. + .075 11 1889 Aug. and Sept. + .226 06 1896 Apr. ard May + . 181 05 1895 Apr. +0.086 0.005 1889 Sept. + .258 07 1896 May and June + . 142 13 1890 May and June + . 166 14 1896 June and July +0. 124 0.008 The relative personal equation of 1890 July + .238 10 these two observers seems to be a 1890 July and Aug. + .237 14 Mean S. P.= +0. 147 function of the time and a mean of 1890 Aug. +0.278 0.006 Prob. error* of a single value 0.020 the above values would therefore have but little meaning. Mean S.-M.=' + 0.241 Prob. error* of a single value 0. 026 * This value may be taken as a measure of the variability of the personal equation. DETERMINATION OF LONGITUDE. 93 Each value in the table depends upon 8 or 10 nights of observation, 4 or 5 nights each before and after the exchange of observers, and may therefore be considered to be a mean value covering a period of from two weeks to a month or more. It is improbable that the variation of the rela- tive personal equation from night to night is as small as would be inferred directly from the above table. The error due to personal equation, remaining in the deduced longitude after the exchange of observers, is one-half the difference between the mean value of the relative personal equation before the exchange of observers and its mean value after the exchange. DISCUSSION OF ERRORS WHEN KEY AND CHRONOGRAPH ARE USED. This discussion is based upon the supposition that the regular program for longitude obser- vations when using an observing key and chronograph, consisting of 5 nights each before and after exchange of observers, has been carried out, and also that the method of selection of stars is the one formerly in use on primary longitude work in this Survey, in which a time set con- sisted of 10 stars, 5 before and 5 after reversal of the horizontal axis. These sources of error are given the same order as those shown on pages 85-87 under the heading : Discussion of Errors when Transit Micrometer is Used. First. An accidental error arising from the accidental errors of observations of 200 stars at each station. If the accidental error of observation of a single star be estimated at 0. 8 10, and this is surely a sufficiently large estimate to cover both the observer's errors and those instrumental errors which belong to the accidental class, the probable error of the final result arising from this cause would be 0. 8 10-^ -JlOQ= 0. 8 010. Second. The statement on page 86 regarding the accidental error arising from the acci- dental errors in the adopted right ascensions of the stars used, is applicable to all methods of observing. Third. For a statement regarding the errors due to the variation of the rate of the chrono- meter see page 86. Fourth. Errors arising from the variation of the relative personal equation from night to night. These are probably among the largest errors involved in longitude determinations. A constant error, not eliminated by the exchange of observers, may possibly arise from this source if the temperature, altitude, moisture conditions, etc., are very different at the two stations. Other than this, the errors arising from this source belong to the accidental class when con- sidered with reference to the computed difference of longitude and are exhibited in the residuals corresponding to the separate nights of observation. Fifth. The statement concerning errors due to lateral refraction on page 86 is equally applicable here. Sixth. No change is necessary in the statement on page 86 regarding the errors due to variation in the transmission time. Seventh. The difference of the transmission time through the two signal relays enters as an error in the final result. This error is made very small in the present work of the Survey by the use of fast-acting signal relays which are as nearly alike as possible. It might be further reduced if each observer carried his own switchboard with him when exchange of stations is made. As stated on page 87, if the difference in longitude which is being measured is large, say more than 30 minutes of time, it is well to abandon the practice of endeavoring to observe the same stars at both stations to such an extent as will bring the exchange of time signals near the middle of the time observations at each station. The error of right ascension thus introduced will be more than offset by the accuracy gained by the proper placing of the exchange. Are there appreciable errors which are constant for the night in the time determinations or in the other operations involved in the determination of a longitude difference by the tele- graphic method; and if so, what is the average magnitude of such errors? The excess of the probable error of a longitude difference computed as indicated on page 89 over its value as de- rived from the computed probable errors of the chronometer corrections at exchange is due to errors which are constant for and peculiar to each night. Using this principle l the error peculiar 1 For the formulae used in applying a similar principle to latitude observations, see pp. 119-123. 94 U. S. COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 14. to a night has been computed from fifteen longitude determinations made since 1890. It was found that the error peculiar to each night, and therefore not capable of elimination by increasing the number of observations per night, expressed as a probable error, was 0. S 022, while the probable error in the result for a night arising from accidental errors of observation, and therefore capable of further elimination by increased observation, was 0. S 013. It should be noted that the errors discussed under all but the first heading above are each capable of con- tributing to the error peculiar to a night. It is likely that variation in the personal equation is the most potent cause of such errors. It is evident from the probable errors given above that very little is lost in ultimate accuracy if clouds interfere so as to cut off a part, say one-fourth, of the regular program of time observations (two sets of ten stars each), and that almost no gain in accuracy would result from lengthening the program. Are there appreciable errors hi a telegraphic determination of a difference of longitude which are constant for the interval of several days over which the determination extends; and, if so, what is the average magnitude of such errors ? We may obtain an answer to tliis question by comparing the probable errors of longitude difference computed as on page 89 with the same probable errors as computed from the residuals developed in adjusting such a longitude net as that given in Appendix No. 2 of the Report for 1897. The excess of the last-named probable errors over the first-named is due to errors which are constant for the station during the time of occupation. From the published adjustment of the great longitude net referred to above (see pp. 246, 247, 255, of Report for 1897), after omitting the first eleven determinations (all made not later than 1872, and several involving trans- Atlantic cables) and the fifty-eighth de- termination (publication incomplete), it follows that the constant error peculiar to each longi- tude determination and not capable of elimination by increasing the number of nights per station, expressed as a probable error, is 0."022, while the accidental error of the deduced difference of longitude, which is capable of further reduction by increasing the number of nights per station (beyond the standard number, ten), is 0. S 011. It follows that a reduction of the number of nights per station to six, or even four, would result in but a slight decrease in accu- racy about 10 per cent. Three sources of errors peculiar to a station in the order of their probable magnitude are those mentioned under the fourth, sixth, seventh, and fifth headings above, namely: Variation in personal equation, variation in transmission time (especially when a repeater interrupts a circuit), the difference of the two signal relay times, and possibly lateral refraction in some cases. REDUCTION TO MEAN POSITION OF POLE. This correction will be applied in the office in accordance with the Preliminary Results published annually by the International Geodetic Association (see p. 85). A STATEMENT OF COSTS. Since 1906 forty-two differences in longitude have been determined in the United States, using the transit micrometer. Forty-one were determined in four seasons. The average cost for the field work and preparing for the field, including all expenses and salaries, was $440. The average cost per difference for the various seasons varied from $360 to $550. The cost of a difference of longitude between two places will vary according to the conditions under wluch work is done, and consequently it should be planned to have the parties in the field when the weather may be expected to be most favorable. The work should be localized for any season as much as is possible. The longer the season the more economically should the work be done. If possible, the stations should be located near the line of the telegraph in order to avoid the delay and the expense of building a long line to the observatory. The determination of longi- tude differences telegraphically in remote regions, such as Alaska, may cost from three to six or more times the average cost of a difference in the United States. No data are readily available showing the cost of the determination of longitudes telegraphically, using the key and chronograph. But owing to the necessity of exchanging DETERMINATION OF LONGITUDE. 95 observers for each difference of longitude and of observing over more nights than when the transit micrometer is used, it is probable that the cost would be from 25 to 50 per cent more than the costs stated above. LONGITUDE BY THE CHRONOMETRIC METHOD. The equipment, program of observations, and methods of computation pertaining to a determination of a difference of longitude by the chronometric method, in which chronometers transported back and forth between stations take the place of the telegraphic signals, may be most conveniently explained by giving a concrete example. The longitude of a station at Anchorage Point, Chilkat Inlet, Alaska, was determined in 1894 by transporting chronometers between that station and Sitka, of which the longitude had previously been determined. At Anchorage Point observations were taken on every possible night from May 15 to August 12, namely on fifty-three nights, by the eye and ear method, using a meridan telescope. The hack or observing chronometer kept sidereal time, and there were also four other chronometers at the station, two keeping mean time and two sidereal. These four chronometers were never removed during the season from the padded double-walled box in which they were kept for protection against sudden changes of temperature and in which the hack chronometer was also kept when not in use. The instrumental equipment and procedure at Sitka was similar to that just described. A sidereal chronometer was the hack, and two other chronometers, one sidereal and one mean time, were used in addition. Nine chronometers, eight keeping mean tune and one sidereal, were carried back and forth between the stations on the steamer Hassler. Aside from the time observations, the programme of operations was as follows : Just before beginning the time observations at Anchorage Point, and again as soon as they were finished on each night, the hack chronometer was compared with the two mean time chronometers by the method of coincidence of beats (described on p. 96). These two were then compared with each of the two remaining (sidereal) chronometers at the station. These comparisons, together with the transit time observations, served to determine the correction of each chronometer to local time at the epoch of the transit observations. Whenever the steamer first arrived at the station, and again when it was about to leave, the hack chronometer was compared with the other station chronometers, as indicated above, was carried on board the steamer and compared with the nine traveling chronometers, and then immediately returned to the station and again compared with the other four station chronometers. On board the steamer the hack was com- pared by coincidence of beats with each of the eight mean time chronometers, and the remaining (sidereal) chronometer was then compared with some of the eight. The comparisons on shore before and after the trip to the steamer served to determine the correction of the hack at the epoch of the steamer comparisons. The steamer comparisons * determined the corrections of each of the traveling chronometers to Anchorage Point time. Similar operations at Sitka deter- mined the corrections of the nine traveling chronometers to Sitka time as soon as they arrived and again just before they departed from Sitka. During the season the steamer made seven and a half round trips between the stations. CARE OF CHRONOMETERS. To secure the greatest possible uniformity of rate a chronometer should be kept running continuously, both when in use and when out of use between seasons of work. When it is allowed to remain stopped for a considerable time, the oil in the bearings tends to become gummy. When started again, the chronometer will tend to have a varying rate for some time until the effects of the stoppage have been worn off. If a chronometer is to be shipped (by express, for example), and therefore is to be subjected presumably to comparatively violent handling and jarring, it should always be stopped and the balance wheel locked by gently inserting small wedge-shaped pieces of clean cork under it. 1 In addition to the chronometer comparisons referred to in this paragraph the steamer chronometers and the station chronometers were each intercompared daily. This was done merely as a check upon their performance. 96 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. A running chronometer should always be protected as carefully as possible against jars, and especially against such sharp quick jars as result from setting it down upon a hard surface. Either the surface upon which it is set should be padded or a cushion should be carried with the chronometer. When it becomes necessary to carry a chronometer in the hand as, for example, when a hack chronometer is carried back and forth between an observatory and a steamer in con- nection with chronometric longitudes the gimbals should be locked to prevent the chronometer from swinging. It is important that the locking should be done in such a way that there will be no looseness and the corresponding tendency to a chucking motion. While the chronometer is being carried, swinging of the arm should be avoided as much as possible. Any swinging of the chronometer in azimuth is especially objectionable, as it tends to make it skip seconds and to damage it. Chronometers have been known to skip seconds, probably from this cause, even in the hands of an experienced and careful officer. On shipboard chronometers should be left free to swing in their gimbals, which should be so adjusted that the face of the chronometer will be approximately horizontal. Any change in this adjustment is apt to produce a change of rate. COMPARISON OF CHRONOMETERS BY COINCIDENCE OF BEATS. The process of comparing a sidereal and a mean time chronometer is analogous to that of reading a vernier. The sidereal chronometer gains gradually on the mean time chronometer, and once in about three minutes the two chronometers tick exactly together (one beat = 0". 5). As one looks along a vernier to find a coincidence, so one listens to this audible vernier and waits for a coincidence. As in reading a vernier one should look at lines on each side of the supposed coincidence to check, and perhaps correct the reading by observing the symmetry of adjacent lines, so here one listens for the approaching coincidence, hears the ticks nearly together, appar- ently hears them exactly together for a few seconds, and then hears them begin to separate, and notes the real coincidence as being at the instant of symmetry. The time of coincidence is noted by the face of one of the chronometers. Just before or just after the observation of the coincidence the difference of the seconds readings of the two chronometers is noted to the nearest half second (either mentally or on paper). This difference serves to give the seconds reading of the second chronometer at the instant of coincidence. The hours and minutes of both chro- nometers are observed directly. When a number of chronometers are to be intercompared, the experienced observer is able to pick out from among them two that are about to coincide. He compares those, selects two more that are about to coincide and compares them, and so on; and thus to a certain extent avoids the waits, of a minute and a half on an average, which would otherwise be necessary to secure an observation on a pair of chronometers selected arbitrarily. At Sitka on July 13, 1894, it was observed that 18 h 30 m 08 8 .00 on chronometer No. 194 (sidereal) = ll h 52 m 30 8 .00 on chronometer No. 208 (mean time); and that ll h 15 m 35 s . 50 on chronometer No. 1510 (mean time) = 14 h 48 m 10 8 .00 on chronometer No. 387 (sidereal). It was known that at the epoch of the comparisons the correction of No. 194 to Sitka sidereal time was -l m 54 8 .01, and of No. 1510 to Sitka mean tune was -6 m 26 8 .34. The required corrections to No. 208 and No. 387 were computed as follows: ft nt A m Time by 194 =18 30 08. 00 Time by 1510 = 11 15 35. 50 Correction to 194 = -01 54. 01 Correction to 1510 = - 6 26. 34 Sidereal time =18 28 13. 99 Mean time = 11 09 09. 16 Sidereal time of mean noon= 7 26 53. 66 Correction mean to sidereal = +01 49. 93 Sidereal interval =11 01 20. 33 Sidereal interval = 11 10 59. 09 Correction, sidereal to mean = 01 48. 34 Sidereal time of mean noon= 7 26 53. 66 Mean time =10 59 31. 99 Sidereal time = 18 37 52. 75 Time by 208 =11 52 30. 00 Time by 387 = 14 48 10. 00 Correction to 208 = -52 28.01 Correction to 387 =+3 49 42.75 The correction to reduce a sidereal to a mean time interval, or vice versa, may be taken from the tables in the back part of the American Ephemeris. The sidereal time of mean noon DETERMINATION OF LONGITUDE. 97 may be taken from that part of the Ephemeris headed "Solar ephemera," and it should not be overlooked that it is the sidereal time of local mean noon that is required, and that, therefore, the longitude (approximate) of the station must be taken into account. The correction to be applied to Washington sidereal time of mean noon to obtain that for the station is the same as the correction to reduce a mean time interval equal to the longitude of the station from Wash- ington to a sidereal interval. COMPUTATION OF LONGITUDE FROM A SINGLE ROUND TRIP. From the operations at Anchorage Point the correction of each station chronometer at the epoch of each set of time observations became known. The intercomparisons on shore before leaving for the steamer and after returning, together with the assumption that each station chronometer runs at a uniform rate between time sets, gave five separate determinations of the correction to the hack at the epoch of the steamer comparisons. Thus, on June 18, 1894, at 3 h .45 by its own face, the middle epoch of the steamer com- parisons, the correction to the hack (No. 380) was By its own rate -2 38. 16 (weight By No. 4969 rated By No. 2490 rated By No. 207 rated By No. 2637 rated 38.30 38.26 38.16 38. 62 (weight f). Mean = -2 38. 30 Weighted mean =-2 38.25 The comparisons of No. 380 with No. 4969 at the station on this date, computed upon the supposition that No. 4969 ran at a uniform rate between preceding and following time observa- tions, showed that the correction to No. 380 at 2 h .64 by its face was -2 m 38 S .34, and at 4 h .36 was 2 m 38 S .25. Assuming it to run uniformly between these epochs, its correction was 2 m 38 8 .30 at 3 h .45, as shown above. An examination of the daily rates of the five chronometers showed that No. 2637 ran very irregularly, and that No. 380 did not run as regularly as the other three. Hence these chro- nometers were assigned less weight than the others, as indicated above. 1 Using the weighted mean value for the correction to No. 380 at the epoch of the steamer comparisons these comparisons give the correction of each traveling chronometer on Anchorage Point time. Similar operations at Sitka gave the correction to each traveling chronometer on Sitka tune on each arrival at and departure from Sitka. Computation of difference of longitude of Sitka and Anchorage Point. FIRST TRIP STARTING FROM ANCHORAGE POINT. Chronomc- Anchorage Point. May 15 Sitka, May 17 Sitka, May 20 Anchorage Point, May 23 M. T. or SU. Chr. epoch Correction Chr. epoch Correction Chr. epoch Correction eph Correction h h m s h h m s h h m s h h m M. T. 231 11.83 -0 03 31.39 7.54 -0 03 02. 93 7.55 -0 03 02. 14 7.65 03 20. -'6 1 607 11.84 -0 01 03.88 7.81 -0 CO 34. 93 7.67 -0 00 33. 73 7.65 -0 01 01.34 1 510 12.15 -0 03 42.50 7.75 -0 03 19. 43 7.52 -003 28.22 7.75 -0 04 05. 90 196 9.49 +2 26 28.51 5.20 +2 2653.00 5.19 +2 26 46.08 5.29 +2 26 1C. 72 1 542 11.92 -0 02 55. 84 7.53 -0 02 29. 37 7.72 -002 31.83 7.81 -0 03 02. 63 1 728 9.38 +2 34 40. 23 5.08 +2 34 59.90 4.91 +2 34 46.00 5.23 +2 34 02. 4(j 208 12.71 -0 42 08. 24 8.17 -0 42 01.19 8.48 -0 42 35.76 8.56 -0 43 35.37 2 167 8.73 +3 18 39. 99 4.39 +3 19 09.98 4.15 +3 19 12.69 4.59 +3 18 47.44 Sid. 387 11.97 +3 46 50.04 7.65 +3 47 22.97 7.7S +3 47 29.07 8.29 +3 47 09.31 i If considered desirable, the relative weights to be assigned to the station chronometers may be determined more accurately by the method outlined in the footnote on p. 100. 8136 13 7 98 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Computation of difference of longitude of Sifka and Anchorage Point Continued. FIRST TRII' STARTING FROM ANCHORAGE POINT Continued. From Anchor- Chro- nometers Total At Sitka Traveling Dailv age Point to Sitka Correction at Sitka on Anchor- Differ- ence of M. T.or Sid. rate age Point longi- tude Time Rate Time Rate Time Rate Time Rat d h s d h s d h s s d h s h m s m s M.T.231 7 19.82 + 1.93 2 24.01 + 0.79 4 19.81 + 1.14 + 0.24 1 19.71 + 0.43 -0 03 30. 96 28. 03 1 507 .81 + 2.54 23.86 + 1.20 19.95 + 1.34 + 0.28 19.97 + 0.51 -0 01 03.37 28.44 1 510 .60 -23.40 23.77 - 8.79 19.83 -14.61 - 3.03 19.60 - 5.50 -0 03 48.00 28.57 196 .80 -17.79 23.99 - 6.92 19.81 -10. 87 -2.25 19.71 - 4.10 +2 26 24.41 28.59 1 542 .89 - 6.79 24.19 - 2.46 19.70 - 4.33 -0.90 19.61 -1.64 -0 02 57.48 28.11 1 728 .85 -37.77 23.83 -13.90 20.02 -23. 87 - 4.94 19.70 - 8.99 +2 34 31.24 28.66 208 .85 -90.13 24.31 -34.57 19.54 -55.56 -11.54 19.46 -20.90 -0 42 29. 14 27.95 2 167 .86 + 7.45 23.76 + 2.71 20.10 + 4.74 + 0.98 19.66 + 1.78 +3 18 41.77 28.21 Sid. 387 20.32 +19.27 24.13 + 6.70 20.19 +12.57 + 2.60 19.68 + 4.73 +3 46 54. 77 28.20 In the form on page 97 the column headed "Chr. epoch" gives the face reading of the chro- nometer, expressed in hours and hundredths (rather than minutes and seconds) for convenience in computation. The corrections at Anchorage Point are to the local time of that station, and at Sitka to Sitka local time. In the form above, the second and third columns give the elapsed chronometer time and the accumulated rate between the Anchorage Point steamer comparisons, and the fourth and fifth columns give the same quantities between the Sitka steamer comparisons. The second column minus the fourth, and the third minus the fifth are the traveling time (both ways) and the accumulated rate while traveling, from which the daily traveling rate as given in the eighth column becomes known. The ninth column gives the traveling time between steamer com- parisons from Anchorage Point to Sitka, and the tenth column gives the accumulated rate dur- ing this interval computed by the use of the eighth column. This accumulated rate being applied as a correction to the chronometer correction on Anchorage Point time at the begin- ning of the trip gives the correction on Anchorage Point time on arrival at Sitka. This difference subtracted from the directly observed correction on Sitka time at that epoch, shown in the upper form, gives the required difference of longitude. It should be noted that in this computation the traveling rate is supposed to be a constant during the round trip, but is not assumed to be the same as the rate while in port. The longitude difference if computed from the return half of the trip, from Sitka to Anchor- age Pjint, would necessarily by this process of computation be identical with that shown above. If the steamer had stopped so short a time at Sitka that only one set of steamer compari- sons had been made while there, as was frequently the case, the above computation would have been simplified in an obvious manner. COMBINATION OF RESULTS. The difference of longitude was thus computed from each traveling chronometer for each round trip, starting from Anchorage Point, the last half trip (iy 2 round trips being made) from Anchorage Point to Sitka, being omitted. A similar computation was also made for each round trip, starting from Sitka, the first half trip, Anchorage Point to Sitka, now being omitted. 1 Each of these computations would be subject to a constant error if the traveling chronometers had uniformly accelerated or uniformly retarded rates, but their mean is free from this error. One half of the computation also serves as a check on the other half. i If the steamer had returned again to Anchorage Point, so as to complete eight round trips, all of the eight would have been used in the first computation; and in the second computation (round trips, starting from Sitka) the last trip from Sitka to Anchorage Point, combined with ihe first trip in the opposite direction, would have been used as the eighth round trip. This principle of computing the difference of longitude from the round trips starting from each station in turn, and combining the two results was used for the first time by Assistant C. A. Schott in 1857 in deriving the difference of longitude of Savannah, Ga., and Fernandina, Fla. (See Coast Survey Report for 1857, pp. 314-324.) DETERMINATION OF LONGITUDE. The method of combining these separate results is shown in the following form . Difference of longitude between Siika and Anchorage Point, ChilJcat Inlet, Alaska. SUMMARY OF RESULTS FROM SEVEN ROUND TRIPS, STARTING FROM ANCHORAGE POINT. 99 Chronometers, M. T. or Sid. l.t 2 d 3 d 4 th 5 th 6 lh 7 lh Means JA Weights S S S S S S S S M. T. 231 28. 03 26. 36 28. 36 28. 19 28. 45 28. 19 28. 18 27.97 3 1507 28. 44 29. 06 29. 18 28. 26 28. 27 28. 20 28. 54 28.56 4 1510 28.57 29.25 29.00 28.52 28.63 28.06 28.58 28.66 7 196 28.59 29.09 29.54 28.59 28.43 28.51 28.92 28.81 3 1542 28. 11 28. 11 28. 66 28. 23 28. 47 28. 38 28. 37 28.33 22 1728 28. 66 28. 94 29. 16 28. 63 28. 58 28. 43 28. 59 28.71 6 208 27. 95 27. 40 28. 21 28. 19 28. 42 28. 42 28. 09 28.10 6 2167 28.21 28.56 28.90 28.55 28.68 28.27 28.64 28.54 17 Sid. 387 28.20 28.44 28.91 27.93 28.41 27.93 28.59 28.34 6 Mean 28.31 28.36 28.88 28.34 28.48 28.27 28.50 28.45 Weighted mean 28.25 28.38 28.82 28.35 28.52 28.28 28.49 28.44 Weight 3122212 Weighted mean 0" O m 28'.440 S .05 SUMMARY OF RESULTS FROM SEVEN ROUND TRIPS, STARTING FROM SITKA. Chronometers, M. T. or Sid. lt 2> 6 th 7"> Means a Weights S S S S S S S S M. T. 231 28.87 28.78 28.74 28.39 28.37 28.71 28.11 28.57 3 1507 27. 69 29. 08 29. 11 27. 76 28. 78 27. 93 28. 64 28.43 4 1510 28.37 28.88 28.82 27.91 28.83 28.10 28.58 28.50 7 196 28.59 29.07 28.95 27.66 28.03 29.56 29.20 28.72 3 1542 28.93 28.57 28.59 28.22 28.50 28.50 28.32 28.52 22 1728 27.59 28.90 28.75 27.99 29.01 28.09 28.75 28.44 6 208 27.71 28.03 28.52 28.58 27.88 28.76 27.65 28.16 6 2167 28.24 28.71 28.80 28.27 28.77 28.31 28.49 28.51 17 Sid. 387 28. 68 28. 80 28. 43 27. 69 28. 97 27. 98 28. 73 28.47 6 Mean 28.30 28.76 28.75 28.05 28.57 28.44 28.50 28.48 Weighted mean 28. 41 28. 69 28. 70 28. 13 28. 61 28. 38 28. 44 28.48 Weight 1222222 Weighted mean C h 0" Final mean AX 2SM80.05 h m = +0 00 28.460.05 Let N be the number of days during which the chronometers are depended upon to carry the time during each round trip, reckoned by adding to the "traveling time," as given in the sixth column of the form on page 98, the interval between each comparison of the hack chro- nometer with the traveling chronometers and the nearest (either before or after) time obser- vation made at that station. The weight assigned to each trip is proportional to the reciprocal of N. This weighting depends upon the assumptions that errors in the computed longitude arising from the time determinations and from the chronometer comparisons are small as compared with those arising from variations in Chronometer rates; that the time is carried by the combined station chronometers over the intervals during which they are depended upon with about the same degree of accuracy (due regard being paid to the length of the interval) as the combined traveling chronometers carry the time during the trip, and, finally, that the errors arising from the variations in the chronometer rates belong to the accidental class and are proportional to the square root of the length of the interval over which the time is carried. 100 U. S. COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 14. WEIGHTS ASSIGNED TO SEPARATE CHRONOMETERS. Even a cursory examination of such a table as that given on the preceding page shows that some chronometers run much more uniformly than others, and therefore furnish determina- tions of the longitude difference which are entitled to greater weight. Let Z 1; 1 2 , 1 3 , . . . l a be the derived values of the difference of longitude as given by one chronometer on the different trips, and let I be their mean. Let n be the number of trips. Then, by the ordinary laws of least squares, assigning equal weights to the separate trips, the probable error of any one of these Z'sis . . q-Q'T 71-1 The weight p, inversely proportional to the square of this probable error to be assigned to a chronometer, is proportional to 71-1 The computation of weights may be put in the following convenient tabular form: COMPUTATION OF WEIGHTS. From the seven round trips starting from Anchorage Point. Chronometer I 231 27.97 1507 2S-.56 1510 2S-.66 196 28-.81 1542 2S-.33 1728 28-.71 208 28'. 10 2167 28-.S4 337 2S-.34 l-l, l-l, 1-1 3 1-1, l-k l-l 1-1 7 (l-J,) (l-J,)' (1-1 3 ? ft-W (l-k)* (l-k? (l-lrf Z(l-W By 2d comb.* Mean of 2 n-1 n-1 - .06 + 1.61 - .39 - .22 - .48 - .22 - .21 + .12 - .50 - .62 + .30 + .29 + .36 + .02 + .09 - .59 - .34 + -14 + .03 + .60 + .08 + .22 - .28 - .73 + .22 + .38 + .30 - .11 + .22 + .22 - .33 + .10 - .14 - .05 - .04 + .05 - .23 - .45 + .08 + .13 + .28 + -12 + .15 + .70 - .11 - .09 - .32 - .32 + .01 + .33 - .02 - .36 - .01 - .14 + .27 - .10 + .14 - .10 - .57 + -41 - .07 + -41 - .25 .00 2.59 .15 .05 .23 .05 .04 .01 .25 .38 .09 .08 . 13 .00 .01 .35 . 12 .02 .00 .36 .01 .05 .08 .53 .05 .14 .09 .01 .05 .05 .11 .01 .02 .00 .00 .00 .05 .20 .01 .02 .08 .01 .02 .49 .01 .01 .10 .10 .00 .11 .00 . 13 .00 .02 .07 .01 .02 .01 .32 . 17 .00 .17 .06 3.11 .47 1.79 6 3.3 .94 2.30 1.62 6 3.7 .87 .89 0.88 6 6.8 .95 2.73 1.84 6 3.3 .24 .30 0.27 6 22. 2 .37 1.78 1.08 6 5.6 .73 1.22 0.98 6 6. 1 .34 .36 0.35 6 17.0 .75 1.32 1.04 6 5.8 2-(l-l n )' * From similar results from seven round trips starting from Sitka. A similar computation was made using the seven round trips starting from Sitka, the results of which are shown in the line marked "by 2d combination," and the weights were derived from the mean results of the two computations. 1 DISCUSSION OF ERRORS. The error in a difference of longitude observed and computed as indicated in the preced- ing sections depends upon the errors in the transit tune observations, errors in the comparison of chronometers, errors arising from variations in the rates of chronometers, and, finally, the relative personal equation of the two observers concerned. i The relative weights to be assigned to the station chronometers when they are used to determine the correction of the hack at the epoch of the steamer comparisons might be computed by an analogous process. Let O be the correction to a chronometer at the epoch of transit time obser- vations as determined from those observations. Let / be its correction at that same epoch interpolated between its observed corrections at the last preceding and first following transit time observations on the assumption that its rate during that interval is constant. For a group of chronome- ters whose corrections are all determined a number of times in succession by the same transit observations, the relative weights are evidently proportional to ^ ij_Q\r DETERMINATION OF LONGITUDE. 101 The errors in the time observations will in general be very small in co.nparison with the other errors affecting the result. For the probable magnitude of the time errors see the first part of this publication. In Appendix No. 3 of the Report for 1894 and in No. 3 of 1895 may be found detailed statements of the results of several determinations of longitude by the chro- nometric method which will serve to give a concrete idea of the magnitude of the errors involved in such determinations. The relative magnitude of the errors arising from the time determi- nations increases as the time, N (see p. 99), required for a round trip decreases. The errors made in comparing chronometers by the method of coincidences are negligible in then- effect upon the final result. The checks obtained during the intercomparisons of chronometers show that the probable error in a single comparison is about 8 .01, correspond- ing to a probable error of about 4 8 in estimating the time of coincidence of ticks. The errors arising from variations in the rates of chronometers are by far the most serious class of errors involved in chronometric determinations of longitude. The table of results given on page 99 gives a fair indication of the magnitude of the errors to be expected from this source. The various traveling chronometers are subjected to variations of temperature, humidity, and barometric pressure, and to disturbances arising from the motion of the ship, which are common to them all. Do these common conditions produce variations in rate which are common to all the chronometers, and therefore introduce a common error into the various values of the longitude difference resulting from any one trip ? An examination of the results of six chrono- metric determinations of longitude in Alaska, printed in the 1894 and 1895 Reports, indicates that such errors in the deduced longitudes, common to all the chronometers on a given trip, are exceedingly small upon an average so small that they are concealed by the accidental errors. Chronometers are compensated for temperature as well as possible by the maker, but such compensation is necessarily somewhat imperfect. In general, however, this compensa- tion is so nearly perfect that little or nothing is gained in accuracy by deriving and using tem- perature coefficients connecting the temperature and the rate. There are occasional excep- tions; for example, the Button chronometer No. 194 (see pp. 77-78 of the Report for 1894) shows a very large variation in rate due to change of temperature. In considering the errors due to variations in chronometer rates it should not be overlooked that the station chronometers are depended upon to carry the time over the interval from the nearest time observations to the steamer comparisons in precisely the same manner in which the traveling chronometers are depended upon during the trip. It is because of this fact that it may be desirable during periods of very bad weather to supplement the transit observations upon stars by transit observations upon the sun, as indicated on page 51, or in low latitudes by theodolite or vertical circle observations for tune, or even by sextant observations for time. Unless the relative personal equation is eliminated from the computed longitude it is apt to be one of the largest errors affecting the mean result, except when the round trips are very long or very few chronometers are carried. It may be eliminated by any of the methods sug- gested on pages 90-93. Assuming that the relative personal equation is eliminated by direct determination or otherwise, the error of the mean result of a chronometric longitude determination will be nearly inversely proportional to the square root of the number of chronometers carried (provided the stations are supplied with a sufficient number of good chronometers to make the shore errors small), to the square root of the number of round trips, and the square root of the average value of N (the interval over which the time is carried by the chronometers). It will depend very inti- mately upon the quality of the chronometers and upon the care with which they are protected from temperature changes and jars. It will be affected very little by an increase in the errors of the time observations proper, resulting from very fragmentary observations on cloudy nights or from substituting some more approximate method for transit observations upon stars. From the above principles and the numerical values given in Appendix No. 3 of the 1894 Report and in No. 3 of the 1895 Report, one may make an estimate of the errors to be expected 102 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. if the above elaborate plan of operations can be carried out only in part, as, for example, when an observer determines the longitude of a new station by making a single trip to it, carrying a few chronometers only and making all time observations at both ends of the trip himself. In connection with any plan of operations which involves long intervals between the arrival at and the departure from a given station, it should be kept in mind that the computation usually involves the assumption that the rates of the traveling chronometers are the same on the trip to the station as on the return trip, and therefore a long stay at the station is apt to increase the error of the final result by giving the chronometers a long time to acquire new rates. Under extreme conditions it may sometimes be well to avoid this assumption and to use a separate traveling rate for each half trip derived from observations just preceding or following that half trip. PART III. THE DETERMINATION OF LATITUDE BY MEANS OF THE ZENITH TELESCOPE. INTRODUCTORY. A measurement of the meridional zenith distance of a known star, or other celestial object, furnishes a determination of the latitude of the station of observation. In the zenith telescope, or IIorrebow-Talcott, 1 method of determining the latitude, there is substituted for the measure- ment of the absolute zenith distance of a star the measurement of the small difference of meridional zenith distances of two stars culminating at about the same time, and on opposite sides of the zenith. The effect of this substitution is the attainment of a much higher degree of precision, arising from the increased accuracy of a differential measurement, in. general, over the corre- sponding absolute measurement; from the elimination of the use of a graduated circle from the essential part of the measurement; and from the fact that the computed result is affected, not by the error in estimating the absolute value of the astronomic refraction, but simply by the error in estimating the very small difference of refraction of two stars at nearly the same altitude. Because of its great accuracy, combined with convenience and rapidity, the Horrebow- Talcott method has become the only standard method of this Survey. For other methods of determining the latitude, involving in most cases absolute measurements of zenith distance or altitude, the reader is referred to treatises on astronomy. The method of determining the latitude by observing the time of transit of a star across the prime vertical, is one which is capable of a very high degree of accuracy and is well adapted to field use, as the effects of instrumental errors may be readily eliminated. To determine the latitude of a station by this method, the times of transit of various stars (of positive declination less than the latitude) across the plane of a transit placed approximately in the prime vertical are observed. The inclination of the transverse axis is determined accurately with a striding level. The effects of error of collimation and pivot inequality are eliminated by reversal of the axis. The effects of azimuth error (deviation of the instrument from the prime vertical) and of constant errors in the observed times (personal equation) are eliminated by observing some stars to the eastward of the zenith and others to the westward. The declinations of the stars observed must be accurately known, as the declination errors enter directly into the latitude at about their full value, but the right ascensions need be known but approximately. This method has been little used by this Survey, perhaps because more time is required to prepare an extended observing list than in the zenith telescope method, but it may be found useful in the future. If the only instrument available is a theodolite having a good striding level, but not equipped for observations by the zenith telescope method, observations in the prime vertical will give the best possible determination of the latitude. (For details in regard to this method, see Chauvenet's Astronomy, Vol. II, pp. 238-271, and Doolittle's Practical Astronomy, pp. 348-377. For an interesting, early test of the method [1827] by Bessel, with a very small portable instrument, see Astronomische Nachrichten, Vol. 9, pp. 413-436.) GENERAL INSTRUCTIONS FOR LATITUDE WORK. 1. In order that the records and computations of the latitude work of this Survey may be uniform in character and that there may be approximately the same accuracy in the results, some general directions are given here which should be carried out by all observers of this Survey, 1 See p. 245 of Appendix 14, Report for 1880, for some general remarks on Talcott's method. 103 104 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION XO. 14. engaged upon this class of work, unless they are directed otherwise by special instructions or unless exceptional circumstances are encountered which make changes necessary or desirable. 2. The Horrebow-Talcott method should be followed, using the zenith telescope or the meridian telescope. (See p. 8 for description of the latter instrument. The zenith telescope is described below.) 3. A pair of stars should be observed only once at a given station, unless some gross error is discovered, in which case the pair may be reobserved. Not more than two stars should be observed at one setting of the instrument. A star may be observed on more than one night, if paired with a different star on each night. 4. A sufficient number of pairs should be observed at a station to make it reasonably certain that the probable error of the mean result is not greater than 0".10 (see directions for procedure in making the office computation). No additional expenditure of time or money should be made in trying to reduce the probable error below this limit. In no case, however, should the number of pairs observed at a station be less than 10. 5. No determination of the micrometer value should be made in the field, as this value is computed at the office from the regular observations for latitude. 6. The pairs observed should be so selected that the algebraic sum of the measured micro- meter differences in turns at a station is less than the total number of pairs. This sum should be made small, in order that the computed latitude may be nearly free from any effect of error in the mean value of the micrometer screw. 7. The stars observed upon should be taken from "The Preliminary General Catalogue of 6188 Stars for the Epoch 1900" by Lewis Boss, which was published by the Carnegie Institution of Washington in 1910. 8. Duplicates of the latitude records, in the form of entries in the latitude computation sheets, should be made and checked as the work progresses. Only such portions of the latitude computations should be made in the field as are necessary to ascertain the degree of accuracy secured. 9. The duplicates and computations, both complete and incomplete, for each station should be sent to the office by registered mail, as soon as practicable after the completion of the occu- pation of the station. Each book of original records should be sent to the office by registered mail soon after the last of the corresponding duplicates and computations have been forwarded, but not so soon as to arrive in Washington by the same mail. It is desirable to have the records and computations sent to the office promptly, in order to avoid their possible loss. 10. Original descriptions of stations should be inserted in the original record of latitude observations and a duplicate description of each station should be written in a volume kept especially for the purpose. This volume should be sent to the office at the close of a season's work. 11. The form of record of observations and of field and office computations of results should conform to those shown in this publication. These General Instructions will be referred to from time to time in the siicceeding text. DESCRIPTION OF THE ZENITH TELESCOPE. Illustration No. 13 shows one of the best zenith telescopes now in use in this Survey. This instrument, Zenith Telescope No. 4, was originally made by Troughton & Simms, of London, in 1849, and was remodeled at the Coast and Geodetic Survey Office in 1891. It carries a telescope with a clear aperture of about 76mm (3*inches), and a focal length of about 116,6cm (46 inches). The magnifying power with the eyepiece ordinarily used is 100 diameters. Two latitude levels are used instead of one, to secure increased accuracy. Each of these levels carries a graduation which is numbered continuously from one end to the other (instead of each way from the middle), the numbering of the upper one running from to 50 and of the lower from 60 to 110. A 2mm division on the upper level has a value of about 1".6 and on the lower about 1".4. The vertical axis of the instrument is in the vertical plane in which the telescope swings. The clamp arm, perforated for the sake of lightness, gives the telescope a No. 13. ZENITH TELESCOPE. DETERMINATION OF LATITUDE. 105 marked degree of stability in so far as changes of inclination are concerned. The eyepiece micrometer, arranged to measure zenith distance, has a value of about 45" per turn, and the micrometer head is graduated to hundredths of a turn. The better known type of zenith telescope, in which the telescope is mounted eccentrically on one side of the vertical axis instead of in. front of it, is also in use in the Survey. The meridian telescopes described on page 8 are extensively used for latitude determinations, as well as for time. In latitude work with the meridian circle at astronomic observatories the instrument is usually fitted with a reversing prism. By rotating this prism the apparent motion of the star is changed from the direction right to left to the direction left to right or vice versa. A pointing is made on the star before it transits, the prism is reversed, and a second pointing is made after the transit. The observer may always place the wire above the center of the star's image (or below) but as the image is reversed by the prism, one of the pointings is made on the south side of the center of the star and the other pointing on the north side. The mean of the two point- ings will be free from any constant or systematic error in the bisection of the star. It is believed that the systematic error of bisection does not affect the results of latitude observations made by the Talcott method, except to a small degree due to the fact that an observer's systematic error of bisection may be slightly different for stars of different magnitude. A pair may be composed of stars of very different magnitudes. The reversing prism need not be used in any latitude observations by the Talcott method which are made for the usual geodetic orgeographic purposes. SUPPORT FOR THE INSTRUMENT. The support for the latitude instrument most frequently used in this survey is a wooden tripod made of lumber about 6 inches square in cross-section, well braced and set firmly in the ground to a depth of from 1 to 3 feet, depending on the nature of the soil. Piers made of brick, of cement blocks, or of concrete are also used. The concrete pier is not as satisfactory as the other types, if it is used very soon after it is constructed. When latitude and azimuth are both observed at a station the same pier may be used for mounting both the latitude instru- ment and the theodolite. A type of pier used by some of the parties of this Survey is shown in illustration No. 24 and is described on page 140. OBSERVATORIES AND OBSERVING TENTS. At the field stations only a temporary structure to protect the instrument from wind during the observations and from rain during the stay at the station is needed. The observer is seldom at a station more than a week after everything has been made ready for the observing, and an observatory such as is shown in illustration No. 14, built of rough lumber, answers every purpose. It is advisable to have 2 doors in the observatory to insure the free circulation of air. No part of the building should touch the ground except at the corners. The roof may be made water-tight by boards or a covering of felt or tar paper. A canvas sheet is sometimes carried with the outfit and the roof is made by stretching this sheet over the rafters and tying it to the sides of the observatory. The canvas may be removed during the observations, thus leaving the whole top of the observatory open to the sky. When a station is located in a town, although for only a short time, the observatory should as a rule be made neatly, of smooth lumber, as shown in illustration No. 15. Buildings at permanent latitude stations need not be discussed here, as this publication deals only with observations made for geodetic or geographic purposes. An observing tent such as is shown in illustration No. 16 or in illustration No. 17 is more frequently used on latitude work than the wooden observatory, and it has the great advantage that it is easily transported and quickly set up. Except on mountain peaks or at other places where transportation is difficult the tent has a floor similar to that used with an observatory. Where a floor or platform is not used, the observer must be extremely careful not to shift his weight during the interval between the pointing on a star and the reading of the levels. 106 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. and in this case the bubble readings must be made by an attendant who must also stand in one place without shifting his weight from the time the observation is made until the level is read. ADJUSTMENTS. When setting up the instrument place two of the foot screws in an east and west line. The level correction may then be kept small during the progress of the observations by using one foot screw only. The vertical axis may be made approximately vertical by use of the plate level, if there is one on the instrument, and the final adjustment made by using the latitude level. The position of the horizontal axis may then be tested by readings of the striding level. If the horizontal axis is found to be inclined, it must be made horizontal by using the screws which change the angle between the horizontal and vertical axes, if the instrument is of the old form. With the new form of instrument (illustration No. 13), or with a meridian telescope, the two axes will always remain so nearly at right angles that no means for making this adjustment is needed. With these instruments the vertical axis may be made vertical by using both the striding level and the latitude level at the same time. The eyepiece and objective should be carefully focused as indicated on pages 14 and 15. It is important that the focus of the objective should be kept constant during the stay at a station, since the angular value of one turn of the eyepiece micrometer is depended upon to remain constant for the station. However, the results of the determination of the value of a turn of the micrometer vary in some cases as much as 0".13, corresponding to a range of about 3.3 millimeters in the distance between the objective and the micrometer lines (see p. 129). In connection with the common habit of carefully keeping the draw tube clamped for the purpose of holding the micrometer value constant, it is interesting to note that while in the field in 1905 Assistant W. H. Burger focused zenith telescope No. 2 five times in rapid succession with a range of only 0.1 millimeter in the position of the sliding tube. The movable micrometer thread with which all pointings are to be made must be truly horizontal. This adjustment may be made, at least approximately, in daylight after the other adjustments. Point, with the movable thread, upon a distant well-defined object, with the image of that object near the apparent right-hand side of the field of the eyepiece, and with the telescope clamped in zenith distance. Shift the image to the apparent left-hand side of the field by turning the instrument about its vertical axis. If the bisection is not still perfect, half the correction should be made with the micrometer and half with the slow-motion screws which rotate the whole eyepiece and reticle about the axis of figure of the telescope. Repeat, if necessary. The adjustment should be carefully tested at night after setting the stops by taking a series of pointings upon a slow-moving star as it crosses the field with the telescope in the meridian. If the adjustment is perfect, the mean reading of the micrometer before the star reaches the middle of the field should agree with the mean reading after passing the middle, except for the accidental errors of pointing. It is especially important to make this adjustment carefully, for the tendency of any inclination is to introduce a constant error into the computed values of the latitude. The line of collimation (see p. 13) as defined by the middle vertical line of the reticle must be very nearly perpendicular to the horizontal axis. If the instrument is a meridian telescope, or of the form shown in illustration No. 13, this adjustment may be made as for a transit (p. 15) by reversing the horizontal axis in the wyes. If the instrument is of the form in which the telescope is to one side of the vertical axis, the method of making the test must be modified accordingly. It may be made by using two collimating telescopes which are pointed upon one another in such positions that the zenith telescope may be pointed first upon one and then upon the other with no intermediate motion except a rotation of 180 about the horizontal axis. It may be made as for an engineer's transit, but using two fore and two back points, the distance apart of each pair of points being made double the distance between the vertical axis and the axis of collimation of the telescope. A single pair of points at that distance apart may ba used and the horizintal circle trusted to determine when the instrument has been turned No. U. OBSERVATORY. DETERMINATION OF LATITUDE. 107 180 in azimuth. Or a single point at an approximately known distance may be used and the horizontal circle trusted as before, and a computed allowance made on the horizontal circle for the parallax of the point when the telescope is changed from one of its positions to the other. Thus, let d = the distance of the vertical axis from the axis of collimation of the tele- scope, D = the distance to the point, and p = the parallax for which correction is to be made ; then, in seconds of arc: 2d p ~Dsml" If one considers the allowable limit of error in this adjustment (see p. 134) it is evident that refined tests are not necessary, and that a telegraph pole or small tree, if sufficiently distant from the instrument, may be assumed to be of radius = d, and the adjustment made accordingly. The stops on the horizontal circle must be set so that when the abutting piece is in contact with either of them the line of collimation is in the meridian. For this purpose the chronometer correction must be known roughly within one second, say. Set the telescope for an Ephemeris star which culminates well to the northward of the zenith, and look up the apparent right ascension for the date. Follow the star with the middle vertical line of the reticle, at first with the azimuth motion free and afterwards using the tangent screw on the horizontal circle, until the chronometer, corrected for its error, indicates that the star is on the meridian. Then clamp a stop in place against the abutting piece. Repeat for the other stop, using a star which culminates far to the southward of the zenith. It is well, if time permits, to test the setting of each stop by an observation of another star before commencing latitude observations. The correction to the chronometer may be obtained by observations on the sun or stars with a sextant or a vertical circle (see pp. 52-56), by observing the time of transit of stars with a theodolite, or by using the zenith telescope as a transit instrument. With the zenith telescope in good adjustment and approximately in the meridian and the sidereal time known within several minutes, the chronometer time of transit of a star near the zenith is noted. This obser- vation gives a close approximation to the chronometer error. Then a north star of high decli- nation is used and the telescope is put more nearly in the meridian by the method explained above. Next the chronometer time of transit of a second zenith star is observed, which will usually give the chronometer correction within a second. With this value of the chronometer correction the telescope may be put closely enough in the meridian for observing. The finder circle must be adjusted to read zenith distances (see p. 16). THE OBSERVING LIST. The Boss catalogue 1 of 6188 stars is now available, and is at present the best list from which to select pairs of stars. (See paragraph 7 of General Instructions, p. 104.) The latitude of the station should be obtained to the nearest minute from a map, a triangulation station, or from preliminary observations on the sun or stars. In the Boss catalogue the declinations of the stars are given and the observing list may be made out like the form shown below. Any other arrangement of the data may be used. To find all available pairs in a given list one may, for each star in succession within the zone of observation, 45 each way from the zenith, sub- tract the declination from twice the latitude and then compare this difference with the decli- nation of each star in the list within the following 20 m of right ascension. Any star whose declination 2 is within 20' of the above difference will combine with the star under considera- tion to make a pair, provided the other conditions stated below are fulfilled. By proceeding thus every available pair will be found. 3 1 Preliminary general catalogue of 6188 stars for the epoch 1900, Lewis Boss, Carnegie Institution of Washington, 1910. 2 Or 180 t for subpolars. 3 At stations in Alaska there are but few stars in the zone extending 45 northward from the zenith as compared with the corresponding zone to the southward, and the above process may be improved by taking in succession only stars to the north of the zenith and comparing each with stars in both the preceding and the following 10. To make the search with a subpolar star subtract 180 3 from twice the latitude and pair with any star whjse declination is within 2ff of this difference, provided its right ascension differs from that of the subpolar anywhere from ll h 40" to 12"> 20. 108 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Observing list (Form 1). [St. Anne, 111., June 23, 1908. Zenith telescope No. 4. ^=41 Ol'.S. Search faetor=2 0=82 03'.] Star No. Boss catalogue Mag. Right ascen- sion Declina- tion S Differ- ence between 3'a = sum of declina- tions Zi-'ij, N-S- 0* (Xi if) Star north or south Setting = j differ- ence of 3's Tunis h m s o / / / f / 4327 4.5 16 55 22 82 11 N 12 4379 4.9 17 11 53 -0 21 82 32 81 50 -13 -17 S 41 16 28 4441 5.9 17 28 13 28 28 S 10 4494 5.8 17 42 04 53 50 25 22 82 18 +15 +20 N 12 41 30 4623 5.1 18 13 22 64 22 N 24 4651 5.4 18 18 45 17 47 46 35 82 09 + 6 + 8 S 23 18 16 4669 5.9 18 22 26 29 47 S 20 4711 5.5 18 31 52 52 17 22 30 82 04 + 1 + 1 N 11 15 20 * a= number of turns of the micrometer screw in one minute of arc=1.34. The value of one turn of the micrometer screw=44".650. The approximate mean right ascensions and declinations for the observing list are obtained for the time of the observations by multiplying the annual variation by the number of years elapsed since the epoch of the catalogue and combining the products algebraically with the right ascension and declination given in the catalogue used. In the above form there is no column for zenith distances. The setting for a pair is one- half the difference between the declinations of the two stars of a pair. To get the values in the column N S subtract double the latitude (for station St. Anne, 82 03') from the sum of the declinations of the two stars and multiply the result in minutes of arc by the number of turns of the micrometer screw in a minute of arc. N S is positive if the. north star has the greater zenith distance and is negative if the south star has the greater zenith distance. The center of the comb in the micrometer eyepiece is called 20, and increasing readings on the graduated head go with increasing zenith distances. Then the setting of the micrometer wire for any north star is 20 H ~ an d for any south star 20 ^2 These settings are given in the last column of the above table. When one star of the pair is a subpolar, the finder circle setting is 90 \Zd. N S in this case is a (180 difference of d's 2) and is positive or negative according as the north star has the greater or lesser zenith distance. The setting of the micrometer wire will be given by the same general expression as above. For the purposes of the observing list it is sufficiently accurate to know the mean right ascensions to within one second and the declinations and derived quantities to the nearest minute of arc. The approximate reading of the turns is given to facilitate identification of the stars and to enable the observer to put the micrometer line approximately in position before the star enters the field of view. The middle reading of the micrometer comb is called 20 to avoid negative readings. If the Ten Year Catalogues for 1880 and 1890 and the Nine Year Catalogue for 1900, by the Royal Observatory at Greenwich, are used, then the form of the observing list could be made to advantage in a manner somewhat different from that shown above, for in those publications the north polar distances are given instead of the declinations. The list may be similar to that shown below, where the settings, etc., are derived from the north polar distances of the stars. In the first column of the example are given the Boss catalogue numbers, though the stars are also in the lists of the Greenwich catalogues mentioned above. They are the same stars as those in the first form of star list. No. 16. OBSERVING TENT. No. 17. OBSERVING TENT. DETERMINATION OF LATITUDE. Observing List (Form 2). [St. Anne, HI., June 25, 1908. Zenith Telescope No. 4. j> 41 01' .3. Search factor- 180*- 2 ^-97 57*.) 109 Sum of Star No., Boss cat- alogue Mag. a North polar distances and difference N. P. D.'s; and search factor minus sum , 8 pairs with plus micrometer difference = 41 01' 20".33. Mean , 7 pairs with minus micrometer difference = 41 01' 20". 12. The mean of 7 pairs with minus micrometer differences minus the mean of 8 pairs with plus micrometer differences = 0".21. Normal equations 15c+13. 5^-0.01=0 13. 5c+2346. 59^+31. 872=0 r,=--0".0187 c=+0". 0130 Observation equations c- 8. 3)-! -0.03=0 c- 0.1^+0.46=0 c+14. S^+0. 21=0 c-14. 1^-0. 17=0 c+ 9.1^-0.01=0 c-13. 8^-0. 31=0 c-16. Srj+0. 08=0 c- 2.3^+0.43=0 c-15. 7^-0. 73=0 c+21. 9^+0. 56=0 c+ 8. 27-j+0. 16=0 c+ 0. 9^+0. 01=0 c- 2. 8r, -0.54=0 c+16. 4^-0. 30=0 c+16. lr,+0. 17=0 Latitude of St. Anne latitude station Reduction to sea level, elevation of station, 206 meters Reduction to mean position of pole * Latitude of St. Anne latitude station, reduced to sea level and the mean position of the pole For an explanation of the above adjustment see page 130. ^W'?= o ^ Corrected value of one-half turn of micrometer screw =22". SllSiO". 0046 eB= ^ / 0455Xl 1 39 =0 . 22 = 41 01' 20".240".06 -0.03 + 0.07 = 41 01' 20".280".06 1 See Astroaomische Nachrichten No. 4414. DETERMINATION OF LATITUDE. 115 GENERAL NOTES ON COMPUTATIONS OF LATITUDE IN THE UNITED STATES COAST AND GEODETIC SURVEY. The result from each pair of stars is given equal weight. This is done upon the supposition that the theoretical weights are so nearly equal that, if they were used, the final value for the latitude of a station would seldom be changed by more than 0".01. A first rejection limit of 3 ".00 from the mean value of the latitude is used. After the 3".00 rejection limit has been applied the probable error of a result from a single pair, e p , is computed from all the remaining values, and then 5e p is used as an absolute rejection limit, and 3.5e p is used as a doubtful limit beyond which rejection is to be made if strong evidence in favor of rejection is found other than the residual itself. Such evidence may consist of positive notes indicating bad conditions during the observation of the particular pair concerned, con- tradictions in the record indicating a probable misreading, or a mean declination of a star with a probable error so large that it might account for the large residual. A new value of one-half turn of the micrometer is to be derived from the latitude observa- tions only in those cases in which the mean latitude from pairs with plus micrometer differ- ences differs by more than 0".20 from the mean latitude from pairs with minus micrometer differences. It is believed that, when the agreement is within 0".20, a new value of one-half turn, if derived from the observations, would differ from the old by less than 0".01 and the final latitude would ordinarily be changed by less than 0".01. It is also believed that the derived correction to the old value would, in these cases, be but little, if any, larger than its own probable error. The formulae used in computing the probable errors, if a correction to the micrometer value is derived from the latitude observations, are: 1(0. ,=Y- (p-2) # V(0.455)2J 2 (p-2)(p-^$ V ' (0.455)2" J^. e r = probable error of r.= - - (p_2)jjf l > The correction for elevation to reduce the mean latitude to sea level is always applied. (See p. 130.) The reduction to a triangulation station or to other points is also applied on the latitude computation and the relation of the latitude station to such point or points is there indicated. Unless the latitude station is within a few meters of the triangulation station and due east or west of it, the latitude computation should show the latitude of both the latitude station and the triangulation station. EXPLANATION OF COMPUTATION. Let and ' equal the true meridional zenith distances of the southern and northern stars, and 8 and 8' the apparent declinations of the same, respectively; then the expression for the latitude is Now, if z, z' denote the observed zenith distances of the south and the north stars; n, s the north and the south readings of the level for the south star, and n' , s' the same for the north star; d the value of one division of level; r and r' the refraction corrections and m and m' the 116 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. reductions of the measured zenith distances to the meridian for the south and the north stars, respectively, then s?2% -jf = -5".510, the secular variation, -~= -".00880, the proper motion, // = -".001; the mean epoch, E, =1875.5, and the probable error, es Bp = 0". 03; 40- 42- 44- 46- 48* 50" 52- 54- 56> 58" 60" I 1 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .02 .02 89 2 .01 .01 .01 .01 .01 .01 .01 .01 .02 .02 .02 .02 .02 .02 .03 .03 .03 .03 .03 88 3 .01 .01 .01 .01 .01 .01 .01 .02 .02 .02 .02 .03 .03 .03 .03 .04 .04 .04 .04 .05 .05 87 4 .01 .01 .01 .01 .01 .02 .02 .02 .02 .03 .03 .03 .04 .04 .04 .05 .05 .06 .06 .06 .07 86 5 .01 .01 .01 .01 .02 .02 .02 .02 .03 .03 .03 .04 .04 .05 .05 .05 .06 .06 .07 .07 .08 .09 85 6 .01 .01 .01 .02 .02 .02 .03 .03 .03 .04 .04 .05 .05 .06 .06 .07 .07 .08 .08 .09 .10 .10 84 7 .01 .01 .02 .02 .02 .03 .03 .03 .04 .04 .05 .05 .06 .06 .07 .08 .08 .09 .10 .10 .11 .12 83 8 .01 .02 .02 .02 .03 .03 .03 .04 .04 .05 .05 .06 .07 .07 .08 .09 .09 .10 .11 .12 .13 .14 82 9 .01 .02 .02 .02 .03 .03 .04 .04 .05 .05 .06 .07 .07 .08 .09 .10 .11 .11 .12 .13 .14 .15 81 10 .01 .02 .02 .03 .03 .04 .04 .05 .05 .08 .07 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 80 12 .01 .01 .02 .03 .03 .04 .06 .05 .06 .06 .07 .08 .09 .1(1 .11 .12 .13 .14 .15 .16 .17 .19 .20 78 14 .01 .01 .03 .03 .04 .04 .05 .06 .07 .07 .08 .09 .10 .11 .12 .14 .15 .16 .17 .19 .20 .22 .23 76 16 .01 .02 .03 .03 .04 .05 .06 .07 .07 .08 .09 .10 .12 .13 .14 .15 .17 .18 .20 .21 .23 .24 .26 74 18 .01 .02 .03 .04 .05 .05 .06 .07 .08 .09 .10 .12 .13 .14 .16 .17 .18 .20 .22 .23 .25 .27 .29 72 20 .01 .02 .04 .04 .05 .06 .07 .08 .09 .10 .11 .13 .14 .1* .17 .19 .20 .22 .24 .26 .28 .29 .32 70 22 .01 .02 .04 .05 .05 .06 .07 .09 .10 .11 .12 .14 .15 .17 .18 .20 .22 .24 .26 .28 .30 .32 .34 68 24 .01 .02 .04 .05 .06 .07 .08 .09 .10 .12 .13 .15 .16 .18 .20 .21 .23 .25 .27 .29 .32 .34 .36 66 26 .01 .02 .04 .05 .06 .07 .08 .10 .11 .12 .14 .15 .17 .19 .21 .23 .25 .27 .29 .31 .34 .36 .39 64 28 .01 .03 .05 .05 .07 .08 .09 .10 .12 .13 .15 .16 .18 .20 .22 .24 .26 .28 .31 .33 .35 .38 .41 62 30 .01 .03 .05 .06 .07 .08 .09 .11 .12 .14 .15 .17 .19 .21 .23 .25 .27 .30 .32 .34 .37 .40 .42 60 32 .01 .03 .05 .06 .07 .08 .10 .11 .13 .14 .16 .18 .20 .22 .24 .26 .28 .31 .33 ..'ill .39 .41 .44 58 34 .01 .03 .05 .06 .07 .09 .10 .11 .13 .15 .16 .18 .20 .22 .24 .27 .29 .32 .34 .37 .40 .42 .45 56 36 .01 .03 .05 .06 .07 .09 .10 .12 .13 .15 .17 .19 .21 .23 .25 .28 .30 .32 .35 .38 .41 .44 .47 54 :i be the mean latitude from observations on pair No. 1, y> 2 from pair No. 2, and so on. Let v be the residual obtained by subtracting 9?,, 9> 2 . . . in turn from the indiscriminate mean for the station of

lt 2 , q> 3 , . . . or An exception to the above weights arises when two or more north stars are observed at one setting of the telescope in connection with the same south star, or vice versa, and the com- putation is made as if two or more independent pairs had been observed. The results of the component pairs in such a combination are not independent, since they involve in common the DETERMINATION OP LATITUDE. 121 error of observation and the error of declination of the common star. The weight to be assigned to each component pair in a doublet is on this account but two-thirds of that given above, 1 and to each component pair in a triplet is one-half. The combination of two stars on one side of the zenith with one on the other side is called a doublet, and three stars on one side of the zenith with one on the other side is called a triplet. The present practice in the United States Coast and Geodetic Survey is not to observe doublets or triplets. (See paragraph 3 of General Instructions, p. 104.) If a combination observed at one setting of the telescope includes two or more stars on each side of the zenith, it may be broken up in the computation into two or more independent doublets or triplets, each of which may be treated as indicated above. If a given star on one side of the zenith is observed in connection with a certain star on the other side of the zenith on a certain night (or nights), and on a certain other night (or nights) is observed in connection with some other star, the two results are independent in so far as the observations are concerned, but involve a common adopted declination for one of the two stars of each pair. The proper weight to be assigned depends in this case upon the relative magnitude of and e, but is for their ordinary values so nearly equal to the weight for an independent pair that it may, with little error, be assumed to be such without going to the trouble of evaluating it. The weight to be assigned to a zenith star observed in both positions of the telescope is (e 2 \~ l 2e 2 - + -JT- ) in which N a is the number of nights' observations upon it. The most probable value

-l)Iw in which A

,, 9> 2 ,

Star Nos. 38 55'+ (2058) 09.81 -.01 .00 4440 09.80 .00 .00 09.80 4513 08.07 -.07 .00 4550 07.92 +.08 .01 08.00 4513 08.12 .00 .00 4555 08.13 -.01 .00 08.12 4526 09.31 -.34 .12 4550 08.40 +.57 .32 08.80 +.17 .03 09.44 -.47 22 09.01 -.04 .00 08.85 +.12 .01 08.97 4526 09.36 -.26 .07 4555 08.62 +.48 .23 09.51 -.41 . 17 08.91 +.19 .04 09. 12 -.02 .00 09.11 -.01 .00 09.10 Sum 6 69 Pair, Star Nos. B. A.C. * V D n w v~<;> i$ j WJ<^ (lOyr.) [c. s.] (2058) 4440 38 55' 09". 80 -1.00 1.00 2 11 19.80 -0.99 0.98 10.78 4513 4550 08 .00 + .80 .64 2 5 0.00 + .81 .66 3.30 4513 4555 08 .12 + .68 .46 2 5 0.60 + .69 .48 2.40 4526 4550 08 .97 - .17 .03 6 4* 3.88 - .16 .03 0.12 4526 4555 09 .10 - .30 .09 6 4* 4.40 - .29 .08 0.32 4577 (2158) 08 .83 - .03 .00 6 8 6.64 - .02 .00 0.00 (2158) 4646 08 .72 + .08 .01 6 8 5.76 + .09 .01 0.08 (2195) 4688 09 .11 - .31 .10 5 12 13.32 - .29 .08 0.96 4706 4726 08 .25 + .55 .30 5 12 3.00 + .56 .31 3.72 4742 (2233) 08 .50 + .30 .09 5 12 6.00 + .31 .10 1.20 (2254) 4847 08 .93 - .13 .02 5 12 11.16 - .12 .01 0.12 4876 4937 08 .92 - .12 .01 5 12 11.04 - .11 .01 0.12 4958 (2341) 08 .83 - .03 .00 5 12 9.96 - .02 .00 0.00 (2350) 5026 09 .15 - .35 .12 5 9* 10.35 - .34 .12 1.08 [12591 (2365) 09 .35 - .55 .30 5 5* 6.75 - .54 .29 1.45 5076 5084 08 .64 + .16 .03 5 12 7.68 + -17 .03 0.36 5115 5153 08 .87 - .07 .00 5 12 10.44 - .06 .00 0.00 5168 5178 08 .62 + .18 .03 5 12 7.44 + .19 .04 0.48 5249 5293 08 .50 + .30 .09 5 12 6.00 + .31 .10 1. 20 5313 5322 09 .22 - .42 .18 5 12 14.64 - .41 .17 2.04 5344 (2537) 08 .44 + .36 .13 5 12 5.28 + .37 .14 1.68 +3.41 Sums 16 .87 -3.48 3.63 203 164 .14 31.41 Means 38 55' 08". 80 08".81 * For explanation of these four weights, see p. 123. DETERMINATION OF LATITUDE. 123 09 = 0.083 -0.009 = 0.074 f (4.97) = . Latitude = 38 55' 08".810".06. In computing the values of w<, 38 55' 08".00 was first dropped from each value of . An independent determination of may be obtained from the probable errors of the mean declinations of the stars observed, as given in the Boss catalogue. For the stars observed at a station the mean value of the probable error of the mean of two declinations is e = 9 in which N a is the total number of stars observed. For a particular pair Ie\ /?*** ; - in which only the two stars of the pair are included in the summation in the numerator. From this formula and from that given on page 120 (viz, e 2 ^=e 2 p e 2 ) two separate values for e^for each pair may be computed. Which should be used in the formula fixing the weight to be assigned to the mean result from a pair ? There are two objections to the rigid use in all cases of the second value (from the latitude computation). That value is a mean for all the pairs of a list, and in using it the fact that some declinations have very much larger probable errors than others in the same list is ignored. Moreover, in practice, the formula e 2 ^ = e 2 p s 2 is sometimes found to give a value for e^ which is so small as to be evidently erro- neous, and sometimes e 2 ^ is even negative, which is an absurdity. On the other hand, when- 2 e z ever the value e 2 ^ = -^fis smaller than e 2 ^ = e 2 s 2 p , and that is usually the case, it indicates that there is in the observations some error peculiar to each star, which combines with the declination error, and so apparently increases it. When such errors exist, the weights should be correspondingly reduced, and therefore the values of 2 = e 2 p s 2 should be used in the weighting. The following method of weighting, therefore, seems to be the best for use in the office (e 2 \~' e \ 4. ) , use for each pair the larger Wn/ Ie 2 of the two available values of e 2 ^, namely, e 2 % = j-* and e 2 ^ = e 2 J> s 2 . By so doing all the dis- advantages of each of the two methods discussed in the preceding paragraph are avoided. To find quickly which of the values of e 2 ** from the mean place computation are greater than e 2 = e 2 p s 2 one may first note on the list of mean places for what stars e 2 t exceeds 2 (e 2 p s 2 ). Only pairs involving such stars need be examined further. To illustrate, of the pairs involved in the latitude computation shown on page 122, there were only four for which the mean place com- putation gave values of e 2 exceeding 0.074. The stars involved in these four pairs were 4526, 4550, 4555, (2350), 5026, [1259], (2365), and the corresponding values of e 2 t were 0.37, 0.08, 0.10, 2e 2 0.18, 0.24, 0.08, 0.73. The weights assigned to these four pairs therefore depend upon e 2 f = j- 1 in each case. 124 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. COMBINATION OF RESULTS WHEN EACH PAIR IS OBSERVED BUT ONCE. It is the present practice of this Survey to observe a pair of stars only once at a station, and in the final computations the resulting latitude from each pair observed is given unit weight. (See the first paragraph under the heading "General Notes on Computations of Latitude in the U. S. Coast and Geodetic Survey" on p. 115.) Whenever the plan of observing each pair but once at a station is carried out the method of combining results and computing probable errors outlined in the preceding pages fails, and for it must be substituted the following procedure, for which little additional explanation is needed: 2 _ 0.455 2V in which e p is the probable error of the result from a pah-, including both the error of observation and the declination errors, v is the residual obtained by substracting the latitude from a single pair from the indiscriminate mean of all the pairs, and p is the number of pairs. In the field computation and also in the final computation this indiscriminate mean is considered to be the final value of the latitude. Its probable error is 0.455 2V >(p-\) No value of the probable error of observation not involving the decimation error is available from such a field computation. But the computed values of e p and e give sufficiently good indications of the accuracy of the observations to enable the observer to decide in the field whether the instrument is in good condition and whether more observations are needed and that is all that is necessary. (See p. 104.) If desired, the office computation may be carried further as the probable error of the decima- tion of a star e* may be obtained from the catalogue. 2$ The probable error of a single observation is given by the formula e* = e 2 p -?, in which N, is the total number of stars observed. If weights were given each pan* (not the present practice in this Survey), the weight to be assigned to a pan- would be e in which for each pair e 2 TJ the summation covering the two stars of that pan- only. * DETERMINATION OF LEVEL AND MICROMETER VALUES. For methods of determining the level value see page 46. Until recently the method most frequently used in this Survey for determining the microm- eter value is as follows: 1 The tune is observed that is required for a close circumpolar star, near elongation, to pass over the angular interval measured by the screw. Near elongation the apparent motion of the star is nearly vertical and nearly uniform. That one of the four close circumpolars given in the Ephemeris, namely, a, d, and A Ursae Minoris and 51 Cephei, may be selected which reaches elongation at the most convenient hour. In selecting the star it may be assumed with sufficient accuracy that the elongations occur when the hour-angle is six hours on either side of the meridian. In planning the observations and in making the computation it is necessary to know the tune of elongation more accurately, and it may be computed from the formula cos t-E = cot d tan < 1 See Appendix No. 3, United States Coast and Geodetic Survey, Report for 19(10, for a full discussion of the determination of micrometer value. DETERMINATION OF LATITUDE. 125 Chronometer time of elongation =ct 4Tt E , the plus sign being used for western elonga- tion and the minus for eastern elongation. t K is the hour-angle at elongation reckoned eastward or westward from upper culmination, and AT is the chronometer correction. If desired E , the zenith distance of the star at elongation may be computed from the formula cos E = cosec d sin $ It is advisable to have the middle of the series of observations about elongation. The observer may obtain an approximate estimate of the rate at which the star moves along the micrometer by a rough observation or from previous record, and time the beginning of his observations accordingly. To begin observations the star is brought into the field of the telescope and to the proper position, the telescope is clamped both in zenith distance and azimuth, the micrometer is made to read an integral number of turns, and the bubble is brought approximately to the middle of the level tube. The chronometer time of transit of the star across the thread is observed and the level read. The micrometer thread is then moved one whole turn in the direction of the apparent motion of the star, the tune of transit again observed and the level read, and the process repeated until a sufficiently large portion of the middle of the screw has been covered by the observations to correspond with what is actually used in the latitude observations. If desired, an observation may be made at every half turn, or even at every quarter turn, by allowing an assistant to read the level. It is well to note the temperature. The form of record and computation is shown below, the first four columns being the record, and the remainder the computation, of the value of one turn of micrometer from observa- tions made at the New Naval Observatory June 18, 1897. = 38 55' 08".S. For the star B. A. C. 8213 at the time of observation _ ++ _. To obtain the sum of the series (J)<+(J)'+ ()'+(l) ( +( i) . . . +#, apply the formula to the series l<+2<+3<+4< . . +(4i)< and divide by 256- 4. See Sammlung von Formilndtr reinfn und a.igewandfen Afathematik von Dr. W. Laska, p. 88 (Braunschweig, 1S88-1S94). DETERMINATION OF LATITUDE. 129 procedure. The focal adjustment is liable to be disturbed more or less when the micrometer box is turned, and a corresponding constant error introduced into the result. In observing at elongation the telescope is depended upon to be stable in zenith distance, the direction in which it is designed to be stable, and the level readings furnish a means of correcting in large p . Let J< have the same meaning as before, viz, 0o~0u o~2> e tc. (See computation on p. 114.) For each pair an observation equation of the form c M^r^ + A = may be written. The resulting normal equations, from which r l may be derived, are 2 Jf,c + ^ M 2 !? 1 ! I M^(j) = The computation will be sufficiently accurate if M^ is carried to tenths of turns only, and as here indicated without assigning weights to the separate pairs. To the preliminary values of < 2 . . . , the results from the separate pairs, may now be applied the corrections M^ and the latitude computation completed as before. REDUCTION TO SEA LEVEL. The reduction of the observed latitude to sea level is given by the expression J0=- 0.000171 h sin 2 in which J is the correction in seconds of arc to be applied to the observed latitude, h is the elevation of the station above sea level in meters, and < is the latitude of the station. This correction may be gotten from the following table: DETERMINATION OF LATITUDE. Reduction of latitude to sea level. [The correction is negative in every case.] 131 * ft 5 85 10" 80 18 75 20 70 25 65 30 60 35 55 40 50 45 Feet Meiers // // // // // // // /' // 100 30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 200 61 .00 .00 .01 .01 .01 .01 .01 .01 .01 300 91 .00 .01 .01 .01 .01 .01 .01 .02 .02 400 122 .00 .01 .01 .01 .02 .02 .02 .02 .02 500 152 .00 .01 .01 .02 .02 .02 .02 .03 .03 600 183 .01 .01 .02 .02 .02 .03 .03 .03 .03 700 213 .01 .01 .02 .02 .03 .03 .03 .04 .04 800 244 .01 .01 .02 .03 .03 .04 .04 .04 .04 900 274 .01 .02 .02 .03 .04 .04 .04 .05 .05 1000 305 .01 .02 .03 .03 .04 .05 .05 .05 .05 1100 335 .01 .02 .03 .04 .04 .05 .05 .06 .06 1200 366 .01 .02 .03 .04 .05 .05 .06 .06 .06 1300 396 .01 .02 .03 .04 .05 .06 .06 .07 .07 1400 427 .01 .02 .04 .05 .06 .06 .07 .07 .07 1500 457 .01 .03 .04 .05 .06 .07 .07 .08 .08 1600 488 .01 .03 .04 .05 .06 .07 .08 .08 .08 1700 518 .02 .03 .04 .06 .07 .08 .08 .09 .09 1800 549 .02 .03 .05 .06 .07 .08 .09 .09 .09 1900 579 .02 .03 .05 .06 .08 .09 .09 .10 .10 2000 610 .02 .04 .05 .07 .08 .09 .10 .10 .10 2100 640 .02 .04 .05 .07 .08 .09 .10 .11 .11 2200 671 .02 .04 .06 .07 .09 .10 .11 .11 .11 2300 701 .02 .04 .06 .08 .09 .10 .11 .12 .12 2400 732 .02 .04 .06 .08 .10 .11 .12 .12 .13 2500 762 .02 .04 .07 .08 .10 .11 .12 .13 .13 2600 792 .02 .05 .07 .09 .10 .12 .13 .13 .14 2700 823 .02 .05 .07 .09 .11 .12 .13 .14 .14 2800 853 .03 .05 .07 .09 .11 .13 .14 .14 .15 2900 884 .03 .05 .08 .10 .12 .13 .14 .15 .15 3000 914 .03 .05 .08 .10 .12 .14 .15 .15 .16 3100 945 .03 .06 .08 .10 .12 .14 .15 .16 .16 3200 975 .03 .06 .08 .11 .13 .14 .16 .16 .17 3300 1006 .03 .06 .09 .11 .13 .15 .16 .17 .17 3400 1036 .03 .06 .09 .11 .14 .15 .17 .17 .18 3500 1067 .03 .06 .09 .12 .14 .16 .17 .18 .18 3600 1097 .03 .06 .09 .12 .14 .16 .18 .18 .19 3700 1128 .03 .07 .10 .12 .15 .17 .18 .19 .19 3800 1158 .03 .07 .10 .13 .15 .17 .19 .20 .20 3900 1189 .04 .07 .10 .13 .16 .18 .19 .20 .20 4000 1219 .04 .07 .10 .13 .16 .18 .20 .21 .21 4100 1250 .04 .07 .11 .14 .16 .19 .20 .21 .21 4200 1280 .04 .07 .11 .14 .17 .19 .21 .22 .22 4300 1311 .04 .08 .11 .14 .17 .19 .21 .22 .22 4400 1341 .04 .08 .11 .15 .18 .20 .22 .23 .23 4500 1372 .04 .08 .12 .15 .18 .20 .22 .23 .23 4600 1402 .04 .08 .12 .15 .18 .21 .23 .24 .24 4700 1433 .04 .08 .12 .16 .19 .21 .23 .24 .24 4800 1463 .04 .09 .13 .16 .19 .22 .24 .25 .25 4900 1494 .04 .09 .13 .16 .20 22 .24 .25 .26 5000 1524 .05 .09 .13 .17 .20 !23 .24 .26 .26 5100 1554 .05 .09 .13 .17 .20 .23 .25 .26 .27 5200 1585 .05 .09 .14 .17 .21 .23 .25 .27 .27 5300 1615 .05 .09 .14 .18 .21 .24 .26 .27 .28 5400 1646 .05 .10 .14 .18 .22 .24 .26 .28 .28 5500 1676 .05 .10 .14 .18 .22 .25 .27 .28 .29 132 TJ. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Reduction of latitude to sea level Continued. Jl 5 85 10" 80 15 75 20 70 25 65 30 60 35 55 40. 50 45 Feet Meters // // // // // // // // // 5600 1707 0.05 0.10 0.15 0.19 0.22 0.25 0.27 0.29 0.29 5700 1737 .05 .10 .15 .19 .23 .26 .28 .29 .30 5800 1768 .05 .10 .15 .19 .23 .26 .28 .30 .30 5900 1798 .05 .11 .15 .20 .24 .27 .29 .30 .31 6000 1829 .05 .11 .16 .20 .24 .27 .29 .31 .31 6100 1859 .06 .11 .16 .20 .24 .28 .30 .31 .32 6200 1890 .06 .11 .16 .21 .25 .28 .30 .32 .32 6300 1920 .06 .11 .16 .21 .25 .28 .31 .32 .33 6400 1951 .06 .11 .17 .21 .26 .29 .31 .33 .33 6500 1981 .06 .12 .17 .22 .26 .29 .32 .33 .34 6600 2012 .06 .12 .17 .22 .26 .30 .32 .34 .34 6700 2042 .06 .12 .17 .22 .27 .30 .33 .34 .35 6800 2073 .06 .12 .18 .23 .27 .31 .33 .35 .35 6900 2103 .06 .12 .18 .23 .28 .31 .34 .35 .36 7000 2134 .06 .12 .18 .23 .28 .32 .34 .36 .36 7100 2164 .06 .13 .19 .24 .28 .32 .35 .36 .37 7200 2195 .07 .13 .19 .24 .29 .33 .35 .37 .38 7300 2225 .07 .13 .19 .24 .29 .33 .36 .37 .38 7400 2256 .07 .13 .19 .25 .30 .33 .36 .38 .39 7500 2286 .07 .13 .20 .25 .30 .34 .37 .38 .39 7600 2316 .07 .14 .20 .25 .30 .34 .37 .39 .40 7700 2347 .07 .14 .20 .26 .31 .35 .38 .40 .40 7800 2377 .07 .14 .20 .26 .31 .35 .38 .40 .41 7900 2408 .07 .14 .21 .23 .32 .36 .39 .41 .41 8000 2438 .07 .14 .21 .27 .32 .36 .39 .41 .42 8100 2469 .07 .14 .21 .27 .32 .37 .40 .42 .42 8200 2499 .07 .15 .21 .27 .33 .37 .40 .42 .43 8300 2530 .08 .15 .22 .28 .33 .37 .41 .43 .43 8400 2560 .08 .15 .22 .28 .34 .38 .41 .43 .44 8500 2591 .08 .15 22 .28 .34 .38 .42 .44 .44 8600 2621 .08 .15 .22 .29 .34 .39 .42 .44 .45 8700 2652 .08 .16 .23 .29 .35 .39 .43 .45 .45 8800 2682 .08 .16 .23 .29 .35 .40 .43 .45 .46 8900 2713 .08 .16 .23 .30 .36 .40 .44 .46 .46 9000 2743 .08 .16 .23 .30 .36 .41 .44 .46 .47 9100 2774 .08 .16 .24 .30 .36 .41 .45 .47 .47 9200 2804 .08 .16 .24 .31 .37 .42 .45 .47 .48 9300 2835 .08 .17 .24 .31 .37 .42 .46 .48 .48 9400 2865 .09 .17 .24 .31 .38 .42 .46 .48 .49 9500 2896 .09 .17 .25 .32 .38 .43 .47 .49 .50 9600 2926 .09 .17 .25 .32 .38 .43 .47 .49 .50 9700 2957 .09 .17 .25 .32 .39 .44 .48 .50 .51 9800 2987 .09 .17 .26 .33 .39 .44 .48 .50 .51 9900 3018 .09 .18 .26 .33 .40 .45 .48 .51 .52 10000 3048 .09 .18 .26 .33 .40 .45 .49 .51 .52 CORRECTION FOR VARIATION OF POLE. The reduction to the mean position of the pole is derived from the provisional results published by the Latitude Service of the International Geodetic Association. (See p. 85.) DISCUSSION OF ERRORS. In discussing the errors of zenith telescope observations it is desirable to consider separately, as on page 48, the external errors, observer's errors and instrumental errors. The principal external errors are those arising from errors in the adopted declinations and those due to abnormal refraction. DETERMINATION OF LATITUDE. 133 The adopted declinations used in the computation necessarily have probable errors which are sufficiently large to furnish much, often a half, of the error of the computed latitude. This arises from the fact that a good zenith telescope gives results but little, if any, inferior in accuracy to those obtained with the large instruments of the fixed observatories which were used in deter- mining the declinations. Of the stars observed at thirty-six latitude stations, nearly on the thirty-ninth parallel, between 1880 and 1898, the average value of e~ derived from the mean place computations was o".16 and the extreme values were 0".12 and 0".23. The average probable error of the declination of a star in 1900 as given for the 6188 stars in the Boss catalogue is about 0".18, and hence the average value of e from the Boss stars would be about 0".13. These figures furnish a good estimate of the accidental errors to be expected from the adopted declina- tions. To estimate the constant errors to be expected from this source is a rather difficult matter. The principal constant error in declination to be feared is that arising from errors in the adopted systematic corrections applied to the separate catalogues of observed places. The three principal researches in regard to these systematic corrections have been made by Profs. Lewis Boss, A. Auwers, and Simon Newcomb. Judging by the differences between the results of these three researches, the constant error in the mean declinations based upon Professor Boss's researches, may possibly be as great as 0".3, but is probably much smaller than that. In regard to errors arising from abnormal refraction it should be noted that only the dif- ference of refraction of the two stars of a pair enters the computed result. The errors in the computed differential refractions are probably very small when all zenith distances are less than 45 and when care is taken to avoid local refraction arising from the temperature inside the observatory being much above that outside, or from masses of heated air from chimneys or other powerful sources of heat near the observatory. If there were a sensible tendency, as has been claimed, for all stars to be seen too far north (or south) on certain nights, because of the existence of a barometric gradient, for example, it should be detected by a comparison of the mean results on different nights at the same station. The conclusion from many such compar- isons made by Prof. John F. Hayford is that the variation in the mean results from zenith telescope measurements from night to night is about what should be expected from the known accidental errors of observation and declination; or, in other words, that if there are errors peculiar to each night they are exceedingly small. 1 The observer's errors are those made in bisecting the star and in reading the level and micrometer. Errors due to unnecessary longitudinal pressure on the head of the micrometer may also be placed in this class. Indirect evidence indicates that the error of bisection of the star is one of the largest errors concerned in the measurement. The bisections should be made with corresponding care. The probable error of a bisection must be but a fraction of the apparent width of the micrometer line if the observations are to be ranked as first class. It is possible to substitute three or more bisections for the one careful bisection recommended in the directions for observing (p. 110), but it is not advisable to do so. On account of the comparative haste with which such bisections must be made, it is doubtful whether the mean of them is much, if any, more accurate than a single careful and deliberate bisection, while the continual handling of the micrometer head, which is necessary when several bisections are made, tends to produce errors. With care in estimating tenths of divisions on the micrometer head and on the level grad- uation, each of these readings may be made with a probable error, of 0.1 division. If one turn of the micrometer screw represents about 60" and one division of the level about I", such reading would produce probable errors of 0".04 and 0".05, respectively, in the latitude from a single observation. These errors are small, but not negligible, for the whole probable error of a single observation arising from all sources is often less than 0".30 and sometimes less than 0".20. 1 See Report of the Boundary Commission upon the Survey and Re-marking of the Boundary between the United States and Mexico West of the Rio Grande, 1891 to 1896 (Washington, 1898), pp. 107-109, for one such comparison. 134 U. S. COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 1.4. While reading the level the observer should keep in mind that a very slight unequal or unnecessary heating of the level tube may cause errors several times as large as the mere reading error indicated above, and that if the bubble is found to be moving, a reading taken after allow- ing it to come to rest deliberately may not be pertinent to the purpose for which it was taken. The level readings are intended to fix the position of the telescope at the instant when the star was bisected. It requires great care in turning the micrometer head to insure that so little longitudinal force is applied to the screw that the bisection of the star is not affected by it. Such a displace- ment of 1-4000 of an inch in the position of the micrometer line relative to the objective produces an apparent change of more than 1" in the position of a star if the focal length of the telescope is less than 50 inches. The whole instrument being elastic, the force required to produce such a displacement is small. An experienced observer has found that hi a series of his latitude observations, during which the level was read both before and after the bisections of the star, the former readings continually differed from the latter, from 0".l to 0".9, nearly always in one direction. 1 Among the instrumental errors may be mentioned those due (1) to an inclination of the micrometer line to the horizon; (2) to error in the adopted value of one division of the level; (3) to inclination of the horizontal axis; (4) to erroneous placing of the azimuth stops; (5) to error of collimation; (6) to the instability of the relative positions of different parts of the instrument; (7) to the irregularity of the micrometer screw; (8) to the error of the adopted mean value of one turn of the micrometer screw. The first of these sources of error must be carefully guarded against, as indicated on page 106, as it tends to introduce a constant error into the computed latitudes. The observer, even if lie attempts to make the bisection in the middle of the field (horizontally), is apt to make it on one side or the other, according to a fixed habit. If the line is inclined, his micrometer readings are too great on all north stars and too small on all south stare, or vice versa. The error arising from an erroneous level value is smaller the smaller are the level correc- tions and the more nearly the plus and minus corrections balance each other. If the observer makes it his rule whenever the record shows a level correction of more than one division to correct the inclination of the vertical axis between pairs, this error will be negligible. Little time is needed for this if the observer avoids all reversals by simply manipulating a foot-screw so as to move the bubble as much to the northward (or the southward) as the record indicates the required correction to be. The errors from the third, fourth, and fifth sources may easily be kept within such limits as to be negligible. An inclination of 1 minute in the horizontal axis, or an error of that amount in either collimation or azimuth, produces only about 0". 01 error in the latitude. All three of these adjustments may easily be kept well within this limit. The errors arising from instability may be small upon an average, but they undoubtedly become large at times and produce some of the largest residuals. One of the most important functions of the observer is to guard against them by protecting the instrument from sudden temperature changes and from shocks and careless or unnecessary handling, and by avoiding long waits between the two stars of a pair. The closer the agreement in temperature between the observing room and the outer air the more secure is the instrument against sudden and unequal changes of temperature. Most micrometer screws now used are so regular that the uneliminated error in the mean result for a station arising from the seventh source named above is usually regligible. Irregu- larities of sufficient size to produce a sensible error in the mean result may be readily detected by inspection of the computation of micrometer value if that computation is made as indicated on pages 126-128. The two forms of irregularity most frequently detected in modern screws on our latitude instruments are those with a period of one turn anil those of such a form that the value of one turn increases continuously from one end of the screw to the other. The periodic irregularity operates mainly to increase the computed probable error of observation and must 1 U. S. Coast and Geodetic Survey Report, 1892, part 2, p. 58. DETERMINATION OF LATITUDE. 135 be quite large to have any sensible effect upon the computed mean value of the latitude. If the value of the screw increases continuously and uniformly from one end to the other, the computed results will be free from any error arising from this source, provided all settings are made so that the mean of the two micrometer readings upon a pair falls at the middle of the screw. If this condition is fulfilled within one turn for each pair, the error in the mean result will usually be negligible. If the settings are not so made, it may be necessary to compute and apply a correction for the irregularity. Evidence has already been presented on pages 126-130 to show that it is difficult to obtain the actual mean micrometer value. It is important, therefore, to guard against errors arising from the eighth source by selecting such pairs that the plus and minus micrometer differences actually observed at a station shall balance as nearly as possible. The final result will be free from error from this source if the weighted mean of the micrometer differences, the signs being preserved, is zero. The only effect of the error in the mean micrometer value in that case is to slightly increase the computed probable errors. The weights are not, however, usually known during the progress of the observations. If the indiscriminate mean of the micrometer differ- ences for each pair, taken with respect to the signs, is made less than one turn at a station, the error of the mean result from this source will usually be less than its computed probable error. THE ECONOMICS OF LATITUDE OBSERVATIONS. Two questions imperatively demand an answer under this heading. What ratio of num- ber of observations to number of pairs will give the maximum accuracy for a given expenditure of money and tune ? What degree of accuracy in the mean result for the station is it desirable and justifiable to strive for'? The answer to the first question depends upon the relative magnitude of the accidental errors of declination and of observation. At 36 stations nearly on the thirty-ninth parallel, at which latitude observations have been made since the beginning of 1880, the average value of e#, the probable error of the mean of two declinations (derived from the mean place com- putations), is 0".16 and the extreme values were 0".12 and 0".23. At 37 stations occupied with zenith telescopes along the thirty-ninth parallel the extreme values of e, the probable error of a single observation, were 0".16 and 0".98, and at about one-half of the stations it was less than 0".42. 1 Similarly, at 43 stations along that parallel occupied with meridian telescopes e was less than 0".45 at one-half the stations, and the extreme values were 0".21 and 1".27. In the light of these figures one may use the following table to determine the most economical ratio of number of observations to number of pairs : Weight to be assigned to mean latitude from a single pair. e^ being assumed to be 0".16. Number of observations on the pair 1 2 3 4 5 6 0.16 20 26 29 31.2 32.6 33.4 0.20 15 22 26 28.1 29.8 31.0 0.30 9 14 18 20.8 22.9 24.6 0.40 5.4 9.5 12.7 15.2 17.4 19.1 0.60 2.6 4.9 6.9 8.7 10.2 11.7 0.80 1.5 2.9 4.2 5.4 6.5 7.6 1.00 1.0 1.9 2.8 3.6 4.4 5.2 1 One thousand two hundred and seventy-seven observations for variation of latitude at San Francisco in two series gave e= 0".19 and e=-= 0".28. A similar series at the Hawaiian Islands in 1891-92, 2434 observations, gave e 0".16. On the Mexican boundary in 1892-93, 1362 observations at fifteen stations gave e= 0".19 to 0".38. All these observations were made with zenith telescopes. (See Coast and Geodetic Survey Reports, 1893, Part 2, p. 494; 1892 Part 2, pp. 54 and 158; 1892, Part 2, p. 50, and Mexican Boundary Report, 1891-1896, p. 101.) 136 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. The measure of efficiency of the first observation is the weight shown in the first column, and of each succeeding observation is the resulting increment of weight. Thus, if e= 0".16, the first observation gives a weight of 20, while the second observation is less than one-third as efficient, the increment of weight being only 6, and the fifth and sixth observations com- bined are about one-ninth as efficient as the first observation. Stated otherwise, the probable error of a single observation being in this case the same as the probable error of the mean of two declinations, little is gained by reducing the observation error while the declination error is allowed to remain. If e= 0".60, the table shows that the second and third observations are each nearly as efficient as the first. The larger is e the less difference there is between the first and succeeding observations, but in every case the first observation is more efficient than any later observation. If each observation after the first involved the same amount of time spent in preparation, observation, and computation as the first, it is evident that to secure a maximum of accuracy for a given expenditure each pair should be observed but once. Additional observations on new pairs require appreciably more time than the same number of observations on pairs already observed only in the following items: Preparing the observing list, computing mean places, and computing apparent places. Several observations per pair save an appreciable amount of time in the apparent place computation only when the successive nights of observation follow each other so closely that the apparent places on certain nights may be obtained by interpola- tion. (The interval over which a straight-line interpolation may be carried with sufficient accuracy is three days.) After balancing this slight increase in labor against the greater efficiency of the first obser- vation upon a pair over any succeeding observation, it is believed that if e is not greater than 0".40, each pah- should be observed but once. If e is much greater than 0".40, two or possibly even three observations per pair may be advisable. It is true that if but a single observation is made upon each pair the observer in the field will not be able to determine his error of observation accurately Qie may do so approximately by assuming <> = 0".16), but the field computation will still perform its essential function of detecting omissions and deficiencies if they exist. What degree of accuracy in the mean result for a station is it desirable and justifiable to strive for? Omitting from consideration stations occupied to determine the variation of latitude, and stations occupied upon a boundary at which one purpose of the latitude observa- tions is to furnish a means of recovering the same point again, the ordinary purpose of latitude observations in connection with a geodetic survey is to determine the station error in latitude, or, in other words, to determine the deflection of the vertical, measured in the plane of the meridian, from the normal to the spheroid of reference at the station. Broadly stated, the purpose of astronomic observations of latitude and longitude (and to a large extent of azimuth also) in connection with a geodetic survey is to determine the relation between the actual figure of the earth as defined by the lines of action of gravity and the assumed mean figure upon which the geodetic computations are based. In determining this relation three classes of errors are encountered: The errors of the geodetic observations, the errors of the astronomic observa- tions, and the errors arising from the fact that only a few scattered astronomic stations can be occupied in the large area to be covered, and that the station errors as measured at these few points must be assumed to represent the facts for the whole area. It suffices here in regard to errors of the first class, which are not within the province of this appendix, to state that they are in general of about the same order of magnitude as those of the second class. The average value of the station error in latitude, without regard to sign, at 381 stations used in the Supplementary Investigation of the Figure of the Earth and Isostasy, is 3".8. An examination of these station errors shows that although there is a slight tendency for their values for a given region to be of one sign and magnitude the values at adjacent stations are nevertheless so nearly independent that the nonpredictable rate of change of the station error per mile is frequently more than 0".l. Six stations within the District of Columbia show an irregular variation of station error in latitude with a total range of 1".8. Stating the result DETERMINATION OF LATITUDE. 137 of the examination in another form, if the station error at a point is assumed to represent the average value of the station error for an area, and if the error of that assumption is to be not greater than 0".10, the area adjacent to the station to which the assumption is applied must not be greater than 10 square miles. If one bears in mind that financial considerations so limit the number of latitude stations that in general the above assumption must be extended over hundreds of square miles, it becomes evident that a probable error of 0".10 in the latitude determination is all that it is desirable or justifiable to strive for. 1 One observation upon each of from 15 to 25 pairs will nearly always secure that degree of accuracy, and the observations may be completed in a single night. As indicated in the General Instructions for Latitude Work, page 104, paragraphs 3 and 4, this Survey has adopted the plan of using such a number of pairs, observed but once, as will make it reasonably certain that the final computation will give a probable error not greater than 0".10 in the resulting latitude. Between 1905 and 1908, Assistant W. H. Burger determined the latitude at 63 stations in the United States, making only one observation on a pair (unless it was found that some mistake was made on a pair, in which case a second observation was made on it if observations were made on a second night). The average number of pairs observed per station was 16.7, with a maximum of 34 pairs and a minimum of 9 pairs. The average e p was 0".38 and the average 6$ was 0".10. The average number of nights on which observations were made at a station was 1.9. Assistant Wm. Bowie occupied 7 stations in 1908. The average number of pairs observed per station was 15, with a maximum of 16 and a minimum of 15 pairs. The, average e p was 0".31 and the average e^ was 0".08. Observations were made on only 8 nights for the 7 stations. At only one station were observations made on more than one night. COST OF ESTABLISHING A LATITUDE STATION. It is difficult to give accurately the cost per station for recent latitude work as usually the parties were also making observations for azimuth. However, a fair estimate of the cost, including salary of the observer, for latitude stations by this Survey in any except mountainous country is about $200 per station. In a rough area where pack animals would be used exten- sively the cost might double this estimate. Where transportation is easy and the stations not distant from each other the stations should cost much less than $200 each if the party remains in the field for long seasons. 1 yhe above discussion also applies, though with less force, to longitude and azimuth observations. In both these cases the errors of observation are necessarily much larger than in latitude observations. PART IV. DETERMINATION OF THE ASTRONOMIC AZIMUTH OF A DIRECTION. GENERAL REMARKS. Various methods are employed in the Coast and Geodetic Survey for determining astro- nomically the azimuth of a triangulation line, or what is the same thing, the direction of that line with respect to the meridian, and there are, perhaps, no other geodetic operations in which the choice of the method, the perfection of the instrument, and the skill of the observer enter so directly into the value of the result. It is intended to give here in a concise form an account of several methods now in use, and to present the formulae as well as specimens of record and examples of computation. If it is proposed to measure a primary or subordinate azimuth, the observer will generally have the choice of the method most suitable and adequate for the pur- pose, and accordingly provide himself with the proper instrument; yet frequently he may find himself already provided with an instrument, in which case that method will have to be selected which is compatible with the mechanical means at hand and at the same time insures the greatest accuracy. The astronomic azimuth, or the angle which the plane of the meridian makes with the vertical plane passing through the object whose direction is to be determined, is generally reckoned from the south and in the direction southwest, etc. However, when circumpolar stars are observed it will be found more convenient to reckon from the north meridian and eastward - that is, in the same direction as before. The geodetic azimuth differs from the astronomic azimuth. The former is supposed free from local deflections of the plumb line or vertical, it being the mean of several astronomic azimuths, all referred geodetically to one station, and it may be supposed that in this normal azimuth the several local deflections will have neutralized each other. The astronomic azimuth is, of course, subject to any displacement of the zenith due to local attraction or deflection. We may distinguish between primary and secondary azimuths the one fixing the direc- tion of a side in primary triangulation, the other having reference to sides of secondary or tertiary triangulations or to directions in connection with the measure of the magnetic decli- nation. For the determination of a primary azimuth the local time (sidereal) must either be known as, for instance, when a telegraphic longitude is at the same time determined or special observations must be made for it. For subordinate azimuths, time and azimuth obser- vations may sometimes be made together, as with the alt-azimuth instrument for magnetic purposes, in which case the sun's limbs are usually observed. In refined work in high latitudes, and for certain rare cases in low latitudes, the transit instrument is needed to furnish the chro- nometer correction. For primary azimuths, in latitudes not greater than those in the United States, the local time may be found with sufficient accuracy by means of an especially con- structed vertical circle, used in the Coast and Geodetic Survey, and shown in illustration No. 8. For secondary azimuths, local time may be found by means of sextants or alt-azimuth instruments. PRIMARY AZIMUTH. The requirements for primary azimuth are that the astronomic azimuth observations and the necessary time observations should be made using such methods, instruments, and number of observations as to make it reasonably certain that the probable error of the astronomic azimuth does not exceed 0".50. It is not desirable to spend much time or money in reducing 138 No. 18. TWELVE-INCH DIRECTION THEODOLITE. No. 19. SEVEN-INCH REPEATING THEODOLITE. No. 20. FOUR-INCH THEODOLITE. DETEBMINATION OF AZIMUTH. 139 the probable error below this amount. At Laplace stations (coincident triangulation, longi- tude, and azimuth stations), however, the astronomic azimuth should be determined with a probable error not greater than 0".30 and the observations should be made on at least two nights. When observations are made to determine the astronomic azimuth of a line of the primary triangulation, the azimuth station should coincide with a station of the triangulation and the mark used should be some other station of the scheme. In this way the azimuth is referred directly to one of the lines of the triangulation. The probable error of the azimuth of a line obtained from an observed astronomic azimuth on a mark separate from the triangu- lation is greater than the probable error of the observed azimuth. The practice in the United States Coast and Geodetic Survey is for the party on primary triangulation to observe all necessary astronomic azimuths during the progress of the triangu- lation. Where a direction instrument is used, the star is often observed upon in the regular series of observations upon the triangulation stations. In such cases the last object observed upon in any one series is the star, and the instrument is reversed immediately after the first pointing upon it. Where the star is observed upon in connection with two or more triangula- tion stations, the station next preceding it is the one to which the astronomic azimuth is referred. INSTRUMENTS. So great a variety of instruments is used for azimuth determinations that it is of little avail to describe any particular instrument in detail. Illustration No. 18 shows a 12-inch ' direction theodolite (No. 146) made at this office and now in use for the measurement of hori- zontal angles and azimuths in primary triangulation. It carries a very accurate graduation, which is read to seconds directly and to tenths by estimation by three microscopes. 2 A glass- hard, steel center also contributes toward making this theodolite and others of identical con- struction furnish results of a very high degree of accuracy. The graduation of the horizontal circle on this instrument is to 5' spaces. An 8-inch repeating theodolite reading to five seconds by two opposite verniers is shown in illustration No. 19. For observations on the sun for azi- muth in connection with magnetic determinations a small 4-inch theodolite is often used. (See illustration No. 20.) This instrument reads to minutes on each of two opposite verniers. The transit instruments and meridian telescopes described in connection with time observations on pages 7-8 are also frequently used for azimuth either in the meridian (p. 160) or in the vertical plane of a circumpolar star at or near elongation (p. 157). When the azimuth is observed during the progress of the primary triangulation the regular triangulation signal lamps shown in illustrations Nos. 21 and 22 are used. The smaller lamp can be seen under average conditions to a distance of about 30 miles. The larger lamp has been observed in the southwestern portion of the United States, where the atmosphere is very clear, up to distances of 120 miles. Where the mark is only a short distance from the station, an ordi- nary lantern, a bull's eye lantern, or an electric hand lamp may be used. In connection with a triangulation along the coast the lantern of a lighthouse can be used as the mark. INSTRUMENT SUPPORTS. While making observations for a secondary azimuth the instrument used is xisually supported upon its own tripod, mounted upon stakes driven firmly into the ground. In primary triangula- tion the theodolite is frequently mounted upon a tripod which may be as much as 25 or more meters above the ground. Where the instrument is not elevated it is mounted upon a specially constructed wooden tripod or stand which has its legs firmly set into the ground and well braced. On the top of the legs is fitted a wooden cap usually 2 inches thick. On this cap are fastened the plates which receive the foot screws of the theodolite. The structure shown in illustration No. 23 is used to elevate the instrument in triangula- tion and azimuth work. It consists of a tripod on which the instrument rests and a four-sided 1 Following the usual practice, the size of the theodolite is here designated by giving the diameter of the graduated horizontal circle. ' For a more complete description of this instrument see Report for 1894, pp. 265-274. 140 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. scaffold on which the observer stands. The tripod and scaffold do not touch each other at any point. The top floor of the scaffold is not needed on azimuth work and is only used on primary triangulation when there are two observing parties working in conjunction. A complete descrip- tion of this type of signal is given on pages 829 to 842 of Appendix 4, Report for 1 903. Most of the azimuth stations are in places where it is difficult to carry lumber, and as a result it is usual to have no platform around the stand when the instrument is only elevated above the ground to the height of the observer's eye. Where no platform is used the observer should be careful not to step close to a leg of the stand while making the observations on the star. Such pre- cautions are not necessary to the same extent while making the observations on the mark (or triangulation station), assuming, of course, that the mark is not far from being in the horizon of the station. As a result of not using an observing platform it may be necessary to make more observations to get the desired degree of accuracy than if a platform had been used. The errors resulting from not having a platform are mainly of the accidental class and their effect on the final azimuth is small. Where both azimuth and latitude are to be observed at a station, but not at the same time as the triangulation observations, a wooden pier similar to that shown in illustration No. 24 has been found satisfactory in every way. It was used to a great extent by former Assistant W. H. Burger and to a limited extent by Assistant W. Bowie. It will be seen that the spread and slope of the legs of the stand make it possible to mount on it each of the instruments in turn, the top section of the pier being removed when used for latitude. The pier is made as if for the azimuth work, and then the top is sawed off at such point as will make the base of the pier of the required height for the latitude instrument. AZIMUTH MARK. When it is necessary to elevate a signal lamp over a triangulation station used as a mark a number of devices may be used. A simple pole well guyed is frequently used, but this is not very satisfactory, for it is difficult to keep the support of the lamp accurately centered over the station mark. A device like that shown in illustration No. 25 may be used, and this has the advantage that the light keeper does not have to climb the pole when posting and inspecting the lamp. A very satisfactory and inexpensive structure frequently used in the United States Coast and Geodetic Survey is shown in illustration No. 26. The legs, of lumber 2 by 4 inches in cross section, are anchored securely in the ground and at intervals the structure is guyed by wire. The light keeper goes up the inside of this signal, and near its top there is an opening leading out to a seat. Such a signal may be built to a height of 140 feet or more. An acetylene lamp, like one of those shown in illustrations Nos. 21 and 22, should be posted at the distant triangula- tion station used as the mark. When the azimuth of a line of the triangu'ation is not measured directly, a special azimuth mark is erected, which is afterwards referred to the triangulation by means of horizontal angles. There has been considerable variety hi the azimuth marks so used, each chief of party adapting the mark to the special conditions in which he finds himself and to his own convenience. A box with open top having in its front face a round hole or a slit of suitable size, through which the light of a bull's eye or common lantern can be shown, makes a satisfactory mark. See illus- tration No. 27. A white or black stripe of paint or signal muslin can be placed on the box, cen- tered over the opening, upon which to make observations during the day in order to refer the astronomic azimuth of the mark to a line of the triangulation. The location of the mark is generally determined, in part at least, by the configuration of the ground surrounding the station, but it should not be placed any nearer than about one statute mile in order that the sidereal focus of the telescope may not require changing between pointings upon the star and upon the mark, since any such change is likely to change the error of collima- tion. Should the mark be closer to the station than one mile and no change be made in the sidereal focus when pointing upon the mark, there would probably be errors caused by parallax. If practicable, the mark should be placed nearly in the horizon of the station occupied, in order that small errors of inclination of the horizontal axis of the instrument may not affect the point- a. 5 < z HI z u I- u u DETEEMINATION OF AZIMUTH. 141 ings upon the mark, and corresponding readings of the striding level will be unnecessary. In choosing the position of the mark it should be kept in mind that the higher the line of sight to it, above the intervening ground the more steady the light may be expected to show and the smaller the errors to be expected from lateral refraction. SHELTER FOR THE INSTRUMENT. An especially designed tent should be used to shield the instrument from the wind. Illus- trations 16 and 17 show two tents which have proved satisfactory. The tent should be only as heavy as is necessary to withstand strong winds and protect the instruments from rain. When not in actual use the instruments used for azimuth observations should be dismounted and placed in their packing cases. Owing to the short time during which an azimuth station is occupied for observations it is usually not necessary or desirable to erect a wooden observatory to protect the instruments. ARTIFICIAL HORIZON. Instead of determining the inclination of the horizontal axis by readings of a striding level, observations are sometimes taken upon the image of the star as seen reflected from the free surface of mercury (an artificial horizon) in addition to the direct observations upon the star. The error in azimuth produced by the inclination of the horizontal axis is of the same numerical value for the reflected observations as for the direct observations, but is reversed in sign, and the mean result is free from error from this source, provided the cross-section of each pivot is circular, or at least that the two pivots have similar cross-sections similarly placed. Considerable care and ingenuity is necessary to protect the mercury effectually against tremors and against wind, either of which will by disturbing the mercury surface make the reflected star image so unsteady as to make accurate pointing upon it difficult or impossible. A glass roof over the mercury to protect it from the wind should never be employed in connection with azimuth observations, since reversal of it does not sufficiently correct for errors arising from refraction at the glass. Large boxes, or tubes of considerable size, with their openings covered with mosquito netting, have proved the most satisfactory protection of the mercury against the wind. It is believed that the lateral refi action of the direct and reflected ray, when the mercury is set on the ground, may introduce uncertain and possibly large errors into the azimuth. This trouble can be avoided by placing the artificial horizon on a stand nearly as high as the theodolite. This, however, can not be done with the direction theodolite (except in very low latitudes). The artificial horizon can not be used in high latitudes when making observations on Polaris, as the horizontal circle of the theodolite would intercept the reflected ray. POINTING LINES. The pointings in azimuth observations are usually taken by using either a single vertical line in a reticle (or attached to a micrometer) or a pair of parallel vertical lines about 20" (of arc) apart. The first has the advantage over the second that it does not involve the necessity of bisecting a space by eye, as the observation consists simply of noting when the star image appears symmetrical with respect to the line. On the other hand, it has the disadvantage that frequently when a very bright star (or light) is observed the line appears to be "burned off" near the star image; that is, it becomes invisible because of its comparative faintness, and the pointing is correspondingly uncertain. So also if a very faint star (or light) is observed its image may nearly or completely disappear behind the line and so make accurate pointing difficult. For many stars of intermediate degrees of brightness one or the other of these diffi- culties exists to a greater or less degree. If two vertical hnes are used and the distance between them is properly chosen these two difficulties will be avoided and both star (or mark) and lines will always be distinctly visible at the same instant. The observation now consists in noting when the image of the star (or mark) bisects the space between the two hnes. This process is probably but slightly less accurate under any conditions of brightness than the direct bisection 142 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. H. of a star image under the most favorable conditions as to brightness. In measuring horizontal angles and azimuths in Colorado, Utah, and Nevada, along the thirth-ninth parallel, and on all primary triangulation on the ninety-eighth meridian since 1901, and on the Texas-California arc of primary triangulation, two vertical lines about 20" apart were used. During the progress of the triangulation along the western part of the thirty-ninth parallel, observations were made at times upon Polaris in daylight to determine the astronomic azimuth, This is a satisfactory method and occasionally is convenient for the observer. GENERAL CONSIDERATIONS. Let the hour angle (<), declination (d), and latitude (cp] at the eastern and at the western elongation, effects of dd and dg> will disappear in the combination of the two results ; this, therefore, is the most favorable condition for observing. In general, effects of dd and d

), and if S is positive, the best time for observing is before the eastern transit, or after the western transit over the prime vertical, when the change in azimuth with respect to time is a minimum, but the star (or sun) should not be too near the zenith nor be so low as to be affected by changes of refraction; if 3 is negative, the star (or sun) should be observed some distance from the meridian. 1 These considerations have led to the plan of making first-class azimuth observations almost exclusively upon the close circumpolars ct, S, and ]. Ursse Minoris and 51 Cephei. The apparent places of these four stars are given in the American Ephemeris for every day of the year. Illus- tration No. 28 will assist in readily finding the two fainter stars ^ Ursse Minoris and 51 Cephei, which barely become visible to the naked eye under the most favorable circumstances; it also shows that when d Ursse Minoris and 51 Cephei culminate on either side of the pole, Polaris is not far from its elongation; and, likewise when the pole star culminates, the other two are on opposite sides of the meridian, near their elongations. A similar approximate relation exists between a and A Ursse Minoris. Polaris offers the advantage of being observable in daytime with portable instruments; hence it may be observed at eastern and western elongations, or at upper and lower culminations, provided the sun be not too high; A Ursse Minoris, from its greater proximity to the pole and its smaller size, presents to the larger instruments a finer and steadier object for bisection than Polaris; 51 Cephei is also advantageously used on account of its small size. The star B. A. C. No. 4165, shown on the diagram, was proposed and used for azimuth work by Assistant G. Davidson. The apparent processional motion of the pole in 100 years is indicated by the direction and length of the arrow. The sun is employed only to determine azimuths of inferior accuracy, generally in connection with the determination of the magnetic declination. ' The statements made in a general and somewhat indefinite form in this paragraph may be stated in accurate mathematical form by deriving dA in terms of it, dip,d3, respectively, from the formula * n cos tana sin p cost (see p. 143), or from the formulae used in its derivation. No. 25. EIGHTY-FOOT SIGNAL. No. 24. WOODEN PIER USED FOR THEODOLITE AND ZENITH TELESCOPE. DETEKMINATION OF AZIMUTH. 143 GENERAL FORMULA. Four methods of determining azimuth will be treated in detail in this publication, namely, (1) the method in which a direction theodolite is used, as in the measurement of horizontal directions; (2) the method of repetitions with a repeating theodolite; (3) the micrometric method, using an eyepiece micrometer; (4) the determination of azimuth from time observa- tions with a transit or meridian telescope approximately in the meridian. 1 Certain formulae wliich are common to the first three of these methods will be stated here for convenient reference. The computation of the azimuth of a terrestrial line of sight from a set of azimuth observa- tions consists essentially of a computation of the azimuth of the star at the instant of observa- tion, a computation of the horizontal angle between the star and the mark, and the combination of these two results by addition or subtraction. In the spherical triangle defined by the pole, the zenith, and a star, the side zenith-pole is the co-latitude, the side star-pole is the polar distance of the star, and the angle at the pole is the hour angle 2 or its explement. Starting from these three as known parts, the spherical triangle may be solved by the ordinary formulae of spherical trigonometry. The solution to obtain the azimuth of the star, which is the angle of this triangle at the zenith, may, without any approximations, be put in the form . sin t cos

If a mean time chronometer is used, the value I ^ 1 ,, T should be increased by its one hundred and eightieth part. This table was copied from pages 634-637 ot Doolittle's Practical Astronomy. These tabular values may be found in various other places. No. 25. STRUCTURE FOR ELEVATING SIGNAL LAMP OVER TRIANGULATION STATION USED AS MARK. No. 26. STRUCTURE FOR ELEVATING SIGNAL LAMP OVER TRIANGULATION STATION USED AS MARK. No. 27. AZIMUTH MARK. DETERMINATION OF AZIMUTH. 145 level before and after reversing it. h is the altitude of the star. It is only necessary to know h approximately an occasional reading of the setting circle will give it with abundant accuracy, If the graduation on the striding level is numbered continuously in one direction the Level Correction = j \(w w') + (e e r ) tan h in which the primed letters refer to readings taken in the position in which the numbering increases toward the east. 1 If the mark is not in the horizon of the instrument a similar correction, if appreciable, must be applied to readings upon the mark, Ti now being the altitude of the mark. Ordinarily the mark is so nearly in the horizon of the instrument that tan Ti is nearly zero and the correc- tions required to pointings upon the mark are negligible. The formula as written gives the sign of the correction to be applied to the readings of a horizontal circle of which the numbering increases in a clockwise direction. This is also the sign of the correction to the computed azimuth (counted clockwise) for level readings in connec- tion with pointings upon the mark, but in connection with pointings upon the star the sign must be reversed to give corrections to the computed azimuth of the mark. DIRECTION METHOD ADJUSTMENTS. The measurement of an azimuth by this method is essentially similar to the process of measuring a difference of two horizontal directions with a direction theodolite. The quantity measured in this case is the difference of azimuth of a circumpolar star and a mark instead of a difference of azimuth of two triangulation signals. The fact that the azimuth of the star is continually changing adds new features to the computation, and makes it necessary to know the time of each pointing upon the star. The fact that the star is at a considerable altitude makes readings of the striding level a necessity and decreases the accuracy of the measurement because errors of inclination of the horizontal axis have a marked influence as contrasted with their comparatively unimportant effects upon the measurements of horizontal angles in a triangulation. The adjustments required are identical with those which are necessary when the instrument is to be used for the measurement of horizontal directions. The adjustments of the focus of the telescope, of the line of collimation, for bringing the vertical lines of the reticle into vertical planes, of the setting circle (if used), and of the strding level may be made as described in connection with a transit on pages 14-16. The vertical axis of the instrument must be made to point as nearly as is feasible to the zenith by bringing the striding level to the proper reading in each of two positions at right angles to each other. The microscopes with which the horizontal circle is read must be kept in adjustment. Ordinarily it will only be found necessary to adjust the eyepiece by pushing it hi or pulling it out until the most distinct vision is obtained of the micrometer lines and of the circle graduation. If the micrometer lines are not apparently parallel to the graduation upon which the pointing is to be made, they should be made so by rotating the micrometer box about the axis of figure of the microscope. If to do this it is necessary to loosen the microscope in its supporting clamp, great caution is necessary to insure that the distance of the objective from the circle of graduation is not changed. The error of run of the reading micrometers should be kept small. In other words, the value of one turn of the micrometer in terms of the circle graduation should not be allowed to differ much from its nominal value. The value of the micrometer may be adjusted by changing the distance of the objective from the gradua- tion. The nearer the objective is to the graduation the smaller is the value of one turn. A change in this distance also necessitates a change in the distance from the objective to the micrometer lines, these lines and the graduation being necessarily at conjugate foci of the ' See footnote on p. 23. 8136 13 10 146 TT. S. COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 14. objective. This adjustment of the micrometer value is a difficult one to make, but when once well made it usually remains sufficiently good for a long period. As stated on page 139, primary azimuths are nearly always observed during the progress of the primary triangulation, and the same instrument is used to make the observations on the azimuth star that is used to determine the horizontal directions of the lines of the triangulation. For a number of years past only the 12-inch (30 cm.) direction theodolites (described in Appen- dix 8, Coast and Geodetic Survey Report for 1894) have been used on primary triangulation. (See illustration No. 18.) Practically all the observations for primary azimuth are made on Polaris. In recent years the azimuth observations have been made at the same time that horizontal observations are being made that is, Polaris is observed at a setting of the instru- ment in connection with one or more of the triangulation stations. The observations on Polaris are made at the end of the position in order that the direct and reversed observations on the star may come close together. Instead of determining the astronomic azimuth of the line used as the initial direction for the horizontal angle work it is considered that the azimuth has been determined of the line observed over just previous to the observations on Polaris. If at any station it is necessary to make the observations for azimuth in connection with two lines of the triangulation, then the probable error of the angle between the two lines must be taken into account in deriving the probable error of the azimuth. When a quadrilateral system is used in the triangulation and both diagonal lines are observed, then at each station there will be five primary directions to observe. Illustration No. 29 shows the lines radiating from such a station. The station A, the first to the east of Polaris, is chosen as the initial and the other stations are observed in turn from left to right, and after observations have been made on E they are made on Polaris. If, for any reason, the line to E is not observed with the other stations during observations for any- one position, then Polaris also should not be observed. Later on the instrument should be set for the missing position, and Polaris should be observed in connection with station E. The observer is instructed to secure an accuracy represented by a probable error of 0".50 for the greater portion of the primary azimuths, and the observations may all be made during one night. This accuracy can usually be secured by observing one set in each of from 12 to 16 positions of the instrument. In no case must an azimuth depend upon less than 10 positions. At some of the triangulation stations where the accumulated twist of the triangulation is to be determined by a coincident longitude and' azimuth station the azimuth is determined with an accuracy represented by a probable error of 0".30, and the observations are made on at least two nights. DIRECTION METHOD EXAMPLE OF RECORD AND COMPUTATION. There are shown below samples of records of azimuth observations on Polaris and the computations. The observations were carried on at the same time that observations of hori- zontal directions were made at the primary triangulation station, Sears, in Texas. The chro- nometer correction and rate were determined from observations with a vertical circle on stars approximately on the prime vertical. Examples of the time observations and computations made at Sears for use in the azimuth observations are shown on pages 54 and 55 of this publication. No. 28. URS.MIN. XII CIRCUMPOLAR STARS. No. 29. Polaris Static DIAGRAM SHOWING DIRECTIONS TO TRI ANGU LATION STATIONS AND POLARIS DETERMINATION- OF AZIMUTH. 147 Form 251 Horizontal directions. [Station, Sears, Tex. (Triangulation Station). Observer, W. Bowie. Instrument, Theodolite 168. Date, Doc. 22, 1908.] Posi- tion Objects observed Time Tel. D or R Mic. Backward For- ward Mean Mean D and R Direc- tion Remarks ft TO , ,, 1 Morrison 8 19 D A B 35 41 35 41 1 division of the striding level = C 36 34 37.0 4".194 R A 180 00 36 35 B 32 31 35 34 33.8 35.4 00.0 Buzzard D A 53 30 43 42 B 41 42 C 34 33 39.2 R A 233 30 39 37 B 34 32 C 38 3S 36.3 37. S 02.4 Allen D A no 14 61 62 B 57 55 C 61 59 59.2 R A 350 14 50 49 B 63 60 53 53 54.7 57.0 21.6 Polaris D A 252 01 54 53 W E km s B 54 53 9.3 28.0 1 48 35.5 C 51 51 52.7 27. 7 9. 1 1 51 06.0 18.4 0.5 18.9 1 49 50.8 R A 72 01 09 09 24.9 6.3 B 02 01 13.0 31.7 C 10 08 06.5 29.0 11.9 -13.5 25.4 - 7.0 148 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Form 380. Computation of azimuth, direction method. [Station, Sears, Tex. Chronometer, sidereal 1769. ^=32 33 31". Instrument, theodolite 168. Observer, W. Bowie.) Date, 1908, position Chronometer reading Chronometer correction Sidereal time a of Polaris t of Polaris (time) t of Polaris (arc) S of Polaris Dec. 22, 1 1 49 50. 8 4 37.5 1 45 13. 3 1 26 41. 9 18 31.4 4 37' 51". 88 49 27. 4 2 2 01 33. 4 37. 5 1 56 55. 5 1 26 41. 9 30 13. 6 7 33' 24". 3 2 16 31.0 4 37.4 2 11 53. 6 1 26 41. 8 45 11. 8 11 17'57".0 4 2 43 28. 8 4 37.3 2 38 51. 5 1 26 41. 8 1 12 09. 7 18 02' 25". 5 log cot 8 log tan log cos t 8. 31224 9. 80517 9. 99858 8. 31224 9.80517 9. 99621 8. 31224 9. 80517 9. 99150 8. 31224 9. 80517 9.97811 log o (to five places) 8. 11599 8. 11362 8. 10891 8. 09552 log cot 8 log sec log sin t log ; 6 1 a 8. 312243 0. 074254 8. 907064 0. 005710 8. 312243 0. 074254 9. 118948 0. 005679 8. 312243 0. 074254 9. 292105 0. 005618 8. 312243 0. 074254 9. 490924 0. 005445 log (tan A) (to 6 places) A= Azimuth of Polaris, from north* Difference in time between D. and R. Curvature correction 7. 299271 06 50. 8 m s 2 30 7. 511124 11 09.2 m s 2 00 7. 684220 16 36. 9 m s 3 18 7. 882866 26 15.0 m s 1 38 Altitude of Polaris=ft 1 tan A=level factor / // 33 46 0.701 / // 33 46 0.701 O / // 33 46 0.701 O / // 33 46 0.701 Inclination f Level correction Circle reads on Polaris -7.0 -4.9 252 01 29. 6 -7.2 -5.0 86 58 11. 2 -7.0 -4.9 281 54 27. -1.8 -1.3 116 45 48. 6 Corrected reading on Polaris Circle reads on mark 252 01 24. 7 170 14 57. 86 58 06. 2 281 54 22. 1 5 15 58.2 200 17 42.4 116 45 47.3 35 18 45. 4 Difference, mark Polaris Corrected azimuth of Polaris, from north * 278 13 32. 3 06 50. 8 180 00 00.0 278 17 52. 11 09. 2 180 00 00. 278 23 20. 3 16 36. 9 180 00 00. 278 32 58. 1 26 15. 180 00 00. Azimuth of Allen (Clockwise from south) 98 06 41. 5 98 06 42.8 98 06 43.4 98 06 43. 1 To the mean result from the above computation must be applied corrections for diurnal aberration and eccentricity (if any) of Mark. Carry times and angles to tenths of seconds only. * Minus, if west of north. t The values shown in thjs line are actually four times the inclination of the horizontal axis in terms of level divisions. DETERMINATION OF AZIMUTH. Summary of azimuth results. [Sears, Tex., Dec. 22, 1908.] 149 Posi- tion Azimuth of Allen V i>' o / // 1 98 06 41. 5 +0.8 .64 2 42.8 -0.5 .25 3 43.4 -1.1 1.21 4 43.1 -0.8 .64 5 39.7 +2.6 6.76 6 42.7 -0.4 .16 7 41.6 +0.7 .49 8 43.3 -1.0 1.00 9 40.0 +2.3 5.29 10 45.0 -2.7 7.29 11 43.3 -1.0 1.00 12 40.7 +1.6 2.56 X\ 2 =27. 29 e= 0.6745 / Iv 2 \ n(n 1) 0".31 The mean observed azimuth 98 06' 42".260".31. Diurnal aberration +0.32. Correction for eccentric light +0.04. Correction for elevation of mark 0.01 . Keduction to mean position of pole * 0.29. Azimuth of the line from Sears to Allen 2 = 98 06 42.32 0.31. DIRECTION METHOD EXPLANATION OF RECORD AND COMPUTATION. The triangulation stations and Polaris which were observed at one setting of the instru- ment (in this case position No. 1) are placed in the record in the order of their azimuths (left to right) from the initial station, "Morrison." The telescope in its direct position is pointed upon each station in turn and finally upon Polaris. The telescope is then reversed, and the first pointing after reversal is upon Polaris; then pointings are made upon the triangulation stations in the reverse order of azimuth (from right to left). The readings in the reversed position of the telescope are placed directly under the direct reading. The mean of the readings in the direct and in the reversed positions of the telescope is used in computing the direction of a line with reference to the initial line. There are three microscope micrometers on the instrument used in making the observations at Sears, and at each pointing a backward and forward reading of each micrometer was made on the two graduations of the circle nearest the center of the comb. The mean run of the micrometers was kept very small and as the micrometer was placed upon a different portion of the five-minute space between successive graduations, the resultant effect of the micrometer run was negligible. The initial positions (minutes and seconds) of the micrometer wire on the circle for the first four positions were 00' 40", 01' 50", 03' 10", and 04' 20". In general, 12 or 16 positions of the circle are used for the initial settings and these readings of the minutes and seconds on the initial are repeated in each group of four positions; that is, in positions 5 to 8, 9 to 12, and 13 to 16. It can be shown that on any object the error due to run is practically zero in each set of four positions of the circle, if the mean run of the three micrometers with regard to sign is less than 1".0 and the run of no one micrometer is larger than 3".0. Special observations are made in primary triangulation to determine whether the run of the micrometers is within these limits. ' See Astronomische Nachrichten No. 4414. 'Sears and Allen are triangulation stations. 150 II. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. The chronometer time of the observations on Polaris and also the level readings are shown in the record. The time of making an observation may be noted by the observer who picks up and carries the beat of the chronometer, or an assistant may note the clock time upon a signal from the observer. When the latter method is used the observer calls "Mark" when the star is bisected. The chronometer corrections shown in the computations resulted from a special series of time observations with the vertical circle at the station (see pp. 54 and 55). The formula used in making the computation is the third form of the azimuth formula shown on page 143. The tables on pages 165 to 173 which give the logarithm of ^ -- were used in i a the computations. Much time is saved in such computations as the above by carrying along all the different sets at one time and thus working along the horizontal lines of the form shown instead of down each column. Also tan and sec (f> are constants for the station, cos t and sin t may be taken out at one opening of the logarithm table, etc. A comparison of corresponding parts of different columns furnishes rough checks which serve to locate any large errors quickly. The value of one division of the striding level is 4". 194. In general, one set like the above, in each of 12 to 16 positions of one of the 12-inch theodolites, will give a probable error of the result less than 0".50. Even where the observations for azimuth are made coincidently with those for horizontal directions in a triangulation there is no difficulty in completing the azimuth observations at a station in one evening. For special stations a probable error of the result of 0".30 or less must be gotten and observations must be made on more than one night. The general practice now in the Coast and Geodetic Survey is to make only one pointing on the star in each of the positions of the telescope and therefore the correction for curvature of the path of the star between the two pointings is usually negligible. When there is a delay in making the second pointing the curvature correction should be computed by the formula shown on page 144. 2 sin 2 -ir .. are given on pages 151-152. The small table shown below gives Sill I Tabular values of the values of the curvature correction direct for values of the interval, 2r, between the two pointings on the star, from 2 to 7 minutes, and azimuths of Polaris less than 2 30', for use with the direction method, when only two observations are made on Polaris for one setting of the instrument. Curvature correction. N^ 2t Azi- N. muthof\ Polaris. \ 2m 3 *. 5 6 m 7" o / // ff // // // // 10 .0 .0 .0 .0 .1 .1 20 .0 .0 .0 .1 .1 .1 30 .0 .0 .1 .1 .2 .2 40 .0 .1 .1 .1 .2 .3 50 .0 .1 .1 .2 .3 .3 1 00 .0 .1 .1 .2 .3 .4 1 10 .0 .1 .2 .2 .4 .5 1 20 .0 .1 .2 .3 .4 .6 1 30 .0 .1 .2 .3 .5 .6 1 40 .1 .1 .2 .4 .5 .7 1 50 .1 .1 .3 .4 .6 .8 2 00 .1 .2 .3 .4 .6 .8 2 10 .1 .2 .3 .5 .7 .9 2 20 .1 .2 .3 .5 .7 1.0 2 30 .1 .2 .3 .5 .8 1.1 DETERMINATION OF AZIMUTH. 2 sin 2 ^ T sin 1" 151 T 1- 2m 3" 4m 5 m 6m ~m 8 I H It n 0.00 1.96 7.85 17.67 31.42 49.09 70.68 96.20 125.65 1 0.00 2.03 7.98 17.87 31.68 49.41 71.07 96.66 126.17 2 0.00 2.10 8.12 18.07 31.94 49.74 71.47 97.12 126.70 3 0.00 2.16 8.25 18.27 32.20 50.07 71.86 97.58 127.22 4 0.01 2.23 8.39 18.47 32.47 50.40 72.26 98.04 127. 75 5 0.01 2.31 8.52 18.67 32.74 50.73 72.66 98.50 128.28 6 0.02 2.38 8.66 18.87 33.01 51.07 73.06 98.97 128. 81 7 0.02 2.45 8.80 19.07 33.27 51.40 73.46 99.43 129.34 g 0.03 2.52 8.94 19.28 33.54 51.74 73.86 99.90 129.87 9 0.04 2.60 9.08 19.48 33.81 52.07 74.26 100.37 130.40 10 0.05 2.67 9.22 19.69 34.09 52.41 74.66 100.84 130. 94 11 0.06 2.75 9.36 19.90 34.36 52.75 75.06 101.31 131. 47 12 0.08 2.83 9.50 20.11 34.64 53.09 75.47 101. 78 132. 01 13 0.09 2.91 9.64 20.32 34.91 53.43 75.88 102.25 132.55 14 0.11 2.99 9.79 20.53 35.19 53.77 76.29 102. 72 133.09 15 0.12 3.07 9.94 20.74 35.46 54.11 76.69 103.20 133.63 16 0.14 3.15 10.09 20.95 35.74 54.46 77.10 103.67 134. 17 17 0.16 3.23 10.24 21.16 36.02 54.80 77.51 104.15 134. 71 18 0.18 3.32 10.39 21.38 36.30 55.15 77.93 104.63 135.25 19 0.20 3.40 10.54 21.60 36.58 55.50 78.34 105.10 135.80 20 0.22 3.49 10.69 21.82 36.87 55.84 78.75 105.58 136. 34 21 0.24 3.58 10.84 22.03 37.15 56.19 79.16 106.06 136.88 22 0.26 3.67 11.00 22.25 37.44 56.55 79.58 106.55 137. 43 23 0.28 3.76 11.15 22.47 37.72 56.90 80.00 107.03 137. 98 24 0.31 3.85 11.31 22.70 38.01 57.25 80.42 107.51 138.53 25 0.34 3.94 11.47 22.92 38.30 57.60 80.84 107.99 139.08 26 0.37 4.03 11.63 23.14 38.59 57.96 81.26 108.48 139.63 27 0.40 4.12 11.79 23.37 38.88 58.32 81.68 108. 97 140.18 28 0.43 4.22 11.95 23.60 39.17 58.68 82.10 109.46 140.74 29 0.46 4.32 12.11 23.82 39.46 59.03 82.52 109.95 141.29 30 0.49 4.42 12.27 24.05 39.76 59.40 82.95 110.44 141.85 31 0.52 4.52 12.43 24.28 40.05 59.75 83.38 110.93 142.40 32 0.56 4.62 12.60 24.51 40.35 60.11 83.81 111.43 142. 96 33 0.59 4.72 12.76 24.74 40.65 60.47 84.23 111.92 143. 52 34 0.63 4.82 12.93 24.98 40.95 60.84 84.66 112.41 144.08 35 0.67 4.92 13.10 25.21 41.25 61.20 85.09 112.90 144. 64 36 0.71 5.03 13.27 25.45 41.55 61.57 85.52 113.40 145.20 37 0.75 5.13 13.44 25.68 41.85 61.94 85.95 113.90 145. 76 38 0.79 5.24 13.62 25.92 42.15 62.31 86.39 114.40 146.33 39 0.83 5.34 13.79 26.16 42.45 62.68 86.82 114.90 146.89 40 0.87 5.45 13.96 26.40 42.76 63.05 87.26 115.40 147.46 41 0.91 5.56 14.13 26.64 43.06 63.42 87.70 115.90 14S. 03 42 0.96 5.67 14.31 26.88 43.37 63.79 88.14 116.40 148.60 43 1.01 5.78 14.49 27.12 43.68 64.16 88.57 116.90 149. 17 44 1.06 5.90 14.67 27.37 43.99 64.54 89.01 117.41 149. 74 45 .10 6.01 14.85 27.61 44.30 64.91 89.45 117.92 150.31 46 .15 6.13 15.03 27.86 44.61 65.29 89.89 118.43 150.88 47 .20 6.24 15.21 28.10 44.92 65.67 90.33 118. 94 151.45 48 .26 6.36 15.39 28.35 45.24 66.05 90.78 119. 45 152.03 49 .31 6.48 15.57 28.60 45.55 66.43 91.23 119.96 152. 61 50 .36 6.60 15.76 28.85 45.87 66.81 .91.68 120.47 153.19 51 .42 6.72 15.95 29.10 46.18 67.19 92.12 120.98 153.77 52 .48 6.84 16.14 29.36 46.50 67.58 92.57 121. 49 154.35 53 .53 6.96 16.32 29.61 46.82 67.96 93.02 122.01 154.93 54 .59 7.09 16.51 29.86 47.14 68.35 93.47 122.53 155.51 55 .65 7.21 16.70 30.12 47.46 68.73 93.92 123.05 156.09 56 .71 7.34 16.89 30.38 47.79 69.12 94.38 123.57 156.67 57 .77 7.46 17.08 30.64 48.11 69.51 94.83 124.09 157.25 58 .83 7.60 17.28 30.90 48.43 69.90 95.29 124.61 157. 84 59 .89 7.72 17.47 31.16 48.76 70.29 95.74 125. 13 158. 43 i 152 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. 2 sin 2 Yi T sin l'~ T 9m 10"' Urn 12m 13" 14'" 15" 16" a ,, 159. 02 196.32 237.54 282. 68 331. 74 384. 74 441.63 502. 46 1 159. 61 196. 97 238. 26 283.47 332.59 385.65 442.62 503.50 2 160.20 197.63 238.98 284.26 333.44 386.56 443.60 504.55 3 160.80 198. 28 239.70 285.04 334.29 387. 48 444.58 505.60 4 161. 39 198.94 240.42 285.83 335. 15 388.40 445.56 506.65 5 161. 98 199.60 241. 14 286.62 336.00 389.32 446.55 507.70 6 162. 58 200.26 241. 87 287.41 336.86 390.24 447.54 508.76 7 163.17 200.92 242.60 288.20 337.72 391. 16 448.53 509.81 8 163. 77 201.59 243.33 289.00 338.58 392.09 449.51 510.86 9 164.37 202.25 244.08 289. 79 339.44 393.01 450.50 511.92 10 164.97 202.92 244.79 290.58 340.30 393.94 451.50 512. 98 11 165.57 203.58 245.52 291.38 341. 16 394.86 452.49 514. 03 12 166.17 204.25 246.25 292.18 342.02 395. 79 453.48 515.09 13 166.77 204.92 246.98 292.98 342.88 3%. 72 454.48 516. 15 14 167. 37 205.59 247. 72 293.78 343.75 397.65 455.47 517. 21 15 167. 97 206.26 248.45 294.58 344.62 398. 58 456.47 518. 27 16 168.58 206.93 249. 19 295.38 345. 49 399.52 457.47 519. 34 17 169. 19 207. 60 249.93 296.18 346.36 400.45 458. 47 520.40 18 169.80 208.27 250.67 296.99 347.23 401.38 459. 47 521. 47 19 170. 41 208.94 251.41 297.79 348. 10 402.32 460.47 522.53 20 171. 02 209.62 252.15 298.60 348. 97 403.26 461. 47 523.60 21 171.63 210. 30 252.89 299.40 349. 84 404.20 462.48 524. 67 22 172. 24 210. 98 253.63 300.21 350.71 405.14 463.48 525.74 23 172.85 211.66 254.37 301. 02 351. 58 406.08 464.48 526. 81 24 173.47 212.34 255.12 301.83 352.46 407.02 465.49 527.89 25 174.08 213. 02 255.87 302.64 353.34 407.96 466.50 528.96 26 174. 70 213. 70 256.62 303.46 354.22 408.90 467.51 530.03 27 175. 32 214. 38 257.37 304.27 355.10 409.84 468.52 531.11 28 175. 94 215. 07 258.12 305.09 355.98 410. 79 469.53 532.18 29 176.56 215. 75 258.87 305.90 356.86 411.73 470.54 533.26 30 177.18 216.44 259.62 306.72 357. 74 412.68 471.55 534. 33 31 177.80 217. 12 260.37 307.54 358. 62 413.63 472. 57 535.41 32 178.43 217. 81 261. 12 308.36 359. 51 414. 59 473.58 536.50 33 179.05 218.50 261. 88 309.18 360.39 415. 54 474.60 537. 58 34 179.68 219. 19 262.64 310.00 361. 28 416. 49 475. 62 538. 67 35 180.30 219.88 263.39 310. 82 362. 17 417.44 476. 64 539. 75 36 180.93 220.58 264.15 311.65 363.07 418.40 477.65 540.83 37 181.56 221.27 264.91 312. 47 363.96 419. 35 478. 67 541.91 38 182. 19 221.97 265.68 313. 30 364.85 420.31 479. 70 543.00 39 182.82 222.66 266.44 314. 12 365.75 421.27 480.72 544.09 40 183.46 223.36 267.20 314. 95 366.64 422.23 481. 74 545.18 41 184.09 224.06 267.96 315. 78 367.53 423.19 482.77 546.27 42 184.72 224.76 268.73 316. 61 368. 42 424.15 483.79 547. 36 43 185.35 225.46 269.49 317.44 369.31 425.11 484.82 548. 45 44 185.99 226.16 270.26 318. 27 370. 21 426.07 485.85 549.55 45 186.63 226.86 271.02 319. 10 371. 11 427.04 486.88 550.64 46 187.27 227.57 271. 79 319.94 372. 01 428. 01 487.91 551.73 47 187. 91 228.27 272. 56 320. 78 372. 91 428.97 488.94 552.83 48 188.55 228. 98 273.34 321. 62 373.82 429.93 489.97 553.93 49 189.19 229.68 274. 11 322.45 374. 72 430.90 491.01 55.5. 03 50 189.83 230.39 274.88 323.29 375. 62 431.87 492.05 556. 13 51 190.47 231.10 275.65 324. 13 376. 52 432.84 493. 08 557. 24 52 191. 12 231.81 276.43 324.97 377.43 433. 82 494.12 558.34 53 191. 76 232.52 277.20 325.81 378. 34 434. 79 495. 15 559.44 54 192. 41 233.24 277.98 326.66 379. 26 435.76 496.19 560.55 55 193.06 233.95 278.76 327.50 380.17 436.73 497.23 561.65 56 193.71 234.67 279.55 328.35 381.08 437.71 498. 28 562.76 57 194. 36 235.38 280.33 329.19 381.99 438.69 499.32 563.87 58 195.01 236.10 281.12 330.04 382.90 439.67 500.37 564.98 59 195.66 236.82 281.90 330.89 383.82 440.65 501.41 566.08 DETERMINATION OF AZIMUTH. 153 METHOD OF REPETITIONS EXAMPLE OF RECORD AND COMPUTATION. Remarks similar to those appearing on page 145 apply here also. The observations required to determine the azimuth of a mark by the method of repetitions are the same as those required to measure a horizontal angle in a triangulation with the same repeating theodolite, with the addition of level readings, and readings of the chronometer at the instants of the pointings upon the star. The adjustments required are those mentioned on page 145, with the exception that a repeating theodolite is ordinarily read by verniers instead of microscopes. Record Azimuth by repetitions. [Station, Kahatchee A. State, Alabama. Date, June 6, 1898. Observer, O. B. F. Instrument, 10-inch Gambey No. 63. Star, Polaris.] [One division striding level=2".67.] Objects Chr. time on star Pos. of tel. Repeti- tions Level read- ings W E Circle readings Angle i 1 A n B Mean Mark D 178 03 22.5 20 21.2 Star 14 h 46 m 30' 1 4. 5 10. 7 9. 2 5. 9 49 OS 2 52 51 D 3 9. 6 5. 6 5. 2 10. 56 10 R 4 11.3 4.0 7. 8 7. 4 Set No. 5 14 59 12 5 15 01 55 R 6 8. 7 6. 6 11. 9 3. 4 100 16 20 20 20 72' 57' 50". 2 ]4 54 17.7 68. 2 53. 6 + 14.6 Star 15 04 44 R 1 11.9 3.4 8. 5 6. 8 07 18 '2 09 54 R 3 7.9 7.3 11. 2 4. 1 Set No. 6 14 15 D 4 9. 6. 1 5. 9 9, 6 16 14 5 15 18 24 6 5. 9 9. 6 9. 1 6. 2 Mark D 177 27 00 00 00 72 51' 46". 7 15 11 48.2 69. 4 53. 1 +16.3 i 154 U. S. COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 14. Computation Azimuth by repetitions. [Kahatchee, Ala. ^-33 13' 40".33.] Date, 1898, set June 6 5 June 6 6 Chronometer reading 14 54 17. 7 15 11 48. 2 Chronometer correction -31.1 -31.1 Sidereal time 14 53 46. 6 15 11 17.1 noi Polaris 1 21 20. 3 1 21 20. 3 t of Polaris (time) 13 32 26. 3 13 49 56. 8 t of Polaris (arc) 203 06' 34". 5 207 29' 12". d of Polaris 88 45 46. 9 log cot S 8. 33430 8. 33430 log tan 9. 81629 9. 81629 log cos t 9. 96367n 9. 9479871 log a (to five places) 8. 11426n 8. 09857n log cot 3 8. 334305 8. 334305 log sec 0. 077535 0. 077535 log sin t 9. 593830w 9.66421171 log q 9. 994387 9. 994584 " 1 a log ( tan ^4) (to 6 places) 8. 00005771 8. 070635?i .A=Azimuth of Polaris, from north* 34' 22". 8 40' 26". 8 m s " TO * " [1 47.7 119.3 7 04.2 98.1 5 09. 7 52. 3 4 30. 2 39. 8 2sin 2 J T 1 26. 7 4. 1 1 54. 2 7. 1 T ana g j n ^// 1 52. 3 6. 9 2 26.8 11.8 4 54. 3 47. 2 4 25.8 38.5 7 37.3 114.0 6 35. 8 85. 4 Sum 343.8 280.7 Mean 57.3 46.8 1 r 2 sin 2 J r 1 7 C >8 1. 670 log if sin 1" -L. t log. cos t log. a Hour-angle, t, in time in arc = 8. 34362 = 9. 78436 = 8. 96108 n 17 39 01 .0 264 45' 15".0 = 7. 08906 n 1 In this instrument increased readings of the micrometer correspond to a movement of the line of sight toward the east when the vertical circle is to the east, and toward the west when the vertical circle is to the west. DETERMINATION OF AZIMUTH. 1.57 log. cot 8 = 8. 343618 log. sec ^ =0. 068431 log. &in t = 9. 998177 n loe. 5 - = 9. 999467 " 1 I* 1 g. (-tan 4) = 8. 409693 n A =+1 28' 16".91 log. 12.67 = 1. 10278 log. curvature corr. = 9. 51247 Curvature corr. = 0. 33 Diur. Aber. corr. = +0. 32 Mean azimuth of star = + 1 28' 16".90 Mark west of star 19 . 76 Azimuth of mark, E. of N.=+l 27' 57",14 The correction for elevation of mark and the reduction to the mean position of the pole are applied to the final result of the separate measures at a station. In the case of this par- ticular station the necessary information is not yet available for reduction to the mean position of the pole. (See p. 85.) MICROMETRIC METHOD EXPLANATION OF RECORD AND COMPUTATION. The compact form of record shown above does not indicate the order in which the obser- vations were taken. The micrometer line is placed nearly in the collimation axis of the tele- scope, a pointing made upon the mark by turning the horizontal circle, and the instrument is then clamped in azimuth. The program is then to take five pointings upon the mark; direct the telescope to the star; place the striding level in position; take three pointings upon the star with chronometer times; read and reverse the striding level; take two more pointings upon the star, noting the times; read the striding level. This completes a half -set. The hori- zontal axis of the telescope is then reversed in its Y's; the telescope pointed approximately to the star; the striding level placed in position; three pointings taken upon the star with observed chronometer times; the striding level is read and reversed; two more pointings are taken upon the star, with observed times; the striding level is read, and finally five pointings upon the mark are taken. Three such complete sets may be observed in from thirty to fifty minutes. The effect of a uniform twisting of the instrument in, azimuth is eliminated from the result. The bubble of the striding level has plenty of time to settle without delaying the observer an instant for that purpose. The zenith distance of the star should be read occasionally, once during each set, say, as it is needed in making the computation. If it is read with one of the star pointings in each set, its value at any other time may be obtained with sufficient accuracy by interpolation. It should be borne in mind in making the computation that the micrometer measures angles in the plane defined by the telescope and its horizontal axis. To reduce the measured angle between the collimation axis and the star to a horizontal angle, it must be multiplied by cosec , as indicated in the computation. To avoid ah 1 approximation in the computation it would be necessary to reduce each pointing upon the star separately, as the zenith distance is constantly changing. It is sufficiently accurate, however, to reduce the mean of the pointings of a half-set with the mean zenith distance of that half-set, as indicated in the computation. To use a single zenith distance for the whole set will sometimes introduce errors which are rather too large to be neglected. The factor cosec will not, in general, be necessary in connection with pointings upon the mark, because the mark will usually be nearly in the horizon of the instru- ment, and cosec therefore nearly unity, and because the collimation axis is purposely placed as nearly as possible upon the mark and the angle concerned is therefore very small. The micrometer value may be determined by observations upon a star near culmination by the process outlined on page 124. If the striding level is read in connection with such obser- 158 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. vations, the correction to be applied to each observed time to reduce it to what it would have been with the transverse axis horizontal is .. 1 dcos rsec d - for upper culmination and for a level of which the graduation is numbered both ways from the middle. For lower culmination the sign of the correction must be reversed. Another convenient way of determining the micrometer value, all in daylight, is to measure a small horizontal angle at the instrument between two terrestrial objects, both with the micrometer and the horizontal circle of the theodolite. This method is less liable to constant errors than the circumpolar method. If the azimuth mark is placed to the southward of the station, the program of observing and the computation are but slightly modified. DISCUSSION OF ERRORS. It is convenient and conducive to conciseness to discuss separately the external errors, observer's errors, and instrumental errors, which together comprise the errors of observation. The external errors affecting azimuth determinations are those due to lateral refraction of the rays of light from the star or mark to the instrument, to errors in the adopted right ascension and declination of the star observed, and to error in the adopted latitude of the sta- tion of observation. Examination of many series of azimuth observations indicates that, in general, they are subject to some error which is peculiar to each night of observation, and constant for that night, but changes from night to night. For example, from 144 sets of micromctric observa- tions of azimuth, made on 36 different nights at 15 stations on the Mexican boundary in 1892-93, it was found that the error peculiar to each night was represented by the probable error 0".38, and that the probable error of each set, exclusive of this error, was 0".54. 1 In other words, in this series of observations the error peculiar to each night, which could not have been eliminated by increasing the number of observations on that night, was two-thirds as large, on an average, as the error of observation in the result from a single set. Similarly, from the published results of 418 sets of micrometric observations on 8 nights at a European station, 2 it follows that the error peculiar to each night was 0".55, while the probable error of a single set was 0".80. The micrometric observations are peculiarly adapted to exhibiting this error, because of their great accuracy and the rapidity with which they may be taken. Azimuth was observed at 73 stations on the transcontinental triangulation along the thirty- ninth parallel. Most of these observations were taken by the direction method, and although they are, for various reasons, but poorly adapted, as a rule, to exhibiting the errors peculiar to the separate nights, there are no less than 16 cases out of the 73 in which a mere inspection indicates that there were errors of that character. The most plausible explanation of the above facts seems to be that there is lateral refrac- tion between the mark and the instrument, dependent upon the peculiar atmospheric condi- tions of each night. Whether that explanation be true or not, the fact remains that an increase of accuracy in an azimuth determination at a given station may be attained much more readily by increasing the number of nights of observation than by increasing the number of sets on each night. If one series of observations is made early in the evening and another series just before dawn on the same night, these series may be considered, in so far as the preceding sen- tence is concerned, to be on different nights, as the atmospheric conditions will have been given an opportunity to change. The line from the station to the mark should not pass close to any objects, such as a smoke- stack, building, clump of trees, or a hill. Even when the line is close to the ground which has 1 See Report of International Boundary Commission, United States and Mexico, 1891-96 (Washington, 1898), pp. 69-72. 1 Station Kampenwand. See pp. 68-92, Veroflentlichung der Konigl. Bayerischen Commission Jiir die Internationale Erdmessung, Astron. omische-Geodatische Arbeiten, Heft 2 (Miinchen, 1897). DETERMINATION OF AZIMUTH. 159 a decided slope normal to the line, there may be decided lateral refraction. During the primary triangulation in the city of Greater New York the errors on the lines which were close to stacks and buildings were much greater than on the clear lines. There was a line in the Texas-Cali- fornia arc of primary triangulation which at one point was very close to the side of a steep hill. The line was observed from the end nearest the hill on several days and nights, with a total range in the means for the several observing periods of 7.7 seconds of arc. It was found that the observations made when the wind was blowing across the line toward the hill gave the more reliable results. (See p. 62 of Special Publication No. 11 of the U. S. Coast and Geo- detic Survey.) The positions of the four principal close circumpolars have been determined by so manj r observations at the fixed observatories under such favorable conditions that it is difficult to believe that the errors in their adopted right ascensions and decimations are sufficiently large to produce errors in the computed azimuths that are otherwise than small in comparison with the other errors involved in the azimuth observations. On the other hand, when Polaris (or some other circumpolar) has been observed at both culminations or both elongations, at a given station, the observations at one culmination (or elongation) have often shown a tendency to differ by a constant from those at the other culmination (or elongation), as if the adopted right ascension (or declination) were in error. It should be borne in mind in such cases that the atmospheric conditions have been reversed, so to speak, between the culminations (or elonga- tions) ; for in one case the temperature will be rising and in the other falling, in general, the two cases occurring at the extreme ends of darkness or of light, or one in the darkness and the other in the light. Hence only a long and careful investigation will determine whether such constant differences are due to defective star places or to changed atmospheric conditions. It is important from a practical point of view to note that if the azimuth observations at a station are all made upon one star and are equally distributed between two hour-angles, differ- ing by about twelve hours, the mean result will be sensibly independent of the errors of the adopted right ascension and declination, and the conditions will be decidedly favorable to eliminating the effects of lateral refraction from the mean result. An error in the adopted latitude of the station produces the maximum effect when the star is observed at elongation and is without effect if the star is observed at culmination. For Polaris at elongation, to produce an error of 0".01 in the computed azimuth the adopted lati- tude must be in error by 0".70 for a station in latitude 30, and by 0".14 for a station in latitude 60. The error in the computed azimuth from this source will be larger for a star farther from the pole. The astronomic latitude (defined by the actual line of gravity at the station) is required for the computation, and not the geodetic latitude. This error, which will in general be very small, will also be eliminated by observing the star at two positions about twelve hours apart. The observer's errors are his errors of pointing upon the mark and star, errors of pointing upon the circle graduation if reading microscopes are used, errors of vernier reading if verniers are used, errors of reading the micrometer heads, errors in reading the striding level, and errors in estimating the times of the star pointings. There is such a large range of difference in the designs of the various instruments used for azimuth observations that little can be said of the relative and absolute magnitude of these errors that will be of general application. Each observer should investigate these errors for himself with the particular instrument in hand. It will be found in general that the observer's errors play a minor part in furnishing the final errors of the results, except perhaps in the micrometric method. The effect of errors in tune, either errors in estimating the times of the star pointings, the personal equation of the observer, or errors in the adopted chronometer correction, may be estimated by noting the rate at which the star was moving in azimuth when the observations were made. Such errors are usually small, but not insensible except near elongation, and will tend to be eliminated by observations of the same star at two hour-angles differing by about twelve hours. 160 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Of the magnitude of the instrumental errors arising from imperfect adjustment and imperfect construction and imperfect stability little of general application can be said, because of the great variety of the instruments. With the larger and more powerful instruments the errors due to instability become rela- tively great and should be guarded against by careful manipulation and rapid observing, by using a carefully planned program of observations, and by protecting the instrument against temperature changes as far as possible. In this connection it should be noted that each of the programs of observation given on the preceding pages is especially adapted to elimination of the effect of any continuous twisting of the instrument in azimuth, and is so planned that the observer will not ordinarily lose time in waiting for the bubble of the striding level to come to rest. That observer of azimuth will be most successful in avoiding errors due to instability who keeps it most clearly and continuously in mind that the instrument and its support are made of elastic material of such a large coefficient of thermal expansion that no part remains of fixed dimensions or shape. He will be especially careful about the thermal conditions and the stresses to which his instrument is subjected and will observe with the greatest rapidity consistent with allowable observer's errors. The errors due to the striding level become more serious the farther north is the station, as may be seen by inspection of the formula for the level correction (p. 144). The errors of graduation of the horizontal circle have the same effect in azimuth observa- tions as in observations of horizontal angles. The number of positions in which the circle must be used in the direction method may therefore be decided upon the same basis as in the angle measurements. The micrometric method gives a higher degree of accuracy than either the method of repetitions or the method of directions. This is probably due largely to the great rapidity with which the observations may be made, a condition which is very favorable to the elimination of errors due to instability of the instrument and its support. The error, in the final result for a station by this method, due to an error in the adopted value of one turn of the micrometer may be made very small by so placing the azimuth mark (or marks) and so timing the observations that the sum of the angles measured eastward from the mark (or marks) to the star shall be nearly equal to the sum of such angles measured westward. STATEMENT OF COSTS. When azimuths are observed with a theodolite during the progress of a triangulation the cost is very small. This is the method now employed in the primary triangulation by the Coast and Geodetic Survey. It is probable that the observations and field computations for an azimuth do not involve an additional cost of more than $50 per azimuth station. If, however, the azimuths are observed by a separate party some time later than the tri- angulation, and when there is more or less building of signals at the stations at each end of the line for which the azimuth is determined, the cost per station will vary during a season's opera- tions from $200 to $500. When an observer must go out in the field to determine a single azimuth at a distant point the expense may be more than $500. A season's work should be planned so that the cost and time of traveling between stations will be a minimum. If prac- ticable, the work in any locality should be done at that time of the year when the most favorable weather conditions may be expected. AZIMUTH FROM TIME OBSERVATIONS. For a number of years azimuths of a secondary degree of accuracy for use in connection with tertiary triangulation in Alaska have been obtained directly from time observations with a transit or meridian telescope, with little additional labor of observing and computing. With the adoption of the transit micrometer the accuracy of the results was greatly increased, approaching primary in character. This method of determining azimuths has proved of great value in high latitudes where slow-moving stars are high in altitude, and, consequently, satis- factory azimuths from observations with a theodolite are difficult to obtain. DETERMINATION OF AZIMUTH. 161 Observations on a mark which is set closely in the meridian are made during each half set of observations for time. See page 80 for description of method of observing time in high latitudes. The azimuth correction, computed from the time observations, is combined with the reading on the mark to get the azimuth. It is necessary, of course, to have the mark near enough to the meridian of the instrument to fall within the field that can be measured by means of the reticle or with the micrometer wire. It is best, in the case of the transit micrometer, to place the mark so nearly in the meridian that its image will fall inside the range of the comb, so that the number of turns of the microme- ter screw may be readily counted between the pointings in the direct and reversed positions. The mark may be placed either to the north or south and should, if practicable, be at least a mile from the instrument. The method of observing is as follows: Just before beginning time observations with the telescope band east, say, a number of observations are taken on the mark; the telescope is reversed to the position band west, and an equal number of observations is made on the mark. The stars of the first half set are then observed, followed by observations on the mark. The telescope is next reversed to the position band east, the mark observed, and then the stars of the second half set are taken. Finally, observations are taken on the mark, the telescope is reversed to position band west, and the same number of observations is made on the mark. This completes the first set of azimuth observations, and the observations on the stars for a full time set. The mean of all the readings on the mark band east, is adopted as the final value in this position of the axis and, similarly, the mean is taken for all readings with band west. The mean of these two adopted values for band east and band west gives the reading of the colli- mation axis, and the difference between either of the two values and the mean is the angle between the mark and the collimation axis of the telescope. This angle, combined with the azimuth constant of the time set, gives the azimuth of the mark. The angle is observed as so many turns of the micrometer head or screw, or spaces of the reticle. This angle is considered to be positive when the mark is east of the colh'mation axis, when pointing south, or west of that axis when pointing north. To this angle (reduced to seconds of time) is added algebraically the azimuth constant, a (see p. 25), derived from the computation of the time set. This azimuth constant is the angle between the meridian and the collimation axis. It is considered to be positive if the collimation axis is east of the meridian, with the telescope pointing south, or if the axis is west of the meridian with the telescope north. If the mark is much out of the horizon of the instrument, readings of the striding level should be made while observing on the mark, and its elevation should be measured roughly with the finder circle. The correction for inclination of axis is applied as on page 145 and the reduction to the horizon, of the angle between mark and collimation axis, is made as on page 157. If readings on the mark are obtained in only one position of the telescope axis, it will be necessary to take into consideration the collimation constant of the time set and the equatorial interval 1 of the assumed zero as well as the azimuth constant. The reading on the mark made with the micrometer screw, or estimated on the reticle, is referred to some assumed zero of the screw or diaphragm. Combining the angle between the mark and this zero with the equatorial interval of the zero gives the angle between the mark and the line of collimation. This latter angle, combined with the collimation constant of the time set, gives the angle between the mark and the collimation axis. This last angle, the angle between the mark and the collimation axis, combined with the azimuth constant of the time set, gives the desired angle between the mark and the meridian. That part of the azimuth angle which lies between the collimation axis of the telescope and the mark must be reduced to the horizon if the mark is not in the horizontal plane of the instrument. Any inclination cf the horizontal axis must be corrected for, as explained on page 145. 1 This is the angle between the mean position of the micrometer wire or the mean lines of the reticle and the assumed zero. See p. 32. 8136 13 11 162 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. The following examples with explanations will show this method of determining azimuth : Example of record Readings on azimuth mark. TRANSIT MICROMETER. [Station, Fairbanks, Alaska. Date, Aug. 9, 1910. Observer, E. Smith. Instrument: Transit No. 18, with transit micrometer. Mark to northward.] Before observations for time on first half-set Between the two half-sets After observations for time on second half-set Band East West West East East West T T T T T T +5. 050 +0. 952 +0. 890 +5. 050 +5. 120 + 1.000 5.070 0.915 0.960 5.070 5.090 0.946 5.110 0.940 0.950 5. 093 5.121 0.985 5.110 0.990 0. 965 5.082 5.120 0.930 5.040 0.920 0.938 5.060 5.068 0.985 5.020 0.990 0.910 5.049 5.140 0.982 5. 055 0.930 0.970 5. 023 5.140 0.960 5.110 0. 930 0.959 5.100 5.110 0.930 5. 090 950 0.960 5.110 5.080 0.959 5. 120 0.985 0.958 5.098 5.090 0.967 Means: +5. 078 +0. 947 +0. 946 +5. 074 +5. 108 +0. 946 Computation of azimuth from time observations. TRANSIT MICROMETER. [Fairbanks, Alaska, 1910. Transit No. 18. Equatorial interval of one turn of micrometer, 2.826. Mark to northward.] Date August 8 August 8 August 9 Band East West East West East West T s T s T s T s T s T s Mean reading on mark Mean reading of E. and W. (reading of collimation axis) 5.074 3.048 1.023 3.048 5. 067 3.032 0.9% 3.032 5.087 3.016 0.94B 3.016 Angle, mark to collimation axis -2. 026- -5. 73 -2. 025= -5. 72 -2.035- 5.75 -2. 036= -5. 75 -2. 071- 5.85 -2. 070- -5. 85 a (from time set) -0.16 -0.36 0.21 0.25 0.04 0.12 Angle, mark to meridian -5.89 -6.08 -5.96 -6.00 -5. 89 -5.97 Mean for set (in time) -5.9S -5.98 -5.93 Mean for set (in arc) -89".7 -89".7 -89".0 Mean azimuth of mark east of north, V 29".5. Correction for elevation of mark, 0.0. Reduction to mean position of pole, 1 +0.8. Azimuth of mark, 180 01' 30".3. The comb should be considered as being numbered from one side to the other and in such a way that the numbers increase with increasing numbers on the micrometer head as the wire is moved across the field. For convenience the first tooth may be given the number 1 rather than zero. The observer in the field must note in the record for one position of the telescope (band west or east) whether the line of sight points farther east or west with increasing readings on the micrometer head. In the example above, with band east, the readings increase on the micrometer head as the line of sight moves toward the east. That is, for the reading of five turns, band east, the line of sight is about two turns east of the collimation axis. With band west increasing readings correspond to a motion of the line of sight toward the west, a reading of one turn, band west, corresponding to a postion of the line of sight of about two turns east of the collimation axis. A set of azimuth observations was made with each of two time sets on August 8. i See Astronomische Nachrlchten No. 4504. DETERMINATION OF AZIMUTH. 163 Computation of azimuth from time observations. DIAPHRAGM. |St. Michael, Alaska, 1898. Meridian telescope No. 13. Equatorial interval of one space of reticle, 3-. 455. Mark to southward.] Date July 13 July 14 July 1.5 Clamp East West East West East West Spaces s Spaces s Spaces s Spaces s Spaces s Spaces s Angle, mark to center line -0.20= 0.69 0.00- 0.00 -0.175= 0.60 -0.025= -0.09 -0. 75= -2. 59 -0. 15= -0. 52 Mean of E and W -0.34 -0.34 -0.34 0.34 -1.56 -1.56 ( Angle mark to collimation axis) a (from time set) +0.39 +0.86 +0.40 +0.72 +1.75 +1.63 Angle, mark to meridian +0.05 +0.52 +0.06 +0.38 +0.19 +0.07 Mean for set (in time) +0-.28 +0-.22 +0". 13 Mean for set (in are) +4". 2 +3". 3 +2".0 Date July 18 Sept. 13 Sept. 17 Clamp East West East West . East West Spaces s Spaces t Spaces 8 Spaces s Spaces s Spaces Angle, mark to center line -0.975- -3. 37 -0.05- -0.17 0.00= 0.00 0.00- 0.00 +0. 25= +0. 86 +0.825- + 2. 85 Mean of E and W 1.77 -1.77 0.00 0.00 +1.86 +1.86 (Angle mark to collimation axis) a (from time set) +2.78 +2.64 +0.41 +0.06 -2.01 -1.42 Angle, mark to rrferidian + 1.01 +0.87 +0.41 +0.06 0.15 +0.44 Mean for set (in time) +0".94 +0. 24 +0>. 14 Mean for set (in arc) + 14".l +3". 6 +2".l Final mean, mark east of south, 00' 04". 9 Correction for elevation of mark 0.0 Azimuth of mark 359 59' 55".l There is no essential difference between the above method and that with the transit microm- eter. The angle between the mark and the center line of the diaphragm is estimated in spaces of the reticle. The accuracy of the resulting azimuth in this case as well as in that of the transit micrometer depends largely on the accuracy with which the azimuth constant is deter- mined from the time observations. The effect of errors of pointing and reading on the mark may be made relatively small by repeated observations. The work of the Latitude Service of the International Geodetic Association began in 1899, so it is only for observations made after that year that a satisfactory reduction can now be made to the mean position of the pole. It is probable that in a few years a reliable value of this reduction can be had, based on theoretical grounds. Computation of azimuth from time observations. DIAPHRAGM. [St. Michael, Alaska, 1898. Meridian telescope No. 13. Readings on mark in only one position of telescope axis. Equatorial interval of one space of reticle, 3>.455. Mark to southward.] Date July 13 July 14 Clamp East East Spaces s Spaces s Mark east of center line -0.20= -0.69 -0.175= -0.60 Eq. interval of center line 0.00 0.00 c +0.12 +0.18 a +0.39 +0.40 Mark east of south -0.18 -0.02 Mark east of south -2". 7 -0". 3 164 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. The above is taken from the example already given for observations in both positions of the telescope. In this case of deriving the azimuth from observations on the mark in only one position of the axis, the equatorial interval of the assumed zero and the collimation constant of the time set must be applied to the reading on the mark. The collimation constant is applied with the same sign as derived from the computation of the time set when the observations on the mark are made with band west, mark south, and with the opposite sign when made witli band east, mark south. The equatorial interval, i, of the assumed zero of the reticle or microm- eter is considered positive when west of the mean line or position, band west. It follows, then, that when i and c are combined in the azimuth angle they are applied with opposite signs. Defining the measured angle between the mark and the assumed zero as positive when the mark is east of the zero, pointing south, and using a, c, and i, with their conventional signs, the follow- ing general expressions cover all cases : M , j . . . = - {a w + (M + c-i) sec h}l5 JBandE . . . = 360- {a, + (M-e+i) sec Mark,,orth BandW ' ' ' "= 180- {a w + (Jf-c + i) sec ^JBandE . . . =180- K + (M+c-i) sec A} 15 a w and a E are the azimuth constants from the time set. M is the angle (in seconds of time) between the mark and the assumed zero of the micrometer or diaphragm. It is assumed to be positive when the mark is east of the zero when pointing south. It is also positive when the mark is west, pointing north, c is the collimation constant of the time set. i is the equato- rial interval, in seconds of tune, between the mean position of the micrometer wire and the assumed zero of the micrometer, or between the mean line of the reticle and the assumed zero. h is the angle of elevation or depression of the mark. The quantity to be subtracted from 360 or 180 is in seconds of arc. CORRECTION FOR ELEVATION OF MARK. When the object used as an azimuth mark is at a considerable elevation, it is necessary to apply a correction to obtain the astronomic azimuth of the projection of the mark on the sphe- roidal surface of reference. This correction, in seconds, is: in which e 2 is the square of the eccentricity and a the semi-major axis of the spheroid of refer- ence; is the latitude of the observing station; a is the azimuth of the line to the mark; and h is the elevation of the mark. For h in meters, and Clarke's 1866 dimensions of the spheroid, as stated in meters, this expression becomes: + 0'^.000109 h cos 2 sin la, or + [ 6.0392] h cos 2 < sin 2a, where the number in brackets is a logarithm, the dash over the characteristic indicating that 10 is to be substracted from it. The sign of the expression shows that when the mark is either southwest or northeast of the observing station the observed azimuth of the mark must be increased to obtain the correct azimuth, while for mark northwest or southeast, the observed azimuth must be decreased. CORRECTION FOR VARIATION OF THE POLE. A correction is necessary to reduce the observed astronomic azimuth to the mean position of the pole. This correction may amount to a half-second or more for points in the northern part of the United States. The secant of the latiude is a factor of the correction, so the value becomes larger for the higher latitudes. (See p. 85.) DETERMINATION OF AZIMUTH. 165 Log 1-a Log a 1 2 3 4 5 6 7 8 9 Proportional parts 9.00 0.045758 5869 5980 6092 6204 6317 6429 6542 6656 6769 111 108 105 102 | 99 8.99 0.044660 4769 4878 4987 5096 5205 5315 5425 5536 5647 1 11.1 10.8 10.5 10 2 9.9 98 3591 3697 3803 3909 4016 4122 4229 4337 4444 4552 2 22.2 21.6 21.0 20.4 19.8 97 2549 2652 2755 2858 2962 3066 3171 3275 3380 34S6 3 33.3 32.4 31.5 30.6 29.7 96 1532 1633 1733 1834 1936 2037 2139 2241 2343 2446 4 44. 4 43.2 42.0 40 g 39 6 95 0.040541 0639 0737 0836 0935 1034 1133 1232 1332 1432 5 55.5 54.0 52.5 51.0 49.5 94 0. 039575 9670 9766 9862 9959 0055 8152 0249 8346 0443 6 7 66.6 77.7 64.8 75 6 63.0 73.5 61.2 71.4 59.4 69.3 93 8633 8726 8819 8913 9007 9101 9195 9290 9385 9480 8 88.8 86.4 84.0 81.6 79.2 92 7714 7805 78% 7987 8079 8171 8263 8355 8447 8540 9 99.9 97.2 94.5 91.8 89.1 91 6818 6907 6996 7085 7174 7263 7353 7443 7533 7624 8.90 0. 035944 6031 6118 6204 6291 6379 6466 6554 6642 6730 96 93 90 87 84 89 5092 5177 5261 5346 5431 5516 5601 5687 5772 5858 88 87 86 85 4261 3451 2660 0.031888 4343 3531 2738 1965 4426 3611 2816 2041 4508 3692 2896 2118 4591 3772 2974 2195 4674 3853 3053 2272 4757 3934 3132 2349 4841 4016 3211 2426 4924 4097 3291 2504 5008 4179 3371 2582 1 2 3 4 5 9. 6 19.2 28.8 38.4 48.0 9. A 18.6 27.9 37.2 46.5 9. 18.0 27.0 36.0 45.0 8. 7 17.4 26.1 34.8 43.5 g. 4 16.8 25.2 33.6 42.0 84 83 82 81 1136 0402 0.029685 8987 1210 0474 9756 9056 1285 0547 9827 9125 1360 0620 9898 9194 1435 0693 9970 9264 1510 0766 0041 9334 1585 0840 0113 9404 1660 0914 0185 9474 1736 0987 0257 9544 1812 1061 0329 %i5 6 7 g 9 57.6 67.2 76.8 86.4 55.8 65.1 74.4 83.7 54.0 63.0 72.0 81.0 52.2 60.9 69.6 78.3 50.4 58.8 67.2 75.6 8.80 0.028305 8372 8440 8508 8576 8644 8712 8780 8849 8918 81 78 75 72 69 79 78 7640 6990 7705 7055 7771 7119 7838 7183 7904 7248 7970 7313 8037 7378 8103 7443 8170 7509 8237 7574 1 8.1 7.8 7.5 7.2 6.9 77 6357 6420 6482 6545 6608 6672 6735 6799 6862 (i'J-'li 2 16.2 15.6 15.0 14.4 13.8 76 5739 5800 5861 r,\m 5984 6046 6108 6170 6232 (12114 3 24.3 23.4 22.5 21.6 20.7 75 0. 025136 5195 5255 5315 5375 5435 5496 5556 5617 5678 4 32.4 31.2 30.0 28.8 27.6 5 40.5 39.0 37.5 36.0 34.5 74 4547 4605 4664 4722 4781 4840 4899 4958 5017 5076 6 48.6 46.8 45.0 43.2 41.4 73 3973 4029 4086 4143 4201 4258 4316 4373 4431 4489 7 56.7 54.6 52.5 50.4 48.3 72 3412 3467 3523 3579 3635 3691 3747 3803 3859 3916 g 64.8 62.4 60.0 57.6 55.2 71 2865 2919 2973 3027 3082 3137 3191 3246 3301 3357 9 72.9 70.2 67.5 64.8 62.1 8.70 0.022331 2383 2436 2489 2543 2596 2649 2703 2757 2811 fiO 1 '11' A OfwjQ 9191 91 7Q 99OK tyvjo H 63 60 57 55 Otf 68 1809 1301 1861 1351 1913 1401 liW>4 1452 2016 1503 ^UDo 1553 xm 1604 Xfa 1655 BQ 1707 lO 1758 1 6.6 6.3 6.0 5.7 5.5 67 0804 0853 0902 0952 1001 1051 1100 1150 1200 1250 2 13.2 12.6 12.0 11.4 11.0 66 0319 0367 0415 0463 0512 0560 0609 0657 0706 0755 3 19.8 18.9 18.0 17.1 16.5 65 0.019846 9893 9940 9987 0034 0081 0128 6176 0223 0271 4 26.4 25.2 24.0 22.8 22.0 5 33.0 31.5 30.0 28.5 27.5 64 9384 9430 9475 9521 9567 9613 9660 9706 9752 9799 6 39.6 37.8 36.0 34.2 33.0 63 8933 8978 9022 9067 9112 9157 9202 9247 9293 9338 7 46.2 44.1 42.0 39.9 38.5 62 8493 8536 85SO 8S24 8667 8711 8755 8800 8844 8888 8 52.8 50.4 48.0 45.6 44.0 61 8063 8105 8148 8191 8233 8276 8319 8363 8406 8449 9 59.4 56.7 54.0 51.3 49.5 8.60 0.017643 7685 7726 7768 7810 7852 7894 7936 7978 8020 53 51 49 47 45 59 7233 7274 7315 7355 7396 7437 7478 7519 7560 7602 58 6833 6873 6913 6952 6992 7032 7072 7112 7153 7193 1 5.3 S.I 4.9 4.7 4.5 57 6443 6482 6520 6559 6598 6637 6676 6715 6755 6794 2 10.6 10 2 9 8 9 4 9.0 56 6062 6099 6137 6175 6213 6251 li2S!) 6328 6366 6404 3 15.9 15.3 14.7 14.1 13.5 55 0.015689 5726 5763 5800 5837 5874 5912 5949 5986 6024 4 21.2 20.4 19.6 18.8 18.0 5 26.5 25.5 24.5 23 5 22 5 54 5326 5362 5398 5434 5470 5507 5543 5579 5616 5653 6 31.8 30.6 29.4 28.2 27.0 53 4971 5006 5041 5077 5112 5147 5183 5218 5254 5290 7 37.1 35.7 34.3 32.9 31.5 52 4624 4659 4693 4727 4762 4797 4831 4866 4901 4936 g 42.4 40.8 39.2 37.6 36 51 4286 4319 4353 4387 4420 4454 4488 4522 4556 4590 9 47.7 45.9 44.1 42.3 40.5 8.50 0.013955 3988 4021 4054 4087 4120 4153 4186 4219 4253 43 41 39 37 35 1 4.3 4.1 3.9 3.7 3.5 2 8.6 8.2 7.8 7.4 7.0 3 12.9 12.3 11.7 11.1 10.5 4 17.2 16.4 15.6 14.8 14.0 5 21.5 20.5 19.5 18.5 17.5 6 25.8 24.6 23.4 22.2 21.0 7 30.1 28.7 27.3 25.9 24 5 g 34.4 32. g 31.2 21.6 28.0 9 38.7 36.9 35.1 33.3 31.5 166 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Loq -j 9 I a Log a 1 2 3 4 5 6 7 8 9 Proportional parts 8.50 0.013955 3988 4021 4054 4087 4120 4153 4186 4219 4253 34 33 32 31 30 49 3633 3665 3697 3729 3761 3793 3825 3858 3890 3923 1 3. 4 3. 3 3.2 3. 1 3.0 48 3318 3349 3380 3411 3443 3474 3506 3537 3569 3601 2 6.8 6.6 6.4 6.2 6.0 47 3010 3040 3071 3101 3132 3163 3194 3225 3256 3287 3 10. 2 9. 9 9.6 9. 3 9. r 46 2709 2739 2769 2799 2829 2S59 2889 2919 2949 2979 4 13.6 13^2 12 8 12.4 12. 45 0. 012416 2445 2474 2503 2532 2562 2591 2621 2650 2680 5 17.0 16.5 16.0 15.5 15.0 6 20.4 19. 8 19.2 18.6 18.0 44 2129 2158 2186 2215 2243 2272 2300 2329 2358 2387 7 23.8 23. 1 22.4 21. 7 21. 43 1849 1877 1905 1933 1961 1989 2017 2045 2073 2101 8 27.2 26 4 25 6 24.8 24.0 42 41 1576 1309 1603 1335 1630 1362 1657 1388 1685 1415 1712 1442 1739 1468 1767 1495 1794 1522 1822 1549 9 30.6 29.7 28.8 27^9 27! o 8.40 0.011048 1074 1100 1126 1152 1178 1204 1230 1256 1283 29 28 27 26 25 39 38 0794 0545 0819 0570 0844 0594 0869 0619 0895 0644 0920 0669 0946 0694 0971 0718 0997 0743 1023 0769 1 2.9 2.8 2.7 2.6 2.5 37 0302 0326 0350 0374 0399 0423 0447 0472 0496 0520 2 5.8 5.6 5.4 5.2 5.0 36 35 0065 0.009833 0088 9856 0112 9879 0135 9902 0159 9925 0183 9948 0207 9972 0230 9995 0254 0018 (127s 0041 3 4 5 8.7 11.6 14.5 8.4 11.2 14.0 g. 1 10.8 13.5 7.8 10.4 13.0 7.5 10.0 12.5 34 33 32 31 9607 9386 9170 8959 9629 9408 9191 8980 9652 9430 9213 9001 9674 9452 9234 9022 9697 9474 9256 9043 9719 9496 9277 9064 9742 9518 9299 9085 9765 9540 9320 9106 9787 9562 9342 9127 9810 9584 9364 9149 g 17.4 20.3 23.2 26.1 16.8 19.6 22.4 25.2 16.2 18.9 21.6 24.3 15.6 18.2 20.8 23.4 15.0 17.5 20.0 22.5 8.30 0. 008753 8773 8794 8814 8835 8855 8876 8897 8917 8938 24 23 22 21 20 29 28 8552 8355 8572 8375 8592 8394 8612 8414 8632 8433 8652 8453 8672 8473 8692 8492 8712 8512 8733 s.vu 1 2.4 2.3 2.2 2.1 2.0 27 26 8163 7976 8182 7994 8201 8013 8220 8031 8050 8259 8069 8278 MISS 8297 8106 8316 8125 8336 8144 3 7.2 6.9 6.6 6.3 6.0 25 0. 007792 7811 7829 7847 7865 7884 7902 7920 7939 7957 4 9.6 9.2 8.8 8.4 8.0 5 12.0 11.5 11.0 10.5 10.0 24 7614 7631 7649 7667 7685 7702 7720 7738 7756 7774 6 14.4 13.8 13.2 12.6 12.0 23 7439 7456 7473 7491 7508 7526 7543 7561 7578 7596 7 16.8 16.1 15.4 14.7 14.0 22 7268 7285 7302 7319 7336 7353 7370 7387 7404 7421 g 19.2 18.4 17.6 16.8 16.0 21 7101 7118 7134 7151 7167 7184 7201 7218 7234 7251 9 21.6 20.7 19.8 18.9 18.0 8.20 0. 006938 6954 6971 6987 7003 7019 7036 7052 7068 7085 COCO 19 18 17 16 15 19 18 6779 6624 6639 6811 6654 6670 6685 oooo 6701 6716 6890 6732 6748 6763 1 1.9 1.8 1.7 1.6 1.5 17 6472 6487 Coon 6502 fl-lSO 6517 6367 6532 6547 6562 6578 6593 6608 2 3.8 3.6 3.4 3.2 3.0 16 15 6323 0.006178 UJOO 6193 Do 0,5 6207 6221 6236 6250 6265 6279 6442 6294 6457 6309 4 7.6 7.2 6.8 6.4 6.0 5 9.5 9.0 8.5 8.0 7.5 14 6037 6051 6065 6079 6093 6107 6121 6135 6150 6164 6 11.4 10.8 10.2 9.6 9.0 13 5898 5912 5926 5940 5353 5967 5981 5995 6009 6023 7 13.3 12.6 11.9 11.2 10.5 12 5763 5777 5790 5803 5817 5830 5844 5857 5871 5885 8 15.2 14.4 13.6 12.8 12.0 11 5631 5644 5657 5670 5684 5697 5710 5723 5737 5750 9 17.1 16.2 15.3 14.4 13.5 8.10 0. 005502 5515 5528 5541 5553 5566 5579 5592 5605 5618 14 13 12 11 10 09 08 5376 5253 5389 5265 5401 5277 5414 5290 5426 5302 5439 5314 5451 5327 5464 5339 5477 5351 5489 5364 1 1.4 1.3 1.2 1.1 1.0 07 5133 5145 5157 5169 5181 5193 5205 5217 521*1 5241 2 2.8 2.6 2.4 2.2 2.0 06 5015 5027 5038 5050 5062 5074 5085 5097 5109 5121 3 4.2 3.9 3.6 3.3 3.0 05 4900 4912 4923 4935 4946 4957 4969 4980 4992 5004 4 5.6 5.2 4.8 4.4 4.0 5 7.0 ' 6.5 6.0 5. 5 5. 04 4788 I 4799 4810 4822 4833 4844 4855 4866 4K7S 4889 6 8.4 7.8 7.2 6. 6 6. 03 4679 4690 4700 4711 4722 4733 4744 4755 4766 4777 7 9.8 9. 1 i 8. 4 7. 7 7. 02 4572 4582 4593 4603 4614 4625 4636 4646 4657 4668 g 11.2 10.4 9.6 8.8 8.0 01 4467 4477 4488 4498 4509 4519 4529 4540 4550 4561 9 12.6 11.7 10.8 , 9.9 9.0 8.00 0.004365 4375 4385 4395 4405 4416 4426 4436 4446 4457 DETEKMINATION OF AZIMUTH. 1 167 Log Ia Log a 1 2 3 4 5 6 7 8 9 Proportional parts 8.00 0.004365 4375 4385 4395 4405 4416 4426 4436 4446 4457 7.99 4265 4275 4285 4295 4305 4315 4325 4335 4345 4355 11 10 98 4167 4177 4187 4196 4206 4216 4226 4235 4245 4255 97 4072 4082 4091 4100 4110 4119 4129 4139 4148 4158 ~^^ 96 3979 3988 3997 4007 4016 4025 4035 4044 4053 4063 1 1.1 1.0 95 0.003888 3897 3906 3915 3924 3933 3942 3951 3961 3970 2 2.2 2.0 3 3.3 3.0 94 3799 3808 3817 3826 3834 3843 3852 3861 3870 3879 4 4.4 4.0 93 3712 3721 3729 3738 3747 3755 3764 3773 3782 3790 5 5.5 5.0 92 3627 3636 3644 3653 3661 3670 3678 3687 3695 3704 6 6.6 6.0 91 3545 3553 3561 3569 3577 3586 3594 3602 3611 1 3619 7 7. 7 7 7.90 0.003463 3472 3480 3488 3496 3504 3512 3520 3528 3536 8 8.8 8.0 9 9.9 9.0 89 3384 3392 3400 3408 341(1 3424 3432 3440 3448 3456 88 3307 3315 3322 3330 333S 3345 3353 3361 3369 : 3377 87 3231 3239 3246 3254 3261 3269 3277 3284 3292 3299 86 3158 3165 3172 3180 3187 3194 3202 3209 3217 3224 g 8 85 0.003086 3093 3100 3107 3114 3121 3129 3136 3143 3150 84 3015 3022 3029 3036 3043 3050 3057 3064 3071 ! 3078 9 8 83 2946 2953 20(10 2967 2974 2980 2987 2994 3001 3008 2 1. 8 1. 6 82 2879 2886 2892 2899 2906 2912 2919 2926 2933 2939 27 24 81 2813 2820 2826 2833 2839 2X46 2852 2859 2xia> 2872 . / 1 39 7.80 0.002749 2755 2762 27<>.x 2774 2781 2787 2794 2800 2807 4 5 3. 6 4.5 . t> 4.0 79 2686 2692 2699 2705 2711 2717 2724 2730 2736 2743 6 5.4 60 4.8 5 6 78 2625 2631 2637 2643 2649 2655 2661 2668 2674 2680 i . o 79 6 A 77 2565 2571 2577 2583 2589 2595 2601 2607 2613 2619 o . ^ 81 1 7 9 76 2506 2512 2518 2524 2530 2535 2541 2547 2553 2559 . 1 1. & 75 0.002449 2455 2460 2466 2472 2478 2483 2489 2495 2501 74 2393 2399 2404 2410 2415 2421 2427 2432 2438 2443 7 C 73 2339 2344 2349 2355 2360 2366 2371 2377 2382 2388 72 2285 2290 2296 2301 2306 2312 2317 2322 2328 2 33 71 2233 2238 2243 2249 2254 2259 2264 2269 2275 22X0 7.70 0.002182 2187 2192 2197 2202 2207 2213 2218 2223 222S 1 2 0.7 1.4 0.6 1.2 69 2132 2137 2142 2147 2152 2157 2162 2167 2172 2177 3 2.1 1.8 68 2084 20XX 2093 20! IS 2103 2108 2113 2118 2122 2127 4 2.8 2.4 67 2036 2041 2046 2050 2055 2060 2085 2069 2074 2079 5 3.5 3.0 66 1990 1994 1999 2003 2008 2013 2017 2022 2027 2031 6 4.2 3.6 65 0. 001944 1949 1953 1958 1962 1967 1971 1976 1980 1985 7 8 4.9 5.6 4.2 4.8 64 1900 1904 1909 1913 1918 1922 1926 1931 1935 1940 9 6.3 5.4 63 1857 1861 1865 1869 1874 1878 1XX2 1887 1891 1896 62 1814 1818 1823 1827 1X31 1835 1840 1844 1848 1852 61 1773 1777 1781 1785 1789 1793 1798 1802 1806 1810 7.60 0. 001732 1736 1740 1744 1748 1753 1757 1761 1765 1769 5 4 59 1693 1697 1701 1705 1709 1713 1716 1720 1724 1728 58 1654 1658 1662 1666 1670 1673 1677 1681 1685 1689 1 0.5 0.4 57 1617 1620 1624 1628 1632 1635 1639 1643 1647 1650 2 1.0 0.8 56 1580 1583 1587 1591 1594 1598 1602 1605 1609 1613 3 1.5 1.2 55 0.001544 1547 1551 1554 1558 1562 1565 1569 1572 1576 4 2.0 1.6 5 2.5 2.0 54 1508 1512 1515 1519 1522 1526 1529 1533 1537 1540 6 3.0 2.4 53 1474 1477 1481 1484 1488 1491 1495 1498 1502 1505 7 3.5 2.8 52 1440 1444 1447 1450 1454 1457 1461 1464 1467 1471 8 4.0 3.2 51 1408 1411 1414 1417 1421 1424 1427 1431 1434 1437 9 4.5 3.6 7.50 0.001376 1379 1382 1385 1388 1391 1395 1398 1401 1404 168 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Log 7 * 1 a Logo 1 2 3 4 5 6 7 8 9 Proportional parts 7.50 0. 001376 1379 1382 1385 1388 1391 1395 1398 1401 1404 49 1344 1347 1350 1354 1357 1360 1363 1366 1369 1372 48 1314 1317 1320 1323 1326 1329 1332 1335 1338 1341 47 1284 1287 1290 1292 1295 1298 1301 1304 1307 1311 46 1254 1257 1260 1263 1266 1269 1272 1275 1278 1281 45 0.001226 1229 1231 1234 1237 1240 1243 1246 1249 1251 44 1198 1201 1203 1206 1209 1212 1214 1217 1220 1223 43 1170 1173 1176 1179 1181 1184 1187 1190 1192 1195 42 1144 1146 1149 1152 1154 1157 1160 1162 1165 1168 41 1118 1120 1123 1126 1128 1131 1133 1136 1139 1141 7.40 0. 001092 1095 1097 1100 1102 1105 1107 1110 1113 1115 39 1067 1070 1072 1075 1077 1080 1082 1085 1087 1090 38 1043 1045 1048 1050 1053 1055 1058 1060 1062 1065 37 1019 1022 1024 1026 1029 1031 1033 1036 1038 1011 4 3 36 0.000996 998 1001 1003 1005 1008 1010 1012 1015 1017 35 34 0.000973 951 976 953 978 956 980 958 982 960 985 962 987 964 989 967 991 969 994 971 1 2 0.4 0. 8 0.3 0.6 33 929 932 934 936 938 940 942 945 947 949 3 1.2 0.9 32 908 910 913 915 917 919 921 923 925 927 4 1.6 1.2 31 888 890 892 894 896 898 900 902 904 906 5 2 1.5 7.30 0.000867 869 871 873 875 877 879 882 884 886 6 2.4 1.8 7 2.8 2. 1 29 848 850 852 854 855 857 859 861 863 865 8 3~2 2.4 28 828 830 832 834 836 838 840 842 844 846 9 3.6 2.7 27 809 811 813 815 817 819 821 823 825 826 26 791 793 795 796 798 800 802 804 806 SOS 25 0. 000773 775 777 778 780 782 784 786 787 789 24 755 757 759 761 762 764 766 768 769 771 23 738 740 742 743 745 747 748 750 752 754 22 721 723 725 726 728 730 731 733 735 736 21 705 707 708 710 711 713 715 716 718 720 7.20 0.000689 690 692 694 695 697 698 700 702 703 2 1 19 673 675 676 678 fiTQ fiS1 COO fifli fiftfi fiS7 I 18 658 659 661 662 o/y 664 001 665 Ooo 667 05* 669 oso 670 DO/ 672 17 643 644 646 647 649 650 652 653 655 656 1 0. 2 0. 1 16 628 630 631 633 634 635 637 638 640 641 2 0. 4 00 0. 2 0"! 15 0.000614 615 617 618 620 821 622 624 625 627 4 . O 0.8 li 0.4 14 600 601 603 604 605 607 608 610 611 612 5 1.0 0.5 13 586 588 589 590 592 593 594 596 597 599 6 1. 2 0. 6 12 573 574 576 577 578 580 581 582 5S4 585 7 1. 4 0. 7 11 560 561 562 564 565 566 568 569 570 572 8 1. 6 0. 8 7.10 0. 000547 548 550 551 552 553 555 556 557 559 9 1.8 0.9 09 535 536 537 538 540 541 542 543 545 546 08 522 524 525 526 527 529 530 531 532 533 07 511 512 513 514 515 516 518 519 520 521 06 499 500 501 502 504 505 506 507 508 509 05 0.000488 489 490 491 492 493 494 495 497 498 04 476 478 479 480 481 482 483 484 485 486 03 466 467 468 469 470 471 472 473 474 475 02 455 456 457 458 459 460 461 462 463 465 01 445 446 447 448 449 450 451 452 453 454 7.00 0.000435 436 437 438 439 440 441 442 443 444 DETERMINATION OF AZIMUTH. 1 169 Log a 1 2 3 4 5 6 7 8 9 Proportional parts 7.00 0.000435 436 437 438 439 440 441 442 443 444 10 9 6.9 345 353 361 370 378 387 396 405 415 425 1 1.0 0.9 8 274 280 287 294 301 308 315 322 330 337 2 2.0 1.8 7 218 223 228 233 239 244 250 256 262 268 3 3.0 2.7 6 173 177 181 185 190 194 199 203 208 213 4 4.0 3.6 5 0.000137 141 144 147 151 154 158 161 165 169 5 5.0 4.5 6 6.0 5.4 4 109 112 114 117 120 122 125 128 131 134 7 7.0 6.3 3 87 89 91 93 95 97 100 102 104 107 8 8.0 7.2 2 69 70 72 74 75 77 79 81 83 85 9 9.0 8.1 1 55 56 57 59 60 61 63 64 66 67 6.0 0.000043 44 45 47 48 49 50 51 52 53 8 7 5.9 34 35 36 37 38 39 40 41 41 42 ~ 0. 8 0. 7 8 27 28 29 29 30 31 31 32 33 34 2 L6 L4 7 6 22 17 22 18 23 18 23 19 24 19 24 19 25 20 26 20 26 21 27 21 3 A 2.t 3 2 2.1 2 8 5 0.000014 14 14 15 15 15 16 16 17 17 5 3! 5 4 11 11 11 12 12 12 13 13 13 13 6 7 4^8 5 6 4.2 4 9 3 9 9 9 9 10 10 10 10 10 11 g 6*4 5.6 2 7 7 7 7 8 8 8 8 8 8 2770 2753 2736 2720 2703 23n 2687 2670 2653 2636 2619 2603 2586 2569 2552 J.1.35 1 2.0 2. 1 2.2 2.3 2. 4 24 n 2518 2501 2483 2466 2449 2432 2414 2397 2380 2362 2 3 6.0 6.3 6.6 6.9 7.2 25n 9.992345 2327 2310 2292 2275 2257 2239 2222 2204 2186 4 8.0 8.4 8.8 9.2 9.6 26 n 2168 2150 2132 2114 2096 2078 '2060 2042 2024 2008 5 10.0 10.5 11.0 11.5 12.0 27 n 1987 1969 1951 1932 1914 1896 1877 1858 1840 1821 6 12.0 12.6 13.2 13.8 14.4 28 n 1803 1784 1765 1746 1727 1709 1690 1671 1652 1633 7 14.0 14.7 15.4 16.1 16.8 29n 1613 1594 1575 1556 1537 1517 1498 1478 1459 1440 8 9 16.0 18.0 16.8 18.9 17.6 19.8 18.4 20.7 19.2 21.6 8.30n 9 991420 1400 1381 1361 1341 1322 1302 1282 1262 1242 31 n 1222 1202 1182 1162 1142 1122 1101 1081 1061 1040 25 ZD 27 28 29 32 n 33 n 1020 0813 0999 0792 0979 0771 0958 0750 0938 0729 0917 0708 0896 0886 0875 0665 OS55 0644 0834 0622 1 2.5 2.6 2.7 2.8 2.9 34 n 0601 05SO 0558 0537 0515 0493 0472 0450 0428 0406 2 5.0 5.2 5.4 5.6 5.8 3 7.5 7.8 8.1 8.4 8.7 35n 9.990385 0363 0341 0319 0297 0274 0252 0230 0208 0186 4 10.0 10.4 10.8 11.2 11.6 36 n 0163 0141 0118 0096 0073 0051 0028 0005 9982 8960 5 12.5 13.0 13.5 14.0 14.5 37 n 9. 989937 9914 9891 9868 9845 9821 9798 9775 9752 9728 6 1.1. 1) 15.6 16.2 16.8 17.4 38 n 9705 9682 9658 9634 9611 9587 9563 9540 9516 9492 7 17.5 18.2 18.9 19.6 20.3 39 n 9468 9444 9420 9396 9372 9348 9323 9299 9275 9250 8 20.0 20.8 21.6 22.4 23.2 9 22.5 23.4 24.3 25.2 26.1 8.40n 9. 989226 9201 9177 9152 9127 9103 9078 9053 9028 9003 41 n .19 n 8978 8725 8953 8928 8673 8903 8647 8877 8622 8852 8596 8827 8570 8801 8544 8776 8518 8750 8492 30 31 32 i - n 43n 8465 8439 8413 8386 8360 8334 8307 8280 8254 8227 1 3.0 3.1 3.2 44 n 8200 8173 8147 8120 8093 8066 8038 8011 7984 7957 2 6.0 6.2 6.4 3 9.0 9.3 9.6 45n 9. 987929 7902 7874 7847 7819 7791 7764 7736 7708 7680 4 12.0 12.4 12.8 46n 7652 7624 75% 7568 7539 7511 7483 7454 7426 7397 5 15.0 15.5 16.0 47 n 7369 7340 7311 7282 7253 7224 7195 7166 T137 7108 6 18.0 18.6 19.2 48 n 7079 7049 7020 6990 6%1 6931 6902 6872 6842 6812 7 21.0 21.7 22.4 49 n 6782 6752 6722 6692 6662 6631 6601 6571 6540 6510 8 24.0 24.8 25. 6 9 27.0 27.9 28.8 8. 50n 9. 986479 6448 6418 6387 6356 6325 6294 6263 6232 6200 DETERMINATION OF AZIMUTH. 1 173 Log Ia Logo 1 2 3 4 5 6 7 8 9 Proportional parts 8.50n 9.986479 6448 6418 6387 6356 6325 ftfll 1 6294 IQttrt 6263 V i is 6232 5916 6200 CBOJ 32 34 36 38 40 51 n 52 n 6169 5852 6138 5820 6106 5788 6075 5756 6043 5723 OU11 5691 OtfoU 5659 oy-io 5626 5593 oo&t 5561 1 3.2 3.4 3.6 3.8 4.0 53n 5528 5495 5462 5429 5396 5363 5330 5297 5263 5230 2 6.4 6.8 7.2 7.6 8.0 54n 5197 5163 5129 5096 5(H>2 5028 4994 4960 4926 4892 3 9.6 10.2 10.8 11.4 12.0 4 12.8 13.6 14.4 15.2 16.0 55n 9.984858 4823 4789 4755 4720 4685 4651 4616 4581 4546 5 16.0 17.0 18.0 19.0 20.0 56 n 4511 4476 4441 4406 4370 4335 4300 4264 4228 4193 6 19.2 20.4 21.6 22.8 24.0 57 n 4157 4121 4085 4049 4013 3977 3941 3904 3868 3831 7 22.4 23.8 25.2 26.6 28.0 58 n 3795 3758 3721 3684 3648 3611 3573 3536 3499 3462 8 25.6 27.2 28.8 30.4 32.0 59 n 3424 3387 3349 3312 3274 3236 3198 3160 3122 3084 8 28.8 30.6 32.4 34.2 36.0 8.60n 61 n 9. 983046 2658 3007 2619 2969 2580 2930 2541 2892 2501 2853 2462 2814 2422 2776 2382 2737 2343 2698 2303 42 44 46 48 50 62 n 63n 2263 1858 2223 1817 2183 1776 2142 1735 2102 1694 2062 1653 2021 1611 1981 1570 1(140 1528 1899 1486 1 2 8.4 8.8 9.2 9.6 10.0 64n 1444 1403 1361 1319 1276 1234 1192 1149 1107 1064 3 12.6 13.2 13.8 14.4 15.0 4 16.8 17.6 18.4 19.2 20.0 65n 9.981022 0979 0936 0893 0850 0807 0763 0720 0677 0633 S 21.0 22.0 23.0 24.0 25.0 66n 0589 0546 0502 0458 0414 0370 0325 0281 0237 0192 6 25.2 26.4 27.6 28.8 30.0 67 n 0147 0103 0058 0013 5968 9923 9878 9832 787 9741 7 29.4 30.8 32.2 33.6 35.0 68n 9.979695 9650 9604 9558 9512 9466 9420 9373 9327 9280 8 33.6 35.2 36.8 38.4 40.0 69 n 9234 9187 9140 9093 9046 8999 8952 8904 8857 8809 9 37.8 39.6 41.4 43.2 45.0 8.70n 9.978762 8714 8666 8618 O1OO 8570 '-N- 1 8522 8473 7QQK 8425 7Q'je 8376 7CGJ% 8328 705ft 52 54 56 58 60 71 n 72 n 8279 7786 8230 7736 8181 7686 O1O6 7636 INISi 7586 8034 7535 1000 7485 MOO 7434 (Sou 7384 /SoD 7333 1 5.2 5.4 5.6 5.8 6.0 73 n 7282 7231 7180 7128 7077 7026 6974 6922 6870 6818 2 10.4 10.8 11.2 11.6 12.0 74 n 6766 6714 6662 6610 6557 6505 6452 6399 6346 6293 3 15.6 16.2 16.8 17.4 18.0 4 20.8 21.6 22.4 23.2 24.0 75 n 9. 976240 6187 6133 6080 6026 5972 5918 5864 5810 5756 5 26.0 27.0 28.0 29.0 30.0 76 n 5702 5647 5593 5538 5483 5428 5373 5318 5262 5207 6 31.2 32.4 33.6 34.8 36.0 77 n 5152 5096 5040 4984 4928 4872 4816 4759 4703 4646 7 36.4 37.8 39.2 40.6 42.0 78 n 4589 4532 4475 4418 4361 4304 4246 4188 4131 4073 8 41.6 43.2 44.8 46.4 48.0 79 n 4015 3957 3898 3840 3781 3723 3664 3605 3546 3487 9 46.8 48.6 50.4 52.2 54.0 8. 80n 9.973428 3368 3309 3249 3189 3129 3069 3009 2949 2888 62 64 66 68 70 ft! -n 9tt98 97R7 97flfi OftJK OKQA neno 9J.fi! 9-infl 2338 o >7f, ol H 82n dOSIO 2215 X/Bf 2153 jffUO 2090 JO*> 2028 *O54 1966 IHM0 1903 invl 1840 ^WU 1777 1714 1651 1 6.2 6.4 6.6 6.8 7.0 83n 1588 1525 1461 1398 1334 1270 1206 1141 1077 1013 2 12.4 12.8 13.2 13.6 14.0 84n 0948 0883 0818 0753 0688 0623 0557 0492 0426 0360 3 18.6 19.2 19.8 20.4 21.0 4 24.8 25.6 26.4 27.2 28.0 85n 9.970294 0228 0161 0095 0028 9962 9895 5828 9760 9693 5 31.0 32.0 33 34.0 35.0 86n 9.969626 9558 9490 9422 9354 9286 9218 9149 9081 9012 6 37.2 38.4 39.6 40.8 42.0 87 n 8943 8874 8804 8735 8666 85% 8526 8456 8386 8316 7 43.4 44.8 46.2 47.6 49.0 88n 8245 8175 8104 8033 7962 7891 7819 7748 7676 7604 8 49.6 51.2 52.8 54.4 56.0 89 n 7532 7460 7388 7316 7243 7170 7097 7024 6951 6878 9 55.8 57.6 59.4 61.2 63.0 8.90n 9.966804 6731 6657 coin 6583 KCQji 6509 C7CQ 6435 CftQO 6360 ^ftft7 6285 ecoi 6211 c < e t 6136 EOyO 72 74 76 78 80 91 n 92n 6061 5301 5985 5224 oyiu 5147 OSo4 5070 u/oy 4992 OOOO 4915 OOU/ 4837 OOol 4759 O-1O4 4681 oo/o 4603 1 7.2 7.4 7.6 7.8 8.0 93 n 4525 4446 4368 4289 4210 4130 4051 3972 3892 3812 2 14.4 14.8 15.2 15.6 16.0 94n 3732 3652 3571 3491 3410 3329 3248 3167 3086 3004 3 21.6 22.2 22.8 23.4 24.0 4 28.8 29.6 30.4 31.2 32.0 95n 9.962922 2840 2758 2676 2594 2511 2428 2845 2262 2179 5 36.0 37.0 38.0 39.0 40.0 96n 2095 2012 1928 1844 1760 1675 1591 1506 1421 1336 6 43.2 44.4 45.6 46.8 48.0 97 n 1251 1165 1080 0994 0908 0822 0735 0649 0562 0475 7 50.4 51.8 53.2 54.6 56.0 98n 9.960388 0301 0213 0126 0038 S950 862 5773 5685 9596 8 57.6 59.2 60.8 62.4 64.0 99 n 9.959507 9418 9329 9239 9149 9059 8969 8879 8789 8698 9 64.8 66.6 68.4 70.2 72.0 9.00n 8607 8516 8425 8334 8242 8150 8058 7966 7874 7781 82 84 86 88 90 1 8.2 8.4 8.6 8.8 9.0 2 16.4 16.8 17.2 17. i; 18.0 3 24.6 25.2 25.8 26.4 27.0 4 32.8 33 6 34.4 35.2 36.0 5 41.0 42.0 43.0 44.0 45.0 6 49.2 50.4 51.6 52.8 54.0 7 57.4 58.8 60.2 61.6 63.0 8 65.6 67.2 68.8 70.4 72.0 9 73.-S 75.6 77.4 79.2 81.0 INDEX. Page. Additions to previous edition .................................... 5 Adjustment and description of the transit micrometer ............ 9 Adjustment and description of the vertical circle ................. 52 Adjustments, direction method of determining azimuth .......... 145 Adjustments of the transit ....................................... 14 Azimuth ..................................................... 16 Collimation .................................................. 15 Finder circle ................................................. 16 Focusing of eyepiece ......................................... 14 Focusing of objective ......................................... 14 Horizontal axis ............................................... 15 Vert icaiity of micrometer wire ............................... 15 Wind ........................................................ 15 Wire illumination ............................................ 15 Adjustments of the zenith telescope ............................. 106 Apparatus for determining longitude by telegraphic method, arrangement of ................................................. 81 Apparent star places for latitude work, computation of ........... 116 Artificial horizon ................................................. 141 Azimuth: Adjustment of transit for ..................................... 16 Correction for elevation of mark in computation of ............ 164 Correction for variation of the pole in computation of ......... 164 Correction in time computation ............................... 25 Curvature correction in computation of ....................... 150 Direction method, adjustments ............................... 145 Direction method, computation of ............................ 148 Direction method, explanation of record and computation ____ 149 Discussion of errors ......................................... 158 Example of record and computation, direction method ....... 146 From time observations ............ .. ......................... 160 From time observations when no transit micrometer is used, computation of ............................................. 163 From time observations with the transit micrometer, computa- tion of ...................................................... 162 From time observations with the transit micrometer, example of record ................................................... 162 General considerations ....................................... 142 Instruments .................................................. 139 Instrument, shelter for ....................................... 141 Instrument support .......................................... 139 Mark ........................................................ 140 Method of repetitions, computation of ........................ 154 Method of repetitions, example of record and computation. . 153 Method of repetitions, explanationof recordand computation. . 155 Methods of determining astronomic ........................... 138 Micrometric method, example of record and computation ..... 155 Micrometric method, explanation of record and computation. 157 Observations made in connection with triangulation ......... 139 Primary ..................................................... 138 Statement of costs ............................................ 160 Summary of results .......................................... 149 Table of log .............................................. 165 Books of reference ................................................ 5 Cape tables, reduction mean to apparent declinations with ....... Ill Care of chronometers ............................................. 95 Chronograph ..................................................... 11 Chronograph, electrical connections for ........................... 12 Chronograpbic observations for tune, table of weights for incom- plete transits ................................................... 38 Chronograph, use of .............................................. 12 Chronometer corrections and rates in longitude determinations with the transit micrometer ................................... 83 Chronometers, care of ............................................ 95 Page. Chronometers, comparison by coincidence of beats.... 96 Chronomctric method of determining longitude 95 Combination of results 93 Computation of 97 Discussion of errors 100 Closing error in longitude between Key West and Atlanta, com- putation of. 85 Collimation adjustment of transit 15 Collimation axis of transit ; 13 Collimation correction in time computation 25 Collimation of transit, line of 13 Combination of latitude results, each pair observed more than once 119 Combination of latitude results, when each pair is observed but once 124 Comparison of chronometers by coincidence of beats 96 Complete least square method, computation of time set by 41 Contact correction for transit micrometer 13 Correction for: Azimuth in time computation 25 Collimation in time computation 25 Curvature in azimuth computation 150 Curvature of apparent path of star in computation of microme- ter value 127 Differential refraction in latitude computation 117 Diurnal aberration in computation of time 24 Elevation of mark in azimuth computation 1C4 Inclination of axis of transit in time computation 22 Inequality of pivots of transit in time computation 23 Rate in time computation 24 Variation of the pole in azimuth computation 164 Variation of the pole in latitude computation 132 Variation of the pole in longitude computation 85 Cost of azimuth determinations, statement of. 160 Cost of establishing latitude station 137 Cost of longitude determinations, statement of. 94 C urvature correction in azimuth computation ISO Curvature of apparent path of star in computation of micrometer value, correction for 127 Derivation of (a.t) in time computation 25 Differential refraction in latitude computation, correction for 117 Differential refraction in latitude computation, table of correc- tions for iig Direction method for determining azimuth 145 Adjustments 145 Computation of 148 Example of record and computation 146 Explanation of record and computation 149 Directions for observing latitude 109 Diurnal aberration in computation of time, correction for 24 Diurnal aberration in computation of time, table of corrections for 24 Economics of latitude observations 135 Electrical connections for chronograph 12 Elevation of mark, correction to azimuth for 164 Equatorial intervals of transit, determination of. 43 Errors in azimuth, discussion of 158 Errors in latitude, discussion of. 133 Errors in longitude: By chronometric method, discussion of 100 When key and chronograph are used, discussion of 93 When transit micrometer is used, discussion of 85 Errors In time determinations: Discussion of. 48 E sternal 48 175 176 T7. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14. Errors in time determinations Continued. Page. Instrumental 48 Observer's 50 Exchange ot signals telegraphic method of determining Iongitud3, record ol 82 Eye and ear method of observing time, directions lor 19 Eye and ear observations Tor time, table of weights for incomplete transits 36 Eyepiece of transit, focusing of 14 Finder circle adjustment of transit 16 Focusing of eyepiece of transit 14 Focusing of objective of transit 14 Horizontal axis of transit, adjustment of 15 Illumination of wires of transit 15 Inclination of axis of transit in time computation, correction for . . 22 Incomplete transits: In chronographic observations for time, table of weights for 38 In eye and ear observations for time, table of weights for 36 In time computation, reduction of 32 Table for use in computation of 32 With transit micrometer 24 Inequality ot pivots of transit in time computation, correction for. 23 Inequality of pivots ol transit, determination of 44 Instructions for determining longitude with the transit micrometer in high latitudes 80 Instructions for determining longitude with the transit micrometer in Jow latitudes 79 Instructions lor latitude work, general 103 Key method of observing time, computation of transit obser- vations 30 Key method of observing time, directions for 18 Latitude: Combination of results, each pair observed more than once ... 119 Combination ol results when each pair is observed but once . . 124 Computation 112 Computation of apparent tarplaces 116 Computation oi value Ji micrometer from observations on a close circumpolar star 126 Correction for curvature ol apparent path of star in computa- tion of micrometer value 127 Correction for differential refraction 117 Cost of establishing station 137 Determination of level and micrometer values 124 Determination of micrometer value from observations of 129 Directions for observing 109 Discussion of errors 132 Economics of observations for 135 Example of record and computation Ill Explanation of computation 115 From a single pair, weight to be assigned to mean 135 General instructions for determining 103 General notes on computation of 115 Methods of determining 103 Observing list (form 1) 108 Observing list (form 2) 109 Reduction for variation of pole 132 Reduction mean to apparent declinations with Cape tables. . . Ill Reduction to sea level 130 Reduction to the meridian 119 Summary of computation 114 Table for reduction to sea level 131 Table of corrections for differential refraction 118 Table of corrections for reduction to the meridian 119 Level and micrometer values, determination of 124 Level value of transit, determination of 46 Line intervals for transit No. 18, table of 33 Line of collimation of transit 13 Longitude: Arrangement of apparatus, telegraphic method of determining 81 By wireless telegraphy 78 Chronometer corrections and rates, In determination of 83 Cnronometric method, computation of 97 Combination of results by chronometric method 98 Combination of results when no transit micrometer is used ... 89 Longitude Continued. Page, Computation of closing error between Key West and Atlanta. 85 Computation of difference, when transit micrometer is used ... 84 Correction for variation of the pole 85 Determination, computation when no transit micrometer is used Determination, program when no transit micrometer is used . . 87 Determination, statement of cost 94 Discussion of errors in chronometric method of determining . . 100 Discussion of errors when key and chronograph are used 93 Discussion of errors when transit micrometer is used 85 Instructions for use of the transit micrometer in high latitudes for determining 80 Instructions for the use of the transit micrometer in low lati- tudes for determining 79 Method of operations when transit micrometer is used 81 Program and apparatus of the telegraph ic method 79 Record of exchange of signals, telegraphic method of determin- ing 82 Three general methods of determining 78 Weights assigned to separate chronometers in chronometric method of determining 100 Mark for azimuth observations 140 Meridian telescope, description of 8 Method of operations for determining longitude, transit micrometer method SI Methods of determining astronomic azimuth 138 Methods of determining latitude 103 Micrometer and level values, determination of 124 Micrometer, transit 8 Micrometer value from latitude observations, determination of 129 Micrometer value from observations on a close circumpolar star, computation of 126 Micrometer wire of transit, test of verticality of 15 Micrometric method of determining azimuth, example of record and computation loo Micrometric method of determining azimuth, explanation of rec- ord and computation 157 Notes on computation of latitude, general 115 Objective of transit, focusing of 14 Observatories and observing tents 105 Observing for determination of time, directions for 18 Observing list for determination of time 17 Observing list (form 1) for latitude 108 Observing list (form ?) for latitude 109 Parallax, table of sun's 60 Personal equation in time determination 90 Personal equation in time determination, table of relative 92 Pivot inequality of transit, determination of 44 Pointing lines 141 Pole variation in azimuth computation, correction for 164 Pole variation in latitude computation, correction for 132 Pole variation in longitude computation, correction for 85 Primary azimuth 138 Rate correction in time computation 24 Record and computation: Direction method of determining azimuth, example of 146 For determination of time, example of 20 Micrometric method of determining azimuth, example of 155 Of latitude determination, example of Ill Of time by the second method, example of 28 Repetition method of determining azimuth, example of 153 Record, azimuth from time observations with the transit microme- ter, example of 162 Record of observations on stars with the vertical circle for determi- nation of time 54 Record of observations on the sun with the vertical circle for deter- mination of time 56 Reduction mean to apparent declinations with Cape tables Ill Reduction to the meridian in latitude computation 119 Reduction to the meridian in latitude computation, table of correc- tions for 119 Reference books 5 Refraction, correction for differential 117 INDEX. 177 Page. Refraction tables 5S Repetition method of determining azimuth: Computation of 154 Example of record and computation 153 Explanation of record and computation 155 Sea level reduction for latitude 130 Sextant observations lor time 52 Shelter for azimuth instrument 141 Star factors for use in computation of time 60 Star factors obtained graphically 61 Star factors, table ot 62 Star list for time determinations 29 Star observatio'ns with the vertical circle to determine time 53 Stars for time observations, selection of 42 Striding level of transit, adjustment of 15 Sun observations with transit to determine time 51 Sun observations with vertical circle to determine time 56 Sun's parallax, table of 60 Support for latitude instrument 105 Supports for azimuth instrument 139 Tables (see list of tables on p. 4). Telegraphic method of determining longitude, program and appa- ratus. 79 Tents and observatories, observing 105 Time: By means of the transit instrument 7 Collimation correction in computation of 25 Computation of observations on stars with vertical circle to determine 55 Computation of observations on the sun with vertical circle to determine 56 Computation of transit observations for 21 Computation of transit observations, key method of observing. 30 Correction for azimuth in computation of 25 Corrrections for diurnal aberration in computation of 24 Derivation of (ct t) in computation of 25 Directions for observing by eye and ear method 10 Directions for observing by key method 18 Directions for observing by transit micrometer method 18 Directions for observing for determination of 18 Discussion of errors in determination of 48 Example of record and computation for determination of 20 Example of record and computation, second method 28 External errors in determination of 48 Instrumental errors in determination of 48 Observations, azimuth from 160 Observations on the sun with transit to determine 51 Observers errors in determination of 50 Observing list for determination of 17 Other methods of determining 51 Personal equation in determination of 90 Rate correction in computation of 24 Record of observations on stars with vertical circle to deter- mine 54 Record of observations on the sun with vertical circle to de- termine 56 Reduction of incomplete transits in computation of 32 Relative weights depending on star's declination in computa- tion of 38 Selection of stars for observations of 42 Set, computation by complete least square method 41 Set, computation by least square method 39 Set, explanation of second method of computation of 34 Set, explanation of usual method of computation of 27 Set, second method of computation of 34 Set, usual method of computation of 26 Sextant observations for 52 Star factors for use in computation of 60 Star list for determination of 29 Table for use in computing incomplete transits in computa- tion of 32 Table of corrections for diurnal aberration in computation of. 24 8136 13 12 Page. T ime Continued. Table ot relative personal equation In determination of 92 Table of star factors tor use in computation ol 61 Table ot weights to transits depending on the star's decima- tion in computation ol 39 Vertical circle observations tor 52 Vertical circle observations on a star to determine 53 Vertical circle observations on the sun to determine 56 Weights for incomplete transits in chronographic observations for 38 Weights for incomplete transits in eyo and ear observations for. 36 Transit, adjustments of: Azimuth 16 Collimation 15 Tinder circle 16 Focusing of eyepiece 14 Focusing of objective 14 Horizontal axis 15 Verticality of micrometer wires 15 Wind 15 Wire illumination 15 Transit: Broken telescope 8 Collimation axis of 13 Correction for inclination of axis of 22 Correction for inequality of pivots of 23 Description of large portable 7 Determination of equatorial intervals of 43 Determination of level value of 46 Determination of pivot inequality of 44 Instrument, determination of time by means of 7 Line of Collimation of 13 Micrometer 8 Micrometer, contact correction for 13 Micrometer, description and adjustment 9 Micrometer, incomplete transits with 24 Micrometer method of observing time, directions for 18 Observations for time, computation of 21 Observations, key method of observing time, computation of. 30 Observations on the sun to determine time 51 Triangulation, azimuth observations made in connection with 139 Variation of pole in azimuth computation, correction for 164 Variation of pole in latitude computation, correction for 132 Variation of pole in longitude computation, correction for 85 Vertical circle: Computation of time from observations on stars with 55 Computation of time from observations on the sun with 56 Description and adjustments 52 Observations for time 52 Record of observations on stars for determination of time with. 54 Record of observations on the sun for determination of time with 56 Time from observations on a star with 53 Verticality of micrometer wire of transit, test of 15 Weights: Assigned to separate chronometers in longitude determination by chronometric method 100 Assigned to separate chronometers in longitude determination by chronometric method, computation of 100 Depending on star's declination in time computation, relative. 38 For incomplete transits in chronographic observations for time, table of 38 For incomplete transits in eye and ear observations for time, table of 36 To be assigned to mean latitude from a single pair 135 To transits depending on the star's declination, table of 39 Wind adjustment of transit 15 Wireless telegraphy, longitude by 78 Xonith telescope, adjustments of 106 Zenith telescope, description of 104 RETURN EARTH SCIENCES LIBRARY TO > 230 Earth Sciences Bldg. 642-2997 LOAN PERIOD 1 1 MONTH ALL BOOKS MAY BE RECALLED AFTER 7 DAYS Books needed for class reserve are subject to immediate recall DUE AS STAMPED BELOW FORM NO. 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