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DEPARTMENT OF (COMMERCE
U. S. COAST AND GEODETIC SURVEY
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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}rpe 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 5s 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
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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 0
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.
</,=24' 33'
Cata-
logue
Star
Magni-
tude
Right ascension
a
Declination
S
Zenith distance
C
Star factors
Diurnal
aberration
K
A
C
B
h m s
0 /
O /
Bt
ft Tauri
1.8
5 20 25
+28 32
N 3 59
-0.08
1.14
1.14
-0.02
At
£ Aurigae
5.0
26 40
+32 07
N 7 34
-0.15
1.18
1.17
-0.02
B
t Orionis
2.8
30 53
- 5 58
S 30 31
+0.51
1.01
0.87
-0.02
B
o Aurigae
5.7
38 42
+49 47
N 25 14
-0.66
1.55
1.40
-0.03
B
£ Leporis
3.5
42 44
-14 51
S 39 24
+0.65
1.04
0.80
-0.02
A
v Aurigae
3.9
45 03
+39 07
N 14 34
-0.32
1.29
1.25
-0.02
B
S Aurigae
3.8
5 51 52
+54 17
N 29 44
-0.85
1.71
1.48
-0.03
B
6 Aurigae
2.7
53 23
+37 12
N 12 39
-0.28
1.26
1.22
-0.02
B
v Orionis
4.4
6 02 16
+14 47
S 9 46
+0.18
1.04
1.02
-0.02
B
i) Geminor.
3.3
09 16
+22 32
S 2 01
+0.04
1.08
1.08
-0.02
B
8 Monocer.
4.5
18 50
+ 4 38
S 19 55
+0.34
1.01
0.94
-0.02
B
10 Monocer.
5.0
23 22
- 4 42
S 29 15
+0.49
1.01
0.88
-0.02
B
5 Monocer.
4.4
6 35 51
+ 9 59
S 14 34
+0.26
1.02
0.98
-0.02
A
</>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
<j> 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 ±0s.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. 0 20. 0
ft Tauri
60. 1 19. 0
S Monocer.
17. 7 59. 5
% Aurigae
<f> 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. 0
£ 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. 0 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. 0
n Geminor.
62. 1 19. 7
<j> Geminor.
N
N
16. 8 58. 7
17. 0 59. 4
61.3 19.4
62. 0 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= bw
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. 0 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. 0 23. 9
1.9
57. 5 64. 6 .1
34.6
38.8
.4
44.7 50.6
.3
44. 0 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. 0
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. 0
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. 0
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. 0 15. 1 .1
20.1 27.0
.1
13. 2 20. 4 .6
31. 0 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. 0 36. 8 .8
41.0
45.1
.1
50. 6 54. 4 5. 0
10. 2 13. 9 .1
21. 6 25. 7
.3
14. 7 19. 0 .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. 0
42.5
43.8
.3
51. 8 53. 0 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. 0
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 + e1) } ~ = [ (w + wf) -(« + «') 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-wf) + («-«') 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 bE = 3e- 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 =<j> + 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. 0 20. 0
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-T0)rh
in which t is the chronometer time of transit of a star, T0 is the mean epoch of the time set, that
is, the mean of ah1 the chronometer times of transit, and rh is the hourly rate of the chronometer
on sidereal time, + when losing and -- when gaining. The quantity (t— T0) 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=08.021 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°
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 ah1 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. <j>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. <j> 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 ow -0.04=0
2. 3.00 <?t+3.89 c-0.99 ow -0.13=0
5. 2.12 3t+2. 75 c-0. 70 aw -0.09=0
6. 5.12 54+5.85 c -0.13=0
9. 4.71 cM+5.38 c -0.12=0
10. 9.53 Si +2.61=0
(2)X.707
(6)X-920
11. 9t= -0.274
^r=+15.00-0.274=+14.726
3. 3.00 5(-3. 15 c+0. 56 a +1.63=0
E
4. 3.00 «-3.47 c-0. 34 a£ +1.74=0
7. 1.82 di-1.91 c+0. 34 a£ +0.99=0
8. 4.82 <5<-5.38 c +2.73=0
12. -1.32 -5.38 c +2.73=0
14. -0.82 +1.02 -0.99 «w -0.13=0
16. -0.82 -0.83 +0.56 a. +1.63=0
(3)X-607
13. c= +0.262
15. aw= +0.071
17.
= +0.036
+ .02
-.02
-.02
+.01
+.01
+.03
.00
+.02
+.03
+ .02
-.02
-.03
* See note below table on p. 18.
DETERMINATION OF TIME. 27
EXPLANATION OF ABOVE COMPUTATION.
The serial numbers indicate the order of the various steps of the computation.
Each equation, for a star, is of the form:
Equation 1 is obtained by adding corresponding terms of the three such observation equa-
tions for the three south stars (1, 3, and 5). Equations 2, 3, and 4 are obtained in a similar
manner, there being two equations in each half set, one involving the three stars farthest south,
the other the remaining stars of the half set, in this case three in number. There are then four
equations, involving four unknowns, which can be solved by simple algebraic elimination. In
the above computation this has been reduced to systematic mechanical operations. The
azimuth constants are first eliminated, next c is eliminated, and then dt is obtained. The
computation is so arranged that the multipliers are always less than unity, which are used
to reduce coefficients in certain equations to equality with corresponding coefficients in other
equations. This makes it possible to carry through the entire computation with the aid of
Crelle's (or other similar) tables. In making substitutions in equations, such as 14 and 16,
where there is a choice between two equations, it is always well to select the equation
having the larger coefficient for the unknown sought. If the computation is followed in
these respects and a sufficient number of whole seconds are dropped from the (oc — f) to insure
that dt will be less than one second, there is no necessity, in any given case, of carrying the
computation to a greater number of decimal places than are shown above.
The checks which must be satisfied, if the computation is correct, are: (1) The algebraic
sum of all the residuals must not in hundred ths of seconds be more than one-half the number
of stars in the complete set; (2) the sum of the two, three, or four residuals corresponding to
each of the four equations designated above as 1, 2, 3, and 4 must seldom be as large as, and
never exceed, Os.02.
If these checks are not satisfied, the following principle may be found useful in detecting
whether the error was made during the process of solution of the four equations. If the work
of solution is correct, the derived values of the unknowns substituted in any one of the equations
should give a residual not greater than CP.Ol (the substitution being carried to thousandths of
seconds), but if any equation shows a residual greater than this, the error in the solution was
made in deriving an equation of a higher serial number, the serial numbers having been assigned
in the order in which the computation was made.
The chronometer correction JJ1 is then equal to dt plus the number of whole seconds
which were dropped from (ce — t) in order to lighten the work involved in making the computa-
tion. In this case it is equal to — 08.274 + 158.00= +148.726. The chronometer epoch for
which this correction applies is the mean of the chronometer times of the observed transits; that
is, the mean of the t's. It is not the mean of the right ascensions — unless, of course, the chronom-
eter correction happens to be zero.
While it is advisable to have the instrumental constants c, a^, and 0% small, it is not
desirable to strive to have them close to zero. For the azimuth constant one second is a good
limit to keep within, while if the collimation constant is less than 0s. 2 it is well not to attempt
further adjustment with a view of reducing it.
The computations are somewhat simpler when the transit is reversed on each star and one-
half the observations on a star are made in each of the positions — band west and band east —
for the collimation is eliminated by the method of observing and the only unknowns are one
azimuth constant and the clock correction, AT.
28 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
A SECOND EXAMPLE OF RECORD AND COMPUTATION.
On page 26 reference is made to a second method of solution for AT, a, and c, without
the use of least squares. This second method is used when a different selection of stars is made
from that shown on page 18. The difference between the two star sets is that in the example
of computation shown on page 26 the instrumental constants c and a are determined from all
the stars, each star being given unit weight, while in the method which follows there is observed
in each half set a slow-moving star, called the azimuth star, from which the azimuth constant
for that half set is principally determined. Besides this azimuth star there are four time stars
in each half set, and it is from the eight time stars in the entire set that the collimation constant
is mainly derived. It seems that the method of having all time stars in a set is preferable to
the other method, in which both time and azimuth stars are used. In the former, the clock
correction depends on all 12 stars instead of being derived mainly from 8 stars only, and
the collimation correction is more accurately determined. The azimuth constants, however,
are not so accurately determined by the first as by the second method, but this is immaterial
if the plus and minus azimuth factors in each half set are about equally balanced.
While this second method has been superseded in the longitude work of the Coast and
Geodetic Survey, it is considered desirable to continue it in this publication.
Using this second method, time acceptable for latitude or azimuth work can be easily
obtained with a meridian telescope, a zenith telescope, or even with an engineer's transit or
theodolite. In its usual form the star set consists of four tune stars and an azimuth star with
the instrument in each position, band west and band east. If greater accuracy is desired the
number of time stars in a half set may be increased, or if less accuracy is needed the number may
be decreased. In the work of the Survey up to the time of the adoption of the transit micrometer
and the method of computation shown on pages 20-27, the standard time set consisted of two
half sets, in each of which was one azimuth star and four time stars.
The following set of observations was made with a small portable transit, using an observing
key to record the observations chronographically. With the record of observations there are
given the readings of the level, the correction for inclination of the horizontal axis of the tele-
scope (which in this case includes a correction for inequality of pivots), and the computation of
(«-<). A correction for rate has been introduced. The correction for diurnal aberration and
the correction for rate are obtained in the same manner as shown on page 24. The form on
which the level readings are recorded is shown on page 20.
DETERMINATION OF TIME.
Star list for Washington, D. C. — Latitude 38 ° 54' N.
29
Star factors
Star
Cata-
logue
Magni-
tude
Right ascen-
sion
a
Declina-
tion
i
Zenith dis-
tance
C
Diurnal
aberra-
tion
Incli-
na-
Colli-
ma-
Azimuth
K
tion
tion
A
B
C
h m s
o /
o /
s
17 H. Can. Yen.
B
5.5
13 30 12
+37 43
+ 1 11
-.02
1.26
1.26
+ .02
t) Ursa Maj.
B
2.0
43 30
+49 50
-10 56
-.02
1.53
1.55
- .30
rj Bootis
B
3.0
49 47
+18 55
+19 59
-.02
0.99
1.06
+ .36
11 Bootis
B
6.0
56 31
+27 53
+11 01
-.02
1.11
1.13
+ .22
a Draconis
B
3.3
14 01 39
+64 52
-25 58
-.04
2.12
2.36
-1.03
d Bootis
B
5.0
05 42
+25 35
+13 19
-.02
1.08
1.11
+ .25
or Bootis
B
1.0
10 58
+19 43
+19 11
-.02
1.01
1.06
+ .35
A Bootis
B
4.0
12 29
+46 34
- 7 40
-.02
1.44
1.46
- .19
d Bootis
B
3.8
21 43
+52 20
-13 26
-.03
1.59
1.64
- .38
5 Ursse Min.
A
4.5
27 51
+76 09
-37 15
-.06
3.33
4.18
-2.53
B-= Berliner Astronomisches Jahrbuch. A=American Ephemeris.
30
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
Following the computation are given any explanations needed to supplement or qualify
the explanations of computations given on pages 22-27.
[Station, Washington, D. C. Date, May 17, 1896. Observer, G. R. P.
Star
17 H. Can. Yen.
, Urs. Maj.
jj Bootis
n Bootis
a Draco.
Position of band
Mrest
West
West
West
West
Direction of ob-
|
jective for level
*
S
JV
reading
J
W E
W E
W E
d d
d d
d d
22.7 24.1
27. 8 20. 0
28. 0 19. 9
Level readings
27. 1 19. 9
22. 9 24. 8
22. 9 24. 9
SVt and JE
49.8 44.0
50. 7 44. 8
50. 9 44. 8
JW-JE
+5.8
+5.9
+6.1
Remarks and
Means of levels
comput a t i o n
d
of b
N +6. 10
1
S +5.85
+5. 98X.0279=
+ . 167=0W
— . 010— pivot ine-
quality
+ . 157=bw
Observed transit
h m t
h m s
h m s
h m s
h m s
Line 1
13 29 56. 90
13 43 10. 60
13 49 34.45
13 56 17.20
14 01 07. 55
2
13 30 00. 10
Mean
14.35
37.05
20.00
13.30
3
03.30
5
18.30
39.70
22.80
19.35
4
09.70
14.20
26.15
45.00
28.40
31.25
5
12.90
Correc-
30.15
j
47.75 l
31.50
$
37.30
s
6
16.00
tion
33.80
33.80
50.25
50.25
34.30
34.30
43.00
43.00
7
19.30
15.15
37.95
68.10
52.95
100.70
36.90
68.40
49.00
86.30
8
22.60
41.70
7.85
55.70
0.70
40.05
8.45
55.20
6.45
9
29.00
10
49.70
8.00
13 50 00. 90
0.60
45.65
X.45
14 02 06. 90
6.25
10
32.20
X 1.26
53.60
7.95
03.75
0.80
48.60
8.60
12.90
6.20
11
-
- + 1.92
57.55
8.15
06.50
0.95
51.50
8.70
18.85
6.40
Mean
16.12
33.99
10.85
50.36
4.00
34.26
2.90
43.15
1.60
R
+ .03
+ .02
+ .01
+ .01
.00
K
- .02
- .02
- .02
- .02
- .04
Bb
+ .20
+ .24
+ .16
+ .17
+ .33
t
13 30 16. 33
13 43 34. 23
13 49 50.51
13 56 34. 42
14 01 43. 44
a
13 30 12. 26
13 43 30. 14
13 49 46. 62
13 56 30.53
14 01 38.92
OL-t
-4.07
-4.09
-3.69
-3.89
-4.52
DETERMINATION OF TIME.
31
Instrument, transit No. 18. Chronometer, Negus, 1836 (daily rate, 1«.51 gaining).]
d Bootis a Bootis
A Bootis
e Bootis
5 Urs. Min.
East
East
East
East
East
S
N
A
W E
W E
W E
d d
d d
d d
27.1 20.9
27. 2 20. 9
22. 2 26. 0
22.7 25.2
22.9 25.3
27.2 21.0
49. 8 46. 1
50. 1 46. 2
49. 4 47. 0
+3.7
+3.9
+2.4
Means of levels
Thin clouds and hazy
d
Temperature 76° F
N. +3. 15
S. +3.70
j
1
+3.42X.0279-
+ .095-#E
+ . 010= pivot inequality
+.105=bE
Am s
14 05 29. 40
Mean
Am j
14 10 45.50
Am s
14 12 11. 15
Am s
14 21 22.30
Am s
14 26 53. 15
32.20
s
48.20
14.80
26.40
27 03.15
34.85
44.76
50.90
18.60
30.60
14.25
40.60
Correc-
56.20
25.95
38.90
35.30
43.35
tion
58.90
1
29.50
J
42.90
1
45.85
s
46.20
48 90
12.69
14 11 01.65
04 30
01.65
03 20
33.45
37 00
33.45
66 50
47.35
51 35
47.35
94 25
57.15
UOQ 1-17 fu-1
S7.15
119 Sfi
51.90
10
07.10
3.30
40.70
6.65
55.40
4.30
£>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:
tm = 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.
tm 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 60s, the maximum error is less than 08.01 for a star of which <? = 70°, and is only
03.3 if 5 = 85°. If the extreme interval is 15s, the maximum error is less than 08.01 if «J = 85°.
The more exact formula for use with circumpolar stars is the same as that given above,
except that for each equatorial interval, i, must be substituted i %j sec r, in which r is the hour
angle of the star at transit across the line, or with sufficient accuracy r = i sec 3 = the actual time
interval from the mean line.
The following table will be found useful in connection with this formula.
T
log Veos t
log V sec T
T
T
log V sec T
log V COS r
log V sec i
log V COS T
m
TO
TO
1
9. 99999
0.00000
16
9. 99965
0.00035
31
9. 99867
0. 00133
2
99
01
17
960
040
32
858
142
3
99
01
18
955
045
33
849
151
4
98
02
19
950
050
34
840
160
5
97
03
20
945
055
35
831
169
6
95
05
21
939
061
36
821
179
7
93
07
22
933
067
37
811
189
8
91
09
23
927
073
38
800
200
9
89
11
24
921
079
39
789
211
10
86
14
25
914
086
40
778
222
11
83
17
26
907
093
41
767
233
12
80
20
27
899
101
42
756
244
13
77
23
28
892
108
43
744
256
14
73
27
29
884
116
44
732
268
15
9. 99969
0.00031
30
9. 99876
0. 00124
45
9. 99719
0. 00281
' The collimation factor C (as given in the star list on p. 29) is the sec i.
DETERMINATION OF TIME.
33
If the chronometer rate exceeds 15s per day it will be desirable to take it into account in
making the reduction of incomplete transits to the mean line.
Another method of reducing incomplete transits is to construct from the known equatorial
intervals a table similar to that of which a portion is printed below showing the interval of each
line from the mean line corresponding to various declinations. The correction of each observed
line to the mean line is then taken out directly from the table and the mean of the various
corrected transits taken.
Intervals of lines of Transit No. 18 from mean line.
[The numbering of the lines is for band west.]
3
Line I
Line II
Line III
Line IV
LineV
Line VI
Line VII
Line VIII
Line IX
LineX
Line XI
o
0
10
15
s
+15. 20
15.43
15.74
s
+12. 69
12.89
13.14
S
+10. 15
10.31
10.51
s
+5.06
5.14
5.24
S
+2.52
2.56
2.61
-0.09
0.09
0.09
S
-2.52
2.56
2.61
*
-5.11
5.19
5.29
t
-10.09
10.25
10.45
-12.65
12.84
13.10
s
-15.15
15.38
15.68
36
38
40
18.79
19.29
19.84
15.69
16.10
16.57
12.55
12.88
13.25
6.25
6.42
6.61
3.11
3.20
3.29
0.11
0.11
0.12
3.11
3.20
3.29
6.32
6.48
6.67
12.47
12.80
13.17
15.64
16.05
16.51
18.73
19.23
19.78
51
52
53
24.15
24.69
25.26
20.17
20.61
21.09
16.13
16.49
16.87
8.04
8.22
8.41
4.00
4.09
4.19
0.14
0.15
0.15
4.00
4.09
4.19
8.12
8.30
8.49
16.03
16.39
16.77
20.10
20.55
21.02
24.07
24.61
25.17
Transit No. 18 was the instrument used for the observations shown on pages 30-31. The
incomplete transit of the star 17 H. Can. Ven., of which the declination is 37° 43', may be
computed as indicated below:
Line
I
II
III
IV
V
VI
VII
VIII
IX
X
Correction
+19. 22
+16.04
+12. 83
+ 6.40
+ 3.19
- 0.11
- 3.19
- 6.46
-12.75
-15.99
Corrected transit
16.12
16.14
16. 13
16.10
16.09
15.89
16.11
16.14
16.25
16.21
Mean =16. 12, agreeing with the result shown in the
example on page 30.
The special advantage of this method of reducing incomplete transits is that a wild observa-
tion upon any one line is at once detected. Such wild observations are apt to occur under the
conditions which produce incomplete transits, viz., clouds, haste, or difficulty with illumination.
CORRECTION FOR RATE.
The method of computing this correction is shown on page 24.
CORRECTIONS FOR DIURNAL ABERRATION, COLLIMATION, AND AZIMUTH.
The correction for diurnal aberration and general expressions, for the collimation and
azimuth corrections are shown on pages 24-25.
8136°— 13 3
34 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
COMPUTATION OF 4 T, a AND c, USING AZIMUTH STARS AND METHOD OF APPROXIMATIONS.
The method of computation shown below was in use in the field by parties of this Survey
for many years.1 It is now replaced by the method shown on page 26.
[Station, Washington, D. C. Date, May 17, 1890.]
Star
Band ! o.-t
C
A
Cc
Aa
tt-t- Cc-Aa
s
s
s
s
s
17 H. Can. Ven.
W
-4.07
+ 1.26
+ .02
+.04
+ .01
-4.12
+ .10
n Urs. Maj.
W
-4.09
+1.55
- .30
+.05
- .17
-3.97
-.05
i) Bootis
W
-3.69
+1.06
+ .36
+.03
+ .20
-3.92
-.10
II Bootis
W
—3.89
+1.13
+ .22
+.04
+ .12
-4.05
+ .03
at 14^ 02m.OO
a Draconis
W
-4.52
+2.36
-1.03
+.08
- .58
-4.02
.00
J T= -04«.024
d Bootis
E
-3.94
-1.11
+ .25
-.04
+ .13
-4.03
.00
a Bootis
E
-3.81
-1.06
+ .35
-.03
+ .18
-3.S6
-.07
1 Bootis
E
-4.23
-1.46
- .19
-.05
- .10
-4.08
+.05
0 Bootis
E
-4.29
-1.64
- .38
-.05
- .19
-4.05
+ .02
S Urs.Min.
E
-5.44
-4.18
-2.53
-.13
-1.28
-4.03
.00
Or-<
C
A
Cc
a-t- Cc
Aa
O.-t- Cc-Aa
First approximation:
s
Mean of time stars
W
-3.94
+1.25
+ .08
+.06
-4.00
+ .05
-4.05
c=+.0ol
Azimuth star
W
-4.52
+2.36
-1.03
+.12
-4.64
— .59
-4.05
ow— + .577
Mean of time stars
E
-4.07
-1.32
+ .01
-.07
-4.00
.00
-4.00
BE-+.484
Azimuth star
E
-5.44
-4.18
-2.53
-.21
-5.23
-1.23
-4.00
Second approximation
Mean of time stars
W
+ .04
-3.98
+ .04
-4.02
C=+.032
Azimuth star
W
+ .08
-4.60
- .58
-4.02
ow= + .559
Mean of time stars
E
-.04
-4.03
.00
-4.03
nE_+.504
Azimuth star
E
-.13
-5.31
-1.28
-4.03
i
1 Thecomplete formula for the chronometer correction is A T=<x.-(tm+R+K+Bb+Cc+Aa). Let t=tm+R+K+£b, then A r=(a— 0— Cc-Aa
so that it will be seen that the corrections Cc and Aa are to be subtracted algebraically from a— (.
EXPLANATION OF THE COMPUTATION.
The first five columns of the upper portion of the computation were compiled from the
record and computation shown on pages 30-31 and from the observing list shown on page 29,
The remaining columns were filled out after the computation of a and c, shown in the lower
portion of the form, was completed.
It should be noted that the five stars of each group, observed in one position of the instru-
ment, have been so selected that one is a slowly moving northern star at a considerable distance
from the zenith, while the other four are all comparatively near the zenith, some transiting to
the northward of it and some to the southward, and at such distances from it that then- mean
azimuth factor, A, is nearly zero. These four stars of each group may be for convenience called
time stars, since the determination of time falls mainly upon them, while the slowly moving
northern star serves to determine the azimuth error of the instrument, and may be called the
azimuth star.
In this computation to derive c and a the time stars in each position of the instrument
are combined and treated as one star by taking the means of their (n — t)'&, and of their star
factors C and A, respectively, these means being written below the separate stars in the form,
together with the azimuth stars. On the assumption that the means of the time stars in the
two positions of the instrument are equally affected by the azimuth correction, the first approxi-
* It was devised in the seventies by Assistant Edwin Smith, then an aid in this Survey. See p. 280, Appendix 4 of the Report for 1904.
DETERMINATION OF TIME. 35
mation to c is found by dividing the difference between tbe mean (a — t)'s by the difference
between the 6"s. In the example,
,. . (.-*-t)w-((Y-t)E -3.94- (-4.07) +0.13
c (hist approximation) = -Q^C^ = + 1.25-7^.32) = +2^7 = + ° -051'
Tsing this approximation to c, the correction Cc is then subtracted from the a — t of each mean
of the time stars and of each azimuth star, and the values of a — t— Cc, in the seventh column
on the fifth to eighth lines from the bottom of the form, are obtained.
Separate values for the azimuth error of the instrument are then derived for each position
of the instrument as follows:
( T - t - (7c)tlme stars ~(ne-t- fle)a,|muth star - 4.00 -(- 4.64) +0.64
'
time stars
=
-A azimuth star. " +0.08 - (- 1.03) = + 1.11
-4.00 -(-5.23) +1.23
«*= VOTOI— C-2T53r +2754= *
With these values of aw and <IE the corrections Aa are applied, giving the values ct — i— Cc — Aa
in the last column but one. If these do not agree for the stars east and west it indicates that
the mean values ce — t, used in deriving c, were not equally affected by the azimuth error, so that
their difference was not entirely due to c, as was assumed. An improved value of c may now
be obtained by treating the difference in the last column as still an error of collimationj and
thus obtaining a correction to the first approximate value of c. Thus, in the example,
-4.05 -(- 4.00) _ -0.05 _
+ 1.25- (-1.32) +2.57
Applying this correction to the first approximate value of c= +0.051, we have for a second
approximation c= +0.032. Proceeding as before, improved values for aw and aE are found.
If the star sets are well chosen and the instrumental errors small, the first approximation will
generally suffice. If the values of a — t— Cc — Aa differ by but a few hundredths, east and west,
there is little gained by making a closer adjustment. The chronometer correction will prob-
ably not be changed at all, but the instrumental errors and star residuals will be slightly altered,
as is apparent from the example, where the closer adjustment is made for the purpose of illus-
trating the method.
In the first approximation the value of c may at once be derived more closely when there
is much difference between the mean A's for the time stars, by estimating the effect of
this difference in A on the A T, and allowing for this effect when deriving c in the first place.
The formula for c then becomes
_
c~
It is here necessary to estimate the azimuth of the instrument, a, roughly in advance, and
this may be done by inspection. Thus, in the example, assuming a= +08.5, we have
_ -3.94-^4.07- (+. 07) X ( + 0.5) _ +.09 _
+ 1.25 + 1.32 =+2.57~
agreeing closely with the value j^iven by the second approximation.
When satisfactory values of c, aw, and «E have been obtained, the corrections Cc and Aa
are applied separately to each star, as shown in the upper part, and the values of the chronometer
correction (AT) derived separately. The residuals are taken for each group from the mean
of that group, and thus furnish a convenient check on the computation, as their sums for each
group should approximate zero. Unusual residuals also point to possible errors in a — t. The
36
TJ. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
mean of the A T's from the separate stars gives the final chronometer correction at the epoch of
the mean of the chronometer times of transit of the stars observed.
This whole computation may be made with rapidity by the use of Crelle's multiplication
tables.
The field computation having been made as outlined above,1 the more refined office com-
putation may be made as indicated on pages 39-41. It is desirable in this office computation
to introduce weights dependent upon the declination of the star and the number of lines of the
reticle upon which the star was observed.
The four equations, solved by successive approximations above, may be solved by direct
elimination, in case the coefficients of aw and aE do not become relatively small in the two equa-
tions gotten by taking the mean of the time stars in the two half sets.
RELATIVE WEIGHTS FOR INCOMPLETE TRANSITS.
Sometimes the transit of a star is observed over some of the lines of the diaphragm and
missed over the others. Obviously the deduced time of transit over the mean line from such
an incomplete transit should be given less weight than that from a complete transit.
For observations made by the eye and ear method the relative weights given by Chauvenet
may be used, viz:
n (N+3)
P~ N (n + 3)
in which p is the weight to be assigned to the computed time of transit over the mean line, N
is the total number of lines in the diaphragm, and n is the number of lines upon which obser-
vations were made.2 This formula is based upon the assumption that (c)2 = 3(£,)2, in which (E) =
the probable error of an observed transit of an equatorial star over a single line and (e,) =the
probable culmination error referred to the equator, a constant for all the fines of the diaphragm
for any one star, but variable from star to star, and supposed to be due mainly to atmospheric
displacement, to outstanding instrumental errors, to irregularities in clock rate, and to changes
in personal equation.
The following table shows the values of p and V? for the two cases of 5 and 7 fines in the
diaphragm :
Table of weights for incomplete transits for use with eye and ear observations.
N=S
N=7
P
Vp
P
Vp
1
0.40
0.63
0.36
0.60
2
0.64
0.80
0.57
0.75
3
0.80
0.89
0.71
0.84
4
0.92
0.96
0.82
0.91
5
1.00
1.00
0. 90 0. 95
6
0: 95 0. 97
7
1.00
1.00
i For a more complete account of this method of computation, see Appendix No. 9, Report for 1896. The above account is largely taken from
that appendix.
» See Chauvenet's Astronomy, Vol. II, p. 198. The derivation of this formula follows the same lines as that given on the following pages for
weights to be assigned to incomplete transits taken by the chronographic method.
DETERMINATION OF TIME. 37
The relative weights to be assigned to incomplete transits observed by the chronograph
method may be derived as follows :
r2=(E1)2 + i^
in which r = the probable error of the time of transit over the mean line, arising from the com-
bined effect of the culmination error referred to the equator (EJ) and of the probable error of
the transit of an equatorial star over a single line (E).
To find r, individual determinations of right ascensions of stars, all referred to the same
epoch (mean place), may be compared with their respective average values; thus, from 558
results of 36 stars observed at the United States Naval Observatory with the transit circle
(using a magnifying power of 186) in 1870 and 1871, it was found that r= ±08.034. To apply
tliis value to our instruments it must be somewhat increased, though not in proportion to the
respective magnifying powers, since some of the errors involved approach the character of
constants ; multiplying it by 1 .5 and 1 .75 for our larger and smaller transits, respectively, there
is obtained r= ±08.051 and r= ±08.060. For the larger transits (E)=±08.063 and for the
smaller («)= ±08.080. (See p. 39.) Substituting these values in the above formula, together
with the values 25 and 15 for n as actually used in the observations cited on page 38, there is
obtained
(0.051)2= (0* + and (0-060)'= (0' +
which give
(O = ±09.049 and (E,) = ±0". 056
for the larger and smaller instruments, respectively.
If the weight for a complete transit is unity, the weight for an incomplete transit is
Hence, for the larger instruments, using the above values for (E,) and (E),
and for the smaller instruments
2.0
n
very nearly. From these expressions the relative weights have been computed for total number
of threads N=25, 17, 13, and 11 for the larger instruments and for N=15, 13, 11, and 9 for
the smaller ones, and are shown in the following table.
38 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
Table of weights for incomplete transits for use with chronograj>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
AT=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
(«)
0
s
o
s
o
s
0
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
0 075
29 7
0 067
170
01 1 n
28.5
0.058
0 7
0 071
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 tan2 d
39
Transit, No. 5 (£)=V(0-066)2+(0.036)2 tan2 d
Meridian telescope, No. 13 (£)=V(0.069)2+(0.078)2 tan2 3
Meridian telescope, No. 13 (s)=V(0.087)2+(0.055)2 tan2 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
0 /
s
s
"
0
±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 aw, aE, 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=Jic — 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 + IpAaw = 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(=-4S.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 aw=- .06
- 05d*+.65aw= + .39
aw=+».601
+3.46 St- .22aE=- .16
- .22 St+1.32 aE= + .72
aE=+«.543
3t=-'.012
+7.29 Q- .27 7=1
- .27 Q+1.97 q=0
At 14h 02m JT=-43.020
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 aw 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 dt1 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
^^
\ n0 -
where 2pJ2 is the sum of the weighted squares of the residuals (last column in form), n0 is the
number of stars and ne is the number of unknown quantities or number of normal equations,
remembering in this example that there are four unknowns, dt, aw, aE, 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 0 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, aw, aE, 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£aE
-Jpdt
The normal equations will be—
IpCdt + IpAECaE
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 aE+3. 10 aw+ 0. 21 c =-7. 70
+2. 81 St +2. 52 <z.E 1. 73 c = -3. 74
+3. 10 St +2. 88 aw+ 2. 07 c = -3. 23
+0. 21 St -1. 73 aE+2. 07 aw+ 11. 86 c =+1. 94
c =+0.183
aE=- 1.342
aw=- 1.233
ot =-0-02
At S.h5 AT = -20.12
Q = .79
e =±. 035
The remarkably large value for Q arises from the fact that the azimuth errors, aw and aE
are but feebly determined, see column headed pA and the normal equations.
Sometimes it is assumed that the azimuth error is the same for both halves of a set, and
the distinction between aw and aE is dropped and a single a derived from the whole set, the
normal equations being modified accordingly. This procedure is entirely justifiable if the
azimuth error during the two half sets is actually the same. If the two azimuths really differ,
some error will be introduced into the computed results by this procedure, and the error so
introduced will be larger the greater is said difference. Experience shows that the instability
of the instrument in azimuth is in general sufficient to make it desirable to distinguish between
the two azimuth errors if accurate results are desired, except when there are but few stars
observed in the set, say, seven or less.
THE SELECTION OF STARS.
The stars shown in the observing list (p. 18) and used in the computation on pages 21,22 and
26 were chosen by the method now used for longitude work in latitudes less than 50°. In each
half set there are five to seven time stars (six stars preferred), a time star being one which has
an A factor less than unity. These stars are so selected that the algebraic sum of the A factors
in a half set shall not be greater than unity. It is desirable to have the algebraic sum of the
A factors of the stars in a half set as small as can be obtained by the use of good judgment
in their selection, 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 the balancing is already within the
specified limit.
In endeavoring to obtain the maximum number of stars per hour, subject to the condition
of the balancing of the A factors, consideration must be given the question of level readings
DETERMINATION OF TIME. 43
and reversals of the instrument. Ample time should be provided for the performance of these
operations. In longitude work allowance must be made for the exchange of time signals,
which, if the stations are not very far apart, usually takes place between the two sets — that
is, between the second and third half sets. The exchange may be made, however, at any time
during the observing period if there is trouble in getting a clear wire between the two observa-
tories or if clouds break up prearranged sets of stars. An observer soon learns from practice
how much time must be allowed for the different operations.
It is desirable, but not necessary, to observe the same stars at both stations when deter-
mining a difference of longitude. This is of less importance, however, than securing rapid
observations with the A factors in each half set well balanced. When the two stations are not
distant, many of the stars observed at one station will necessarily be observed at the other.
In longitude work the observations each night consist normally of four half sets of six
stars each, with a reversal of the instrument between each two consecutive half sets. The
reversal of the instrument after each of the half sets is a precaution which experience has
justified, for should only three half sets be observed (through interference of clouds or for other
reasons) two sets can still be obtained by combining the first and second and the second and
third half sets, thus obtaining two corrections to the chronometer and its rate.
Where it is desired to use the azimuth star method of solution shown on pages 34 and 40, a dif-
ferent selection of stars is to be made. A half set consists of five stars following each other in rapid
succession, so chosen that the algebraic sum of the A factors of the four time stars (each near
the zenith) will be nearly zero, and that the azimuth star of each half set will have its A factor
greater than unity, and yet not be so near the pole as to render the star's transit across the
field of observation so slow as to produce long waits between observations. In a time set,
chosen as above, observation upon the azimuth star in each half set serves principally to
determine the azimuth error of the instrument, but has little effect upon the computed time,
since this is almost independent of the azimuth error (the sum of the A factors of the time
stars being nearly zero for each half set). Where only approximate time is required, the
number of time stars in a half set may be reduced to two, one north and one south of the zenith.
In high latitudes (more than about 50°), it is not feasible to secure time sets with well-
balanced A factors, since the stars between the zenith and the pole have comparatively small
A factors, which become relatively still smaller after weights are assigned. This condition
prevents any but a comparatively weak determination of the azimuth error of the 'instrument.
In such latitudes it is therefore desirable to select sets of stars which will be solved by rigid
least-square methods. Under normal conditions there should be six stars in each half set,
and while the algebraic sum of the A factors in each half set should be kept as small as can be
conveniently done, no very slow-moving stars should be introduced for this purpose. One
azimuth star with a declination between 55° and 75° should be selected and observed below
the pole.
The preliminary or field computations may be made like that shown on page 26. The
final least square computations are made at the office.
As has already been stated (p. 25), the preference is now given to the American Ephemeris
over other star lists, as it contains the apparent places of more stars than other available cata-
logues. It is well to obtain all stars, when possible, from a single catalogue, but this is not
essential. It may be considered as almost essential, certainly so from an economic standpoint,
to use only stars for which apparent places are published. The time and labor consumed in
computing the apparent right ascension of stars for which only mean places are available
add to the cost of both the field and office work. Furthermore, it will be found that sufficient
stars can be selected for all time work in the northern hemisphere from such catalogues as the
American Ephemeris and Nautical Almanac or the Berliner Astronomisches Jahrbuch, and the
selection of mean place stars is unnecessary.
DETERMINATION OF EQUATORIAL INTERVALS.
The equatorial intervals of the lines of the diaphragm are needed to reduce incomplete
transits. (See p. 32.)
44
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
To determine these, select complete transits of stars of large declination.
Let tlt i2, t3 ...... in be the observed times of transit over the successive lines, tm, their
mean, and -iu i2, i, ...... in their equatorial intervals from the mean line and d the declination
of the star:
\ = (t1— tm) cos d
i2=(t2-tm) cos d
etc.
"in = (<n — <m) COS d
also 0 = i1+i2+'i3 ...... +in.
The intervals of the lines j eas , i of the mean line will then be | > 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 = (<, — tm) cos d -/ cos TJ
etc. ^___
^n = (<n - tm) COS d $ COS Tn
where TU T2, TS ...... rn are the hour angles of the circumpolar star for the successive lines.
When it is necessary to use the more exact formula for circumpolars as given above, the
table on page 32 will be found convenient.
If the chronometer rate exceeds 15s per day it will be desirable to take it into account in
computing the equatorial intervals.
A convenient form for the computation of equatorial intervals follows. The observations
used were made by Assistant Fremont Morse at Sitka, Alaska, in 1894, with Meridian Telescope
No. 7, and by the eye and ear method.
K Draconis. 3=70° 22' 27". Log. cos 3=9.52618. Clamp West.
Line
May 14
May 15
May 16
May IS
Mean
Log. mean
Log. i
(equatorial
interval)
S
S
S
S
S
S
1
^1 — ^m
-87. 60
-88.00
-87. 10
-87.60
-87. 575
1. 94238
1. 46856
-29. 414
2
tz — tm
-44. 60
-44.00
-44. 60
-44. 60
-44. 450
1. 64787
1. 17405
-14. 930
3
t3 — tm
- 0.10
0.00
+ 0.40
+ 0.40
+ 0. 175
9. 24304
8. 76922
+ 0. 059
4
ti — t,a
+43. 90
+44.00
+43. 90
+43. 40
+43. 800
1. 64147
1. 16765
+14. 711
5
<5-<m
+88. 40
+88.00
+87. 40
+88. 40
+88. 050
1. 94473
1. 47091
+29. 574
The quantities (^-<m), (t2-tm), etc., for each date were taken directly from the record of
observations.
The equatorial intervals were thus computed from observations upon three different stars
and the means taken.
It is not necessary to make special observations to determine the equatorial intervals.
Complete transits observed during the regular progress of time observations may be utilized
for that purpose. If observations upon stars of large declination are not available, observa-
tions upon stars of small declination may be used, and will be found to give almost as accurate
values for the equatorial intervals.
When pressed for time in the field an incomplete transit of a star may be reduced by assuming
that actual intervals between lines on that star are the same as on some preceding date on
which a complete transit of that star was observed at that station. The formulse on page 32
may then be used by dropping the factor sec d and substituting actual intervals for equatorial
intervals.
PIVOT INEQUALITY.
The pivot inequality should be determined with the instrument mounted upon a very
stable pier in a room in wlu'ch the rate of change of temperature is small during the observa-
tions. The observations consist of a series of readings of the striding level as indicated in the
DETERMINATION OF TIME.
45
example of record and computation given below. The notation is the same as on pages 22-23;
that is, /?,„ and /?e indicate the apparent inclination of the telescope axis in each of its two posi-
tions as given directly by the readings of the striding level. Then the pivot inequality
and is to be expressed in seconds of time.
Observations for inequality of pivots of transit, No. 19.
[Station, Atlanta, Ga., MaA 12, 18%. G. R. P., observer.]
""
Band west
Band east
Object glass south
Object glass north
Zenith
distance
Time
Temper-
ature
Zw-2t
Sw — Ze
0e — @u>
4
Level
4
= $W
Level
4
= Pc
= P
W. end
E.end
W.end
E.end
0
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 0° 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
£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.S03, 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 0.805.
' 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.S01 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 0.81. The probable error of a single record (position of micrometer wire)
from this source is about ±0.S02, but the number of such records obtained on a star makes the
probable error of the mean of these observations less than ±0.801, 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 0s. 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 constant2 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 0
1 06 02 5
d
14 1
14 4
d
12 0
11 8
23.8
11.8
14.3
R
L
L
K
R
L
L
R
0
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 0° 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
<f> 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
0 , „
h m s
0 , „
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 <f>, $,
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
0
E
A
B
C
D
Mean
Sun's upper limb 0
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
0
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
0
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
0
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
0
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
0 , „
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 ^, <j>
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 CB, 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 CT 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 = ru X CB X CT in which ru is the refraction under standard
conditions obtained from the first table and CB and CT 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" 25m.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 (rM)
[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
0 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
0 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
0 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
0 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
0 42.3
0.03
40
27 18. 2
9.20
40
6 49.1
0 81
20
2 35.9
0.14
20
1 25.0
0.05
30
0 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
0 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
0 40.0
0.03
10
23 13.5
7.17
10
6 26.0
0 73
20
2 28.1
0.13
20
1 22.0
0.05
56 00
0 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
0 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
0 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
0 35.0
0. 023
50
19 09.8
5.20
50
5 58.9
0 63
40
2 18.6
0.11
37 00
1 17.1
0.04
60 00
0 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
0 04
61 00
0 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
0 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
0 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
0 28.4
0.021
40
15 32. 1
3.62
20
5 09.4
0 48
20
2 08.1
0.10
30
1 10.5
0.04
65 00
0 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
0 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
0 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
0 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
0 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
0 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
0 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
0 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
0 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
0 16.7
0.018
20
11 04.2
1.98
40
3 55.1
0.28
40
1 50.6
0.08
30
0 59.2
0.03
75 00
0 15.6
0.018
30
10 44.9
1.88
14 00
3 49.5
0.27
28 00
1 49.1
0 08
45 00
0 58.2
0.03
76 00
0 14.5
0.018
40
10 26. 5
1.79
20
3 44.2
0.26
20
1 47.6
0.07
30
0 57.2
0.03
77 00
0 13.5
0.018
50
10 09.1
1.70
40
3 39.1
0.25
40
1 46.1
0.07
46 00
0 56.2
0.03
78 00
0 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
0 55.2
0.03
79 00
0 11.3
0.018
10
9 36.9
1.54
20
3 29.4
0.23
20
1 43.2
0.07
47 00
0 54.2
0.03
80 00
0 10. 3
0.018
20
9 21 9
1.46
40
3 24.8
0.23
40
1 41.8
0.07
30
0 53.3
0.03
81 00
0 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
0 52.5
0.03
82 00
0 08.2
0.018
40
8 54.0
1.33
20
3 16.1
0.21
20
1 39.1
0.07
30
0 51.6
0.03
83 00
0 07.2
0.018
50
8 41.0
1.27
40
3 12.0
0.20
40
1 37.8
0.06
49 00
0 50. 7
0.03
84 00
0 06. 1
0 018
6 00
8 28.6
1.22
17 00
3 08.2
0.19
31 00
1 36.6
0.06
30
0 49.8
0.03
85 00
0 05. 1
0.018
10
8 16.7
1.16
20
3 04 5
0.19
20
1 35.3
0.06
50 00
0 48.9
0.03
86 00
0 04. 1
0.017
20
8 05.3
1.12
40
3 00.9
0.18
40
1 34.1
0.06
30
0 48.0
0.03
87 00
0 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
0 47.2
0.03
88 00
0 02.0
0.017
40
7 43. 9
1.02
20
2 54. 0
0.17
20
1 31.8
0.06
30
0 46.3
0.03
89 00
0 01. 0
0.017
50
7 33.9
0.98
40
2 50. 7
0.16
40
1 30.6
0.06
52 00
0 45.5
0.03
90 00
0 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
CT
Temperature
CT
Temperature
CT
Temperature
CT
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
0 890
-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
0
-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
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
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
0
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
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°-<?) for stars observed at
upper or lower culmination respectively.
The signs of the factors are as follows :
A is plus except for stars between the zenith and the pole.
B is plus except for stars observed at lower culmination.
G is plus for stars at upper culmination and minus for stars at lower culmination, when
observations are made with the instrument in the position, band (clamp or illumination) west.
G is minus for stars at upper culmination and plus for stars at lower culmination when obser-
vations are made with the instrument in the position, band (clamp or illumination) east.
These factors are given to two decimal places in the tables on pages 62 to 77, and will be
found sufficiently accurate whenever the errors of adjustment, a, b, and c, are not allowed to
exceed one second of time. In 1874 this Survey published more extended tables, giving these
factors to three decimal places. Where, from any cause, observations are made with an instru-
mental error abnormally large it is desirable to take the corresponding star factors from the
more extended table or to compute them.
DETERMINATION OF TIME. 61
STAR FACTORS OBTAINED GRAPHICALLY.
For a number of years there has been in use in the Survey a nomogram for obtaining graph-
ically the star factors A, B, and C, and also K, the correction for diurnal aberration. This
nomogram was devised by Mr. C. R. Duvall, a computer in the Survey. It is not only more
expeditious than the tables, but the elimination of the double interpolation which the use of
the tables necessitates adds to the accuracy of the derived factor in many cases.
The nomogram is shown in illustration No. 9, reduced in size. It consists of two systems
of equidistant parallel lines perpendicular to each other, a system of arcs of equidistant concen-
tric circles, and a transparent arm, carrying a graduated straight line which revolves about the
common center of the circles. The decimeter has been the unit of length in the nomograms
used. The three systems of lines are drawn at a common distance apart of 1 centimeter. The
estimated tenth of this centimeter space gives the second decimal place in the required factors.
The graduated line on the under surface of the transparent arm passes through the center
of the axis about which the arm revolves. A secant graduation is made upon this line, measured
from the center of the axis of revolution. That is, the graduation corresponding to any angle
is at a distance from the center equal to the secant of the angle in question. This center of the
axis of revolution is the common center of the concentric circles and also the origin of the two
systems of parallel lines.
The graduations on the arm are for the declinations. In the nomograms used the gradua-
tions have not been carried beyond three decimeters from the center, which limits the use of
the instrument to declinations from 0° to slightly over 70°.
The zenith distances are graduated on one of the concentric circles at a convenient dis-
tance from the center. In the instrument shown in the illustration the distance is 25 centime-
ters. Since stars are never observed at zenith distances approaching 90°, the upper part of
the quadrant is not used.
To determine the factors A, B, and C of a given star, revolve the transparent arm until
the graduated line of the arm coincides with the star's zenith distance on the graduated arc.
Holding the arm in this position, place a needle point at that point of the graduated line which
corresponds to the star's declination. The position of this point in the three systems of equi-
distant lines gives the three factors, A being the ordinate, B the abscissa, and C the radius
vector.
The nomogram shown in the illustration is of thin bristol board pasted smoothly on thick
cardboard. The transparent arm is of celluloid one-sixteenth of an inch thick. The axis of
the arm is a solid metal cylinder with a«head which fits against the back of the cardboard.
The axis is made long so that the arm can be placed on it and revolved without being made
fast.
The correction for aberration may be taken from the same nomogram, as follows: Set the
revolving arm at that angle on the graduated circle which is equal to the latitude of the given
station. From the graduated line of the arm read off the decimation at each intersection with
a broken-line ordinate. These declinations are the limits between which « has the values
08.00, 08.01, 08.02, etc., for the latitude of the station in question. By means of these limits
the K of any star can be immediately written down from its declination. The broken-line ordi-
, . . .005 .015 .025
nates are drawn at distances from the origin equal to -TV>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
0°
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
0°
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.]
:
0°
10°
15°
20"
22°
24°
26°
28°
30°
32°
34°
36°
38°
40°
41°
42°
C
0
0
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
0
0°
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
0
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
0
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
0
0
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
0
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
0
65°
65j°
66°
66J°
67°
67j°
68°
esr
69°
69° 107
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'
71°4CK
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
70°10'
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
0
70° 10'
70°20'
70° 30'
70° 40'
70° 50'
71°
71° 10'
71° 20'
71° 30'
71° 40"
71° 5V
72°
72° 107
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'
73°20'
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
0
72-40'
72° 50'
73°
73° 10*
73° 20'
73° 30!
73° 40'
73-50'
74°
74°10/
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
0 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
0
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° SO1
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 method1 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 key1 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:
0
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 (T0) 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 Tl} and AT2 and ±r2 are the same quantities, respec-
tively, for the second set at epoch T2.
Computation of difference of longitude.
BETWEEN MIAMI AND KEY WEST, FLA.
T0
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
= 1°36'50".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
- 0
+ 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
- 0
+ 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^ V12 = ±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
JV
(from western
signals)
Key West
*J
Atlanta
TW
Key West
"M
Atlanta
*TW
Difference
J TE-J Tw
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 pl and p3 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 pt = — -2 in which ^ and r2 are the probable errors of the chronometer corrections.
r\~ r2
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 error1 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
Mw
-""
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
0 328
.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
0.8013, 0.8013, and 0.S012.
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 0.8009. 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 0.8008 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 0.8038 and the smallest was 0.8003. Four of the corrections
obtained were less than 0.S010 and seven were less than 0.8015. 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 ±0
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 ±0.810,
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 ±0.810-^ -JlOQ= ±0.8010.
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.S022, 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.S013. 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.S011. 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 18h 30m 088.00 on chronometer No. 194
(sidereal) = llh 52m 308.00 on chronometer No. 208 (mean time); and that llh 15m 35s. 50 on
chronometer No. 1510 (mean time) = 14h 48m 108.00 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 -lm 548.01, and of No. 1510 to Sitka mean tune was -6m 268.34. 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 3h.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 2h.64 by its face was -2m 38S.34, and at 4h.36
was — 2m 38S.25. Assuming it to run uniformly between these epochs, its correction was — 2m
388.30 at 3h.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
ep«h 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
—0 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
0 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 (iy2 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 2d 3d 4th 5th 6lh 7lh
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" Om 28'.44±0S.05
SUMMARY OF RESULTS FROM SEVEN ROUND TRIPS, STARTING FROM SITKA.
Chronometers,
M. T. or Sid.
l»t 2<i 3d 4'h 5«>> 6th 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 Ch 0"
Final mean AX
2SM8±0«.05
h m
= +0 00
28.46±0.05
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 Z1; 12, 13, . . . la 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,
1-13
1-1,
l-k
l-l«
1-17
(l-J,)»
(l-J,)'
(1-13?
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-ln)'
* 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 ±08.01, correspond-
ing to a probable error of about ±48 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 0 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 catalogue1 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 20m of right ascension. Any star whose
declination2 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 llh 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 /
0 /
0 /
f
0 /
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 ~ — and 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 <if
N-S*
Star
north
or
south
Setting
=J dif. of
N. P. D.'s
Turns
N. P. D.'s
o /
0 /
0 /
h ffi s
4327
4.5
16 55 22
7 49
N
12
4379
4.9
17 11 53
90 21
98 10
S
41 16
28
82 32
-13
-17
4441
5.9
17 28 13
61 32
S
10
4494
5.8
17 42 04
36 10
97 42
N
12 41
30
25 22
+15
-1-20
4623
5.1
18 13 22
25 38
N
24
4651
5.4
18 18 45
72 13
97 51
S
. 23 18
16
46 35
+ 6
+ 8
4669
5.9
18 22 26
60 13
S
20
4711
5.5
18 31 52
37 43
97 56
N
11 15
20
22 30
+ 1
+ 1
* N— S— a (search factor— sum of N. P. D.'s), where 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.
When a subpolar star is used slight changes will be necessary, similar to those described
for the case where the observing list is prepared in terms of the decimations.
Among the requisites for a pair of stars for an observing list, are, that their right ascensions
shall not differ by more than 20m, or 12b±20m when a subpolar is used, to avoid too great errors
arising from instability in the relative positions of different parts of the instrument; nor by
less than about lm, that interval being required to take the readings upon the first star and
prepare for the second star of a pair; that their difference of zenith distances shall not exceed
the half length of the micrometer comb, 20' for many instruments; that each star shall be
bright enough to be seen distinctly, not fainter than the seventh magnitude for the larger instru-
ments; and that no zenith distance shall exceed 45°, to guard against too great an uncertainty
in the refraction. The third of the above conditions may be used more converiently in this
form; the sum of the two declinations must not differ from twice the latitude by more than 20'.
The total range of the list in right ascension is governed by the hours of darkness on the pro-
posed dates of observation.
In the list of pairs resulting directly from the search there will be many pairs which overlap
in time. A feasible observing list may be formed by omitting such pairs that among the
remainder the shortest interval between the last star of one pair and the first star of the next
is not less than 2m. In that interval a rapid observer can finish the readings upon one pair and
set for the next, under favorable circumstances. The omitted pairs may be included in a list
prepared for the second or third night of observation. It will frequently be found that the
same star occurs in two or more different pairs. Such pairs may be treated like those which
overlap in time.1
DIRECTIONS FOR OBSERVING.
All adjustments having previously been made, set for the first star and await it with the
bubble of the latitude level nearly in the middle of the tube, and with the micrometer line at
that part of tho comb at which the star is expected, as shown by the observing list. Watch
the chronometer so as to know when to expect the star. When the star enters the field, place
the micrometer line approximately upon it. As soon as the star comes within the safe observing
limits of the field bisect it carefully. As the star moves along watch the bisection and correct
1 Past records furnish abundant evidence that observations made by pointing twice upon a close zenith star, once In each position of the
Instrument, give results of a low order of accuracy, probably because of the hurry with which the observations must bo made, and of the fact that
one or both of the observations must be made out of the meridian. It is therefore not advisable to make such observations.
110 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
it if any error is detected. Because of momentary changes in the refraction, the star will
usually be seen to move along the line with an irregular motion, now partly above it and now
partly below. The mean position of the star is to be covered by the line.1 It is possible, but
not advisable, to make several bisections of the star while it is passing across the field. As
soon as the star reaches the middle vertical line of the diaphragm read off promptly from the
comb the whole turns of the micrometer, read the level, and then the fraction of a micrometer
turn, in divisions, from the micrometer head. Set promptly for the next star, even though it
is not expected soon. In setting for the second star of a pair all that is necessary is to reverse
the instrument in azimuth and set the micrometer line to a new position. The abutting piece
must be brought gently against the stop and the circle securely clamped in that position.
Especial care should be taken in handling the micrometer screw, as any longitudinal force
applied to it produces a flexure of the telescope which tends to enter the result directly as an
error. The last motion of the micrometer head in making a bisection should always be in one
direction (preferably that in which the screw acts positively against its opposing spring), to insure
that any lost motion is always taken up in one direction. The bubble should be read promptly,
so as to give it as little time as possible to change its position after the bisection. The desired
reading is that at which it stood at the instant of bisection. Avoid carefully any heating of
the level by putting the reading lamp, warm breath, or face any nearer to it than necessary.
During the observation of a pair the tangent screw of the setting circle must not be touched,
for the angle between the telescope and the level must be kept constant. If it is necessary
to relevel, to keep the bubble witliin reading limits, use the tangent screw which changes the
inch' nation of the telescope. Even tliis may introduce an error, due to a change in the flexure
of the telescope, and should be avoided if possible. It is desirable to relevel the instrument
from time to time between pairs, so as to keep the level correction small, less than one division
of the level if possible.
Occasionally the approximate time should be noted at which the star being observed
crosses the middle vertical line of the diaphragm, so as to make sure that the adjustment of the
stops in azimuth remains satisfactory. It is desirable (though not necessary) to have a
recorder. He, should be a man above the average in intelligence, and should be able to pre-
pare an observing list after a little practice and to assist in computing the results. It is not
economical to take a man from place to place unless he can assist in the computations. The
recorder may count seconds aloud from the face of the chronometer in such a way as to indicate
when the star is to culminate. Such counting aloud serves a double purpose. It is a warning to
the observer to be ready and it indicates where to look for the star if it is faint and difficult to
find. It also gives for each star a rough check upon the position of the azimuth stops. It is
only a rough check, because the observing list gives mean right ascensions instead of apparent
right ascensions for the date, but it is sufficiently accurate (see p. 1 19). The observer, or recorder,
can easily make allowance for the fact that all stars (except circumpolars) will appear to be too
early or too late, according to the observing fist, by about the same interval, 0s to 5s, the differ-
ence between the mean and apparent right ascension. If a star can not be observed upon the
middle fine, on account of temporary interference by clouds or tardiness in preparing for the
observation, it may be observed anywhere witliin the safe limits of the field (often indicated
by vertical fines on the diaphragm) and the chronometer tune of observation recorded. In
practice a star is seldom observed off the meridian.
It is desirable to make all settings with such accuracy that the mean of the two micrometer
readings on a pair shall not differ from 20 turns by more than 1 turn. It is not infrequently
true that the value of a micrometer screw increases slightly but steadily from one end to the
other. In such cases the correction to each observed value of the latitude, due to this irregu-
larity of the screw, will be insensible if the settings are made with the indicated accuracy, but
not otherwise.
1 This wording must be modified to correspond if, in accordance with the considerations stated on p. 141, two close parallel lines are used
Instead of a single line.
Form 255.
DETERMINATION OF LATITUDE.
EXAMPLE OF RECORD AND COMPUTATIONS.
Zenith telescope record for latitude.
[Station, St. Anne. Date, June 25, 190S. Chronometer, 2637. Observer, W. Bowie.]
Ill
No. of
pair
Star
number
Boss
Cat.
N.or
S.
Micrometer
Level
Chronome-
ter time of
culmina-
tion
Chronome-
ter time of
observa-
tion
Meridian
distance
Remarks
Turns
Div's.
North
South
t
d
4327
N
11
69.0
6.0
39.1
(*)
16 55 24
(*)
+30f
9
67.8
99.5
+46
4379
S
27
34.4
40.2
7.2
17 11 47
Struck instrument
100.5
68.7
4441
S
9
61.0
40.3
7.2
17 28 07
10
101.2
69.4
+24
4494
N
31
47.0
7.1
40.4
17 41 58
69.4
101.3
4623
N
24
88.2
9.2
42.6
18 13 18
11
71.6
103.8
+ 16
4651
S
16
66.0
42.2
8.7
18 18 39
103.2
71.0
4669
S
19
62.5
44.2
10.9
18 22 20
12
106.0
73.8
+ 15
4711
N
20
55.4
11.2
44.7
18 31 45
Mean of double star
74.4
106.5
Form 32a.
* These columns are only used when a star is observed off the meridian.
t This is the continuous sum, up to this pair, of the south minus the north micrometer turns.
Reduction, mean to apparent declination, with Cape tables.
[Station, St. Anne triangulation latitude station.]
Order
1
2
Date
Star No.
June 25, 19C
4327
8
4379
4441
4494
4623
4651
4669
3
a.
16 55. 4
17 11.9
17 28. 2
17 42.1
18 13. 4
18 18. 7
18 22.4
7
11
4
G^r0
H+aa
So
18 33. 8
4 40.0
82 11
18 50. 3
4 56.5
-0 20
19 06. 6
5 12.8
28 28
19 20. 5
5 26.7
53 50
19 51. 8
5 58.0
64 21
19 57. 1
6 03.3
17 46
20 00.8
6 07.0
29 46
8
9
P'
P'x
+2.95
-2.49
+4.36
-3.68
+5.74
-4.84
+6.90
-5.82
+9.40
-7.92
+9.81
-8.27
+10. 08
- 8.50
12
13
<?'„
8Q'
+6.26
0.00
0.00
-0.03
+1.78
+0.03
+2.14
+0.02
+0.14
0.00
-0.08
0.00
- 0.27
0.00
14
15
Q'
Q'y
+6.26
+0.66
-0.03
0.00
+1.81
+0.19
+2.16
+0.23
+0.14
+0.01
-0.08
-0.01
- 0.27
- 0.03
5
10
16
17
18
*o(")
P'+P'x
Q'+Q'y
v-' V
23.30
+0.46
+6.92
+0.08
-.001 .00
-30. 91
+ 0.68
- 0.03
+ 0.58
-.059-. 03
24.93
+0.90
+2.00
+0.52
+. 024+. 01
23.54
+1.08
+2.39
+0. 35
-.035-. 02
57.64
+1.48
+0.15
+0.25
+ .029+.01
46.61
+1.54
-0.09
+0.56
+.007 .00
31.42
+ 1.58
- 0.30
+ 0.51
-.033 -.02
19
»(")
30.76
29.71
28.36
27.34
59.53
48.62
33.19
log ^=l
ffo
log g=0. 49813
log (70=1.30216
(l+z)=9. 19597
l+z=+0. 157
log h=l. 31041
log /!„=!. 26717
log ^=log (1+2,)=0. 04324
l+y=+1.105
h m
G= 1 38.4
H=U 44.6
j= +0.585
T= 0. 483
Make computation by horizontal lines in the order indicated. For explanation of (?„' and S Q', see pages (2) and (5)
of Cape tables. Opposite S0 in the sixth line place the degrees and minutes, and opposite <50 (") the seconds of the
mean declination. The quantities x, y, i, and T may be assumed constant fora night, and should be taken for an epoch
midway between the first and last stars. The quantities G and H may be assumed constant for periods not exceeding
four hours each, and should be taken for the midway epoch of each such period. Use aa, G, H.G-\-n0. and H-\-a0, to tenths
of minutes of time; x, y, and T to three significant figures; and all other quantities to two decimal places.
112
Form 33.
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
Latitude
[Station, St. Anne. State, Illinois.
Date
Catalogue Micrometer Level
Meridian
distance
Declination
Star No.
Xor
s
Reading
Diff. Z.D.
n
s
Diff.
190S.
t d
t d
d
d
d
S
o / //
June 25
4327
N
11 69.0
06.0
39. 1
82 .1 1 30. 76
+ 15 65.4
67.8
99.5
4379
S
27 34.4
40.2
07.2
+2.1
-0 20 29. 71
100.5
68.7
4441
S
9 61.0
40.3
07.2
28 28 28. 36
-21 86. 0
101.2
69.4
4494
N
31 47.0
07.1
40.4
-0.05
53 50 27. 34
69.4
101.3
4623
N
24 88. 2
09.2
42.6
64 21 59.53
- 8 22.2
71.6
103.8
4651
S
16 66.0
42.2
08.7
-1.05
17 46 48.62
103.2
71.0
4669
S
19 62. 5
44.2
10.9
29 46 33. 19
92.9
106.0
73.8
4711
N
20 55. 4
11.2
44.7
-0. 95
52 16 49.44
74.4
106.5
DETERMINATION OF LATITUDE.
113
computation,
Observer, W. Bowie. Instrument, zenith telescope No. 4.]
Sum and half sum
Corrections
Latitude
Remarks
Micrometer
Level
Refraction
Meridian
o / //
/ //
//
//
//
0 / //
81 51 01. 05
40 55 30. 52
+5 49.48
+0.78
+0.18
41 01 20. 96
Struck instrument
82 18 55. 70
41 09 27. 85
-8 08.02
-0.02
-0.14
41 01 19. 67
82 08 48. 15
41 04 24. 08
-3 03. 56
-0.39
-0.06
41 01 20. 07
82 03 22. 63
41 01 41.32
-0 20. 74
-0.35
-0.01
41 01 20. 22
Value of one division of latitude level : Upper — 1. 600
Lower -1.364
Mean -1.482
8136°— 13 8
114
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
Summary of latitude computation.
[St. Anne, 111., June 25, 1908.]
Star No.
Mic. diff.
*
u. —:"•
Corrected
— 2
Boss catalogue
M
41° 01'
6$ J0
*
jiji
3667
3729
+ 8.3
20.26
-0.03
0.00
-0.11
20.15
+0.09
0.01
*(2265)
3803
+ 0.1
19.77
+0.46
0.21
0.00
19.77
+0.47
0.22
3842
3856
-14.3
20.02
+0.21
0.04
+0.20
20.22
+0.02
0.00
3949
3979
+14.1
20.40
-0.17
0.03
-0.19
20.21
+0.03
0.00
4063
4072
- 9.1
20.24
-0.01
0.00
+0.12
20.36
-0.12
0.01
4081
4090
+13.8
20.54
-0.31
0.10
-0.19
20.35
-0. 11
0.01
4112
4129
+16.3
20.15
+0.08
0.01
-0.22
19.93
+0.31
0. 10
4161
4201
+ 2.3
19.80
+0.43
0.18
-0.03
19.77
+0.47
0.22
4327
4379
+15.7
20.96
-0.73
0.53
-0.22
20.74
-0.50
0.25
4441
4494
-21.9
19.67
+0.56
0.31
+0.30
19.97
+0.27
0.07
4623
4651
- 8.2
20.07
+0. 16
0.03
+0.11
20.18
+0.06
0.00
4669
4711
- 0.9
20.22
+0.01
0.00
+0.01
20.23
+0.01
0.00
4745
4758
+ 2.8
20.77
-0.54
0.29
-0.04
20.73
-0.49
0.24
*(3019)
4799
-16.4
20.53
-0.30
0.09
+0.22
20.75
-0.51
0.26
4824
4892
-16.1
20.06
+0.17
0.03
+0.22
20.28
-0.04
0.00
73 4
2 08
1 85
1 73
1 39
86 9
2 09
1 77
Algebraic sum
-13.5
-0.01
Mean
- 0.9
20.23
20 24
* 2285 and 3019 are ten-year 1880 numbers. The mean declinations for these stars were obtained from several sources.
O. 455X1.85 , n/, 9f-
^ -—
The value of one-half turn of the micrometer as used in the field = 22".325.
Mean <f>, 8 pairs with plus micrometer difference = 41° 01' 20".33.
Mean <f>, 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=±^/0455Xl139=±0.22
= 41° 01' 20".24±0".06
-0.03
+ 0.07
= 41° 01' 20".28±0".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, ep, is
computed from all the remaining values, and then 5ep is used as an absolute rejection limit,
and 3.5ep 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^.
er = probable error of r.= - —- — —
(p_2)jjfl>
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
<p=
and if JWand M' be the micrometer readings of the south and the north stars, increased microm-
eter readings corresponding to increased zenith distances, and R the value of one turn, then
The details of the computation of the second and third terms in the above formula are
sufficiently indicated in the computation shown above. The first, fourth, and fifth terms are
explained more fully on the following pages (117-119).
Tenths of divisions of the micrometer head are usually estimated.
COMPUTATION OF APPARENT PLACES.
The data given in the Boss preliminary general catalogue of stars for 1900 in regard to a
star, from which its apparent place at the time of observation is to be computed, are the mean
right ascension and declination, «m and 8m for the year 1900, tm; the annual variation in right
ascension, "Tr2; the annual variation in declination —5?, (the annual precession and proper
motion together constitute the annual variation) ; and the secular variation of the precession
d?d
in declination, given for 100 years, which, by moving the decimal point, becomes ~^jr- There
are also given the proper motion in declination, /*'; the mean epoch E; the probable error of
the declination at the mean epoch eaEp', efi/j the probable error of 100 //'; and the probable
error of the declination for 1910, es. The probable error of the declination for any date, T, is
The reduction to the apparent place at observation is made in two steps; first, the given
mean place is reduced to the mean place at the beginning of the year of observation, and upon
that as a basis the apparent place computation is then made.
Let the mean right ascension and declination at the beginning of the year of observation be
called a0 and 80
Then
§0 = §m + (to - tf +y2(t0- «j'
The Boss catalogue shows that for the star 4327, <Tm = rt',9oo = 16h 56m 12s, with an
annual variation ~j^ = -6".304. Also dm=dlMO =82° 12' 07".6G. The annual variation,
7<> s?2%
-jf = -5".510, the secular variation, -~= -".00880, the proper motion, // = -".001; the
mean epoch, E, =1875.5, and the probable error, esBp= ±0". 03; <v, the probable error of 100//'
= ±0".13, and the probable error of the declination for 1910= ±0".05.
i The correction for inclination as here given is for a level of which the graduation is numbered in both directions from the middle. If the
graduation is numbered continuously from one end to the other with numbers increasing toward the objective, the level correction is
(Compare this with the similar formula for a striding level on page 23.) If the numbering on the level graduation increases toward the eyepitcc this
formula becomes
DETERMINATION OF LATITUDE. 117
This star was observed for latitude in June, 1908, at St. Anne, 111., Oh 43m west of Washington.
n-0 = 16h 56m 12s -8 (68.304) = 16h 55m 22s, which is sufficiently close to the apparent right
ascension for use in connection with latitude observations.
£0 = 82° 12' 07".66 + 8[-5".510+K(8)(-".00880)]=82°ll'23".30. The probable error
of the declination for 1908 = V(0"-03)2 + { .325(0". 13) j- 2 = ± 0".05.
The apparent declination,1 d, at the instant of observation may now be computed by the
formula given on page 526 of the American Ephemeris for 1908, namely,
d= d0 + TfjL'+g cos (G + <x0) + h cos (H+a0)sin. d0 + icos d0,
in which g, G, h, H, and i are quantities called independent star numbers which are functions
of the tune only and are given in the Ephemeris (pp. 532 to 539, 1908) for every Washington
mean midnight during the year, r is the elapsed decimal fraction of the fictitious year and is
given in the Ephemeris with the independent star numbers.
This formula has been put in a more convenient form, conducive to more rapid compu-
tation, and adapted to the use of natural numbers and Crelle's Rechentafeln, in an appendix
to the Cape Meridian Observations, 1890-91, entitled "Star-Correction Tables," by W. H.
Finlay, M. A.
The formula is
in which /, P' , and Q' are tabulated in the Finlay tables.
P' = ga cos (G + a0) and is tabulated with respect to the argument G + a0 and can be obtained
from one opening of the tables for all stars and dates.
Q' = h0 cos (H+ <TO) sin d0 and is tabulated with respect to the arguments (H + <TO) and d0.
I = i cos £0 and is tabulated with respect to i and d0. Q' and 7 can be obtained from the
same opening of the tables for any given star and date, and all interpolations involve such
small tabular differences that they may be made mentally.
The values chosen for g0 and h0 are 20".0521 and 18".500, respectively, so that x is generally
negative and never greater numerically than unity, while y is always positive and never greater
than 0.11; thus the multiplications by x and y can be easily effected by Crelle's Rechentafeln.
x and y are functions of the time only, and with sufficient accuracy may usually be considered
constant for a single night.
If the period over which the observations extend on any night is not more than four hours
long, the quantities g, 7i, G, H, i, and r may be taken from the Ephemeris for the middle of the
observing period and assumed to be constant for the night. The errors from this assumption
will be small and of both algebraic signs.
The computation of the apparent places of seven stars observed at the St. Anne latitude
station is shown on page 111.
When a given star is observed on several nights in succession it is not necessary to compute
the apparent place for every night of observation. The apparent place may be computed
for certain nights at intervals of not more than three days and the declination for intermediate
nights may be obtained by interpolation.
CORRECTION FOR DIFFERENTIAL REFRACTION.
The difference of refraction for any pair of stars is so small that we may neglect the varia-
tion iii the state of the atmosphere at the time of the observation from that mean state supposed
in the refraction tables, except for stations at high altitudes. The refraction being nearly
proportional to the tangent of the zenith distance, the difference of refraction for the two stars
will be given by
r-r' = 57".7sin (z-z') sec2z,
1 In the comparatively rare cases in which it is n?eessary to compute the apparent right ascension of a star it may be done by the use of the
formula given in Finlay's tables.
118
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
and since the half difference of zenith distances, as measured by the micrometer, is the quantity
applied in the computation, the following table of corrections to the latitude for differential
refraction has been prepared with the argument one-half difference of zenith distance at the
side, and the argument zenith distance at the top.
If the station is so far above sea level that the mean barometric pressure at the station is
less than 90 per cent of the mean barometric pressure at sea level (760mtn) it may be desirable
to take this fact into account by diminishing the values given in the following table (computed
for sea level) to correspond to the reduced pressure. That is, if the mean pressure is 10 per
cent less than at sea level diminish each value taken from the table by 10 per cent of itself, if 20
per cent less diminish tabular values by 20 per cent, and so on. This need only be done roughly,
since the tabular values are small.
Correction to latitude for differential refraction =% (r — r').
[The sign of the correction is the same as that of the micrometer difference.]
One-half
diff.of zenith
distances
Zenith distance
0°
10"
20"
25°
30°
35°
40°
45°
0.0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.5
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.02
1.0
0.02
0.02
0.02
0.02
0.02
0.03
0.03
0.03
1.5
0.03
0.03
0.03
0.03
0.03
0.04
0.04
0.05
2.0
0.03
0.03
0.04
0.04
0.04
0.05
0.06
0.07
2.5
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.08
3.0
0.05
0.05
0.06
0.06
0.07
0.08
0.09
0.10
3.5
0.06
0.06
0.07
0.07
0.08
0.09
0.10
0.12
4.0
0.07
0.07
0.08
0.08
0.09
0.10
0.11
0.13
4.5
0.08
0.08
0.09
0.09
0.10
0.11
0.13
0.15
5.0
0.08
0.09
0.10
0.10
0.11
0.13
0.14
0.17
5.5
0.09
0.10
0.10
0.11
0.12
0.14
0.16
0.18
6.0
0.10
0.10
0.11
0.12
0.13
0.15
0.17
0.20
6.5
0.11
0.11
0.12
0.13
0.14
0.16
0.19
0.22
7.0
0.12
0.12
0.13
0.14
0.16
0.18
0.20
0.23
7.5
0.13
0.13
0.14
0.15
0.17
0.19
0.21
0.25
8.0
0.13
0.14
0.15
0.16
0.18
0.20
0.23
0.27
8.5
0.14
0.15
0.16
0.17
0.19
0.21
0.24
0. 29
9.0
0.15
0.16
0.17
0.18
0.20
0.23
0.26
0.30
9.5
0.16
0.16
0.18
0.19
0.21
0.24
0.27
0.32
10.0
0.17
0.17
0.19
0.20
0.22
0.25
0.29
0.34
10.5
0.18
0.18
0.20
0.21
0.23
0.26
0.30
0.35
11.0
0.18
0.19
0.21
0.22
0.25
0.28
0.31
0.37
11.5
0.19
0.20
0.22
0.23
0.26
0.29
0.33
0.39
12.0
0.20
0.21
0.23
0.25
0.27
0.30
0.34
0.40
12.5
0.21
0.22
0.24
0.26
0.28
0.31
0.36
0.42
13.0
0.22
0.22
0.25
0.27
0.29
0.33
0.37
0.44
13.5
0.23
0.23
0.26
0.28
0.30
0.34
0.39
0.45
14.0
0.23
0.24
0.27
0.29
0.31
0.35
0.40
0.47
14.5
0.24
0.25
0.28
0.30
0.32
0.36
0.41
0.49
15.0
0.25
0.26
0.29
0.31
0.34
0.38
0.43
0.50
15.5
0.26
0.27
0.29
0.32
0.35
0.39
0.44
0.52
16.0
0.27
0.28
0.30
0.33
0.36
0.40
0.46
0.54
16.5
0.28
0.29
0.31
0.34
0.37
0.41
0.47
0.55
17.0
0.29
0.29
0.32
0.35
0.38
0.43
0.49
0.57
17.5
0.29
0.30
0.33
0.36
0.39
0.44
0.50
0.59
18.0
0.30
0.31
0.34
0.37
0.40
0.45
0.51
0. 00
18.5
0.31
0. 32 0. 35
0.38
0.41
0.46
0.53
0.62
19.0
0.32
0. 33 0. 36
0.39
0.43
0.48
0.54
0.64
19.5
0.33
0. 34 0. 37
0.40
0.44
0.49
0.56
0.65
20.0
0.34
0. 35 ! 0. 38
0.41
0.45
0.50
0.57
0.67
DETERMINATION OF LATITUDE.
REDUCTION TO THE MERIDIAN.
119
If a star is observed off the meridian while the line of collimation of the telescope remains in
the meridian, the measured zenith distance is in error on account of the curvature of the
apparent path of the star. Let m be the correction to reduce the measured zenith distance to
what it would have been if the star had been observed upon the meridian.
Then,
in which T is the hour-angle of the star. The signs are such that the correction to the latitude
( = -Q) is always plus for the stars of positive declination and minus for stars of negative decli-
nation (below the equator), regardless of whether the star is to the northward or to the southward of
Tfk 77?
the zenith. ^~ or -^- is, then, always applied as a correction to the latitude, with the sign of the
right-hand member of the above equation. For a subpolar 180°— d must be substituted for d,
making the correction negative in this case just as for stars of southern declination. The follow-
ing table gives the corrections to the latitude computed from the above formula. If both stars
of a pair are observed off the meridian, two such corrections must be applied to the computed
latitude.
Correction to latitude for reduction to meridian.
[Star off the meridian but instrument in the meridian. The sign of the correction to the latitude is positive except for stars south of the equator
and subpolars.]
I
10-
15"
20-
22-
24*
26-
28-
30-
32"
34-
36«
3S>
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
:<s
.01
.03
.05
.06
.08
.09
.10
.12
.13
.15
.17
.19
.21
.23
.26
.28
.30
.33
.36
.39
.41
.44
.48
52
40
.01
.03
.05
.07
.08
.09
.11
.12
.14
.16
.17
.19
.21
.24
.26
.28
.31
.34
.36
.39
.42
.45
.48
5C
45
.01
.03
.05
.07
.08
.09
.11
.12
.14
.16
.18
.20
.22
.24
.26
.29
.31
.34
.37
.40
.43
.46
.49
45
The catalogues now available contain so many stars which may be observed for latitude
that it is not desirable to move the instrument out of the meridian to observe a star which is
missed as it crosses the meridian.
COMBINATION OF RESULTS, EACH PAIR OBSERVED MORE THAN ONCE.
Separate values of the latitude being computed from each observation upon each pair,
it remains to combine these in such a way as to obtain the most probable value of the latitude
and to obtain certain probable errors.
120 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
Let p be the total number of pairs observed. Let the number of observations upon pair
No. 1 be 7i,, upon pair No. 2, n2, and so on, and let the total number of observations at the sta-
tion be 710 = 711 + 712 + 72.3 . . . Let A be a residual obtained by subtracting the result from
a single observation on a certain pair from the mean result from all the observations upon that
pair. Let e be the probable error of a single observation of the latitude, excluding the error
arising from defective adopted declinations.
The various values of J depend upon and are a measure of the probable error of observation,
but are independent of the errors of the adopted declinations. According to the principles of
least squares,
0.455JJ2 0.4552"J2
e' =
No. obs. — No. unknowns
Let g>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 <px, <p2, <p3 . . . There will be p such residuals, and they are a meas-
ure of the probable error of the mean result from a pair, which will be called ep, arising from
both errors of observation and errors of declination.
, 0.455 Iv2
a* —
~
p-1
Let epl, ep2 . . . be the probable errors, respectively, of g>lt <p.f, <ps . . . Let e«
be the probable error of the mean of two decimations. Then
e i —
These various values e2pl, e2^, . . . differ from each other because of the various
values of %, n2, . . . even though e2^ and e2 are assumed to be constant, and the value
derived above for e*p is their mean value. Adding these various equations, p in number, and
taking the mean, member by member, there is obtained
e2 e2 e2
p
Placing
e
gfi+i+i
PL»i "» «i J
rfl 1 1 "1
- — -| — +— =£2
p\_n,n,n3 J
to abbreviate the notation, and solving for e2« there is obtained
Having determined the values of ez» and e2, the proper relative weights, wlt w2, inversely
proportional to the squares of their probable errors, may now be assigned to <plt 9>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
(e2 \~l
2e2- + -JT- ) in which Na is the number of nights' observations upon it.
The most probable value <p0 for the latitude of the station is the weighted mean of the
mean results from the various pairs, or
_Wi<
The probable error of <p0 is
"
>-l)Iw
in which A<p is the residual obtained by subtracting 9>,, 9>2, <p3 . in turn from <p0.
A concrete illustration of the processes indicated by the above formulas is furnished by
the following reproduction of certain parts of the computation of the latitude of the New Naval
Observatory from observations made in 1897 with a zenith telescope.
1 This may be made evident as follows: Let a\ and as be respectively the declination plus the measured zenith distance of a first and second
south star, and 03 the declination minus the measured zenith distance of a north star observed in combination with them. Let the probable errors
of QI, a», aabeei, ei, e$, respectively. Note that ei, €3, e s each include errors both of declination and observation. If the two component pairs are com-
puted separately and the mean taken, the result is of the form f-^~^+"^^SU— •y+'j+'f and its probable error squared is f-j J + (^) + ("if)'
Assuming that fi— fj— fs, this becomes fai1, the square of the probable error of the mean result from the combination. By the same reasoning li
may be shown that the square of the probable error of the result from a single independent pair is (-rrj + (~^) =id2- The weights to be assigned
to the combination and to an independent pair are then in the ratio of (|fi!)— ' and ( jci1)— ', or of j to 1 . If the weight for an independent pair is unity
the weight of each component of a doublet is therefore two-thirds.
122
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
Pairs
<t>
Star Nos.
38° 55'+
(2058)
09.81
-.01
.00
4440
09.80
.00
.00
09.80
4513
08.07
-.07
.00
4550
07.92
+.08
.01
08.00
4513
08.12
.00
.00
4555
08.13
-.01
.00
08.12
4526
09.31
-.34
.12
4550
08.40
+.57
.32
08.80
+.17
.03
09.44
-.47
22
09.01
-.04
.00
08.85
+.12
.01
08.97
4526
09.36
-.26
.07
4555
08.62
+.48
.23
09.51
-.41
. 17
08.91
+.19
.04
09. 12
-.02
.00
09.11
-.01
.00
09.10
Sum
6 69
Pair, Star Nos.
B. A.C.
*
V
D«
n
w
v~<;>
i$
j<t>
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) = 0.
Latitude = 38° 55' 08".81±0".06.
In computing the values of w<£, 38° 55' 08".00 was first dropped from each value of <f>.
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 Na 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, e2^=e2p — e2) 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
e2^ = e2p — s2 is sometimes found to give a value for e^ which is so small as to be evidently erro-
neous, and sometimes e2^ is even negative, which is an absurdity. On the other hand, when-
2ez
ever the value e2^ = -^fis smaller than e2^ = e2 — s2p, 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« = e2p— s2 should be used in the
weighting.
The following method of weighting, therefore, seems to be the best for use in the office
(e2 \~'
e\ 4. — ) , use for each pair the larger
Wn/
Ie2
of the two available values of e2^, namely, e2% = — j-* and e2^ = e2J> — s2. 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 e2** from the mean place computation are greater than e2» =
e2p — s2 one may first note on the list of mean places for what stars e2t exceeds 2 (e2p — s2). 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 e2» exceeding 0.074. The stars involved in these four pairs were 4526,
4550, 4555, (2350), 5026, [1259], (2365), and the corresponding values of e2t were 0.37, 0.08, 0.10,
2e2
0.18, 0.24, 0.08, 0.73. The weights assigned to these four pairs therefore depend upon e2f = — 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 ep 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 ep and e<t> 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* = e2p — «-»?, 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 e2 —— 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 — 4T±tE, the plus sign being used for western elonga-
tion and the minus for eastern elongation. tK 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 <x = 23h 27™ 458.6 and 5 = 86° 44'
13". 4. The chronometer correction at the time of the observations was known to be + 28.3.
Whence the chronometer time of eastern elongation was computed to be 17h 38m 168.5 and the
zenith distance 51° 00'. 5.
126
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
Computation of value of micrometer.
Station New Naval Observatory, Washington, D. C. Observer, O. B. F. Star, B. A. C. 8213 E. E. Date, June 18, 1897. Instrument, Zenith
telescope, No. 4.]
Mi-
crome-
ter
read-
ing
Chronom-
eter time
of observa-
tion
Level
Time
from
elonga-
tion
Reduc-
tion to
mean
state o
level
Corrections
Reduced
time
Time at 20
turns
J
J'
n
s
Time
Level
t
Am s
d
d
m
t
s
s
A m s
Am s
s
s
35
17 15 08.5
|13. 1
168.4
39.9
101.9
)„
—0.10
+2.3
-0.1
17 15 10.7
17 28 10.7
+4.7
-0.3
34
16 02.0
22.2
+2.1
-0.1
16 Ot.O
12.0
+3.4
-1.2
33
16 53.5
21.4
+ 1.9
-0.1
16 55.3
11.3
+ 4.1
-0.2
32
17 45. 0
T13.2
168.6
40.0
102.0
I 20.5
+0.15
+ 1.6
+0.1
17 46. 7
10.7
+4.7
+0.7
31
18 37.5
19.6
+ 1.4
+0.1
18 39. 0
11.0
+4.4
+0.7
30
19 30. 0
18.8
+ 1.3
+0.1
19 31.4
11.4
+4.0
+ 0.7
29
20 22.5
17.9
+ 1.1
+0.1
20 23.7
11.7
+3.7
+0.7
28
21 16.0
17.0
+0.9
+0.1
21 17.0
13.0
+2.4
-0.3
27
22 07.5
16.1
+0.8
+0.1
22 08.4
12.4
+3.0
+0.7
26
23 00.0
15.2
+0.7
+0.1
23 00.8
12.8
+ 2.6
+0.6
25
23 53.0
14.4
+ 0.6
+0.1
23 53.7
13.7
+ 1.7
0.0
24
24 45.5
13.5
+0.5
+0.1
24 46. 1
14.1
+ 1.3
0.0
23
25 37.5
12.6
+0.4
+0.1
25 38.0
14.0
+ 1.4
+ 0.4
22
26 30.5
11.8
+0.3
+ 0.1
26 30. 9
14.9
+0.5
-0.2
21
2V 23.0
[13.2
[68.3
40.0
101.8
' 10.9
-0.10
+0.2
-0.1
27 23.1
15.1
+0.3
0.0
20
28 16.0
10.0
+0.2
-0.1
28 16.1
16.1
-0.7
-0.7
'13.2
40. 0
19
29 08.0
9.1
+0.15
+0.1
+0.1
29 08. 2
16.2
-0.8
-0.5
k68. 5
102.1
18
3000.5
8.2
+0.1
+0.1
30 00.7
16.7
-1.3
-0.6
17
30 53. 0
7.4
+0.1
+0.1
30 53. 2
17.2
-1.8
-0.8
16
31 44.5
6.5
+0.1
+0.1
31 44. 7
16.7
-1.3
0.0
15
32 37. 0
5.7
0.0
+0.1
32 37. 1
17.1
-1.7
0.0
14
33 29.5
4.8
0.0
+0.1
33 29.6
17.6
-2.2
-0.2
13.2
40. 1
13
34 22.0
68.3
101.9
3.9
0.00
0.0
0.0
34 22.0
18.0
-2.6
-0.3
12
35 14.0
3.0
0.0
0.0
35 14.0
18.0
-2.6
+ 0.1
11
36 06.5
2.2
0.0
0.0
36 08.5
18.5
-3.1
-0.1
10
36 58. <i
1.3
0.0
0.0
36 58. 5
18.5
-3.1
+0.2
9
37 50.5
0.4
0.0
0.0
3750.5
18.5
-3.1
+ 0.6
8
38 43.5
-0.4
0.0
0.0
38 43. 5
19.5
-4.1
-0.1
7
39 35.5
-1.3
0.0
0.0
39 35. 5
19.5
-4.1
+0.2
6
40 28.0
-2.2
0.0
0.0
40 28. 0
20.0
-4.6
0.0
5
41 19.5
-3.0
0.0
0.0
41 19.5
19.5
-4.1
+0.9
Mean
17 28 15.4
Assumed value of Rl = 52s.
2480 r,
r,
log 15
log cos d
= + 820.3
= + 0.3308
528.3308
= 1.7187573
1.1760913
= 8.7552522
1.6501008
44".679
Corr. for refraction — 0 .030
One turn 44".649
For explanation of notation, see page 128.
DETERMINATION OF LATITUDE.
127
Because of the curvature of the apparent path of the star its rate of change of zenith distance
is not constant, even near elongation. The rate of change at elongation may readily be com-
puted. It is at that instant in seconds of arc 15 cos d per second of sidereal time. The table
of curvature corrections given below enables one to correct the observed times to what they
would have been if in the place of the actual star there were substituted an ideal star whose
motion was vertical at a constant rate 15 cos d and which coincided with the actual star at
the instant of elongation.
Correction for curvature of apparent path of star, in computation of micrometer value.
[The correction tabulated is - (15 sin I")2 13— y™ (15 sin I")1 t5 in which t is the time from elongation. Apply the corrections given in the
table to the observed chronometer times, adding them before either elongation, and subtracting them after either elongation.]
T
Corr.
T
Corr.
T
Corr.
T
Corr.
T
Corr.
T
Corr.
m
»
m
s
m
s
m
s
m
«
m
»
6
0.0
16
0.8
26
3.3
36
8.9
46
18.5
56
33.3
7
0.1
17
0.9
27
3.7
37
9.6
47
19.7
57
35.1
8
0.1
18
1.1
28
4.2
38
10.4
48
21.0
58
37.0
9
0.1
19
1.3
29
4.6
39
11.3
49
22.3
59
39.0
10
0.2
20
1.5
30
5.1
40
12.2
50
23.7
60
41.0
11
0.2
21
1.8
31
5.7
41
13.1
51
25.2
61
43.1
12
0.3
22
2.0
32
6.2
42
14.1
52
26.7
62
45.2
13
0.4
23
2.3
33
6.8
43
15.1
53
28.3
63
47.4
14
0.5
24
2.6
34
7.5
44
16.2
54
29.9
64
49.7
15
0.6
25
3.0
35
8.2
45
17.3
55
31.6
65
52.1
In the computation the fifth column shows the values of T, and the seventh column the
resulting curvature corrections.
When the reading of the level changes, it indicates, upon the usual assumption that the
relation between the level vial and the telescope remains constant, that the inclination of the
telescope has changed. The effect of the movement of the telescope may be eliminated in the
computation by applying to each observed time the correction in seconds of time,
± {(»-«)-(»'-*')}
30 cos d
to reduce it to what it would have been if the readings of the north and south end of the bubble
had been n' and s', respectively.
If, as in the present case, the level graduation is numbered continuously from one end to
the other with the numbers increasing toward the eye end, instead of being numbered in both
directions from the middle, the required correction becomes
d
'30 cos d
In each of these formula? the plus sign is to be used for western elongation and the minus
sign for eastern elongation. It is convenient to take for the assumed n' and s' the actual
readings at some one moment during the set of observations.
Zenith telescope No. 4 had two latitude levels, and the correction was computed by taking
the mean of the two and using the mean value of d (= 1".482). The sixth column shows the
mean values of (n' + s') — (n + s) and the eighth column the resulting corrections, the factor
on d . being 0. 87.
30 cos d
Let RI be an assumed approximate value of one turn in time and let rl be a required cor-
rection to .Bj. Let jT0 be an approximate value of the chronometer time of transit of the star
across the micrometer line set at 20 turns (the middle of the screw) and t0 a required correction
to T0. Then, upon the assumption that the screw has a uniform value throughout the part
128 U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
observed upon and that the star moves in the direction of increasing readings (western elongation) ,
for each observed time an observation equation may be written of the form
t+(20-R0] (R1 + r1}-(Tl,+t,}=0
in which t is the observed time of transit across the line set at the reading /?„ after correction
for curvature and level. After transposition this may be written
(20-/?0)r1-^0 = J
in which
J-T0-p+(20-12(,)JBJ
whence the normal equations become
1(20- R.V-r, -2(20- R0}t0 = 2(20- R0}d
= - IJ.
If the turns observed upon are symmetrical about 20, 1(20 — R0) becomes zero. If, more-
over, as in the numerical case here shown, T0 is purposely taken equal to the mean value of
t+ (20 — R0)Rlt 2 A is zero and t0 derived from the second normal equation is necessarily zero.
Also the first normal equation reduces to the working form
If the star is observed at eastern elongation it moves in the direction indicated by decreasing
micrometer readings and throughout the preceding formulae R,, — 20 must be substituted for
20 -R0.
In the computation form printed above, the values of t + (R0 — 20) R{ are shown in the column
headed "Time at 20 turns," Rl being assumed = 52s. T0 was assumed= 17h 28m 15.84, the mean
for this column, and the J's written accordingly.
The equation 2(R9-20)2r1 = 2(R9-20)J reduces numerically to 2480r11 = 820.3.
A' is the residual obtained by substituting the derived value rt in each observation equation,
or J'-J-(B0-20)rt.
The remainder of the computation needs no explanation except that the correction for refrac-
tion to be applied to the value of one turn is the change of refraction for a change of zenith
distance equal to one turn, or in the most convenient form for use, it is the value of one turn in
minutes of arc times the difference of refraction for 1' at the altitude at which the star was
observed (approximately =<j&). The difference of refraction for 1' may be obtained from any
table of mean refractions with sufficient accuracy. The correction for refraction is always
negative, since the change of refraction is always such as to make a star appear to move slower
than it really does.
It will sometimes be necessary to apply a correction for rate. This correction, to be applied
to the computed value of one turn, is in seconds of arc
(rate of chronometer in seconds per day) (value of one turn in seconds of arc)
86400"
The correction is negative if the chronometer runs too fast.
The micrometer value is sometimes determined by turning the micrometer box 90° and
observing upon a close circumpolar near culmination. There are two serious objections to this
' In this computation it becomes necessary to find the sum ol the series l»+2*+3!+4* .... +15*. It is convenient for this purpose to
use the ,'ormula ls+21+3'+4! . . . +i*— 3 + 2+5- Occasionally in least square computations it becomes necessary to compute the sum of a
similar series of fourth powers. One may then use the formula l'+2<+3'+4< . . +i>_++_£. 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<irt for small movements in that direction. But when the observations are made at culmination
the instrument is depended upon to remain fixed in azimuth, the direction in which, because of
its peculiar design, it is weakest, and there is no check upon changes in azimuth corresponding
to the level readings. Hence, it is not advisable to observe for micrometer value at culmination.
The only modifications in the computations are that there are no corrections for level or
refraction, and that in computing the curvature correction r is now the hour-angle. The
curvature correction is additive before either culmination, and subtractive after it.
It is decidedly questionable whether it is advisable to determine the mean value of the
micrometer screw by observations upon close circumpolars either at culmination or elongation.
Such observations consume a great deal of time both in observation and in the subsequent
computation, and experience shows that they are subject to unexpectedly large and unexplained
errors. For example, during the observations for variation of latitude at Waikiki, Hawaiian
Islands, in 1891-92, the micrometer value was thus determined twelve times. The results
show a range of about 0".13 or one three-hundred-and-thirtieth of the mean value, corresponding
to a range of about 3.3 millimeters in the distance between the objective and the micrometer
line, though the draw tube was kept clamped continuously, and the range of temperature during
the entire year was only about 11° C. (Coast and Geodetic Survey Keport, 1892, Part II, p. 61.)
Similaily, sixteen determinations of the value of a micrometer used at fifteen stations on the
Mexican Boundary Survey of 1892-93 showed a range of 0".33 or one one-hundred-and-ninetieth
of the mean value.1 In this case the draw tube was undamped and the telescope refocused
at the beginning of the observations at each station. The observed value was apparently not a
function of the temperature. The San Francisco series of observations for variation of latitude
also show a similar large range in the observed micrometer value (viz: 0".17). (Coast and Geo-
detic Survey Report, 1893, Part II, p. 447.) In general, whenever the micrometer value is
determined repeatedly by the circumpolar method so large a range of results is developed as to
force one to suspect that large constant errors are inherent in this method of observation. It
can. hardly be urged that the differences between the results represent actual changes in the
micrometer value, for such differences are developed even when successive determinations are
made during a single evening. Moreover, whenever the mean micrometer value is determined
from the latitude observations themselves it is frequently found to differ radically from that
derived from circumpolar observations on the same nights. So marked and so frequent has the
latter form of disagreement been, that many of the office latitude computations have actually
been made during the last few years by rejecting the micrometer value from circumpolar observa-
tions, when there is a marked difference between it and the value computed from the latitude
observations as indicated below, and using the latter value in the latitude computation.
DETERMINATION OF MICROMETER VALUE FROM LATITUDE OBSERVATIONS.
After considering the above facts and conclusions this Survey decided to adopt the method
of computing the micrometer value from the latitude observations, and since the beginning of
the year 1905 no observations have been made on close circumpolar stars for that purpose.
The total range in the values of one turn of the micrometer screw of zenith telescope No.
2, as determined from the latitude observations for 36 of the 63 stations established by Assistant
W. H. Burger, from 1905 to 1908, is 0".17. This is one two hundred and seventy-third of the
mean value.
As to the accuracy of the micrometer value determin'ed from the latitude observations,
it may be noted that if it be assumed that the probable error of a single observation of latitude
1 Report of the International Boundary Commission, United States and Mexico, 1891-1896 (Washington, 1898), p. 103.
8136°— 13 9
130 U. S. COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 14.
is ±0".40, of the mean of two declinations is ±0".16 (see p. 133) and of the latitude as
derived from independent pairs is ±0".10, the probable error of the micrometer value, as
determined from a single observation upon a pair having a difference of zenith distance of ten
turns would be
'.40)2 + 4(0.16)2+(0.10)2 = ±0".05.
There can be little doubt, therefore, that the mean micrometer value determined from
all the latitude observations at a station is more accurate than that determined from even
three or four sets of circumpolar observations each requiring an hour or more of time.
It has been urged that to determine an instrumental constant from the observations in
the computation of which it is to be used is a questionable procedure; that it "smooths out"
the results, but probably does not give real accuracy. The force of this objection disappears
when one contrasts the proposed practice of deriving a single instrumental constant from ob-
servations on twelve or more pairs with the usual and unquestioned practice in transit time
computations of deriving three instrumental constants (two azimuth and one collimation con-
stant) from only ten to twelve observations on as many stars.
It should be noted that the form of the computation of circumpolar micrometer obser-
vations given on page 126 is especially adapted to the detection of irregularities and periodic
errors, as they will at once become evident from an inspection of the values of J'. One com-
mon form of irregularity in screws is a continuous increase in the value from one end to the
other, in which case J' tends to have the same sign at the two ends of the set and the opposite
sign in the middle.
To derive the mean micrometer value from the latitude observations let M^ be the differ-
ence, in turns, of the micrometer readings on the two stars of a pair, taken with the same sign
as in the latitude computation, let r, be the required correction to the assumed value of one-half
turn with which the computation of the latitude was made, let p be the number of pairs, and
let c be the correction to the mean latitude <f>0. Let J<£ have the same meaning as before,
viz, 0o~0u <f>o~<j>2> etc. (See computation on p. 114.) For each pair an observation equation
of the form c — M^r^ + A<j> = 0 may be written. The resulting normal equations, from which rl
may be derived, are
— 2 Jf,c + ^ M2!?1! — I M^(j) = 0
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 <£„ <j>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<f>
in which J<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.
0
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 ep 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 ep 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
0
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 (<p) be slightly in error by the quantities
dt, dd, and dtp, and let dA equal their effect upon the azimuth (A) ; then, in general, it will be
seen that, all other circumstances being equal, dA increases as the zenith distance (£) decreases;
for a star near the pole and for a latitude not too high a small error in time and in latitude has
but a slight effect upon the azimuth, and in the case of a circumpolar star at elongation (when
the parallactic angle is 90°) a small error in time, dt, will not affect the azimuth; but small
errors in declination, dd, and in latitude, d<p, then attain nearly their maximum effect upon the
azimuth. If observations are made upon a circumpolar star (8>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<p
disappear in mean results of observations of equal and opposite azimuths. In observations
on a circumpolar star in the meridian the effect of a small error in time and in right ascension
may be eliminated by a combination of results from upper and lower culminations; for a star
in the meridian the quantities dd and d<p do not enter in the azimuth. If the object to be
observed, star (or sun), is of great polar distance (also d< <f>), 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 <f> 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 <p tan d — sin <p cos t
in which A is the azimuth of the star counted from the north in a clockwise direction,3 and
the hour angle t is counted westward from upper culmination continuously to 24h, or 360°, at
the next upper culmination. This is the most convenient formula for use with either of the
first three methods. The first term of the denominator changes very slowly and may be tabu-
lated for slightly different values of d during the period of observation. The second term, for
a close circumpolar star, may be computed with sufficient accuracy by five-place logarithms.
The computation of the azimuth from this formula may be considerably shortened by
transforming it as indicated below and using the table given on pages 165-173: 4
sin t
tan A = —
cos <p tan d — sin <p cos t
cot d sec <p sin t
1 — cot d tan (p cos t
= — cot d sec <p sin 1 1 ^ • J
in which a = cot d tan <p cos t.
The second form of this formula is about as convenient as the first. It involves the same
number of logarithms as the first and one less reduction from logarithms to numbers.
The third form in connection with the tables given on pages 165-173 gives a much quicker
computation process than either of the other two. In using this form and the tables, log cot
3 sec cp sin t must be carried to six places and log cot d tan <p cos t to five places. The most con-
venient arrangement of the computation is shown on page 148. The formula and tables involve
no approximations, and the only errors resulting from their use are those arising from the cast-off
decimal places (logarithms limited to six places). These errors are of the accidental class, and
i The method of determining azimuth by observations upon the sun at any hour angle is not treated in this publication, because it is used
mainly in making observations for magnetic declinations and a description of it, with tables for making the parallax and refraction corre rtions, is
given in "Principal Facts of the Earth's Magnetism" published in 1909, and also in " Directions for Magnetic Measurements" published in 1911,
both issued by the Coast and Geodetic Survey.
1 In this publication the hour angle will be reckoned westward from zero at upper culmination (increasing with the lapse of time) to 360° or 24\
» In astronomic computations it is more convenient to count the azimuth from the north instead of from the south, as in geodetic computa-
tions. If the direction of the count is clockwise, as here stated, to change from one reckoning to the other it is only necessary to add or subtract
180°.
« The formula and the table are both copied from Formiln und Hulfsta/dn fiir Geographische Ortabestimmunyen von Prof. Dr. Th. Albrecht,
Leipzig, 1894. The range of the table has, however, been considerably extended.
144 U. S. COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 14.
will seldom exceed 0''.04 for any case covered by the table, and for most observations made
below latitude 50° the error will not exceed 0".01. These quantities are so small in comparison
with the errors of observation as to be negligible. A few observations made in Alaska may be
beyond the range of the tables on pages 165-173, and when that is found to be the case, one
may easily substitute the second formula on page .143 for the third.1
To compute the azimuth of a star at the time of each pointing made upon it during a set
of observations is an unnecessarily laborious process. If for the hour angle, t, of the azimuth
formula is taken the mean of the hour angles of the set, the computed azimuth is that corre-
sponding to the mean hour angle, but is not the required mean of the azimuths corresponding to the
separate hour angles, since the rate of change of the azimuth is continually varying because of
the curvature of the apparent path of the star. The difference between the two quantities indi-
cated by the italics is small, though not usually negligible, for the interval of time covered by a
set of observations. The most convenient way of making the computation for a set of observa-
tions is to use the mean hour angle in the azimuth formula and apply to the result a
. 1 _2 sin2 i T
Curvature Correction = tan A-Z
n sin 1"
in which n is the number of pointings upon the star in the set and r for each observation is the
difference 2 between the time of that observation and the mean of the times for the set. The
sign of this curvature correction is always such as to decrease numerically the azimuth reckoned
from the north, or in other words, if azimuths are counted clockwise its algebraic sign will be +
when the star is west of north and — when the star is east of north. If the star crosses the
meridian during the progress of a set the curvature correction will ordinarily be zero. The
formula is approximate, but for circumpolars and for the interval of time usually covered by
2 sin2 \ T
a set of observations its errors are negligible. The value of the term— — -TTJ — may be found
sm i
on pages 151-152 of this publication.3
If the star observed is Polaris, a convenient rough check on the computation may be
obtained from Table V of the American Ephemeris and Nautical Almanac, entitled Azimuth of
Polaris at all Hour Angles.
Because of the rapid motion of the observer, due to the rotation of the earth on its axis,
a star is seen slightly displaced from its real position. The required
Correction for Diurnal Aberration = 0". 32 °°S A COS *
cos h
The sign of the correction is always positive when applied to azimuths counted clockwise.
The greatest variation of the correction from its mean value, 0".32, for the four circumpolars
ordinarily observed and for latitudes not greater than 50°, is 0".02. The correction for diurnal
aberration need not be applied to the separate sets but simply to the mean result for a station.
If the horizontal axis is inclined when the pointings are made upon either the star or the
mark the corrections indicated below must be applied.
Level Correction = - \(w + w') — (e + e') tan h
if the striding level carries a graduation numbered in both directions from the middle, d is
the value of one division of the level and w, e and w', e' are the west and east readings of the
1 Various other formulas for computing the azimuth of circumpolar stars have been proposed and used. Each of them requires either the same
or a greater time for the computation than that here given, when the whole computation, including the preparation of the auxiliary tables required
with some of them, is taken into account. As uniformity of practice is conducive to rapid computation, it is considered desirable that all should
use the formula; given, and therefore no others are here stated. It should be noted that the formula given is accurate and general; that is, it
applies to any of the close circumpolars at any hour angle.
> 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 — er) 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
0
,
,,
„
„
1
Morrison
8 19
D
A
B
0
0
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
0
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
0
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
0 18 31.4
4° 37' 51". 0
88 49 27. 4
2
2 01 33. 0
4 37. 5
1 56 55. 5
1 26 41. 9
0 30 13. 6
7° 33' 24". 0
3
2 16 31.0
4 37.4
2 11 53. 6
1 26 41. 8
0 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 <j>
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 0
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
0 06 50. 8
m s
2 30
0
7. 511124
0 11 09.2
m s
2 00
0
7. 684220
0 16 36. 9
m s
3 18
0
7. 882866
0 26 15.0
m s
1 38
0
Altitude of Polaris=ft
1 tan A=level factor
0 / //
33 46
0.701
0 / //
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. 0
-1.8
-1.3
116 45 48. 6
Corrected reading on Polaris
Circle reads on mark
252 01 24. 7
170 14 57. 0
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
0 06 50. 8
180 00 00.0
278 17 52. 0
0 11 09. 2
180 00 00. 0
278 23 20. 3
0 16 36. 9
180 00 00. 0
278 32 58. 1
0 26 15. 0
180 00 00. 0
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
/ Iv2
\ n(n —
1)
±0".31
The mean observed azimuth
98° 06' 42".26±0".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 <f> 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 sin2 -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»
6m
7"
o /
//
ff
//
//
//
//
0 10
.0
.0
.0
.0
.1
.1
0 20
.0
.0
.0
.1
.1
.1
0 30
.0
.0
.1
.1
.2
.2
0 40
.0
.1
.1
.1
.2
.3
0 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 sin2 ^ T
sin 1"
151
T
0»
1-
2m
3"
4m
5m
6m
~m
8»
I
„
„
„
„
H
„
„
It
n
0
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 sin2 Yi T
sin l'~
T
9m
10"'
Urn
12m
13"
14'"
15"
16"
a
„
„
„
„
„
,,
„
„
0
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
0
178
03
22.5
20
21.2
Star
14h 46m 30'
1
4. 5 10. 7
9. 2 5. 9
49 OS
2
52 51
D
3
9. 6 5. 6
5. 2 10. 0
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. 0 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". 0
d of Polaris
88 45 46. 9
log cot S
8. 33430
8. 33430
log tan <j>
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 <j>
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*
0° 34' 22". 8
0° 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
2sin2J T
1 26. 7 4. 1
1 54. 2 7. 1
T ana gjn ^//
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 r2 sin 2J r
1 7C>8
1. 670
log if sin 1"
-L. t <JO
log (curvature corr.)
9.758
9.741
Curvature correction
-0.6
-0.6
Altitude of Polaris=A
32° 07'
-T tan A=level factor
.419
.419
Inclination t
+3.6
+4.1
Level correction
—1". 5
-1". 7
Angle, star — mark
72 57 50. 2
72 51 46. 7
Corrected angle
72 57 48. 7
72 51 45. 0
Corrected azimuth of star*
0 34 22. 2
0 40 26. 2
Azimuth of mark E of N
73 32 10. 9
73 32 11. 2
180 00 00. 0
180 00 00. 0
Azimuth of mark
253 32 10. 9
253 32 11. 2
(Clockwise from south)
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 See footnote on p. 148.
DETERMINATION OF AZIMUTH. 155
METHOD OF REPETITIONS— EXPLANATION OF RECORD AND COMPUTATION.
Throughout the observations the instrument was always turned in a clockwise direction
about its vertical axis. In set No. 5 the swing from the mark to the star was made with the
upper motion loose and lower motion clamped, and therefore with the circle reading changing,
and in set No. 6 the reverse was the case. In set No. 5 the explement of the small angle between
the star and the mark was really measured, while in No. 6 the angle itself was measured. Both
results may be computed directly in terras of the angle by making the subtractions thus, in set
No. 5.
, (360° + 178° 03' 21//.2)-100° 16' 20".Q , „
angle =— fi =72 57 50 .2
in set No. 6,
, (3600 + 177°27/00//.0)-1000 16' 20". 0 790 «/ x«// 71
angle = — — ^ — =72" 51 4o .7 .
If the clamp on the horizontal circle produces a constant error, either by dragging or
overrunning, these two results will be equally in error with opposite signs, and their mean will
be free from the constant part of the clamp error. Hence, it is desirable to observe the sets
alternately in the order Mark-Star, Star-Mark, as here indicated.
The summary of results for this station shows 37 sets of observations were made on four
nights. From the 18 sets observed in the order Star-Mark the mean azimuth was 73° 32' 12".07,
and from the 19 sets observed in the order Mark-Star the mean was 73° 32' 12".89, showing
that the clamp error was very small. The adopted indiscriminate mean of all the 37 sets was
73° 32' 12".49. The correction for diurnal aberration ( + 0".31) being applied, the resulting
azimuth of the mark, E. of N. equals 73° 32' 12".80±0".16. The probable error of a single
0.455 „ QS
n_1}
During these observations the instrument was supported upon its tripod, the legs of which
were set upon large stakes driven solidly into the ground.
The level readings were taken with the first, third, fourth, and sixth pointings upon the
star, that is, at the beginning and end of the set and just before and just after the reversal of
the telescope. In each case the level was read in one position just before perfecting the pointing
upon the star, and in the other position immediately after the pointing upon the star. The
value of one division of the level was 2".67.
The computation needs no further explanation. The formula
tan A = — cot d sec <p sin t ( _ }
was used.
The correction for elevation of mark, when appreciable, is applied in the final summary
of results, just as in the case of the direction method. The reduction to the mean position of the
pole is also applied to the final result, but for observations previous to the year 1900 no such
reduction can now be made. (See p. 85.)
MICROMETRIC METHOD— EXAMPLE OF RECORD AND COMPUTATION.
In the micrometric method 2 the small difference of azimuth of the star and the mark is
measured with an eyepiece micrometer, independently of the graduated horizontal circle of
the instrument, even if it has one. The mark must therefore be placed nearly in the vertical
of the star at the time at which it is to be observed. The method may be used with the star at
any hour-angle, but unless the star is near elongation it will pass beyond the safe range of the
micrometer after but two or three sets of observations have been taken, whereas if the mark
1 The computer should notice the convenient fact that in dividing an angle by six the remainder, when the degrees are divided, is the tens
figure in the minutes, and the remainder in the minutes is the tens figure in the seconds.
* For an account of this method, together with some historical notes, see Appendix No. 2 of the Report for 1891.
156
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. H.
is placed nearly under the star at elongation (preferably one or two minutes of arc inside) the
observations may be continued for two hours or more and the results will also be nearly inde-
pendent of the chronometer error. The instrument used may be a theodolite, a meridian
telescope, a transit, or, in fact, any instrument having a stable horizontal axis and furnished
with an eyepiece micrometer capable of measuring angles in the plane defined by the telescope
and its horizontal axis.
Record and computation — Azimuth ~by micrometric method.
[Station No. 10, Mexican Boundary. Date, October 13, 1892. Observer, J. F. H. Instrument, Fauth Repeating Theodolite, No. 725 (10 in.).
Star, Polaris near eastern elongation.)
Circle
Level readings
W E
Chronometer
time
T
2 sin ! $ T
Micrometer readings —
sin 1"
On star
On mark
E
E
W
w
8. 0 9. 9
10. 0 7. 3
h m s
9 06 38. 0
07 32. 0
08 05. 5
09 13.0
09 48. 0
9 12 01. 8
12 24. 7
12 48. 3
13 36. 3
13 58. 1
m s
3 58.6
3 04.6
2 31. 1
1 23.6
0 48.6
1 25.2
1 48.1
2 11.7
2 59.7
3 21.5
31. 05
18.59
12.45
3.82
1.29
3.96
6.37
9.46
17.61
22. 11
18'. 379
.388
.400
.424
.430
18'. 310
.315
.315
.311
.316
;.=2» 12m W. of
Washington
A. 01 o -in/ oc//
+ 18.0 -17.2
+0.8
9. 0 9. 0
7.0 10.9
1 div. of level
=3".68
1 turn of mic.
m// 70
18. 4042
18. 100
.100
.090
.086
.080
18.3134
18. 290
.275
.279
.281
.279
Means
Means
+16.0-19.9
— 3 9
Mean ld. 55
9 10 36. 6
12.67
18. 0912
18. 2808
£ of star at middle of first half of set=58° 48'.
£ of star at middle of second half of set=58° 46'.
a=lh 20m 07-.4.
cosec C=1.1691 . cot 58° 47'=0. 606.
cosec C=1.1695.
<J=88° 44' 10". 4.
Collimadon axis raads 4 (18.3134+18.2808)1 =18t.2971
Mark east of colHmation axis 18.3134-18. 2971 =0.0163= 02".02
Circle E., star E. of collimation axis (18.4042-18.2971) (1.1691)= 0 .1252
Circle W., star E. of collimation axis (18.2971-18.0912) (1.1695)= 0 .2408
Mean, star E. of collimation axis
Mark west of star
Level correction (] .55) (0.92) (0.606)
= 0 .1830= 22 .64
= 20 .62
= - 0 .86
Mark west of star, corrected = 19 .76
Mean chronometer time of observation= 21h 10m 36S.6
Chronometer coirection =—2 11 28 .2
Sidereal time = 18 59 08 .4
a = 1 20 07 .4
log. cot S
log. tan <j>
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 ah1 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 manjr
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
0 950
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, 0° 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 . . . = - {aw+ (M + c-i) sec h}l5
JBandE . . . « = 360°- {a, + (M-e+i) sec
Mark,,orthBandW ' ' ' "= 180°- {aw+ (Jf-c + i) sec
^JBandE . . . «=180°- K + (M+c-i) sec A} 15
aw and aE 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 e2 is the square of the eccentricity and a the semi-major axis of the spheroid of refer-
ence; <j> 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 cos2 0 sin la, or
+ [ 6.0392] h cos2 <£ 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
0
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. 0
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 0
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
0
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. 0
r 46
2709
2739 2769
2799
2829
2S59
2889
2919
2949
2979
4
13.6
13^2
12 8
12.4
12. 0
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. 0
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. 0
04
4788 I 4799
4810
4822
4833
4844
4855
4866
4K7S
4889
6
8.4 7.8 7.2
6. 6 6. 0
03
4679
4690 4700
4711
4722
4733
4744
4755
4766
4777
7
9.8
9. 1 i 8. 4
7. 7 7. 0
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
I—a
Log a
0
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 0
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
0 9
0 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
0
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 0
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
0
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
<j
7 2
6 3
1
5
6
6
6
6
6
6
6
7
7
5.0
0.000004
4
5
5
5
5
5
5
5
5
6
5
4
0.000000
1
1
1
1
1
2
2
3
3
1
0.6
0.5
2
1.2
1.0
3
1.8
1.5
4
2.4
2.0
5
3.0
2.5
6
3.6
3.0
4 n
1.000000
9999
9999
9999
9999
9999
9998
9998
9997
9997
7
8
4.2
4.8
3.5
4.0
5.0 n
9.999996
96
95
95
95
95
95
95
95
95
9
5.4
4.5
1 n
95
94
94
94
94
94
94
94
93
93
2 n
93
93
93
93
92
92
92
92
92
92
4
3
3 n
4 n
91
89
91
89
91
89
91
88
90
88
90
88
90
87
90
87
90
87
89
87
1
0.4
0.3
2
0.8
0.6 •
5 n
9.999986
86
86
85
85
85
84
84
83
83
3
1.2
0.9
6 n
83
82
82
81
81
81
80
80
79
79
4
1.6
1.2
7 n
78
78
77
77
76
78
75
74
74
73
5
2.0
1.5
8 n
73
72
71
71
70
69
69
68
67
til!
6
2.4
1.8
9 n
66
65
64
63
62
61
60
59
59
58
7
2.8
2.1
8
3.2
2.4
6.0 n
9.999957
56
55
53
52
51
50
49
48
47
9
3.6
2.7
1 n
45
44
43
41
40
39
37
36
34
33
2 n
31
30
28
26
25
23
21
19
17
15
2
1
3 n
13
09
07
05
03
01
898
8%
893
4 n
9. 999891
888
886
883
880
878
875
872
869
866
1
0.2
0.1
2
0.4
0.2
5 n
9.999863
KM
856
853
849
846
842
839
835
831
3
0.6
0.3
6 n
827
823
819
815
810
806
802
797
792
7S7
4
0.8
0.4
7 n
782
777
772
767
71)1
756
750
744
738
732
5
1.0
0.5
8 n
726
720
713
706
700
693
685
678
671
663
6
1.2
0.6
9 n
655
647
639
631
622
613
604
595
585
576
7
1.4
0.7
8
1.1
0.8
7.00n
9.999566
565
564
563
562
561
560
559
558
557
9
1.8
0.9
170
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. H.
1
Log a
0
1
2
3
4
5
6
7 8
9
Proportional parts
7.00 n
9. 999566
565
'564
563
562
561
560
559
558
557
01 n
556
555
554
553
552
551
550
549
548
547
02 n
545
544
543
542
541
540
539
538
537
536
03 n
535
534
533
532
531
530
528
527 526
525
04n
524
523
522
521
520
519
517
516
515
514
05n
9. 999513
512
511
510
508
507
506
505
504
503
06n
502
501
499
498
497
4%
495
494 ; 492
491
07 n
490
489
488
487
485
484
483
482
481
479
08 n
478
477
476
475
473
472
471
470
469
467
09n
466
465
464
462
461
460
459
457
456
455
7. 10 n
9. 999454
452
451
450
449
447
446
445
443
442
11 n
441
440
438
437
436
434
433
432
430
429
1 2
12 n
428
427
425
424
423
421
420
419
417
416
13 n
415
413
412
410
409
408
406
405
404
402
1 0. 1 0. 2
14 n
401
400
398
397
395
394
393
391
390
388
2 0. 2 0. 4
15 n
9. 999387
386
384
383
381
380
378
377
3/6
374
3 0.3 0.6
4 0.4 0.8
16 n
373
371
370
368
367
365
364
363
361
360
5 0.5 1.0
17 n
358
357
355
354
352
351
349
348
346
345
6 0. G 1.2
18 n
343
342
340
339
337
336
334
333
331
329
7 0.7 L4
19 n
328
326
325
323
322
320
319
317
315
314
7.20n
9.999312
311
309
307
306
304
303
301
299
298
9 0.9 L8
21 n
296
295
293
291
290
288
286
285
283
282
22n
280
278
277
275
273
272
270
268
266
265
23n
263
261
260
258
256
255
253
251
249
247
24 n
246
244
242
241
239
237
235
234
232
230
25n
9.999228
227
225
223
221
219
218
216
214
212
26 n
210
209
207
205
203
201
199
198
196
194
2', n
192
190
188
186
185
183
181
179
177
17:,
28 n
173
171
169
168
166
164
162
160
158
1511
29n
154
152
150
148
146
144
142
140
138
136
7.30n
9.999134
132
130
128
126
124
122
120
118
116
31 n
114
112
110
108
Hid
104
102
100
09S
096
3 4
32 n
094
091
089
087
(is;,
083
081
079
077 075
33 n
072
070
C68
066
064
062
060
057
055 ! 053
34 A
051
049
047
044
C42
040
038
036
033 i 031
1 0. o 0. 4
2 0. 6 0. 8
35 n
9.999029
027
024
022
020
018
015
013
Oil 009
3 0.9 1.2
36 n
006
004
002
8999
8997
8995
8992
8990
8988 8985
4 1.2 1.6
37 n
9.998983
8981
8978
8976
8974
8971
8969
8967
8964 8962
5 1.5 2.0
38 n
8959
8957 8955
8952
8950
8947
8945
8943
8940 8938
6 1.8 2.4
39 n
8935
8933 ' 8930
8928
8925
8923
8920
8918
8915 8913
7 2. 1 2.8
8 2.4 3.2
7.40n
9.998910
8908
8905
8903
8900
8898
8895
8893
8890 8888
9 2.7 3.6
41 n
8885
8883
8880
8877
8875
8872
8870
8867
8864 8862
42 n
8859
8857
8854
8851
8849
8846
8843
8841
8838 8835
43 n
8833
8830
8827
8825
8822
8819
8816
8814
8811 8808
44 n
8805
8803
8800
8V97
8794
8792
8789
8786
8783
8781
45 n
9. 998778
8775
8772
8769
8766
8764
8761
8758
8755
8752
46 n
8749
8746
8744
8741
8738
8735
8732
8729
8726
8723
47 n
8720
8717
8714
8711
8708
8705
8702
8699
8696
8693
48 n
8690
8687
8684
8681
8678
8675
8672
8669
8666
8663
49 n
8660
8657
8654
8651
8648
8644
8641
8638
8635
8632
7.50n
9. 998629
8626
8622
8619
8616
8613
8610
8607
8603
8600
DETERMINATION OF AZIMUTH.
1
171
LOKO
0
1
2
3
4
5
6
7
8
9
Proportional parts
7. 50n
9.998629
8626
8622
8619
8616
8613
8610
8607
8603
8600
51 n
8597
8594
8590
8587
8584
8581
8577
8574
8571
8568
4
5
52 n
8564
8561
8558
8554
8551
8548
8544
8541
8538
8534
53 n
8531
8528
8524
8521
8517
8514
8511
8507
8504
8500
54 n
8497
8493
8490
8486
8483
8479
8476
8472
8469
8465
1
0.4
0.5
2
0.8
1.0
55n
9.998462
8458
8455
8451
8448
8444
8440
8437
8433
8430
3
1.2
1.5
56 n
8426
8422
8419
8415
8411
8408
8404
8400
8397
8393
4
1.6
2.0
57 n
8389
8386
8382
8378
8375
8371
8367
8363
8360
8356
5
2.0
2 5
58 n
8352
8348
8344
8341
8337
8333
8329
8325
8321
8318
6
2.4
3.0
59 n
8314
8310
8306
8302 ' 8298
8294
8290
8286
8282
8278
7
2.8
3.5
g
3.2
4.0
7.60 n
9.998274
8271
8267 8263 8259
8255 8251
8246
8242
8238
9
3.6
4.5
61 n
8234
8230
8226 , 8222 8218
8214 8210
8206
82.02
8197
62 n
8193
8189
8185
8181 , 8177
8172 8168
8164
8160
8156
63 n
8151
8147
8143
8139
8134
8130
8126
8121
8117
8113
6
7
64n
8108
8104
8100
8095
8091
8087
8082
8078
8073
8069
65n
9.998064
8060
8055
8051
8047
8042 8038
8033
8028
8024
0 a
0 7
66 n
8019
8015
8010 8006 8001
7997 , 7992
7987
7983
7978
. O
67 n
7973
7969
7964 ' 7959 : 7955
7950
7945
7941
7936
7931
2
1.2
1 ft
1. 4
21
68 n
7926
7922
7917 j 7912 7907
7902
7898
7893
7888
7883
I. o
. 1
69 n
7878
7873
7868
7863 7859
7854
7849
7844
7839
7834
4
5
2. 4
3.0
2.8
3.5
7. "On
9.997829
7824
7819
7814 7809
7804
7799
7794
7789
7783
6
3.6
49
4.2
J Q
71 n
7778
7773
7768
7763 7758
7753
7748
7742
7737
7732
. £.
t. a
72 n
7727
7722
7716
7711 7706
7700
7695
7690
7685
7679
8
4.8
5 A
5.6
60
73 n
7674
7669
7663
7658 7652
7647
7642
7636
7631
7625
. 4
. o
74 n
7620
7614
7609
7603 7598
7592
7587
7581
7576
7570
75 n
9. 997565
7559
7553
7548
7542
7537
7531
7525
7519
7514
<j
76 n
7508
7502
7497
7491
7485
7479
7473
7468
7462
7456
77 n
7450
7444
7438
7433
7427
7421
7415
7409
7403
7397
78 n
7391
7385
7379
7373
7367
7361
7355
7349
7343
7337
79 n
7330
7324
7318
7312
7306
7300
7293
7287
7281
7275
i
0.8
0.9
2
1.6
1.8
7.80 n
9. 997268
7262
7256
7250
7243
7237
7231
7224
7218 7211
3
2.4
2.7
81 n
7205
7199
7192
7186
7179
7173
7166
7160
7153 7147
4
3.2
3.6
82 n
7140
7134
7127
7120
7114
7107
7100
7094
7087 7080
5
4.0
4.5
83n
7074
7067
7060
7053
7047
7040
7033
7026
7019 7013
6
4.8
5.4
84n
7006
6999
6992
6985
6978
6971
6964
6957
6950 6943
7
5.6
6.3
8
6.4
7.2
85n
9. 996936
6929
6922
6915
6908
6901
6894
6887
6880 6872
9
7.2
8.1
86 n
6865
6858
6851
6844
6836
6829
6822
68U
6807 ; 6800
87 n
6792
6785
6778
6770
6763
6755
6748
6740
6733 6725
88 n
6718
6710
6703
6695
6688
6680
6672
6665
6657
6650
89 n
6642
6634
6626
6619
6611
6603
6595
6587
6580
6572
10
11
7.90n
9.996564
6556
6548
6540
6532
6524
6516
6508
6500
6492
91 n
6484
6476
6468 i 6460
6452
6444
6435
6427
6419
6411
1
1.0
1.1
92n
6403
6394
6386 6378
6369
6361
6353
6344
6336
6328
2
2.0
2.2
93 n
6319
6311
6302 ; 6294
6285
6277
6268
6260
6251
6242
3
3.0
3.3
94 n
6234
6225
6217
6208
6199
6190
6182
6173
6164
6155
4
4.0
4.4
5
5.0
5.5
95n
9. 996146
6138
6129
6120
6111
6102
6093
6084
6075
6066
6
6.0
6.6
96 n
6057
6048
6039
6030
6021
6012
6003
5993
5984
5975
7
7.0
7.7
97 n
5966
5956
5947
5938
5929
5919
5910
5900
5891
5882
8
8.0
8.8
98 n
5872
5863
5853
5844
5834
5825
5815
5805
5796
5786
9
9.0
9.9
99 n
5777
5767
5757
5747
5738
5728
5718
5708
5698
56S9
S.OOn
9.995079
5669
5659
5649
5639
5629
5619
5609
5599
5589
172
U. S. COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 14.
/
Logo
0
1
2
3
4
5
6
7
8
9
Proportional parts
S.OOn
1995679
5669
5659
5649
5639
5629
5619
5609
5599
5589
10
11
12
13
14
01 n
5578
5568
5558
5548
5538
5528
5517
5507
5497
5486
02n
5476
5466
5455
5445
5434
5424
5413
5403
5392
5382
1
1.0
1.1
1.2
1.3
1.4
03n
5371
5361
5350
5339
5329
5318
5307
5296
5286
5275
2
2.0
2.2
2.4
2.6
2.8
in n
5264
5253
5242
5231
5220
5209
5198
5187
5176
5165
3
3.0
3.3
3.6
3.9
4.2
4
4.0
4.4
4.8
5.2
5.6
05n
9.995154
5143
5132
5121
5110
5098
5087
5076
5065
5053
5
5.0
5.5
6.0
6.5
7.0
08n
5042
5031
5019
5008
4996
4985
4973
4962
4950
4939
t;
6.0
6.6
7.2 7.8
8.4
07 n
4927
4916
4904
4892
4<81
4869
4857
4845
4833
4822
7
7.0
7.7
8.4 91
9.8
08 n
4810
4798
4786
4774
4762
4750
4738
4726
4714
4702
8
8.0
8.8
9.6
10.4
11.2
09n
4690
4677
4665
4653
4641
4628
4616
4604
4591
4579
9
9.0
9.9
10.8
11.7 12.6
8. 10 n
9.994567
4554
4542
4529
4517
4504
4492
4479
4466
4454
15
16
17
18
19
11 n
4441
4428
4415
4403
4390
4377
4364
4351
4338
4325
12 n
4312
4299
4286
4273
4260
4247
4234
4220
4207
4194
t
1 5
1 6
1 7
18 19
13 n
4181
4167
41.14
4141
4127
4114
4100
4087
4073
4080
2
3 0
3 2
3 4
3*6 ! 38
14 n
4046
4032
4019
4005
3991
3978
3964
3950
3936
3922
3
4^5
4^8
S.I
5.4 i 5^7
15 n
9. 993908
3894
3880
3866
3852
3838
3824
3810
3796
3782
4
g
6.0
7.5
6.4
8.0
6.8
8 5
7.2
9 0
7.6
9.5
16 n
17 n
3767
3623
3753
3609
3739
3594
3725
3579
3710
3565
3698
3550
3681
3535
3667
3521
3652
3.106
3638
3491
6
9^0
10 5
9^6
11 2
10.3
11 9
10.8
12 6
11.4
13 3
18 n
19 n
3476
3325
3461
3310
3446
3295
3431
3279
3416
3264
3401
3248
3386
3233
3371
3218
3356
3202
3340
3186
8
9
12^0
13.5
12! 8
14.4
13^6
15.3
U.4
16.2
15^2
17.1
8.20n
9.993171
3155
3140
3124
3108
3092
3077
3061
3045
3029
20
21
22
23
24
21 n
3013
2997
2981
2965
2949
2933
2917
2900
2884
2868
22 n
2852
2835
2819
2803
?7SI>
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
I—a
Logo
0
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 0
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
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