A TEXT-BOOK OF GEODETIC ASTRONOMY BY JOHN F. HAYFORD, C.E., Associate Member American Society of Civil Engineers; Expert Computer and Geodesist U. 5. Coast and Geodetic Survey. FIRST EDITION, FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS. LONDON : CHAPMAN & HALL, LIMITED. 1898. Copyright, 1898, BY JOHN F. HAYFORD. ROBERT DRUMMOND, PRINTER, NEW YORK. PREFACE. ( To be read by th: student as well as the teacher.} THE purpose of this book is to furnish a text which is sufficiently short and easy to be mastered by the student of civil engineering in a single college term, but which shall give him a sufficiently exact and extensive knowledge of geodetic astronomy to serve as a basis for practice in that line after graduation. Though the book has been prepared primarily for students, the author has endeavored to insert such sub- ject-matter, tables, and convenient formulae as would make it of value as a manual for the engineer making astronomical observations. In order to make the book sufficiently short it has been necessary to omit all mathematical processes except those actually necessary for developing the working formulae. And as the object of the work is to teach a certain limited branch of astronomy, rather than to teach mathematics, the simpler and special means of deriving the working formulae have been chosen, in every case in which there was a chance for choice, instead of the more difficult and general derivation that would naturally be chosen by the mathematician. The occasion for the book is the fact that in the course of study prescribed for students of civil engineering at Cornell University but five hours per week for one term can be devoted to the text-book work and lectures on astronomy. Under these conditions it is out of the question to use Chauvenet's standard work. Even Doolittle's Practical IV PREFACE. Astronomy contains more mathematics than a student can be expected to master thoroughly in that period. Of various other text-books available none seem to fit the special condi- tions. In the wording of the book it is tacitly assumed that the observer is in the northern hemisphere. To make the word- ing general would require too many circumlocutions. It is assumed that the student has a knowledge of least squares. If, however, he has not such knowledge, it will not debar him from following nearly every part of the text except 107-113, dealing with the treatment of transit time obser- vations by least squares, and 154-157, giving the process of combining the results for latitude with a zenith telescope by that method. If he reads carefully 283-285, stating the technical meaning of the phrase " probable error," the statement of the uncertainty of a given observation in terms of the probable error, or the statement of the errors to be ex- pected from certain sources in such terms, should convey to him a definite meaning. Considerable space has been devoted in the text to a dis- cussion of the various sources of error in each kind of obser- vation treated. Two separate considerations seem to the author to justify this. One is that the special value of geodetic astronomy as a part of the course of training of an engineer depends largely upon the fact that in studying it he is brought face to face with the idea that instruments are fallible, and that therefore their indications must be carefully scrutinized and interpreted; and that if the best results are to be secured from them, the sources of the various minute errors which combined constitute the errors of observation must be carefully studied. The other consideration is that an observer's success in securing accurate results with moderate effort depends to a considerable extent upon his PREFACE. V power to estimate rightly the relative importance of the various errors affecting his final result. The accuracy of a man's thoughts, as well as of his speech, when dealing with a given subject depends largely upon the precision of his understanding of the special vocabulary of that subject. With that idea in view the finder list of definitions given in 312 has been prepared. The student who is not sure of the exact meaning of a word may turn to this list and so find the exact definition quickly. In reading definitions the context should also be read. When a word is defined in the text it is printed in italics. The effort has been made to select the formulae which have been found in practice to lead to accurate and rapid computa- tions. They have been gathered at the end of the volume for convenient reference, and adjacent to each formula will be found references to the corresponding portion of the text, so that for those who may use the book as a manual the list of formulas with these references may serve as an index or finder for the text. In the five principal chapters the instrument has first been described, and the adjustments given, as well as directions for observing, and an example of the record. The derivation of the formulae, the computation, etc., follow. If the text-book work and the practical work of the observatory are carried on together during the same term one naturally wishes the students to become familiar with the instruments and their manipulation as soon as possible. In that case it is recom- mended that the first portions only of certain chapters be taken and the later portions omitted temporarily. The fol- lowing order may then be used: 1-27, 37, 51-63, 83-91, 134-146, 177-187, 201-203, 205 to middle of 210, 273-276, 28-50, 64-82, 92-133, 147-176, 188-272, 277 to the end. During the preparation of this volume the text-books on VI PREFACE. astronomy written by Chauvenet, Doolittle and Loomis have been freely consulted, as well as various reports of the Coast and Geodetic Survey, of the Northern Boundary Survey, of the U. S. Lake Survey, and the report of the Mexican Boundary Survey of 1892-93. Appendix No. 14 to the Coast Survey Report for 1880, which is written by Assistant C. A. Schott and is used as a manual by the officers of that survey, has been extensively drawn upon as the best exposi- tion of good field methods known to the author. Several tables have been taken from that source, notably the table of factors for the reduction of transit time observations given in 299. The Superintendent of the Coast Survey has very kindly furnished certain data, and photographs of instru- ments. JOHN F. HAYFORD. WASHINGTON, D. C., April 23, 1898. CONTENTS. CHAPTER PAGE I. INTRODUCTORY i Apparent Motions I The Earth 3 Precession and Nutation 5 Planets, Satellites, Stars 6 Diurnal Motion 8 Definitions 10 Time 18 Questions and Examples 27* II. COMPUTATION OF RIGHT ASCENSION AND DECLINATION 30 Position of Sun and Planets 31 Interpolation 31 Position of Moon 39 Position of Stars 39 Aberration 41 Mean Places 44 Proper Motion 48 Computation of Apparent Places 50 Questions and Examples 55 III. THE SEXTANT 59 Description of Sextant 59 Adjustments 62 Directions for Observing for Time 65 Record and Computation 71 Parallax 73 Refraction 75 Derivation of Formula 80 Discussion of Errors 82 Eccentricity 88 Other Uses of the Sextant 90 Questions and Examples 94 vii Vlii CONTENTS. IV. THE ASTRONOMICAL TRANSIT 96 Descriptions of Transits 96 Adjustments 99 Directions for Observing 103 The Chronograph 105 Record and Computation. 107 Reduction to Mean Line 108 Inclination Correction 112 Correction for Diurnal Aberration 1 16 Azimuth, Collimation, and Rate Corrections 117, 118, 119 Computation Without Least Squares 120 Least Square Computation 126 Unequal Weights 131 Auxiliary Observations 133 Value of the Level 135 Discussion of Errors 140 Miscellaneous 144 Questions and Examples 146 V. THE ZENITH TELESCOPE AND THE DETERMINATION OF LATITUDE. 150 Description of Zenith Telescope 151 Adjustments 153 Observing List 156 Directions for Observing 160 Derivation of Formula 163 Computation 166 Combination of Results 168 Micrometer Value 174 Discussion of Errors 181 Other Methods of Determining Latitude 187 Questions and Examples 194 VI. AZIMUTH. 197 Description of Instrument 197 Adjustments 200 Directions for Observing with Direction Instrument 201 Record Direction Instrument 205 The Circle Reading 2of Level Correction 209 Azimuth Formula 211 Curvature Correction 213 Correction for Diurnal Aberration 216 Computation Direction Instrument 218 Method of Repetitions 220 CONTENTS. . IX CHAPTER PACK Micrometric Method 223 Discussion of Errors 230 Other Instruments and Methods 233 Questions and Examples 238 VII. LONGITUDE 241 Telegraphic Method Apparatus and Observations 242 Telegraphic Method Computation 244 Telegraphic Method Discussion of Errors and Personal Equation 248 Longitude by Transportation of Chronometers 253 Observations upon the Moon 262 Observations upon Jupiter's Satellites 267 VIII. MISCELLANEOUS 268 Suggestions about Observing 268 Suggestions about Computing 270 Probable Errors 272 Variation of Latitude 274 Economics of Observing 276 TABLES ( 279 NOTATION AND PRINCIPAL WORKING FORMUL/E 328 LIST OF DEFINITIONS 343 INDEX 345 FIGURES 352 GEODETIC ASTRONOMY, CHAPTER I. INTRODUCTORY. 1. THIS book is limited to the treatment of astronomy as applied to surveying, or to what might be called geodetic astronomy. Only such matters are treated as are pertinent to this particular limited branch of the subject. Moreover, the subject as thus limited is treated from the point of view of the engineer who wishes to obtain definite results, rather than from that of a mathematician more interested in the processes concerned than in their final outcome. 2. The bodies considered by the engineer in geodetic astronomy are the stars; the Sun; the planets, including the Earth; the Moon, the Earth's satellite; and to a very limited extent some of the satellites of the other planets. The engineer from his standpoint upon the surface of the Earth sees these different bodies moving about within the range of his vision, aided by a telescope if necessary. Their apparent motions in the sky as seen by him are quite complicated. His success in locating and orienting himself upon the Earth 2 GEODETIC ASTRONOMY. 2. by observations upon these heavenly bodies for that is his particular purpose in observing them depends first of all upon his having a clear and accurate conception of their apparent motions, and then upon his possession of, and ability to use efficiently, the instruments with which the observations are made. Much of the complexity in the apparent movements of these heavenly bodies is due to the fact that the observer sees them not from a fixed station in space, but from a standpoint upon one of the planets, the Earth, which is moving rapidly through space with a motion which is in itself quite complicated. He sees then in the apparent motion of each heavenly body upon which he gazes not only the actual motion of that body, but also, reflected back upon him, so to speak, he sees the actual motion of the seemingly solid and immovable earth upon which he stands. He is like a passenger upon a train at night who looks out upon the many lights of a town. He sees the lights all apparently in motion. In one case the apparent motion of the particular light may be entirely due to his own motion with the train upon which he is riding, the light itself being at rest. In another case the light may be upon another moving train and its apparent motion will then be due to the actual motion of each of the trains. If the darkness is sufficient to conceal the landscape, he may be at a loss to determine what portions of the apparent motions of the lights are due to his own change of position and what to the motions of the lights themselves. He is then in the position of a man when he first begins to study the apparent movements of the heavenly bodies. Let us first form concrete conceptions as to the actual motion of each of the bodies under consideration, including the Earth itself. We will then be in a position to understand the apparent motions. 4- THE EARTH. 3 3. Conceive the Sun to be a very large self-luminous mass of matter. For the present let it be supposed to be fixed in space. Around this central Sun revolve eight planets, namely, in order of their distance from the Sun, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. All these planets move nearly, but not exactly, in the same plane passing through the Sun. The orbit, or path, of any one of them in its own orbital plane is very nearly a perfect ellipse with one focus at the Sun, and the velocity with which the planet moves varies at different parts of its orbit in such a way that the line joining the planet and Sun describes equal areas in equal times. This orbit and law of velocity result from the fact that each planet is pursuing its path in obedi- ence to a single force, gravity, continually directed toward a fixed center, the Sun. 4. The Earth may be taken as a representative planet. It is the most important of the planets for our present purpose. It moves about the Sun in an elliptical orbit at a mean dis- tance from the Sun, in round numbers, of 92 800000 miles.* Though the orbit is an ellipse, its major and minor axes are so nearly equal that if it were plotted to scale the unaided eye could not distinguish it from a circle. The greatest dis- tance of the Earth from the Sun exceeds the least distance by but little more than 3$. The Sun is in that focus of the ellipse to which the Earth is nearest during the winter (of the northern hemisphere). The eccentricity of the ellipse, and therefore the difference of the two axes, is very slowly decreasing. The plane of the Earth's orbit is not absolutely fixed in direction in space. It changes with exceeding slow- ness so slowly, in fact, that it is used as one of the astronomical reference planes. Moreover, the position of the * See " The Solar Parallax and its Related Constants," Harkness, p. 140. 4 GEODETIC ASTRONOMY. 5- elliptical orbit in the plane is slowly changing; that is, one focus necessarily remains at the Sun, but the direction of the major axis of the ellipse gradually changes. Roughly speaking, the Earth makes one complete circuit of its orbit in the period of time which is ordinarily called one year. At different portions of the orbit its linear velocity varies according to the law (common to all the planets) that the line joining it and the Sun describes equal areas in equal times. Each portion of the nearly circular path being almost perpendicular to the line joining the Earth and Sun at that instant, the linear velocity is nearly inversely proportional to the distance of the Earth from the Sun. Evidently the angular velocity varies still more largely than the linear, since the greatest linear velocity comes at the same time as the least distance from the Sun, and vice versa. At the same time that the Earth, as a whole, is swinging along in its orbit it is rotating uniformly about one of its own diameters as an axis.* This rotation is so nearly uniform in rate that it is assumed to be exactly uniform and is used to furnish our standard of time. Roughly speaking, the interval of time required for one rotation of the Earth on its axis is what is called one day. The more exact statement will be made later. 5. The axis of rotation of the Earth points at present nearly to the star called Polaris, or North Star, and makes an angle of about 66 with the plane of the Earth's orbit, or 23^ with the perpendicular to that plane. The direction of this axis of rotation is not fixed in space, but changes just as * The diameter about which the rotation takes place is, however, not strictly fixed with respect to the Earth, is not, in other words, always the same diameter, but varies through a range of a few feet only on the sur- face of the Earth. See 286-7. For present purposes, however, it will be considered as fixed. 5- PRECESSION AND NUTATION. 5 the axis of a rapidly spinning top is seen to wabble about. This change is quite slow, but extends through a large range of motion. It is compounded of two motions called respec- tively precession and nutation. By virtue of the motion called precession the axis of the earth tends to remain at an angle of about 23!- with the perpendicular to the plane of the Earth's orbit (usually known as the plane of the ecliptic], and to revolve completely around it, describing a cone of two nappes with an angle of about 47 (twice 23^) between opposite elements. The time required to make one such complete revolution is, at the present rate, about 26000 years.* The motion of the Earth's axis called nutation is compounded of several periodic motions, the principal one of which is such as to cause the axis to describe a cone of which the right sec- tion is an ellipse, and of which the greatest angle between opposite elements is about eighteen seconds of arc and the least about fourteen. f * The change of seasons is caused by the inclination of the Earth's axis to the plane of its orbit. At present the northern end of the axis is in- clined directly away from the Sun at about Dec. 2ist; the Sun then appears to be farther south than at any other time, and it is winter in the northern and summer in the southern hemisphere. At about June 2Oth the reverse is true, namely, it is summer in the northern and winter in the southern hemisphere. On account of the precession the winter of the northern hemisphere will occur in June, July, and August about 13000 years hence. f All the various motions of the planets and their satellites the peculiar mathematical properties of their orbits, the variability of the planes of the orbits and of the orbits in the planes, etc. are by celestial mechanics shown to be due simply to the action of gravitation. Or, stating the matter from the converse point of view, given these various bodies in their actual positions and having their actual motions at a given instant, and given the law that gravitation acts between each pair of them with an intensity inversely proportional to the square of their distance apart and directly proportional to the product of the two masses, the position and motion of each one of them, their orbits, etc., at any other stated time may be computed from the principles of celestial mechanics alone. Even the precession and nutation are caused by gravitation and are thoroughly 6 GEODETIC ASTRONOMY. 6. 6. The Earth was taken as representative of the planets. Most or all of the various phenomena which have been indi- cated in the motion of the Earth are repeated in each of the other planets. Each has an orbital plane of its own which is slightly variable and does not in any case at present make an angle of more than about 7 with the plane of the Earth's orbit. Each moves in that plane in an ellipse of which the eccentricity and position are slowly changing. Each has a rotation about its own axis, with which, however, the en- gineer is not concerned. The Moon is a satellite of the Earth, revolving about it under the action of gravity just as the Earth revolves about the Sun, and in its orbit are again found the same peculiarities as in the orbit of the Earth itself. The orbit of the Moon is an ellipse of variable eccentricity and position, with the Earth in one focus, and lying in a variable plane making an angle of about 5 with the plane of the Earth's orbit. The several variations mentioned are much greater in the case of the Moon than of the Earth, and the motion, moreover, is subject to other perturbations. Its motion is therefore a most diffi- cult one to compute. Each one of the other planets, except Mercury and accounted for by principles of celestial mechanics derived from the above law of gravitation. The luni-solar precession is due to the fact that the Earth is not a sphere, but a spheroid, having an excess of matter in the equatorial regions. One component of the attraction of the Moon and Sun acting upon this equatorial excess tends continually to shift the position of the equator in one direction without changing its angle with the eclip- tic. The action of the planets upon the Earth as a whole tends to draw it out of the plane of its orbit, or rather to change the orbital plane. This change is called the planetary precession. The luni-solar and planetary precessions together constitute what is often called simply the precession. Nutation is made up of periodic motions which are due to regular periodic fluctuations in the forces which produce precession. Nutation might be described as the periodic part of precession 7- THE STARS. 7 Venus, has one or more satellites bearing the same relations to it that the Moon does to the Earth. The comparatively erratic motions of the numerous asteroids, or small planets, moving in orbits between that of Mars and Jupiter, and of the comets and meteors which occasionally visit the solar system, prevent their use by the engineer.* The Sun, the eight planets and their satellites, and the asteroids together constitute the solar system. 7. The stars are self-luminous bodies at great distances from the solar system. Their remoteness is to a certain extent indicated by the fact that with the best telescopes and with the highest magnifying powers at present available the image of a star cannot be magnified. It remains with all powers and telescopes a point of light of which the apparent size is merely a measure of the imperfection of the telescope and eye. But the best evidence of the immense distance even to the nearest of the fixed stars is the fact that even though the diameter of the Earth's orbit, 186 million miles, be taken as the base of a triangle of which the vertex is at the star, it is only with the greatest difficulty, if at all, that the angle at the star can be detected even though the instruments used be of the highest order of accuracy and a long series of observa- tions are used. In the few cases in which this angle at the star has been successfully measured it has been found to be not greater than one second of arc. For the purposes of the engineer, then, it may be assumed that each and every star is at so great a distance from the Earth that the true direction in space of the straight line from the Earth to the star is the same at all times of the year notwithstanding the widely separated positions the Earth may occupy in its orbit. * For an interesting treatment of comets, meteors, and asteroids, see Young's General Astronomy, and Chamber's Astronomy, pp. 104-109, 278-430, 780-816. 8 GEODETIC ASTRONOMY. 8. If the stars had no motion relative to each other or to the solar system as a whole, the true direction of the line from the Earth to any one star would not vary from year to year. As a matter of observation, however, it is known that in general the true direction of such a line does change, although the change is exceedingly slow in every case. This change will be treated more in detail in a later chapter. The apparent motion of any particular heavenly body as seen by an observer upon the Earth is the compound result of the motion of the Earth and of that body. 8. In the case of a star, the object observed is for most purposes at what may be considered an infinite distance. The line joining the observer and star preserves, therefore, a sensibly constant direction in spite of the motion through space of the observer upon the Earth. The apparent motion of the star is caused by the rotation of the Earth about its axis and the change in the direction of that axis in space. The rotation of the Earth causes the line of sight to a star to seem to describe at a uniform rate a right circular cone, of which the axis is the line joining the observer with a point in the sky at an infinite distance in the axis of the Earth pro- duced. In other words, the axis of the cone is a line from the observer parallel to the axis of the Earth. Such a line, for any point in the northern hemisphere, pierces the sky in a point not far from the North Star, Polaris. The angle between any element of the cone and its axis is the angle between the line joining observer to star, and the axis of the Earth. This angle is called the polar distance of the star, north polar distance if measured from the north end of the Earth's axis. So long as these two lines are fixed in direction in space, the line of sight to the star continues to describe the same right circular cone once for every turn which the Earth makes on its axis. For example, the line of sight to 8. APPARENT DIURNAL MOTION OF THE STARS. 9 Polaris makes an angle of about i J with the axis of the Earth, and describes a corresponding right circular cone. Or it may be said that Polaris seems to describe a circle in the sky of which the radius subtends an angle at the eye of ij. With a good telescope Polaris may be followed completely around the circle, all of which would be above the horizon for any point in the United States. With the naked eye only that portion of the apparent motion which occurs during the hours of darkness could be observed. For an observer at Ithaca, in latitude 42^, the cone for a star having a north polar distance less than 42^ is entirely above the horizon. Given one view of the star and an idea of the position of the vanishing point of the Earth's axis in the sky, an observer is able to trace out the whole apparent path of the star. For Ithaca, a star of north polar distance of 42^ has its cone tangent to the horizon; and if greater than that value, a part of the cone must be below the horizon, and the star is neces- sarily invisible on that portion. If the north polar distance is 90, the cone becomes a plane. Stars still farther south describe a right circular cone about the southern portion of the Earth's axis produced, the angle of the cone being the south polar distance of the star. In every case the diurnal rotation of the Earth causes the line of sight to a star to describe a right circular cone. But, as has already been stated ( 5), the direction of the Earth's axis is continually changing slowly, and hence the north polar distance or angle between the Earth's axis and the line join- ing the observer and star is continually changing. The cone of revolution therefore slowly changes from day to day. If the object observed is not a star at a practically infinite distance, but a planet, the Moon, or the Sun, at a finite dis- tance, the line joining observer to object describes a surface any small portion of which may be considered to be a portion IO GEODETIC ASTRONOMY. IO. of the surface of a right circular cone. But the north polar distance of the object now continually changes, not only on account of the change in the direction of the Earth's axis, but still more largely on account of the change in the true direc- tion of the line joining observer and object, two points which are at a finite distance from each other and both in motion. This last cause also makes the rate at which the surface is described variable. 9. The two principal reference planes of astronomy are the plane of the equator and the plane of the Earth's orbit, or, as it is generally called, \.\\.z plane of ecliptic. The plane of the equator is a plane passing through the center of the Earth and perpendicular to its axis of rotation. Neither of these two planes, from what has already been written, are fixed in space, nor fixed relatively to each other. Their changes of position are, however, very slow. 10. To avoid the necessity of using cumbersome expres- sions and circumlocutions, it is convenient to make use of the celestial sphere as an arbitrary conception. The celestial sphere is a sphere of infinite radius, the eye of the observer being supposed to be at its center. Any celestial object is consid- ered to be projected along the line of sight to the surface of this sphere and is referred to as occupying that position upon the sphere. Then for convenience one may speak of arcs, angles, and triangles upon the celestial sphere instead of using the complicated expressions necessary in speaking always of the actual lines and planes which are under consideration. The sphere is assumed to be of infinite radius so that lines which are parallel and at a finite distance apart will intersect the sphere in the same point, or at least what is sensibly one point, since two points at a finite distance apart must appear as one when seen from an infinite distance. So also parallel planes which are at a finite distance apart intersect the 12. DEFINITIONS. II celestial sphere in the same arc. For example, the axis of the Earth and a line parallel to it through the eye of the observer both intersect the celestial sphere in the same pair of points called the poles of the equator, or more briefly the poles, north and south respectively. Also the plane of the equator, and a plane parallel to it through the eye of the observer, intersect the celestial sphere in the same great circle which is called the equator of the celestial sphere, or more fre- quently simply the equator. 11. The equator, the ecliptic, hour-circles, and the horizon are all great circles of the celestial sphere formed by the intersection of various planes with that sphere. The ecliptic is the intersection of the plane of the ecliptic, or, in other words, the plane of the Earth's orbit, with the celestial sphere. The Sun, therefore, is always seen pro- jected on some point of the ecliptic. An hour-circle is the intersection of a plane passing through the Earth's axis with the celestial sphere. All hour- circles are then great circles passing through the poles. The horizon is the intersection with the celestial sphere of a plane passed through the eye of the observer perpendicular to the plumb-line, or line of action of gravity, at the observer. All horizontal lines at a given point on the Earth's surface pierce the celestial sphere in the horizon of that point. In each of these cases it is evident that the great circle on the celestial sphere would not be changed if the intersecting plane were moved parallel to itself a finite distance, for instance, to pass through any other point in or upon the sur- face of the Earth. For example, the horizon may be consid- ered to be the intersection with the celestial sphere of a plane passing through the center of the Earth and perpendicular to the observer's gravity line, instead of that given above. 12. The angle between a line joining the center of the 12 GEODETIC ASTRONOMY. 12. Earth to a star (or other celestial object) and the plane of the equator is called the declination of that object. It is meas- ured upon the celestial sphere by that portion of the hour- circle passing through the star which is between the star and the equator. The declination is considered positive when measured north from the equator. It follows from the definition of polar distance (given in 8) that the declination and polar distance are complements of each other. The equator and the ecliptic intersect each other at an angle of about 23 27'. Their two points of intersection on the celestial sphere are called the equinoxes. That one at which the Sun is found in the spring is called the vernal equinox, and that at which it is found in the fall the autumnal equinox. As both the equator and ecliptic move slowly in space the equinoctial points slowly shift in position upon the celestial sphere. The right ascension of a star, or other celestial object, is the angle, measured along the equator, between the two hour-circles which pass through the star and the vernal equi- nox respectively. In other words, the righ't ascension is the angle between two planes, one passing through the Earth's axis and the star and the other through the Earth's axis and the vernal equinox. It is reckoned in degrees from o to 360, in the direction that would appear counter-clockwise if one looked toward the equator from the north pole, from west to east. Right ascensions are still more frequently expressed in time, 24 hours being equivalent to 360 degrees. The zenith is the point in which the action-line of gravity produced upward intersects the celestial sphere. The oppo- site point on the celestial sphere is called the nadir. The intersection with the celestial sphere of a plane passed through its center, the zenith, and the pole is called the meridian, and the plane itself is called the meridian plane* 13- DEFINITIONS. 13 The intersection of the meridian plane with the plane of the horizon is called the meridian line. It connects the north and south points of the horizon. The intersection with the celestial sphere of a plane through the zenith perpendicular to the meridian plane is called the prime vertical. The east and west points of the horizon are in the prime vertical. 13. The angle, measured along the equator, between the meridian and the hour-circle passing through a star (or other celestial object) is the hour -angle of the star. In other words, the hour-angle is the angle between the meridian plane and a plane passing through the Earth's axis and the star. Hour- angles are reckoned like right ascensions, either in degrees, minutes, and seconds of arc or in hours, minutes, and seconds of time. In this book hour-angles will be measured for 180 each way from the upper branch of the meridian and will always be considered positive. The student should distinguish carefully between an hour- angle and a right ascension. Each is an angle between two planes. In each case one of the two planes is defined by the Earth's axis and the star, and therefore changes direction but slowly in space. The second plane concerned in the case of a right ascension is defined by the Earth's axis and the vernal equinox. This plane changes its direction very slowly. So the right ascension of a star is an angle which is slowly chang- ing, at a rate of less than one minute of arc per year for nearly all the stars. The second plane concerned in the measurement of an hour-angle is the plane of the meridian. This accompanies the Earth in its diurnal rotation. Hence the hour-angle of a celestial object varies rapidly, 360 for each rotation of the Earth on its axis. The right ascension and declination are spherical co-ordinates locating a celestial object with reference to the hour-circle through the vernal equinox and the equator. The hour-angle and declination 14 GEODETIC ASTRONOMY. 1 5. are two spherical co-ordinates locating a celestial object with reference to the meridian and equator. 14. It is convenient for some purposes to refer the posi- tion of a heavenly body by spherical co-ordinates to the planes of the meridian and horizon, the two co-ordinates in this case being the altitude and azimuth. The altitude of a heavenly body is its angular distance above the horizon, or the angle between the line joining the observer to the star, and the horizontal plane. Any great circle of the celestial sphere passing through the zenith is called a vertical circle. The altitude of a star is measured by that portion of the verti- cal circle passing through the star which is included between the star and the horizon. The azimuth of a star, or other celestial body, is the angle between the plane of the meridian and the vertical plane passing through the star. The same definition applies to a line joining two terrestrial points. The azimuth at station A on the Earth's surface, of the line join- ing stations A and B, is the angle between the vertical plane at A passing through the line AB and the meridian plane of A. The azimuth of a star is measured on the celestial sphere by that portion of the horizon included between the star's vertical circle and the meridian line. In general the altitude and azimuth of a celestial object are both changing rapidly because of the Earth's rotation. The zenith distance of a star is its angular distance from the zenith, measured, of course, along a vertical circle. The zenith distance and altitude are complements of each other. 15. The astronomical latitude of a station on the surface of the Earth is the angle between the line of action of gravity at that station and the plane of the equator. It is measured on the celestial sphere along the meridian from the equator to the zenith. The astronomical longitude of a station on the surface of 15- LATITUDE AND LONGITUDE. 1 5 the Earth is the angle between the meridian plane of that station and some arbitrarily chosen initial meridian plane. Usually the meridian of Greenwich, Eng., is taken as the initial meridian, but sometimes that of Paris or of Berlin, or in the case of detached surveys some arbitrary meridian plane to which all points of the survey may be conveniently referred. Unless otherwise stated astronomical latitude or astronomical longitude is meant when the word latitude or longitude is used in this book. The student should distinguish astronomical latitude and longitude from geodetic latitude and longitude, and should be careful not to confuse either one of these with celestial latitude and longitude. The geodetic latitudes and longitudes differ from the astronomical in that, instead of being referred to the actual action-line of gravity at the station, they are referred to a gravity line which has been corrected for local deflection, or station error.* Celestial latitudes and longitudes form a * In the operations of geodesy the action-line of gravity has been found to be nearly perpendicular at all stations to the surface of an imaginary ellipsoid of revolution generated by the revolution of an ellipse about its minor axis, the minor axis coinciding with the axis of rotation of the Earth. This is the form which a rotating liquid mass necessarily assumes under the action of no other forces than the action of gravitation between its component parts. Values for the polar and equatorial diam- etersi respectively, of this ellipsoid having been determined such that its surface is as nearly as possible perpendicular at all points to the action- lines of gravity, the outstanding difference of direction between the normal to the surface of the ellipsoid at any point and the actual action-line of gravity at that point is called the station error, or local deflection of the vertical at that point. The station error is supposed to be due to varia- tions of density in the interior of the Earth near the station, and to the local irregularities of the surface. The operation of determining the station error at a given place is as follows: The astronomical latitude and longitude of each of a number of stations are determined. The stations are connected by an accurate geodetic survey. All the latitudes and longitudes are then reduced to one of the stations by use of the known elements of the ellipsoid. The mean 1 6 GEODETIC ASTRONOMY. 1 6. system of spherical co-ordinates, frequently used by the astronomer but seldom by the engineer. In this the ecliptic and vernal equinox play the same part as do the equator and vernal equinox in the case of declinations and right ascen- sions. 16. In general, when the engineer observes a heavenly body he has one of four objects in view, namely, to determine his astronomical latitude, the azimuth of a line joining his station with some other terrestrial point, the true local time at the instant of observation, or the longitude of his station. The determination of longitude always involves a determina- tion of the true local time together with additional operations which are in some cases quite complicated. The instrument used in any case for the determination of time, latitude, or azimuth indicates the position of the horizon, and conse- quently of the zenith, by means of attached spirit-levels, or of the various values of the latitude of this single station as thus obtained is called its geodetic latitude. The corresponding statement applies to the longitude. It is evident that the greater the number of stations and the more widely scattered they are the nearer will the vertical as given by the geodetic latitude and longitude coincide with the normal to the ellipsoid. The difference between the astronomical and geodetic latitude at a given point is therefore usually called the station error in latitude. A similar statement defines station error in longitude. For further information on this subject see Clark's Geodesy, pp. 287-288, Merriman's Geodetic Sur- veying, pp. 79-88, or any extended treatise on geodesy. Station errors in longitude, or deflections of the vertical at right angles to the meridian, change the plane of the meridian from the position it would otherwise occupy and so change all azimuths from the values they would otherwise have. Hence there arises the same distinction between the astronomical azimuth of a line and its geodetic azimuth as is drawn above between the astronomical and geodetic latitudes and longitudes. On account of station error the line of gravity at a station and the axis of the Earth do not, in general, intersect. Hence to be exact the meridian plane must be said to be defined, not by the line of gravity and the Earth's axis of rotation, but by the line of gravity and the point in which the axis of rotation produced intersects the celestial sphere. 1 6. PURPOSE OF OBSERVATIONS. 1 7 by a basin of mercury having a free horizontal surface. The star, or other celestial object, is usually observed with a tele- scope. The two points on the celestial sphere always observed are, therefore, the zenith and the object. The right ascension and declination of the object observed become known, independently of the observations, by the methods indicated in the next chapter. The process most frequently used is to acquire, by instrumental observation and by the means indicated in the next chapter, a knowledge of three of the elements, arcs and angles, of some triangle on the celes- tial sphere of which one of the unknown elements, now capable of computation, is the quantity sought, or is one from which the required quantity can be readily derived. For example, suppose that the latitude of the station of observation is known, and that the zenith distance of a certain star is accurately observed. Let the true local sidereal time (see 1 8) at the instant of observation be required. In the triangle on the celestial sphere defined by the pole, the zenith, and the instantaneous position of the star, the arc from the zenith to the pole is known, being the complement of the latitude of the station. The arc from the star to the pole becomes known by the methods indicated in the next chapter, since it is the complement of the declination of the star at the instant of observation. The arc from the zenith to the star, the zenith distance, was directly observed. Hence in the spherical triangle pole-zenith-star all three arcs are known and any of the angles may be computed. The angle at the pole of that triangle is the hour-angle of the star at the instant of observation. This being computed by the methods of spherical trigonometry, a mere addition to or subtraction from the right ascension of the star (which becomes known by the methods of the following chapter) gives the true local sidereal time (as will .be shown later). 1 8 GEODETIC ASTRONOMY. 1 8. 17. On account of the rapid apparent motion of most celestial objects, time enters as an important element into almost every astronomical problem with which the engineer has to deal. Three kinds of time are in use in astronomy: sidereal time, apparent solar time, and mean solar time. The passage of a star or other celestial object across the meridian is called its transit or culmination. The meridian (a great circle of the celestial sphere) is divided into two half-circles by the poles. If the whole of the meridian be considered, a star has two transits for each complete rotation of the Earth on its axis : one over that half of the meridian stretching from pole to pole which includes the zenith, and the other over that half which passes through the nadir. The first of these is called the upper transit or upper culmination, and the second the lower transit or lower culmination. The word transit or culmination unmodified usually means the upper transit. The expression " the passage of a star across the meridian " refers, of course, to the apparent motion of the star. It would be more accurate to say that the meridian passes the star. But to refer directly to the apparent motion as if it were real saves circumlocution, is more clear in many cases, and is not misleading if one keeps in mind that this is merely a mode of speech. 18. A sidereal day is the interval between two successive transits of the vernal equinox across the same meridian. Its hours are numbered from o to 24. The sidereal time is o h oo m oo s at the instant when the vernal equinox transits across the meridian. The sidereal time at a given station and instant is the right ascension of the meridian, or is the same as the hour-angle of the vernal equinox, counted in the direc- tion of the apparent motion of the stars, at that station and instant. Right ascensions being reckoned .from west to east, ig. APPARENT SOLAR TIME. 1 9 opposite to the apparent motion of the stars, it follows from the above definition that the sidereal time at the instant of transit of a star is the same as the right ascension of that star. The sidereal day is substantially the interval of time required for one rotation of the earth on its axis, and the uniformity of the rotation of the earth is depended upon to furnish the ultimate measure of time. Because of the motion of the vernal equinox on the celestial sphere, about 50" per year, the sidereal day and the time of one rotation of the earth on its axis differ by about one one-hundredth of a second. 19. The interval between two successive transits of the Sun across the meridian is called an apparent solar day. The apparent solar time for any instant and station is the hour- angle of the Sun, at that instant, from that meridian. " But the intervals between successive returns of the Sun to the same meridian are not exactly equal, owing to the varying motion of the Earth around the Sun, and to the obliquity of the ecliptic." Let Fig. i represent a section of the universe on the plane of the Earth's orbit as seen from some position in space on the side on which the north pole is situated. The Earth is seen moving around its orbit in a counter-clockwise direction, while at the same time its rotation about its own axis appears to be counter-clockwise. The figure is not to scale, but is merely a diagram in which certain dimensions are exaggerated for the sake of clearness. Suppose that A is the position of the Earth at a certain time, about March 21, when the Sun is seen projected against the celestial sphere upon the vernal equinox. Let B be the position of the Earth one sidereal day later. Then Aa and Ba are parallel lines, the vernal equinox being at an infinite distance (on the celestial sphere). The Earth has made one complete rotation on its axis between 20 GEODETIC ASTRONOMY. 19. the two positions, and the vernal equinox has returned to the same meridian. The Earth having moved a distance AB along its orbit, the Sun is now seen projected against the celestial sphere at b instead of a. Before the Sun will return to the meridian of position A again the Earth must rotate through the additional angle represented by aBb reduced to the plane of the Earth 's equator. (The figure represents a section in the plane of the ecliptic.} The apparent solar day will then be longer than the sidereal day by the time required for the Earth to rotate through this angle, on an average a little less than four minutes. Let the angle governing the excess of the apparent solar over the sidereal day be examined further. As the Earth proceeds forward along its orbit the Sun will apparently move backward on the celestial sphere along the ecliptic to points , c, d, etc. One of the laws of gravitation governing the motion of the Earth in its orbit is that the line joining the Earth to the Sun sweeps over equal areas in equal times. The linear velocity then varies nearly inversely as the distance to the Sun, and the angular velocity varies still more than the linear. The angular velocity is about 7$ greater during the winter (of the northern hemisphere) than during the summer. The various arcs ab, be, cd, etc., along the ecliptic, each corresponding to one sidereal day, will vary in value through that range. But the excess of the apparent solar over the sidereal day depends upon these arcs projected upon the equator along hour-circles, the rotation of the Earth being uniform when measured along the equator. When such a small arc as ab near either equinox is projected upon the equator, it will be considerably reduced, being at an angle of 23^ to the equator, the angle between the equator and ecliptic at the equinoxes. On the other hand, when a portion of the ecliptic about 20. TIME. 21 midway between the equinoxes is projected along its limiting hour-circles upon the equator, the projected length will be greater than the original. In short, the difference between the sidereal and apparent solar day varies by a rather compli- cated law from- about 4 m 26 s to 3 m 35% being on an average 3 ra 56 s . 55 5 (in sidereal time). 20. Apparent solar time is a natural and direct measure of duration, inasmuch as it is indicated directly by the hour- angle of the Sun, the most conspicuous of all the heavenly bodies. But a clock or chronometer cannot be regulated to keep this kind of time accurately, since the different days are of unequal length. To avoid the difficulties thus arising from the direct use of the Sun as a measure of time, a fictitious mean Sun is used. The mean Sun is supposed to move in the equator with a uniform angular velocity, and to keep as near the real Sun as is consistent with perfect uniformity of motion. This mean Sun makes one complete circuit around the equator at a uniform rate while the Earth is making a com- plete circuit around its orbit, at a variable rate. It is some- times as much as 16 minutes ahead of the real Sun, and sometimes behind it by that amount. A mean solar day is the interval between successive transits of the mean Sun over the same meridian. The mean solar time for any instant and station is the hour-angle of the mean Sun at that instant from that meridian. For brevity mean solar time is often called simply mean time. The mean solar day is about 3 m 56" longer than the sidereal day, that being the amount by which the apparent solar day exceeds the sidereal day on an average. Stated more exactly, 24 hours of mean solar time is the same interval as 24 h 03 $6\$$$ of sidereal time. The sidereal and mean solar time coincide for an instant about March 2 1 each year. The former gains 24 hours on the latter in a year. 22 GEODETIC ASTRONOMY. 21. The equation of time is the correction to be applied to apparent time to reduce it to mean time. It is the interval of time by which the mean Sun precedes or follows, or is fast or slow of, the real Sun at a given instant. Its limiting values are about -f- i6 m and i6 m . It is given in the American Ephemeris and Nautical Almanac for every noon at Washing- ton (and Greenwich). It can be obtained for any intermediate instant with an error not greater than o s . I, usually much less, by a simple straight-line interpolation. 21. The civil day, according to the customs of society, commences and ends at midnight. The hours from midnight to noon are counted from o to 12 and are marked A.M. The remaining hours from noon to midnight are again num- bered from o to 12 and marked P.M. The astronomical day commences at noon on the civil day of the same date. Its hours are numbered from o to 24, from noon of one day to noon of the next. The astronomical time as well as the civil time may be either apparent solar or mean solar. The convenience of the astronomical day for the astronomer arises from the fact that he does not have to change the date on his record of observations in the midst of a night's work as he would be obliged to if he used civil dates. The zeros of sidereal, apparent solar, and mean solar time are, by definition, the instants of transit, across the meridian, of the vernal equinox, the Sun, and the mean Sun, respec- tively. The time, therefore, (of any of the three kinds,) will be the same for two stations at a given instant only in case those stations are on the same meridian. If the stations are not on the same meridian the difference of their times (of any of the three kinds) is a difference of two hour-angles measured from the respective meridians to the same object, and is therefore the angle between the meridians or the differ- 22. TIME. 23 ence of longitude of the two stations. A difference of longi- tude is then a difference of time. 22. In the United States, excluding Alaska, for every mile of distance, east or west along a parallel of latitude, the longitude changes by about four or five seconds of time. If each city and town used its own local mean solar time, the traveller would find himself at considerable inconvenience, on the modern railroads which transport him from 500 to 1000 miles per day, to keep his watch regulated to the time of his various stopping points. Even when the railroads and the general public used one particular time for considerable areas, that time being usually that of some large city or important railroad division terminus, as was the case a few years ago, there was still confusion and annoyance arising from the fact that each kind of time was changed to the next by the addi- tion or subtraction of some irregular number of minutes, which was apt to be forgotten when most needed. These, and other reasons, have led to the general adoption in this country of what is called standard time. The standard time for each particular locality is the mean solar time of the nearest meridian which is an exact whole number of hours, four, five, six, seven, etc., west of Greenwich. The standard meridians for this country are thus: 75 or 5 h west of Greenwich, running near Utica, N. Y., Philadelphia, Pa., and off Cape Hatteras. 90 or 6 h west of Greenwich, running near St. Louis, Memphis, and New Orleans. 105 or 7 h west of Greenwich, running near Denver, Colorado. 120 or 8 h west of Greenwich, running along the east line of the northern part of California and near Santa Barbara, Cal. 135 or 9 h west of Greenwich, running near Sitka, Alaska. To reduce the local mean solar time to standard time it 24 GEODETIC ASTRONOMY. 23. is merely necessary in each case to apply as a correction the difference of longitude of the station and the standard meridian. Only the astronomer or engineer, however, is obliged to use this process. The traveller has occasion simply to change from one kind of standard time to another which differs from it by exactly one hour, an interval which is easy to remember. To Convert Mean Solar to Sidereal Time. 23. To convert mean solar to sidereal time or vice versa, it is necessary to take account logically of two facts, that the zeros of the two kinds of day differ by a certain interval, to be derived from the Ephemeris, and that the two kinds of hours bear a fixed ratio to each other which is nearly, but not quite, unity. The local mean solar time at St. Louis, Mo., 52 37 8 .o/ west of Washington, is Q h 2i m 23 S .35 A.M., July 29, 1892. What is the local sidereal time ? Local mean solar time = g h 2i m 23". 35 Time of mean noon = 12 oo oo .00 Mean solar interval to nearest mean noon = 2 38 36.65 Reduction to sidereal interval (see 290) = -f- o oo 26 .06 Sidereal interval to nearest mean noon 2 39 02.71 Sidereal time of mean noon, July 29, 1892, at Washington = 8 h 3i m 14". 23 Correction due to longitude to re- duce to St. Louis (see below). ... = -f- o oo 08 .64 Sidereal time of mean noon, July 29, 1892, at St. Louis = 8 31 22.87 Required sidereal time at St. Louis = 5 52 20 . 16 The first step is to obtain the mean solar interval between the given time and the nearest mean noon, and to reduce it to an equivalent sidereal interval by use of the tables in 290 23. CONVERSION OF TIME. 2$ (reprinted from the back part of the Ephemeris). The derivation of these tables from the equation given at the end of each is sufficiently obvious. The next step is to derive from the American Ephemeris and Nautical Almanac the sidereal time of that mean noon, or, in other words, the difference of the zero points of the two kinds of time at noon of that day. The Ephemeris, in the part headed " Solar Ephemeris " (pp. 377-384 in the volume for 1892), gives directly the sidereal time of every Washington mean noon for the year. What is required is the sidereal time of St. Louis mean noon. The vernal equinox, marking the zero of sidereal time, shifts 3 m 56 S .555 per mean solar day with respect to the mean Sun, marking the zero of mean solar time. The sidereal time of mean noon for a given point then increases 3 m 56 3 .555 per day. St. Louis being 52 37 8 .O7 west of Washington, its mean noon occurs at that interval of mean solar time later than the mean noon of Washington. Its sidereal time of mean noon is evidently that of Washing- ton increased by the motion of the vernal equinox relative to the mean Sun in 52 37 S .O7, or This proportional part is precisely that given by the table, 290, for the reduction of mean solar to sidereal time, and hence the correction is taken directly from that table. Having now the sidereal interval to the nearest local mean noon, and the local sidereal time of that mean noon, the required sidereal time is obtained by a simple subtraction (or addition, as the case may call for). Note that the longitude of the station is used only in reducing the sidereal time of mean noon at Washington to the local sidereal time of mean noon. An error of 4 s in the longi- tude produces an error of only O 8 .oi in this reduction. 26 GEODETIC ASTRONOMY. 2$. Example of the Reduction from Sidereal to Mean Time. 24. At a certain instant in the evening of May 21, 1892, at Harvard Observatory, it was found by an observation upon a star that the sidereal time was I3 h 4i m 27 s . 34. What was the mean time at that instant? Harvard Observatory is 23 m 41" east of Washington. Given sidereal time = i3 h 41 27". 34 Sidereal time of mean noon, May 21, 1892, at Washington = 3 h 59 H 8 .73 Correction, due to longitude, to reduce to Harvard Obser- vatory ( 290) = o oo 03 .89 Sidereal time of mean noon, May 21, 1892, at Harvard Observatory '= 3 59 07.84 Sidereal interval after mean noon = 9 42 19.50 Reduction to mean time interval ( 291) = o or 35.40 Required mean time at Harvard Observatory = 9 40 44 .10 P.M. The Ephemeris, 25. The American Ephemeris and Nautical Almanac referred to in the above computation is an annual publication of the United States Government. It can be obtained at any time by sending one dollar to the Nautical Almanac Office, Washington, D. C. It, or its equivalent, is a necessity to an engineer making astronomical determinations, as will be seen by the many references to it in the following chapters. As it forms a part of the outfit of the astronomical observer and computer, the student should become familiar with its general arrangement, should acquire a general understanding of all parts of it, and should obtain a thoro'ugh grasp of those par- ticular portions to which he finds especial reference in the text of this book. To gain familiarity with the most frequently used portions of the Ephemeris, it is especially desirable that the following pages of the text at the back of 26. QUESTIONS AND EXAMPLES. 2/ the Ephemeris headed " On the Arrangement and Use of the American Ephemeris" be read; viz., the first four pages of the explanation of Part I (pp. 493-496 in the volume for 1892), and the first three pages of the explanation of Part II (pp. 501-503 of the volume for 1892). The Governments of Germany, France, and England, and some others, issue similar publications. QUESTIONS AND EXAMPLES. 26. i. The position of the Sun projected upon the celes- tial sphere is always at some point of the ecliptic. Explain why this statement is not true in regard to a planet. 2. What is the relation between the latitude of a station and the altitude of the pole at that station ? 3. Given the latitude of a station and the declination of a star, how may the zenith distance of the star at the instant of upper culmination be determined ? 4. In the case of a circumpolar star how may the zenith distance at lower culmination be determined, the declination and latitude being given ? A circumpolar star is one comparatively near the pole, say within ten degrees. 5. How would you determine the zenith distance at upper culmination, and also at lower, for a circumpolar star of which the polar distance is given ? The latitude of the station is supposed to be known. 6. The hour-angle of the star Vega, east of the meridian, at a certain instant on the evening of June 30, 1892, at the Cornell Observatory was 2 h I i m 14 s . The right ascension of Vega at that instant was i8 h 33 m 19 s . What was the local sidereal time? Also, what was the Washington sidereal time, Cornell being 2 m i6 s east of Washington ? 28 GEODETIC ASTRONOMY. 26. 7. At a certain instant the hour-angle and zenith distance of a star are observed. The declination of the star is known. In the spherical triangle star-zenith-pole what parts are known and how may the latitude of the station be computed ? 8. What was the hour-angle of the Sun on September 29, 1892, at a station 4 h I9 m 46 s . 3 west of Washington, when a clock which was 3 I s . 9 fast of local mean time indicated 2 h 4I m l8 s <9 P.M.? The equation of time for apparent noon at Washington on September 29 was 9 57 s ./i and for the 3Oth, io m I7 S .05. Ans. 2 h 5 What was the mean time? The equation of time at Washington apparent noon on that date was 38 S .95, and on the i8th was 52 S .49. Ans. io h 32 m 35 S .2 A.M. 12. The hour-angle of the Sun as observed at a certain instant, at a station 2 h 14 34" east of Washington, on the forenoon of May 21, 1892, was found to be 2 h 48* 19". 3. What was the sidereal time ? The equation of time for apparent noon at Washington was 3 m 3 7 s . 96 on May 20, 26. QUESTIONS AND EXAMPLES. 29 and 3 m 33 S .96 on the 2ist. The sidereal time of mean noon at Washington on the 2ist was 3 h 59 m n s .73. Ans. I L o6 m 27 s . 3. 13. Suppose you are carrying a watch which is 2O 8 fast of standard (75th meridian) time, and that you wish to start a sidereal clock on correct time to within I s at Cornell (2 m i6 8 east of Washington or 5 h O5 m 56 s west of Greenwich). Sup- pose the date to be Sept. 30, 1892, and the sidereal time of mean noon for Washington on that date to be I2 h 39 37 s . 2. What time should the sidereal clock indicate when your watch reads f oo m oo 8 ? Ans. 19* 34 m 29". 14. Explain why the " sidereal time of mean noon" as given in the last column of the " Solar Ephemeris " (pp. 377-384 in the volume for 1892), in the American Ephemeris and Nautical Almanac is not the same as the "apparent right ascension " at " mean noon " as given in the second column. 15. Why is not the " equation of time for apparent noon " as given in the eighth column, the same as the differ- ence of the two columns mentioned in the preceding example? 1 6. What is the relation between the right ascension of the Sun at mean noon, the equation of time at mean noon, and the sidereal time of mean noon ? 17. Look up the sidereal time of mean noon for to-day in the Ephemeris. Then, knowing the time of day and your latitude, hold two sheets of paper parallel respectively to the plane of the equator and the plane of the ecliptic. 30 GEODETIC ASTRONOMY. 2 7. CHAPTER II. COMPUTATION OF RIGHT ASCENSION AND DECLINATION. 27. In the astronomical practice of the engineer the right ascension and declination of the object observed are usually known quantities determined from sources external to his own observations. The object of this chapter is to show how the right ascension and declination for the instant of observation are obtained from the available sources of information. The various heavenly bodies which the engineer is called upon to observe have all been observed frequently at the various fixed observatories with large instruments and at many different times extending over a long period of years. From these observations the positions, that is, right ascen- sions and declinations, at various stated times are determined, and the motions are carefully computed. This makes it pos- sible to compute the position of each of these bodies at any stated future time with an accuracy depending on the pre- cision of the observations and the remoteness of the future time. The results of such computations of positions made in advance, and also the data for such computations, are given in the ephemerides issued by various governments: the American Ephemeris, Berliner Jahrbuch, Connaissance du Temps (Paris), British Nautical Almanac, etc. Various other occasional publications also give the data for such computa- tions. The engineer uses these computations of position made in advance, and the published data for such computa- 2Q. INTERPOLA TION. 3 1 tions, to obtain the right ascension and declination at the instant of his observation. When references are given in the following text to data in the American Ephemeris, it should be understood that sub- stantially the same data may also be obtained from the other national ephemerides. Position of the Sun and Planets. 28. The right ascension and declination of the Sun are given for Washington mean and apparent noon in the American Ephemeris (pp. 3/7-384 of the volume for 1892) for every day of the year, together with some other data that are frequently needed for computation purposes. The corre- sponding data are also given for Greenwich in first part of the Ephemeris. The right ascension and declination are also given in the American Ephemeris for each planet for every day of the year when its transit is visible at Washington (on pp. 393-411 of volume for 1892). The corresponding data are given in more complete form for Greenwich in the first part of the Ephemeris (pp. 218-249 f tne volume for 1892). For the methods by which the right ascension and declination of the Sun, or a planet, at any given intermediate time, are to be derived from the values stated in the Ephemeris, see the following sections, Nos. 29-34. Interpolation. 29. By interpolation is meant the process by which, hav- ing given a series of numerical values of a function corre- sponding each to a stated value of the independent variable, the value of the function for any other intermediate value of the variable is found independently of a knowledge of the analytical form of the function. The independent variable is often called the argument. For example, the right ascension 32 GEODETIC ASTRONOMY. 30. of the Sun is a known function of time as the independent variable. It is given in the Ephemeris for certain stated times. When it is required for any other time, instead of computing it directly from the known function, it is much more convenient and rapid to deduce it by interpolation from the stated numerical values. Interpolation always leads to approximate results which may be made more exact as the process of interpolation is made more complicated and laborious. The error of inter- polation is the difference between an interpolated value and the value which would be found if one resorted to direct computation from the known function. Of the multitude of methods of interpolation, with widely varying degrees of con- venience, rapidity, and accuracy, three methods will be found sufficient for the ground covered by this book. These three may be described briefly as interpolation along a chord, inter- polation along a tangent, and interpolation along a parabola. Interpolation along a Chord. 30. In interpolation along a chord the rate of change of the function, between the two stated values of the variable which are adjacent to the value for which the interpolation is to be made, is assumed to be constant and equal to the total change of the function between those points divided by the interval between the stated values of the variable. If the actual values of the function were represented graphically, all inter- polated values would lie along chords of the function curve, connecting points on the curve corresponding to stated values of the variable. For example, the right ascension of Jupiter at I2 h 35 m .5 mean time at Washington, on Oct. i, 1892, was (Ephemeris, p. 405). Required its right ascension at I5 h I4 m .2 Washington mean time, on Oct. i? The interval 31- INTERPOLATION. 33 between stated values of the variable is 23 h 55 m .6 = 23^93. The change in the value of the function is 29 8 .oi. The rate of change is then 29 8 .oi -r- 23". 93 = I 8 .2I2 per hour. The interval over which the interpolation is carried from the nearest given value is I5 h I4 m .2 I2 h 35.$ = 2 h 38 m .7 = 2 h .64. The change during that interval is (2.64) (1.212) = 3 s . 20. The required right ascension is i b 2i m 07 8 .8i 3'. 20 = i h 2i m 04 s .6i. The result would have been identical with this had the interpolation been made from the other adjacent value, namely, that at I2 h 31. ion Oct. 2d. In algebraic form this interpolation maybe expressed thus: VrV . or Fl= F t -(F t -Ffj. . . . (i) the first form being used when the interpolation is made forward from the value F lt and the second when it is made backward from F. 2 . Fj is the required interpolated value corresponding to the value F/ of the independent variable. V l and F, are the adjacent stated values of the argument to which correspond the given values F l and F 9 of the function. Interpolation along a Tangent. 31. Interpolation along a tangent is, in general, more accurate than interpolation along a chord, but can only be used conveniently when the rates of change, or first differen- tial coefficients of the function, are given at the stated values of the variable, in addition to the values of the function itself. In this interpolation the rate of change, for the inter- val from the nearest stated value of the variable to the value 34 GEODETIC ASTRONOMY. 31. for which the interpolation is to be made, is assumed to be constant and equal to the given rate of change at the stated value of the variable. The interpolated points represented graphically would lie on a tangent, at the nearest stated value of the variable, to the curve representing the function. For example, let it be required to find the declination of the Sun on Sept. 5, 1892, at 9 h 30 m A.M., Washington mean time. On page 382 of the Ephemeris for that year, the nearest time for which the declination is given, is Washington mean noon of that day. For that instant the declination is + 6 28' i8".6. Its rate of change for that instant is stated to be 5 5". 92 per hour. The interval over which the interpolation is to extend is 2 h .5 backward from noon. Then by interpolation along the tan- gent to the curve (representing declinations) at noon of Sept. 5th, there is obtained as the declination at 9 h 30 A.M., 6 28' i8".6 + (2.5)(55 // .92) = 6 30' 38".4. In this method of interpolation the shorter the tangent the smaller the error of interpolation, and therefore care should be taken to inter- polate from the nearest stated value of the variable. The formula for this interpolation is Fj and Vj are the required interpolated value and the corre- sponding given argument, V l and F : are the nearest tabular value of the argument and the corresponding value of the function, and (-77?) is the given first differential coefficient corresponding to V^ 33- INTERPOLATION. 35 Interpolation along a Parabola. 32. In interpolation along a parabola it is assumed that the second differential coefficient of the function is constant between adjacent stated values of the independent variable, or, in other words, that the rate of change of slope of the function curve is constant between those points. This assumption places the interpolated points along a parabola, with axis vertical, passing through two points of the function curve, the uniform rate of change of slope being a property of such a parabola. There are two cases arising under this method, depending upon whether the first differential is, or is not, given for the stated values of the variable. 33. For an example of the first case take the problem proposed in the preceding section, in which it is required to find the declination of the Sun at g h 30 A.M., Washington mean time, Sept. 5, 1892. The data given in the Ephemeris for 1892 for Washington mean noon Sept. 4 are declination = +6 50' 37". 5, and the first differential coefficient = 55".66; and for Sept. 5, declination = 6 28' i8".6, and first differential coefficient = 5 5". 92. It is proposed to place the interpolated value on a parabola (with axis vertical) coin- ciding with the curve of declinations at the two given points, and also having a common tangent at each of these points. To make the interpolation, the principle will be used that a chord of such a parabola is parallel to the tangent at a point of which the abscissa is the mean of the abscissae of the two ends of the chord. The slope of the chord (of the parabola) corresponding to the interval 9 h 3O m to I2 h , on Sept. 5, is then the same as the slope of the tangent at the middle of that interval, io h 45. The slope of the tangent changes by ( 5 5". 92) ( 55".66) = o".26 in 24 hours, or o".oio8 per hour. The slope at io h 45* = $$".92 36 GEODETIC ASTRONOMY. 34. + (c/.oio8)(i.25) = 55".9i. The interpolated value at 9 h 3 o m is 6 28' i8".6 + (55". 9 i)(2. 5 ) = 6 30'' 38". 4 . This method of interpolation, though most easily remem- bered, perhaps, in the geometrical form, may be put in con- venient algebraic form as follows: dF\ ((dF\ ldF\ 1 fi(F>- F,)l ~1 or Fi = according to whether the interpolation is made forward from F t or backward from F 2 . The notation is the same as in the preceding paragraphs. The two results are identical, but the arithmetical work will be shorter if the interpolation is made from whichever of the given points happens to be the nearer. 34. The second case of interpolation along a parabola occurs when the first differential coefficients are not given. The assumptions involved are just as before. As an example, take the problem proposed a few paragraphs back, of finding the right ascension of Jupiter, at I5 h 14. 2, Washington mean time, on Oct. i, 1892. The Ephemeris gives the right ascension = i h 2i m 368.59 at I2 h 39 m .9 on Sept. 30; = i h 2i m o; 8 .8i at I2 h 35 m .5 on Oct. i ; = i h 20 m 38 s .8o at I2 h 31. i on Oct. 2. It is proposed to interpolate the required point on a pa- rabola, with axis vertical, passing through these three given r (dF\ fdF * If the second derivative is constant, then ridF\ - \_\dVh is really ,. Call Vi V\* dV. Then (3) put in the calculus notation dF becomes FI = Fi-\- -. dV ' -\- , ' in which - and TTTT are values dV dfy* 2 dV * corresponding to the point F\ . V\. 34- INTERPOLATION. 37 points. Again, using the principle that in such a parabola, a chord, and the tangent at a point of which the abscissa is the mean between the abscissae of the two ends of the chord, are parallel, the slope of the parabola at any point may be com- puted. The slope of the parabola at the middle of the first interval, at o h 37 m -7 = o h .63 on Oct. i, is (07". 8 1 36". 59) -i- 23.93 = I B .2O3 per hour. At the middle of the second interval, at o h 33^3 = o h .56 on Oct. 2, it is (38 8 .8o 67 8 .8i) -f- 23.93 = i 8 . 212 per hour. The interval over which the interpolation is made, from the nearest given value, is I2 h 35 m .5 to I5 h I4 m .2 on Oct. i, or 2 h 38 m .7 = 2 h .64. The slope of the chord for this interval is that of the tangent at its middle, I3 h 54. 8 = I3 b .9i. This slope is, assum- ing the rate of change of the slope constant, I 8 .2O3 -j- s 56ll|E<- I-.2I2) - (- I-.203)] = - I-.208 per hour. The right ascension at I5 h I4 m .2 is i h 2i m O7 8 .8i ( i 8 . 208X2.64) = i h 2i m 04 s . 62. This sample interpolation is made in the present form simply for the purpose of illustrat- ing the principles involved. The numerical work of inter- polation should ordinarily be done as indicated in formula (4) of the following section. Putting this method in the algebraic language it takes the following form: Let F^ F w and F % be three successive given values of the function corresponding to the values F lf F 3 , and F s of the independent variable ; and let F a be the stated value of the variable nearest to which lies the value for which the interpolation is to be made. Let F 2 be the required value of the function corresponding to V f . Then y . I rr y i y / -Pi+l Fa _ ^+ 1 r> _ ^ 3 - Fa _ ri j Fa _|_ ra FT+F! K Fi - r O; ~T~ ~~^~ 38 GEODETIC ASTRONOMY. 35 or, in simplified form, FF ^FF F F 1 f Vr Fi 3 l I I s *^1 -*!| *l- l I */ *i If, as is usually the case, the successive differences between F,, F a , and F 8 are all the same and equal to D, this may be further simplified to the form ,- V,\; ( 4 a) in which d^ is the second difference, or (F^ F^ (F^ F^. The second term in the square bracket will usually be com- paratively small, and therefore easy to compute. For a more complete discussion of interpolation, giving other more complex and accurate formulae, see Chauvenet's Spherical and Practical Astronomy, vol. I. pp. 79-91 ; Doolittle's Practical Astronomy, pp. 69-98; and Loomis* Practical Astronomy, pp. 202-212. Accuracy of Interpolation of Position of Sun and Planets. 35. An interpolation along a tangent, the first differen- tial coefficients or hourly changes being given, from the values given for noon of each day in the Ephemeris (pp. 377-384 of the volume for 1892), will give the right ascension of the Sun at any time with an error of interpolation not exceed- ing o s .6, and the declination with an error of interpolation not exceeding i /r .8. For nearly all cases the error of interpola- tion will be much less than these extreme limits. Approxi- mately, the extreme error of interpolation along a tangent is one-eighth of the second difference at that point,* meaning * The interpolation along a tangent will evidently give the greatest error when the interpolated point is midway between the tabulated values, 37- STAR PLACES. 39 by a second difference the difference between successive first differences. If greater accuracy is desirable, which will often be true of declinations, but seldom of the right ascen- sions, an interpolation along a parabola will always give all needful accuracy. In dealing with the planets an interpola- tion along a tangent, or along a chord in those cases in which the first differential coefficients are not given, will in many cases give a sufficient degree of accuracy, and interpolation along a parabola will give all needful precision in every case. Position of the Moon. 36. In the first part of the Ephemeris, in which the standard meridian is that of Greenwich (pp. 2-217 f the volume for 1892), the Moon's right acension and declination are given for every hour during the year, together with the corresponding first differential coefficients. An interpolation along a tangent, from the nearest hour, will give the Moon's right ascension at any time with an error of interpolation not exceeding O 3 .O5. The corresponding limit for declination interpolated along a tangent is i" . This will usually be a sufficient degree of accuracy. But if for some special reason a greater precision is required, an interpolation along a parab- ola will give the results far within the limits of error of the tabular values themselves. Positions of Stars. 37. The American Ephemeris gives the right ascension and declination of four close circumpolar stars for every upper the tangent then used, corresponding to one-half of a tabular interval, being longer than is necessary in any other case. If for this case the interpolation along a parabola be used, the interpolated value will differ from that found by using the tangent by one-eighth the second difference, as may be seen by inspection of the formulae (2), 31, and (3), 33. If, then, the second interpolation be assumed to be exact, this value is the error of the first interpolation. 40 GEODETIC ASTRONOMY. 3/. transit at Washington (pp. 302-313 of 1892); of every tenth transit for about 200 stars; and the right ascension only for every tenth transit visible at Washington of about 200 more. Other national Ephemerides contain similar lists, which often comprise about the same stars. This list is made up of stars whose positions are well determined by many observations at various observatories. They are also chosen with especial reference to the needs of the engineer and navigator as regards brightness and distribution on the celestial sphere. An idea of the care with which their positions have been determined may be gained from the mere statement of the fact that in computing many of these declinations fifty cata- logues of recorded observations, at many different observa- tories, made at various times during a total interval of a century and a quarter, were consulted, and the various observations upon any one star combined in each case in a single least-square computation.* The positions of the close circumpolars at any time may be obtained with all needful accuracy by interpolation along a chord from the values given in the Ephemeris. For the other stars given in the Ephemeris (at lO-day intervals) an interpolation along a parabola will usually be necessary. When other stars must be observed than these Ephemeris stars of which the places are given at frequent intervals, a complicated procedure is necessary to obtain the position of the star at the time of the observation. This process forms the subject of the remainder of this chapter. The position or place of a star is usually given in one of * See "Survey of the Northern Boundary from the Lake of the Woods to the Rocky Mountains " (Washington, 1878), pp. 409-615, for a complete report on the computation of star places for that survey by Lewis Boss (pp. 421-424 give catalogues consulted). Many of the star places given in the Ephemeris are from this computation. 3$. A BERRA TION. 4 1 three ways, which should be carefully distinguished. Either its apparent place, true place, or mean place is given. The right ascension and declination, as defined in 12, indicate the true place of a star or other celestial object. But the apparent direction of a star, even aside from the refraction of the line of sight by the terrestrial atmosphere, is affected by aberration. The apparent place of a star is its true place modified by the aberration of light. An observer sees a star in a position which differs from what is technically called its apparent place by the effect of refraction only.* It should be carefully noted that the word " apparent " is not here used in the ordinary sense, but in the special technical sense which it must be understood to have hereafter throughout this book. Aberration. 38. Aberration is an apparent displacement of a star resulting from the fact that the velocity of light is not infinite as compared with the velocity of motion through space of the observer, stationed at a point on the Earth's surface. If one is standing in a rain which is falling in vertical lines, the umbrella must be held directly overhead. If, how- ever, one is riding rapidly through such a rain-storm, the umbrella must be inclined forward. In the first case a drop of rain entering at the centre of one end of a straight open tube held with its axis vertical would pass along the axis of the tube to the other end without touching the tube. In the second case, however, if it is desired that drops which enter the tube at the upper end shall continue down the tube without touching the sides, it will be necessary to incline the tube forward from the vertical to a certain angle which is dependent on the relative velocity of the horizontal motion of the tube and the vertical motion of the rain. So when a * For a detailed consideration of refraction see 67-69. 42 GEODETIC ASTRONOMY. 3Q. telescope is to receive along its axis the light undulations from a star, it must be inclined forward in the direction of the actual motion of the telescope in space so as to make a slight angle the aberration with the actual line joining telescope to star. This small angle, the aberration, amounting at most to about 20", is evidently dependent upon the relative velocity of light and of the telescope, and the angle between those two velocities. (The student may easily draw a diagram for himself showing the geometrical relations con- cerned.) The motion of the telescope is compounded of that due to the diurnal rotation of the Earth on its axis and the annual revolution of the Earth about the Sun. These give rise to the diurnal aberration and annual aberration, respec- tively. The diurnal aberration evidently affects right ascen- sions directly, but has no effect upon declinations. The annual aberration in general affects both. For an example of the way in which diurnal aberration is taken into account in computations, see 96. The effect of annual aberration is included in the apparent place computation created later in this chapter. The velocity of light is, according to the best determi- nations, about 186300 miles per mean solar second.* It requires about eight minutes for light to travel from the Sun to the Earth. An observer, then, does not see a celestial object in its true position at the instant when the light enters the eye, but in the position which it occupied when that light left the object an appreciable interval earlier, for all celestial objects. This phenomenon is called planetary aberration. With this form of aberration the engineer is not concerned. 39. The mean place of a star is its position referred to the mean equator and mean ecliptic, as distinguished from its * See " The Solar Parallax and its Related Constants," Wm. Harkness, Washington, 1891, pp. 142 and 29-32. 39- MEAN- PLACES. 43 position as referred to the actual or true equator and ecliptic. The equator and ecliptic as they would be if unaffected by periodic variations, in other words by nutation, are called the mean equator and mean ecliptic. The mean place of a star, then, at a given instant, differs from the true place by the effect of nutation at that instant, and from the apparent place by the effects of both nutation and aberration. To avoid inconveniences arising in the course of computa- tions of star places, if any other form of year is employed in reckoning time, the astronomer uses what is called the Besselian fictitious year. The beginning of the fictitious year is the instant at which the celestial longitude of the mean Sun is 280, or, in other words, when the mean Sun is 280 from the vernal equinox measured along the ecliptic.* The beginning of the fictitious year differs from the beginning of the ordinary year by a fraction of a day, which varies for different years. The places given in the Ephemeris, referred to in 37, for every day or every ten days, are apparent places, and are so marked. When the engineer is obliged to have recourse to stars which are not so given in the Ephemeris, he consults one or more of the various available star catalogues or star lists. f These catalogues and lists give the mean places of the stars at the beginning of some stated fictitious year, together with other data relative to each star. The problem which then confronts the engineer is to derive, from that given mean place, the apparent place at the time at which his observation was made. This is done in two steps. Firstly, the mean place of the star is reduced from the epoch of the catalogue to the beginning of the fictitious year at some * See definition of celestial longitude, 15. f For references to a few of such catalogues and lists sec 141. 44 GEODETIC ASTRONOMY. 41. part of which its apparent place is desired. Secondly, the apparent place of the star at the time of observation is deduced from the mean place at the beginning of the ficti- tious year. Reduction of Mean Places from Year to Year. 40. To serve as a concrete example, let it be supposed that the star ^ Hercules was observed at its transit across the meridian at St. Louis, Mo., on July 16, 1892; and the authority depended upon for its position is Boss's Catalogue of 500 Stars for 1875.0.* This star is No. 312 in that cata- logue and its mean place as there given for the beginning of the fictitious year 1875 * s ^1875.0 = i? h 4i m 34 s - o = mean right ascension; #1875.0 + 2 7 47' 4 2 "' 1 ? mean declination. (Throughout this book a and 6 will be used to indicate the apparent right ascension and declination, respectively, at the time of the observation under consideration. The same letters with the subscript m , thus, a m , d m , will be used to indi- cate the mean place. With a year as a subscript as above, they will be understood to indicate the mean place at the beginning of that fictitious year.) 41. The reduction from the mean place at 1875.0 to that at 1892.0 involves simply the change in the mean equator and mean ecliptic during that time. The determination of the laws of change of these two fundamental reference circles, and the method of computing the effect of those changes upon right ascensions and declinations, belong rather to the prov- ince of the astronomer than to that of the engineer. It * Survey of the Northern Boundary from the Lake of the Woods to the Rocky Mountains (Washington, 1878), pp. 592-615. 41. ME A N PL A CES. 45 suffices for the engineer to accept the results of the investi- gations of the astronomer in the following form : da m f . = *+. sin <* tan * + /!; ... (5) - = n . cos a m // ; (6) d*d m dn da in which m = 46". 062 3 -f- o". 0002 849^ 1800) (/ being ex- pressed in years), and n = 20". 0607 o // .oooo863(/ 1800). The numerical values for m and n as here given are those most extensively used, and are the result of exhaustive investigations by the astronomers Peters and Struve. /* and // are proper motions per year in right ascension and declina- tion respectively, for an account of which see 44, 45. For the present these proper motions may be considered simply as changes at a uniform rate in each of the two co-ordinates, without any reference to their meaning or method of deriva- da m dd m tion. j~ and -j- are rates of change per year. The formulae given above are neither complete nor exact, many terms of the exact formulae having been dropped, and those which are retained having been somewhat modified. But they furnish the complete basis for a reduction, with sufficient accuracy for the purposes of the engineer, from the mean place given in a catalogue to the mean place at the beginning of any other fictitious year within thirty or perhaps fifty years. For the formulae in complete form adapted to the use of the astronomer to bridge over long intervals of time, sometimes more than a century, see " Survey of the Northern Boundary from the Lake of the Woods to the 46 GEODETIC ASTRONOMY. 42. Rocky Mountains," pp. 416-420; and for a detailed discus- sion of them see Doolittle's Practical Astronomy, pp. 560- 578 and 583-5^9. 42. The engineer is, however, relieved of the necessity for performing the numerical operations indicated by formulae (5), (6), and (7). For star catalogues and lists give in addi- j 7j\ 7 2 j\ tion to a m and S m the values of , -jr, and the term -j~ is tabulated,* in 292 of this book, for the arguments a m and ji-, of which it is evidently a function. Students will find slight differences between different authorities in regard to the nomenclature of this part of the * This table is, so far as the author knows, a new one. It was com- puted from formula (7) above, for the date 1900, to six places of decimals and afterward reduced to five. It is hoped that all the tabular values are, in so far as the computation is concerned, within 0.6 of a unit in the fifth d*d m place. The formula used for 7-5- is, however, approximate in itself in having omitted the effect of proper motion. Theory indicates that this omission should produce an error so small as to be negligible for our present purpose. To test that conclusion, as well as the accuracy of com- 2 j> putation of the table, for fifty stars (every tenth) of Boss' List, Northern Boundary Report, was derived from the table and compared with that given by Boss in the list. Boss' values were computed from the exact formulae. The greatest difference found was o". 00003. This would cause an error of only o".oi in a reduction extending over 30 years, and only o".O4 in 50 years. It is believed, therefore, that the table is abundantly accurate within the limits over which its arguments extend. It should not, however, be assumed to hold good beyond those limits. The table does not cover the comparatively rare cases in which is negative (for stars near the pole). For these cases the formula (7) must be used. The table is computed for the year 1900. The same computation made for any date between 1700 and 2100 would give values differing from those of the table by not more than one unit in the last decimal place given in the table. 43- MEAN PLACES. 47 S subject. p is usually known as the " annual variation in right ascension/' ^ is sometimes given under the heading " annual variation in declination." Sometimes the terms n cos a and // are given separately as " annual precession " TJ and " proper motion," respectively. . a m is sometimes given in the form of a " change per 100 years " in the annual pre- cession. Having given the mean place of a star, a m and 6 m for a date t m at the beginning of some fictitious year, the place at time t at the beginning of any other fictitious year is (8) ~(t, - o f d ^r- (9) 43. To return to the numerical case in hand, the following data are also given in Boss* list for the star /* Hercules: ~7j annual variation in right ascension = + 2 s - 345 \ -r = annual variation in declination, including proper motion all for the date 1875. The mean place for 1892 is then, by (8) and (9), l892 = 27 47' 42". 17 + (i7)(-2 // .370i)+Ki7) a (o // .oo 3 38o) = 27 47 r 42". 17 - 40".2 9 + o".49 = 27 47' O2 // . 3 7. 48 GEODETIC ASTRONOMY. 44. d*d m If TT had not been given in the star list, as frequently it is not, it could have been obtained by entering the table 292 with the arguments a m = i? h 4i m .6 and ~ = -|- 2 s . 345. The value as then found from the table would have been = o". 0034 1, and the final value for d l892 would have been identical with that given above. Proper Motion. 44. When the co-ordinates of a star, as observed directly at widely separated times, are reduced to the same epoch, it is usually found that, aside from discrepancies arising from accidental errors of observation, there are systematic differ- ences in the various values indicating a steady movement of the star in some one direction with the lapse of time. Observations on another star indicate usually that it has also such a motion peculiar to itself, which is without any apparent relation to the motion of the first star. So each star is, in general, found to have an unexplained motion peculiar to itself, called its proper motion. This proper motion is always exceedingly small, and is assumed to take place along an arc of a great circle of the celestial sphere, and at a uniform rate in each case. Probably neither of these assumptions are strictly true; but the accumulated proper motion for several centuries even would be so small that observations of the highest degree of accuracy now obtainable would not be sufficient to prove the path of star to be curved, or its motion to be other than uniform. When the mean position of a star for a given date is to be derived from the results of many observations at various times in the past by the process indicated briefly in 37, an unknown annual proper motion in declination, and another in 45- PROPER MOTION. 49 right ascension, are introduced into the least square adjust- ment. The annual proper motions in declination and in right ascension as thus derived are then used in deducing the place of the star at any future date in the manner indicated in formulae (8), (9), (5), and (6), 41, 42. For a full discus- sion of the treatment of proper motion, from the astronomer's point of view, with the refinements necessary when reductions are to be made covering very long periods of time, see Doolittle's Practical Astronomy, pp. 578-583, and Chauve- net's Astronomy, vol. I. pp. 620-623. A concrete idea of the magnitude of the proper motion usually found may be gained from the fact that in the Boss Catalogue of 500 Stars for the epoch 1875.0 there are only 8 stars out of the 500 for which the annual proper motion in declination exceeds o".5O. In 367 cases it is less than o". 10. Proper motions in right ascension are of the same order of magnitude, keeping in mind, of course, that I s of right ascension represents, for a star near either pole, a much smaller displacement upon the celestial sphere than I s for a star near the equator. 45. That the so-called proper motion is not really due to an erroneous determination of the precession, is put in evi- dence by the fact that the various proj>er motions for different stars do not show the systematic relation which they must necessarily have if due to a shifting of the reference circles. Precession does not change the relative positions of the stars. Proper motions do. To what are the proper motions due ? rst. If they are due to a motion of the solar system as a whole through space, the stars which are ahead in the direc- tion of motion must seem to be separating in all directions from the point toward which we are moving, must seem to be going backward at the sides, and apparently closing together behind us, just as points of the landscape seem to a traveller 50 GEODETIC ASTRONOMY. 46. to move. 2d. If the proper motions are due to actual motions of the stars themselves, acting as entirely independent bodies, the proper motions should seem to be without any relation to each other. 3d. If, on the other hand, they are due to actual motions of the stars, which are not, however, independent, one would expect to find laws connecting the proper motions, laws, however, which would differ from those called for by the last supposition above. A close study of the proper motions seems to indicate that there is some truth in each of the three suppositions. Computations based upon hundreds of observed proper motions, made by different astronomers at various times, have all agreed, in a general way, in indicating that there is a slow motion of the solar system as a whole through space toward a point in the neighborhood of a = i/ h , d = -j- 35. For details in regard to the computations, see Chauvenet's As- tronomy, vol. I. pp. 703-708. In regard to the third sup- position, it may be noted that in a few rare cases of double stars, two stars apparently very near to each other, the observed proper motions indicate that the two revolve about some common centre are linked together by gravitation. But though some laws connecting the various proper motions have been thus discovered, the salient fact to keep in mind is that the second supposition is very largely true, that the discovered laws only account for an extremely small fraction of the actually observed proper motions. Reduction from Mean to Apparent Place. 46. To reduce from the mean place at the beginning of the year to the apparent place at a given date, it is necessary to reduce the mean place up to date, and then apply to that result the effect of nutation and aberration at that date. 47- MEAN TO APPARENT PLACE. 5 1 This computation, if made directly from the known laws of nutation and aberration, is very laborious.* But such a direct computation is not necessary. This is again one of the cases in which it is advisable for the engineer simply to accept the results of the astronomer's investigations in the convenient form in which they are given in the Ephemeris, without going through all the details of the derivation of those results. 47. Suffice it to say that this reduction has been put in the following convenient form : sin (G + a ) tan tf o sin C^+ <*<>) sec <$o ( m time); (10) = t --r/- cos -f- h cos (H ' + ) = 9.8185* log tan S = 9.7217 log T5? sin (G + <*o) tan S = 9.3263* y 1 ^ sin (G -f <*o) tan = 0.21 log iV = 8.8239 log h 1.3043 H 156 52', (#" + a ) = 62 25', log sin (ff+ a ) = 9-9475 log sec 5 = 0.0532 log && sin (#" + <*<>) sec d = 0.1290 /fc sin H-- o sec S a, at i6 h io m 14*.!. St. Louis Sidereal Time, July 16, 1892 =17'' 42 i6 8 .oo The computation for 5, following the order of (n) is d = 27 47' 02".37 TJJ.' = (o.54)( o".76), [n f is given = o".76 in Boss' list] = o .41 log S o-9 OI 9 log cos (G -f a o) = 9.8766* log g cos (G -f tfo) = 0.8382 -cos(+ o- ) = - 6 .89 log h = 1.3043 log cos (/f + <*o) = 9.6661 log sin <5 = 9.6685 log h cos (If-}- cto) sin d =. o 6384 h cos (//+ a ) sin d Q = -j- 4 .35 log f = 0.5363 log cos d = 9.9468 log i cos 5 = 0.4831 * cos d =4-3 .04 6, at i6 h io m i4.i. St. Louis Sidereal Time, July 16, 1892 = 27 47' O2".46 The above example shows how far out the computation needs to be carried. Where many star places are to be com- puted, the computation is materially shortened by using printed blank forms so arranged as to facilitate the work. 54 GEODETIC ASTRONOMY. . 49. Especially convenient forms of that nature are in use in the Coast and Geodetic Survey. 49. In computing a number of stars on a single night which will usually be the case in dealing with latitudes observed with a zenith telescope considerable time will be saved at an exceedingly small sacrifice of accuracy by the following procedure. First interpolate the values of the independent star-numbers for every whole hour from Wash- ington mean midnight for the period over which the oberva- tion extends. Then for each star use the interpolated value of each star-number for the nearest hour as interpolated, instead of making a special interpolation for each. For an account of the method of computation of the independent star-numbers, and the method of computing star places by the use of the Besselian star-numbers, see Doolittle's Practical Astronomy, pp. 609-617; Chauvenet's Astronomy, vol. I. pp. 645-651 ; and the Ephemeris, pp. 280-284 (f tne volume for 1892). The Besselian star-numbers are not ordi- narily so convenient for the engineer as the independent star- numbers. If one has a great number of star places to compute, under certain conditions, the work may be abridged somewhat by using differential and graphic methods. For the details of a differential method which reduces the labor of computa- tion about one-half in case the place of each star is to be computed on three or more nights, see Coast and Geodetic Survey Report, 1888, pp. 465-470. A somewhat similar method to be used when the places are to be computed for a few stars on many nights will be found in the Coast and Geodetic Survey Report for 1892, Part II, pp. 73-75. For a graphic method of reducing from the mean to the apparent place in declination, see Coast and Geodetic Survey Report, 1895, pp. 371-380. 50. QUESTIONS AND EXAMPLES. 55 QUESTIONS AND EXAMPLES. 50. I. At a certain instant in the forenoon of May 27, 1892, the observed hour-angle of the Sun at Cornell Observa- tory was 2 h 30 41 s . What was its apparent right ascension and declination at that instant ? Cornell is 2 m i6 s east of Washington. The Ephemeris for 1892 (p. 380) gives for Washington apparent noon May 26th a = 4 h 15 47*.93, d = 2i if 45". 3, hourly motion in right ascension = -f- io s .i4i, hourly motion in declination =-\-2$".i6; and for Washington apparent noon May 2/th a = 4 h 19 5i B .57, 6 = 21 27' 38 // .3, hourly motion in right ascension = -f- l6 8 . 160, in declination = -f- 2 4 // - 2 3- Ans. By interpolation along a tangent a = 4 h 19 25 8 .67, 6 = 21 26' 36".5. By interpolation along a parabola a = 4 h I9 m 25 s . 67, 6 = 21 26' $6". 4. 2. What was the apparent declination of the Sun at 7 h 4i ra 32 s A.M. Greenwich mean time on Dec. 21, 1892 ? The Ephemeris for 1892 (p. 201) gives the apparent declina- tion of the Sun at Greenwich mean noon Dec. 2ist = 23 27' i8".6, and its hourly motion = + o". 18; and for Dec. 20th <$ = 23 27' 08". 7, with an hourly motion = i".oo. Ans. 23 27' i8".9. 3. Work the preceding problem, as a check, from the fol- lowing data from the Ephemeris (p. 384): Declination of Sun at Washington mean noon Dec. 2ist = 23 27' 17". o, hourly motion = -\- o".43. Hourly motion for Washington mean noon Dec. 2Oth = o".75. Washington is 5 h 8 m I2 8 west of Greenwich. Ans. 23 27' i8".9. 4. At a station 2 h 58 west of Washington the south zenith distance of Jupiter, at the instant of its meridian transit on July 16, 1892, was observed to be 29 22' 17". 4. $6 GEODETIC ASTRONOMY. 50. What was the latitude of the station ? The apparent declination of Jupiter at its meridian transit at Washington is given in the Ephemeris (p. 404) as follows: July I5th -f- 7 56' u".8, July 1 6th + 7 57' 53".8, and July i;th + 7 59' 32 // .o. ^,y. 37 21' 23".5. 5. What was the right ascension and declination of the Moon at 8 h 30 09" P.M. local mean time at Cornell April 8, 1892 ? Cornell is 5 h 5 m 56 s west of Greenwich. In the Ephemeris (p. 62) the position of the Moon is given for 2 h A.M. Greenwich mean time April 9th, a = ii h 33 43 8 .9i, 3 = -f- 7 34' 02". 2; " difference for i minute" in right ascension = + I s - 7980; " difference for I minute " in decli- nation = 1 3". 192. The differences for I minute in right ascension and declination respectively at i h A.M. are -f- I 8 .8oo3 and 13". 167. Ans. By interpolation along a tangent a = n h 33 oo s .9i, S = 7 39' i7".7- By interpolation along a parabola a = i i h 33 m oo s .9O, 6=7 39' I 7 ".6. 6. What was the apparent right ascension of the star A Aquarii at transit at Mount Hamilton, Cal., Sept. i, 1892 ? For upper transit at Washington (Ephemeris, p. 362) on August 27th a = 22 h 46 6i s .5i, and =- = + o s . 10 per ten days. Also for Sept. 6th a 22 h 46 6i 8 .66, and -^ + O 8 .O5 per ten days. The longitude of Mount Hamilton is 2 h 58 m 22 s west of Washington. Ans. By interpolation along a chord a 22 b 47 oi s .63. By interpolation along a tangent from Sept. 6th a '= 22 h 47 m oi 8 .64. By interpolation along a parabola a = 22 h 47 oi 8 .63. 7. For star BAG 5706 * I875 = i6 h 50 4?. 9, d l875 = 46 SO- QUESTIONS AND EXAMPLES. 57 44' 3 1 ".22. Its annual variation in right ascension for that date = -f- I s . 72 1, and in declination (including proper motion) = 6''. 0105. What was its mean declination for 1895.0 ? Ans. S^. If each is 5', the maximum error introduced into a measured angle of 120 is 4 /x .o, and for other angles in the ratio indicated above. This error is therefore small provided the adjustments are carefully made and frequently verified, but it is sensibly a constant affecting the mean of a set of observations made at nearly the same reading of the arc. If the telescope is parallel to the plane of the sextant, but, in observing, the contacts are made with the images out of the center of the field, the sight line is inclined to the plane of the * Chauvenet's Astronomy, vol. n. p. 116. 73- ZKXORS. 85 sextant, and the effect on the measured angle is the same as if the telescope were so inclined. It is important, therefore, that every observation should be made nearly in the middle of the field of the telescope. If the horizon-glass is not perpendicular to the plane of the sextant, the error introduced is greater the smaller the angle observed, and will ordinarily be appreciable only in the determination of the index error. In determining the index error by observing the Sun's semi-diameter, the error in any one reading from this cause will be less than \" even if the horizon-glass is inclined as much as 50" to its normal position (which is about the maximum error of this adjustment made as indicated in 55). This error is eliminated from the derived index correction, for both positive and negative read- ings are numerically too small by the same amount. If the center about which the index-arm swings does not coincide with the center about which the graduated arc is described, an error due to this eccentricity will be introduced into every reading. The magnitude of this error will evi- dently depend upon the size of the angle measured as well as upon other conditions. See 76 for the method of deter- mining, and correcting for, eccentricity. . The errors treated in the last three paragraphs are func- tions of the angle measured, but are constant for a given reading of the sextant so long as the condition of the instru- ment remains unchanged. Their effect may therefore be eliminated almost wholly from the final result in determining time by measured altitudes of the Sun, by making observa- tions both in the forenoon and afternoon at about the same altitude. The computed altitude will be too great, or too small, by the same amount in both cases, if the two altitudes are equal, and one computed time will be as much too late as the other is too early. This procedure will also ^eliminate the 86 GEODETIC ASTRONOMY. 74. error in the computed time arising from an error in the assumed latitude. The errors arising from changes in the relative position of different parts of the sextant due to stresses or to changes of temperature are probably small in comparison with the other errors considered under the next heading. So also are the errors of graduation of the sextant arc. The error which may arise from either or both glasses of the horizon roof being prismatic instead of plane, may be eliminated by reversing the roof when half the observations have been taken. To avoid errors arising from the prismatic form of the shades, some instrument-makers provide a contrivance by which the colored shades may be rotated 180 from their original position; but it is better to use colored shades between the eyepiece and the eye instead of the colored shades in front of the index and horizon glasses. A shade in this position may be of a prismatic form without vitiating the observed results. Observer's Errors. 74. The errors which are classed as instrumental depend to a considerable extent upon the care and judgment with which the sextant is manipulated. But aside from the manipulation, which is an important as well as a difficult portion of the observer's duty, the final result also depends upon his estimates of the positions of contact of the two images and of the chronometer times of those contacts. His estimate of the position of contact is subject to both an accidental and a constant error.* The accidental error * A constant error is one which has the same effect upon all the observa- tions of the series, or portion of a series, under consideration. Accidental errors are not constant from observation to observation; they are as apt *o 74- XAOAS. 87 depends mainly upon the personality and experience of the observer and the care with which he observes, but also to a certain extent upon the steadiness of the refraction, the power of the telescope, the brightness and definition of the images, and the physical conditions affecting the observer's comfort. A probable error of 14" seems from various recorded series of observations to be a fair estimate of the accidental error in a single measurement of the Sun's double altitude by an experienced observer with an ordinary sextant under average conditions. The accidental error in the mean of twelve observations constituting a set is on this basis 14" -f- 4/i2 = 4". This corresponds, for observations taken in the United States when the Sun is observed from two to four hours from the meridian, to o 8 . 15 to O 8 .4O. The con- stant error of the observer's estimate of the position of contact is eliminated from the mean for a set if half the observations are taken upon the Sun's upper limb and half upon the lower. The observer's estimate of the time of contact is also subject to both an accidental and a constant error. The accidental error, judging from time observations made with a transit instrument, is about o 8 . 1 for a single observation, or 8 .03 for a mean of twelve observations constituting a set, a small error as compared with that arising from the uncertainty of the position of contact. The constant error made in estimating the time, or personal equation* of the observer, may be as great as O 8 .5 for some men. It affects all the observations of a set alike. be minus as plus, and they presumably follow the law of error which is the basis of the theory of least squares. It is then the effect of accidental errors upon the final result, which may be diminished by continued repe- tition of the observations and by the least square methods of computa- tion, whereas the effect of constant errors must be eliminated by other processes. * See 125. 88 GEODETIC ASTRONOMY. 76. Error of the Computed Time. 75. The mean result from a set of observations such as that given in 63 is subject, then, to an accidental error of about O 8 .25, on an average, arising almost entirely from the observer's accidental errors. It is also subject to an error which is constant for the set, arising from uneliminated instru- mental errors and the error of the computed refraction (neglect- ing for the time being the personal equation of the observer). A fair estimate of this constant error under average field con- ditions seems to be O 8 .25. This makes the probable error of the result from the set about V 0.25" -f- 0.25' = o s .35, aside from personal equation. It is evident from the above estimate that increasing the number of observations in a set, or number of sets taken under the same circumstances, diminishes the final error but little (only one term under the radical above being reduced). The constant instrumental error may, however, be almost entirely eliminated by making observations at about the same altitude in both forenoon and afternoon as indicated in 73. There is no feasible way of eliminating the personal equation error in the field. The above estimate of the errors from various sources is believed to be a fair one for average conditions. A special investigation for a particular observer and set of con- ditions may show errors either somewhat smaller or much larger than those indicated. Correction for Eccentricity. 76. Unless the center about which the index-arm swings coincides exactly with the center of the graduation, every sextant reading will be in error by the effect of this eccen- tricity (as noted in 73), which effect is different for readings taken on different parts of the arc. To eliminate the effect 76. ECCENTRICITY. 89 of eccentricity upon the sextant readings one may proceed as follows : First. The values of angles as measured with the sextant may be compared with their true values determined in some other way. For example, the angles between certain terrestrial objects may be measured with the sextant and then with a good theodolite. In making this comparison it must be kept in mind that a theodolite as ordinarily used measures horizontal and vertical angles, while the sextant measures directly the angle between the two objects in the plane (horizontal, oblique, or vertical) passing through the two objects and the sextant. Also, in this case, the sextant parallax, 53, must be taken into consideration unless the objects are very distant. The angular distance between two known stars may be observed and compared with its value as computed from the known right ascensions and declinations of the stars, corrected for the effect of refraction at the time of observation. This computation will be found, unfortunately, to be rather laborious. Or, the altitude of a known star (or of the Sun) may be measured at a known time at a station of which the latitude is known. The true altitude of the star may be computed, and becomes comparable, after correction for refraction, with that measured with the sextant. Second. For each sextant observation which is compared with a known angle an observation equation of the form J x + K y + v = D A ..... (17) is formed, in which x y y, and v are unknowns to be deter- mined, and D A B t Q m is the difference between the true 90 GEODETIC ASTRONOMY. 7/. value of the angle 6 t and the measured value m . 6 m is the reading of the sextant corrected for index error. e 8 e J = sin -- cos - and K = sin 2 . . . (18) Third. The most probable values of x, y, and v are determined from these observation equations by the method of least squares. Fourth. These values of x, y, and v may now be substi- tuted in equation (17) and a table of corrections computed by substituting o, 10, 20, . . . successively for 6 m , the corre- sponding computed values of D A being evidently the correc- tions for eccentricity which must be applied to measured angles.* So many and such accurate observations are required for a satisfactory determination of the eccentricity of a sextant that it will usually be found more convenient to eliminate the effect of eccentricity upon time observations by observing both in the forenoon and afternoon with the Sun at about the same altitude, as indicated in 73. But in sextant observa- tions for latitude a special determination of the eccentricity is necessary if the highest attainable degree of accuracy is desired. Other Uses of the Sextant. 77. In determining time with a sextant by the preceding method the latitude is supposed to be known. If, conversely, the time of observation of the altitude of the Sun (or a star) * This method of determining the corrections to be applied for eccen- tricity, which is here given in condensed form, and without the derivation of the formulae, will be found treated in full in Doolittle's Practical Astronomy, pp. 196-206. Certain refinements there given, which add much to the labor of computation and little to the accuracy of the computed result, have here been omitted. 79- MISCELLANEOUS. 9! is known, the latitude of the station may be computed by the method of 171, or by that of 172, if the observation is made near the meridian. The most favorable time for thus determining the latitude by observations upon the Sun is about apparent noon, for then the altitude is changing very slowly, and hence an error in time will have but little influence on the computed result. The refraction is also a minimum at apparent noon. 78. At sea, observations with the sextant for time are usually made in the middle of the forenoon and of the after- noon, and for latitude at apparent noon. To make this lati- tude observation the Sun is watched with a sextant for a few minutes before apparent noon, as its altitude increases slowly at a diminishing rate. The observation is made when the altitude stops increasing and is at its maximum. With suffi- cient accuracy it may be assumed that the Sun is then on the meridian, and that therefore the latitude is the declination of the Sun plus its south zenith distance. In observing at sea the natural horizon is used, and an allowance must be made for the dip of the horizon, or downward inclination of the line of sight to the apparent horizon, due to the height of the sex- tant above the surface of the sea (see table, 298 *). The observation of latitude and of local time serves to locate the observer at sea, provided he also knows the Greenwich time of the observation. JjjThis ^ ast he dbtains from the known rate of his chronometer and its known error on Greenwich time at some previous date. 79. An observation at sea of the altitude, at a known instant of Greenwich time, of any celestial object (Sun, Moon, planet, or star) serves to locate the observer upon an *This table is reproduced, with a slight extension, from Chauvenet's Astronomy. It is computed for a mean state of the atmosphere. 92 GEODETIC ASTRONOMY. 80. arc of a small circle on the Earth's surface, of which the pole is at a point in the line joining the object and the Earth's center, and of which the polar distance is equal to the observed zenith distance of the object. This small circle, or such a portion of it as is necessary, may be plotted on a sphere or chart. A second such observation on an object in some other azimuth serves to locate the observer on another small circle intersecting the first in two points. These two points of intersection are usually so far apart that there is no difficulty in discriminating between them, and the observer's position becomes definitely known. This process of deter- mining a position at sea is known as Sumner's method. For a more complete statement of this method see Chauvenet's Astronomy, vol. I. pp. 424-428. 80. If, for the purpose of determining local time, an obser- vation is taken upon a star east of the meridian, and the observation is repeated west of the meridian at the same reading of the sextant, the computation of time may be made independently of any knowledge of the index error, eccen- tricity, or other errors of the sextant, and independently of any computation of the refraction, upon the assumption that these quantities retain the same values at the second observa- tion which they had at the first, and that therefore the two observations are made at the same (unknown) altitude. Dur- ing such an interval of a few hours, the declination of any star is for the present purpose sensibly constant. Upon these assumptions it may be shown that the mean of the two observed chronometer times is the chronometer time corre- sponding to the transit of the star across the meridian. (Let the student prove this.) If the object observed is a planet, the Moon, or the Sun, the same method of computation may be used, but it will be necessary to apply a correction for the 8 1. MISCELLANEOUS. 93 change of declination during the interval between the two observations.* The advantages of this method are the ease and simplicity of the computation. Its disadvantage is the liability of losing the second observation on account of clouds or other hindrances. If observations are taken within one hour (say) of the same hour-angle east and west of the meridian respec- tively, are computed as indicated in 64, and the mean taken, the elimination of errors is almost as complete, advan- tage may be taken of temporary breaks in the clouds, and the observer is not put to the inconvenience of being ready at some particular moment. The computation will consume a little more time. 81. The Covarrubias method of observing, developed by the Mexican astronomer of that name, serves to eliminate the instrumental errors, and accomplishes that purpose without the necessity of the long wait between observations which is required in the method stated above. Two stars are selected which are several hours apart in right ascension, and have declinations not very different. At a certain time each night, which is first estimated roughly by the observer, these two stars will for an instant be at the same altitude, one east and the other west of the meridian. A few minutes before this time he observes one of the stars, noting the chronometer time and the sextant reading. He then turns to the second star, which he finds approaching the same altitude, and observes the chronometer time at which the sextant reading, and therefore the altitude, is the same for this second star as that before observed upon the first star. From this observa- tion of the two chronometer times at which the two stars reach * For the computation of this correction, see Doolittle's Practical Astronomy, pp. 230, 231; Chauvenet's Astronomy, vol. I. pp. 198-201; or Loomis' Practical Astronomy, pp. 126-130- 94 GEODETIC ASTRONOMY. 82. the same altitude the error of the chronometer may be com- puted independently of any knowledge of the exact absolute value of that altitude. This method will sometimes be found desirable, especially in case the only available sextant is an inferior one, or has been damaged to such an extent that its indications are unreliable. It is not developed in detail here for lack of space. For a complete statement of the method see " Nuevos Metodos Astronomicos para determinas la hora, el azimut, la latitude y la longitude"; F. D. Covarrubias, Mexico, 1867. QUESTIONS AND EXAMPLES. 82. i. Prove, using figures if necessary, that the test as given in 54 for determining whether the index-glass is per- pendicular to the plane of the sextant is valid. 2. Explain why two images of the same object cannot be made to coincide in the sextant telescope if the index-glass is in perfect adjustment but the horizon-glass is inclined to the plane of the sextant (see 55). Explain also how it is possi- ble that such coincidence may be secured if both the index and horizon glasses are inclined. Why is it advisable to make certain of the index-glass adjustment before adjusting the horizon-glass ? 3. Suppose that the index correction of a certain sextant is found to be 15" '. Through what angle and in what direction must the horizon-glass be rotatecl to make the cor- rection zero ? 4. Show that the errors due to the inclination to the plane of the sextant of the sight line and of the index glass are not eliminated by the process of eliminating the index error indicated in 62. 5. The radius of the graduated circle of a certain sextant is 4 in. What linear movement of the vernier corresponds 82. QUESTIONS AND EXAMPLES. 95 to a change of 10" in its reading ? Explain the need of the caution in the last sentence of 58. 6. Explain, by diagrams if necessary, why the images behave as stated in 58 when the sextant is moved in the various ways there described. 7. Show that the limb of an image of the Sun seen in a sextant telescope, which is preceding with respect to the apparent motion of the image, corresponds necessarily to the preceding limb of the Sun, regardless of the number of reversals to which said image may have been subject in its progress to and through the telescope. By the use of this principle show that when observations are made with ap- proaching images it is the upper limb of the Sun which is being observed if it is forenoon and the lower limb if it is afternoon. 8. What is the error of the chronometer on local mean time from the last half of the set of observations given in 63 ? Ans. 2 h i i m 53 8 .5.* * In this example the student may find that his computation gives a result differing by as much as o 8 .2 from the one here given on account of calling o".5 a whole second, where it has in this computation been called zero, or vice versa. The fact that such a difference may exist may be used as an argument for carrying the computation to one more decimal place. A careful investigation indicates, however, that such a procedure would add so much to the labor of computation, especially in making the various interpolations, that it is not considered advisable. If the computation were carried one decimal place farther, the computed result from a complete set of observations would seldom be changed by more than o'.i, whereas the probable error of that result is o*.3 or o'.4. For a further discussion of the question of the number of decimal places to which a computation should be carried, see 277. 96 GEODETIC ASTRONOMY. 83. I CHAPTER IV. THE ASTRONOMICAL TRANSIT. 83. THE astronomical transit is designed primarily to be used for the determination of time with its telescope in the plane of the meridian. Its essential parts are a telescope, an axis of revolution fixed at right angles to the telescope, a suitable support for said axis, such that it shall be stable in azimuth and inclination, and a striding level with which the inclination of the axis may be determined. Fig. 10 shows an astronomical transit which is now and has been for several years past in use in the U. S. Coast and Geodetic Survey for time determinations of the highest order of accuracy. The focal length (distance from the lines of the eyepiece diaphragm to the optical center of the objective) of the telescope AB is 94 cm. (37 in.). The clear aperture of the object-glass is 7.6 cm. (3 in.), and the magnifying power with the diagonal eyepiece, A, ordinarily used is 104 diameters. In the focus of the eyepiece is a thin glass diaphragm upon which are ruled lines which serve the same purpose as the spider lines or cross wires more commonly placed in that position in a telescope. The system of lines consists of two horizontal lines near the middle of the field, and thirteen vertical lines. The milled head shown at C con- trols, by means of a rack and pinion, the distance of the diaphragm from the object-glass, and serves therefore to focus the object-glass, or, in other words, to bring the image 83. DESCRIPTION OF TRANSIT. 97 formed by the object-glass into coincidence with the dia- phragm. The diaphragm, and the corresponding image formed by the object-glass, are much larger than the field of view of the eyepiece. To enable the observer to see various parts of the diaphragm, and the corresponding portions of the image, the whole eyepiece proper is mounted upon a horizontal slide controlled by the milled head shown at D. The lines of the diaphragm are seen black against a light field. The illumination of the field at night is obtained from one of the lamps shown at E. The light from the lamp passes in through the perforated end of the horizontal axis, and is reflected down to the eyepiece by a small mirror in the interior of the telescope, fastened to a spindle of which the milled head G is the outer end. The perforated disk shown at F carries plain, ground, and colored glasses, to be used by the observer to temper the illumination. The horizontal axis FF is 51.5 cm. (2oJ in.) long, ending in pivots of bell metal. HH is the striding level, in position, resting upon the pivots of the horizontal axis. The iron sub-base, a portion of which shows at /, is cemented firmly to the pier. The transit base is carried by three foot-screws resting upon this sub-base. The device shown at J serves to give the instrument a slow motion in azimuth. The lever K actuates a cam to raise the cross-piece L y and with it the columns MM. The horizontal axis is then raised sufficiently upon the forks at the upper end of M and M to clear the Ys. The cross-piece L is then free to turn (180) until arrested by the fixed stops, and thus to reverse the horizontal axis FF in the Ys. The setting circles NN are 10 cm. (4 in.) in diameter, are graduated to 20' spaces, and are read to single minutes by verniers. They are set to read zenith distances. 98 GEODETIC ASTRONOMY. 84. Fig. 1 1 shows another type of transit in use in the U. S. C. & G. S. Its peculiarities are the folding frame, the graduated scale at Q to facilitate putting the telescope in the meridian, and the fact that the eyepiece is furnished with a movable line carried by a micrometer screw, while one of the setting circles carries a sensitive level so that the instrument may be used both as a zenith telescope (see Chapter V) and a transit. The screw Amoves the upper base 55 in azimuth with respect to the lower base RR. The Theory of the Transit. 84. If a transit were in perfect adjustment the line of collimation * of the telescope as denned by the mean line of the reticle would be at right angles to the transverse axis upon which it revolves, and that transverse axis would be in the prime vertical, and horizontal. Under these circumstances the line of collimation would always lie in the meridian plane * The line of collimation of a telescope is that line of sight to which all observations are referred. In an engineer's transit the line of collimation is the line of sight on which all observations are made, and is defined by a vertical line in the middle of the field of view of the telescope. In an astronomical transit the observations are made on the several lines of sight defined by the several lines of the reticle. These various observations are all referred to an imaginary line of sight, or line of collimation, which is defined, however, by the mean of all the lines, and not by the middle line. The mean line is, of course, near to the middle line, the spacing of the lines in the reticle being made as nearly uniform as possible. Imagine a plane passed through any line of the reticle of a telescope and through the center of the object-glass. Imagine the plane produced indefinitely beyond the object-glass. Evidently every point of which the image is seen in the telescope in apparent coincidence with this line of the reticle must lie in this plane in space. The line of the reticle may be said to define this plane in space, or a point of the reticle line may be said to define a line in space. For convenience a line of the reticle is ordinarily spoken of as defining a line of sight rather than a plane of sight, it being tacitly understood that one point only of the reticle line is referred to ordinarily the middle point. 85. ADJUSTMENTS. 99 and the local sidereal time at which any star might be seen in the line of collimation of the telescope would necessarily be the same as the right ascension of that star. In observing meridian transits for the determination of time, these condi- tions are, by careful adjustment of the instrument, fulfilled as nearly as possible. The time observations themselves, and certain auxiliary observations, are then made to furnish determinations of the errors of adjustment; and the observed times of transit are corrected as nearly as may be to what they would have been if the observations had been made with a perfectly adjusted instrument. The observed time of transit of any star, as thus corrected, minus the right ascension of the star, is the error (on local sidereal time) of the chronometer with which the observation was made. Adjustments of the Transit. 85. Let it be supposed that observations are about to be commenced at a new station at which the pier and shelter for the transit have been prepared. By daylight make the fol- lowing preparations for the work of the night. By whatever means are at your disposal determine the direction of the meridian, mark it upon the top of the pier, and put the foot-plates of the transit in such positions that the transit telescope will swing nearly in the meridian (true, not magnetic). A compass needle will serve for this purpose if no other more accurate and equally convenient means is at hand. An accurate determination of the meridian is not yet needed. To give the foot-plates a good bearing upon the pier and to fix them rigidly in position, it is well to cement them in position with plaster of Paris. Set up the transit and inspect it. Focus the telescope carefully if it is not already in good focus. The eyepiece may be first focused upon the reticle with the telescope IOO GEODETIC ASTRONOMY, 86, turned up to the sky. The focus for most distinct vision of the reticle lines is what is required. Now direct the telescope to some distant object (at least a mile away, and preferably much farther) and focus the object-glass, by changing its dis- tance from the reticle, so that when the eye is shifted about in front of the eyepiece there is no apparent change of rela- tive position (or parallax) of the lines of the reticle and of the image of the object. If the eyepiece itself has been properly focused, this position of the object-glass will also be the position of most distinct vision. The focus pf the object-glass will need to be inspected again at night, and corrected if necessary, using a star as the object. None but the brightest stars will be seen at all, unless the focus is nearly right. Bisect some well-defined distant object, using the apparent upper part of the middle vertical line of the reticle. Rotate the telescope slightly about its horizontal axis until the object is seen upon the apparent lower part of this same line. If the bisection is still perfect, no adjustment is needed. If, how- ever, the bisection is no longer perfect, the reticle must be rotated about the axis of figure of the telescope until the line is in such a position that this test fails to discover any error. 86. Now bisect the distant object with the middle line of the reticle. Reverse the telescope axis in its Ys. If the bisection still remains perfect, the line of sight defined by the middle line of the reticle is at right angles to the horizontal axis and the mean line may be assumed to be sufficiently near to that position. If necessary, however, make the adjust- ment by moving the reticle sidewise, so as to make the error of collimation small. By error of collimation is meant the angle between the line of sight defined by the mean line of the reticle and a plane perpendicular to the horizontal axis of the telescope. Level the horizontal axis of the telescope. Adjust the 8/. ADJUSTMENTS. IOI level so that when it is reversed its reading will change but little. Test the finder circles to see that they have no index error. Point upon an object and read one of the finder circles. Reverse the telescope, point upon the object again, and read the same finder circle as before. The mean of these two readings is evidently the zenith distance of the object (if the circle is graduated to read zenith distances) and their half difference is the index error of the circle. This index error may be made zero by raising or lowering one end or the other of the level attached to the vernier of the finder circle. This same process will evidently serve if the circle reads elevations instead of zenith distances. If the circle is to be made to read declinations directly, the same process is still applicable. For if the circle be made to read zenith distances with an index error equal to the latitude of the station, its readings will be declinations for one position of the telescope (though not for the other, after reversal). The other circle may be made to read declinations for the other position of the tele- scope. A " finder list" of stars, showing for each star to be observed its name, magnitude, setting of finder circle, and the right ascension or the chronometer time of transit to the nearest minute, will be found to be a convenience in the night work. In making out a finder list the refraction may be neglected, not being sufficient to throw a star out of the field of the telescope. The zenith distance of a star is then tf, south zenith distances being reckoned as positive. The Azimuth Adjustment. 87. In the evening, before the regular observations are commenced, it will be necessary to put the telescope, more accurately in the meridian. Having estimated the error of IO2 GEODETIC ASTRONOMY. 87. the chronometer in any available way, within say five minutes, and having carefully levelled up the axis, set the telescope for some bright star which is about to transit within 10 (say) of the zenith. Observe the chronometer time of transit of the star. This star at transit being nearly in the zenith, its time of transit will be but little affected by the azimuth error of the instrument. The collimation error and level error have been made small by adjustment. Therefore the difference between the right ascension of the star and its chronometer time of transit will be a close approximation to the error of the chronometer. Now set the telescope for some slow- moving star which will transit well to the northward of the zenith (let us say, of declination greater than 60 and a north zenith distance of more than 20). Compute its chronometer time of transit, using the approximate chronometer error just obtained. As that time approaches bisect the star with the middle line of the reticle, and keep it bisected, following the motion of the star in azimuth by the use of whatever means have been furnished on that particular transit for that purpose. Keep the bisection perfect till the chronometer indicates that the star is on the meridian. The telescope is now approxi- mately in the meridian. The adjustment may be tested by repeating the process, i.e., 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 its transit. If the star transits, apparently, too late, the object-glass is too far west (for a star 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 amount of movement which 88. DIRECTIONS FOR OBSERVING. 10$ must be given to the azimuth-screw, one computes roughly what fraction of a turn must be given to it. This may be done by computing the azimuth error of the instrument roughly by the method indicated in 102, having previously determined the value of one turn of the screw. An experi- enced observer will usually be able on the second or third trial to reduce the azimuth error to less than i s (= 15"). The table given in 310 will be found convenient in making the first approximation to the meridian. Directions for Observing. 88. The instrument being completely adjusted and the axis levelled, set the telescope for the first star. It is not advisable to use the horizontal axis clamp during observa- tions, for its action may have a slight tendency to raise one end or the other of the axis. See to it, loading one end if necessary, that the centre 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. When the star enters the field, bring it between the horizon- tal lines of the reticle, if it is not already there, by rapping the telescope lightly. Center the eyepiece so that the vertical line nearest the star is in the apparent middle of the field of view. As the star approaches the line pick up the beat of the chronometer.* Observe the chronometer time of transit across the line, estimating to tenths of seconds. Then center, the eyepiece on the second line and observe the transit there, and so on, until observations have been made upon all the lines, taking care always to keep the eye- piece centered upon the line which is in use. The directions and suggestions given in 60 for observing time apply here with equal force. In order to estimate * The eye and ear method of observing without a chronograph is here referred to. For a description of the chronograph and the method of using it in observing, see 89. 104 GEODETIC ASTRONOMY. 88. tenths of seconds it is necessary to divide the half-second interval given directly by the chronometer into fifths by some mental process. To secure accuracy and ease in mak- ing this estimate it is advisable to transform it into a proc- ess of estimating relative distances. The apparent motion of the star image is nearly uniform (not quite so on account of disturbance by irregular refraction). Let us suppose that at the instant when the chronometer tick which indicates the time to be I s . 5 is heard the star is seen at A (Fig. 12). Let the observer retain a mental picture of this relative position of the star and the line. When the chronometer tick for 2 s . o is heard, he sees the star at B. If he has retained (for o s . 5 only) the mental picture referred to above, he has before his mind's eye exactly what is indicated in the figure. He esti- mates the ratio of the distances from A to the line and from A to B, and concludes that the ratio is nearer to f than to , or f, and calls the time of transit of the star across line /, i 1 .?. Though this mental process may seem awkward at first, it will ultimately be found to be both easier and more accurate than the direct process, for all cases in which the star has an apparent motion which is sufficiently rapid to make the dis- tance AB appreciable in a half-second. An experienced observer, using this process, is able to estimate the time of transit of a star's image across each line of the reticle with a probable error of about o 8 . i . It is well here to bear in mind the suggestion given in 60, that he who hesitates is inaccurate. The successful observer decides promptly, but without hurry, upon the second and tenth at which the transit occurred. At convenient intervals between stars the striding level should be read in each of its positions upon the horizontal axis. At about the middle of the observations which are to constitute a set the telescope should be reversed, so that the effect of the error of collimation (and inequality of pivots) 89. THE CHRONOGRAPH. 10$ upon the apparent time of transit may be reversed in sign. Each half-set should contain one slow-moving star (of large declination) to furnish a good determination of the azimuth error of the instrument. The telegraphic longitude parties of the Coast and Geodetic Survey make the best time determinations that are made with portable instruments at present in this country. In their practice ten stars are observed in each set, five before and five after reversal of the telescope. From two to four readings of the level are taken in each of its positions in each half-set. Care is taken to have the telescope at different inclinations during the different readings of the level, inclined sometimes to the south and sometimes to the north, so that the level may rest in turn upon various parts of the pivots. This is done to eliminate, in part at least, the effect of irregularity in the figure of the pivots upon the determination of the inclination of the axis. The Chronograph. 89. The preceding directions for observing were given on the supposition that the eye and ear method of observing the times of transit is to be used. If, instead, the time obser- vation proper is made with a chronograph, the method is changed in that one particular only. A common form of the chronograph is shown in Fig. 13. The train of gear-wheels partially visible through the back glass of the case at F is driven by a falling weight, and drives the speed governor at AECCDD, the screw /, and the cylinder H. As the speed of rotation of the governor increases, the weights CC move farther from the axis until a small projection on one of them strikes the hook at E and carries it along. This hook carries with it in its rotation the small weight A. The result of the impact and of the added friction at the base of A is to cause the speed of the governor IO6 GEODETIC ASTRONOMY. 89. to decrease until the hook E is released. The speed then increases until the hook is engaged, decreases again 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 clock breaks. By moving the adjusting nuts DD upon their screws, the critical speed at which the hook is engaged may be adjusted. The carriage M is moved parallel to the axis of" the cylinder H by the screw /. The pen G, carried by an arm projecting from the carriage M, tends to trace a helix at a uniform rate upon the paper, or " chronograph sheet," stretched upon the cylinder. The magnets KK are in an electric circuit (through the wires Z), with a break-circuit clock or chronometer. Whenever the circuit is broken, the armature A 7 " is released and the back portion of the arm carrying the pen is drawn back, by a spring, to contact with the stop at J. The pen then makes a small offset from the helix. It returns to the helix as soon as the current is renewed. As a result the equal in- tervals of time between the instants at which the chronometer breaks the electric circuit are indicated by equal linear in- tervals between offsets on the line drawn by the pen, the speed being kept constant by the governor. The chronome- ter is usually arranged to break the circuit every second or every alternate second, and to indicate the beginning of each minute by omitting one break. The hours and minutes may be identified by recording at some point upon the sheet the corresponding reading of the face of the chronometer. The electric circuit passing through the magnets KK^ the chronometer, and battery also passes through a break-circuit key in the hand of the observer. To record the exact time of occurrence of any phenomenon he presses the key at that instant, breaks the circuit, and produces an additional offset in the helix, of which the position indicates accurately the go. RECORD AND COMPUTATION. IO? time at which it was made. In observing a star, to determine a chronometer error the instant of transit of the star image across each line of the reticle is so recorded. To read the fractions of seconds from the chronograph sheet it is con- venient to use a scale divided into intervals corresponding to tenths of seconds. The process of reading is also facilitated by writing the number of the second at the head of each col- umn, and of the minute on each line, of the chronograph sheet before beginning to read. The experienced observer gains very little in accuracy by substituting the chronographic method of observing time for the eye and ear method. He gains somewhat in the con- venience and rapidity with which his night's record is made. These small gains are not usually sufficient to justify the use of the chronograph in the field, except in connection with telegraphic determinations of longitude. In that case the chronographic method has special advantages (see 234-242). Example of Record and Computation. 90. The following time set was observed as a part of the telegraphic longitude work of the Coast and Geodetic Survey in May, 1896. The observations were made with a chrono- graph. The explanation of each separate portion of the computation is given in detail under the appropriate heading in the following sections. The constants of the instrument used in this example are here inserted for convenient reference. One division of the striding level = i".674. Pivot inequality = O 8 .oio with band west. Equatorial intervals of lines with band west : Line i. 15". 20 Lines. 2 s . 52 Line 9. -)- IO "-9 " 2. i2 s .69 " 6. + o 8 .09 " 10. + i2 8 .65 " 3. io s .i5 " 7. +2 8 .52 " ii. 4-158.15 " 4. 5 8 .o6 " 8. + 5 s . 1 1 IO8 GEODETIC ASTRONOMY. Q2. Time of Transit Across Mean Line. 92. If the transit of the star across every line of the reticle is observed, the time of transit across the mean line, or line of collimation, is evidently obtained by taking the mean of the several observed times. In obtaining the sum of the several times for this purpose an error of a whole second in any one observed time, which might otherwise remain unnoticed, will be detected by the use of the auxiliary sums shown 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. The unexpressed minute for each is the same as that for the middle line. The sum of these auxiliary sums and of the middle time is the total sum required in comput- ing the mean. It will frequently happen, especially on partially cloudy or hazy nights, that the transits of a star across several lines of the reticle will be successfully observed, and yet the observer may fail utterly to secure the transits across the remaining lines. It then becomes necessary to reduce the mean of the observed times of transit across certain of the lines to the mean of all of the lines. Let t^ /, / 8 , . . . be the observed times of transit across the successive lines, and let t m be their mean, or the time of transit across the mean line. Let z,, z a , z,, ... be the equatorial intervals of the succes- sive lines from the mean line, or the intervals of time which elapse for an equatorial star (star of zero declination) between transits across the separate lines and the transit across the mean line. Then for an equatorial star ^ = t^ t m , * 3 = / 3 t m> *' = *, *m> 92. RED UCTION TO MEAN LINE. 109 To determine the relation for any other star between i nj the equatorial interval for any line, and t n t m (in which t n is the time of transit over that line), deal with the spherical triangle defined by the pole, the star at the instant when it is on any line, and the star at the instant when it is on the mean line. In Fig. 14, let P, A, and B represent these points respectively. The sides PB and PA are the polar distance of the star ( 90 #). The angle at P expressed in seconds of arc is i$(t H t m }, (t n t m ) being expressed in seconds of time. The side AB, expressed in seconds of arc, may be taken equal to 15^, i n being expressed in seconds of time. Using the law that the sines of the sides of any spherical tri- angle are proportional to the sines of the opposite angles, there is obtained sin 154 : sin (90 d) = sin i$(t n Q : sin A. (19) The arc AB corresponding to the equatorial interval for a line will seldom exceed 15' in any transit, and is usually much less. For such an isosceles spherical triangle as this, AB being short, angles A and B are necessarily nearly equal to 90. Assuming as an approximation that A = 90, (19) may be written sin 154 sin i$(t n t m ) cos d. Again, assuming that the small angles i$i H and i$(t n t m ) are proportional to their sines, and dividing both members by 15, we obtain 4 = (^ O cos $ (20) In the derivation of (20) besides the approximations men- tioned, there is another in the assumption that AB corre- sponds to the perpendicular distance between the two lines of the reticle concerned, whereas none but images of equatorial stars will pursue a path across the reticle which is perpendic- ular to all the lines. Every other star image follows an HO GEODETIC ASTRONOMY. 93. apparent path which is curved (the apparent radius of curva- ture being less, the greater the declination), and therefore not perpendicular to more than one line of the reticle. So AB corresponds, except for an equatorial star, to an oblique dis- tance between lines of the reticle, the obliquity being exceed- ingly small. In spite of all these approximations it has been found by comparing (20) with the formula expressing the exact relation between i n and (t n t m ) that for an extreme value of i n = 6o 8 the computed value of (t n t^) will be in error by less than O 8 .oi for any star of declination less than 70, and that for a star of declination 85 the error is only 8 .3.* 93. Let us deal now with such a case as that of star 17 H. Can. Ven. in the preceding computation, in which the star image was observed to transit across the first ten of the eleven lines of the reticle and the transit across the eleventh line was missed. Suppose that the equatorial intervals *,, z a , z,, . . . have previously been determined by special observations as indicated in 114. From (20) we may write sec sec sec (21) *m = *io *io sec tf whence (*,+vH+ +*'..) sec o * For the exact treatment of this problem, see Chauvenet's Astronomy, vol. II. pp. 146-149; or Doolittle's Astronomy, pp. 291-293. 93- REDUCTION TO MEAN LINE. . Ill in which all quantities are known. By the use of (22) the time of transit t m across the mean line may be computed even though the transits across certain of the lines are missed. Adding the corresponding terms of the separate equations of (21), together with the equation t m = / t n sec 6 (which is true though t n is unknown), and dividing by 1 1, there is obtained sec cos sec # = o".3IQ cos sec # 186000 tan i = O 8 .02i cos sec tf. . (35) For convenience k is tabulated in terms of and d in 301. As the aberration causes the star to appear too far east, the observed time of transit is too late, and k is negative when applied as a correction to the observed times (except for sub- polars). Azimuth Correction. 97. If the transit is otherwise in perfect adjustment but has a small error in azimuth, the line of collimation will describe a vertical circle, i.e., a great circle passing through the zenith, at an angle with the meridian (measured at the zenith) which we will call a. In Fig. 1 8, let z be the zenith, B be the position of a star when it is on the meridian, and B' its position when observed crossing the line of collimation of a transit which has an azimuth error a. By applying the process of the latter part of 92 to this spherical triangle it may be shown that BB' = a sin . Il8 GEODETIC ASTRONOMY. 98. Applying the same process again to the spherical triangle defined by B, B' , and the pole, it may be shown that the corresponding hour-angle is BB' sec d. Let the angle a be expressed in seconds of time, and be called positive when the object-glass is too far east with the telescope pointing southward. Then the required correction to the observed times, equal to the time elapsed between position B' and B of the star, may be written Azimuth correction = a sin sec d = Aa, . (36) in which A is written for sin C, sec tf, and is tabulated in terms of and d in 299. The methods of deriving a from the time observations will be treated later ( 100-110). Collimation Correction. 98. If the instrument is otherwise in perfect adjustment but has a small error of collimation ( 86), the mean line describes a small circle parallel to the meridian, at an angular distance c, the error of collimation, from it, when the telescope is rotated about its horizontal axis. By the same line of reasoning that was used in 92 in dealing with the intervals of the various lines from the mean line, it may be shown that if c be expressed in time, then the Collimation correction * = c sec 8 = Cc, . . (37) * Objection may be made to the methods used in deriving the formulae of 92-98 because of the many approximations involved in them. For, in addition to the stated approximations that have been made in the derivations, the fact that the formulae for inclination, azimuth, and col- limation are not independent has been neglected, and each treated as if entirely independent of the other. Is such objection valid ? Our present purpose is to furnish the engineer with such mathematical formulas (together with an intelligent 99' CORRECTION FOR RATE. IIQ in which C is written for sec 8 and is tabulated in terms of in 299. The collimation error necessarily changes sign when the telescope axis is reversed in its Ys. Let c be the error of collimation with lamp or band east, and let it be called positive when the star (above the pole) is observed too soon. To take account of the change in sign of the collimation error with lamp west, let the sign of C be reversed (so that C sec 6) whenever the lamp is west. With this convention as to the sign of C the algebraic sign of the product Cc in (37) will always be correct. The methods of deriving c from the time observations will be treated later ( 100-112). Correction for Rate. 99. If the rate of the chronometer is known to be large, it may be necessary to apply a correction to each observed time to reduce it to the mean epoch of the set. The correction required is the change in the error of the chronometer in the interval between the observation and the mean epoch of the set. If the rate of the chronometer is less than I s per day and the interval in question is not more than 3O m , the greatest correction for rate will be o s .O2. In such a case, if the cor- rection for rate is ignored, the computed correction to the understanding of them) as will serve him most efficiently in making cer- tain astronomical determinations with portable instruments. The formulae furnished are sufficiently accurate for his purpose. The degree of accu- racy is roughly indicated to give him a basis for confidence. The alterna- tive procedure is to derive the exact formulae, at a large expenditure of time and mental energy; to find that said formulae are too complicated for actual use in computation; to simplify them by dropping terms and making transformations that are approximate; and to arrive finally, when ready for actual numerical computation, at the same simple formula as are here derived directly (or their equivalents in simplicity and inaccuracy). This procedure furnishes more mathematical training than that adopted in the text. But mathematical training is not the primary object of this treatise. I2O GEODETIC ASTRONOMY. IOI. chronometer will be sensibly exact, but the computed prob- able errors will be slightly too large. If the rate of the chronometer is very large, it may even be necessary to apply a rate correction to the reduction to the mean line, in case of an incomplete transit, as derived in 92. Computation of Azimuth, Coll imat ion, and Chronometer Correc- tions, without the Use of Least Squares. 100. Having corrected each observed time of transit for inclination and aberration, the azimuth error a and the colli- mation error c, as well as the required chronometer correction, may be derived from the observations by writing an observa- tion equation of the following form for each star observed, AT c + aA + cC- (a - 37) = o, . . . (38) forming the corresponding normal equations, and solving for the required quantities, AT C the chronometer correction, a, and c. In (38) a is the apparent right ascension of the star (reduced to mean time if a mean-time chronometer is used), and T e f is the observed chronometer time of transit of the star corrected for diurnal aberration, inclination , and rate of chronometer. This least square process is rather laborious, and a shorter method is desirable for obtaining approximate results. Such a short method * without least squares will now be treated. It is a method of successive approximations to the required results. 101. The exact form of the computation is shown below in a numerical example dealing with the observations shown in 9'- * This method, which has been in continual use in the field on the longitude parties of the Coast and Geodetic Survey for many years, was devised in the '7o's by Mr. Edwin Smith, then an aid on that Survey. 101. COMPUTATION WITHOUT LEAST SQUARES. 121 STATION : Washington, D. C. DATE : May 17, 180,6. I ?? SI *5 <^ f^ vn- o 6 b + H-li II ^ v- Q o CO Tj- o' II ft) Q o vn N vn ?1 + + 11 II ^ v> Q I 6 + II N & O ""> O co O M O M O O o^^^o ? i s vn vn O O 4 4 1 1 88 4 4 I 1 N M O O 4 4 1 1 5*5 4 4 1 1 b o o o o 1 ++ 1 o o o o o 1 + 1 II ^ "3 ^ I * i i W t^ w vn N M 0^ O O S 1 ^^^? vn o O ""> SCO w too O vn *f 800 N M 1 a i rr 1 1 * CO -^J- rt- rf 1 1 1 1 1 ^ O M | - ^ M t^ O N CO O W CS M vn en co O Oco C5 i TJ- 8 to N CO O O^vO CO 4 1 1 CO M O CO 4 vn 1 1 b o o o o -H.HHH O O O O M ++ 1 1 1 S i i Tf -Tf 1 1 rf vn 1 I & 3- vn co Tj-co O O O O O ^- en u-> vn co O O O O w O M I"*** M M Tf co O'O ++ ?? 1 1 o o o o o +++++ 00000 1 1 1 1 1 C5 o o +H- O O 1 1 ^ o* O O w co O co co N o o vn o co co M CO CJ CO m ^ 8? 6 M + 1 SCO *? 6 cJ 1 O O O O M JLj O O O O N JLLJ O M co Z^^g? <*> ".^ M C-J ++ N CO CO M M 4 I | ++ + + + T T T T i ^j K, 1 CS r^ o o o N O O O co m Tf M CO & *t OCO W W Tf ^ $ * W O^ ^o CO rf 1 I ?t II ^- vn 1 1 II 00 ]_U_JJ CO CO Tj- Tj- IT) 1 1 1 1 1 II ^^^^^ y y y y y MM * MM B i: .a rt^ ^ M u ,-ss ^ v* o E^papQQ M 5>R-M .5 n w to 2 O O O in O O O O * mpQMffl^ >4 tt^D vn 'Mean of 1 stars Azimuth sta Mean of \ stars Azimuth sta Mean of 1 stars Azimuth sta Mean of t stars ^ Azimuth sta xoaddy Jsi xojddy pz 122 GEODETIC ASTRONOMY. 104. 102. The first five columns of the main portion of the computation are compiled from 91, and from the table of 299 (factors Ay B, C). The remaining columns are filled out after the computation of a and c, shown in the lower part of the tabular form, is completed. 103. It should be noted that the five stars of each group, observed in one position of the instrument, have been so selected that one is a slowly-moving northern star at a con- siderable 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 so placed that their mean azimuth factor (A) is nearly zero. These four stars of each group are for convenience called time stars, since the determination of the time falls mainly upon them r while the slowly-moving star serves to determine the azimuth error of the instrument and is called the azimuth star. 104. In the computation * to derive c and a, the four time stars in each position of the instrument are combined and treated as one star, by taking the means of their (a 77)'s and of their factors C and A, respectively, the means being written below the separate stars in the computation form, together with the azimuth starsj On the assumption that the means of the time stars in the two positions of the instrument are equally affected by the azimuth error, the first approximation to c is found by dividing the difference between the two mean values of a T c ' by the difference between the two mean <7s. Or (<* T c '} w (a T C ) E W (39) *This example of the method of computing a and c without least squares, and much of the explanation of it, is taken with little modifica- tion from Appendix No. 9 of the Coast and Geodetic Survey Report for 1896, by Asst. G. R. Putnam, 105- COMPUTATION WITHOUT LEAST SQUARES. 12$ In the example in hand - 3-94- (-4-07) +0.13 Using this approximation to c, the correction Cc is then subtracted from the a T c ' of the means of the time stars and of the azimuth stars, and the values of a 27 Cc obtained. Separate values for the azimuth error of the instrument are then derived for each position of the instrument as fol- lows, upon the assumption that the difference between the (a TJ Cc) for the mean of the time stars and for the azimuth star of a group is due entirely to azimuth error. Upon this assumption, for each position of the instrument \ a *< -^)time stars (** * ^jazimuth stars *= . "time stars -^azimuth stars and Numerically, in the present case _ 4.00 ( 4.64) _ + 0.64 _ " _|_ 0.08 ( 1.03) ~~ -f- 1. 1 1 _ 4 .oo_(_ 5.23) _ + 1.23 _ --- - - + '486. o.oo ( 2.53) +2.53 (42) With these approximate values of a w and a B the correc- tions Aa are applied, giving the values a T c f Cc Aa in the last column in the lower part of the computation form. 105. If these do not agree for the two positions of the instrument, it indicates that the mean values of a T c f used in (40) in deriving c were not equally affected by the azimuth error, so that their difference was not entirely due to c, as was assumed in using (39). A second approximation to the true value of c may now be obtained by considering the differences 124 GEODETIC ASTRONOMY. 10$. in the last column to be due to error in the first approximate value of c\ substituting from that column in formula (39); and thus obtaining a correction to the first approximate c. Thus in the present case the second member of (39) becomes 4.05 ( 4.00) 0.05 + 1.25 -(- 1.32) =+^7= - OI 9- ' () The second approximation to the true value of c is then -f- s . 05 1 o s . 019 = + o 8 . 032. Proceeding as before, improved values for a w and a B are found by the use of formula (41). Thus in the present case there are obtained as second approximations and 3.98 ( 4.60) + 0.62 +0 .o8- (-i.-^) = +TTl -4.03-(- 5.31) +1.28 (44) This process of making successive approximations to the values of c and a may be continued until the values (a T c ' Cc Ad) show a sufficiently good agreement^Th general, with a well-chosen time set, the final value for AT C will not be changed by as much as O 8 .oi by any number of approxima- tions made after the above agreement has been brought within the limit o s .O5. When satisfactory values for c, a w , and a E have been obtained, the corrections Cc and Aa are applied separately to each star, as shown in the sixth, seventh, and eighth columns of the upper part of the computation form, and the values of the chronometer correction ( A T c ) derived separately from each star. The residuals furnish a check on the computation. IO6. COMPUTATION WITHOUT LEAST SQUARES. 12$ Any large error in observation or computation will be indi- cated by the residuals, and may often be located by a careful study of them. The mean value of AT C is the required chronometer correction at the epoch of the mean of the observed chronometer times. 106. A study of the above process of successive approxi- mation to the values of c, a w , and a B shows that the rapidity with which the true values are approached depends upon three .conditions. The mean A for the time stars for each position of the instrument should be as nearly zero as possi- ble. In each position of the instrument the A for the azimuth star should differ as much as possible from the A for the mean time star, while corresponding C's should differ as little as possible. The last two conditions' are difficult to satisfy simultaneously, but the fact that both must be considered leads one to avoid observing sub-polars. The conditions here stated show why the stars for the above time set were chosen as indicated in 103. It is not advisable to spend time in observing more than one azimuth star in each half set. It should be noted that the choice of stars indicated above also insures the maximum degree of accuracy in the determi- nation of A,T C for a given expenditure of time, regardless of the method of computation. The two things which especially commend this approxi- mate method of computing time to those observers who have used it much in the field are the rapidity with which the computation may be made (especially when Crelle's multipli- cation-tables are used), and the accuracy which results from the fact that the derived values o.f a and c depend upon all the observations, and not upon observations upon a few stars only, as is frequently the case with other approximate metnods. 126 GEODETIC ASTRONOMY. 107. Computation of the Azimuth, Collimation, and Chronometer Corrections by Least Squares. 107. We start with the observation equations * indicated in (38). For lamp west each of these equations are of the form Cc-(a- 7*/) = o, . . (45) and for lamp east, of the form AT C + A E a E + Cc - (a - T c f ) = o. . . (46) The subscripts added to A discriminate between factors applying to stars observed with lamp east and those observed with lamp west. This is done for the purpose of avoiding confusion in the normal equations. The parenthesis (a T c ') is an approximate value for the clock correction after taking account of inclination, rate, and aberration. It is the observed quantity. The coefficients A and C may be obtained from the table in 299. Treating the observation equations all together as a single group, as many equations as stars, the four derived normal equations are of the form + 2Cf - 2(a - T c ') = o; + 2A w Cc-2A w (a -7y) = o;. Tc + 2A* a E + 2A E Cc - 2A (a - T/) = o, c + 2CA w a w + ^CA E a E + 2C*c -2C(a- T c ') = o. The solution of these equations gives the required quan- tities AT e , a w , a E , and c. To obtain the probable error of a single observation sub- * Observation equations are also called conditional equations by some authors. 108. LEAST SQUARE COMPUTATION. I2/ stitute these values back in the observation equations, (45) and (46), and obtain residuals v lt v v v^ . . . , one for each equation. The probable error of a single observation is = 0.674^7- _, .... ( 4 8) in which n is the number of observations, and n v is the num- ber of normal equations (and of unknowns). To obtain the probable error, e , of the computed dT c , proceed as follows : Rewrite the normal equations (47), putting Q in the place of AT C , i in the place of 2(a T e '), and o in the place of the other absolute terms. Solve the resulting equations for Q. Notice that since all the coeffi- cients in these new equations are just as before, it is only that part of the computation which deals with the absolute terms that is changed in the solution of the normal equations. e. = eVQ. (49) The form of the normal equations, and method of com- puting the probable error, are here stated for convenience of reference. For the corresponding reasoning the student must depend upon his knowledge of least squares, the methods here given being the ordinary least square methods for dealing with a set of observation equations in case there are no rigid conditions to be satisfied.* 108. As a concrete illustration of this least square adjust- ment for determining AT e we may take the set of observations given in 91, from which AT C has been computed without *See Wright's Adjustment of Observations (Van Nostrand, New York), or Merriman's Least Squares (John Wiley & Sons, New York). 128 GEODETIC ASTRONOMY. IO8. the use of least squares in 106. The observation equations are V V* AT C + Q.02a Hr + .26; -f- o 8 . 07 = o OMo 0.0100 AT C - 0.300^ + .56^ -|- o .09 = o 4~ -05 0.0025 4T e + o.36a,y + .06^ 0.31=0 -f- o .09 0.0081 AT C + o.22a + .13^ .11 = o .03 0.0009 AT c -i^a w + .36, + o .52 = o .00 0.0000 ^7V +0.250^ .!!<: o .06 = o .01 O.OOOI ^T; +0.35^- .06^ o .19 = o H~ -7 0.0049 Z/7V o.20a E .46. + o .23 = o o .06 0.0036 AT C o.3Sa E - .64* -f- o .29 = o .02 0.0004 ATc 2.53^ L j.i8r -f i .44 = o -f- o .01 O.OOOI Sum = 0.0306 = 2v* From the absolute term in each equation 4 s . oo has been dropped, as is frequently the case in least square computa- tions, for the purpose of shortening the numerical work. The true value of (a T e ') is then, in each case, that written above, -f- 4 s .oo. The four normal equations formed from the above obser- vation equations in the usual way are + iQ.oojT]. 0.73^^ 2.$ia B 2.08^+1.97 = 0; 0.73^7;+ i. 33*^ 2.24^0.70 = 0; 2.51/77; + 6.77^+ 10. 84<; 3.88 = o; 2.08J7; 2.24^^+ 10.84^ + 36.64;: 5.56 = o. The solution of these equations for the unknowns gives a w = + o s . 568, 0* = + o".5ii, c=+o s .034, and 4T C = 8 .020, which combined with the 4 s . oo which was dropped to ease the numerical work gives AT C = 4 s . 020. If these values are now substituted in the observation equations, 108, the residuals (v) there shown are obtained. HO. LEAST SQUARE COMPUTATION. 12$ From these the probable error of a single observation, see formula (48), is The modified normal equations being solved for Q as indi- cated in 107, its value is found to be 0.1158. Hence the probable error of the result (4T C ) is, see formula (49), e = 0.048 1/0.1158 = o s .oi6. 109. If, instead of computing a separate value for the azimuth error, a, for each of the positions of the telescope axis, before and after reversal, the azimuth error is assumed to be the same throughout the whole set, the principles involved in the computation are the same as before; the dis- tinction between a w and a E is dropped; there are but three unknowns and three normal equations instead of four; and the work of solving the normal equations is correspondingly shortened. The loss of accuracy in the computed result depends upon the magnitude of the actual change in the azimuth error at reversal. If no more than six stars are observed in a set, it may be advisable to use this process so as to reduce the number of unknowns. 110. Experience shows that the process outlined in 105 106 gives such an accurate value for c that the value subse- quently derived from a least square adjustment is found to be substantially identical with it. When such a preliminary com- putationhasbeen made,theleast squareadjustmentisshortened considerably, with little loss of accuracy, by accepting this preliminary value of r, applying the collimation corrections (as well as the inclination, rate, and aberration corrections) before the least square adjustment, and treating the clock GEODETIC ASTRONOMY. I IO. correction and the two azimuth errors as the only unknowns. It is well in this case to treat each half set separately. The discrepancy between the two values for the clock correction thus derived, when reduced for clock rate to the same epoch, indicates the amount of error in the assumed value for c. To illustrate this method we may use the same set of observations as in 91, 101. Let it be assumed that the preliminary computation shown in 101 has been made, and let the value -f- o s .O32 for c given there be accepted as a basis for this computation. The observation equations now become - 0.02a^ + OMI = O AT C 0.30^ + o .14 = o AT C + 0.360^ o .28 = o \ For the first half of set. AT c -\- o.22a w o .07 = o A T c 1.030 + o '6 = o .10 = o AT C + o.35a o .22 = o AT C o.2oa -f- o .18 = o } For the second half of set. A T c 0.380^ -f o .24 = o AT C 2.$3a E -f i .31 = o The normal equations for the first half of the set are 0.730^ + 0.50 = o; 0.73.47;+ 1.330^ 0.75 = o; and for the second half of the set 5.00J7; 2.51^ + 1.41 =o; 6.770* 3-54 = O. The solution gives for the first half-set AT C = o s .oi9, a w = + o 8 . 5 5 3, and Q 0.217, and for the second half-set AT C = o s . 024, a R = + o 8 . 5 14, and Q = 0.246. The probable error of a single observation derived from III. UNEQUAL WEIGHTS. 131 the first half set is o s .O58, and from the second o s .O32. The probable error of AT C from the first half set is o s .O27, and from the second O 8 .oi6. The final result from the complete set is, by this method of computation, AT C = 4 S .022 8 .0l6. The difference between the two values for AT C derived from the two halves of the set serves to indicate the degree of accuracy of the assumed value of c. Introduction of Unequal Weights. 111. In the preceding treatment it has been tacitly assumed that all observations are of equal weight. But incomplete transits should be given less weight than complete transits. For if only a few lines of the reticle are observed upon, evidently the accidental errors made in estimating the times of transit across the separate lines will not be eliminated to as great an extent as if all the lines were observed. Then, too, the image of a star of large declination moves much more slowly across the reticle than does the image of an equatorial star, and it is therefore more difficult to estimate the exact time of its transit across each line. If it is found by the investigation of many records for such slow-moving stars that the error of observation is larger for such stars than for equatorial stars, it is proper to give them less weight in the computation of time. An extended discussion of this matter of weights may be found in the Annual Report of the Coast and Geodetic Survey for 1880, pp. 213, 235-237. It suffices for our purpose here to give, in slightly abridged form, the tables of relative weights which were derived from that dis- cussion (see 302, 303). The weights as given in these tables were deduced for transit instruments having a clear aperture of object-glass from if to 2f inches, and a magnifying power from 70 to 100 diameters. It may be extended with 132 GEODETIC ASTRONOMY. 112. little error to instruments of the same nature which are con- siderably larger or smaller. 112. To introduce the unequal relative weights, w t into the least-square adjustment, it is necessary to multiply each observation equation by Vw, and to make the usual subse- quent modifications in the least square computation. These modifications are indicated in the following example, the same problem as that treated in 108, without the use of unequal weights. If an incomplete observation is made upon a slow star, so that both the tables of 302 and of 303 must be used, first multiply the two relative weights w together, and then take the square root of that product as the multi- plier for the observation equations. The square roots of the weights given to the ten stars, in order of observation, are respectively 0.9, 0.8, i.o, 0.9, 0.6, 0.9, i.o, 0.8, 0.8, and 0.3. The observation equations shown in 108 are multiplied by these factors respectively. The normal equations resulting from the weighted observation equations so obtained are -\- 0.53^ + o s . 14 = o; 0.57^: o s .34 = o; 0.04/47; + o.S?a E + 0.93*: s . 50 = o; + 0.53^7;- The solution of these equations gives AT C = o*.oi9 > a w = + s . 583, a E = + s . 544, and c = + o s .O33. The probable error of an observation of weight unity is / ^W V /O.O223 0.674. A'/'- -0.674 A/ -==:fco*.O4i. \ n n v * V 10 4 The probable error of AT C = e = e VQ = o s .04i Vo.i47 = o s .oi6. 114- AUXILIARY OBSERVATIONS. 133 113. Unless an extreme degree of accuracy is required, the assumption that all observations are of equal weight is suffi- ciently exact. The introduction of unequal weights adds so little to the accuracy of the computation that economic con- siderations will often indicate that the least square adjustment should be made on the basis of equal weights.* Auxiliary Observations. 114. Aside from the observations and computations which have been treated in detail, certain others are necessary for the determination of the instrumental constants which have been assumed in the preceding treatment to be known. The equatorial intervals of the lines of the reticle may be determined from any series of complete transits, i.e., observa- tions in which the transit of each star was observed across every line. % A special series of observations is not required, for the complete transits of the particular series of time observations under treatment may be utilized for this purpose in addition to using them to determine the clock correction. For every complete transit every term in equation (20) (see 92), namely, i n (t n t m ) cos #, is known except i n . Every * The computation of a series of time observations taken by the author on the shore of Chilkat Inlet, Alaska (in latitude 59 10'), in 1894, was made in the field by least squares, giving all stars equal weight, regardless of their declinations and of the number of missed lines. In the final com- putation subsequently made at the Coast and Geodetic Survey Office in Washington unequal weights were assigned. In the series there were 46 sets, each consisting, generally speaking, of observations upon 10 stars. The average difference, without regard to sign, between the chronometer corrections as computed in the two ways from the same set of observations was o.04. This is about equal to the probable error of the clock correc- tion computed from a set. But it must be remembered that the conditions were extreme. On account of the high latitude of the station many of the stars were slow-moving stars (even those observed in the zenith). There was so much interference by clouds that complete observations on all the stars were secured on only 10 nights out of the 46, and observations on a single line only of the reticle were not infrequent. 134 GEODETIC ASTRONOMY. complete transit observed furnishes, then, a determination of the equatorial interval of every line. The transit of a slow- moving star gives a more accurate determination of the equatorial intervals than the transit of a star of small declina- tion, for the errors in observing t n do not increase so rapidly, with increase of declination, as cos d decreases. For this reason some observers prefer to make a special series of observations for equatorial intervals using stars of large decli- nation only. In computing and using the equatorial intervals it must be borne in mind that when the telescope axis is reversed in its Ys the order in which the star transits across the lines is reversed, and also the algebraic sign of the equa- torial interval of each line. 115. The portion of a set of observations given below will serve to show how the pivot inequality, p i (see 94), is determined by a series of readings of the striding level, upon the telescope axis placed alternately in each of its two possi- ble positions with clamp west and clamp east (the clamp instead of the lamp being here used to indicate the position of the axis). OBSERVATIONS FOR INEQUALITY OF PIVOTS OF TRANSIT NO. 4. STATION : Seaton, Washington. G W. D., observer. June 19, 1867. u CLAMP WEST. CLAMP EAST.V a g Object-glass S. Object-glass N. *.-* V I Level. i(2/ - 2,e\ Level. K2? - Ze) 4 o V a W. end. E.end. b v> W.end. E.end. be < H h. m. o d. d. d. d. d. d. d. 33 10.30 A.M. 73 60 o 64.0 + 0.600 59-o 65.2 -0.425 o. 256 65.2 58.8 64.0 59'5 SO 45 A.M. 72 65.0 59-o + 0.950 64 o 59-5 0.250 o . 300 60.8 63.0 59'Q 64.5 45 50 A.M. 72.5 60.8 66.0 63-0 58.0 + 1.450 59-5 04.0 64.0 60.0 0.125 - 0.394 40 11.00 A.M. 72.8 65.0 58.8 + 1.050 64.0 60.0 0.175 -0.306 61 .0 63.0 59 3 64.0 IS 05 A.M. 73 60.5 63.0 -f 1.200 59-2 64 o -0-575 - o 444 65-5 58 2 63.0 60.5 Il6. THE LEVEL VALUE. 135 The value of one division of this striding level was known to be I ".05. The whole set, of which this is a part, gave for a mean value of pi 0.337 divisions of the level = o // .354 = O s .024. 116. The most accurate way of determining the value of one division of a level is by means of what is known as a level-trier or level-tester. The level-trier is essentially a bar supported at one end upon two pivots so that it is free to rotate about that end in a vertical plane, and carried at the other end by a good micrometer screw with its axis vertical. The length of the bar between supports and the pitch of the screw being known, the change of inclination of the bar corresponding to one turn of the screw is known. To deter- mine the value of a division of a level it is placed upon the bar, and movements of the bubble corresponding to successive small movements of the micrometer screw are observed. Both ends of the bubble must of course be read, as its length is apt to change rapidly with changes of temperature. By comparing successive movements of the bubble corresponding to equal successive movements of the screw the uniformity of the value of a level division, or, in other words, the constancy of the radius of curvature of the upper inner surface of the level tube,* may be inferred. Observations of the length of time required for the bubble to come to rest in a new position also give an indication of the value of the level as an instru- * The longitudinal section of the upper inner surface of a level tube is made as nearly a perfect circle as possible. If the student will consider how great is this radius of curvature in a sensitive striding level, he will appreciate to a certain extent the wonderful accuracy with which this sur- face must be ground. He will also understand why small deformations of the level tube by unequal changes of temperature have such a marked effect upon the movement of the bubble. The radius of curvature for a level of which each division is two millimeters long and is equivalent to one and a quarter seconds of arc, a common type of level, is more than three hundred meters (about a thousand feet). 136 GEODETIC ASTRONOMY. H9- ment of precision. This time is greater the smaller is the value of a division (of a given length) expressed in arc, but for levels of the same division value, is less the more perfect is the inner upper surface. If the level tube 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. It may be well also to determine whether changing the length of the bubble, by changing the amount of liquid in the chamber at the end of the level tube, changes the apparent value of one division. 117. If an observer is forced to determine the value of a level division in the field, remote from a level-trier, after some accident let us say, which leads to replacing an old, well-known (but broken) level by another of which the value is unknown, his ingenuity will lead him to devise a method of utilizing whatever apparatus is at his disposal. The three methods given below will be found suggestive. 118. If a telescope having an eyepiece micrometer similar to that of a zenith telescope ( 135), measuring altitudes or zenith distances, is available, the unknown angular value of a division of the level may be found by comparison with the known angular value of a division of the micrometer. Place the level in an extemporized mounting fixed to the telescope. 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 telescope as a whole, and note the new reading of the level. Every repetition of this routine gives a determination of the value of a level division. 119. If in his instrumental outfit he has another well- 120. THE LEVEL VALUE. 137 determined level of sufficient sensibility the observer may use it 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. 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 both levels. 120. The following method * gives fully as great precision as either of the other two outlined above, and is especially valuable because the required means are apt to be at hand in the field even when the apparatus required for the other two methods is wanting. For this method the only instrument required is a theodo- lite, or an engineer's transit, or any other instrument having both horizontal and vertical circles (not necessarily with a fine graduation) and a good vertical axis. Mount the level on the plate of the instrument parallel to the plane of the telescope and adjust it as if it were a plate level. Make the vertical axis truly vertical in the usual way. Measure the zenith dis- tance of some well-defined stationary object, taking readings with (vertical) circle right and circle left to eliminate index error. Now incline the vertical axis directly toward or from the object, from i to 3, by use of the foot-screws. The direction of this inclination may be assured by use of the plate level which is at right angles to the plane of the tele- scope. Measure the apparent zenith distance of the object again. The apparent change in the zenith distance is evi- dently the inclination of the axis to the vertical, which we will call y. If, now, the instrument is revolved completely * Described in full by Prof. G. C. Comstock in the Bulletin of the Uni- versity of Wisconsin, Science Series, vol. I, No. 3, pp. 68~74 ; and said by him to be due originally to Braun. Those desiring further details are referred to that article, from which this statement is condensed. 138 GEODETIC ASTRONOMY. 121. around its vertical axis, two positions will be found at which the bubble of the level is in the middle of the tube. For positions near these two the bubble is within such limits that it may be read. It is from readings of the bubble in such positions, in connection with readings of the horizontal circle and the above outlined determination of y, that the value of a division of the level is derived. 121. Let Fig. 19 " represent a portion of the celestial sphere adjacent to the zenith, Z, and let V and S be the points in which the axis of the theodolite, and the line drawn from the center of curvature of the level tube through the middle of the bubble, respectively, intersect the sphere." 14 Since the bubble always stands at the highest part of the tube, its position, 5, and the corresponding value of q are found by letting fall a perpendicular from the zenith upon the arc VS, and in the right-angled spherical triangle thus formed we have the relation " tan q = tan y cos ft. . . . < . (50) " Since the level tube turns with the theodolite when the latter is revolved in azimuth, while the positions of the points Fand Z remain unchanged, it appears that the angle ft must vary directly with the readings of the azimuth circle." " If we represent by A Q the reading of the circle when the arc VS is made to coincide with VZ, we shall have corresponding to any other reading A' " tan q = tan y cos (A A 1 ). . . . (51) The value of A may be obtained by taking the mean of any two readings of the circle for which the bubble stands at the same part of the tube. " If A' and A" denote slightly different readings of the azimuth circle, b' and b" the corresponding readings of the 121. THE LEVEL VALUE. 139 middle of the bubble on the level scale, we may write two equations similar" to (5 1 )? "and taking their difference obtain " sin (q'-q"} A' - A" . / A' + A"\ ^ -- H$ = 2sm- - smL4 - - tan y. (52) cos q cos q 2 \ ' / lt Since q' q" is the distance moved over by the bubble, we may write q q" = (b' b"}d, where d is the value of a division of the level, and transform " (52) into _ 2 tan yc^qsm^(A f - A") sin [A - \(A' + A")] sin i" b'-b" In this equation cos 2 q may usually be placed equal to unity. For greater accuracy the average value of q from equation (51) may be used. A Q may be determined as indi- cated just below equation (51). Only an approximate value for it is required. All other quantities in the second member of ($3) are known. Hence d may be computed. It simplifies the computation to take the readings at equidistant points on the circle. Sin \(A' A") will then be constant. It would seem at first sight that a value of d derived in this way by use of vertical and horizontal circles reading to half-minutes only (say) must necessarily be crude. If, how- ever, the inclination of the vertical axis is made 3, y and its tangent, and therefore d, may be determined within one four-hundredth part. The readings of the horizontal circle do not need to be very refined, because for the positions used a comparatively large change in the circle reading is necessary to produce an appreciable change in the position of the bubble. This method, then, serves to determine the level value with an accuracy which bears little relation to the fine- ness of the circle graduations. 140 GEODETIC ASTRONOMY. 123. Discussion of Errors. 122. Following the same general plan as in discussing the errors of sextant observations, the external errors, instru- mental errors, and observer's errors will be discussed sepa- rately, and then their combined effect will be considered. The two principal external errors are the error in the assumed right ascension of the star, and the lateral refraction of the light from the star. If only such stars as are given in the va-ious national ephemerides are observed for time, the probable errors in the right ascensions will usually be on an average O 8 .O4 or o s .O5, and no appreciable constant errors need be appre- hended from this source. From considerations which need not be stated in detail here, one is led to the conclusion that the effect of lateral refraction upon transit time observations must be quite small in comparison with the other errors ; but it is difficult to estimate, because it is always masked by other errors follow- ing about the same law of distribution. For further consid- eration of this matter see 219. 123. Among the instrumental errors may be mentioned those arising from change in azimuth, collimation, and inclina- tion, from non-verticality of the lines of the reticle, from poor focusing and poor centering of the eyepiece, from irregularity of pivots, and from variations in the clock rate. The errors of azimuth and collimation being determined from the observations themselves are quite thoroughly can- celled out from the final result, provided they remain constant during the period over which the observations extend, and provided also that the stars observed are so distributed in declination as to furnish a good determination of these con- stants. Their changes, however, during that interval, arising 123. ERRORS. 141 from changes of temperature, shocks to the instrument, or other causes, produce errors in the final result. It is in this connection that the stability of the pier is of especial impor- tance. Such changes will evidently be smaller the more rapidly the observations are made and the more carefully the instrument is handled. In general they are probably small but not inappreciable. To a considerable extent the same remarks also apply to the inclination error. The changes in inclination during each half-set evidently produce errors directly. Hence again the desirability of rapid manipulation. But the mean value of the inclination is determined from readings of the striding level, not from the time observations, and the level may give an erroneous determination of the mean inclination. Differ- ent observers seem to differ radically as to the probable mag- nitude of errors from this source, but the best observers are prone to use the striding level with great care. However small this error may be under the best conditions and most skilful manipulation, there can be no doubt that careless handling and slow reading* of the striding level, or a little heedlessness about bringing a warm reading lamp too near to it, 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 such an instrument as that described in 83 produces an error of o 8 . 1 or more in the deduced time of transit of a zenith star. If the lines of the reticle are not carefully adjusted so as to define vertical planes ( 85), stars will be observed too early or too late if observed above or below the middle of the reticle. Such errors may be made very small by careful * It is here assumed that before attempting to read the level it has been in position long enough for the bubble to come to rest in the position of equilibrium. 142 GEODETIC ASTRONOMY. 124. adjustment and by always observing within the narrow limits given by the two horizontal lines of the reticle. Poor focusing of either the object-glass or the eyepiece leads to increased accidental errors because of poor definition of the star image. But poor focusing of the object-glass is especially objectionable, because it puts the reticle and the star image in different planes, and so produces parallax. The parallax error may largely be avoided by centering the eyepiece each time over the line of the reticle upon which the star is next to be observed. This repeated centering should never be omitted even though the observer may be confident that the focusing is perfect. It also serves in a measure to avoid errors which might otherwise be produced by the imperfec- tions of the eyepiece. If the inequality of the two pivots has been carefully determined as indicated in 115, the errors arising from defects in their shapes may ordinarily be depended upon to be negligible. Changes in the rate of the timepiece during a set of observations evidently produce errors in the deduced clock 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 very poor timepiece, or if clouds interfere so as to extend the interval required for a set of observations over several hours, this error may become appreciable. It is less the more rapidly the obser- vations are made. The errors introduced by irregularity in the action of a chronograph of the form described in 89 are too small to be considered, especially if its speed is assumed to be constant simply during the interval between successive clock breaks, and the chronograph sheet is read accordingly. 124. The observer s errors are by far the most serious in 124. ERRORS. 143 transit time observations. He is subject to both accidental and constant errors in his estimate of the time of transit . From computations based upon thousands of observed transits it is known that an experienced observer is subject to an accidental error of from o s .o6 to o 8 . 15 in estimating the time of transit of a star of declination less than 60 across a single line of the reticle of such instruments as those de- scribed in 83. For slower stars his error, expressed in time, is of course still greater. If the observations of the transits of a given star across the different lines of the reticle were not subject to any error common to all the lines, the probable error of the deduced time of transit across the mean line would vary inversely as the square of the number of lines in the reticle. But experience, as above, indicates that there is an error common to all the lines of O 9 .O5 to O*.I2. This error, sometimes called the culmination error, is an observer's error, which is constant for the interval during which the star is transiting across the reticle, but which may change before the next star is observed. Prom the method by which this value (o s .O5 to O 8 .I2) was deduced it also necessarily includes the small errors due to lateral refraction and irregularities in clock rate, as well as some small outstanding instrumental errors. The probable error, r, of the time of transit of a star (of declination less than 60) across the mean line of a reticle is, therefore, given by an equation of the form (o 8 .o6) a to (o s . 1 5)' r 9 = (o 8 . 05 to o s . 12)' + v - -- in which n is the number of lines in the reticle. (Compare in, 302, 303.) For a more extended discussion see Coast and Geodetic Survey Report for 1880, Appendix No. 14, pp. 235, 236, or Doolittle's Practical Astronomy, pp. 318-322 144 GEODETIC ASTRONOMY. I2/. 125. In addition, still, to these errors there is another which is constant for all the observations of a set. Every experienced observer, though doing his best to record the time of transit accurately, in reality forms a fixed habit of observing too late, or too early, by a constant interval. This interval between the time when the star image actually tran- sits across a line of the reticle and the recorded time of transit is called the absolute personal equation of the observer. The difference between the absolute personal equations of two observers is called their relative personal equation. The rela- tive personal equation of two experienced observers has been known to be as great as I s . 2, and values greater than o s ,25 are common. For a more detailed discussion of personal equation, see 243, 244. 126. To sum up, it may be stated that the accidental errors in the determination of a clock correction from obser- vations with a portable astronomical transit upon ten stars may be reduced within the limits indicated by the probable error o s .O2 to o s . 10, but that the result is subject to a large constant error, the observer's absolute personal equation, which may be ten times as great as this probable error. Miscellaneous. 127. In the field it is often necessary to use other instru- ments as transits for the determination 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 as compared with the astronomical transits whose per- formance has just been discussed, unless, indeed, one has it firmly fixed in mind that the principal errors in a transit time determination are those due directly to the observer. On the other hand, a zenith telescope of the common form in which the telescope is eccentric with respect to the vertical axis has 130. MISCELLANEOUS. 145 been found to give rather disappointing results, 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. 128. The mathematical theory for the determination of time by the use of the transit in any position out of the meridian has been thoroughly developed. That practice has been advocated. But the additional difficulty of making the computation, 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 unnecessary. The transit is generally used in the meridian for time, at least in this country. 129. The use of the transit for time in the vertical plane passing through Polaris at the time of observation has also been advocated and has been used to a considerable extent in Europe. " The obvious advantage which this mode of observing possesses lies in the shorter period of time during; which the observer depends upon the stability of his instru- mental constants. For meridian observations this period is rarely much less than half an hour, while by the method sug- gested " in which the whole time set consists of a pointing; upon Polaris immediately followed by an observation of the. transit of a zenith or southern star across that vertical plane " it need never exceed five minutes."* This method is open, to a less extent, to the same objections as that of the preceding paragraph. This, in connection with the fact that it is rarely used in this country, makes its extended discussioa inadvisable here. 130. If the transit is turned at right angles to the plane of the meridian, in other words, is put in the prime vertical, * See Bulletin of the University of Wisconsin, Science Series, vol. !., No. 3. pp. 81-93. 146 GEODETIC ASTRONOMY. 133. an observation of the time of transit of a star across the mean line of its reticle furnishes a good determination of the lati- tude of the station if the clock correction is known. Or, if both transits, east and west of the zenith, are observed, the latitude may be computed without a knowledge of the clock correction. Formerly this method * was often used for the determination of latitude. Now it is almost entirely super- seded by the use of the zenith telescope for latitude. 131. The Sun or a planet may sometimes be observed for time. In the case of the Sun the transit of both the preced- ing and the following limb may be observed, and the mean taken as the time of transit of the center. Both limbs of a planet may possibly be observed if a chronograph is used. Otherwise the preceding and following limbs may be observed alternately on successive lines of the reticle, taking care that the number of observations on each limb is the same, and the mean of all taken as the transit of the center across the mean line. 132. It is not advisable to observe the Moon for time, for its place is not well determined. Usually but one limb can be observed, the other being either obscure or invisible; and the observation of the limb on a side line of the reticle is affected by the rapid change in the Moon's right ascension and by a parallax due to its comparative nearness to the Earth. QUESTIONS AND EXAMPLES. 133. i. An observer who is trying to get his transit into the meridian to begin observations for time finds that an observation upon rf Draconis (6 = 61 45 ') indicates that his chronometer is 4*. 2 fast of local sidereal time, while an obser- vation upon /3 Herculis (6 = 21 43') indicates that his * For the detail of this method see Doolittle's Practical Astronomy, pp. 348-377, or Chauvenet's Astronomy, vol. u., pp. 238-271. 133- QUESTIONS AND EXAMPLES. 147 chronometer is I s . 3 slow. Assuming that the instrument is in perfect adjustment with respect to collimation and inclina- tion, how much must he turn the slow-motion screw which shifts his instrument in azimuth, if one turn produces a change of 200" in azimuth ? The latitude of the station is 39 58'. Is the object-glass too far east, or too far wesc, when the telescope is pointing northward ? Ans. 0.37 turn. Too far west. 2. The star 5 Ursae Minoris (tf = 76 09') was observed to transit as follows: Line I, 9 h 48 4i s .O3: II, 49 38*. 52; III, 50 m 35 S .I2; IV, 5i m 3i s .70; V, 52 m 28 S .79; VI, 53 26 s . 40; VII, 54 m 22 s . 5 8. Derive the equatorial intervals of the various lines from the mean line. Ans. I, 40 s . 93; II, -27 s . 17; III, - I3 S .62; IV, -o s .o8; V, + I 3 S . 59; VI, +27 S .38; VII, + 4O S .8 3 . 3. While observing a transit of the star 24 Comae (6 = 18 57') clouds interfered so that observations upon the first and second lines of the reticle were missed. The observed times of transit across the remaining lines were as follows: III, 8 h I3 m 20 s . 4 o; IV, I3 m 34 8 .93; V, i 3 m 4 9 s .2S; VI, I4 m o 3 8 .88; VII, I4 m i8 s .io. The known equatorial interval of the first line is 40*. 86, and of the second 27 s .3i. Deduce the time of transit across the mean of the seven lines. Ans. 8 h I3 m 34 S .90. 4. Draw two diagrams illustrating the geometric relations from which formulae (34) and (35) of 96 are derived. 5. The following ten stars were observed for time with a Troughton and Simms transit at Cornell University on May 23, 1896. Given the partially reduced results as indicated below, compute the correction to the Howard clock (keeping mean time) with which the observations were made. GEODETIC ASTRONOMY. 133. Position Corrected Transit Right Ascension Star. o. of Across Reduced to Lamp. Mean Line.* Mean Time. e Virginis 11 31' W 8 h $6 m i8'.84 8 h 48 ra 22'.66 43 Comae. . . . ....... 28 24 W g 06 18.93 8 58 22.48 20 Can, Ven 41 07 W g 12 09.90 9 04 13.35 Ursae Maj.. . . 55 28 W g 19 00.84 9 n 04.30 Gr. 2001. o 72 56 IV 9 22 47.60 9 14 50.19 17 H. Can. Ven 37 43 E g 29 23.87 9 21 26.84 7? Ursae Maj 49 50 E g 42 39.58 9 34 42.52 V Bootis 18 55 E g 48 54.94 9 40 58.23 ii Bootis 27 53 E g 55 37.63 9 47 40,83 a Drac 64 52 E 10 oo 45.68 9 52 48.27 Ans. By the method of 104, AT C ; m 56 S .73, c = + O 8 .i7, a w = + o s .73, and a E = +o s .3O. By the method of 107, AT C = j m $6*. 74, 0.02, a w = + o 8 . 69, a E = + o 8 . 34, c + o s . 17. 6. The following ten stars were observed for time at Washington, D. C. (0 38 54'), on June 22, 1896, with a sidereal chronometer. Given the following data, compute the chronometer correction on local sidereal time: Position Corrected Transit Right Ascension Star. 5. of Across Reduced to Lamp. Mean Line.* Mean Time. 3 Serpentis 5 19' E i5 h io rn 03'. 96 i5 h io m O4.i4 i H. Urs. Min 67 46 E 15 13 32.01 15 13 30.38 /w Bootis. 37 44 E 15 20 37.07 15 20 36.65 * Draconis 59 20 E 15 22 41.21 15 22 40^29 ^'Bootis 41 ii E 15 27 15.11 15 27 14.70 C Cor. Bor. seq.... 36 58 W 15 35 30.18 15 35 30.68 K Serpentis 18 28 W 15 44 05 .86 15 44 06.54 C Urs. Min 78 07 W 15 47 52-19 15 47 51.32 e Cor. Bor 27 ii W 15 53 19.42 15 53 19.92 6 Draconis 58 51 W 15 59 59-56 15 59 59-92 Ans. By the method of 104, AT C = + o s .o8, c~ + O 8 .32, a s = + o s .6g, and a w = + o s .82. By the method of 107, AT C = + o s .o8 o s .O3, c = + O 8 .32, a E = + o s .66, and a w + o s .8o. * Transit corrected for diurnal aberration, pivot inequality, and inclination. 133- QUESTIONS AND EXAMPLES. 149 7. Suppose that a striding level carries a continuous graduation of one hundred divisions each one-twentieth of an inch long, and that each division represents one second of arc. By about how much does the arc which is the longitudinal section of the upper inner surface of the level tube depart from the chord of that arc joining the end graduations ? Ans 0.00030 inch. ISO GEODETIC ASTRONOMY. 134. CHAPTER V. THE ZENITH TELESCOPE AND THE DETERMINATION OF LATITUDE. The Principle of the Zenith Telescope. 134. The zenith distance pf a star when on the meridian is the difference between the latitude of the station of obser- vation and the declination of the star. Hence a measurement of the meridional zenith distance of a known star furnishes a determination of the latitude. In the zenith telescope, or Horrebow-Talcott, method of determining the latitude there is substituted for this measurement of the absolute zenith distance of a star .the measurement of the small difference of zenith distances of two stars culminating * at about the same time on opposite sides of the zenith. The effect of this sub- stitution is the attainment of a much higher degree of pre- cision, arising from the increased accuracy of a differential measurement, in general, over the corresponding absolute measurement; from the elimination of the use of a graduated circlef in the measurement; and from the fact that the com- puted result is affected, not by the error in estimating the absolute value of the astronomical refraction, but simply by the error in estimating the very small difference of refraction of two stars at nearly the same altitude. * A star is said to culminate at the instant when it crosses the meridian. \ The zenith telescope carries a graduated circle, but it is used simply as a finder or setting circle, and its readings do not enter the computed result. 135- THE ZENITH TELESCOPE. I$I One may form a concrete conception of the relation between the latitude and the measured difference of zenith distance as follows: Suppose an observer, A, measures the difference of the meridional zenith distances of two stars and finds one to be i farther south of his zenith than the other is north of it. Suppose that another observer, B y is stationed just i' due north of A, and measures the difference of zenith distances of those same stars at the same times. For B the southern star will evidently be \' farther from the zenith than for A, and the northern star i' nearer the zenith. Hence B will find the difference of the zenith distances to be i 02' '. Or, a given change in the position of the observer, along a meridian, produces double that change in the difference of ^zenith distances of two stars which culminate on opposite sides of the zenith. (Let the student draw a figure, in the plane of the meridian, to illustrate this paragraph.) Description of the Zenith Telescope. 135. Fig. 20 shows a zenith telescope which is the prop- erty of the Coast and Geodetic Survey. The arm A turns with the superstructure of the instru- ment and may be clamped to the horizontal circle, which is fixed to the base. At B and B are two stops which may be clamped to the circle in such positions that the telescope will be in the meridian when the arm A is in contact with either of them. One end of the horizontal axis is shown at C. The striding level is shown at D. It is counterweighted so as to make it balance on the horizontal axis. By means of the vernier and tangent screw at E, the levels FF (called latitude levels) can be set at any required angle with the telescope. These levels each carry a 2-mm. graduation of 50 divisions, numbered continuously from one end. The value of one division is about 1.5 seconds. One level would serve the 152 GEODETIC ASTRONOMY. 135. purpose, but two were placed upon this instrument so that increased accuracy might be secured by reading both. By means of the clamp at G and the tangent screw at H, operat- ing upon the sector /, the telescope may be brought to any desired inclination. The object-glass has a clear aperture 7.6 cm. (= 3.0 in.) in diameter, and its focal length is 116.6 cm. (= 45.9 in.). The eyepiece has a magnifying power of 100 diameters. The focal plane of the object-glass lies in the rectangular brass box shown aty. The micrometer screw, of which the graduated head is shown at K, controls a rectangular brass frame sliding in parallel guides within this box. The movable line with which the star bisections are made is stretched across the sliding frame. While in use the object-glass is so focused as to make the focal plane coincide with the plane in which this line moves. To facilitate counting the whole turns of the micrometer screw a small brass strip is placed in one side of the field of view of the eyepiece nearly in the plane of the micrometer line. The edge of the strip is filed into notches o.oi in. apart. The pitch of the screw being o.oi in., the micrometer line appears to move one notch along this comb for each com- plete turn of the screw. The whole turns are thus read from the comb, and the fractions are read from the head of the screw, which is graduated into one hundred equal divisions. In Fig. 21, drawn in a vertical plane through the center of the telescope, let O be the optical center * of the object- glass. Let 5 be the position of a star. The star image is formed at the focus T y which is necessarily in the line SO produced. If the star is to appear bisected, the micrometer line must be placed at T. If another star later occupies the * The optical center of a lense is that point through which all incident rays pass without permanent change of direction. 136. ADJUSTMENTS. 1 53 position S', its image will be formed at T' , in S'O produced, and to make a bisection the micrometer screw must be turned until the micrometer line is at T' . The recorded number of turns of the micrometer screw required to move the line from T to T' gives a measurement of the linear distance TT' . For the small angles concerned this linear distance is propor- tional to the angle TOT' , the equal of SOS' . Hence the observed movement of the micrometer screw gives a measure- ment of the difference of zenith distances, SOS' , of the two stars. In this particular instrument, the pitch of the screw being about o.oi in. and the focal length OT about 45.9 in., o.oi one turn of the screw measures an angle * of about sin" 1 , 45-9 or about 40". The more common form of zenith telescope differs from the one here shown in having the telescope mounted eccen- trically on one side of the vertical axis instead of in front of it, as in this case; in having a clamp which acts directly upon the horizontal axis in the place of the clamp at G acting on the sector /; and in having only one latitude level instead of two. Adjustments. 136. The vertical axis must be made truly vertical. In adjusting and using the instrument it will be found conven- ient to have two of the three foot-screws in an east and west direction. The vertical axis may be made approximately vertical by use of the plate level, if there is one on the instru- ment, and the final adjustment made by using the latitude * The value of one turn cannot be determined with sufficient accuracy by such linear measurements. They are given here merely to illustrate the principle involved. The indirect process by which the value is ordi- narily determined will be found described in 158-164. 154 GEODETIC ASTRONOMY. 138. level. The process in each case is precisely the same as that of using the unadjusted plate levels of an engineer's transit to adjust its vertical axis. The horizontal axis must be perpendicular to the vertical axis. This may be tested, after the vertical axis has been adjusted, by reading the striding level in both of its positions. If the horizontal axis is inclined, it must be made horizontal by using the screws which change the angle between the hori- zontal and vertical axes. 137. The line of collimation must be perpendicular to the horizontal axis. If the instrument is of the form shown in Fig. 20, this adjustment may be made as for an astronomical transit ( 86) by reversing the horizontal axis in the Ys. If the instrument is of the form in which the telescope is eccen- tric with respect to the vertical axis, the method of making the test must be modified accordingly. 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 from the vertical axis to the axis of the telescope. Or, a single pair of points at that distance apart may be used and the horizontal circle trusted to determine when the instrument has been turned 180 in azimuth. If one considers the allowable limit of error in this adjustment (see 167), it becomes evident that a telegraph pole or small tree, if sufficiently distant from the instrument, may be assumed to be of a diameter equal to the required distance between the two points. Or, a single point at a known dis- tance may be used and a computed allowance made on the horizontal circle for the parallax of the point when the tele- scope is changed from one of its positions to the other. 138. During daylight the object-glass should be carefully focused on the most distant well-defined object available, to insure that stars may be seen at night. A neglect to do this 139- ADJUSTMENTS. 155 may cause the observer, especially if inexperienced, much annoyance while he is trying to find out why stars for which the settings are properly made do not appear in the telescope. At the first opportunity the focus should be tested upon a star. When once the focus has been satisfactorily adjusted at a station, so that there is no parallax, it should not again be changed at that station. For any change in the object- glass focus changes the angular value of a division of the micrometer. It is well to clamp the slide so as to make an accidental change of focus impossible. 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, and throughout the observations, 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 horizontal motion free, and afterward 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 to test the setting of each stop again by an observation of another star before commencing latitude observations. 139. The movable line, attached to the micrometer, with which 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 mov- able line, upon a distant well-defined object, with the image of that object near the apparent right-hand side of the field of the eyepiece. Shift the image to the apparent left-hand side 156 GEODETIC ASTRONOMY. 140. 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. 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 its 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 Observing List. 140. Before commencing the observations at a station, an observing list should be prepared, showing, for each star to be observed, its catalogue number or its name, its magnitude, mean* right ascension and declination (at the beginning of the year), zenith distance, whether it culminates north or south of the zenith; and for each pair the setting of the ver- tical circle (the mean of the two zenith distances), the differ- ence of the zenith distances with its algebraic sign as given by formula (54); and finally the micrometer comb setting for each star. For the purposes of the observing list the right ascen- sions to within one second of time, and the declinations and derived quantities within one minute of arc, are sufficiently accurate. If the micrometer comb reading is one minute per notch, and the middle notch is called 20, the comb setting for * For the definition of the mean place of a star see 37, 39. 141. OBSERVING LIST. 1 57 one star is 20 + the half-difference of zenith distances for one star, and 20 that half-difference for the other star of a pair. The requisites for a pair of stars for this list are that their right ascensions shall not differ by more than 2O m , to avoid too great errors from instability in the relative positions of different parts of the instrument; nor by less than i m , 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 half the length of the micrometer comb, 20' for the usual type of instrument; that each star shall be bright enough to be seen distinctly not fainter than the seventh magnitude for the instruments here described; and that no zenith distance shall exceed 35, to guard against too great an uncertainty in the refraction. The selection of a series of such pairs from the stars of a catalogue requires much time and patience. The total range of the list in right ascension is governed by the hours of darkness on the proposed dates of observa- tion, and by the convenience of the observer. The third of the above conditions may perhaps be used more conveniently in this form: the sum of the two declinations must not differ from twice the latitude by more than 20'. To prepare the list the latitude of the station should be known within a minute. It may possibly be secured from a map; if not, then from a sextant observation of the Sun, or from an observation of the meridional zenith distance of a star with the finder circle of the zenith telescope. 141. The stars selected should be such that their com- puted mean places may be made to depend in each individual case upon observations at several different observatories. The declination of a star as derived from observations at a single observatory will not in general be sufficiently accurate for the purpose in hand. 1 58 GEODETIC ASTRONOMY. 141. If the observer knows that in making the computation an ample collection of catalogues * of original observations at each of various observatories will be available, he may select all the suitable pairs he can find in any extensive list of stars, say the British Association Catalogue, or any of the Green- wich Catalogues, and trust to finding afterward in the various catalogues a sufficient number of observations upon each star at various observatories to give an accurate determination of its place. This is the usual procedure in the Coast and Geodetic Survey. A computer in the office at Washington calculates the declinations of the stars which have been observed for latitude by bringing together in a least square adjustment all the observations upon each star that he finds in his large collection of star catalogues. If the necessary collection of catalogues of original obser- vations is not known to be available, the observing list had best be made up from star lists in which the declinations given are the result of the compilation and computation of original observations at various observatories as outlined above. Among such available lists of mean places are those in the various national ephemerides; Preston's Sandwich Island List in the Coast and Geodetic Survey Report for 1888, Appendix No. 14, pp. 511-523; the list given in Appendix No. 7, pp. 83-129, C. & G. S. Report for 1876; and Boss' list in the re- port of the Survey of the Northern Boundary from the Lake of the Woods to the Rocky Mountains, pp. 592-615. These lists are given in about the order of the accuracy of their star places when reduced to the present time, the more recently computed places being more accurate, if other conditions are about the same. In the report of the Mexican Boundary Survey of 1892-93, which is about to be published, there will * An indication of what is meant by "an ample collection of catalogues " may be gained by reading 37. 142 OBSERVING LIST. 1 59 be given an unusually accurate list of declinations prepared for that survey by Prof. T. H. Safford. 142. If an observer finds difficulty in securing a sufficient number of pairs from the available lists of accurately com- puted places, he may extend his list in two ways. Firstly, and preferably, if he has a good instrument, by extending the limits given in 140. The limit of difference of right ascen- sions may not safely be extended much ; the limit of difference of zenith distances may be extended to the full length of the micrometer comb, say 40" ; and zenith distances as great as 45 may be allowed. Secondly, he may have recourse in his extremity to a catalogue of original observations at the Greenwich observatory for enough pairs to complete his list. The number of pairs required may be estimated by the con- siderations dealt with in 169. In the list of pairs resulting directly from the search in the star catalogues 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 shall not be less than 2 m . In that interval a rapid observer can finish the readings upon one pair, set and be ready for the next, under favorable circumstances. The omitted pairs may be included in a list prepared for the second or third night of observation if one uses the second plan outlined in 169. Also, 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, or the three or more stars form- ing what might be called a compound pair may all be observed at one setting of the telescope and then treated in the com- putation as two or more separate, but not independent, pairs. It is desirable to so select the pairs that the algebraic sum of all the differences of zenith distances for a station shall be l6b GEODETIC ASTRONOMY. 143. nearly zero, so as to make the computed latitude for the station nearly free from any effect of error in the mean value of the micrometer screw. Directions for Observing. 143. The instrument being adjusted, set the vertical circle to read the mean zenith distance, or " circle setting" as marked in the observing list, of the first pair. Direct the telescope to that side of the zenith on which the first star of the pair will culminate. Put the bubble of the latitude level nearly in the middle of the tube by using the tangent screw which changes the inclination of the telescope. Place the micrometer thread at that part of the comb at which the star is expected, as shown by the observing list. Watch the chronometer to keep posted as to when the star should appear. When the star enters the field place the micrometer thread approximately upon it. and center the eyepiece over the thread. As soon as the star comes within the limits indi- cated by the vertical lines of the reticle bisect it carefully. As the star moves along watch the bisection and correct it if any error can be detected. Because of momentary changes in the refraction, the star will usually be seen to move along the thread with an irregular motion, now partly above it, now partly below. The mean position of the star is to be covered by the line. An attempt is being made to secure a result which is to be in error by much less than the apparent width of the thread, hence too much care cannot be bestowed upon the bisection. 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 reticle 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 145' DIRECTIONS FOR OBSERVING. l6l 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 thread to a new position. 144. The instrument must be manipulated as carefully as possible. Especial care should be taken in handling the micrometer screw, as any longitudinal force applied to it pro- duces 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 the same 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 level and telescope must be kept constant. If it is necessary to relevel, to keep the bubble within reading limits, use the tangent screw which changes the inclination of the telescope. Even this may introduce an error, due to a change in the flexure of the telescope, and should be avoided if possible. 145. For first-class observing it is desirable to have a recorder. He may count seconds from the face of the chronometer for a minute before culmination in such a way as to indicate when the star is to culminate according to the right ascension given on the observing list, taking the known chronometer correction into account. Such counting aloud serves a double purpose. It is a warning to be ready and indicates where to look for the star if it is faint and difficult 1 62 GEODETIC ASTRONOMY. I 4 6. to find. It also gives for each star a rough check upon the position of the azimuth stops and warns the observer when they need readjustment. It is only a rough check, because the observing list gives mean right ascensions (for the begin- ning of the year) instead of apparent right ascensions for the date. But in view of 167 it is sufficiently accurate. The observer can easily make allowance for the fact that all stars will appear to be fast or slow according to the observing list by about the same interval, o 8 to 5" (the difference between the mean and the apparent place). If a star cannot be observed upon the middle line, on account of temporary interference by clouds or tardiness in preparing for the obser- vation, observe it anywhere within the safe limits of the field as indicated by the vertical lines of the reticle and record the chronometer time of observation. EXAMPLE OF RECORD. 146. Station No. 8, near San Bernardino Ranch, Arizona. Instrument Wurdemann Zenith Telescope No. 20. Observer J. F. H. Dale August 9, 1892. Micrometer. Level. No. of Star No. N. or Pair. B. A. C. S. Turns. Divisions. N. S. Remarks. OQ 7528 S. 22 82.0 19.9 54-9 Sky perfectly clear. 7544 N. 16 98.9 56.0 20.9 Chronometer 21* fast. 7566 N. 24 71.2 52.9 17.9 9 1 7586 S. 13 68.9 17-9 52.8 7631 N. 30 29.0 52-5 17.6 93 7662 S. 9 13-0 I6. 4 Sr-7 148. FORMULAE. 163 Derivation of Formula. 147. Let C and C x be the true meridional zenith distance, and d and 6' the declination, of the south and north star of a pair, respectively. Then the latitude of the station is *=i(*+*') + KC-C') (54) Let the student draw the figure and prove this formula. Let (z z') be the observed difference of zenith distances of the two stars, the primed letter referring to the north star, (z ^) is in terms of the observed micrometer readings = (M M')r, in which M and M r are the micrometer read- ing upon the south and north star, respectively, expressed in turns, and r is the angular value of one turn. Before this observed difference of zenith distances may be used in (54) it must be corrected for the inclination of the vertical axis as given by the level readings, for refraction, and for reduction to the meridian if either star is observed off the meridian. 148. Let d be the value of one division of the latitude level. Let n and s be the north and south reading, respec- tively, of the level for the south star, and n' and s' the same for the north star. Then, if the level tube carries a graduation of which the numbering increases each way from the middle, the inclination of the vertical axis, considered positive if the upper end is too far south, is jK + * ! )-(*+*')i (55) If the level tube carries a graduation which is numbered UNIVERSITY sSLCAUFOaH^ I&J. GEODETIC ASTRONOMY. 149. continuously from one end to the other, with the zero nearest the eyepiece, the inclination of the vertical axis is '' ;;V K' + *')-( + *)} ..... (56) If the zero is nearest the object-glass the algebraic sign must be changed from that given above. The inclination of the vertical axis makes the south zenith distance too small by the amount indicated by (55) or (56) and the north too large by the same amount. Hence the d correction to \(z z') is \(n -f- ') (s -f- /)}, or the corre- 4 sponding expression (56). 149. The refraction makes each apparent zenith distance too small. If R and R' represent the refraction for the south and north star, respectively, the correction to (z z') is (R - R'\ and to J(* - /) is (R - R'). Let m be the correction to the apparent zenith distance of a south star observed slightly off the meridian to reduce it to what it was when on the meridian, and m' the correspond- ing reduction for a north star observed off the meridian. The correction to \(z 2') will then be J(/# m'). m and m' are of course zero in the normal case, when the observation is made in the meridian. Formula (54) may now be written, for an instrument with the level graduated both ways from the middle, = * + .. (57) !$!. FORMULAE. 165 This is the working formula for the computation, but the values of the last two terms may be conveniently tabulated. 150. The difference R R being very small, the variation of the state of the atmosphere at the time of observation from its mean state (see refraction tables, 294-297) may be neglected, except for stations at high altitudes. It has been shown, by the investigations of the laws of refraction which have been referred to in 67-69, that this differential refrac- tion, for the mean state of the atmosphere, is, with sufficient accuracy for the present purpose, R-R'= $7". 7 sin (* *>) sec* *. . . (58) By computation from this formula the value of the term %(R R f ) of formula (57) has been tabulated, in 304, for the arguments %(z z'} as directly observed With the microm- eter, and the zenith distance. If the station is so far above sea-level that the mean barometric pressure is less than y 9 ^, say, of 760 mm. ( 294), the mean pressure at sea-level, it is necessary to take this fact into account by diminishing the values of the differential refraction given in 304 in the ratio of the mean pressures. That is, if the ,mean pressure is 10$ less than at sea-level diminish the values of 304 by io#; if 20$ less subtract 20$, and so on. Inspection of the table shows that this allowance need only be made roughly, since the tabular values are small. 151. The value of , and of its equal , is tabulated in 305. The table gives directly the correction to the latitude for any case of a star observed off the meridian, but within one minute of it. If both stars of a pair are observed off the meridian two such corrections must be applied, one for each star. For the difficult derivation of the formula from which i66 GEODETIC ASTRONOMY. 152- this table is computed see Chauvenet's Astronomy, vol. II. pp. 346, 347; or Doolittle's Practical Astronomy, pp. 505, $06. EXAMPLE OF COMPUTATION. 152. Station No. 8, near San Bernardino Ranch, Arizona. Instrument Wurdemann Zenith Telescope No. 20. Observer J. F. H. Date August 9, 1892. [Left-hand page of Computation.'} o 1. Micrometer, i turn = ioo div. Level, i div. = i".28. i 5**J VH Level JO to c5ed Reading. Diff. Z. D. N. S. Corr. in Q * in fc Div. t. d. t. d. 7528 S. 22 82.0 19.9 54-9 19 46' 48". 62 7544 N. 16 98.9 + 5 83.1 56.0 20.9 + 0.52 42 47 05 .83 7S66 N. 24 71.2 52.9 17.9 37 47 26 .64 9 1 7586 S. 13 68.9 ii 02.3 17.9 52.8 4~ 0.03 25 3 55 -38 93 7631 7662 N. S. 30 29.0 9 13-0 52.5 16.4 17.6 4-0.50 55 17 2 3 -93 7 44 26 .71 [Right-hand page of Computation.] Sum and Mean of Declinations. Corrections. Latitude. Remarks. Micrometer. Level. Refraction. Meridian. 62 33' 54"-45 31 16 57 .22 4- 3' oi".o5 + o".6 7 4- o".o4 31 19' 5 8".98 62 51 22 .02 31 25 41 .01 - 5 42 .26 -f o .04 o .08 58 .71 63 oi 50 .64 3i 3 55 -3 10 57 .01 + o .64 - o .18 58 .77 153- EXPLANATION OF COMPUTATION. Explanation of Computation. 153. The first seven columns of the computation need no explanation. The eighth column gives the values of i{(' + -0 ( + s)\, 148, the level tube of this instru- ment being graduated continuously from end to end with the zero nearest the eyepiece. The ninth column gives the meridian distance of such stars as were not observed upon the meridian. It is the hour-angle of the star expressed in seconds of time. To obtain the apparent right ascension within one second, which is sufficiently accurate for the determination of the required hour-angle for the present pur- pose, proceed as follows: Select a star from the mean place list of the Ephemeris which has nearly the same right ascen- sion and declination as the star in hand. Compare its mean right ascension with its apparent right ascension for the date and assume that the corresponding change for the star in hand is the same. The declinations given in column ten are those resulting from the apparent place computation made as indicated in 46-49, or from the Ephemeris by interpolation in case the star is one of which the apparent place is there given. The computation of the values in the second column of the right-hand page of the computation is facilitated, if there are many observations, by first constructing a table giving 10, 20, 30, etc., turns of the micrometer reduced to arc by multi- plying by : then of I, 2, 3, 4, 5, 6, 7, 8, 9 turns reduced to arc: of 10, 20, 30, 40, 50, 60, 70, 80, 90 divisions thus reduced to arc: of I, 2, 3, ... etc.: and of o. I, 0.2, 0.3, . . . etc. Such a table reduces the multiplication process otherwise required to a process of adding five tabular quan- tities. l68 GEODETIC ASTRONOMY. 154. The corrections for refraction were obtained by subtract- ing 20$ from the values of 304, the barometric pressure being only about four-fifths as great at San Bernardino as at sea-level. The latitude is obtained from each pair by adding the various corrections algebraically to the mean declination, as indicated in formula (57). With sufficient accuracy for some purposes the indiscrimi- nate mean of all the individual values may be taken as the final value of the latitude. If the best, or most probable, value is desired the procedure outlined below must be fol- lowed. Combination of Individual Results by Least Squares. 154. Let us first deal with the simplest case. Namely, let it be supposed that / separate pairs have been observed on each of ri nights at a station, each pair being observed on every night. For this case it will be found that the indis- criminate mean is, after all, the most probable value of the latitude, but the principles developed will be found useful in dealing with other more difficult cases in which this is not true. The differences A obtained by subtracting the mean result for any one pair from the result on each separate night for that pair are evidently independent of errors of declination. We may compute from these differences, or residuals, the probable error of a single observation e. This error of observation includes the observer s errors, instrumental errors, and all external errors except the errors of the assumed declinations. Then by least squares e = *This square bracket [ j is here used to indicate summation, as it fre- quently is in text-books on least squares. 154- COMBINATION OF RESULTS. l6g in which \_A A\ stands for the sum of the squares of all the residuals A obtained from all the pairs. The probable error e p of the mean result from any one pair may also be computed from the observations by the formula (6o) in which v is .the residual obtained by subtracting the mean result for the station from the mean result for each pair. There are / such residuals. \yv\ stands for the sum of the squares of these residuals. Let e B be the probable error of the mean of the two declinations. e p evidently includes the declination errors of the two stars of a pair. From the ordinary law of transmis- sion of accidental errors Whence e$ may thus be obtained, from the observations for lati- tude, by substituting the values e and e p computed by (59) and (60) in (62). n f being the same for all pairs, it is evident from (61) that the means from the various pairs have equal weight. The most probable value for the latitude is then the indiscriminate mean of the results from the separate pairs, or, what is numerically the same in this case, the indiscriminate mean of all the individual results for latitude. I?O GEODETIC ASTRONOMY. I55 The probable error of the final result for latitude is (63) 155. The simple case just treated seldom occurs in prac- tice. Observations upon certain pairs are missed on some of the nights by accident or by cloud interference; work may be entirely stopped by clouds after half the observations of an evening have been made; or on the later evenings of a series the observer may purposely, with a view to more effectual elimination of declination errors, include in his observing list certain pairs which have not before been observed at that station, in the place of pairs which have already been observed once or more. In the usual case, then, a total of p pairs are observed, pair No. I being observed n l times (i.e., on n l nights), pair No. 2 7z a times, . . . , and the total number of observations is n = n l -{- n^ -\- n z . . . By the same reasoning as before we have, by the ordinary least square formula, the probable error of a single observa- tion = Ao.455) " V - P To obtain the probable error e p of the mean result from any one pair with rigid exactness it is necessary to take into account the fact that different pairs must now be given different weights, since some are observed more times than others. To do this would make the computation consider- ably longer than is otherwise necessary. Fortunately, inves- tigation of the numerical values concerned shows that the results are abundantly accurate if formula (60) is here used 1 56. COMBINATION OF RESULTS. I? I and the fact that the pairs are of unequal weight neglected in deriving e p . With sufficient accuracy, then, (65) According to the laws of accidental errors the probable errors of the mean results from the separate pairs are The values e p . lt e p .^ . . . differ from each other because of the various values of ,, n^ n aJ . . . For use in deriving e& an average value of the second term under the radical must be obtained. Again neglecting the unequal weights of the pairs, this average value, which will be called e% is + ' + ' * V Corresponding to (62), there is now obtained (68) from which e s may be computed from the latitude observa- tions. 156. The proper weights for the mean results from the separate pairs are inversely proportional to the squares of their probable errors. Hence these weights w lt w^ w t , . . . , are proportional to (69) 1/2 GEODETIC ASTRONOMY. 156. and may now be computed from the known values of e& and e. The most probable value for the latitude of the station is the weighted mean of the mean results from the various pairs, or . . _ |>0] ' ' in which 0, is the mean result from the first pair, a from the second pair, and so on. Also, the probable error of this result is /^455)^3 ' V (/ - ow ' in which [wz/ a ] stands for the sum of the products of the weight for each pair into the square of the residual obtained by subtracting from the mean result for that pair, and [w] is the sum of the weights. In case two north stars are observed in connection with the same south star, or vice versa, and the computation is made as if two independent pairs had been observed, the weight of each of these pairs as given by (69) should be mul- tiplied by \ to take account of the fact that they are but partially independent. Similarly if three north stars have been observed in connection with the same south star the weights from (69) for each of the three resulting pairs should be multiplied by J.* If, however, a given north star is observed in connection with a certain south star on a certain night or nights, and * Coast and Geodetic Survey Report, 1880, p. 255 ; or Professional Papers of the Corps of Engineers, No. 24 (Lake Survey Triangulation), p. 625. 157- COMBINATION OF RESULTS. 173 on a certain other night or nights is observed in connection with some other south star, the case is different, and the com- putation is sufficiently accurate, though not exact, if each of such pairs is given the full weight resulting from (69). If very few pairs are observed more than once at a sta- tion the determination of e s from the latitude observations obviously fails, and it must be estimated in some other way from the star catalogues, for example. 157. As an example of the application of formulae (64) to (71), the process of combining the various values for the lati- tude of station No. 8 on the Mexican Boundary Survey may be given. At this station TOO observations were made on 75 pairs, 25 of the pairs being observed twice each, and the other 50 once each. The observations extended over four nights. The sum of the squares of the fifty residuals, A, obtained by subtracting the mean for each pair which was observed twice, from each of the two values from that pair, was 2.50 square seconds. The probable error of a single observation was then, from (64), The indiscriminate mean of the 75 results, one from each pair, was found to be 31 19' 59". 02. By subtracting this value from each of the 75 separate values, squaring, and adding, it was found that \vv~\ = 11.62 square seconds. From (65) it followed that = o". 2 6 7 . 1/4 v GEODETIC ASTRONOMY. 158. The term ( -- 1 --- 1 -- . . .) of formula (6?) is here \n l n^ n s I (25X0.5)+ 50= 62.5, and 2 . 5) = 0.0379. , ' Whence from (68) ( ^' = 0.07140.0379 = 0.0335 or e s =o".i$$. Substituting the values of ef and e* in (69), the weights for the pairs observed twice was found to be 17.8, and for those upon which one observation only was made, 12.7. Since it is the relative weights only which affect the final result these weights were for convenience written i.oand 0.7, respectively. The resulting weighted mean as indicated in (70) was found to be 31 19' 59". 01. From the residuals corresponding to this value it was found that \wv'*~\ 9.26 square seconds. Hence Determination of Micrometer and Level Values. 158. The most advantageous method of determining the screw value is to observe the time required for a close circum- polar star near elongation to pass over the angular interval measured by the screw. Near elongation the apparent motion of the star is very nearly vertical and uniform. That one of the four close circumpolars given in the Ephemeris, namely, # , #, and ^ Ursae Minoris, and 5 1 Cephei, 158. MICROMETER VALUE. 175 may be selected which reaches an elongation at the most con- venient hour. In selecting the star it may be assumed that the elongations for such stars occur when the hour-angle is six hours, on either side of the meridian. Having selected the star, it is necessary both for planning the observations and for the computation to compute the time of elongation more accurately. The spherical triangle defined by the pole, the star, and the zenith is necessarily, for a star at elongation, right-angled at the star. Hence it may be shown that cos t E = tan cot d, . ... (72) in which t E is the hour-angle of the star at elongation (always less than 6 h ). This hour-angle added to or subtracted from the right ascension of the star, for a western or eastern elon- gation respectively, gives the sidereal time of elongation, whence the chronometer time of elongation becomes known by applying the chronometer error. It is advisable to have the middle of the observations about at elongation. The observer should obtain an approximate estimate of the rate at which the star moves along his micrometer by a rough obser- vation or from previous record, and time the beginning of his observations accordingly. Everything being ready for the observations, the star is brought into the field of the telescope, the telescope clamped, the star image so placed as to be approaching the micrometer line with the micrometer reading some exact integral number of turns, and the level bubble brought near to the middle of the tube. The chronometer time of transit of the star across the line is observed, and the level read. Then the micrometer line is moved one whole turn in the direction of the motion of the star, the time of transit is again observed and the level read, and the process repeated until as much of the middle portion of the screw has been covered by the observations as GEODETIC ASTRONOMY. 159. is considered desirable. If desired, an observation may be made at every half-turn, or even every quarter-turn by allow- ing an assistant to read the level. 159. In Fig. 22, let P be the pole, 5 the position of the star when observed transiting across the micrometer line, and S B its position at elongation. The small circle SS E is a por- tion of the apparent path of the star. Let SK be a portion of the vertical circle through 5 limited by the great arc PSs* Let the length (in seconds of arc) of SK, measuring the change in zenith distance of the star in passing from the posi- tion 5 to elongation, be called z. Let the sidereal interval of time, in seconds, from position 5 to elongation be called r. In the spherical triangle SKP the angle at K is 90, and that at Pis, in seconds of arc, I$T. Therefore sin z cos d sin (I$T) ..... (73) The various values of z corresponding to observed values of r might be directly computed from (73). But the compu- tation becomes much easier and shorter, though based upon a more difficult conception, if one proceeds as follows: (73) may be written sn (74) From the known expansion of the sine in terms of the arc there is obtained sin(iSr) = (ijr) sin i"-i(i 5 rsin i")' +T ^(i 5r s in i")'. (75) By substitution from (75) in (74) * = 15 cos<5Sr-i(l5 sin i")V + Th94 30 cos d Below is a portion of the record and computation. i8o GEODETIC ASTRONOMY. 163. Level | c/5 c c Readings. I fl g !T (J V H J.f? . Chronometer o rt O - Corrected JM Time. i; c 3 o > ^2 z u Times. O rt i- &&H N. S. 13 us Is 6 i II H u J $ 22.5 2I h 20 m 51 5-4 I 3 m 39- o + o".5 + O.2O + 0-.4 2I h 20 m SI 8 . 9 22 22 23 O 50.5 17. 12 07 + o .3 + 0.10 + .2 22 2 3 .5 21-5 21 23 57 5 25 32 5 50-4 SO. 6 17- 17- 10 32 5 8 57 5 + .2 + O . I + 0.30 + 0.10 + o .6 + 0.2 23 58 -3 25 32 .8 20.5 27 05 o 50.6 17- 7 25 o + .1 + O.IO + O .2 27 05 .3 12.5 52 08 5 Si-* 17- '7 38 5 I .0 - 0.40 -o .8 52 06 .7 3 i m i 4 .8 12 II 53 44 o 55 16 o 56 50 o 51-2 51-3 51.2 17- 19 14 o 20 46 22 20 - 1 .3 - 1 .7 2 .1 0.50 0.60 0.50 1 .0 I .2 I .O 53 4i -7 55 13 -i 56 46 .9 18 .2 14.8 14 .1 I0. 5 58 25 .0 51.2 17- 23 55 -0 -- 2 .6 0.50 I .0 58 21 .4 16 .1 Mean for 10 turns (from all the observations) = 31 Mean time for i turn = 187*. 465. 14". 65 o. Log 187.465 = 2.2729202 Log cos d = 8.3438467 Log 15 = 1.1760913 Log 62". 067 = 1.7928582 = 62".o67 Correction for refraction * = o .051 " " rate = o .000 One turn Final value = 62 .016 o".on 163. If the values of both the level and the micrometer are unknown, one may observe for micrometer value as out- lined above, and also derive the value of the level in terms of the micrometer as indicated in 118. We may first compute the micrometer value, omitting the level corrections; then derive the value of the level division from this approximate value of micrometer. The level corrections being now intro- duced into the micrometer computation will be found to modify it so slightly that a second approximation for the level value will not ordinarily be required. * Refraction at this station was only | of that at sea-level. 1 65. EXROKS. l8l 164. If no special observations for micrometer value have been made, or if such observations have proved defective, the micrometer value may be derived directly from the latitude observations. Let P be the mean latitude, as deduced with an approximate micrometer value, from all pairs for which the micrometer difference (taken S N] was positive, 4> N the mean latitude from pairs with minus micrometer differences, D P the mean of the positive micrometer differences, and D N the mean of the negative differences. Then the correction to the approximate value of one turn is * For methods of determining the level value alone, see II6-I2I. Discussion of Errors. 165. The external errors affecting a zenith telescope observation are those due to defective declinations and those due to abnormal refraction. The declinations used in the computation have probable errors which are sufficiently large to furnish much, often more than one-half, of the error in the final computed result. This arises from the fact that a good zenith telescope gives results but little inferior in accuracy to those obtained with the large instruments of the fixed observatories which are used in determining the declinations. The following three examples will serve to indicate the * This formula is not exact, from the least square point of view, that is, it does not give the most probable value of the required correction. But it gives so nearly the same numerical results as the exact least square treat- ment, and leads to so short and simple a computation, that its use is advis- able. 1 82 GEODETIC ASTRONOMY. 1 66. improvement in the available declinations during the last few years, and the magnitude of the declination errors to be expected. The probable error of the mean of two declina- tions, e s , was found * to be 0^.5 5 for the list of stars furnished to the U. S. Lake Survey by Prof. T. H. Safford in 1872. Similarly, for the list of stars furnished from the Coast and Geodetic Survey Office for use in determining the variation of latitude at the Hawaiian Islands in 1891-92^ e s o".i8; for the list furnished to the Mexican Boun- dary Survey by Prof. Safford in 1892-93, e s = o". 18. Such a high degree of precision as that of the last two exam- ples is only attainable by an up-to-date computation from many catalogues of many observatories. By the time such lists are available in print their accuracy has ordinarily diminished considerably with the lapse of time. The errors in the computed differential refractions are probably very small, and it is not likely that they increase much with an increase of the mean zenith distance of a pair, up to the limit, 45. If there were a sensible tendency, as has been claimed, for all stars to be seen too far north, or south, on some nights, because of the existence of a barometric gradient for example, it should be detected by a comparison of the mean results for different nights at the same station. Many such comparisons made by the writer indicate that in zenith telescope latitudes there is no error peculiar to the night. The variation in the mean results from night to night was found in all the cases examined to be about what should be expected from the known probable errors of obser- vation and declination. 166. The observer s errors are those made in bisecting the star, and in reading the level and micrometer. Here also may * Professional Papers of the Corps of Engineers, No. 24, pp. 622-638. f Coast and Geodetic Survey Report, 1892, Part 2, p. 158. 1 66. EXROXS. 183 perhaps be classed the errors due to unnecessary longitudinal pressure on the head of the micrometer. Indirect evidence indicates that the error of bisection of the stars is one of the largest errors concerned in the measure- ment. It probably constitutes the major part of the computed error of observation, and the bisection should be made with corresponding care. With care in estimating tenths of divisions on the micro- meter head and on the level tube, each of these readings may be made with a probable error of o. I division. For the ordinary case of a micrometer screw of which one turn repre- sents about 60", and of a level of which the value is about i" per division, such reading would produce probable errors of o /x .O4 and 0^.05, respectively, in the latitude from a single observation. These errors are small, but by no means insignificant when it is considered that for first-class observing the whole probable error of a single observation, arising from all sources except declination, is less than o".3O, and sometimes even less than o".2O. 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 read- ing error indicated above; and that if the level bubble is found to be moving, a reading taken after allowing 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 posi- tion 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. A dis- placement of 3-gVir part of an inch in the position of the micrometer line relative to the object-glass produces in the telescope of Fig. 20 a change of more than i" in the apparent 1 84 GEODETIC ASTRONOMY. 1 67. position of the star. The whole instrument being elastic, the force required for even such a displacement is small. An experienced observer has found that in a series of his latitude observations, during which the level was read both before and after the star bisections, the former readings continually differed from the latter, from o". I to C/'.Q, always in one direction.* 167. Among the instrumental errors may be mentioned those due: 1st, to an inclination of the micrometer line to the horizontal; 2d, to an erroneous level value; $d, to inclina- tion of the horizontal axis; 4th, to erroneous placing of the azimuth stops; 5th, to error of collimation; 6th, to irregu- larity of micrometer screw; 7th, to an erroneous mean value of the micrometer screw; 8th, to the instability of the relative positions of different parts of the instrument. The first-mentioned source of error must be carefully guarded against, as indicated in 139, as it tends to introduce a constant error. The observer, even if he 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 stars, or vice versa. The error from using an erroneous level value is smaller the smaller are the level corrections and the more nearly the plus and minus corrections in a series balance each other. To insure that it shall be negligible it is necessary to relevel every time the correction becomes more than two seconds, at most. The errors from the third, fourth, and fifth sources may easily be kept negligible. An inclination of one minute of arc in the horizontal axis, or an error of that amount in either collimation or azimuth, produces only about o".oi error in the * Coast and Geodetic Survey Report, 1892, Part 2, p. 58. 1 68. EXXOXS. 185 latitudes. All three of these adjustments may easily be kept far within this limit. Most micrometer screws are so regular that the unelimi- nated error in the mean result for a station from the sixth cause is usually very small. But it should not be taken for granted that a given screw is regular. Large irregularities may be detected by inspection of the computation of the micrometer value. Errors with a period of one turn may be detected by making the observations for micrometer value at every quarter-turn, and then deriving the value of each quarter, o to 25 divisions, 25 to 50 divisions, etc., of the head separately. The four mean values thus derived should agree within the limits indicated by their probable errors. 168. To guard against error from the seventh source the pairs must be so selected as to make the plus and minus micrometer differences at a station balance as nearly as possi- ble,* For example, at the fifteen astronomical stations occupied on the Mexican Boundary Survey of 1892-93 the mean micrometer difference, taken with regard to sign, never exceeded 0.36 turn at any station, and was less than o. 10 turn at nine of the stations. If the plus and minus micrometer differences balance exactly at a station, an erroneous microm- eter value does not affect the computed latitude, but merely increases the computed probable errors. * It seems an easy matter to make an accurate determination of mi- crometer value. But experience shows that such determinations are subject to unexpectedly large and unexplained errors. For example, in the Hawaiian Island series of observations, mentioned above, the microm- eter value was carefully determined twelve times. The results show a range of nearly ^ of the total value. This corresponds to a range of about one-sixth of an inch in the focus of the object-glass. In the San Francisco series, and in general wherever the micrometer value has been repeatedly measured, the same large discrepancies have been encoun- tered. Hence the need of carrying out the suggestions of the above para- graph. 1 86 GEODETIC ASTRONOMY. 169. The errors from the eighth source may be small on an average, but they undoubtedly produce at times some of the largest residuals. They may be guarded against by protect- ing the instrument from sudden temperature changes, and from shocks and careless handling, and by avoiding long waits between the two stars of a pair. The closer the agreement in temperature between the instrument room and the outer air the more secure is the instrument against sudden and unequal changes of tempejature. The computed probable error of a single observation, e, including all errors except those of declination, was found to be as follows in three recent first-class latitude series: In the observations for variation of latitude at San Francisco * in 1891-92, 1277 observations (in two series) gave e = o". 19 and e = o".28 ; in a similar series at the Hawaiian Islands f for the same purpose in 1891-92, 2434 observations gave e= o".i6; from 1362 observations at fifteen stations on the Mexican Boundary in 1892-93, e = o" '.19 to O 7/ .38.J 169. When an observer begins planning a series of obser- vations to determine the latitude of a given point two ques- tions at once arise. How many observations shall be made ? How many separate pairs shall be observed ? Increasing the number of observations increases the cost of both field work and computation. An increase in the total number of separate pairs adds proportionally to the work of computing the mean places, but otherwise has little effect on the total cost. The economics of the problem demand that the ratio of observations to pairs shall be such as to give the maximum accuracy for a given expenditure. Two extremes * Coast and Geodetic Survey Report, 1893, Part 2, p. 494. f Coast and Geodetic Survey Report, 1892, Part 2, pp. 54, 158. \ Transactions of the Association of Civil Engineers of Cornell Uni- versity, 1894, p. 58. 170. ERRORS. 187 of practice are to take 210 observations on 30 pairs, each pair being observed on 7 nights; and to take 100 observations on 100 pairs, each pair being observed but once. The first is the old practice of the Coast Survey.* The recent practice of that Survey is intermediate between these extremes. The latter extreme was approached, but not quite reached, on the Mexican Boundary Survey of 1892-93. Let it be supposed that e o".2i and e$ = o". 18, as in the example given in 157. Then for the former extreme method the effect of the errors of observation on the result would be reduced to o".2i -r- 1/2 10 o".oi4, and the effect of the declination errors to o". 18 ~ V$o = Q^.033, giving for e^ the probable error of the result, 4/(o.oi4) 2 + (0.033)' = o".036. In the latter extreme case the error of observation would be reduced to o".2i -r- Vioo = o // .O2i, the declination error to o".i8 -f- ^100 = it o".oi8, and the probable error of the result to 4/(o.02i) 2 + (0.018)' = o".028. To look at the matter in another light : if with the above data as to e and e s the weight for a pair observed once is called unity, that for a pair observed twice is 1.40, by formula (69), 156; observed seven times is 1.98; and for a pair observed an infinite number of times 2.36. Little is gained in accuracy from the second observation on a pair, and less from each succeeding observation. 170. The zenith telescope furnishes a latitude determina- tion which is so far superior to that given by any other portable instrument that it should always be used where great accuracy is desired. A theodolite, or an astronomical transit, may be used as a zenith telescope if furnished with a suitable eyepiece micrometer, and with a sufficiently sensitive level parallel * C. & G. S. Report, 1893, Part 2, p. 301. 188 GEODETIC ASTRONOMY. I/I. to the plane in which the telescope rotates upon its horizontal axis. For the convenience, however, of those who may desire to determine the latitude with a sextant on explora- tions or at sea, and of those who may be forced by circum- stances to determine the latitude by a measurement of the altitude of the Sun, or a star, with a theodolite or an alt- azimuth, the following formulae are here collected: To compute the latitude from an observed altitude of a star, or the Sun, in any position, the time being known. 171. The requisite formulae are tan D = tan tf sec /, (81) cos (0 D) = sin A sin D cosec 6 ; . . (82) in which 6 is the declination and / the hour-angle of the star (or Sun) at the instant of observation; D is an auxiliary angle introduced merely to simplify the computation ; A is the altitude resulting from the measurement after applying all instrumental corrections, the correction for refraction, and, if the Sun is observed, the corrections for parallax and semi- diameter (see 65, 66). D is to be taken less than 90, and -|- or according to the algebraic sign of the tangent. Formula (82) is ambiguous in that Z>, determined from the cosine, may be either positive or negative. But the latitude of the station is always known beforehand with suffi- cient accuracy to decide between these two values. These formulae are exact, no approximations having been made in deriving them.* * For this derivation see Doolittle's Practical Astronomy, pp. 236, 237; or Chauvenet's Astronomy, vol. I. pp. 229, 230. 172. LATITUDE FROM ZENITH DISTANCES. 189 To compute the latitude from zenith distances of a star, or the Sun, observed near the meridian, the time being known. 172. The rate of change of zenith distance (or of altitude) of a given star is smaller the nearer the star is to the meridian. Hence the effect of a small error in the time, which is assumed to be known, is less the nearer the observation is made to the meridian, and is zero for an observation made precisely on the meridian. Only a single pointing can be made when the star is on the meridian, whereas it is desirable to take several pointings so as to decrease the effect of errors of observation. Hence the desirability of a rapid method for computing the latitude from circummeridian observations. Let be the required latitude of the station; 6 the declination of the star at observation; ,, C f , C 3 , . . . succes- sive observed values of the zenith distance of the star corre- sponding to the hour-angles t^ t^ A,, . . . ; and C the meridional zenith distance of the star. Then . (83) 2 sin' \t 2 sin 6 \t = sin 77-. . (84) i" A, B, and C are evidently constant for a series of obser- vations made near a given meridional passage of the star. Let m lt m v m . . . , n lt a , *,,'..., = 41 29' 03". 5 Instrument T. & S. Altazimuth No. 72. Date Sept. 13 1877. i div. of striding level = 2". 12. Star d Ursae Minoris. Chronometer Negus 1431 (Sidereal). a = i8 h n m 47*. 5 Chronometer correction = + 4*. 5. d = 86 36' 4i".o H "o Level Readings. Horizontal Circle. Object. c Chronometer Times. 1 W. E. Index. Mic. A. Mic. B. Mic. C. IX t d d d d d d Mark D 142 25' 3 19-7 19.0 20.5 20. o 21.7 20.5 * 4 44 142 25 3* 21. 19.7 20.0 21.0 20.0 18.3 Star 44 d d O h og m OQ..O 158 20 o 24. 23.0 25-0 26.5 24-5 24-3 44 60.3 47- 1 09 oi .0 158 20 o 26. 25-5 27-5 28.2 117.1 26.7 44 65.6 09 50 .5 158 20 o 26. 26.8 28.0 28.8 28.0 28.2 Mark 44 142 25 3 23. 22.0 22.0 23.0 22.4 2O. 2 * 4 142 25 3 22. 21.4 20.7 22. O 21.4 18.9 R 322 2 5 3 9- 08. I 07-3 08.8 08.4 06.2 kfc 322 25 3 07- 06.6 5-3 07.2 06.3 04.7 Star * ; 20 03 .5 338 20 o 56. 55-2 57-5 58.2 55-7 M ti 62.4 46.0 21 33 -5 22 46 .0 33 8 20 338 20 i 07. I 19. 06.6 18.9 08.3 20.7 09.2 2O.9 07-H 19.0 07.4 19.0 Mark 44 322 25 3 07.8 07.4 06.4 07.9 07.6 05.5 322 25 3 07.9 07.4 06.2 08.2 07.6 06.5 The mean zenith distance of the star during the observations, from two approximate read- ings of the vertical circle, was found to be 40 52'. * The reading of the whole turns, for the other micrometers, is not repeated in the record, but in making the readings it is called out to the recorder. If he finds it the same as for micrometer A, the record is as above. If it falls a unit below that for A, he indicates it by writing a minus sign over the recorded reading of the head; and if it is a unit above, he calls attention to it and records it as 60 + the given reading. 206 GEODETIC ASTRONOMY. 1 88. The Circle Reading. 188. In the record above, two readings are given for each position of each reading microscope. When commencing to read microscope A for the first time, for example, in the above set of observations, the field of view looked as shown in Fig. 25.* The reading was evidently 25' plus the angle represented by the interval from the zero of the microscope (the position in which the micrometer lines are shown) to the 25' graduation. This plus quantity is read directly from the micrometer when the 25' graduation is bisected (called the forward reading), namely, 3' 1 9". 7, provided there is no error of pointing, and provided each turn of the micrometer repre- sents exactly \' . But neither of these conditions are realized in practice, and a more reliable result may be secured if a reading is also made on the 30' graduation (called the back- ward reading). With perfect pointing and perfect adjustment of the micrometer, the screw would necessarily be turned exactly five revolutions backward to pass from a pointing on the 25 line to a pointing on the 30 line, and the reading of the head of the micrometer would be the same in both cases. It is actually 0.7 less. Neglect for a moment all considera- tion of possible errors in pointing and reading. These two readings would then indicate that one turn of the micrometer c' represents! = 59". 856. Hence the measured interval of 3' 19". 7 from the zero of the micrometer to the 25' line represents ^3-|^j(59 // .856) = 3' 19". 2. This procedure does * The field of view is here shown as it actually appears to the observer. The microscopes invert, and therefore the graduation really increases in the opposite direction from that here shown. f 0.7 division = 0.012 turn. (6o d = i turn.) 189. THE CIRCLE READING. 2O? not involve any assumption as to the exact value of one turn, but in fact derives the circle reading from each pair of mi- crometer readings upon the assumption merely that the grad- uated interval on the circle is exact. This process of making the correction for the run * of the micrometer may be put in convenient form for rapid computation for the above-described instrument as follows. The above figure and explanation will serve as a sufficient proof of the formulae given. 189. Let F' be the forward reading of the micrometer, both comb and head, expressed in turns, this reading being taken on the line of the graduation which is adjacent to the zero of the micrometer in the direction of increasing readings of the micrometer; and let F be the corresponding reading of the micrometer head, expressed in turns. Let B be the backward reading of the micrometer head, expressed in turns, taken on that line of the graduation which is adjacent to the zero of the micrometer in the opposite direction. Let the true reading of the circle to be derived be called T. Let the interval between lines of the graduation be called / (5' in the preceding illustration). Then one turn of micrometer '/ 5' F-H-S+F-J3- Strictly, the required value of r' T is then T = \ & _ gC^O* But remembering that F B is ordinarily only one or two sixtieths of a turn, and is therefore small as compared with the complete interval (5 turns), we may write T= (F') = (i' + ^) (F'), in which the difference B F is now taken in divisions of * The run of a micrometer is the amount by which one turn exceeds, or falls short of, its nominal value, 0.7 in the above example. The error of runs is the error in the result which is introduced by neglecting the run of the micrometer. 2O8 GEODETIC ASTRONOMY. 1 90. the head, and considered to represent seconds. The nominal reading of the micrometer being (i 7 ) (F'), the correction, C rt to this nominal reading, or correction for run to be applied to the forward reading, is (88) The values of C r will be found tabulated in 309 for the arguments F', the nominal forward reading, and B F, ex- pressed in divisions, or nominally in seconds. This table applies of course only to a reading microscope of the above type in which five turns are nominally equal to 5', one space of the circle graduation, and each division of the head is nominally i". The table may be used to correct each forward reading of a series, or the mean value of C r taken out from the table for each reading may be applied to the mean of the forward readings. A similar table may be constructed on the same principle for any other micrometer. 190. In developing the preceding method of computing the true reading of the circle, the accidental errors of pointing upon the graduation have been entirely ignored, it being tacitly assumed that they are small as compared with the error of runs. Let us now make the converse supposition that the error of runs is small as compared with the error of pointing. On this supposition the forward and back readings of the head differ simply because of errors of pointing. Hence they are equally good determinations of the seconds of the reading, and their mean is to be taken. On this sup- position the true reading, so far as the seconds are concerned, is . , . . . ..'. (89) IQ2. LEVEL CORRECTION. 209 191. The use of (89) instead of (88), or the corresponding tables, leads to quite a saving of time. Two other considera- tions also point to such use as advisable. Firstly, the true result sought is in reality between the results given by these two methods, since errors of run and errors of pointing both exist, and in general neither are insensible as compared with F' the other. Secondly, in (88) is as apt to be greater than F r . J as it is to be less than . If is J, the use of (88) gives numerically the same result as (89). Hence the results from the use of (88) are as apt to be greater as to be less than those from (89), and the greater the number of observations treated the nearer the results from the two formulae agree. Hence, in general, there is not a sufficient gain in accuracy over the procedure indicated in (89) to justify the time required to correct for errors of run.* The Level Correction. 192. Any inclination of the horizontal axis affects the circle reading corresponding to the pointing upon the star, and necessitates a correction which is to be determined from the readings of the striding level. In Fig. 26 let NES W represent the horizon. Let s be the star, and Z the zenith. If the instrument is in perfect *' Sometimes the mean value for the run of a given micrometet is de- rived from a special series of observations for that purpose: the run is assumed to be constant; and a correction based upon this mean value is applied to the mean results computed by (89). This procedure shortens the work of applying the correction for run, after the mean value of the run has been computed. The validity of the assumption that the run is a constant is so doubtful, however, that it seems that if the correccion foi run is to be applied at all, it should be based upon a value for the run de- rived from the very readings that are to be corrected. 2IO GEODETIC ASTRONOMY. IQ2. adjustment, when the telescope is pointed upon the star the plane in which the telescope is free to swing about its hori- zontal axis is defined by the arc ZsP, in which P is the pole of the great circle passing through Z, and A, the point in which the horizontal axis produced pierces the celestial sphere. If, now, the horizontal axis be given an inclination b, the west end being placed too high, A will move to a point A' , at a distance b along Az. The zenith of the instrument will virtually be shifted to Z' (such that the arc AZZ' = 90, and ZZ' = &), that being the nearest point to the true zenith to which the telescope can be pointed. The telescope is now free to swing in the arc Z'P. But this arc does not pass through s. To bisect the star it is necessary to turn the instrument about its vertical axis, which now passes through Z\ until the telescope swings in the arc Z's. A' will then be in such a position as A" ' . The change in the circle read- ing, due to the inclination of the axis, is evidently measured by the angle PZ's. The circle, if graduated clockwise, now reads too small by that amount, which will be called C L . Consider the spherical triangle Z'Ps. In this triangle the angle at Z' is the required C LJ the angle at P is b, the side Ps is the altitude of the star A, and the side Z's is the zenith distance of the star as measured with the displaced instru- ment. But, the displacement being small, Z's may be for the present purpose considered equal to Zs, the zenith dis- tance, C, of the star. From the proportionality of the sines of angles and opposite sides in the triangle Z'Ps, we may sin C L sin b . . write -r- = -. Replacing sin C L and sin b by C L and sin A sin C b, those angles being small, and solving for C L , there is obtained 193- FORMULA. 211 Expressing the inclination b in terms of the readings of the striding level, there is obtained the complete formula for the level correction, C L = \(w + - (e + S)\j tan A, , ', (91) for a level having its divisions numbered both ways from the middle. C L =\(w + e)- (W + O} j tan A . . . (92) 4 for a level numbered continuously in one direction, the primed letters referring to the readings taken in the position in which the numbering increases toward the east. C L as given by these formulae is the correction to the circle reading on the supposition that the numbers on the circle graduation increase in a clockwise direction. Similar corrections to the circle readings upon the mark, derived from corresponding readings of the striding level, are necessary if the line of sight to the mark is much inclined. Ordinarily the line of sight to the mark is so nearly horizontal that such corrections are negligible, and the corresponding level readings may be dispensed with, provided that care is taken to keep the instrument well levelled up. Azimuth of the Star. 193. The preceding formulae suffice for the computation of the horizontal angle between the star and mark. It remains to compute the azimuth of the star. The detail of the process of computing the hour-angle of the star from the chronometer reading need not be stated here. The hour-angle t and the declination 8 of the star being known, as well as the latitude of the station 0, the azimuth of the star may be computed from the spherical 212 GEODETIC ASTRONOMY. 193. triangle defined by the star, the zenith, and the pole. Certain sides and angles of this spherical triangle have the values indicated in Fig. 9, in terms of the angles, z the azimuth of the star reckoned from the north, A its altitude, and /, 0, and d. From the principle that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides, we may write sin z sin / Also, from the principle that in any spherical triangle the cosine of any side is equal to the product of the cosines of the other two sides plus the product of their sines into the cosine of the opposite angle, we may write the two formulae, sin tf = sin sin A -f- cos cos A cos z ; . (94) sin A = sin sin d + cos cos d cos t. . . (95) By substituting sin A from (95) in the first term of the second member of (94), and solving the resulting equation for cos Z, there is obtained (i sin* 0) sin sin cos cos d cos t cos (96) By substituting cos 9 for i sin 2 0, and dividing both numerator and denominator by cos 0, (96) reduces to cos sin # sin cos tf cos t , x - cos .4 ' ' ' (97) 195- CURVATURE CORRECTION. 21$ If (93) is now divided by (97), and both denominators of the resulting equation are divided by cos 6, there results sin t tan z = -7 ^ : -7 -, . . (98) cos tan o sin cos / from which z may be computed from the known values of 0, tan d is much larger than sin cos /, and the quantity : -T may be assumed constant for the cos tan 6 sin

= 9.8211300 log sin / = 9.9999497 log tan S = 1.2275941 log cos / = 8.1826260;* log 12.66198 = 1.1025016 1.1021555 -f- 12.65189 -f- 0.01009 12.66198 2 sin 2 At 0037500W .01009 log tan z 8.8974481 z =4 30' 54"-47 j 8.o< ( o. i /2 sin 2 \Ai \ o- : + ...)= -6 . n \ sin i / tan - 20 Correction for diurnal aberration = o .32 sin i " 7 m 6 S ii 21 4 9 101' 75 56 '97 .22 51 4 6 7 51 21 33 .1 .1 .6 46 79 112 .21 .20 .21 6)471 32 Azimuth of star, west of north Mark west of star = 4 30 47 .95 = 15 52 30 .82 Azimuth of mark, west of north =20 23 18 .77 Mean = 78". 55 log 78.55 = 1-89515 log tan z 8.89745 Curvature (0.79260 correction ( 5". 20 220 GEODETIC ASTRONOMY. 2O2. Program of Observing for the Method of Repetitions. 201. To measure a horizontal angle by repetitions one must use an instrument having a clamp and tangent screw to control the motion of the lower or graduated circle, in addi- tion to a similar clamp and tangent screw to control the rela- tion between the upper circle carrying the verniers and the lower graduated circle. At the beginning of the measurement the circle is read. By a suitable manipulation of the two motions, upper and lower, the angle to be measured is multi- plied mechanically three, five, or more times. The circle being read again gives the measured value of the multiple angle, from which the required angle is readily derived. This process serves to greatly decrease the errors arising from erroneous readings of the verniers, and errors of graduation. But any lost motion, or false motion, in clamps and tangent screws, affects the measured angle directly. To eliminate this error as far as possible one may measure both the required angle and its explement, * always revolving the instrument in a clockwise direction with either motion loose, and making all pointings with either tangent screw so that the last motion of that screw is in the direction in which the opposing spring is being compressed. With this procedure, unless the action of the clamps and tangent screws is variable, the derived values of both the angle and its explement will be too large or too small by the same amount. The mean of the measured angle and 360 minus the measured explement will be the correct value of the angle unaffected by the constant errors of the clamps and tangent screws. 202. The following is a convenient program for the measurement of an azimuth by repetitions. After all adjust- ments have been made and the instrument carefully levelled, * 360 minus a given angle is called the explement of that angle. 202. METHOD OF REPETITIONS. 221 clamp the upper circle to the lower in any arbitrary position; point approximately upon the star; place the striding level in position and read it; reverse it, read it again, and remove it; point accurately upon the star, noting the chronometer time, and using the lower clamp and tangent only; read the horizontal circle; unclamp the upper motion and point upon the mark using the upper clamp and tangent screw; unclamp the lower motion and point upon the star, using the lower clamp and tangent screw and taking care to note the chro- nometer time of bisection ; loosen the upper motion and point again upon the mark, using the upper clamp and tangent screws; take another pointing upon the star with the lower motion, noting the time; point again upon the mark, using the upper motion; read the horizontal circle. This completes the observations of a half-set if three repetitions are to be made. In passing from the star to the mark, and vice versa, the instrument should always be rotated in a clockwise direc- tion, and the precaution stated in the preceding paragraph as to the use of the tangent screws must be kept in mind. Before commencing the second half-set the lower motion should be undamped, the telescope reversed in altitude, and the instrument reversed 180 in azimuth. The program will be as for the first half-set, except that now the first pointing is to be upon the mark; #// pointings on the mark are to be made with the lower clamp and tangent screw, and upon the star with the upper clamp and tangent screw ; and the strid- ing level is to be read just after the last pointing upon the star. The direction of motion of the instrument must always be clockwise as before, and the tangent screws must be used as before. 222 GEODETIC ASTRONOMY. 204. 203. EXAMPLE OF RECORD; METHOD OF REPETITIONS. Station Dollar Point, Texas. Observer A. F. Y. cf) = 29 26' 02". 6. Instrument Gambey Theodolite. Date April 5, 1848. I div. of striding level = 3". 68. Star Polaris. Chronometer Hardy No. 50 (Sidereal). a = i h 04 04". 7. Chronometer correction = I 8 .8. d = 88 29' 57".82. Object. Pos. of Tel. Level Readings. No. of Repeti- tions. Chronometer Times. Circle Readings. W. E. Vernier A. Vernier B. Star. D. I2Q.O 7 r -5 3 9 b 3 m 33'-5 91 10' 30" 271 10' 40" 8l.O 119.0 04 47 .5 06 07 .5 Mark. D. 128 14 50 308 14 50 Mark. R. 3 128 14 50 308 14 50 9 08 06 . 5 I2I-5 79.0 09 24 .5 Star. R. 80.0 I2O.O 10 23 .5 91 13 40 271 11 50 204. No reading of the altitude was taken. The altitude may be derived with sufficient accuracy for use in computing the level corrections from the table in 310. The level cor- rection must here be applied to the angle between the star and mark, not directly to- the circle reading. Formulae (91) and (92) will not give the sign of the level correction; that must be derived from the consideration that the star appears to be farther west than it really is if the west end of the horizontal axis is too high, and vice versa. The angle between the star and mark, computed from the first half-set, is (128 14' 50"- 9' 10' 35")* = I2 2I/ 2 5"-0, and from the second half-set is (128 14' 50"- 91 13' 45")* = 12 20' 21".;.* * Evidently this method of computing the second value of the angle necessarily always gives the same numerical result as first computing the explement and then subtracting from 360. 206. MICROMETRIC METHOD. 22$ The remainder of the computation may be made as indi- cated in 200. Directions -for Observing Azimuth with a Micrometer. 205. If the instrument is provided with a good eyepiece micrometer measuring angles in the plane defined by the telescope and its horizontal axis, the most accurate as well as the most rapid way of determining azimuth with it is to place the azimuth mark nearly in the vertical plane of a close cir- cumpolar star at elongation, and then to measure the hori- zontal angle between the star and mark with the micrometer, independently of the graduated horizontal circle of the instrument. 206. To place the azimuth mark with sufficient accuracy in the required position one may take a single pointing upon Polaris on the first night after the station is ready for obser- vations, noting the sidereal time and the reading of the hori- zontal circle. The instrument may then be left standing, with the lower motion clamped, until the next day. During the next day the instrument may be set to such a reading of the horizontal circle computed roughly from the observations of the night before, by the table of 310, or by formula (98), as would place the telescope in the vertical plane of the star about 3O m before or after the elongation at which the obser- vations are to be made. An assistant may then, by previously arranged signals, be aligned at the proposed site of the azimuth mark so as to place it in the direction defined by the tele- scope. The "alignment" of the mark may be made at night, as soon as the pointing is made upon Polaris, instead of waiting until the next day, if necessary; but it is usually easier to pick out a good location for the mark and to trans- mit signals from the station to the mark in daylight than at 224 GEODETIC ASTRONOMY. 2O8. night. The mark may be placed either to the northward or to the southward of the station. 207. The adjustments of the vertical axis, of the levels, of focus (see end of 216), and for bringing the movable micrometer line into a vertical plane, must be made as indi- cated in 1 80. 208. The following is a good program for the observa- tions. Place the micrometer line at such a reading that it is nearly in the line of collimation of the telescope. If this reading is not already known, it may be determined by taking the mean of two readings upon the mark with the micrometer, the instrument being rigidly clamped in azimuth, and the horizontal axis of the telescope reversed in its Ys between the two readings. Clamp the lower circle, and point upon the mark by use of the upper clamp and tangent-screw. Then, with the instrument clamped rigidly in azimuth, take five pointings with the micrometer upon the mark, direct the telescope to the star; place the striding level in position; take three pointings upon the star with the micrometer, noting the chronometer time of each; 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. Reverse the horizontal axis of the telescope in its Ys; point approximately to the star; place striding level in position ; take three pointings upon the star, noting the chronometer times; read and reverse the striding level ; take two more pointings upon the star, noting the times; read the striding level; and finally, make five pointings upon the mark. Such a set of observations may be made very quickly; the effect of a uniform twisting of the instrument in azimuth is eliminated from the result; and the bubble of the striding level has plenty of time to settle without delaying the observer for that purpose. 210. MICROMETRIC METHOD. 22$ 209. With the instrument used for the following observa- tions, 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 if the ver- tical circle is to the west. 210. EXAMPLE OF RECORD AND COMPUTATION. Station No. 10. = 31 19' 35".o. Date October 13, 1892. Star Polaris, near eastern elongation. Observer J. F. H. Instrument Fauth Theodolite No. 725. One division of striding level = 3". 68. Chronometer Negus No. 1716 (Side- real). One turn* of micrometer = 123". 73. Chronometer corr. = 2 h n m 2S-.2. Cir. E. or W. Level Readings. Chronometer Time. A^ 2 sin 3 *A* Micrometer Readings. W. E. sin i" On Star. On Mark. E. E. W. W. d. 8.0 IO.O d. 9.9 7-3 9 h o6 m 38.o 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 3 m 58'-6 3 4 -6 2 31 -i i 23 .6 o 48 .6 I 2 5 .2 I 48.1 2 H . 7 2 59 -7 3 21 -5 3 l -5 18.59 12.45 3-82 1.29 3.96 6.37 9.46 17.61 22.14 t. 18.379 -388 .400 .424 43 t. 18.310 3^5 3 1 5 3" .316 Longitude 2 h 12 m we st of Wash- ington. Means. Means. -f 18.0 + 9.0 7.0 9.0 10.9 18.4042 18.100 .100 .090 .086 .080 18.0912 18.3134 18.290 275 .279 .281 ,279 18.2808 + 16.0 - 3 Mean = - 19-9 9 - i-55 9 10 36 .6 12.67 a of Polaris = i h 2o m O7".4 d of Polaris = 88 44' io".4 Altitude of star at the mean epoch of the observations, per section 308, = 31 Altitude of star at the middle of the first half-set, per section 308, = 31 12 Altitude of star at the middle of the second half-set, per section 308, = 31 14 Collimation reads, ^(18.3134 -f- 18.2808) = 18^.2971 at the time of observation. 13' * The head of this micrometer was graduated to TOO equal parts, the meaning of an increased reading see 209. For 226 GEODETIC ASTRONOMY. 211. Mark east of collimation, 18.3134 18.2971 = o .0163 = 02". 02 Circle E., star E. of collimation, (18.4042 18.2971) -f- 0.8554 * o .1252 Circle W., star E. of collimation, (18.2971 18.0912) -f- 0.8551 f = o .2408 Mean, star E. of collimation = o .1835 = 22 .70 Mark west of star = 20 .68 Level correction, (1.55X0.92X0.606) = 0.86 Mark west of star, corrected = 19 .82 Mean chronometer time of observation = 2i h io m 36". 6 Chronometer correction = 2 n 28.2 Mean sidereal time of observation = 18 59 08 .4 a = i 20 07 .4 Hour-angle (= /) east of upper culmina- tion = 95 14' 45". o = 6 20 59 .o log cos / = 9.9315695 Jog sin (f> = 9.71593 ] og sin / = 9.9981771 log tan 6" = 1.6563815 log cos / = 8. 96108?* log 38.76893 = 1.5884838 ( 1.5879510 j 8.6770IW log tan z n = 8.4096933 ( 38.72140 t 0.04753 = i 28' i6".92 + 0.04753 j / 2 S } n 2 i4 fl \ tan z a - . \- etc. ) = o .33 38.76893 n\ sini J Correction for diurnal aberration = + o .32 Star east of north = z = i 28' i6".9T Mark west of star = log 12.67 ^ 1.10278 from above = 19 .82 log tan z = 8.40969 Mark east of north = i 27 57 .09 9-51247 o"-33 211. Here again the sign of the level correction as applied to the angle between the star and mark must be derived directly from the fact that the star appears to be farther west * 0.8554 = cosine 31 12' (natural), f 0.8551 = cosine 31 14' (natural). 21$. MICROMETRIC METHOD. 227 than it really is if the west end of the axis is too high, and vice versa. 212. The micrometer measures angles in the plane defined by the telescope and its horizontal axis. In Fig. 29 let Z be the zenith, s the star at the instant when a pointing is made upon it with the micrometer, and Zn the vertical circle described by the line of collimation of the telescope. Then sm, a great circle through s perpendicular to Zn, is the arc measured directly with the micrometer. In the right spheri- cal triangle smZ the side Zs is 90 A, the complement of the altitude of the star, and from Napier's rules sin sm = sin mZs cos A . . . . t . (115) Or, writing the angles for the sines of sm and mZs, and solv- ing for mZs, mZs = sm sec A ( IJ 6) mZs, the angle at the zenith, is the required angle between the vertical plane through the star and the vertical plane described by the line of collimation. The computation form shows how this factor, sec A, is most conveniently applied. To be absolutely exact, this factor should be applied to every pointing upon the star. But the computation as given is abundantly accurate, the factor being applied to the mean angle between the line of collimation and the star for each half-set. In fact, the com- putation will often be sufficiently exact, if the factor is applied to the mean value of this angle for the set. 213. The use of the table given in 308 is the most con- venient way of securing the required values of the altitude of the star, unless they are read approximately from the vertical circle during the observations. First compute the mean 228 GEODETIC ASTRONOMY. hour-angle of the star and take out the corresponding altitude for use in deriving the level correction, then the other two angles may be derived by interpolating over the interval to the middle of each half-set with the rate of change of altitude taken from 308. The altitude need only be known within one minute, ordinarily. For any other star than Polaris, the table of 308 not being available, one must either read the required altitude from the vertical circle of the instrument, if it has one, or else resort to the computation upon which the table of 308 is founded. The use of the factor sec A is not necessary with the pointings upon the mark, both because the line of collimation was purposely placed nearly upon the mark, and because sec A is very nearly unity for the small altitude of the mark. 214. Inspection will show that there is nothing in this method of observing or computing which limits its use to the time near elongation. The micrometer may be used in this way and the azimuth computed as above with the star at any hour-angle, even at culmination. But if the star is not near elongation, its motion in azimuth is more rapid, it remains near the vertical plane of the mark a shorter time, and larger angles must be measured with the micrometer or else the series of observations made shorter. Errors in the time also have less effect the nearer the star is to elongation. If the azimuth mark is placed to the southward of the station, the program of observing and the computation are not materially modified. Micrometer Value. 215. To determine the value of one turn of the microm- eter the observer may use a process similar to that used in determining the value of the zenith telescope micrometer. 2 1 6. MICROMETRIC METHOD. 22$ That is, one may observe the times of transit of a close cir- cumpolar star near culmination across the micrometer line set at successive positions one turn apart (or one-half a turn), the instrument being rigidly clamped in azimuth. The correc- tions for curvature may be made by use of the same table, 306, as for the zenith telescope, but now using in the place of r the hour-angle of the star reckoned from the nearest culmination, and making the corrections to the observed times positive before culmination and negative after. The striding level may be read during the observations and a corresponding correction applied. The correction in seconds of time to be applied to each observed time to reduce it to what it would have been with the axis level is d.s'mA.secd , - - -, (117) for a level with a graduation numbered both ways from the middle. The observer must depend upon his instrument to remain fixed in azimuth, unless, fortunately, he has an azimuth mark so nearly in the meridian that he can occa- sionally take a pointing upon it, without unclamping the horizontal circle, during the progress of the observations, and so determine the twist of the instrument. 216. Another convenient way of determining the microm- eter value, without doing any work at night, is to measure a small horizontal angle at the instrument between two terrestrial objects, both with the horizontal circle and the micrometer. If the two objects pointed upon are much above or below the instrument, the measured angle between them 230 GEODETIC ASTRONOMY. 2 19. may be reduced to the horizon for comparison with the circle measurement by use of the factor sec A, as indicated in 2 12. As the micrometer value is depended upon to remain con- stant for the station the focus must be left undisturbed if possible after the micrometer value has been determined. Discussion of Errors. 217. The external errors are those due to errors in the right ascension and declination of the star observed, to lateral refraction of the rays of light from the star or mark to the instrument, and to error in the assumed latitude of the station of observation. 218. Errors of declination enter the computed azimuth with full value when the star is observed at elongation, and errors of right ascension enter with a maximum effect when it is observed at culmination. At intermediate positions both errors enter the computed result with partial values. The errors arising from this source are usually small as compared with the errors of observation, but are nearly constant if all the observations at a station are taken with the star at about the same position in its diurnal path, say near eastern elonga- tion. They may be eliminated to a considerable extent by observing the same star at various positions of its diurnal path, or by observing upon two or more different stars. 219. When the computed results of a long series of accu- rate azimuth observations at a station are inspected it is usually found that they tend to group themselves by nights. That is, the results for any one night agree better with each other than do the results on different nights. They thus appear to indicate that some source of error exists which is constant during each night's observations, but changes from night to night. For example, from 144 sets of micrometric observations of azimuth, made on 36 different nights, at 1 5 221. ERKOKS. 231 stations on the Mexican Boundary in 1892-93, it was found that the error peculiar to each night was represented by the probable error o".38, and the probable error of the result from a single set exclusive of this error was o". 54. 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, was two-thirds as large on an average as the error of observation in the result from a single set. The most plausible explanation seems to be that there is lateral refraction between the mark and the instrument, and that this lateral refraction is dependent on the peculiar atmos- pheric conditions of each night. But whether that explana- tion 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. 220. The accuracy with which the latitude must be known when observing upon Polaris may be inferred from an inspec- tion of the table of 310. It must be known with greater accuracy when a star farther from the pole is used. 221. The observer 's errors are his errors of pointing upon the mark and star, errors of pointing upon the circle gradua- tion 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 bisection. There is such a large range of difference in the designs of the various instruments used for azimuth work that little can be stated in regard to the relative and absolute magnitude of these different errors that will be of general application. Each observer may investigate these various errors for himself 232 GEODETIC ASTRONOMY. 22$. with his own instrument In designing instruments the attempt is often made to so fix the relative power of the telescope and means of reading the horizontal circle that the errors arising from telescope pointings and circle readings shall be of the same order of magnitude. The effect of errors in time may be estimated by noting the rate at which the azimuth of the star was changing at the time it was being observed. The table of 310 will serve this purpose for Polaris. Such errors are usually small, but not insensible except near elongation. 222. Of the relative magnitude of the instrumental errors arising from imperfect adjustment and imperfect construction little of general application can be said, because of the great variety of instruments used. With the more powerful instru- ments, however, it may be stated that the errors due to instability of the instrument become relatively great, and must be guarded against by careful manipulation and rapid observing. The errors due to the striding level become more serious the farther north is the station (see formula (91)). If the level is not a good one, it may be advisable to take more level readings than have been suggested in the preceding programs of observation. With a very poor level, or at a station in a high latitude, it may be well to avoid placing any dependence upon the level by taking half of the observations upon the star's image reflected from the free surface of mercury (an artificial horizon). The effect of inclination of the axis upon the circle reading will be the negative for the reflected star of what it is for the star seen directly. Considerable care will be necessary to protect the mercury from wind and from tremors transmitted to it through its support. 223. The micrometric method treated in 205-2 14 gives a higher degree of accuracy than the other methods described, 22$. MISCELLANEOUS. 233 if a good micrometer is available. It avoids several sources of error, and the observations may be made so rapidly that the conditions are quite favorable for the elimination of errors due to instability. The error, in the final result for a station, due to an error in the value of the micrometer screw may be made as small as desired by so placing the azimuth mark and so timing the observations that the sum of the angles meas- ured eastward from the mark to the star shall be nearly equal to the sum of the angles measured westward from the mark to the star. Other Instruments and Methods. 224. An astronomical transit furnished with an eyepiece micrometer is especially well adapted to give results of a high degree of accuracy in determining azimuths by the micrometric method. 225. If the transit has no micrometer, a secondary azimuth may be determined incidentally to time observations with little extra expenditure of time. Put an azimuth mark as nearly as possible in the meridian of the transit. At the beginning of each half-set of the time observations point upon the mark with the middle line of the reticle. If the mark is nearly in the horizon of the instrument, the collimation and azimuth errors of the transit as derived from each half-set, reduced to arc and combined by addition and subtraction with each other and with the equatorial interval of the middle line, give the azimuth of the mark. The azimuth of a certain mark was so determined from the time observations required for a determination of the longitude of a station at Anchorage Point, Chilkat Inlet, Alaska, in 1894. The computed azimuth of the mark from 38 nights of observation varied through a range of 12". 8. The probable error of a single determination was 2".!. 234 GEODETIC ASTRONOMY. 228. 226. The transit may also be made to furnish a good determination of azimuth by observations in the vertical of Polaris by the method already referred to in 129. 227. If the allowable error of a given azimuth determina- tion exceeds 2", a convenient method is to observe upon Polaris at any hour-angle and use the table given in 310 to compute its azimuth at the time of each observation. The tabulated values * were computed by formula (98). The only correction to be applied to the value as taken from the table is that due to the difference between the apparent declination of Polaris at the time of observation and the value 88 46' with which the table was computed. This may be computed by use of the columns, given at the right-hand side of the table, headed " Correction for i f increase in declination of Polaris," by assuming that the correction is proportional to the increase, and must be changed in sign if the declination is less than 88 46'. The table may also be used as a con- venient rough check on computations made by formula (98). If a star which is not a close circumpolar, or the Sun, is observed for azimuth at a known hour-angle, its azimuth may be computed by formula (98) for each observation, or the observations may be treated in groups covering short intervals of time. But formula (103) will not apply, since certain approximations were made in its derivation which are only allowable when d is nearly 90. 228. For rough determinations of azimuth in daylight, say within 30", when the time is only approximately known, the Sun may be observed with a small theodolite, or with an engineer's transit, as follows: Point upon the mark and read the horizontal circle; point upon the Sun, making the hori- zontal line of the transit tangent to the upper limb and the * See Coast and Geodetic Survey Report, 1895, Appendix No. 10, for the original of this and the following table. 228. MISCELLANEOUS. vertical line tangent to the western limb; note the time, and read both horizontal and vertical circles; repeat this pointing upon the Sun twice more, noting the times and reading the circles; reverse the instrument 180 in azimuth and the tele- scope in altitude; again take three readings upon the Sun, but now make the horizontal line tangent to the lower limb and the vertical line tangent to the eastern limb ; finally, point upon the mark again and read. To compute the azimuth of the Sun one may use the formula sin (s 0) sin (s A) tan 2 4* = - / px i 1 1 8) cos s cos (s P) in which z is the azimuth counted from the north, P is the Sun's north polar distance (= 90 #), and s = 4(0 -|~ A +f)' If desired, the hour-angle of the Sun, and thence the chronometer error, may also be computed from the observa- tions by the formula cos s sin (s A) tan 4* = - 7 -7T- 1 7 -=. . . (i 19) sin (s 0) cos (s P) These formulae may readily be derived from the ordinary formulae of spherical trigonometry as applied to the triangle defined by the Sun, the zenith, and the pole. 236 GEODETIC ASTRONOMY. 229. 229. Observations of Sun for Azimuth, Niantilik, Cumberland Sound, British America, Sept. 18, 1896, P.M. Instrument Theodolite Magnetometer No. 19. = 64 53'. 5. Chronometer correction on Greenwich Mean Time -f 2 m og'.S. Object. c Time. Chronome- ter, 1842. Horizontal Circle. Vertical Circle. A. B. Mean. A. B. Mean. D. R. R. D. D. R. R. D. 7 h 39 m 20" 40 40 4 1 34 42 58 44 04 45 33 7 42 21 .5 7 46 34 47 33 48 31 51 04 S 2 05 53 21 53 58' 233 55 53 229 46 230 05 19 51 16 33 54 50 S 2 09 11 232 38 52 233 I2 59' 55 44 1 16 17 33 55 09 24 38 36 50 09 55 58 58'. s 55 - "56.8 45 -o 04 .0 17 -5 16.5 33 -o 54 -5 ~&~ og .0 23 -5 38 .0 37 -o 51 .0 10 .5 17 17' ii 06 73 33 3 2 46 16 73 50 55 74 02 16 16 ii 03 18' 13 09 33 39 47 57 03 17 07 17'. 5 12 .0 07 -5 33 -o 39 .0 40 .5 46.4 50.5 56 .0 02 . 5 I6. 5 12 .0 05 .0 Sun's first and upper limb.. Sun's second and lower limb Means Sun's second and lower limb Sun's first and upper limb. . . 7 49 5i -3 52 233 56 53 58 53 38.2 55 -5 58.0 56 .8 16 07 .4 Azimuth mark. . . . ... Means . . 230. MISCELLANEO US. Computation. 237 Chronometer time * .......................... Chronometer corr. on Greenwich Mean Time.. Greenwich Mean Time ....................... Sun's Apparent Declination, <5, interpolated from Ephemeris ..................... Observed Altitude ..................... Correction for parallax ................ Correction for refraction ............... Corrected Altitude, A ................. P(=go-d) .......................... Latitude,

32 .5' 28 39.5 30 oi.o 28 25' oo" 28 37 45 29 13 30 208 25' oo" 208 38 15 209 14 oo 58 29' oo" 58 14 45 57 36 oo 58 29' 30" 58 14 30 57 35 45 Sun's lower and second limb. Telescope R. 209 oi ' 30" 209 12 45 209 27 oo 29 oo' 30 ' 29 12 15 29 26 30 57 48 oo" 57 34 30 57 19 15 57 47' 30" 57 34 15 57 18 30 38 53' 18" A = 5 h o8 m oi'.o west of Greenwich. 6 (at mean of the times) = 13 55' 16". (Interpolated from Ephemeris.) 10. If both are available, which should be used in formula (98) the geodetic or the astronomical latitude ? 11. Prove formulae (118), (119), and (120). 233- LONGITUDE. 241 CHAPTER VII. LONGITUDE. To determine the longitude of a station on the Earth's surface, referred to the meridian of Greenwich, is to determine the angle between the two meridian planes passing through the station and Greenwich respectively. (See 15.) This angle between the two meridian planes is the same as the difference of the local times* of the two stations, con- sidering 24 h to represent 360. (See 21.) Hence to deter- mine the longitude of a station is to determine the difference between the local time of that station and the local time of Greenwich. In general the longitude of an unknown station is not referred to Greenwich directly, but to some station of which the longitude is already known. The astronomical determination of the longitude of a station consists, then, in a determination of the local time at each of two stations, the longitude of one which is known and of the other is to be determined, and the comparison of these two times. Their difference is the difference of longitude expressed in time. This may be reduced to arc by the relations 24** = 360, I* = 15, i m = 15', and I 8 = 15". 233. The principal methods of determining differences of * The times may be eifher sidereal or mean solar. The vernal equinox apparently makes one complete revolution about the earth in 24 sidereal hours, and the mean Sun apparently makes one complete revolution in 24 mean solar hours. 242 GEODETIC ASTRONOMY. 235. longitude are by the use of the telegraph, by transportation of chronometers, by observations of the Moon's place, and by observations of eclipses of Jupiter's satellites. The methods of making the necessary determinations of the local time in each of these methods need not be considered here, as they have already been exploited in Chapters III and IV. We need here consider only the methods by which some signal is transmitted between the stations to serve for the comparison of the times. The Observing Program and Apparatus of the Telegraphic Method. 234. The telegraphic method has been used very exten- sively in this country by the Coast and Geodetic Survey, and during the fifty years of its use has been gradually modified. The method and apparatus at present used will be here described. The nightly program at each station is to observe two sets of ten stars each for time with a transit of the type shown in Fig. 10. Each half-set consists in general of four stars having a mean azimuth factor A (see 299) nearly equal to o, and one slow star (of large declination) observed above the pole. Two such half-sets, with a reversal of the telescope in the Ys between them, give a strong determination of the time. The same sets of stars are by previous agreement observed at each station. Between the two time sets, or rather at about the middle of the night's observations, certain arbitrary signals are exchanged by telegraph between the two stations, which serve to compare the two chronometers, and therefore to compare the two local times which have been determined from the star observations.* 235. Fig. 30 shows the arrangement of the electrical apparatus at each station during the intervals when no arbi- 236. TELEGRAPHIC LONGITUDE OBSERVATIONS. 243 trary signals are being sent or received, and each observer is busy taking his time observations. In the local circuit, which is now entirely independent of the Western Union lines, are placed the break-circuit chronometer* (or clock), battery, chronograph, and the break-circuit observing keys. All the time observations are recorded on the chronograph. Mean- while the telegraph operator has at his disposal the usual telegrapher's apparatus upon the main line connecting the two stations, namely, his key, and the sounder relay which con- trols the sounder in a second local circuit. The operator, a few minutes before the time for exchange of signals, secures a clear line between stations, ascertains whether the observa- tions at the other station are proceeding successfully, and finally an agreement is telegraphed between the two observers as to the exact epoch at which the exchange of signals will be made. 236. When that epoch arrives, time observations are stopped at each station, and by suitable switches the electrical apparatus at each station is arranged as shown in Fig. 31. The only change is that now a relay, called a signal relay, is used to connect each local chronograph circuit with the main line in such a way that the local circuit will be broken every time the main circuit is broken, in addition to the regular breaks made in it by the local chronometer. The observer at station A now takes the telegrapher's key (in the main cir- cuit), and sends a series of arbitrary break-circuit signals over the main line by holding the key down except when a dot is sent by releasing the key for an instant the reverse of the ordinary usage of the telegrapher. He listens to his own chronograph, and sends a signal once in each two-second interval at such an instant as will not conflict with his own * Or the chronometer maybe placed in a separate local circuit, breaking this one through a relay. 244 GEODETIC ASTRONOMY. 2 37 chronograph record. Each signal is transmitted by the signal relay at station A to that local circuit, and its time of receipt recorded automatically by the chronograph. At the same instant, except for the time required for the electrical wave to be transmitted over the main line between stations, the signal relay at station B transmits the signal to the local circuit and chronograph there. If these signals coincide with the clock breaks on the chronograph at B at any time, the observer at B breaks into the main circuit with his teleg- rapher's key, and produces a rattle at A's sounder which informs him that he must change his signals a fraction of a second to another part of the intervals given him by his chronograph beat. Thirty signals are sent from station A at intervals of about two seconds. The observer at A then closes his key and the observer at B proceeds to send thirty similar signals from B to A.* The Western Union line is then released, the apparatus at each station is again arranged as shown in Fig. 30, and each observer proceeds to finish his time observations. For a first-class determination this program is carried out for five nights at each station; the observers then change places (to eliminate the effect of personal equation), the instrumental equipment of each station being left undisturbed ; and the same program is again followed for five nights. Example of Computation. 237. A determination of the difference of longitude of Cambridge, Mass., and of Ithaca, N. Y., was made May 16- June 3, 1896. The following is a portion of the field com- putation: * Thirty signals at two-second intervals keep each chronometer in use timing signals for just one revolution (i m ) of the toothed wheel which breaks the circuit in the chronometer, and thus any errors in the spacing on that wheel are eliminated from the final result. 238. COMPUTATION OF TELEGRAPHIC LONGITUDES. 24$ Arbitrary Signals, May 27, 1896. From Ithaca to Cambridge. From Cambridge to Ithaca. Cambridge Record. Ithaca Record. Cambridge Record. Ithaca Record. I4 h l6 m 46 s . 34 48.39 50.32 52.48 9 h 38 19". 52 21 .54 23.50 25.63 I4 h I7 m 56 9 -55 58.51 oo .56 02 .50 gh 39 m 29 s . 63 31 .61 33.63 35-57 42.41 44.31 46.35 48.52 15 -45 17.30 19.36 21 .52 50.37 52.40 54 -44 56.45 23-3I 25 .36 27-39 29-39 I4 h i7 m i7 8 -44i - 25 .702 14 16 51 .739 Difference ; 9 h 38'" 50'. 532 -07 57-107 9 30 53 .425 i 33.783 4 22 59.370 13 55 26.578 Ji m 25*.i6i I4 h i8 m 26. 426 - 25 .700 14 18 00.726 Difference : 9 h 39 m 59M40 07 57 .109 9 32 02 .331 i 33.971 4 22 59 .370 13 56 35 -672 2i m 25*.054 The heading shows which way the signals were sent over the main line, and the four columns give the times of the signals as read directly from the chronograph sheets at the stations indicated. There were 31 or 32 signals in each series, of which only a portion are here printed. The means are given for the whole series in each case. A mean-time clock was used in the chronograph circuit at Ithaca, and a sidereal chronometer at Cambridge. 238. The first time set of the evening at Cambridge gave for the chronometer correction, on local sidereal time, at the mean epoch of the set, when the chronometer read I3 h 3O m .2, 2 5 s . 787. The second set gave the chronometer correction = 2 5 ".677 at the epoch when the chronometer read I4 h 3i m .3. By taking the means of the epochs and corrections, 246 GEODETIC ASTRONOMY. 239. on the assumption that the chronometer rate was constant during this interval, it was found that the correction was 2 5 s . 732 at the chronometer reading I4 h oo" 3 ./. Also from the differences of epochs and corrections it was found that the rate of the chronometer during this interval was o 8 . 00180 per minute. Applying this rate for the interval (i4 h 17. 3 I4 h oo m .7) to the value 2 5 s . 732 of the correction, there is obtained for the chronometer correction at the mean epoch of the signals sent from Ithaca to Cambridge 2 5 s . 702. Similarly, from the time observations at Ithaca it was found that when the clock read Q h 38 m .8 its correction was 7 m 57 s . 107 (on local mean solar time). The computation * shows how the mean epoch of the signals was derived in Cambridge sidereal time (i4 h i6 m 5 I s . 739), and in Ithaca sidereal time (i3 h 55 m 26 8 .578)f. The difference of these two, 2i m 25 8 .i6i is the difference of longitude of the stations, affected by the transmission time of the electric wave, and by the relative personal equation of the two observers. 239. The longitude difference as computed from the other set of signals shown in the computation is evidently affected in the reverse way by the transmission time. Hence the mean of the two derived values, namely, (2i m 25 s . 161 -f- 2i m 2 5 8 -54) = 2i m 25 s . 108, is the longitude difference unaffected by transmission time, provided such time remained constant during the two minutes of the exchange. Also, the transmis- sion time:); itself is J(2i m 25 s .i6i 2i m 25".O54) = * This computation would be simplified in an obvious manner if sidereal timepieces had been used at both stations. f 4 h 22 59". 370 is the sidereal time of mean moon at Ithaca May 27, 1896. \ The mean value of the transmission time on nine nights over this line was 0.070, and the separate values varied from O".O54 to o'.o84. The tele- graph line from Ithaca to Cambridge, by way of Syracuse, New York, and Boston, was 592 miles long, and passed through one repeater (at New York). 240. COMPUTATION OF TELEGRAPHIC LONGITUDES. 247 An inspection of Fig. 31 will show that this is merely the transmission time between the two signal relays, and does not include the transmission time through the relays and the chronograph circuit, as this part of the transmission is always in one direction, no matter where the signal starts from in the main circuit. 240. In the regular program of observation * five values of the longitude would thus be obtained, and then five more similar results after the observers exchanged places. From these two means the effect of relative personal equation would then have been eliminated by computation, as shown in the portion of a field computation given below. DIFFERENCE OF LONGITUDE. Albany, N. Y., west of Montreal. Date, 1896. Sept. 16 " 20 " 24 . . " 28 . Oct. 9... . Oct. 10. . " I 5-- ' 19.. '* 21 .. " 26.. . Observer at Longitude Difference. A Signals. Longitude Difference M Signals. A-M 0.044 0.039 0.038 0.035 0.036 Mean of A and M Signals. d* 4i.o28 41 .066 41 .017 4 -005 41 .034 Per- sonal Equa- tion. + 0.268 -0.268 Difference of Longitude. A M F F F F F S S S s s S S s s s F F F F F o m 4i'-5 41 .086 41 .036 41 .022 41 .052 o 41 .525 41 -569 41 .617 41 -639 4i -578 o m 4i.oo6 41 .047 40 .998 40 .987 41 .016 Means o 41 .491 41 -538 4 i .580 4 1 -599 41 .526 Means = O m 4 I.2 9 6 4i -334 41 -285 4i -273 41 .302 41 .240 41 .285 4i -330 4i -351 41 .284 0.038 0.034 0.031 0.037 0.040 0.052 o 41 .030 o 41 .508 4 1 -553 41 .598 41 .619 4i -55 2 0.039 o 41 .566 o 41 .298 Transmission time = K"- 3 8 ) = o 8 .oig. Relative personal equation, S F = $(41.566 41.030) = -(- o .268. Difference of longitude, A M = o h oo m 41*. 298. * The regular program was not carried out at this station, hence the following illustration is taken from another source. 248 GEODETIC ASTRONOMY. 242. Discussion of Errors. 241. From the final computed result there has thus been eliminated the average relative personal equation during the series of observations, and the average value of the transmis- sion time during the short interval covered by the exchange of signals on each evening. The errors of the adopted right ascensions are also eliminated from the result, because the same stars have been observed at both stations.* 242. The final computed result is subject to the following errors: 1st, that arising from the accidental errors of observa- tions of 200 stars at each station, which must be quite small after the elimination due to 400 repetitions; 2d, that arising from the variation of the relative personal equation of the two observers from night to night, of which the magnitude may be estimated from the following paragraphs; 3d, that due to lateral refraction, to which reference will be made in 245 ; 4th, that due to variations in the rates of the chronometers during the period covered by the observations, which must usually be quite small, as the chronometers are not disturbed in any way during the observations and are protected as far as possible against changes of temperature; 5th, that arising from the variation of the transmission time, between the two halves of the exchange of signals, on each night, which is probably insensible, as this interval is usually only a minute ; 6th, the difference of transmission time through the two signal relays, since this difference always enters with the same sign, as may be seen by an inspection of Fig. 31. This last error is made very small by using specially designed relays which act very quickly, by adjusting the two relays to be as * Where the difference of longitude is very large, the observers may be forced to use different star lists to avoid depending upon their chronometer rates for too long an interval. 243- PERSONAL EQUATION 1 . 249 nearly alike as possible, by controlling the strength of the current passing through the relay so that it shall always be nearly the same, and by exchanging relays when the observers change places, or by a combination of these methods.* Personal Equation. 243. The extent to which the relative personal equation may be expected to vary may be estimated from the follow- ing statement of the experience of two observers who have made the major portion of the primary longitude determina- tions of the Coast and Geodetic Survey during the period indicated. The plus sign indicates that Mr. Sinclair observes later than Mr. Putnam. PERSONAL EQUATION BETWEEN C. H. SINCLAIR AND G. R. PUTNAM, ASSISTANTS C. AND G. SURVEY, RESULTING FROM OR CONNECTED WITH THE TELEGRAPHIC LONGI- TUDE WORK OF THE SURVEY. f By direct comparison at Washington, D. C., 1890, Sept. 17, 18, 19 S P = + o s .266 By direct comparison at St. Louis, Mo., 1890, Nov. 4, 15 -f- o .278 From interchange of observers during longitude determinations, after' one- half of the work was completed, generally from 4 or 5 days' results: Cape May, N. J., and Albany, N. Y., 1891, May and June S P = -\- OM84 o".oii Detroit, Mich., and Albany, N. Y., 1891, June and July -+- o .140 o .008 Chicago, 111., and Detroit, Mich., 1891, July. -f- o .172 o .006 Minneapolis, Minn., and Chicago, 111., 1891, Aug +o .161 o .010 Omaha, Neb., and Minneapolis, Minn., 1891, Aug. and Sept -f- o .176 o .on Los Angeles, Cal., and San Diego, Cal., 1892, Feb. and Mar -j- o .160 o .006 * See Coast and Geodetic Survey Report, 1880, p. 241. f For these data the author is indebted to the Superintendent of the Coast and Geodetic Survey. 250 GEODETIC ASTRONOMY. 243. San Diego, Cal., and Yuma, Ariz., 1892, March -|- o .192 o .004 Los Angeles, Cal., and Yuma, Ariz., 1892, Mar. and April -(- o .140 o .002 Yuma, Ariz., and Nogales, Ariz., 1892, April -j- o .150 o .005 Nogales, Ariz., and El Paso, Tex., 1892, April and May -|- o .126 o .004 Helena, Mont., and Yellow Stone Lake, Wyo., 1892, June and July -f- o .109 o .010 El Paso, Tex., and Little Rock, Ark., 1893, Feb. and March -j- o .082 o .010 The following values depend on unrevised field com- putation : Key West, Fla., and Charleston, S. C., 1896, Feb. and March S P = + o'.i47 Atlanta, Ga., and Key West, Fla., 1896, March. . -f o .121 Little Rock, Ark., and Atlanta, Ga., 1896, April. -f o .130 Charleston, S. C.. and Washington, D. C., 1896, April and May -)- o .183 Washington, D. C., and Cambridge, Mass., 1896, May and June -f o . 142 o".oi3 Washington,, D. C., Naval Observatory and Washington, D. C., Coast and Geodetic Survey Office, 1896, June and July -f- o .117 o .008 Note that the period covered by this record is nearly six years, and that the localities show that the observers were surely submitted to a great variety of climatic conditions. Yet if the first two determinations, made when Mr. Putnam was comparatively new to the work, be omitted, the total range of the results is only o 8 . no, It must be remembered, however, that each of these results, except the first two, depend upon from eight to ten nights of observation, four or five nights each before and after the interchange of observers. It is quite probable that the actual variation of the relative personal equation from night to night is somewhat greater than that shown above. 245- DISCUSSION OF ERRORS. 25 1 244. The absolute personal equation is the time interval required for the nerves and portions of the brain concerned in an observation to perform their offices. Although the per- sonal equation has been studied by many, little more can be confidently said in regard to the laws which govern its magni- tude than that it is a function of the observer's personality, that it tends to become constant with experience, and that probably whatever affects the observer's physical and mental condition affects its value. But so little is known in regard to it, that no observer will predict, before the observations of a night have been computed, that his personal equation was large or small on that particular night. Discussion of Errors. 245. Returning to a consideration of the errors of the telegraphic longitudes, it may be said that the ten results for a station, after eliminating the personal equation, still show a range, ordinarily, in primary work, of from o 8 . 10 or less, to o s .2O. This range is larger than is to be accounted for by the accidental errors of observation, or by any of the other errors enumerated in 242, except perhaps those of the second and third classes. Those most familiar with the observations are apt to account for the large range as arising either from varia- tion in the personal equation or from lateral refraction. One observer of long experience is inclined to suspect the striding level of giving errors which tend to be constant for the night. To whatever these errors may be due, they seem to be fairly well eliminated from the mean for the station. For in the great network of longitude determinations, made by the Coast and Geodetic Survey, covering the whole United States, the discrepancies arising in closing the various " longitude tri- angles " are always less than OMO.* * For a good example of a check showing the degree of accuracy of this network see Coast and Geodetic Survey Report, 1894, p. 85. GEODETIC ASTRONOMY. 247. Personal Equation. 246. If, in making a longitude determination circum- stances prevent the interchange of observers, the effect of the relative personal equation upon the computed longitude may still be eliminated, in part at least, by a special determination of the equation by joint observations at a common station. The two observers may place their instruments side by side in the same observatory, observe the same stars, and record their observations upon the same chronograph. The differ- ence of the two chronometer corrections computed by them, corrected for the minute longitude difference corresponding to the measured distance between their instruments, is then their relative personal equation. Or, they may observe with the same transit as follows: On the first star A observes the transits over the lines of the first half of the reticle, and then quickly gives place to B, who observes the transits across the remaining lines. On the second star B observes on the first half of the reticle, and A follows. After observing a series of stars thus, each leading alternately, each observer com- putes for each star, from the known equatorial intervals of the lines and from his own observations, the time of transit of the star across the mean line of the whole reticle. 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 mean result is inde- pendent 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 instru- ment than in the other method. 247. In certain cases in which it is not feasible to use a telegraph line for a longitude determination, the same prin- ciples may be used with the substitution of a flash of light 248. LONGITUDE BY CHRONOMETERS. 2$$ between stations 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. Longitude by Chronometers Equipment. 248. If a telegraph line is not available between the two stations, the next method in order of accuracy, aside from the flash method alluded to above, is that of transporting chronometers back and forth between them. The transported chronometers then perform the same duty as the telegraph, namely, that of comparing the local times of the two stations. The chronometric method may perhaps be best explained by giving a concrete example. The longitude of a station at Anchorage Point, Chilkat Inlet, Alaska, was determined in 1894, by transportation of chronometers between that station and Sitka, Alaska, of which the longitude was known. At Anchorage Point observations were taken on every possible night from May I5th to August I2th, namely, in 53 nights, by the eye and ear method, with a transit of the type shown in Fig. n, using as a hack for the observations chronometer Bond 380 (sidereal). At the station there were also four other chronometers, two sidereal and two mean. These four 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 instru- mental equipment at Sitka was similar. A sidereal chro- nometer was used as an observing hack, and two other chronometers, one sidereal and one mean, were used in addition. Nine chronometers, eight keeping mean time and one sidereal time, were carried back and forth between the stations on the steamer Hassler. 254 GEODETIC ASTRONOMY. 249. Longitude by Chronometers Observations. 249. Aside from the time observations the procedure 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 No. 380 (sidereal) was compared with the two mean time chronometers by the method of coincidence of beats, to be described later (250). These two were then each compared with each of the two remaining (sidereal) chronometers at the station. These com- parisons, together with the transit observations, served to determine the error of each chronometer on 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 No. 380 was compared with the other station chronometers as indicated above, was carried on board the steamer and compared with the nine steamer chronometers, and then immediately returned to the station and again compared with the four stationary station chro- nometers. As an extra precaution both the station observer and the observer in charge of the steamer chronometer made each of these comparisons. In the comparisons on the steamer, the hack (380) was compared 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 error of the hack (380) at the epoch of the steamer comparisons. The steamer comparisons determined the errors of each of the * The station chronometers were also intercompared on days when no observations were made. But this was merely done to ascertain their per. formance, and these comparisons were not used in computing the longi- tude. 250. LONGITUDE BY CHRONOMETERS. 255 steamer chronometers on Anchorage Point time. Similar observations were made at Sitka to determine the errors of the nine steamer chronometers on Sitka time as soon as they arrived, and again just before they departed from Sitka. During the season the steamer, which was also on other duty, made seven and a half round trips between the stations. The distance travelled was about 400 statute miles for each round trip. 250. 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 chro- nometers tick exactly together (one beat = o s .5). Just as one looks along a vernier to find a coincidence, so here one listens to this audible vernier and waits for a coincidence. As in reading a vernier one should also 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 an approaching coincidence, hears the ticks nearly together, apparently 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 the 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. The hours and minutes are observed directly. When a number of chronom- eters 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 256 GEODETIC ASTRONOMY. 252. 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. Computation of a Longitude by Chronometers. 251. The following example (taken from another set of observations) will show how the chronometer comparisons are computed. A certain Dent mean time chronometer was compared with a certain Negus sidereal chronometer on Oct. 14, 1892, at a station 2 h I2 m west of Washington. It was found that u h 2O m 23 s . o A.M. Dent = I2 h 54 41*. o Negus. The correction to the Dent on local mean time was known to be 2 h n m 53 s . 41, and the correction of the Negus to local sidereal time was required. Time by Dent chronometer 23* 2O m 23". oo Correction to Dent 2 n 53.41 Local mean solar time 21 08 29.59 Reduction to sidereal interval ( 290) +03 28 .38 Sidereal interval from preceding mean noon 21 n 57.97 " time of preceding mean noon (Oct. 13). . 13 31 14 .05 Local sidereal time 10 43 12 .02 Time by Negus chronometer 12 54 41.00 Correction to Negus chronometer 2 n 28.98 The computation is modified in an obvious manner if it is the error of the sidereal chronometer that is known. 252. This process of comparing chronometers is so accu- rate, that it was found that the two values of the error of either of the station sidereal chronometers, as derived from the comparisons described above in two different ways from the hack chronometer, seldom differed by more than o s .O3. This corresponds to an error of n 8 in noting the time of 254- LONGITUDE BY CHRONOMETERS. coincidence of beats, on the supposition that all the error was made in one of the four comparisons concerned. If two chronometers of the same kind, both sidereal or both mean time, are compared directly it requires very careful observing to secure their difference within o s . 10 of the truth. 253. The comparisons of the other four station chrono- meters with the hack chronometer immediately before and after transit observations gave the errors of each of those four chronometers. To compute the errors of the steamer chro- nometers at the time of the comparisons made on the steamer, it is first necessary to secure as good a determination as possi- ble of the error of the hack chronometer at the epoch of those comparisons. One value for that error was obtained in an obvious manner by assuming that the hack chronometer ran at a uniform rate between the last preceding and the next following transit observations. Four other determinations of the error of the hack at that epoch were obtained, by making that same assumption for each of the other four station chronometers, and deriving the error of the hack from the comparisons made with that chronometer at the station before and after the steamer comparisons. The weighted mean of these five values of the error of the hack was used. For the method of deriving the relative weights which were assigned to these five results see 260. At Anchorage Point 13 comparisons were made with the steamer chronometers. In six cases out of the thirteen the range of the five derived values of the error of the hack was less than o s .2. 254. Having now the errors of the steamer chronometers on the local time of each station at the time of arrival at and departure from each station, the difference of longitude was computed in the manner indicated by the following illustra- tion. Suppose chronometer No. 231 to have been found to have the following errors on a certain round trip: 258 GEODETIC ASTRONOMY. 256. Anchorage Point, at departure, May 15, Q h oo m A.M. 40" fast of A. P. time. Sitka, on arrival May 16, 9 oo A.M. n " " Sitka " Sitka, at departure May 22, 9 oo A.M. 4 " "Sitka " Anchorage Point May 23, 9 oo A.M. 32 " " A. P. " From the two Anchorage Point observations it appears that the chronometer has lost 8 8 in the eight days it was gone from there. From the two Sitka observations it appears that 7 s were lost while at Sitka. Hence the chronometer lost I s only while travelling both ways between the stations, or its travelling rate was O 8 .5 per day, losing. Applying this rate to the errors as determined at Anchorage Point, we find that the errors of the chronometer on Anchorage Point time at the epochs of the Sitka steamer comparisons were 39 s . 5 fast and 32 s . 5 fast, and that the difference of longitude required is 398.5 n s .o = 32 s . 5 4 s . o = 28*. 5, A. P. west of S. We have thus derived the longitude difference on the supposition that the steamer chronometers have a travelling rate which is constant during the round trip, and without any assumptions as to the rates while in port. The assumptions as to the station chronometers have been simply that each preserves a constant rate between successive transit observations. 255. The longitude was thus computed from each round trip starting from Anchorage Point, and the mean taken. If the chronometers had continually accelerated (or retarded) rates, this mean was subject to an error arising from that fact. To eliminate such a possible error, and to serve as a check upon the computation, a second computation was made from each round trip starting from Sitka, and the mean taken. The error from acceleration (or retardation) of rates was necessarily of opposite sign in this mean. The mean of these two results is then subject only to accidental errors, in so far as the chronometers are concerned. 256. The following table shows the separate results obtained and the manner of combining them : 257- LONGITUDE BY CHRONOMETERS. 259 DIFFERENCE OF LONGITUDE, IN SECONDS, BETWEEN SITKA AND ANCHORAGE POINT, CHILKAT INLET, ALASKA. SUMMARY OF RESULTS FROM SEVEN ROUND TRIPS, STARTING FROM ANCHORAGE POINT, CHILKAT INLET. Chronometers, M. T. or Sid. i't 2 d 3 d 4 th 5 th 6 th ? th Means. AA Weights. 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 2T6 7 28 21 28.56 28.90 28.55 28.68 28.27 2 8.64 28.54 *7 Sid. 387 28.20 28.44 28.91 27.93 28.41 27.93 28.59 28.34 6 Mean Weighted mean 28.31 28.36 28.88 28.34 28.48 28.27 28.50 28.45 28.44 Weight. 3I222I2 Weighted mean o h c m 28*. 44 o".o5 SUMMARY OF RESULTS FROM SEVEN ROUND TRIPS, STARTING FROM SITKA. Chronometers, M. T. or Sid. , 2 d 3 d 4 th 5 tn 6 .h ? th Means. AA Weights. 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.^0 7 196 28.59 29.07 28.95 27.66 28.03 29.56 9.20 28.72 3 1542 28.93 28.57 28.59 28.22 28.50 28.50 8.32 28.52 22 1728 27.59 28.90 28.75 27.99 29.01 28.09 8.75 28.44 6 208 27.71 28.03 28.52 28.58 27.88 28.76 7.65 28.16 6 2167 28 24 28.71 28.80 28.27 28.77 28.31 8.49 28.51 17 Sid. 387 28.68 28.80 28.43 27.69 28.97 27.98 8.73 28.47 6 Mean Weighted mean 28.30 28.76 28.75 28.05 28.57 28.44 28.50 28.41 28.69 28.70 28.13 28.61 28.38 28.44 28.48 28.48 Weight 1222222 Weighted mean o h o m 28.48 o s .os Final mean AA = -f- o h oo m 28". 46 . o 8 .os Longitude of Sitka, 9 01 21 .48 o .13 Longitude of Anchorage Point 9 01 49 .94 o .14 or 135 27' 29". 10 2".io 257. The steamer started from Anchorage Point at the beginning of the season, and finished at Sitka at the season's end, after 7^ round trips. The last half-trip was omitted in 260 GEODETIC ASTRONOMY. 259. the first part of the above computation, and the first half-trip omitted in the second part. If there had been simply seven round trips starting from Anchorage Point the procedure would have been to deal regularly with all trips in the first half of the computation; and in the last half in addition to the six regular round trips starting from Sitka, the last half- trip (S. to A. P.) and the first half-trip (A. P. to S.) would have been used together as a seventh round trip from Sitka. 258. Let N be the number of days during which the chronometers were depended upon to carry the time during each round trip, reckoned as follows: Add together the two intervals between comparisons of the steamer chronometers with the shore chronometer at the beginning and at the end of each half-trip, and increase this by adding the interval from each comparison of the observing chronometer and steamer chronometers to the nearest transit observations made at that station. The weight assigned to each trip in the above computation is proportional to i/N. 259. What relative weights shall be assigned to the results from the different chronometers ? Some evidently run at a more nearly constant rate than others. Let /,, / 2 , / 3 , . . . 4 be the separate values of the longitude as given by any one chronometer, and l m their mean, and let n be the number of such values, or the number of trips. Then by least squares the probable error of any one value is '. - 4J' + i/ a - LY . (4 r== ") r ] n i By the rule that the weight of a result is inversely propor- tional to the square of its probable error, the relative weights to be assigned to the chronometers are proportional to n i v - ( I21 ) [(A - /)' + tt - T ... (4 - 26 1. LONGITUDE BY CHRONOMETERS. 26 1 The factor 0.455 is dropped for simplicity since we are deal- ing with relative weights only. In the above computation the sum [(/, - l^ + (/,-/)* ...(/,- / w ) a ] was determined from each half of the computation, and the mean used in the denominator of (121). The remainder of the computation needs no explanation. 260. The relative weights assigned to the station chro- nometers as indicated in 253 may be determined by an analogous process. Let o be the error of a chronometer at the epoch of the transit time observations as determined from those observations. Let / be its error at that same instant interpolated between its errors as determined at the last pre- ceding and first following transit time observations on the assumption that its rate during that interval is constant. Then / o is a measure of the behavior of the chronometer. It is the amount by which the chronometer has gone wrong on the supposition that the transit observations may be con- sidered exact. The chronometer apparently indicates that the station at the middle observation was at a distance / o in longitude from its position at the preceding and following observations. For a group of chronometers whose errors are all determined a number of times in succession by the same transit observations, the relative weights are evidently pro- portional to the quantities 261. The above example serves to illustrate the principles involved in the computation of a longitude by chronometers. The accuracy of the derived longitude is greater, the greater the number of chronometers used, the greater the number of trips, the smaller the average value of N ( 258), and of course depends intimately upon the quality of the chronom- 262 GEODETIC ASTRONOMY. 263. eters and the care with which they are protected from jars and from sudden changes of temperature. Unless the round trips are quite short the errors of the transit time observations will be small as compared with the other errors of the process. If considered necessary the relative personal equation of the observers may be eliminated from the result by the same methods that are used in connection with telegraphic determinations of longitude. 262. If the trips are very long, it may possibly be advis- able to determine, by a special series of observations, the temperature coefficient of each chronometer and also a coefficient expressing its acceleration (or retardation) of rate, and to apply corresponding computed corrections to the travelling rates.* The chronometers are compensated for temperature as far as possible by the maker, of course, but such compensation cannot be perfect. The thickening of the oil in the bearings tends to increase the friction with lapse of time, and by diminishing the arc of vibration of the balance- wheel to increase the rate of running. Attempts to use rate corrections depending upon the computed coefficients of a chronometer have usually been rather unsatisfactory, and should not be made except in extreme cases. Longitude Determined by Observing the Moon. 263. If none of the preceding methods are available, one is forced to use those methods which depend upon the motion of the Moon, or perhaps to observe upon Jupiter's satellites. The place of the Moon has been observed many times at the fixed observatories. From these observations its orbit and the various perturbations to which it is subject have been computed. In the American Ephemeris and similar publica- * For details of this process see Doolittle's Practical Astronomy, pp. 383-388. 265. LONGITUDE BY THE MOON. 263 tions, tables will be found giving the Moon's right ascension and declination for every hour, and also other tables giving its place as defined in other ways. Suppose now that an observer at a station of which the longitude is required determines the position of the Moon 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. Among the processes by which the position of the Moon may be determined for this purpose are the following. 264. The local sidereal time of transit of the Moon acrbss the meridian of the station may be observed with a transit, and a chronometer of which the error is determined in the usual way by observations upon the stars. Or, what is in principle the same thing, the right ascension of the Moon may be derived by comparing its time of transit with that of four stars of about the same declination as the Moon, two transit- ing shortly before it, and two soon after it. In either case the right ascension of the Moon at the instant of its transit may be computed, and from the Ephemeris the Greenwich time at which the Moon had that right ascension becomes known. 265. The lunar distance of a heavenly body is the angle between two lines drawn from the center of the Earth one to the center of the Moon and the other to the center of the body considered. Or, in other words, it is the angle between the two objects as seen from the Earth's center. The Ephem- eris gives the lunar distances of the Sun, the four larger planets, and of certain stars, at intervals of three hours, Greenwich mean time. An observer anywhere may measure the angular distance from the Moon to any one of these 264 GEODETIC ASTRONOMY. 268. objects, with a sextant or other suitable instrument. His measurement reduced to the Earth 1 s center gives the lunar distance; from which with the use of the Ephemeris the Greenwich time of the observation becomes known ; and also his longitude if he noted the local time of the observation. 266. A star is said to be occulted during the time it is out of sight behind the Moon. The beginning and end of the occupation, that is, the instants of disappearance and re- appearance of the star, called its immersion and emersion, are phenomena capable of being observed with considerable accuracy. The Ephemeris gives the necessary elements for computing the Washington times of occultation of various stars as seen from any point upon the surface of the Earth. The local time of the occultation being observed, either of the immersion or emersion, and the Washington time being com- puted, the longitude becomes known. An observation of the local times of the phenomena of an eclipse of the Sun or Moon furnishes a similar determination of longitude. 267. The computations required in the last two methods are quite long and complicated, and the theories involved require much study for their mastery. The method of cul- minations gives rise also to rather difficult computations, though not so difficult as those just mentioned. However, the time and labor expended would be fully rewarded if accurate results were obtained. But any of these methods give rise to results which are crude in comparison with those given by the telegraphic method, or by transportation of chronometers. The method of occultations requires the greatest amount of computing, but also gives the greatest accuracy, of the methods named. 268. Three conditions stand in the way of the attainment of accuracy by any method involving the Moon. Firstly, the Moon requires about 2/J days to make one complete circuit 2/O. LONGITUDE B Y THE MOON. 265 in its orbit about the Earth. The apparent motion of the Moon among the stars is then about one-twenty-seventh as fast as the apparent motion of the stars relative to an observer's meridian, which furnishes his measure of time. Any error in determining the position of the Moon is then multiplied by at least twenty-seven when it is converted into time in the progress of the computation. If then the time of transit of the Moon, for example, could be observed as accurately as that of a star, one would expect the errors in a longitude computed from Moon culminations to be twenty- seven times as great as the errors of the local time derived from the same number of star observations. 269. Secondly, the motion of the Moon is so difficult to compute that its positions at various times as given in the Ephemeris, and also of course the data there given in regard to lunar distances and occultations, are in error by amounts which become whole seconds when multiplied by the factor twenty-seven. This source of error is often avoided in the method of Moon's transits (culminations) by using in the computation for each night the Moon's right ascension as corrected at Greenwich, or some other station of known longi- tude, by direct observation on that same night. Thirdly, the limb, or edge of the visible disk of the Moon, is necessarily the object really observed, and this is a "ragged edge" rather than a perfect arc, for purposes of accurate measurement. 270. The determination of the points at which the boundary between Alaska and British America (i4ist meri- dian) crosses the Yukon and Porcupine rivers was one of the comparatively few instances in late years in which it was necessary to resort to observations upon the Moon to deter- mine an important longitude. To determine the longitude by transportation of chronometers would have been exceed- 266 GEODETIC ASTRONOMY, 271. ingly difficult and costly, for there is more than a thousand miles of slow river navigation between the mouth of the Yukon River and either station. At a station near the point where the Yukon crosses the boundary, Moon culminations were observed on 23 nights. Four of the results were rejected as worthless. The other 19 gave results ranging from 9 h 22 m 3O s .o to 48*. 9, with a weighted mean of 38 S .5. Fourteen of these computed results depend upon the Moon's place as corrected by corresponding observations at Greenwich or San Francisco. At the same station two observed occul- tations gave for the seconds of the longitude 3 5 s . 5 and 37 8 .2, and a solar eclipse gave 32 s . 2. At a station near the point where the Porcupine River crosses the boundary, 13 observed Moon culminations computed by the use of corresponding observations on the same nights at San Francisco, Washing- ton, or Greenwich, gave longitudes varying from 9 h 23"" 45 s . 5 to 63 s . 8, with a weighted mean of 5 5 s . 4. One observed occupation, both immersion and emersion, gave for the seconds of the longitude 63 s . 6. These examples* will serve to indicate roughly the possibilities of the lunar methods of determining longitude. Experienced observers took the observations at both stations. It should be noted, however, that in such high latitudes (the stations were near the Arctic Circle), the trigonometric conditions are unfavorable to accurate time determinations, and the climatic conditions were such as to make observing difficult. 271. Those wishing to study these lunar methods of determining the longitude are referred for details to Doo little's Practical Astronomy; to Chauvenet's Astronomy, vol. I. ; and in the American Ephemeris (aside from the tables) * For a more complete account of these observations see Coast and Geodetic Survey Report, 1895, pp. 331-336. 2^2. LONGITUDE B Y JUPITER. 267 especially to the pages, in the back of the volume, headed " Use of Tables." 272. The Ephemeris gives, for each night of the year when Jupiter is not too near the Sun to be observed, the Washington mean time of the occultations and eclipses of Jupiter's satellites by that planet, and also the transit of the satellites and their shadows across the face of the planet. An eclipse may be observed at a station of which the longi- tude is required. By comparison of the computed Washing- ton mean time of the eclipse as given in the Ephemeris, and the observed mean time, the required longitude may be derived. The times of the other phenomena mentioned are given to the nearest minute only. They may be observed simultaneously by two observers using the Ephemeris merely to indicate when to be on the alert. The difference in the local times of observation of the same phenomena is the difference of longitude of the observers, the transit or occulta- tion serving merely as a signal that may be seen at the same instant by both. The difficulty of accurately observing these phenomena (including the eclipses) makes the derived longi- tudes only rough approximations. The time of a satellite may, for example, be observed a whole minute sooner than it actually occurs, if a low-power telescope is used. Such errors may be partially eliminated by observing the reappearance as well as the disappearance. 268 GEODETIC ASTRONOMY. 2/3. CHAPTER VIII. MISCELLANEOUS. Suggestions about Observing. 273. Among the characteristics of a good observer, that is, of an observer who will secure the maximum accuracy with a given expenditure of time and money, in making such astronomical determinations as are treated in this book, may be mentioned the following: He is without bias as to the results to be obtained, his prime motive being always to come as near as possible to the truth. He has that kind of self-control which makes it possi- ble for him to prevent the knowledge that the result he is securing is too small (or too large) to check with other determinations, from having the slightest effect upon his observations. For example, he may know that his observa- tions, in making a telegraphic determination of the longitude of a station, are placing that station o s .5 farther west than it has been fixed by a primary triangulation, and yet have no tendency to observe stars earlier or later than usual. Or, when in reading a micrometer upon an azimuth mark several times in quick succession he secures three or four readings which agree almost exactly, and then one which differs from them by two seconds (say), thus making a bad looking break in his record, he will not suppress or " spring " this reading, though it may serve to make him more careful with following readings. 2/6. OBSERVING. 269 274. He is well aware of the minuteness of the allowable errors. A student, when warned that he must not apply any longitudinal force to the head of the micrometer of a zenith telescope, will perhaps experiment for himself, by purposely applying a little pressure while making a bisection, and not being able to see any appreciable motion, will become in- credulous as to the necessity of the warning. A good observer, on the other hand, knows that he can secure obser- vations such that the combination of errors from all sources produce an error in each result which is as apt to be less than o".3 as greater than that value (e = (/'.jo), although o".3 is a fraction of the apparent width of the line with which he makes the bisection. In other words, he knows that he can make pointings under good conditions, of which the errors are so small as to be invisible in the telescope. He knows that he can make pointings with a probable error of o".5, say, with a telescope with which it would be hopeless to try to see a rod one-sixth of an inch in diameter placed one mile away Q- inch subtends o".$ at one mile). 275. He is conscious that the most delicate manipulation is required. He knows that his instrument is built of elastic material, and that unless he is exceedingly careful to apply only such forces as are necessary he may readily produce deformations in his instrument, which though strictly in accordance with the modulus of elasticity of the material composing it, are yet as large as the largest allowable errors of observation. One may sometimes secure striking ocular evidence of this by watching a bisection, in a reading micro- scope on a horizontal circle (or in the telescope), while a poor observer makes his pointings with another of the reading microscopes on the instrument. 276. A good observer does not consider his instrument to be of fixed dimensions or shape, even when no external 2/0 GEODETIC ASTRONOMY. 277. forces are applied to it. He knows that it is constantly undergoing changes of shape due to changes of temperature ; that these changes even under the best conditions that he can secure may produce errors of the same order of magnitude as the observer's errors; and under adverse conditions may pro- duce errors which are larger than all the others concerned in the measurement. With respect to movements under stress and under thermal changes, the support of the instrument (tripod, block, or pier) should be considered as a part of the instru- ment. Suggestions about Computing. 277. Almost the first question that arises on commencing a given kind of computation for the first time is " To how many decimal places must each part of the computation be carried ?" If too few figures are used the errors from the cast away decimal places become larger than is allowable. If too many places are used the computation becomes slower than is necessary, the work required for interpolations in whatever tables are used being especially liable to increase rapidly with an increase of decimal places. A good general guide in this matter is to carry each part of every computation to as many decimal places as correspond to two doubtful fig- ures in the final result. That is, when the computation is finished, and the probable error is computed, there should in general be two significant figures in the probable error. Or, in other words, the probable error should be between 10 and 100 units in the last place. It may be allowable to drop one more figure than above indicated if to do so decreases the work of computation very much, as in computing sextant observations for time (see foot-note to Example 8, at the end of Chapter III). It is important, after deciding upon the 2/9- COMPUTING. 271 number of places to use in a computation, to adhere to that number strictly. To carry some numbers one place farther than others, in a column to be added, is useless: and worse than useless, for it leads to mistakes such as adding tenths and hundredths, for instance, as if they were in the same column. If a number ends in a five and the last figure is to be cast away, shall the five be called ten or zero ? Both are equally near the truth. A good rule is, in such cases, to make the last retained figure even (not odd). This will mean calling the five a ten about half of the time, and avoids the constant tendency to make the result too large (or too small) that would exist if the five were always called ten (or zero). 278. If many astronomical computations are to be made, Barlow's tables of squares, etc., Crelle's four-place multipli- cation tables, and a machine for multiplying will be found convenient aids, the last two for checks, especially. For example, apparent star places may be computed by loga- rithms in the usual way, and then checked by a separate computation by natural numbers and the use of Crelle's tables (or a computing-machine). The check will be much more efficient than repeating the logarithmic work because an entirely different set of figures are used, and it will take about the same amount of time. 279. The difference between a good computer and a poor one lies largely in the industry and ingenuity with which a good computer applies such checks to his work to find what- ever mistakes he makes. Rough checks should not be despised, such as comparing two computations which are nearly alike (computations of two successive sets of observa- tions for example), or such as checking an exact computation by formula (98), 193, by the use of the table in 310. Means should always be checked by residuals. If resid- 272 GEODETIC ASTRONOMY. 283. uals be obtained by subtracting a mean from each of the separate values, the sums of the positive and of the negative residuals so obtained must not differ by more than units corresponding to the last place of the mean, where n is the number of the separate values. 280. In converting angles into time, or vice versa, it is about as rapid to use the relations 360 = 24'', 15 = i h , i = 4 m , i' = 4 s , 15" = I s , as it is to use the tables given for that purpose on page 560 of. Vega's Logarithmic Tables, and elsewhere. The tables may be used as a check. 281. When several computations of the same kind are to be made, it usually saves time to carry along corresponding portions together. For example, in computing apparent places all the star numbers may be taken out at one time, later all the values of log cos (G -f- ), and so on. The use of a fixed form for a computation saves time and mistakes. The form should represent a logical order of work, and should involve as little repetition of figures as possible. All scribbling, multiplying, dividing, interpolating, etc., should be done on separate sheets of paper from the regular computation. Probable Errors. 282. The reader who does not understand the principles of least squares cannot hope to understand the logic of the formulae given in Chapters IV and V for certain least-square computations. But after a careful perusal of 283-285 a statement of the uncertainty in a certain value in terms of the so-called probable error should not be unintelligible to him. 283. In the expression "probable error" the word " probable " is not used in its ordinary sense, but in a special 283. PROBABLE ERROR. ?73 technical sense. To assert that the probable error of a cer- tain stated value is ^, is to assert the chances are equal for and against the truth of the proposition that the stated value does not differ from the truth by more than e. Thus, to assert that the azimuth of a certain line west of north as derived from a certain series of observations is 59". o o".5, is to assert that it is as likely that the true value of that azimuth is between 58". 5 and 59". 5 W. of N. as that it is some value outside of these limits. To assert that the prob- able error of a single observation in a series = o".5, is to assert that it is an even chance that any particular observation is within o".5 in either direction of the truth. Or, what is the same thing, it is to assert that if a long series of such observations were made, the chances are that one-half of the observations would give results within o".5 of the truth, and one-half would give results differing from the truth by more than o".5. More accurately, perhaps, the probable error should be regarded as referring to accidental* errors only, without reference to possible constant errors. Thus the above state- ments should be modified to read as follows: To assert that the azimuth of a certain line west of north, as derived from a certain series of observations, is 59". o 0^.5, is to assert that if an infinite number of such observations were taken, under the same average conditions, their mean would be as likely to lie between 58". 5 and $9".$ W. of N. as to fall outside those limits. And to assert that the probable error of a single observation = o".$, is to assert that it is an even chance that that particular observation is within o".5 in either direc- tion of the mean which would result from an infinite number of such observations, made under the same average condi- * For the distinction between accidental and constant errors see foot- note to 74. 274 GEODETIC ASTRONOMY. 286. tions. The second form of the statement is non-committal as to possible constant errors affecting all the series alike, which would not be eliminated by increasing the number of observations. Such a constant error would be introduced into an observed azimuth by placing the azimuth light, unknowingly, a little to one side of the monument which it is supposed to indicate. 284. There seems to be some confusion between these two conceptions of the probable error. It is a common mistake among those who use least squares to derive a probable error by methods which correspond to the second form of state- ment above, and then to assume that the first form of state- ment is true. Hence one is always on the safe side to assume the second form of statement to give the true mean- ing of the probable error, and to form an estimate of the possibility of a constant error from other sources of informa- tion. If there is no possibility of a constant error in the observations, the two forms of statement are identical. 285. The relation between the probable error of a single observation, and the total range between the largest and smallest values given by such observations, is as follows: If the probable error of a single observation is e, one is to expect that if a large number of such observations were made, only about one per cent will fall outside a total range of 7^ times e. Or, if the probable error of a single observa. tion is o".5, only about one observation in one hundred would be expected to fall outside a total range of 3 ".8. The Latitude Variation. 286. Until a few years ago it was supposed that the lati- tude of a given station was invariable. During the last few years a vigorous investigation of that assumption has been made, both by means of new series of observations of the 28/. VARIATION OF LATITUDE. 2?$ highest degree of accuracy planned especially for the purpose, and by the re-examination of various old series of observa- tions at the fixed observatories. The result of these investi- gations may be briefly stated as follows: The axis of rotation of the Earth does not coincide exactly with its axis of figure. By axis of figure is meant that line about which its moment of inertia is a maximum. Roughly speaking, the axis of figure describes a cone, with its vertex at the centre of the Earth, about the axis of rotation once in 428 days. The motion of the pole of figure about the pole of rotation during that interval is roughly an ellipse with a major diameter of about 60 feet, described in the direction of decreasing west longitudes, that is, in a counter-clockwise direction, as seen from above at the north pole. This motion is combined also with one of a period of one year, and is variable as to the diameter and position of the ellipse, and otherwise, so that the above statement serves simply as an approximate descrip- tion of the motion. The general law governing the motion is not yet known, and all formulae as yet derived for predict- ing the future motion are empirical. The direction of gravity at a given station is sensibly con- stant as referred to the axis of figure, that is, as referred to the solid Earth. But the latitude as measured is referred to the axis of rotation the plane perpendicular to that line, the equator, being the plane to which the declinations of the stars are referred. Hence the latitude of every station on the Earth varies through a range equal to twice the angle between the two axes, a range of about o".6. It changes from its maximum to its minimum value, and back again to the maximum, once, roughly speaking, every 428 days. This motion can be traced in the past, but not as yet pre- dicted for the distant future. 287. Three examples of the long series of latitude obser- 276 GEODETIC ASTRONOMY. 288. vations made with zenith telescopes for the special purpose of determining the latitude variation may be found in Coast and Geodetic Survey Reports for 1892, part 2, pp. 1-159, and for 1893, part 2, pp. 440-508. The principal investi- gations of the variation by means of latitude observations not specially planned for the purpose have been made by Prof. S. C. Chandler. Indeed, his investigations first proved satis- factorily that such variations are a fact. His results will be found published in various numbers of the Astronomical Journal for several years past. A general statement of the " Mechanical interpretation of the variations of latitudes," by Prof. R. S. Woodward, will be found in the Astronomical Journal No. 345, May 21, 1895. Station Errors and the Economics of Observing. 288. The author cannot close this book without calling attention briefly to one phase of geodetic astronomy to which little attention has apparently been paid, but which is of great importance in planning the astronomical work in connection with a geodetic survey, namely, the relation between the economics of observing and station errors. Broadly stated, the purpose of" the astronomical observa- tions made in connection with a geodetic survey is to deter- mine the relation between the actual figure of the Earth as defined by the lines of gravity and the assumed mean figure upon which the geodetic computations are based.* This is the purpose, whether the astronomical observations be used simply as a check upon the geodetic operations, or whether they be used as a means of determining the mean figure of the Earth. In determining the relation between the actual figure and the assumed mean figure three classes of errors are * See foot-note to 15. 289. ECONOMICS OF OBSERVING. encountered: the errors of the geodetic observations; the errors of the astronomical observations; and the errors due to the fact that only a few scattered stations can be occupied on the large area to be covered, and that the station errors as derived for these few points must be assumed to represent the facts for the whole area. Neglect the first class of errors, as being in the province of geodesy rather than astronomy. The duty of the engineer when planning the astronomical work of a survey is to so fix the number and character of the observations at each station, and the number and position of the stations, as to make the combined errors of the second and third classes a minimum for a given expenditure. By increasing the number of obser- vations at a station the errors of the second class may be diminished, the relation between the number of observations and the error of the result being that said error is inversely proportional to the square of the number of observations in the most favorable case (of no tendency to constant errors in the series of observations). If there are any constant errors affecting the series, then the increase in accuracy with increase in the number of observations is slower than that stated above. The third class of errors may be reduced by increas- ing the number of stations, and distributing them as uniformly as possible, so as to diminish the area to which the result from each station is assumed to apply. 289. To illustrate, suppose the latitude observations for a geodetic survey of a State are being planned. Let us sup- pose that the engineer knows that with the available zenith telescopes and star places an observer can secure a latitude with a probable error of about o". 10 from observations on a single evening, and that he can reduce this to o".o6 by observing on four evenings. Let us assume that he estimates that it will cost the same, on an average, to observe on four OP THE UNIVERSITY 2?8 GEODETIC ASTRONOMY. 289. nights at one station, as to observe at three stations on three different nights.* Should he plan to observe at ten different stations distributed uniformly over the State on four nights at each station, or at thirty stations uniformly distributed for one night only at each ? Obviously the answer depends mainly on the magnitude of the station errors to be expected. If he estimates the station error by consulting the results obtained on the U. S. Lake Survey, he will expect an average station error of nearly 4", with a maximum exceeding io".f If he consults the report J of the " Survey of the Northern Boundary from the Lake of the Woods to the Rocky Moun- tains " he finds that the average station error there was 2" , with a maximum of 8", and that in one case six successive stations on a total distance of 100 miles along the line showed a nearly uniform change of about o". 14 per mile in one direc- tion. If he consults the published results of still other surveys his estimate of the station errors to be expected will not be materially altered. Does it not seem evident that under such circumstances the thirty stations should be occu- pied on one night each ? Yet the usual practice of geodetic surveys in this country corresponds rather to the plan of observing at 10 stations on 4 nights each, even though the observation error in the result from a single night is upon an average only one-twentieth, say, of the station error. With longitudes and azimuths it will be found that the ratio of the errors of the astronomical observations to the station errors is somewhat larger, but not enough larger to materially modify the above economic problem. * It being expected that the observations are to be taken at triangula- tion stations by the same observers who measure the horizontal angles of the triangulation. f See Professional Papers of the Corps of Engineers No. 24 (Lake Sur- vey Report), p. 814. \ Plate opposite page 267 of that report. 290. CONVERSION TABLES. 279 290. CONVERSION OF MEAN SOLAR TIME INTO SIDEREAL. Correction to be added to a mean solar interval to obtain the corresponding sidereal interval. (See 23.) Mean ^h , h _h _h h h For Solar. O I B 2 3 4 5 h Seconds. O m 00*. 000 o m og'. 856 o" 1 19*. 713 o 29'. 569 o m 39 s . 426 o 49 s . 282 0* o'.ooo i o oo .164 o 10 .021 o 19 .877 o 29 .734 o 39 -590 o 49 .447 i o .003 2 o oo .329 o 10 .185 o 20 .041 o 29 .898 39 -754 o 49 .611 2 o .005 3 o oo .493 o 10 .349 o 20 .206 o 30 .062 o 39 .919 o 49 .775 3 o .oo{ 4 o oo .657 o 10 .514 20 .370 o 30 .227 o 40 .083 o 49 .939 4 .Oil 5 o oo .821 o 10 .678 20 .534 o 30 .391 o 40 .247 o 50 .104 5 o .014 6 o oo .986 o 10 .842 20 .699 o 30 -555 o 40 .412 o 50 .268 6 o .016 7 o 01 .150 II .006 20 .863 o 30 .719 o 40 .576 o 50 .432 7 o .019 8 o 01 .314 o ii .171 O 21 .027 o 30 .884 o 40 .740 o 50 -597 8 o .022 9 o oi .478 o ii -335 21 .igi o 31 .048 o 40 .904 o 50 .761 9 o .025 10 o oi .643 o ii .499 21 .356 O 31 .212 o 41 .069 o 50 .925 10 o .027 ii o oi .807 o ii .663 O 21 .520 o 31 .376 o 41 .233 o 51 .089 ii o .030 12 o oi .971 II .828 21 .684 o 31 .541 o 41 .397 o 51 .254 12 o .03-: '3 o 02 . 1 36 o ii .992 O 21 .849 o 31 .705 o i .561 o 51 .418 13 o .03^ 14 o 02 .300 12 .156 22 .013 o 31 .869 o i .726 o 51 .582 J 4 o .038 5 o 02 .464 12 .32! 22 .177 o 32 .034 o i .890 o 51 .746 15 o .041 16 *7 o 02 .628 02 .793 12 .485 12 .649 22 .341 o 32 .198 2 .054 o 51 .911 16 o .044 18 19 02 .957 03 .121 12 .978 O 22 .834 o 32 .691 2 .383 o a -547 o 52 .404 18 19 o .052 20 o 03 .285 o 13 .142 22 .998 o 32 -855 O 2 .711 o 52 .568 20 o .055 21 o 03 .450 o 13 .306 o 23 .163 o 33 - OI 9 o 42 .876 o 52 .732 21 o .057 22 o 03 .614 o *3 -471 o 23 .327 o 33 -'83 o 43 .040 o 52 .896 22 o .060 2 3 o 03 .778 o 13 .635 o 23 .491 o 33 -348 o 43 .204 o 53 .061 2 3 o .063 24 o 03 .943 o n .799 o 23 .656 o 33 -512 o 43 .368 o 53 -225 24 o .066 25 o 04 .107 o 13 .963 o 23 .820 o 33 -676 43 -533 o 53 -389 25 o .068 26 o 04 .271 o 14 .128 o 23 .984 o 33 .841 o 43 .697 o 53 -554 26 o .071 27 o 04 .435 o 14 .292 o 24 .148 o 34 .005 o 43 .861 o 53 -7i8 27 o .074 28 o 04 .600 o 14 .456 o 24 .313 o 34 -169 o 44 .026 o 53 .882 28 o .077 29 o 04 .764 o 14 .620 o 24 .477 o 34 -333 o 44 .190 o 54 .046 29 o .079 3 o 04 .928 o 14 .785 o 24 .641 o 34 .498 44 -354 o 54 .211 30 o .082 3 1 o 05 .093 o 14 .949 o 24 .805 o 34 .662 o 44 .518 54 -375 3 1 o .085 32 o 05 .257 o 15 .113 o 24 .970 o 34 .826 o 44 .683 o 54 -539 32 o .088 33 o 05 .421 o 15 .278 o 25 .134 o 34 .990 o 44 .847 o 54 .703 33 o .090 34 o 05 .587 o 15 .442 o 25 .298 o 35 -155 o 45 .on o 54 .868 34 o .093 35 o 05 .750 o 15 .606 o 25 .463 o 35 -3*9 o 45 .176 o 55 .032 o .096 36 o 05 .914 o 15 .770 o 25 .627 o 35 -483 45 -340 o 55 .196 36 o .099 37 o 06 .078 o 15 -935 o 25 .791 o 35 .648 o 45 .504 o 55 -361 37 O . IOI 38 o 06 .242 o 16 .099 o 25 .955 o 35 .812 o 45 .668 o 55 -525 38 o .104 39 o 06 .407 o 16 .263 o 26 .120 o 35 .976 45 -833 o 55 .689 39 o .107 40 o 06 .571 o 16 .427 o 26 .284 o 36 .140 o 45 -997 o 55 -853 40 .110 4 1 o 06 .735 o 1 6 .592 o 26 .448 o 36 .305 o 46 .161 o 56 .018 4i O . 112 42 o 06 .900 o 16 .756 o 26 .612 o 36 .469 o 46 .325 o 56 .182 4 2 o .115 43 o 07 .064 o i 6 .920 o 26 .777 o 36 .633 o 46 .490 o 56 .346 43 o .118 44 o 07 .228 o 17 .085 o 26 .941 o 36 .798 o 46 .654 o 56 .510 44 o .120 9 o 07 .392 o 07 .557 o 17 .249 o 17 .413 o 27 .105 o 27 .270 o 36 .962 o 37 .126 o 46 .818 o 46 .983 o 56 .675 o 56 .839 45 46 0.123 O . 1 2O 47 o 07 .721 o 17 .577 o 27 .434 o 37 .290 47 -47 o 57 .003 47 o . 129 48 o 07 .885 o 17 .742 o 27 .598 o 37 -455 o 47 -3" o 57 . 168 48 o .131 49 o 08 .049 o 17 .906 o 27 .762 o 37 .619 47 -475 o 57 -332 49 -134 5 o 08 .214 o 18 .070 o 27 .927 o 37 -783 o 47 .640 o 57 .496 50 o .137 5i o 08 .378 o 18 .234 o 28 .091 o 37 -947 o 47 .804 o 57 .660 5' o .140 5 2 o 08 .542 o 18 .399 o 28 .255 38 .112 o 47 .968 o 57 -825 52 o .142 53 54 o 08 .707 o 08 .871 9 18 .563 o 18 .727 o 28 .420 o 28 .584 o 38 .276 o 38 .440 o 48 .132 o 48 .297 o 57 .989 o 58 .153 53 54 0.145 0.148 55 o 09 .035 o 18 .892 o 28 .748 o 38 .605 o 48 .461 o 58 .317 55 o .151 S^ o 09 .199 o 19 .056 o 28 .912 o 38 .769 o 48 .625 o 58 .482 56 .153 57 o 09 .364 o 19 .220 o 29 .077 o 38 -933 o 48 .790 o 58 .646 57 o .156 58 o 09 .528 o 19 .384 o 29 .241 o 39 .097 o 48 .954 o 58 .810 58 o .159 59 o 09 .692 o 19 .549 o 29 .405 o 39 .262 o 49 .118 o 58 -975 59 o .162 Mean h h h H ,h h For Solar. o n I* 2 n 3 4 5 Seconds. 280 GEODETIC ASTRONOMY. 290. CONVERSION OF MEAN SOLAR TIME INTO SIDEREAL. Correction to be added to a mean solar interval to obtain the corresponding sidereal interval. Mean h oh T h |j For Solar. 7 o 9 IO Seconds. om o" 59'. 139 - 08'. 995 m iS.8 5 2 28". 708 m 38' .565 "> 48'. 421 0" o'.ooo I o 59 '303 09 .160 19 .016 28 .873 38 .729 48 .585 i o .003 2 o 59 .467 09 .324 19 .186 29 .037 38 .893 48 .750 2 o .005 3 o 50 .632 09 .488 '9 -345 29 .201 39 -058 48 .914 3 o .008 4 5 o 59 .796 o 59 .960 09 .652 09 .817 19 -509 19 -673 29 -3 6 5 29 -530 39 .222 39 -386 49 -078 49 -243 4 5 .Oil o .014 6 oo . 124 09 .981 19 -837 29 .694 39 -550 49 .407 6 o .016 7 oo .289 10 .145 20 .002 29 .858 39 -715 49 -57 1 7 o .019 8 oo .453 10 .310 20 .166 30 .022 39 -879 49 -735 8 o .022 9 oo .617 10 .474 20 .330 30 .187 40 .043 49 -900 9 o .025 10 oo .782 10 .638 20 .495 30 .351 40 .207 50 .064 10 o .027 ii oo .946 10 .802 20 .659 3 '5 T 5 40 .372 50 .228 ii o .030 12 .110 10 .967 20 .823 30 .680 40 .536 So -393 12 o .033 13 o .274 II .131 20 .987 30 .844 40 .700 50 -557 13 o .036 14 o .439 II .295 21 .152 31 .008 40 .865 50 .721 14 o .038 15 o .603 ii -459 21 .316 31 .172 41 .029 50 .885 15 o .041 16 o .767 ii .624 .480 3 T -337 4i .193 51 .050 16 o .044 17 o .932 n .788 .644 31 .501 4i -357 5i -214 17 o .047 18 02 .096 ii .952 .809 31 -665 4i -522 5i .378 18 o .049 ig 02 .260 12 .117 973 31 -829 41 .686 Si -542 i9 o .052 20 02 .424 12 .28l 3i -994 41 -850 5 1 -707 20 o .055 21 02 .589 12 .445 .302 32 .158 42 .015 5t -871 21 o .057 22 02 .753 12 .609 .466 32 .332 42 .179 52 .035 22 o .060 23 02 .917 12 .774 .630 32 -487 42 .343 52 .200 23 o .063 24 03 .081 12 .938 22 -794 32 .651 42 .507 52 .364 24 o .066 25 03 .246 13 .IO2 22 .959 32 .815 42 .672 52 .528 2 5 o .068 26 03 .410 I 3 .266 23 .123 32 -979 42 .836 52 .692 26 o .071 27 03 .574 13 '43i 2 3 .287 33 -144 43 .000 52 .857 27 o .074 28 03 -739 13 -595 23 -451 33 -308 43 -164 53 -021 28 o .077 2 9 03 -903 13 -759 23 .616 33 -472 43 -329 53 -185 2 9 o .079 30 04 .067 13 -924 23 .780 33 '637 43 -493 53 -349 3 o .082 31 04 .231 14 .088 2 3 -944 33 -801 43 -657 53 -5M 3 1 o .085 32 04 .396 14 -252 24 .109 33 -965 43 -822 53 -678 32 o .088 33 04 .560 14 .416 24 -273 34 -129 43 -986 53 -842 33 o .090 34 04 .724 14 081 24 -437 34 -294 44 -150 54 -007 34 o .093 04 .888 14 -745 24 .601 34 .458 44 -314 54 I 7i 35 o .096 36 05 -053 14 .909 24 .766 34 -622 44 -479 54 -335 36 o .099 37 05 .217 15 -073 24 -93 34 .786 44 -643 54 -499 37 O . IOI 38 05 .381 15 -238 25 -94 34 -95* 44 -807 54 .664 38 o .104 39 05 -546 15 -402 25 -259 35 ."5 44 -97 1 54 .828 39 o . 107 40 05 .710 15 -566 25 -423 35 -279 45 -136 54 .992 40 o .no 41 05 .874 15 -731 25 -587 35 -444 45 .300 55 -156 4' .112 4 2 06 .038 15 .895 25 -751 35 .608 45 -464 55 -321 4 2 o .115 43 06 .203 16 .059 25 -9 6 35 '77 2 45 -629 55 -485 43 o .118 44 06 .367 16 .223 26 .080 35 -936 45 -793 55 -649 44 o .120 45 46 06 .531 06 .695 16 .388 16 .552 26 .244 26 .408 36 .101 36 .265 45 -957 46 .121 55 -814 55 -978 45 46 o .123 o .126 47 06 .860 16 .716 26 .573 36 .429 4 6 .286 56 .142 47 o . 129 48 07 .024 16 .881 26 .737 36 -593 46 .450 56 .306 48 o .131 49 07 .188 17 .045 26 .901 36 .758 46 .614 56 -47' 49 o 1 34 So 07 -353 17 .209 27 .066 36 .922 4 6 .778 56 .635 5 o -137 07 -5'7 17 -373 27 .230 37 -086 46 .943 56 .799 51 o .140 52 07 .681 17 .538 2 7 -394 37 -251 47 -107 56 .964 52 o .142 53 07 .845 17 .702 27 .558 37 '415 47 -271 57 -128 53 o .145 54 08 .010 17 .866 27 -723 37 -579 47 .436 57 -292 54 o .148 08 .174 18 .030 27 .887 37 -743 47 .600 57 .456 55 o .151 56 08 .338 18 .195 28 .051 37 -908 47 -764 57 -621 56 58 08 .502 08 .667 '8 -359 18 .523 28 .215 28 .380 38 .072 38 .236 47 -928 48 .093 57 -785 57 -949 57 58 o . 156 o .159 59 08 .831 1 8 .688 a8 .544 38 .400 48 .257 58 .113 59 o .162 Mean Solar. 6" 7 h 8" 9 h io* ii For Seconds. 290. CONVERSION TABLES. 281 CONVERSION OF MEAN SOLAR TIME INTO SIDEREAL. Correction to be added to a mean solar interval to obtain the corresponding sidereal interval. Mean h h F 'or Solar. I if H 15 16 17 Sec onds. i 58'. 278 2 m o8.i 34 m 27'. 847 28 .on __ Q/:Q 2 47'- 56o o" o'.ooo 2 58 .606 08 .463 18 .319 28 .176 37 - 808 38 .032 47 -724 47 .889 2 o .003 o .005 3 58 -771 08 .627 18 .483 28 .340 38 .196 48 .053 3 o .008 4 58 -935 08 .791 18 .648 28 .504 38 .361 48 .217 4 .Oil 5 59 -099 08 .956 18 .812 28 .668 38 025 48 .381 5 o .014 6 59 -263 09 . 12O 18 .976 28 .833 3 8 .689 48 .546 6 o .016 7 59 .428 09 .284 19 .141 28 .997 38 -854 48 .710 7 o .019 8 59 -592 09 .448 19 -305 29 .161 39 -018 48 .874 8 o .022 9 59 -756 09 .613 19 .469 29 .326 39 182 49 -039 9 o .025 10 59 -920 09 .777 19 .633 29 .490 39 -346 49 .203 10 o .027 it 12 oo .085 oo .249 09 .941 10 .105 19 .798 19 .962 29 -654 29 .818 39 -5" 39 -675 49 -367 49 -S3 1 n 12 o .030 -33 13 oo .413 10 .270 20 .126 29 .983 39 -839 49 .696 13 o .036 14 oo .578 10 .434 2O .290 3 -!47 40 .003 49 .860 14 o .038 IS oo .742 10 .598 20 .455 30 -3" 40 .168 50 .024 15 o .041 16 oo .906 10 .763 2O .619 30 .476 4 -332 50 .188 16 o .044 17 oi .070 10 .927 20 .783 30 .640 40 .496 5 '353 17 o .047 18 oi .235 II .091 20 .948 30 .804 40 .661 50 .517 18 o .049 19 oi .399 " -255 21 .112 30 .968 40 .825 50 .681 T 9 o .052 20 oi .563 II .420 21 .276 3 -133 40 .989 50 .846 20 -055 21 oi .727 II .584 21 .440 3i -297 4 1 -153 51 .010 21 o .057 22 oi .892 II .748 21 .605 31 .461 41 .318 51 .174 22 o .060 23 02 .056 II .912 21 .769 31 -625 41 .482 5i .338 2 3 o .063 24 02 .220 12 .077 21 -933 3 1 -79 41 -646 5 1 -53 2 4 o .066 25 02 .385 12 .241 22 .098 3i -954 41 .810 5 i -667 25 o .068 26 02 .549 12 .405 22 .262 32 .118 4i -975 51 .831 26 o .071 27 02 .713 12 .570 22 .426 32 .283 42 .139 5 1 -995 27 o .074 28 02 .877 12 .734 22 .590 32 -447 42 .303 52 .160 28 o .077 29 03 .042 12 .898 22 -755 32 .611 42 .468 52 .324 2 9 o .079 3 03 .206 13 .062 22 .919 32 -775 42 .632 52 .488 30 o .082 3 .370 13 -227 23 .083 32 .940 42 .796 S 2 .653 o .085 S 2 03 -534 13 -391 23 -247 33 -104 42 .960 52 .817 32 o .088 33 03 .699 13 -555 23 .412 33 -268 43 -125 52 .981 33 o .090 34 03 .863 13 -720 23 -576 33 -43 3 43 -289 53 -MS 34 o .093 35 04 .027 13 -884 23 -740 33 -597 43 -453 53 -310 35 o .096 3 6 04 .192 14 .048 2 3 -905 33 -761 43 - 6l 7 53 -474 36 o .099 37 04 .356 14 .212 24 .069 33 -925 43 -782 S3 -638 37 o .101 38 04 .520 M -377 24 -233 34 090 43 -946 53 -803 38 o .104 39 04 .684 14 -54' 24 -397 34 -254 44 '"o 53 -967 39 o .107 4 04 .849 M -70S 24 -562 34 -418 44 -275 54 -!3i 40 o .no 4 1 05 .013 14 .869 24 .726 34 -582 44 -439 54 -295 41 .112 42 5 -77 '5 -034 2 4 .890 34 -747 44 -603 54 -460 42 o .115 43 05 -342 15 -198 25 .054 34 -9" 44 -7 6 7 54 -624 43 o .118 44 05 .506 15 .362 25 .219 35 -075 44 -932 54 -788 44 .120 45 05 .670 *5 -527 25 .383 35 -239 45 -096 54 -952 45 o .123 46 05 .834 15 .691 25 -547 35 -404 45 .260 55 -"7 46 o .126 47 05 .999 15 -855 25 -712 35 .568 45 -425 55 -281 47 o .129 48 06 .163 16 .019 25 .876 35 -732 45 -589 55 -445 48 o .131 49 06 .327 16 .184 26 .040 35 -897 45 -753 55 -610 49 o .134 5 06 .491 16 .348 26 .204 36 .061 45 -9'7 55 -774 5 o . 137 06 .656 16 . 5 !2 26 .369 36 .225 46 .082 55 .938 o .140 S 2 06 .820 16 .676 26 .533 36 .389 46 .246 50 .102 5 2 o .142 53 06 .984 16 .841 26 .697 36 -554 46 .410 56 -267 53 o .145 54 07 .149 17 .005 26 .861 36 .718 46 -574 56 .431 54 o .148 55 07 -313 17 .169 27 .026 36 .882 46 -739 56 -595 55 o .151 56 57 07 .477 07 .641 17 -334 17 .498 27 .190 27 -354 37 -047 37 -211 46 .903 47 -067 56 -759 56 .924 56 57 o -153 o .156 58 07 .806 17 .662 27 -5*9 37 -375 47 -233 57 .088 58 o .159 59 07 .970 17 .826 27 .683 37 -539 47 -396 57 -252 59 o . 162 Mean .h F 'or Solar. I2h i3 h 14 I5 h 16 1? h Sec onds. 282 GEODETIC ASTRONOMY. 2 9 0. CONVERSION OF MEAN SOLAR TIME INTO SIDEREAL. Correction to be added to a mean solar interval to obtain the corresponding sidereal interval. Mean h For Solar. i8 h 19" 2O h 2I h 22 n 23 h Seconds. o m 2 m 57* -4^7 3 m 07.273 3 m 17*. 129 3 m 26". 986 3 m 36'.8 4 2 3"" 46'. 699 o- o'.ooo i 57 -581 3 7 -437 3 17 -294 3 27 .150 3 37 -007 3 46 -863 i o .003 2 57 -745 3 07 .602 3 17 .458 3 27 .315 3 37 -171 3 47 ^027 2 o .005 3 57 -99 3 7 -766 3 17 .622 3 27 .479 3 37 -335 3 47 .i9 2 3 o .008 4 5 8 -074 3 7 -93 3 17 -787 3 27 .643 3 37 -500 3 47 .356 4 o .on 5 58 .238 3 08 .094 3 17 -95 1 3 27 .807 3 37 -664 3 47 -520 5 o .014 6 58 .402 3 8 -259 3 18.115 3 27 .972 3 37 -828 3 47 -685 6 o .016 7 5 8 .566 3 08 .423 3 18 .279 3 28 .136 3 37 -992 3 47 -849 7 o .019 8 58 .731 3 8 .587 3 18 .444 3 28 .300 3 38 .157 3 48 .013 8 o .022 9 58 .895 3 8 -75 r 3 18 .608 3 28 .464 3 38 -321 3 48.177 9 o .025 10 59 .59 3 08 .916 3 18 .772 3 28 .629 3 38 -485 3 48 .342 10 o .027 it 59 -224 3 09 .080 2 18 .937 3 28 .793 3 38 -649 3 48 .506 ii o .030 12 59 -388 3 09 .244 3 19 - 101 3 28 .957 3 38 -814 3 48 .670 12 o .033 13 59 -552 3 09 .409 3 19 -265 3 29.122 3 38 .978 3 48 .834 J 3 o .036 14 59 .7 J 6 3 09 -573 3 19 -429 3 29 .286 3 39 -142 3 48 -999 14 o .038 15 59 -881 3 9 -737 3 iQ -594 3 29 .450 3 39 -37 3 49 -*63 15 o .041 16 3 oo .045 3 09 ,901 3 19 .758 3 29 .614 3 39 -47 1 3 49 -327 16 o .044 T 7 3 oo .209 3 10 .066 3 19 -922 3 29 .779 3 39 -635 3 49 -492 J 7 o .047 It 3 oo .373 3 10 .230 3 20 .086 3 29 .943 3 39 -799 3 49 -656 18 o .049 19 3 oo .538 3 I0 -394 3 20 .251 3 30.107 3 39 -964 3 49 -820 '9 o .052 20 3 oo .702 3 1 -559 3 20 .415 3 30 .271 3 40 .128 3 49 -984 20 o .055 21 3 oo .866 3 I0 -723 3 20 .579 3 30 -43 6 3 40 .292 3 50 .149 21 o .057 22 3 o .031 3 10 .887 3 20 .744 3 30 .600 3 40 .456 3 50 .313 22 o .060 2 3 3 o .195 3 1J -051 3 20 .908 3 30 .764 3 40 .621 3 5 -477 2 3 o .063 2 4 3 o -359 3 ii .216 3 21 .072 3 3 -929 3 4 -785 3 50 .642 2 4 o .066 25 3 o -5 2 3 3 -380 3 21 .236 3 3i -093 3 4 -949 3 50 .806 25 o .068 26 3 o .688 3 Ix -544 3 21 .401 3 3 1 - 2 57 3 4 1 1 i4 3 5 -970 26 o .071 27 3 o .852 3 " -70S 3 21 .565 3 3i -421 3 4i -278 3 5i .134 27 o .074 28 3 02 .016 3 " -873 3 21 .729 3 3i -586 3 4i .442 3 5i -299 28 o .077 29 3 02 .181 3 .037 3 21 .893 3 3i -750 3 41 .606 3 5i -463 29 o .079 30 3 02 .345 3 12 .201 3 22 .058 3 3i -9M 3 4i -771 3 5i -627 30 o .082 3 1 3 02 .509 3 .366 3 22 .222 3 32 -078 3 4 1 -935 3 5* -791 31 o .085 32 3 2 -673 3 " -530 3 22 .386 3 3 2 .243 3 42 .099 3 5i .956 32 o .088 33 3 02 .838 3 I2 .694 3 22 .551 3 32 -407 3 4 3 -264 3 52 .120 33 o ,090 34 3 03 .002 3 12 .858 3 22 .715 3 32 -57 1 3 42 .428 3 S 2 .284 34 o .093 35 3 03 .166 3 '3 -023 3 22 .879 3 32 .736 3 4 2 .592 3 S 2 -449 35 o .096 36 3 03 .330 3 *3 -187 3 23 .043 3 3 2 .9 3 42 .756 3 52 .613 36 o .099 37 3 03 .495 3 X 3 -SSI 3 23 .208 3 33 -064 3 42 .921 3 52 -777 37 O . IOI 38 3 03 .659 3 3 -515 3 23 .372 3 33 -228 3 43 .085 3 52 .941 38 o .104 39 3 03 .823 3 13 -680 3 23 .536 3 33 -393 3 43 .249 3 53 106 39 o .107 40 3 03 .988 3 13 -844 3 23 -700 3 33 -557 3 43 -4^3 3 53 -270 40 o . no 4 1 3 04 .152 3 14 .008 3 23 .865 3 33 -72i 3 43 -57 8 3 53 -434 4i .112 4* 3 04 .316 3 14 -173 3 24 .029 3 33 -886 3 43 -742 3 53 .5Q8 42 o .115 43 3 04 .480 3 M -337 3 24 .193 3 34 -050 3 43 -96 3 53 -763 43 o .118 44 3 04 .645 3 J 4 -501 3 24 .358 3 34 -2H 3 44 -07 1 3 53 -927 44 .120 45 3 04 .809 3 *4 -665 3 24 .522 3 34 .378 3 44 -235 3 54 '9 l 45 o .123 46 3 4 -973 3 M -830 3 24 .686 3 34 -543 3 44 -399 3 54 .256 46 o .126 47 3 05 . 137 3 4 -994 3 24 .850 3 34 .707 3 44 -5 6 3 3 54 -420 47 o .129 48 3 05 .302 3 15 -158 3 25 .015 3 34 -87* 3 44 -728 3 54 -584 48 o .131 49 3 05 .466 3 ^5 -322 3 25 .179 3 35 -035 3 44 -892 3 54 .748 49 o .134 50 3 05 .630 3 J 5 -487 3 25 .343 3 35 .200 3 45 -056 3 54 -913 50 o .137 Si 3 5 -795 3 *5 -651 3 25 -508 3 35 -364 3 45 -220 3 55 -77 Si o .140 5 2 3 05 .959 3 15 -815 3 25 .672 3 35 -528 3 45 .385 3 55 -24* 52 o . 142 53 3 6 .123 3 15 -980 3 25 .836 3 35 -^93 3 45 -549 3 55 -45 53 o .145 54 3 06 .287 3 16.144 3 26 .000 3 35 -857 3 45 -713 3 55 -570 54 o .148 55 3 06 .452 3 16 .308 3 26 .165 3 36 .021 3 45 -878 3 55 -734 it o .151 56 3 06 .616 3 16 -472 3 26 .329 3 36.185 3 46 .042 3 55 -898 56 .153 57 3 06 .780 3 6 -637 3 26 -493 3 36 -350 3 46 .206 3 56 -063 57 o .156 58 59 3 06 .944 3 07 .109 3 16 .801 3 1 6 .965 3 26 .657 3 26 .822 3 3 6 -5*4 3 36 .678 3 4 6 -370 3 46 -535 3 56 -227 3 56 .391 58 59 o .159 o . 162 Mean Solar. 18" i9 h 20" It* 22" 23" For Seconds. 24 mean solar hours = 24 h o3 m s6 .555 of sidereal time. 291, CONVERSION TABLES. 28 3 291. CONVERSION OF SIDEREAL TIME INTO MEAN SOLAR. Correction to be subtracted from a sidereal interval to obtain the corresponding mean time interval. (See 23.) Sid. O h I h 2 h ,h .h -h For 3 4 5 Seconds. o" o m oo.ooo o m 09*. 830 o m i9 8 . 659 o 111 29. 489 0-39'- 3i8 o 49*. 148 o o'.ooo i 2 o oo .164 o oo .328 o 09 .993 o 10 .157 o 19 .823 o 19 -98.7 o 29 .653 o 29 .816 o 39 .482 o 39 .646 o 49 .312 o 49 -475 i 2 o .003 o .005 3 o oo .491 o 10 .321 20 .151 o 29 .980 o 39 .810 o 49 .639 3 o .008 4 o oo .655 o 10 .485 20 .314 o 30 .144 o 39 -974 o 49 .803 4 .Oil 5 o oo .819 o 10 .649 o 20 .478 o 30 .308 o 40 .137 o 49 .967 5 o .014 6 7 o oo .983 o o .147 o 10 .813 o 10 .976 20 .642 o 20 .806 o 30 .472 o 30 .635 o 40 .301 o 40 .465 o 50 .131 o 50 .295 6 7 o .016 o .019 8 o o .311 o ii .140 20 .970 o 30 ,799 o 40 .629 o 50 .458 8 o .022 9 o o .474 o ii .304 21 .134 o 30 .963 o 40 .793 o 50 .622 9 o .025 o o o .638 II .468 O 21 .297 o 31 .127 o 40 .956 o 50 .786 10 o .027 i . 802 o ii .632 21 .461 o 31 .291 41 .120 o 50 .950 ii o .030 2 o o .966 o ii .795 21 .625 o 3 1 -455 o 41 .284 o 51 .114 12 o .033 3 o 02 . 1 30 o ii .959 O 21 .789 o 31 .618 o 41 .448 o 51 .278 13 o .035 4 02 .294 12 .123 21 .953 o 31 -782 o 41 .612 o 51 .441 14 o .038 5 02 .457 12 .287 O 22 .H7 o 31 .946 o 41 .776 o 51 .605 15 o .041 6 02 .621 12 .451 O 22 .280 o 32 .no o 41 .939 o 51 .769 16 o .044 7 02 .785 12 .615 22 .444 o 32 .274 o 42 .103 5i '933 *7 o .046 8 02 .949 12 .778 22 .608 o 32 .438 o 42 .267 o 52 .097 18 o .049 9 o 03 .113 12 .942 22 .772 o 32 .601 o 42 .431 o 52 .260 19 .052 20 o 03 .277 o 13 .106 22 .936 o 32 .765 o 42 .595 o 52 .424 20 055 21 o 03 .440 o 13 .270 o 23 .099 o 32 .929 o 42 .759 o 52 588 21 .057 22 o 03 .604 o 13 -434 o 23 .263 o 33 -93 o 42 .922 o 52 .752 22 .060 23 o 03 .768 o 13 .598 o 23 .427 o 33 .257 o 43 .086 o 52 .916 23 .063 24 o 03 .932 o 13 .761 o 23 .591 o 33 =420 o 43 .250 o 53 .080 24 .066 25 o 04 .096 o 13 .925 o 23 .755 o 33 -584 o 43 -4H o 53 - 2 43 25 .068 26 o 04 .259 o 14 .089 o 23 .919 o 33 .748 o 43 -578 o 53 -4>7 26 .071 27 o 04 .423 o 14 .253 o 24 .082 o 33 .912 o 43 .742 o 53 -571 27 .074 28 o 04 .587 o 14 .417 o 24 .246 o 34 .076 o 43 .905 53 -735 28 .076 29 o 04 .751 o 14 .581 o 24 .410 o 34 .240 o 44 .069 o 53 .899 29 -079 30 o 04 .915 o 14 .744 o 24 .574 o 34 -403 44 -233 o 54 .063 30 082 31 o 05 .079 o 14 .908 o 24 .738 34 -567 o 44 -397 o 54 .226 3* .085 3 2 o 05 .242 o 15 .072 o 24 .902 o 34 -73 1 o 44 .561 o 54 -390 32 .087 33 o 05 .406 o 15 .236 o 25 .065 o 34 -895 o 44 .724 o 54 -554 33 .090 34 o 05 .570 o 15 .400 o 25 .229 o 35 -059 o 44 .888 o 54 .718 34 093 35 o 05 .734 o 15 -5 6 3 o 25 .393 o 35 .223 o 45 .052 o 54 .882 .096 3 6 o 05 .898 o 15 .727 o 25 .557 o 35 .386 o 45 .216 o 55 -046 36 .098 37 o 06 .062 o 15 .891 o 25 .721 o 35 -55 45 '3 8 o 55 .209 37 .101 38 o 06 .225 o 16 .055 o 25 .885 o 35 .714 o 45 -544 o 55 -373 38 .104 39 o 06 .389 o 16 .219 o 26 .048 o 35 .878 o 45 -707 o 55 -537 39 .106 4 o 06 .553 o 16 .383 26 .212 o 36 .042 o 45 .871 o 55 .701 4 .109 4 1 o 06 .717 o 16 .546 o 26 .376 o 36 .206 o 46 .035 o 55 .865 4 1 .112 42 o 06 .881 o 16 ,710 o 26 .540 o 36 .369 o 46 .199 o 56 .028 42 .115 43 o 07 .045 o 16 .874 o 26 . 704 o 36 -533 o 46 .363 o 56 .192 43 .117 44 o 07 ,208 o 17 .038 o 26 .867 o 36 .697 o 46 .527 o 56 .356 44 .120 45 o 07 .372 17 .202 o 27 .031 o 36 .861 o 46 .690 o 56 .520 45 "3 46 o 07 .536 o 17 .366 o 27 .195 o 37 .025 o 46 .854 o 56 .684 46 .126 47 o 07 .700 o 17 .529 o 27 .359 o 37 .188 o 47 .018 o 56 .848 47 .128 48 o 07 .864 o 17 .693 o 27 .523 o 37 -352 o 47 .182 o 57 .on 48 131 49 o 08 .027 o 17 .857 o 27 .687 o 37 -56 o 47 .346 o 57 -175 49 134 50 o 08 .191 o 18 .021 o 27 .850 o 37 .680 o 47 .510 o 57 -339 5 137 5 1 o 08 .355 o 18 .185 o 28 .014 o 37 .844 o 47 .673 o 57 -503 5 1 139 S 2 o 08 .519 o 18 .349 o 28 .178 o 38 .008 o 47 -837 o 57 .667 52 .142 53 o 08 .683 o 18 .512 o 28 .342 o 38 .171 o 48 .001 o 57 .831 53 H5 54 o 08 .847 o 18 .676 o 28 .506 o 38 -335 o 48 .165 o 57 -994 54 .147 $ o 09 .010 o 09 .174 o 18 .840 o 19 .004 o 28 .670 o 28 .833 o 38 .499 o 38 .663 o 48 .329 o 48 .492 o 58 .158 o 58 .322 II .150 T 53 57 o 09 .338 o 19 .168 o 28 .997 o 38 .827 o 48 .656 o 58 .486 57 .156 58 o 09 .502 o 19 .331 o 29 . 161 o 38 .991 o 48 .820 o 58 .650 58 .158 59 o 09 .666 o 19 .495 o 29 .325 o 39 .154 o 48 .984 o 58 .814 59 . 161 Sid. o" I* 2 h -,h 4 h 5" For Seconds. 284 GEODETIC ASTRONOMY. 2 9 I. CONVERSION OF SIDEREAL TIME INTO MEAN SOLAR. Correction to be subtracted from a sidereal interval to obtain the corresponding mean time interval. Sid. 6 h y h 8" 9 h I0 11" For Seconds. Oin o m 58". 977 m o8V8o 7 m iS".6 3 6 ra 28". 466 38'. 296 m 48". 125 o o*.ooo I o 59 .141 08 .971 18 .800 28 .630 38 .459 48 .289 i o .003 2 o 59 -35 09 -135 18 .964 28 .794 38 .623 48 -453 2 .005 3 o 59 .469 09 .298 19 .128 28 .958 38 .787 48 .617 3 o .oo 4 59 - 6 33 09 .462 19 .292 29 .121 38 .951 48 .780 4 O .Oil 5 o 59 .796 09 .626 19 .456 29 .285 39 -"5 48 .944 5 o .014 6 o 59 .960 09 .790 19 .619 29 .449 39 - 2 79 49 .108 6 o .016 7 oo .124 09 -954 19 .783 29 .613 39 -442 49 -272 7 .010 8 oo .288 i .118 19 .947 29 -777 39 .606 49 >4?6 8 o .022 9 00 -45 2 i .281 20 .III 29 .940 39 -770 49 .600 9 o .025 10 oo .616 i .445 20 .275 30 . 104 39 -934 49 -763 10 o .027 ii oo .779 i .609 20 .439 30 .268 40 .098 49 -927 ii o .030 12 oo .943 i -773 2O .602 30 -432 40 .261 50 .09! 12 .033 '3 oi .107 1 937 20 .766 30 -596 40 .425 So .255 13 o .035 *4 oi .271 I .100 2O .930 30 .760 40 .589 50 .419 M o .038 IS oi -435 i .264 21 .094 30 .923 4o -753 50 .483 15 o .041 16 oi .599 i .428 21 .258 3i -087 40 .917 50 .746 16 o .04^ J 7 oi .762 i .569 21 .422 3* -251 41 .081 50 .910 J 7 o .046 18 oi .926 I .756 21 .585 3i -415 41 .244 Si -074 it o .049 *9 02 .090 I .920 21 .749 3i -579 41 .408 5i -238 19 o .052 20 02 .254 I .083 21 .913 3i -743 4i .572 51 .402 20 o .055 21 02 .418 I .247 22 .077 31 -906 4i .736 5i -565 21 o .057 22 02 .582 I .411 22 .241 32 .070 41 .900 5i -729 22 o .oo 2 3 02 -745 * -575 22 .404 32 .234 42 .064 5i .893 2 3 o .06; 24 02 .909 i -739 22 .568 32 .398 42 .227 52 .057 24 o .066 25 03 .073 i .903 22 .732 32 .562 42 .391 52 .221 25 o .068 26 03 -237 13 .066 22 .896 32 .726 42 .555 52 -385 26 o .071 27 03 .401 13 .230 23 .060 32 .889 42 .719 52 .548 2 7 o .07. 28 03 .564 3 -394 23 22 4 33 -053 42 .883 52 .712 28 o .076 29 03 .728 13 588 23 -387 33 -217 43 -47 52 .876 29 o .079 30 03 -892 I 3 .722 23 -55' 33 -381 43 .210 53 -4o 30 o .082 31 04 .056 13 .886 23 -715 33 -545 43 -374 53 -204 3 1 o .085 32 04 .220 14 .049 23 .879 33 -708 43 -538 53 .368 32 o .087 33 04 .384 14 .213 24 .043 33 -872 43 -7 02 53 -531 33 o .090 34 04 -547 T 4 -377 24 .207 34 -036 43 -866 53 -695 34 o 093 35 04 .711 14 -54 1 24 .370 34 -200 44 -029 53 -859 35 o .096 36 04 .875 14 -75 2 4 -534 34 -364 44 -193 54 -23 36 o .098 37 05 .039 14 .868 24 .698 34 -528 44 -357 54 -187 37 o .101 38 05 .203 15 .032 24 .862 34 -691 44 -521 54 -351 38 O . IO^ 39 05 -367 15 .196 25 .026 34 .855 44 -685 54 .5H 39 o .106 40 05 .53 15 .360 25 .190 35 -019 44 -849 54 -678 40 o .109 4 1 05 .694 J 5 -524 2 5 '353 35 -183 45 .012 54 -842 4 1 .112 42 05 -858 15 .688 25 -517 35 -347 45 -176 55 .6 42 o .115 43 06 .022 15 .851 25 .681 35 -5" 45 -340 55 1 7Q 43 o . 117 44 06 .186 16 .015 25 -845 35 -674 45 -504 55 -333 44 .120 45 06 .350 16 .179 26 .009 35 -838 45 -668 55 -497 45 o .123 46 06 .513 1 6 .343 26 .172 36 .002 45 -832 55 -661 46 . I2 57 -299 56 o .153 57 08 .315 18 .145 27 -975 37 -804 47 -634 57 -463 57 o .156 58 08 .479 18 .309 28 .138 37 -968 47 -797 57 -627 58 o .158 59 08 .643 *8 .473 28 .302 38 .132 47 -9 61 57 -79 1 59 o .161 Sid. 6 h 7 h 8 h 9 h I0 h ii> For Seconds. 2 9 I. CONVERSION TABLES. 28 S CONVERSION OF SIDEREAL TIME INTO MEAN SOLAR. Correction to be subtracted from a sidereal interval to obtain the corresponding mean time interval. Sid. nt I3 h i4 h * 1 6" .7" For Seconds. I m 57 -955 58 .119 07 .948 2 m 17*. 614 I 7 .778 m 27'. 443 27 .607 10 37* -273 37 -437 m 4 7*. 102 47 .266 o- i oVooo o .003 2 58 .282 08 .112 17 .941 27 .771 37 -601 47 -400 2 o .005 3 58 .446 08 .276 18 .105 27 -935 37 .764 47 -594 3 o .008 4 58 .610 08 .440 18 .269 28 .099 37 .9*8 47 .758 4 O .Oil 5 58 -774 08 .603 18 .433 28 .263 38 .092 47 .922 5 o .014 6 58 .938 08 .767 18 .597 28 .426 38 .256 48 .085 6 o .016 7 59 i 01 08 .931 18 .761 28 .590 38 .420 48 .249 7 o .019 8 59 -265 09 .095 18 .924 28 .754 38 .584 48 .413 8 .022 9 59 -429 09 .259 19 .088 28 .918 38 .747 48 .577 9 o .025 10 59 -593 09 .423 19 .252 29 .082 38 .911 48 .741 10 o .027 ii 59 -757 9 .586 19 .416 29 -245 39 -075 48 .905 ii o .030 12 59 -921 09 .750 19 .580 29 .409 39 .239 49 .068 12 o .033 13 oo .084 oo .248 09 .914 10 .078 19 -744 19 .907 29 .573 29 -737 38:3 49 -232 49 -396 13 14 o .035 o .038 I e oo .412 10 .242 20 .071 29 .901 39 -730 49 -560 o .041 16 oo .576 10 .405 20 .235 30 .065 39 .894 49 .724 16 o .044 17 oo .740 10 .569 20 .399 30 .228 40 .058 49 .888 17 o .046 18 oo .904 1 -733 20 .563 30 .392 40 .222 50 .051 18 o .049 19 oi .067 10 .897 20 .727 30 .556 4 .386 50 .215 ig o .052 20 oi .231 ii .061 20 .890 30 .720 40 -549 50 -379 20 o .055 21 oi -395 ii .225 21 -054 30 .884 40 .713 50 .543 21 o .057 22 oi -559 ii .388 21 .218 31 .048 40 .877 50 .707 22 o .060 2 3 oi .723 ii -552 21 .382 31 .211 41 .041 50 .870 23 o .063 24 oi .887 ii .716 21 .546 3i -375 41 -205 5i .034 24 o .066 25 02 .050 ii .880 21 .709 3i -539 41 .369 51 .198 25 o .068 26 02 .214 12 .044 21 .873 3i -73 4i .532 51 .362 26 o .071 27 02 .378 12 .208 22 .037 3i .867 41 .696 5i -526 27 o .074 28 02 .542 12 .371 22 .201 32 .031 41 .860 51 .690 28 o .076 29 O2 .706 12 -535 22 .365 32 .194 42 .024 5i .853 29 o .079 30 02 .869 12 .699 22 .529 32 -358 42 .188 52 .017 3 o .082 31 03 -033 12 .863 22 .692 32 .522 42 .352 52 .181 o .085 32 3 -!97 13 -27 22 .856 32 .686 42 .5!5 52 -345 32 o .087 33 03 .361 13 .igi 23 .020 32 .850 42 .679 52 .509 33 o .090 34 35 03 .525 03 .689 13 -354 13 .518 23 .184 23 -348 33 -013 33 .!77 42 .8*3 43 .7 52 .673 52 .836 34 35 o .093 o .096 36 03 .852 13 .682 23 .512 33 .341 43 i7i 53 -00 36 o .098 37 04 .016 13 .846 23 - 6 75 33 -505 43 -334 53 - I(5 4 37 o .101 38 04 .180 14 .010 23 -839 33 -669 43 -498 53 -328 38 o .104 39 04 .344 T 4 -!73 24 -003 33 .833 43 -662 53 -492 39 o . 106 40 04 .508 T 4 -337 24 .167 33 .966 43 -826 53 -656 4 o .109 04 .672 14 .501 2 4 -331 34 .160 43 -990 53 -819 .112 42 04 -835 14 .665 24 -495 34 -324 44 .154 53 -983 42 o . 115 43 04 .999 14 .829 24 .658 34 -488 44 -317 54 -M7 43 o .117 44 05 .163 14 .993 24 .822 34 .652 44 .481 54 -3" 44 .120 45 5 -327 15 .156 24 .986 34 -816 44 .645 54 -475 45 o .123 46 05 .491 15 -320 25 .150 34 -979 44 .809 54 -638 46 o . 126 47 05 -655 15 .484 25 .314 35 .143 44 -973 54 -802 47 o .128 48 05 .818 15 -648 25 -477 35 -307 45 -137 54.966 48 o .131 49 05 .982 15 .812 25 .641 35 -471 45 '3 55 .!30 49 -134 50 06 .146 15 -976 25 .805 35 -635 45 -464 55 -294 50 .137 51 06 .310 16 .139 25 -9 6 9 35 .798 45 -628 55 -458 o .139 52 06 .474 16 .303 26 .133 45 -792 55 -621 S 2 o . 142 53 06 .637 16 .467 26 .297 36 .126 45 -95 6 55 -785 53 o .145 54 06 .801 16 .631 26 ,460 36 .290 46 .120 55 -949 54 o .147 55 06 .965 T <5 -795 26 .624 36 -454 46 .283 56 .113 55 o .150 56 57 07 .129 07 .293 16 -959 17 .122 26 .788 26 .952 36 .618 36 .781 46 .447 46 .611 56 .277 56 .441 56 57 .153 o .156 58 7 -457 17 .286 27 .116 3 6 945 46 -775 56 .604 58 o . 158 59 07 .620 17 .450 27 .280 37 -109 46 -939 56 .768 59 o . 161 Sid. U 13 h 14" :5 h , '7 h For S< conds. 286 GEODETIC ASTRONOMY. 2 9 I, CONVERSION OF SIDEREAL TIME INTO MEAN SOLAR. Correction to be subtracted from a sidereal interval to obtain the corresponding mean time interval. Sid. 18" i9 h 20 h 2l" 22 h 23" For Seconds. o m 2 m 56*. 932 3 m o6. 7 62 3* i6.59i 3-a6. 42 1 3"> 3 6.2 5 o 3 m 46". 080 o'.ooo i 2 57 .096 57 -260 3 6 -925 3 07 .089 3 l6 -755 3 16 -9!9 3 26 .585 3 26 .748 3 36 .4H 3 36 .578 3 46 -244 3 46 .407 2 o .003 o .005 3 57 -424 3 7 -253 3 17 -083 3 26 .912 3 36 .742 3 46 .571 3 o .008 4 57 -587 3 07 -417 3 i7 -246 3 27 .076 3 36 -906 3 46 '735 4 o .on 57 .751 3 07 .581 3 i7 -4*0 3 27 -240 3 37 -069 3 46 -899 5 o .014 g 57 -9*5 3 7 -745 3 17 -574 3 27 .404 3 37 .233 3 47 -063 6 o .016 7 58 .079 3 07 .908 3 i7 -738 3 27 .568 3 37 -397 3 47 -227 7 o .019 8 58 .243 3 08 .072 3 17 -902 3 27 -73i 3 37 .561 3 47 '390 8 .022 9 58 .406 3 08 .236 3 18 .066 3 27 .895 3 37 -725 3 47 -554 9 o .025 10 58 .570 3 08 .400 3 1 8 .229 3 28 .059 3 37 -889 3 47 -7'8 10 o .027 ii 58 .734 3 08 .564 3 18 -393 3 28 .223 3 38 .052 3 47 -882 ii o .030 12 58 .898 3 08 .728 3 *8 .557 3 2 8 -387 3 38 .216 3 48 .046 12 o .033 13 59 .062 3 08 .891 3 18 .721 3 28 .550 3 38 -380 3 48 .210 T 3 o .035 14 59 -226 3 9 -055 3 18 .885 3 28 .714 3 38 -544 3 48 -373 14 o .038 15 59 -389 3 9 -219 3 19 -049 3 28 .878 3 38 .708 3 48 -537 15 o .041 16 59 -553 3 9 -383 3 19 -212 3 29 .042 3 38 -871 3 48 .701 16 o .044 17 59 >7 1 7 3 9 -547 3 19 -37 6 3 29 .206 3 39 -035 3 48 -865 17 o .046 18 59 .881 3 09 .710 3 19 -540 3 2 9 -370 3 39 -'99 3 49 -029 18 o .049 *9 3 oo .045 3 09 .874 3 i9 -74 3 29 .533 3 39 -363 3 49 -193 *9 o .052 20 3 oo .209 3 10 .038 3 19 -868 3 29 .697 3 39 -527 3 49 '35 6 20 -055 21 3 oo .372 3 10 .202 3 20 .032 3 29 .861 3 39 -691 3 49 -520 21 -057 22 3 oo .536 3 10 .366 3 20 .195 3 3o .025 3 39 -854 3 49 -684 22 o .oo 23 3 oo .700 3 10 .530 3 20 .359 3 30 .189 3 40 .018 3 49 -848 2 3 o .063 24 3 oo .864 3 10 - 6 93 3 20 .523 3 30 -353 3 40.182 3 50 .012 2 4 o .066 2 5 3 01 .028 3 10 .857 3 20 .687 3 30 .5J6 3 40 .346 3 5 . J 75 25 o .068 26 3 01 .192 3 ii .021 3 20 .851 3 30 .680 3 4 .5 3 50 -339 26 o .071 27 3 OI -355 3 ii .185 3 21 .014 3 3o .844 3 40 .674 3 50 -503 27 o .074 28 3 oi .5*9 3 " -349 3 21 .178 % 3 31 .008 3 40 .837 3 5 - 66 7 28 o .076 29 3 oi .683 3 " 5 1 3 3 21 .342 3 3i -'72 3 41 .001 3 50 .831 29 o .079 3 3 oi .847 3 ii .676 3 21 .506 3 3i .336 3 4 1 -'OS 3 50 -995 30 o .082 3 1 3 02 .on 3 ii -84 3 21 .670 3 3' -499 3 4i -329 3 5i -158 31 o .085 3 2 3 2 .174 3 12 .004 3 21 .834 3 3i - 66 3 3 4i -493 3 5i -322 32 o .087 33 3 2 .338 3 12 .168 3 21 .997 3 3i -827 3 4 1 - 6 57 3 5i -486 33 o .090 34 3 02 .502 3 12 .332 3 22 .l6l 3 3i -99* 3 41 .820 3 51 -650 34 o .093 35 3 02 .666 3 12 .496 3 22 ; 3 2 5 3 32 .155 3 4i -984 3 5 T -814 35 o .096 36 3 02 .830 3 I2 .659 3 22 .489 3 22 .653 3 32 -318 3 32 .482 3 42 .148 3 5i -978 36 o .098 38 3 02 .994 3 3 .57 3 12 .823 3 I2 .987 3 22 .817 3 32 .046 3 4 2 .476 3 52 -305 38 o .104 39 3 03 .321 3 13 -151 3 22 .980 3 32 .810 3 42 .639 3 S 2 -469 39 o . 106 4 3 3 -485 3 J 3 -? 1 5 3 23 .144 3 32 -974 3 42 .803 3 52 .633 40 o .109 41 3 03 .649 3 13 -478 3 2 3 .308 3 33 -'33 3 42 -967 3 52 -797 4i .112 42 3 03 .813 3 13 -642 3 23 .472 3 33 -301 3 43 -131 3 S 2 .9i 42 o . 115 43 3 3 -977 3 i3 -806 3 23 .636 3 33 -465 3 43 -295 3 53 .124 43 o .117 44 3 04 .140 3 13 -970 3 23 .800 3 33 -29 3 43 -459 3 53 -288 44 O . I2O 45 3 4 .304 3 14 - X 34 3 23 .963 3 33 -793 3 43 -622 3 53 -452 45 .121 46 3 04 .468 3 14 .298 3 2 4 .127 3 33 -957 3 43 -786 3 53 -616 46 o .126 47 3 04 .632 3 14 -461 3 24 .291 3 34 -"I 3 43 -950 3 53 -780 47 o . 128 48 3 4 .796 3 T 4 -6*5 3 24 .455 3 34 -284 3 44 -"4 3 53 -943 48 o .131 49 3 04 .960 3 14 -789 3 24 .619 3 34 -448 3 44 -278 3 54 -i7 49 o .134 5 3 5 -123 3 T 4 -953 3 24 .782 3 34 -612 3 44 .442 3 54 -271 50 .137 51 3 5 .287 3 15 -"7 3 24 .946 3 34 -776 3 44 -605 3 54 -435 5i o .139 S 2 3 05 .451 3 15 -281 3 25 .no 3 34 -940 3 44 -769 3 54 -599 52 o .142 53 3 5 .615 3 15 -444 3 2 5 -274 3 35 - 10 4 3 44 -933 3 54 -763 53 .145 54 3 5 -779 3 15 .608 3 2 5 .438 3 35 -267 3 45 -097 3 54 -926 54 o .147 55 3 5 .942 3 15 -772 3 25 .602 3 35 .43i 3 45 .261 3 55 -090 55 o .150 5 6 3 06 .106 3 15 .936 3 2 5 -7^5 3 35 -595 3 45 -425 3 55 .254 56 .153 57 3 06 .270 3 16 .100 3 2 5 -929 3 35 -759 3 45 .588 3 55 '4i8 57 o . 156 58 59 3 6 -434 3 06 -598 3 16 .264 3 l6 -427 3 26 .093 3 26 .257 3 35 -923 3 36 .086 3 45 -752 3 45 -9i6 3 55 -582 3 55 -746 58 59 o .158 o .161 Sid. 18" 19" 20" 2t h 22 h 1 23 h For Seconds. 24 sidereal hours = 24* -[3* ssVgog] of mean solar time. 29*. TABLES. 287 292. CHANGE PER YEAR IN THE ANNUAL PRECESSION IN DECLINATION. In units of the fifth decimal place (unit = o".ooooi). (da. m ~\ ) is the annual variation in right ascension. (See 42.) da m Right Ascension ( w ). dt oh com oh 20* oh 4om jh oom ih 20 m ih 40 m 2 h oom 2 h 20 m 2 h 40 m s . 9 - 9 - 8 - 8 -8 - 8 -8 - 7 - 7 .O 9 21 34 46 58 6 9 80 91 IOO . I 9 23 36 So 63 76 88 99 110 .2 9 24 38 53 68 82 95 107 119 3 9 25 41 57 73 88 102 116 128 4 9 26 44 61 78 94 no 124 138 5 9 28 46 65 83 IOO 117 132 147 .6 9 29 49 69 88 1 06 124 141 156 7 9 30 51 72 93 112 131 149 1 66 .8 9 31 54 76 98 119 139 158 175 9 9 33 56 80 103 125 146 166 185 2 .0 9 34 59 84 1 08 131 153 174 194 2 .1 9 35 62 88 "3 137 161 183 203 2 .2 9 36 64 91 118 143 168 191 213 2 -3 9 38 67 95 123 150 175 199 222 2 .4 9 39 69 99 128 156 182 208 232 2 -5 9 40 72 103 133 162 190 216 241 2 .6 9 42 74 106 138 168 197 224 2 5 2 .7 9 43 77 no 143 174 204 233 260 2 .8 9 44 79 114 148 180 212 241 269 2 .9 9 46 82 118 153 187 219 250 2 7 8 3 -o 9 47 84 121 158 193 226 258 288 3 -i 9 48 87 125 163 199 234 266 297 3 -2 9 49 89 129 168 205 2 4 I 275 306 3 -3 9 51 92 133 173 211 248 283 316 3 -4 9 52 94 136 178 218 255 291 325 3 -5 9 53 97 140 183 224 26 3 300 335 3-6 9 54 99 144 188 230 270 308 344 3 -7 9 56 102 148 193 236 277 316 353 3.8 9 57 104 151 197 242 28 4 325 363 3 -9 9 58 107 155 202 248 292 333 372 4 .0 9 60 no 159 207 254 299 342 38i 5 -o 9 72 135 197 257 316 372 425 475 6 .0 9 85 1 60 234 307 377 445 509 569 7 -o 9 98 186 272 357 439 5i8 593 663 8 .0 -9 no 211 -310 - 407 - 501 - 591 -676 -756 , 2 h oom I2 h ao m i ah 40"! I3 h oo , 3 h 20! I 3 h 40 m I 4 h oom I 4 h 20 m I 4 h 40 m Right Ascension (a m ). Change all signs when using this lower argument. 288 GEODETIC ASTRONOMY. 2 9 2. CHANGE PER YEAR IN THE ANNUAL PRECESSION IN DECLINATION. In units of the fifth decimal place (unit = c/'.ooooi). The side argument (r is the annual variation in right ascension. dm Right Ascension (a w ). dt 3 h oom 3 h 2o m 3 h 4 m 4 h oom 4 h 2om 4 h -jam 5 h oom 5^ 2om 5 h 40 in O S .O -6 - 6 - 5 -4 -4 -3 2 2 I .0 109 117 124 131 I 3 6 140 143 145 I 4 6 .1 120 128 136 143 149 154 157 1 60 161 .2 130 I4O 148 156 162 167 171 174 175 3 140 151 160 168 I 7 6 181 186 188 190 4 ISO l62 172 181 189 195 200 203 204 .5 161 173 184 194 202 208 214 217 219 .6 171 184 iq6 206 215 222 228 231 233 .7 181 195 208 219 228 2 3 6 242 246 248 .8 192 206 220 232 242 250 256 260 262 9 202 218 232 244 255 263 270 274 277 2 .0 212 229 244 257 268 277 284 289 291 2 .1 222 240 2 5 6 270 281 291 298 303 306 2 .2 233 251 268 282 294 304 312 317 320 2 -3 243 262 280 295 308 3i8 326 332 335 2 .4 254 274 292 307 321 332 340 346 349 2 .5 264 285 304 320 334 346 355 361 364 2 .6 274 296 316 333 347 359 369 375 379 2 .7 284 37 327 345 361 373 383 389 393 2 .8 295 318 339 358 374 387 397 404 408 2 .9 305 330 351 370 387 400 411 418 422 3 -o 315 340 363 383 400 414 425 433 437 3 .1 326 352 375 396 414 428 439 447 45i 3 -2 336 363 387 408 426 441 453 461 466 3 -3 346 374 399 421 440 455 467 475 480 3 4 357 385 4ii 434 453 469 481 490 495 3 -5 367 396 423 446 466 482 495 504 509 36 377 408 435 459 480 496 509 5i8 524 3 -7 388 419 447 472 493 5io 524 533 538 3-8 398 430 459 484 506 524 538 547 553 3 -9 408 441 47i 497 519 537 552 562 567 4 .0 418 452 483 5io 532 55i 566 576 582 5 .0 522 564 602 636 665 688 707 720 727 6 .0 625 676 722 762 797 825 847 863 872 7 -o 728 787 841 888 929 962 989 1007 1018 8 .0 -831 -899 -961 - 1015 - 1061 - 1099 1129 - 1150 1163 \^ 00 I5h 2om I 5 h 40 m i6h ocm 16^ 2om i6h 40111 17^ oora I 7 h 20 m I 7 h 40 m Right Ascension (a w .) Change all signs when using this lower argument. TABLES. 289 CHANGE PER YEAR IN THE ANNUAL PRECESSION IN DECLINATION. In units of the fifth decimal place (unit = o".ooooi). (da- m \ J is the annual variation in right ascension. *m Right , \scensior 1 (am). dt 6h om 6h 2om 6h 4om 7 h oom yh 2om 7 h 40 m gh oom gh 2om 8h 40 O S .O + 1 + 2 + 2 + 3 + 4 + 4 + 5 + 6 .O 146 144 - 142 -139 - 134 -128 122 - H5 - 106 .1 160 159 156 153 148 142 135 127 117 .2 175 174 171 167 161 155 147 139 129 3 190 188 185 181 175 168 1 60 150 140 4 204 203 2OO 195 189 181 172 162 151 5 219 217 214 209 202 195 I8 5 174 162 .6 233 232 228 223 216 208 I 9 8 1 86 173 7 248 246 243 237 230 221 210 198 184 .8 262 261 257 251 244 234 223 210 196 9 277 275 271 265 257 247 2 3 6 222 207 2 .0 292 290 286 280 271 26l 248 234 218 2 .1 306 304 300 294 285 274 26l 2 4 6 229 2 .2 321 319 315 308 298 287 274 258 240 2 -3 335 333 329 322 312 300 286 270 251 2 .4 350 348 343 336 326 314 299 282 263 2 -5 364 362 358 350 340 327 3" 294 274 2 .6 379 377 372 364 353 340 324 306 285 2 -7 394 39i 386 378 367 353 337 318 296 2 .8 408 406 401 392 38i 366 349 330 308 2 -9 423 420 415 406 394 380 362 342 319 3 -o 437 435 429 420 408 393 375 354 330 3 -i 452 450 444 434 422 406 387 366 34i 3 -2 467 464 458 449 435 419 400 378 352 3 -3 481 478 472 463 449 432 412 389 363 3 -4 496 493 487 477 463 446 425 401 374 3 -5 5io 508 501 491 477 459 438 413 385 3-6 525 522 515 505 490 472 450 425 397 3 -7 540 537 530 519 504 485 463 437 408 3 -8 554 55i 544 533 5i8 499 476 449 419 3 -9 569 566 559 547 531 512 488 461 430 4 .0 583 580 573 56i 545 525 5oi 473 441 5 -o 729 726 717 702 682 657 627 593 553 6 .0 875 871 860 843 819 790 754 712 665 7 .0 1021 1016 1004 984 956 922 880 831 776 8 .0 - 1166 1161 114? 1125 1093 - 1054 1006 -951 - 888 i8fe oom IS* 2on> i8 h 40^ I9h oom igh 2om igh 4 o"i 2oh oom 2oh 2om 2oh 4001 Right Ascensio 1 (>*). Change all signs when using this lower argument. 290 GEODETIC ASTRONOMY. 2 9 2. CHANGE PER YEAR IN THE ANNUAL PRECESSION IN DECLINATION. In units of the fifth decimal place (unit = o".ooooi). fd "5 l V fll J5 9 ".o 9".o 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 -i 7 -i 7 .0 54 39 7 -o 7 -o 6 .9 6 .9 6 .8 6 .8 6 .8 5i 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 -i 5 -r 36 56 5 -o 5 -o 5 -o 5 -o 4 -9 4 .9 4 .9 34 58 4 .8 4 .8 4 -7 4 -7 4 -7 4 .6 4 .6 32 60 4 -5 4 -5 4 -5 4 -4 4 -4 4 -4 4 -4 30 62 4 .2 4 .2 4 .2 4 .2 4 -i 4 -i 4 -i 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 -i 3 -i 3 -i 3 -o 3 -o 3 -o 3 -o 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 I .q I .9 I .9 I .8 i .8 i .8 i .8 12 80 I .6 I .6 I .6 i -5 i -5 i -5 i -5 10 82 I .2 I .2 I .2 I .2 I .2 I .2 I .2 8 84 o .9 o .9 o .9 o .9 o .9 o .9 o .9 6 86 o .6 o .6 o .6 o .6 o .6 o .6 o .6 4 88 o .3 o -3 o -3 o -3 o -3 o -3 o -3 2 90 .0 .0 o .0 .0 .0 .0 .0 292 GEODETIC ASTRONOMY. 294. 294. MEAN REFRACTION (& M ) BAROMETER 760 MILLIMETERS = 29.9 INCHES. Temperature 10 C. = 50 F. (See 68.) Altitude. Mean Refraction. & ! % u Altitude. Mean Refraction. Change per Minute. Altitude. Mean Refraction. \\ 2 u Altitude. Mean Refraction. Change per Minute. ooo 34' 08". n".66 7 oo 7' 24". o".95 19 oo ' 47"-6 o".i 33 oo ' 29". o".o6 10 32 15 -9 10 .88 10 7 14 -9 o .91 20 44 '6 .1 20 28 . o .06 20 30 31 10 .10 20 7 06 .0 o .88 40 41 .6 O . Ij 40 27 o .05 3 28 53 -9 9 -64 3 6 57 -4 o .84 20 00 38 -7 O .1* 34 oo 26 . o .05 40 27 18 .2 9 .20 40 6 49 .1 o .81 20 35 -9 .!< 20 25 .0 o .05 5 25 49 .8 8 .50 50 6 41 .2 o .78 40 33 -2 o .i; 40 24 .0 o .0; I 00 24 28 .3 7 .82 8 oo 6 33 -5 o .76 21 00 30 .6 O .1; 35 oo 23 .0 o .05 10 2 3 13 -5 7 -!7 10 6 26 .0 o .73 2O 28 . o . i; 20 22 .0 o .0; 20 22 4 . 9 6 .58 20 6 18 .9 o .70 4 2 25 .6 .12 40 I 21 .O o .05 3 21 01 .8 6 .06 3 6 12 .0 o .68 22 00 2 2 3 .2 .12 36 oo I 20 .0 o .05 40 20 03 .7 5 -60 40 6 05 .3 o .66 2O 2 2O .9 .12 30 i 18 .5 o .05 So 9 09 .8 5 -20 50 5 58 -9 o .63 4 2 18 .6 O . II 37 oo I 17 . .0, 2 00 8 19 .7 4 -84 9 oo 5 52 .7 o .61 23 oo 2 16 .4 .11 30 I IS -7 o .04 10 7 33 -I 4 -So 20 5 40 .8 o .58 20 2 I 4 .2 O .11 38 oo I 14 .4 o .0. 20 6 49 .7 4 .18 40 5 29 .7 o -54 40 2 12 .1 .10 30 I 13 -I O ,O< 3 6 09 .5 3 -88 10 00 19 .2 o .51 24 oo 2 IO .1 .10 39 oo I II .i O .04 40 5 32 .1 3 -62 20 09 .4 o .48 20 2 08 .1 o .10 30 I IO .5 .0, 5 4 57 -i 3 -39 40 00 .1 o .46 40 2 06 .1 .10 40 oo I 09 .3 .0. 3 oo 4 24 .3 3 -18 II 00 51 -2 o -43 25 oo 04 .2 o .09 30 I 08 .1 o .04 10 3 53 -6 2 .98 20 42 .8 o .40 20 02 .4 o .09 41 oo I 06 .9 .0. 20 3 24 .8 2 .79 40 35 -o o .38 40 oo .6 O .O( 30 I 05 .7 .0,, 30 2 57 -8 2 .6l 2 00 27 -5 o -37 26 oo 58 .8 o .09 42 oo i 04 .6 o .04 40 2 32 -5 2 .46 20 20 .3 o -35 20 57 i o .09 30 03 -5 .04 50 2 08 .7 2 -33 40 13 -5 -33 40 55 -4 o .08 43 oo 02 .i .0* 4 oo I 46 .0 2 .20 3 oo 07 .1 o .32 27 oo 53 -8 o .08 3 01 .3 o .04 10 i 24 .6 2 .09 20 oo .9 o .30 20 5 2 .2 o .08 44 oo OO .2 o .03 20 I 04 o2 I . 9 8 40 55 - 1 o .28 40 50 .6 o .08 30 59 -2 o .0; 3 o 44 .9 i .88 4 oo 49 -5 o .27 28 oo 49 -i o .08 45 oo S 8 .2 o .03 4 o 26 .5 i .79 20 44 -2 o .26 20 47 -6 o .07 30 57 -2 o .03 50 o 09 .1 i .70 40 39 ' o .25 40 46 .1 o .07 46 oo 56 .2 o .03 5 oo 9 52 .6 i .61 5 oo 34 .1 o .24 29 oo 44 -6 o .07 30 55 .2 o .03 10 9 3 6 -9 i -54 20 29 -4 o .23 20 43 -2 o .07 47 oo 54 -2 o .03 20 9 21 .9 i .46 40 24 .8 o .23 40 41 .8 o .07 30 53 -3 o .03 3 9 07 .6 i .40 6 oo 20 .4 O .22 30 oo 40 .5 o .07 48 oo 52 -5 o .03 40 8 54 .0 i -33 20 16 .1 O .21 20 39 -i o .07 30 51 .6 o .03 5 8 41 .0 i .27 40 12 .0 .20 40 37 -8 o .06 49 oo 50 .7 o .03 6 oo 10 8 28 .6 8 16 .7 I .22 I .16 7 oo 20 08 .2 04 -5 o .19 o .19 31 oo 20 3 6 .6 35 -3 o .06 o .06 30 50 oo 49 -8 48 .9 o .03 o .03 20 8 05 .3 I .12 40 oo .9 o .18 40 34 > o 06 30 48 .0 o .03 30 7 54 -3 I .07 8 oo 57 -4 o .17 32 oo 32 .0 o .06 51 oo 47 .2 o .03 40 7 43 -9 I .02 20 54 -o o .17 20 31 .8 o .06 30 46 -3 o .03 5 7 33 -9 o .98 40 5 -7 o .16 40 30 -6 o 06 52 oo 45 -5 o .03 295. TABLES. 293 MEAN REFRAC- TION 295. CORRECTION TO MEAN REFRACTION AS GIVEN IN 294, DEPENDING UPON THE READING OF THE BAROMETER. R = (R M ^ C B)(C D }(C A ). (See 68.) Altitude. Mean Refraction. 1 &; 11 s s 52 30' 53 oo 30 o' 44"- o 43 - o 43 o".03 o .03 o .03 54 oo 3 55 oo o 42 . o 41 . o 40 . o .03 o .03 o .03 , 30 56 oo 57 oo o 40 . 39 o 37 o .03 o .025 o .02, 58 oo 59 oo 60 oo o 36 -4 o 35 -o o 33 .6 o .02; o .02; .022 61 oo 62 oo 63 oo o 32 .3 o 31 .0 o 29 .7 .022 o .022 .022 64 oo 65 oo 66 oo o 28 .4 o 27 .2 o 25 .9 O ,O2I .021 O .O2I 67 oo 68 oo 69 oo o 24 .7 o 23 .6 22 .4 o .020 o .020 .020 70 oo 71 oo 72 oo 21 .2 20 .1 o 18 .9 o .019 o .019 o .019 73 o 74 oo 75 oo o 17 .8 o 16 .7 o 15 .6 o .018 o .018 o .018 76 oo 77 oo 78 oo o 14 -5 o !3 -5 12 .4 o .018 o .018 o .018 79 oo 80 oo 81 oo II .3 10 .3 09 .2 o .018 o .018 o .018 82 oo 83 oo 84 oo 08 .2 07 .2 06 .1 o .018 o .018 o .018 85 oo 86 oo 87 oo 05 .1 04 .1 03 .1 o .018 o .017 o .017 88 oo 89 oo 90 oo 02 .0 01 .0 oo .0 o .017 o .017 o .017 . u'S ai . j 4J + *-> if 8 O w Barome Inches, 11 85 C B *J 00 $a || C B Baromet Inches. Baromet Millime C B 20. o 20. i 20.2 508 5" 0.67 0.67 0.67 24.2 24-3 24.4 615 6i 7 620 0.809 0.81 0.81 28.4 28.5 28.6 721 724 726 0.949 0-953 0.956 20.3 20.4 20.5 5 0.67 0.68 0.68 24-5 24.6 24.7 622 6 25 627 0.82 0.82 0.82 28.7 28.8 28.9 729 732 734 0-959 0.963 0.966 20.6 20.7 20.8 I 0.688 0.692 0.696 24.8 24.9 25.0 6 3 o 632 635 0.82 0.83 0.83 19. o 29.1 29.2 737 739 742 0.970 0.973 0.976 20.9 21.0 53* 533 0.699 0.703 25-1 25.2 637 640 0.83 0.84 29-3 29-4 744 747 0-979 0.983 21. I 536 0.706 25-3 643 0.84 29-5 749 0.986 21.2 21.3 21. 4 538 544 0.709 0.712 0.716 S.I 6 45 6 4 8 650 0.84 0.85 0.85 29.6 29.7 29.8 752 754 757 0.989 0.992 0.996 21-5 21.6 546 549 0.719 0.722 2 5-7 25-8 6 53 655 0.85 0.86 29-9 30.0 759 762 0.999 -1.003 21.7 55i 0.725 25-9 6 5 8 0.86 30.1 765 i .007 21. S 21.9 22.0 554 556 559 0.729 0.732 0-735 26.0 26.1 26.2 660 66 3 66 5 0.869 0.87 0.875 30.2 30.3 30.4 767 770 772 1. 010 1.013 1.016 22.1 22.2 22.3 1 o-739 0.742 0.746 26.3 26.4 26.5 668 671 673 0.879 0.882 0.885 30.5 30.6 30-7 775 777 780 1.020 1.023 1.026 22.4 22.5 22.6 569 572 574 0.749 0.752 0-755 26.6 26.7 26.8 676 678 68 1 0.889 0.892 0.896 30.8 30.9 782 785 787 1.029 1 -33 1.036 22.7 576 0.759 26.O 683 0.800 22.8 579 0.765 27.0 686 v. wyv 0.902 22.9 582 0.766 27.1 688 0.905 23.0 584 0.770 27.2 691 0.909 23.1 587 0-773 27-3 693 0.912 23.2 589 0.776 27.4 696 0.916 23-3 592 0.779 27.5 690 0.920 23-4 594 0.783 27.6 701 0.923 23-5 597 0.786 27.7 704 0.926 2 3 .6 599 0.789 27.8 706 0.929 23-7 2 3 .8 602 605 0.792 0.796 3:1 709 711 0.936 23.9 607 0-799 28.1 7 T 4 o-939 24.0 610 0.803 28.2 716 0.942 24.1 612 0.806 28.3 719 0.946 2 9 4 GEODETIC ASTRONOMY. 297. 296. CORRECTION TO MEAN REFRACTION AS GIVEN IN 294. DEPENDING UPON THE READING OF THE DETACHED THERMOMETER. R = R M (C B )(C D )(C A ). (See 68.) d 14 sg c il Q..J a s CD IN is CD d.- E ag CD H H b 3 H fe 1 3 1.16 i .20 23 25 27 52 39 .00 .02 05 0? no .10 .12 I ' 1 | 1.19 1.22 .26 .28 30 5 1 40 4 1 42 .04 .06 .04 .07 .09 .07 .09 .11 uy . 12 M 17 .20 1.20 1-23 \\ll 1.27 1.30 3* 33 36 33 1 5 49 48 43 .08 .11 T 3 .16 .19 .22 1-25 1.29 1.32 36 39 47 44 .10 13 IS .18 .21 .24 1.28 i-35 39 43 46 45 . 12 .20 23 .26 1.30 i'33 '37 .41 44 46 45 299- TABLES. 301 FACTORS FOR THE REDUCTION OF TRANSIT TIME OBSERVATIONS. This top argument is for the star's declination 6. 5-' 52 53 54 55 56 57 58 59 60 *P 61 4 .14 j>i7 .19 1.22 1-25 1.29 32 1.36 1.40 44 I. 4 6 .48 44 8 .16 .18 .19 .21 .21 23 1:3 1.27 1.30 *-33 $ 1.38 1.40 1.42 '44 3 1.49 1.50 53 43 42 49 .20 23 25 1.28 1.32 39 1.42 1.47 5 1 "S3 56 50 .22 24 27 1.30 i-34 1-37 .41 1.44 1.49 53 1.56 -58 40 5I 23 .26 29 1.32 !. 35 i-39 43 i.47 i.S 55 '58 .60 39 5 2 25 .28 3' i 34 i-37 1.41 45 1.49 1-53 58 i. 60 63 38 53 .27 3 33 i 36 J -39 47 1-55 .60 1.62 .65 37 54 55 .29 3 33 1.38 1.41 I'll 49 50 1-53 1-55 i-57 1-59 .62 .64 1.64 1.66 67 .69 36 35 56 11 59 3* 33 35 39 38 39 .41 .42 1.41 '44 1.46 1-45 1.46 1.48 1.49 1.48 1.50 1.52 i-53 52 31 57 I'A i. 60 1.62 i.6r 1.63 1.65 1.66 .66 .68 .70 1.68 1.70 1.72 73 75 77 34 33 32 3* 60 [38 .41 44 1-47 1.51 I -55 59 i.6 3 1.68 73 1.76 79 30 61 39 42 45 1.49 1-53 1.56 .61 i.6 S 1.70 75 1.78 .80 29 62 63 .40 4 2 43 45 :!i 1.50 i-54 !'55 1.58 i-59 .62 .64 1.67 1.68 1.71 1-73 77 78 1.79 t.8i .82 .84 28 27 64 43 .46 49 1 -53 1.61 65 1.70 i.7S .80 1.83 85 26 65 44 47 51 1-54 xip 1.62 .66 1.71 1.76 .81 1.84 .87 2 5 66 45 .48 52 1-55 1-59 1.63 .68 1.72 1.77 83 1.85 .88 24 67 .46 5 53 i. 60 1.65 .69 i-74 1.79 .84 1.87 .90 23 68 47 5 1 54 lip 1.62 1.66 .70 i. 80 1.88 22 69 .48 52 55 1-59 1.63 1.67 7 1 1.76 x.8i .87 1.90 93 21 70 .49 53 .56 i. 60 1.64 1.68 73 1.77 1.82 .88 1.91 94 2O 7 2 5 5 1 54 54 57 58 1.61 1.62 1.65 1.66 1.69 1.70 74 75 1.78 i. So 1.84 1-85 .89 .90 1.92 95 .96 18 73 52 55 59 1.63 1.67 1.71 .76 i. 80 1.86 .91 1.94 97 17 74 53 56 .60 1.63 1.68 1.72 .76 1.81 1.87 92 i-95 .98 16 75 53 57 .60 1.64 1.68 i-73 77 1.82 1.88 93 1.96 99 15 76 54 .58 .61 1.65 1.69 1-73 .78 1.83 1.88 94 1.97 .00 M 77 55 58 .62 i .66 1.70 1.74 79 1.84 1.89 95 1.98 .01 13 78 55 59 .62 1.66 1.70 1.75 .80 1.85 1.90 .96 1.99 .02 12 79 56 59 63 1.67 1.71 1.76 .80 1.85 1.91 .96 1.99 .02 IT 80 .56 .60 .64 1.67 1.72 1.76 .81 1.86 1.91 97 2. GO 3 IO 81 57 .60 .64 1.68 1.72 1.77 .81 1.86 1.92 .98 2.01 04 9 82 57 .61 .64 T.68 *-73 1.77 .82 1.87 1.92 .98 2.OI .04 8 83 -58 .61 65 1.69 1.77 .82 1.87 i-93 99 2.02 05 7 84 58 .62 65 1.69 J -73 1.78 83 1.88 99 2. O2 .05 6 8 5 58 .62 6 5 1.69 1-74 1.78 .83 1.88 1-93 99 2.02 05 5 86 59 .62 .66 1.70 1.74 1.78 83 1.88 1.94 2.00 2.03 .06 4 87 59 .62 .66 1.70 1.74 1.79 83 1.88 1.94 2.00 2.03 .06 3 88 59 .62 .66 1.70 1.74 1.79 83 1.89 1.94 2.OO 2.03 .06 2 89 59 .62 .66 1.70 1.74 1.79 .84 1.89 1.94 2.00 2.03 .06 I 90 59 .62 .66 1.70 1.74 1.79 .84 1.89 1.94 2.OO 2.03 .06 O The bottom line on this page is the collimation factor C (= sec 8). 302 GEODETIC ASTRONOMY. 299. FACTORS FOR THE REDUCTION OF TRANSIT TIME OBSERVATIONS. This top argument is the star's declination 5. *P 62 62* 63 63* 64 64* 6 5 6 5i 66 66* 67 (. .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 .04 89 2 .07 .07 .08 .08 .08 .08 .08 .08 .08 .09 .09 .09 88 3 . ii . ii .11 . 12 . 12 .12 .12 .12 13 .13 87 86 5 .18 .19 .19 .19 .19 .20 .20 .21 .21 .21 .22 .22 85 6 .22 .22 23 23 23 .24 24 25 25 .26 .26 .27 84 7 .26 .26 .26 27 .27 .28 .28 29 29 30 3 1 31 83 8 .29 30 30 31 31 32 32 33 34 34 35 36 82 9 33 33 34 35 35 36 36 37 38 39 39 .40 81 10 36 37 38 38 39 .40 .40 .41 .42 43 43 44 80 ii .40 .41 .41 42 43 44 44 45 .46 47 .48 49 79 12 44 47 3 45 49 .46 50 47 5 47 .48 52 49 53 50 54 55 'It 53 58 78 77 14 5* 52 53 54 55 56 58 59 .61 .62 76 15 54 55 56 57 58 59 .60 .61 .62 .64 65 .66 75 i6 58 59 .60 .6! .62 .63 .64 65 .66 .68 .69 7 1 74 17 .61 .62 6 3 .64 .66 .67 .68 .69 .70 72 73 75 73 18 65 .66 67 .68 .69 7o 72 73 74 .76 77 79 72 ig .68 .69 .70 72 73 74 .76 77 .78 .80 .82 83 7 1 20 .72 73 74 75 77 79 79 .81 83 .84 .86 .88 70 21 75 .76 78 79 .80 .82 83 85 .86 .88 .90 92 69 22 .78 .80 .81 .82 .84 85 87 .89 .90 92 94 .96 68 23 .82 83 85 .86 .88 .89 .91 .92 94 .96 .98 I.OO 67 24 85 87 .88 .90 .91 93 94 .96 .98 .00 1.02 1.04 66 25 .89 .90 .92 93 95 .96 .98 i .00 i .02 .04 1. 06 i. 08 65 26 .92 93 95 97 .98 I.OO 1.02 1.04 .06 .08 I.IO I. 12 64 27 28 95 08 97 .98 I.OO 1.02 1.04 i. 05 1.07 .09 .12 1.14 1.16 ll 9 I.OO I .O? 3 I.O5 T no 1.07 1 .09 *' .20 61 29 30 1.05 i .03 1.07 i. 08 07 .10 i .oy 1. 12 1.14 1.16 1:3 .21 .19 23 1-25 1.24 1.28 60 31 1. 08 I.IO i. ii 13 1. 15 . I? .20 1.22 .24 .27 1.29 1.32 59 32 i. ii 1. 13 1 - I S 17 I.ig .21 23 1.25 .38 3 1.36 58 33 34 1.17 1. 19 I. 21 2 3 1-25 2 7 30 1.29 1.32 3 1 35 34 37 I -37 1.40 *'39 56 35 1.20 1.22 1.24 .26 1.29 31 33 1.36 38 .41 1.44 1.47 55 36 1.2 3 1-25 1.27 30 1.32 34 37 i-39 .42 45 1.47 i-Si 54 37 1.26 1.28 1.30 33 i-35 37 .40 1.42 45 .48 1 54 53 38 1.29 1.31 i-33 36 1.38 .40 43 1.46 .48 5* 1.54 1.58 S 2 39 1.32 1.34 1.36 39 1.41 43 .46 1.49 52 1.58 i.6x 40 i-35 i-37 *-39 .42 1.44 47 49 1.52 55 58 1.61 1.65 50 4i 1-37 1.40 1.42 45 1.47 50 53 1-55 58 .61 1.64 1.68 49 42 1.40 1.42 i45 47 1.50 53 55 .61 .64 1.68 1.71 48 43 1.48 50 56 58 1.61 .64 .68 1.71 I -75 47 44 i '.46 xij 1.50 53 i's6 58 .61 1.64 .67 , 7 i 1.74 1.78 46 45 1.48 1.51 S6 1.58 .61 .64 1.67 .70 74 1.77 1.81 45 299- TABLES. 303 FACTORS FOR THE REDUCTION OF TRANSIT TIME OBSERVATIONS. This top argument is the star's declination 8. 61* 62 62* 63= 6 3 i 64 <* 6 5 6 5 i 66 66* 67 4*6 1.56 1.58 1.61 1.64 1.6 7 1.70 1.74 i. 80 1.84 44 47 1.53 1.56 1.58 1.61 1.64 1.6 7 I. 7 1.76 i. 80 1.83 1.87 43 48 I -55 1-58 x. 60 1.63 1.66 1.6 9 1.72 1.75 1.79 1.82 1.86 1.90 4 2 49 1.58 1.61 1.63 1.66 1.69 1.72 1-75 1.79 1.82 1.86 1.89 4 1 5 i. 60 1.63 1.66 1.69 1.72 1.78 1.81 1.85 1.88 1.92 l: 9 9 6 40 5 1 1.63 1.66 1.68 1.71 i-74 1.77 i. 80 1.84 1.87 1.91 I-9S 1.99 39 52 1.65 1.68 1.71 1.77 I. 80 1.83 1.86 1.90 1.94 1.98 2.02 38 C| J 53 1.67 1.70 1.76 1.79 1.82 1.85 1.89 1.93 1.96 2.00 2.04 37 3 54 1.69 1.72 I -75 1.78 1.81 1.8 5 1.88 1.91 i-95 1.99 2.03 2.07 36 JJ I 55 1.72 1.74 1.77 i. 80 1.84 1.8 7 1.90 1.94 1.98 2.OI 2.05 2.10 35 1 c 56 1.74 1.77 i. 80 1 1 3 1.86 1.89 i. 93 1.96 2.OO 2.04 2.08 2.12 34 JO* 7. 57 1.76 1.79 1.82 1.88 I.9I i-95 1.98 2. O2 2.06 2.10 2.15 33 S- H 58 1.78 1.81 1.84 I'.S* 1.90 1.93 1.97 2.01 2.05 2.08 2.1 3 2.17 3 2 J 59 i. 80 1.83 1.86 1.89 1.92 1-95 1.99 2.03 3.07 2. II 2.15 2.19 EL ^! 60 x.8i 1.84 1.88 1.91 1.94 1.97 2.01 2.05 2.09 2.1 3 2.17 2.22 3 n S 61 1.83 1.86 1.89 i-93 1.96 2.00 2.03 2.07 2. II 2.15 2.19 2.24 29 A rt 62 1.85 1.88 1.91 1.94 1.98 2.01 2.05 2.09 2.13 2.17 2.21 2.26 28 g "^ 63 1.87 1.90 1.93 1.96 2.00 2.03 2.07 2. II 2.15 2.19 2.23 2.28 27 n "5 64 1.88 x.gx i-95 1.98 2. 02 2.05 2.09 2.13 2.1 7 2.21 2.2 S 2.30 26 3 S 65 1.90 1.93 1.96 2.00 2.03 2.07 2. II 2.14 2.19 2.2 3 2.27 2. 3 2 25 "o 3 66 67 1.91 1.95 1.96 1.98 1.99 2.01 2.03 "3 2.08 2.10 2.12 2.14 2.16 2.18 2.20 2.22 2.25 2.26 2.29 2.31 2.34 2. 3 6 2 4 23 f v " 68 1.94 1.97 2.OI 2.04 2.08 2. II 2.15 2.19 2.24 2.28 2.32 2-37 22 ' s 69 1.96 1.99 2. 02 2.06 2.09 2.13 2.21 2.25 2. 3 2.34 2-39 21 P I 7O 1.97 2.03 2 .40 ir 71 1.98 2.OI 2.05 2.08 2.12 2.l6 2.20 2.24 2.28 2.32 2-37 2.42 J 9 ET 9 72 1.99 2.03 2.06 2.09 2.13 2.17 2.21 2.25 2.29 2-34 2.38 2.43 il 4) e 73 74 2.00 2.OI 2.04 2.05 2.07 2.08 2. II 2.12 2.14 2.15 2.l8 2.19 2.22 2.2 3 2.26 2.27 2.31 2.32 I'll 2.40 2.41 l'.\l 16 to J: 75 2.02 2.06 2.09 2.13 2.16 2.24 2.29 2-33 2-37 2.42 2.47 *5 ~ 76 2.03 2.07 2.10 2.1 4 2.17 2.21 2.25 2. 3 2-34 2-39 2-43 2.48 M o o JE 77 2.04 2.07 2. II 2.15 2.18 2.22 2.26 2.31 2-35 2.40 2.44 2.49 13 ** 78 2.05 2.08 2.12 2.15 2.19 2.23 2.27 2.31 2.36 2.40 2-45 2.50 12 X 1 79 2.06 2.09 2.13 2.16 2. 2O 2.24 2.28 2.32 2-37 2.41 2.46 2.51 II S _> 80 2.06 2.IO 2.13 2.17 2.21 2.2 5 2.29 2-33 2.38 2. 4 2 2.47 2.52 IO Bi 2.07 2.10 2.14 2.18 2.21 2.2$ 2.29 2-34 2.38 2-43 2.48 2.53 9 82 2.08 2. II 2.IS 2.18 2.22 2.26 2.30 2.34 2-39 2.43 2.48 2.53 8 83 2.08 2.12 2.15 2.19 2.22 2.26 2.3 1 2-35 2-39 2-44 2.49 2 54 7 84 2.08 2.12 2.15 2.19 2.23 2.27 2.31 2-35 2.40 2.45 2-49 2-55 6 85 2.09 2. 12 2.l6 2.1 9 2.23 2.27 2. 3 I 2.36 2.40 2-45 2.50 2-55 5 86 2.09 2.13 2.16 2.20 2.24 2.28 2.32 2.36 2.41 2-45 2.50 2-55 4 87 2.09 2.13 2.16 2. 2O 2.24 2.28 2.32 2.36 2.41 2.46 2.50 2.56 3 88 2.09 2.13 2.16 2.20 2.24 2.28 2.32 2.36 2.41 2.46 2.51 2.56 2 89 2. IO 2.13 2.17 2. 2O 2.24 2.28 2.32 2-37 2.41 2.46 2.51 2.56 I 00 2. IO 2.13 2.17 2. 2O 2.24 2.28 2. 3 2 2-37 2.41 2.46 2.51 2.56 O The bottom line on this page is the collimation factor C (= sec S). 304 GEODETIC ASTRONOMY. 299- FACTORS FOR THE REDUCTION OF TRANSIT TIME OBSERVATIONS. This top argument is the starts declination 5. 68 68* 69 w 70 H- 70* 70* 71 f* 71* 1 5 5 5 5 5 5 80 2 .09 .09 .10 .10 .10 .10 . 10 . 10 .11 . II . II . II 88 O_ 3 4 .18 .19 .19 .20 .20 .20 .21 .21 .21 .21 .22 .22 07 86 5 23 23 24 24 25 25 .26 .26 .26 .27 27 27 85 6 27 .28 .28 29 3 31 31 3 1 32 32 33 33 84 7 32 33 33 34 35 36 .36 37 37 37 38 38 83 8 36 37 38 39 .40 .41 .41 42 .42 43 43 44 82 9 .41 .42 43 44 45 .46 .46 47 47 .48 49 49 81 10 45 .46 47 49 50 51 51 52 53 53 54 55 80 ii 12 50 54 :S 52 57 53 58 54 59 .56 .61 t 57 .62 58 59 .64 $ .60 .66 79 78 X 3 59 .60 .61 63 .64 .66 .67 .67 .68 .69 7 7 1 77 14 63 65 .66 .68 .69 .71 .72 .72 73 74 75 76 76 15 .68 .69 7 1 .72 74 .76 77 78 .78 79 .80 .81 75 16 72 74 75 77 79 .81 .82 83 .84 85 .86 87 74 '7 .76 .78 .80 .81 83 85 .86 .88 .89 .90 .91 .92 73 18 .81 83 .84 .86 .88 .90 .91 93 94 95 .96 97 72 19 85 87 .89 91 93 95 .96 .98 99 1. 00 i .01 1.03 20 .89 .91 93 95 .98 1. 00 I. 01 1.02 1.04 1.05 i. 06 i. 08 70 21 94 .96 I'f 1. 00 .02 1.05 .06 1.0 7 1.09 1. 10 i. ii 69 68 23 .02 .06 .04 .07 1.09 .12 . 1.14 .16 I. I 7 1.19 1.20 I .21 1.23 67 fif. 2 4 25 .10 .09 13 15 i . 14 1.18 . ID .21 1.24 25 1.2 7 1.28 1.30 '33 oo 65 26 IS 17 .20 1.22 25 1.28 30 I 31 1-33 i-35 I. 3 6 1.38 64 2 7 .19 .21 .24 1.27 30 i '33 34 1.36 1.38 i-39 63 28 23 25 .28 1.31 34 39 I.4I 1.42 1.44 i! 4 6 i! 4 8 62 2 9 27 .29 .32 J -35 38 1.42 43 1 -45 1.47 1.49 i 5i J -53 61 30 33 36 1.39 43 1.46 .48 1.50 1.52 1-54 1.56 1.58 60 31 35 38 .40 1.44 47 i-Si 52 1-54 1.56 1.58 i. 60 1.62 59 32 33 39 .42 .42 45 45 49 1.48 5' 55 .1-59 1.61 1.65 1.63 1.67 1.65 1.69 1.67 1.72 58 57 34 .46 49 53 i. '56 .60 1.63 .65 lies 1.70 1.72 1.74 1.76 56 35 50 53 .56 i. 60 .64 1.68 .70 1.72 1.74 1.76 1.78 1.81 55 36 54 57 .60 1.64 .68 1.72 74 1.76 1.78 i. 80 1.83 1.85 54 37 57 .61 64 1.68 7 2 1.76 78 1. 80 1.83 1.85 1.87 1.90 53 38 .61 .64 .68 1.72 .76 i. 80 .82 1.84 1.87 1.89 1.91 1.94 52 39 65 .68 .72 .80 1.84 .86 1.88 1.91 1.96 1.98 40 .68 72 75 1.79 .84 1.88 .90 i-93 1.95 1.97 2.OO 2.03 50 4 1 7i 75 79 1.83 .87 1.92 94 1.96 1.99 2.OI 2.04 2.07 49 4 2 75 79 83 1.87 .91 1.96 .98 2.00 2.03 2.05 2.08 2. II 48 43 .78 .82 .86 1.90 95 i 99 .02 2.04 2.07 2.09 2.12 2.15 47 44 .82 -85 Q- .90 1.94 .98 2.03 .06 2.08 2. II 2.13 2.l6 2.19 46 45 1.85 .09 93 i .97 2.07 .09 2.23 45 299- TABLES. 30$ FACTORS FOR THE REDUCTION OF TRANSIT TIME OBSERVATIONS. This top argument is the star's declination 5. w 68 681 69 69* 70 * 70*. * 7i *r * 4 6 C 1.88 1.92 1.96 2.01 2.05 2.IO 2. I" ...5 2.18 2.21 2.24 2.27 44 47 1.91 1.95 2.OO 2.04 2.09 2.14 2.16 .19 2.22 2.25 2.27 2.30 42 48 1.94 1.98 2.02 2.07 2. 12 2.1 7 2.19 .22 2.2 S 2.28 2.31 2.34 42 49 1.97 2.01 2.06 2. II 2.16 2.21 2-23 .26 2.29 2.32 2.35 2.38 50 2.OO 2.04 2.09 2.14 2.19 2.24 2.27 .29 2.32 2-35 2.38 2.41 4 51 2.03 2.07 2.12 2.17 2.22 2.27 2.30 33 2. 3 6 2-39 2.42 2-45 39 5= 2.06 2.10 2.15 2.20 2.2 S 2. 3 2-33 36 2-39 2.42 2.45 2.48 38 (-1 3-36 3-39 3.41 3-44 3-47 3-50 1:3 3-59 3.62 3-65 3-68 3-72 3-75 3-82 3-85 3-89 3-92 4.00 4.04 28 27 1 64 3.36 3-42 3-47 3'53 3-59 3.65 3-72 3.78 3-85 3-92 4.00 4.07 26 pi 65 3-39 3-45 3.50 3.56 3-62 3.68 3-75 3-8i 3-88 3-95 4-03 4.11 25 cT 66 3-42 3-47 3-53 3-59 3-65 3.7i 3.78 3-84 3.91 3-99 4.06 4*4 24 5 67 3-44 3-50 3-56 3.62 3-68 3-74 3-87 3-94 4.02 4.09 4-17 2 3 68 3-47 3-53 3-58 3-64 3-70 3-77 l'.*l 3-90 3-97 4.05 4.12 4.20 22 5' 69 3-49 3-55 3-61 3-67 3-73 3-79 3-86 3-93 4.00 4.07 4-15 4-23 21 P 70 3-52 3 et 3-57 3 60 3-63 o.ge 3-69 3-75 3 ?8 3.82 , 84 3.89 3-95 4-03 4.10 4.18 4.25 2O 5* p 72 54 3-56 j.UO 3-62 3 P 3-67 3-74 j./o 3-8o I:S 3-93 4.00 4.07 4.23 11 73 3.58 3.69 3.76 3.82 3.89 3-95 4.02 4.10 4.17 4-25 4-33 1 7 5 74 3.6o 3*65 3.78 3.84 3-91 3-97 4.04 4.12 4.19 4.27 4.36 16 Sj 75 3.6! 3-67 3-73 3-79 3-86 3-92 3-99 4.06 4.14 4-21 4.29 4-38 15 ^f 76 3-64 3.69 3-75 3-82 3-88 3.94 4.01 4.08 4.16 4-23 4-3' 4.40 14 r> 77 3-65 3-70 3.76 3-83 3-89 3.96 4.03 4.10 4-17 4-25 4-33 4-41 13 t/> 78 3-66 3'72 3.78 3.84 3.97 4.04 4.11 4.19 4-27 4-35 4-43 12 'us* 79 3.67 3-73 3-79 3-86 3-92 3-99 4.06 4-13 4.21 4.28 4.36 4.45 II 80 3-68 3-74 3.87 3-93 4.00 4.07 4.14 4.22 4-3> 4.38 4.46 10 ^ Si 82 3-70 3-7 1 3-75 3-76 3.82 3-83 3-88 3.89 3-94 4.01 4.02 4.08 4-09 4.16 4-17 4-23 4-24 4-32 4-39 4.40 4.48 4-49 I 83 3-72 3-77 3-84 3.90 3- '96 4-3 4.10 4.18 4-25 4-33 4.41 4-50 7 84 3-72 3.78 3.84 3.91 3-97 4.04 4.11 4.18 4.26 4-34 4-42 4-Si 6 85 3-73 3-79 3-85 3-98 4-05 4.12 4.19 4.27 4-35 4-43 4-Si 5 86 3-73 3-79 3.85 3.92 3.98 4-05 4.12 4.20 4.27 4-35 4-43 4-52 4 87 3-74 3 79 3-86 3-92 3-99 4.06 4.13 4.20 4.28 4-36 4-44 4-52 3 88 3-74 3.80 -, g 3-86 3 92 3-99 4.06 4 06 4.^3 4.20 4.28 4.36 4.44 4.53 2 90 3-74 3-74 3.00 3.80 3^86 3-93 3 93 3*99 3-99 4-06 4 *3 4-13 4.21 4.28 4.36 4 44 4-44 4 53 4.53 The bottom line on this page is the collimation factor C (= sec 6). GEODETIC ASTRONOMY. 2 99- FACTORS FOR THE REDUCTION OF TRANSIT TIME OBSERVATIONS. This top argument is the star's declination 5. 77i 77t 78 78} * 78* 79 79* 79* 791 80 ^ .08 .08 .08 .09 .09 .09 .09 .09 .10 .10 .10 80 2 .16 .16 .17 17 .18 .18 .18 .19 .19 .20 .20 88 3 .24 25 2 5 .26 .26 27 27 .28 29 .29 3 87 4 33 34 34 35 36 37 37 38 39 .40 86 5 .40 .41 .42 43 44 45 .46 47 .48 49 So 85 6 49 49 5i 5i S 2 54 55 S6 57 59 .60 84 7 .56 57 59 .60 .61 .62 .64 6 5 .67 .69 .70 83 8 .64 .66 .67 .68 .70 7i 73 75 .76 .78 .80 82 9 72 74 75 77 .78 .80 .82 .84 .86 .88 .90 81 10 .80 .82 .84 85 .87 .89 .91 93 95 .98 i .00 80 ii .88 .90 92 94 <* 98 1. 00 i .02 1.05 .07 .10 79 12 .96 98 1. 00 i 08 1.02 1.09 i. ii 1.14 .20 78 M i .04 I . 12 I .OO 1.14 I. 21 1.27 1.30 i-33 36 3 39 76 15 1.20 1.22 1.25 1.27 1.30 i-33 1.36 i-39 1.42 46 49 75 i6 1.28 1.30 1-33 1.35 1.38 1.41 1.44 1.48 1.51 55 59 74 17 j . 75 1.38 1.40 1.44 1.47 1.50 1 -53 I -57 i. 60 . 64 .68 73 18 1 -43 1.46 1.49 1.52 I -55 1.58 1.62 1.66 1.70 74 .78 72 jg T '5 X T -57 I. 60 1.63 1.67 1.71 '75 1.79 83 .87 20 1.58 I'.ll 1.65 1.68 1.72 1.79 1.83 1.88 .92 97 70 21 1.65 1.69 1.72 1.76 i. 80 1.84 1.88 1.92 1.97 .01 .06 69 22 1.77 i. 80 1.84 1.88 1.92 1.96 2.01 2.06 . ii .16 68 2 3 I'.li 1.84 1.88 1.92 1.96 2.00 2.05 2.09 2.14 .20 25 67 24 1.88 1.92 1.96 2.OO 2.04 2.08 2.13 2.18 2.23 .29 34 66 25 i-95 1.99 2.03 2.07 2.12 2.17 2.22 2.2 7 2.32 38 43 65 26 2.02 2.07 2. II 2.15 2.20 2.25 2.30 2-35 2.41 .46 52 64 s 2.10 2.17 2.14 2.21 2.l8 2.26 2.23 2.31 2.28 2. 3 6 2-33 2.41 2. 3 8 2.46 2-43 2-49 2.58 55 .64 .61 .70 63 62 29 2.24 2.28 2-33 2. 3 8 2-43 2.48 2-54 2^60 2.66 73 79 61 3 2.31 2.36 2.40 2.46 2.SI 2.56 2.62 2.68 2-74 2.81 2.88 60 3 1 2.38 2-43 2. 4 8 2-53 2.58 2.64 2.70 2.76 2.83 2.89 2-97 59 3 2 2-45 2.50 2-55 2.60 2.66 2.72 2. 7 8 2.84 2.91 2.98 3-05 58 33 2.52 2.57 2.62 2.6 7 2-73 2.79 2.8 5 2.92 2-99 3.06 57 34 2.64 2.69 2-75 2.80 2.87 2.93 3-oo 3-07 3-14 3.22 56 35 I'.ls 2.70 2. 7 6 2.82 2.88 2.94 3-o8 3-iS 3-23 3-30 55 36 2.72 2.77 2.8 3 2.89 2-95 3.01 3-08 3-15 3-23 3-3 3.38 54 37 2.78 2.84 2.90 2.95 3-02 3-o8 3-15 3-23 3-3 3.38 3-47 53 38 2.85 2.90 2. 9 6 3-02 3-og 3-16 3-23 3-3 3.38 3-46 52 39 2.91 2-97 3-3 3-09 3.16 3-23 3-30 3-37 3-45 3-53 3-62 5 1 40 2-97 3-03 3-09 3-i6 3.22 3-29 3-37 3-45 3-53 3.61 3-70 5 4* 3-3 3-9 3-16 3.22 3-29 306 3-44 3-52 3-6o 3-69 3-78 49 42 3-9 3-22 3-29 3.36 3-43 3-51 3-59 3-67 3.76 3-85 48 43 3.21 3-28 3-35 3-42 3-50 3-66 3-74 3-83 3-93 47 44 3-21 3-27 3-34 3.41 3.48 3-56 3-64 3-72 3.81 3-9 1 4.00 46 45 3-27 3-33 3-40 3-47 3-55 3.62 3-71 3-79 3-88 3-97 4.07 45 TABLES. FACTORS FOR THE REDUCTION OF TRANSIT TIME OBSERVATIONS. This top argument is the star's declination . 774 77l 78 78* 78* 78* 79 79i 79i HT 80 t , 46 47 3-32 3.38 3-39 3-45 3.46 3-52 3-53 3-59 3 '-67 3.69 3 75 3-77 383 3-86 3-95 4.04 4.14 44 43 48 49 3-43 3-49 3-50 3-57 3-63 3-71 3-73 3-79 3-8r 3-87 3-89 3.96 3-98 4-05 4.14 4.24 4-35 50 3-54 3-6i 3-68 3.76 3-84 3-93 4.02 4.11 4.20 4-30 4.41 40 Si 3- e 9 3-66 3 7* 3-74 3" 79 3-82 0.87 3-90 3ne 3.98 4.07 4-17 4.26 4-37 .48 39 _6 53 3*69 3-77 3.84 o* w / 3-92 yj 4.01 4.09 4.19 4.28 4-38 4-49 iio" 3 37 54 55 3-74 3-78 3.81 3-86 3-89 3-94 3-97 4.02 4 .o6 4.11 4-i5 4.20 4.24 4.29 4-34 4-39 4-44 4-50 4-55 4.60 .66 .72 36 35 56 3-83 3.91 3-99 4.07 4.16 4-25 4-34 4.44 4-55 4.66 4-77 34 57 3.88 3-95 4.04 4.12 4.21 4-3 4-39 4-50 4.60 4.72 4-83 33 58 3-92 4.00 4.08 4.16 4-25 4-35 4.44 4-55 4-65 4-77 4.88 B 3.96 4.00 4.04 4.08 4.12 4.17 4.21 4-25 4-30 4-34 4-39 4.44 49 54 eg 4.60 4.64 4.70 4-75 A. 80 4.82 4-87 4.94 4-99 3 T 30 ori 62 4.08 4.16 4.25 4-34 4 "39 4-43 "53 t . 50 .63 4-73 4-00 4.85 4.92 4.96 1:3 2 9 28 6 3 4.12 4.20 4.29 4-38 4-47 57 .67 4.78 4.89 5-oi 5 5 ll 26 65 4.19 4.27 4.36 4-45 4-55 65 75 4.86 4-93 4-97 5-5 5-09 5.10 5-22 25 66 4.22 4-3i 4.40 4-49 4.58 .68 79 4.90 S.oi S-I4 5.26 24 67 4.26 4-34 4.43 4-52 4.62 .72 .82 4-94 5-S 5-18 5-3 23 68 4.28 4-37 4.46 4-55 4-65 75 .86 4-97 5-09 5-21 5-34 22 69 4-32 4.40 4.49 4-58 4.68 79 4.89 S-oo 5.12 516 5-25 5.28 5.38 21 70 71 4-34 4-37 4-43 4.46 4-55 4.64 4-74 4-85 4 '93 5-<>7 IU 5-19 5-32 S-4 1 5-45 r 7 2 4-39 4-48 4-57 4.67 4-77 4.88 4.98 5.22 5-34 5.48 18 73 4.42 4-5 1 4.60 4.70 4.80 4.90 5.01 5-13 5-25 5-37 *7 74 4-44 4-53 4.62 4.72 4.82 4-93 5-4 5- T 5 5-27 5-4 5-53 16 75 4.46 4-55 4-65 4-74 4.84 4-95 5-06 5-30 5-43 5.56 15 76 4.48 4-57 4.67 4.76 4.87 4 -97 S-9 5-20 5-3 2 5-45 5-59 14 77 4-50 4-59 .68 4.78 4.89 4-99 5-" 5-22 5-35 5-47 5.61 13 78 4-52 4.61 .70 4.80 4.91 5.01 5-13 5-24 5-37 5-50 5-63 12 79 4-54 4-63 .72 4.82 4.92 5-03 5-14 5.26 5-39 5-52 5.65 II 80 4-55 4.64 74 4.84 4-94 5-S 5-i6 5-28 5-4 5-54 5-67 10 Si 4.56 4-65 75 4-85 4-95 5-o6 5.18 5-3 5-42 5-55 5-69 9 82 4-57 4.67 i .76 4.86 4-97 5-o8 5-*9 5.- 3* 5-43 5.56 5-70 8 83 4-59 4.68 .78 4.87 4-98 5-09 5-20 5-32 5-45 5.58 5-72 7 84 4.60 4-69 79 4.88 4.99 5-21 5'33 5-46 5-59 5-73 6 85 4.60 4.69 79 4.89 5-00 5-" 5-22 5-34 5-47 S-6o 5-74 5 86 4.61 4.70 .80 4.90 5-oo 5-" 5-23 5-35 5-47 5-6i 5-74 4 87 4.62 4-7 1 .81 4.90 5.01 5-12 5-23 5-35 5-48 5.61 5-75 3 88 go 4.62 4 62 4-71 .81 4-91 5-oi SOI *" 5-12 5.12 5-24 5 ***4 5-36 5.48 5 *49 5.6. 5.62 5-75 2 I y 90 4-"-* 4.62 4.71 4.81 4.91 5-02 5.13 5-24 5^6 5-49 5-62 SO'6 The bottom line on this page is the collimation factor C (= sec S). 3 I2 GEODETIC ASTRONOMY. 300. 300. FACTORS FOR THE REDUCTION OF TRANSIT TIME OBSERVATIONS AT CORNELL UNIVERSITY. = 42 27'. a v4 B A11 + C All 4- for lamp west, and for lamp east. ' 4 B AH 4 C All 4- for amp west, and for lamp east. 40 35 4- 1.29 + 1.19 0.17 0.26 1.31 1.22 4-62* 63 -74 -77 :3 .20 3 4- 1 .10 o-35 15 63* .Bo .09 24 25 20 tl.02 0.94 0.42 0.49 IO 06 64 64* ~ '%7 .12 .28 32 IO 4-0.87 4-0.80 0.56 0.62 04 02 65 65* -91 -94 .19 .22 37 41 8 4-0.78 0.64 01 66 -98 25 .46 6 4-0.75 0.67 OI 66* .02 29 5 1 4 4-0.73 0.69 00 67 .06 33 56 2 4-0.70 0.71 oo 67* .10 37 .61 4- 2 --0.67 -- 0.65 0.74 0.76 00 oo 68 68* .20 .41 45 .67 73 4 -- 0.62 o.79 00 69 -25 .50 79 6 - - o . 60 0.81 01 69* -30 54 .86 8 -- o-57 0.83 01 7 '35 59 92 4-0.54 0.86 02 - -38 .62 .96 12 0.88 .02 70* -4 1 .64 3.00 14 4-0.49 0.91 03 7* -44 .67 3-03 16 20 4-o. 4 6 4-0.43 4-0.41 o-93 0.96 0.98 04 a 5 -47 -SO -53 .70 3-07 3-" 3-15 22 -4-0.38 I. 01 .08 7 1 * - .56 79 3-19 3 + 0.35 4- 0.32 1.04 1.07 .09 .11 72 72* .60 - .63 .82 85 3-24 3-28 28 4-0.28 1. 10 13 72* .66 .88 3-33 3 4- 0-25 1-13 15 72* .70 .91 3-37 S 2 4- O.22 !.I6 .18 73 -74 95 3-42 -j-o.i8 4-o-i4 1.19 1.23 .21 .24 73* 73* .78 .82 3-02 3-47 3-52 38 4- o.io 1.27 .27 73* .86 3 '05 3-57 40 -j-o. 06 1.30 .31 74 .90 3-09 3-63 42 4-0-03 4- o.oi 1.32 i -35 33 35 74* 74* 3-13 3-17 3-68 3-74 43 O.OI i 37 37 74* 03 3-21 3-8o 44 0.04 i-39 39 75 .08 3-26 3-86 45 0.06 .41 75* 13 3.30 3-93 46 47 0.09 O. 12 !'. Jo 44 47 1 - .18 -23 3-35 3-40 3-99 4.06 48 0.14 1.49 49 76 - .28 3-45 4-13 49 0.17 i-Si S 2 76* -34 3-50 4.21 5 0.20 54 56 76* .40 3-55 4.28 5 1 0.24 76* _ .46 3-6o 4-36 S 2 0.27 i .60 .62 77 -52 3-66 4-44 53 54 0.30 0.34 1.63 1.67 .66 .70 i 9 -0.38 0.42 1.70 1.74 74 79 77* 78 .72 .80 3-85 3-9 1 JiSJ ii -o.Je 0.51 It 78* - -87 -95 3-98 4.06 4 9* 5-02 59 0.55 1.86 94 78* 3-03 4-13 5-13 60 0.60 1.91 .00 79 4.21 5-24 60* 61 0.63 0.66 1.93 1.96 a i 3-21 4.29 4-38 5.36 5-49 61 J 0.68 1.98 . IO 79* 3-4 1 4-47 5-62 + 62 0.71 2.01 13 4-8o 3-51 4-57 5.76 This table is computed for latitude 42 27'. It may be used, however, for any station whose latitude does not differ from that value by more than 7'. For a station in latitude 42 20', or in latitude 42 34', the maximum error in the table is two units in the last place. 303- TABLES. 313 301. CORRECTION TO TRANSIT OBSERVATIONS FOR DIURNAL ABERRATION. The correction is negative when applied to observed times, except for sub-polars. (See 96.) Declination = 5. Latitude = *. o 10 20 30 40 50 60 70 80 o 0.02 0*.02 0*.02 O 8 .O2 o*.o 3 o 8 . 03 o 8 . 04 o*.o6 0*.I2 IO .02 o .02 .02 o .02 o .03 o .03 o .04 0.06 0.12 20 o .02 o .02 .02 .02 o .03 o .03 o .04 o .06 0.12 30 o .02 o .02 o .02 o .02 o .02 o .03 o .04 o .05 O .11 40 o .02 .02 o .02 o .02 o .02 o .03 o .03 o .05 o .09 50 O .OI .01 O .OI o .02 o .02 o .02 o .03 0.04 0.08 60 .01 .01 .01 O .OI .01 .02 o .02 o .03 o .06 70 .01 .01 .01 .01 .01 .01 o .or o .02 o .04 80 o .00 o .00 .00 .00 o .00 o .01 o .01 o .01 o .02 302. RELATIVE WEIGHTS FOR TRANSIT OBSERVATIONS DEPENDING ON THE STAR'S DECLINATION. (See m.) s iu Vw w y-w a TV Vw o IO 20 30 40 I.OO 0.99 0.95 0.87 o. 76 I.OO 0.99 0.97 o-93 0.87 45 50 55 60 65 0.69 0.61 0.52 0.42 0-33 0.83 0.78 0.72 0.65 0.57 70 75 80 85 O.24 0.14 0.07 0.02 0-49 0.37 0.26 0.14 .In the application of the multiplier \'~^ it generally suffices to employ but one significant figure. 303. RELATIVE WEIGHTS FOR INCOMPLETE TRANSITS. (See i IT.) For eye and ear observations. For observations with a chronograph. No. of Lines 5 Lines in Reticle. 7 Lines in Reticle. 9 Lines in Reticle. ii Lines in Reticle. 13 Lines in Reticle. Obs. w Vw w MIV w y-iv IV 4/w IV Vw I 0.40 0.63 0.36 0.60 0.42 0.65 6.42 0.65 0.41 0.64 2 0.64 0.80 0-57 0-75 0.62 0.79 0.62 0.79 O.6O 0.77 3 0.80 0.89 0.71 0.84 o.73 0.85 0.73 0.85 0.71 0.84 4 0.92 0.96 0.82 O.gi 0.82 0.91 0.81 0.90 0.79 0.89 5 I.OO I.OO 0.90 0-95 0.87 0.93 0.86 o.93 0.84 0.92 6 0.95 0.97 0.92 0.96 0.90 o.95 0.88 0.94 7 I.OO I.OO 0-95 0.97 0-93 0.96 0.91 0.95 8 0.98 0-99 0.95 0.97 0-93 0.96 9 I.OO I.OO 0.97 0.98 0-95 0.97 10 0.99 o.99 0-97 0.98 ii I.OO I.OO 0.98 0.99 12 0.99 0.99 13 I.OO I.OO GEODETIC ASTRONOMY. 304. 304, CORRECTION TO LATITUDE FOR DIFFERENTIAL REFRACTION. The sign of the correction is the same as that of the micrometer difference (See 150.) i Diff . of Zenith Distances. Zenith Distance. 10 20 25 3 35 40 45 o'.o o".oo o".oo o".oo o".oo o".oo o".oo o".oo o".oo 0.5 .01 .01 o .01 o .01 .01 o .01 o .01 .02 I .O o .02 .02 .02 o .02 .02 .02 o .03 o .03 I -5 o .03 o .03 o .03 o .03 o .03 o .03 o .04 o .05 2 .0 o .03 o .03 o .04 o .04 o .04 o .05 o .06 o .07 2-5 o .04 ' o .04 o .05 o .05 o .05 o .06 o .07 o .08 3-0 o .05 o .05 o .06 o .06 o .07 o .08 o .09 o .10 3.5 o .06 o .06 o .07 o .07 o .08 o .09 o .10 .12 4.0 o .07 o .07 o .08 o .08 o .09 o .10 O .11 o .13 4.5 o .08 o .08 o .09 o .09 .10 .11 o .13 o .15 5-0 o .08 o .09 .10 o .10 .11 o .13 o .14 o .17 5 -5 o .09 .10 .10 O .11 .12 o .14 o .16 o .18 6.0 .10 .10 .11 O .12 o .13 o .15 o .17 o .20 6.5 .11 .11 .12 o .13 o .14 o .16 o .19 O .22 7-0 .12 .12 o .13 o .14 o .15 o .18 .20 o .24 7-5 o .13 o .13 o .14 o .15 o .16 o .19 O .21 o .25 8.0 o .13 o .14 o .15 o .16 o .18 O .21 o .23 o .27 8.5 o .14 o .15 o .16 o .17 o .19 O .22 o .24 o .29 9.0 o .15 o .16 o .17 o .18 .20 o .23 o .26 o .30 9-5 o .16 o .17 o .18 .20 .21 o .24 o .27 o .32 10 .0 o .17 o .18 o .19 .21 o .23 o .26 o .29 o -34 10.5 o .18 o .19 .20 O .22 o .24 o .27 o .30 o .35 II .0 o .18 o .19 .21 o .23 o .25 o .28 o .31 o .37 ii .5- o .19 .20 .22 o .24 o .26 o .30 o -33 o -39 12 .0 .20 .21 o .23 o .25 o .27 o .31 o -34 o .40 12.5 O .21 .21 o .24 o .26 o .28 o .32 o .36 o .42 13.0 O .22 .22 o .25 o .27 o .29 o .33 o .37 o .44 13.5 o .23 o .23 o .26 o .28 o .30 o -34 o .39 o .45 14.0 o .23 o .24 o .27 o .29 o .31 o -35 o .40 o .47 14.5 o .24 o .25 o .28 o .30 o .32 o .36 o .41 o .49 15-0 o .25 o .26 o .28 o .31 o .34 o .38 o -43 o .50 15.5 o .26 o .27 o .29 o .32 o -35 o -39 o .44 o .52 16.0 o .27 o .28 o .30 o -33 o .36 o .40 o .46 o .54 16.5 o .28 o .29 o .31 o .34 o -37 o .41 o .47 o .55 17.0 o .28 o .29 o .32 o .35 o .38 o .42 o .49 o -57 17.5 o .29 o .30 o -33 o .36 o -39 o .44 o .50 o .59 18 .0 o .30 o .31 o -34 o .37 o .40 o -45 o .52 o .60 18.5 o .31 o .32 o -35 o .38 o .41 o .46 o .53 o .62 19 .0 o .32 o -33 o .36 o .39 o .43 o .48 o .54 o .64 19.5 o .33 o -34 o .37 o .40 o .44 o .49 o .56 o .66 20.0 o .34 o -35 o .38 o .41 o -45 o .50 o .57 o .67 306. TABLES. 315 305. CORRECTION TO LATITUDE FOR REDUCTION TO MERIDIAN. The sign of the correction to the latitude is positive except for stars of negative declination (south of the equator). (See 151.) Hour-angle. s 10" is 9 20 s 25' 30' 35* 40' 45* 50" 55* 6o> 5 o".oo .00 o".oo .01 o".oo .01 o".oo .01 o".oo .02 o".oo o .03 o" . oo o" . oo o .040 .05 o".oo o .06 o".oo o .07 o".oo o .09 90 85 10 o .oolo .01 o .02 o .03 o .04 o .060 .08 o .09 .12 o .14 o .17 80 15 .01 .02 o .03 o .04 o .06 o .08 .11 o .14 o .17 O .21 o .24 75 20 .01 o .02 o .04 o .05 o .08 O . II o .14 o .18 .22 o .27 o .32 70 25 o .01 o .02 o .04 o .07 o .09 o .13 o .17 .21 .26 o .32 o .38 65 30 o .01 o .03 o .05 o .07 .11 o .140 .19 o .24 o .30 o .36 o .42 60 35 .01 o .03 o .05 o .08 .12 o .16 .21 .26 o .32 o -39 o .46 55 40 45 .01 .01 o .03 o .03 o .05 o .06 o .08 o .08 6 .12 0. 12 .160 .22 .1710 .22 o .27 .28 o .340 .41 o .340 .41 o .48 o .49 50 45 306. CORRECTION FOR CURVATURE OF APPARENT PATH OF STAR, IN MICROMETER VALUE DETERMINATIONS. The correction as tabulated is (15 sin i") 7 T s ^(15 sin i") 4 T a . Apply the corrections given in the table directly to the observed chronometer times, adding them before either elongation to the times, and subtracting them after either elongation. (See 159.) T Corr. r Corr. T Corr. T Corr. T Corr. 6 m o'.o i8 m I s . I 30 m 5M 42" I 4 M 54 m 29 s . 9 7 .1 19 I -3 31 5 -7 43 15 .1 55 31-6 8 .1 20 I -5 32 6 .2 44 16 .2 56 33 -3 9 .1 21 I .8 33 6 .8 45 17 -3 57 35 -I 10 .2 22 2 .O 34 7 -5 46 18.5 58 37 -o ii O .2 23 2-3 35 8 .2 47 19 .7 59 39 -o 12 0-3 24 2 .6 36 8 .9 48 21 .O 60 41 .0 13 0.4 25 3 -o 37 9.6 49 22 .3 61 43 -i 14 0.5 26 3 -3 38 10 .4 50 23 -7 62 45 -2 15 o .6 27 3 -7 39 ii -3 51 25-2 63 47 -4 16 o .8 28 4 .2 40 12 .2 52 26.7 64 49 -7 i? 0.9 29 4.6 41 13 -I 53 28 .3 65 52 .1 GEODETIC ASTRONOMY. 307. 307. 2 sin \t (See g 172.) sin i" t OP i m 2 m 3 m 4 m 5 m 6 m 7 m 8 m o".oo ".96 ~^8s~ i 7". 67 31". 42 49". 09 70". 68 9 6". 20 I2 5 ". 6 5 I .00 7 .98 17 .87 31 .68 49 .41 71 -07 96 .66 126 .17 2 o .00 .10 8 .12 18 .07 3i -94 49 -74 7i -47 97 '12 126 .70 3 .00 .16 8 .25 18 .27 32 .20 50 .07 71 .86 97 -58 127 .22 4 .01 23 8 -39 18 .47 32 -47 50 .40 72 .26 98 .04 "7 -75 5 .01 31 8 .52 i3 .67 S 2 -74 50 .73 72 .66 98 .50 128 .28 6 .02 .38 8 .66 18 .87 33 -oi 5i -07 73 .06 98 .97 128 .8l 7 o .02 '45 8 .80 19 .07 33 -27 51 *4 73 -46 99 -43 129 .34 8 9 o .03 o .04 : 8 .94 9 .08 19 .28 19 -48 33 -54 33 -8i 5i -74 52 .07 73 .86 74 .26 99 -90 100 .37 129 .87 130 .40 10 n 12 o .05 o .06 o .08 .67 9 .22 9 .36 9 -So 19 .69 19 .90 20 .11 34 -09 34 .|6 34 -64 52 .41 52 -75 53 -09 74 -66 75 -06 75 -47 100 .84 ioi .31 101 .78 I 3 .94 131 -47 132 .01 j. o .09 .91 9 -64 20 .32 34 -91 53 -43 75 -88 IO2 .25 132 .55 14 .11 .99 9 -79 20 .53 35 -19 53 -77 76 .29 102 . 72 133 .09 j- .12 3 -07 9 -94 20 .74 35 -46 54 -ii 76 .69 103 .20 133 .63 10 o .14 3 '15 10 .09 20 .95 35 -74 54 -46 77 .10 103 .67 *7 o .16 3 ' 2 3 10 .24 21 .16 36 .02 54 -80 77 -Si 104 .15 134 .71 18 o .18 3 -32 10 .39 21 .38 36 . T.O 55 -15 77 -93 104 .63 135 .25 19 o .20 3 -4 10 .54 21 .60 36 .58 55 -So 78 -34 105 .10 135 .80 20 .22 3 '49 10 .69 21 .82 36 .87 55 .84 78 -75 105 .58 136 .34 21 o .24 3 -S8 10 .84 22 .03 37 .15 56 .19 79 -16 106 .06 136 .88 22 o .26 II .00 22 .25 37 -44 S 6 -55 79 -58 106 .55 *37 -43 23 o .28 3 -76 II .15 22 .47 37 -72 56 .90 80 .00 107 .03 137 .98 24 o .31 3 -85 II .31 22 .70 38 .01 57 -25 80 .42 107 .51 J38 .53 2 5 o -34 3 -94 II .47 22 .92 38 .30 57 -60 80 .84 107 -99 139 -08 26 11 29 o .37 o .40 o .43 o .40 4 -03 4 .12 4 .22 4 .32 II .63 II .79 ii '95 12 .11 23 .14 23 -I! 23 .60 23 .82 38 -59 38 .88 39 .17 39 .46 57 -96 58 .32 58 .68 59 -03 81 .26 81 .68 82 .10 82 .52 108 .48 108 .97 109 .46 109 .95 139 -63 140 .18 140 .74 141 -29 3 o .49 4 -42 12 .27 24 .05 39 -76 59 .40 82 -95 no .44 141 -85 3i o .52 12 .43 24 .28 40 .05 59 -75 83 -38 no .93 142 .40 33 34 o .56 o -59 o .63 4 .62 4 .72 4 .82 12 .60 12 .76 12 -93 24 .51 24 -74 24 -98 40 -35 40 .65 40 -95 60 .11 60 .47 60 .84 83 .81 84 -23 84 .66 in .43 in .92 112 .41 142 .96 M3 -52 144 .08 35 o .67 4 -92 13 .10 25 .21 4i .25 61 .20 85 -09 112 .90 144 > 6 4 36 37 o .71 o .75 5 .3 5 ' X 3 13 '27 13 "44 25 -45 25 .68 :: -M 61 -57 61 .94 85 -52 85 .95 113 -40 "3 .9 145 -20 145 -76 38 39 o .79 o .83 5 -24 5 -34 13 .62 13 -79 26 . 16 42 .15 42 .45 62 .31 62 .68 86 .39 86 .82 114 .40 114 .90 146 -33 I 4 6 .89 40 41 o .87 o .91 5 -45 13 .96 14 -13 26 .40 26 .64 42 .76 43 -06 63 .05 63 .42 87 .26 87 .70 115 .40 "5 -90 147 -46 148 .03 42 o .96 5 -67 14 .31 26 .88 43 .37 63 -79 88 .14 116 .40 148 .60 43 I .01 5 -78 14 .49 2 7 .12 43 -68 64 .16 88 .57 116 .90 149 '17 44 I .06 5 -90 14 -67 27 "37 43 99 64 -54 89 .01 117 .41 149 -74 45 I .10 6 .01 14 .85 27 .6l 44 -30 64 .91 89 -45 117 .92 150 .31 46 i -*5 6 .13 IS -03 27 .86 44 '61 65 - 2 9 89 .89 118 %43 150 .88 47 I .20 6 .24 15 -21 28 .10 44 .92 65 .67 90 -33 118 .94 151 -45 48 I .26 6 .36 IS -39 28 .35 45 -24 66 .05 90 .78 "9 -45 152 .03 49 I .31 6 .48 15 -57 28 .60 45 -55 66 .43 91 .23 119 .96 152 .61 5 I .36 6 .60 15 -76 28 .85 45 -87 66 .81 91 .68 120 .47 153 -19 51 I .42 6 .72 15 -95 29 .10 46 .18 67 .19 92 .12 120 .98 153 -77 5 2 I .48 6 .84 16 .14 29 .36 46 .50 67 .58 92 -57 121 .49 154 -35 53 1 -53 6 .06 16 .32 29 .61 46 .82 67 .96 93 .02 122 .OI 154 -93 54 i -59 7 -09 16 .51 29 .86 47 .14 68 .35 93 -47 122 .53 55 -Si 55 i .65 7 .21 16 .70 30 .12 47 '46 68 .73 93 9 2 123 .03 156 .09 56 i .71 7 34 16 .89 30 .38 47 -79 69 .12 94 -38 123 .57 156 -67 57 i .77 7 -46 17 .08 30 .64 48 .n 69 . 5 I 94 -83 T24 .09 157 -35 58 i .83 7 .60 17 .28 30 .90 48 .43 69 .90 95 -29 12 4 .61 1^7 -8.1 59* i .89 7 -72 17 -47 31 .16 48 .76 7 .29 95 -74 125 -'3 158 -43 307- TABLES. 317 _ 2 sin 2 \t sin i" t 9 m 10 n m I2 m I3 I4 m i 5 m | 16- 0* 159" -02 196". 32 237" -54 282". 68 33i". 74 384". 74 44i". 63 502". 46 I 159 .6l 196 .97 238 .26 283 -47 S3 2 -59 385 -65 442 .62 53 -5 2 3 160 .20 160 .80 197 .63 198 .28 238 .98 239 -70 284 .26 285 .04 333 -44 334 -29 3f6 .56 387 .48 443 .60 444 -58 54 -55 505 -60 4 161 .39 198 .94 240 .42 285 .83 335 ^5 388 .40 445 -56 506 .65 5 161 .98 199 .60 241 .14 286 .62 336 -oo 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 39i -16 448 .53 509 -81 8 163 .77 201 .59 243 -33 289 .00 338 .58 392 .09 449 -Si 510 .86 9 164 -37 202 .25 244 .06 289 .79 339 -44 393 - OI 450 .50 511 .92 10 164 .97 202 .Q2 244 -79 290 .58 34<> -30 393 -94 451 .50 5 -98 ii 165 .57 20 3 .58 245 -52 291 .38 341 .16 394 -86 452 -49 5M -03 12 166 .17 204 .25 246 .25 292 ,l8 342 .02 395 -79 453 -48 5'5 -09 J 3 166 .77 204 .92 246 .98 292 .98 342 .88 396 .72 454 .48 5*6 .15 H 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 45<5 -47 518 .27 16 168 .58 206 .93 249 !9 295 .38 345 -49 399 -52 457 -47 519 -34 17 169 .19 207 .60 249 -93 346 -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 2IO .30 252 .89 299 .40 349 -84 404 .20 462 .48 524 -67 22 23 172 .24 172 .85 210 .98 211 .66 253 -63 254 -37 300 .21 3OI . O2 350 -71 35i -58 405 .14 406 .08 463 -48 464 .48 5 2 5 -74 526 .81 24 J 73 -47 212 .34 255 -12 3I -S3 35 -46 40 7 .02 465 .49 527 -89 2 5 174 .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 -Si 53 -03 2 7 175 -32 2I 4 .38 257 -37 3<>4 -27 355 -"o 409 .84 468 .52 531 " 28 175 -94 215 7 258 .12 305 .09 410 .79 469 -53 532 .18 29 176 .56 215 -75 2 5 8 .87 305 .90 356 -86 4" -73 470 -54 533 -26 3 177 .18 216 .44 259 .62 306 .72 357 -74 412 .68 47i -55 534 .33 3 1 177 .80 217 .12 260 .37 307 -54 358 .62 4i3 -63 472 -57 535 -4i 32 178 -43 217 .81 26l .12 308 .36 359 -Si 4M -59 473 -58 536 -5 33 179 .05 218 .50 261 .88 309 .^8 360 -39 4*5 -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 4*7 -44 476 .64 539 -75 36 180 .93 220 .58 264 .15 311 .65 363 .07 418 .40 477 - 6 5 540 .83 37 181 .56 221 .27 264 .91 3 12 -47 363 .96 4*9 -35 478 -67 54t -9i 38 182 .19 221 .97 265 .68 3i3 -30 364 .85 420 .31 479 .70 543 -oo 39 182 .82 222 .66 266 .44 314 .12 3 6 5 -75 421 .27 480 .72 544 -9 40 183 .46 22 3 .36 267 .20 3H -95 366 .64 422 .23 481 .74 545 -18 4i 184 .09 224 .06 267 .96 3'5 -78 3 6 7 -53 423 -19 482 .77 546 .27 42 184 .72 224 .76 268 .73 316 .61 368 . 42 424 -5 483 -79 547 -36 43 185 -35 225 .46 269 .49 317 -44 3^9 -3i 425 " 484 .82 548 -45 44 185 .99 226 .l6 270 .26 318 -27 370 -21 426 .07 485 -85 549 -55 45 186 .63 226 .86 271 .02 319 '10 37i " 427 -4 486 .88 55 -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 -9 1 428 .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 " 322 .45 374 -72 43 -9 491 .01 555 -03 50 189 .83 230 .39 274 .88 323 -29 375 -62 431 -87 49 2 .5 556 -13 51 190 .47 231 .10 275 -65 324 - T 3 376 -52 432 -84 493 .08 557 -24 52 I 9 I .12 231 .81 276 -43 324 -97 377 -43 433 -82 494 .12 558 -34 53 igi .76 232... 52 277 .20 325 -8i 378 -34 434 -79 495 -i5 559 -44 54 192 .41 233 -.24 277 -98 326 .66 379 .26 435 -76 496 .19 56o .55 55 193 .06 233 -95 2 7 8 .76 327 -5 380 .17 436 -73 497 -23 5i -65 56 193 .71 234 -67 279 -55 328 .35 381 .08 437 -7 1 498 .28 562 .76 57 194 -36 35 -38 280 .33 329 -19 381 .99 438 -69 499 -32 563 -87 58 59 195 '*A 195 .66 236 .10 236 .82 28l .12 28l .90 33 -04 330 .89 382 .90 383 -82 439 -67 440 .65 5o -37 501 . 4 x 564 .98 566 .08 GEODETIC ASTRONOMY. 2 sin 2 \t sin i" 17- ,8- 19- 20 m ., 22 m 23" 24- 25- o 1 567". 2 ~635".9~ 70S". 4 784". 9 865". 3 949". 6 1037". 8 1129". 9 1225". 9 I 568 .3 637 .0 709 .7 7 86 .2 866 .6 951 .0 1039 .3 1131 .4 1227 .5 2 569 -4 638 .2 710 .9 787 -5 868 .0 952 -4 1040 .8 "33 -o 1229 .2 3 570 -5 639 -4 712 .1 788 .8 869 .4 953 -8 1042 .3 "34 -6 1230 .8 4 57 1 -6 640 .6 7^3 -4 790 .1 870 .8 955 -3 1043 .8 1136 .2 1232 .5 s 572 .8 641 .7 714 .6 79 1 -4 872 .1 956 -7 1045 .3 "37 -8 1234 .1 6 573 -9 642 .9 715 .9 792 .7 873 -5 958 .2 1046 .8 "39 -3 I2 35 -7 7 575 - 644 .1 717 .1 794 -o 874 .9 959 -6 1048 .3 1140 .9 I2 37 .3 8 576 .1 645 -3 718 .4 795 -4 876 .3 961 .1 1049 -8 "42 -5 1239 .0 9 577 -2 646 .5 719 .6 796 .7 877 .6 962 .5 1051 .3 1144 .0 1240 .6 10 578 -4 647 .7 720 .9 798 .0 879 .0 963 -9 1052 .8 H45 - 6 1242 .3 ii 579 -5 648 .9 722 .1 799 -3 880 .4 965 .4 I0 54 -3 "47 -2 1243 .9 12 580 .6 650 .0 723 -4 800 .7 881 .8 966 .9 I0 55 -9 1148 .8 1245 - 6 13 581 -7 651 .2 724 -6 802 .0 883 .2 968 .3 1057 .4 "So .4 1247 .2 14 582 .9 652 .4 725 -9 803 -3 884 .6 969 .8 1058 .9 1152 .0 1248 .9 15 584 -o 653 -6 727 .2 804 .6 886 .0 971 .2 1060 .4 "53 -6 1250 .5 16 17 585 .1 586 .2 654 -8 656 .0 728 .4 729 .7 806 .0 807 .3 887 .4 888 .8 972 -7 974 -i 1062 .0 1063 .5 "55 .2 1156 .8 1252 .2 1253 .8 18 587 -4 657 -2 730 -9 808 .6 890 .2 975 -5 1065 .0 1158 .3 1255 -5 X 9 588 .5 658 .4 732 -2 809 .9 891 .6 977 -o 1066 .5 "59 '9 1257 .1 20 589 -6 659 -6 733 -5 8n .3 893 .0 978 -5 1068 .1 1161 .5 1258 .8 21 590 .8 660 .8 734 -7 812 .6 804 -4 979 -9 1069 . 6 "63 .1 1260 .5 22 591 -9 662 .0 736 .0 813 .9 895 .8 981 .4 1071 .1 1164 .7 1262 .2 23 593 - 663 .2 737 -3 8i<; .2 897 .2 982 .9 1072 .6 1166 .3 1263 .8 24 594 -2 664 .4 738 -5 816 .6 898 .6 984 .4 1074 .2 1167 .9 1265 .5 25 595 -3 665 .6 739 -8 817 .9 900 .0 985 .8 1075 .7 1169 .5 1267 . i 26 596 -5 666 .8 741 .1 819 .2 901 .4 987 -3 1077 .2 1171 .1 1268 .8 27 597 -6 668 .0 742 -3 820 .5 902 .8 988 .8 I0 7 8 .7 1172 .7 1270 .5 28 598 -7 669 .2 743 -6 821 .9 904 .2 990 -3 1080 .3 "74 -3 1272 ,i 29 599 -9 670 .4 744 -9 823 .2 905 -6 991 .8 1081 .8 "75 -9 1273 .7 3 601 .0 671 .6 746 .2 824 .6 907 .0 993 -2 1083 -3 "77 -5 1275 -4 3 1 602 .2 672 .8 747 -4 825 .9 908 .4 994 -7 1084 .8 "79 - 1 1277 .1 3 2 33 34 603 -3 604 .5 605 .6 674 .1 675 -3 676 .5 748 .7 750 .0 751 .3 827 .3 828 .6 829 .9 909 .8 911 .2 912 .6 996 .2 997 .6 999 -i 1086 .4 1087 .9 1089 .5 1180 .7 1182 .3 "83 .9 1278 .8 1280 .4 1282 .1 35 606 .8 677 .7 752 .6 831 .2 914 .0 1000 .6 1091 .0 "85 -5 1283 .8 36 607 .9 678 .9 753 -8 832 .6 9i5 -5 1002 .1 1092 .6 1187 .1 1285 .5 11 609 .1 610 .2 680 .1 681 .3 755 -i 756 -4 833 -9 916 .9 918 .3 1003 .5 1005 .0 1094 .1 1095 -7 1188 .7 "9 -3 1287 .1 1288 .8 39 6n .4 682 .6 757 -7 836 .6 919 .7 1006 .5 1097 .2 1191 .9 1290 .5 40 612 .5 683 .8 759 -o 838 .0 921 .1 1008 .0 1098 .8 "93 -5 1292 .3 4 1 613 -7 685 .0 760 .2 839 -3 922 .5 1009 .4 IIOO .3 "95 * 1293 .8 42 614 .8 686 .2 76l -5 840 .7 923 -9 1010 .9 nor .9 1196 .7 1295 -5 43 616 .0 687 .4 762 .8 842 .0 925 -3 IOI2 .4 1103 .4 "98 .3 1297 .2 44 617 .2 688 .7 764 .1 843 -4 926 .8 1013 .9 1105 .0 "99 -9 1298 .9 45 618 .3 689 .9 765 -4 844 .7 928 .2 1015 .4 1106 .5 1201 .5 1300 .5 46 619 .5 691 .1 766 .7 846 .1 929 .6 1016 .9 1108 .1 1203 .1 I 302 . 2 47 620 .6 692 .4 768 .0 847 -5 931 .0 ioiS .4 1109 .6 1204 .7 1303 .9 48 621 .8 693 -6 769 -3 848 .9 932 .4 1019 .9 IIII .2 I206 .4 1305 -6 49 623 .0 694 .8 770 .6 850 .2 933 -8 IO2I .4 III2 .7 I2O8 .O 1307 -3 50 624 .1 696 .0 771 .9 851 .6 935 -2 936 .6 1022 .8 III4 .3 1209 .6 1309 .0 5 1 52 625 .3 626 .5 697 .3 698 .5 '773 -I 774 -5 854 938 .1 1025 .8 III7 .4 1212 .9 1312 .4 53 627 .6 699 .7 775 -7 855 -7 939 -5 1027 .3 1118 .9 1214 .5 1314 .1 54 628 .8 701 .0 777 .1 857 -I 940 .9 1028 .8 1120 .5 1216 .1 1315 -7 55 630 .0 7O2 .2 778 .4 858 .4 942 -3 1030 . 3 1122 .O 1217 .7 1317 .4 56 631 .2 703 -5 779 -7 859 .8 943 .8 1031 .8 1123 .6 1219 .4 1319 .1 57 632 .3 704 .7 781 .0 861 .1 945 -2 1033 .3 1125 .1 1221 .O 1320 .8 58 6 33 -5 705 -9 782 .3 862 .5 946 .6 1034 .8 1126 .7 1222 .6 1322 .5 59 6 34 -7 707 .1 783 .6 863 .9 948 .1 1036 .3 1128 .3 1224 .2 1324 -2 307- TABLES. 319 2 sin* 2 sin< it sin i" 2 sin* |/ sin i" t 26- 27" 28 m 29" OP 1325". J429". I537"-- 1649". o I 1327 1431 . 1539 .; 1650 .9 2 1329 . 1433 1541 .: 1652 . 3 1 33 1 . 1434 1654 .7 4 1332 . 1436 . 1544 . 1656 .6 5 1334 1438 . 1546 1658 .5 6 1336 . 1440 . 1548 .4 1660 .4 7 1337 I 44 2 . 1550 .2 1662 .3 8 H43 1664 .2 9 1341 M45 1553 -9 1666 .1 10 1342 . 1447 1555 -8 1668 .0 i 1344 1449 1557 -6 1669 . 9 2 1346 - *559 -5 1671 .( 3 1348 . 1452 1561 .'. 1673 -8 4 1349 . 1454 1563 -2 1675 -7 1 1353 1456 . 1458 . 1565 .0 I 5 66 . 9 1677 .6 l6 79 -5 7 1354 1459 1568 .7 1681 .4 8 1461 . I 57 -5 1683 .3 9 1358 '. 1463 . 1572 .4 1685 .2 20 1360 . 1465 . J 574 -3 1687 -2 21 1361 . 1466 . 1576 .1 1689 .1 22 1363 -5 1468 . 1578 .0 1691 .0 23 1365 .2 1470 . 1579 -8 1692 .9 24 1367 .0 1472 .3 1581 .7 1694 .8 25 1368 .7 1474 . 1583 -5 1696 .7 26 1370 .4 1475 -9 1585 -3 1698 .6 27 1372 . 1477 -7 1587 -2 1700 .5 28 1373 -9 T 479 -5 1589 -i 1702 .5 29 1375 -6 1481 .3 1590 .9 1704 .4 30 1377 -3 1483 .1 1592 .7 1706 .3 31 1379 -0 1484 .9 1594 .6 1708 .2 32 1380 .8 1486 .7 *596 -5 1710 .2 33 1382 .5 1488 .5 1598 .3 1712 .1 34 1384 .2 H90 .3 1600 .2 1714 .0 11 1385 -9 1387 -7 1492 .1 l6o2 . I 1604 .0 1715 -9 1717 .9 37 1389 .4 T 495 '7 1605 .9 1719 .8 38 1391 -2 J 497 -5 1607 .7 1721 .7 39 1392 .9 1499 -3 1609 .6 1723 -6 40 1394 -7 1501 .i 1611 .5 1725 .6 41 1396 .4 1502 .9 1613 .3 1727 -5 4 2 1398 -2 1504 -7 I6l 5 .2 43 X 399 -9 1506 .5 1617 .1 1731 ' 44 1401 .7 1508 .4 1619 .0 1733 -4 n 1405 .2 1510 .2 1512 .0 1620 .8 1622 .7 *735 -3 1737 .2 47 1406 .9 1513 -8 1624 .6 1739 -2 48 1408 .7 1515 .6 1626 .5 1741 .2 49 1410 .4 57 -4 1628 .3 1743 .1 So 1412 .2 519 -2 630 .2 1745 -I 51 I 4 I 3 -9 521 .O 632 .1 1747 .0 52 1415 -7 522 .9 634 .0 1749 -0 53 54 1417 -4 1419 -2 524 -7 526 .5 635 -9 637 '7 1750 .9 1752 .8 55 1420 .9 528 .3 639 -6 1754 -8 5 6 1422 .7 530 .2 641 .5 1756 .8 57 1424 .4 S3 2 .0 6 43 -3 1758 -7 58 59 1426 .2 1427 .9 533 .8 535 - 6 645 .2 647 .1 I7 ^ 'I 1762 .6 t_ t it o m o".oc 20 "4g I .OC 10 2 O .OC 20 J 3 o .00 30 65 4 o .00 40 5 o o .0 50 .76 6 o .0 21 82 7 o o .0 IO A 8 o .0 20 .< 9 o o .o< 3 ,j 10 o .09 40 .0 II . I 50 .12 12 O .1 22 O .19 10 .2 10 .2< 20 .2 2O t - 30 .2 3 .'. 40 o .2) 40 4 50 .2 5 6 10 O .2 IO '.69 20 o .30 20 77 30 o .3 30 85 4 o .3 40 9; So o -3 5 .01 10 o .3? IO 3 IO 3 -18 20 20 3 -27 30 4O o .41 3 3 -36 50 -45 50 3 -55 15 o o .47 25 o 3 -64 10 o .49 10 3 -74 20 o .52 20 3 -3 4 30 o .54 30 3 -94 40 o .56 40 4 -05 50 -59 5 4 '*5 16 o o .61 26 o 4 .26 IO o .64 TO 4 -37 20 o .67 20 3 o .69 3 4 .60 50 -75 5 4 -83 10 o .81 10 5 .08 20 .84 20 5 -20 3 o .88 30 5 -33 40 o .91 40 5 -46 50 o -95 5 5 .60 18 o o .98 8 o 5 -73 IO I .02 10 5 -87 20 I .06 20 ' -OI 3 .09 30 6 .15 40 11 40 3 50 iS 5 44 19 o .22 9 59 IO .26 10 75 20 3 20 .90 3 35 30 .06 40 .40 40 .22 50 44 So .38 20 o 49 o o 55 t O I 4 " o' '.000 *5 o .001 16 o .001 17 O .001 18 o .002 19 .002 20 .003 21 o .004 22 .005 23 o .007 2 4 o .009 11 o o .Oil .014 27 .017 28 o .021 29 o .026 30 o .032 3 20 GEODE TIC A S TRONOM Y. 308. 308. (See 175.) Hour- angle be- fore or after The Correction to be applied to the Latitude of the station to obtain the apparent altitude of Polaris. Computed for the declination 88 46' and the mean refraction. Correc- tion for i' in- crease upper Culmina- Latitude Latitude Latitude Latitude Latitude Latitude Latitude in the declina- tion. 3 o 35 40 45 50 55 60 Polaris. o h oo m 4-1 is'- 6 4-i i5'-3 4-i 15'- 1 4-i H'-9 + 1 o I4 , 8 4-i i4'-6 4-i 14'- 5 I'.Q 15 4-i 15 -4 4-i 15-2 4-i 14 -9 + i 14.8 4-1 14^6 4-i 14 .4 4-i 14 .3 I .0 3 4-i 14-9 4-i 14 .7 4-i 14.5 4-i 14-3 4-i M .2 4-i 14 -o 4-1 13 -8 I .0 45 oo 4-1 14 .2 4-1 13 .0 + 1 13 -9 4-i 12 .8 4-i 13 -7 4/i I2 .5 - 3 :l + '3-3 + 1 12 .2 4-1 I 3 .2 4-1 12 .0 j-i 13 -o 4-i ii .9 i .0 I .0 15 4-i ii .6 4-i ii .3 4-i ii .1 -t-i 10.9 4-x io .8 4-i io .6 4-i io .4 o .9 30 45 00 4-1 09 .9 4-i 07 .9 4 i 05 .6 4-i 09 .6 + 1 07 .6 4- 1 05 .3 4-1 09 .4 4-i 07 .3 4-i 05 .0 + 1 09 .2 4-i 07 .2 + 1 04 .8 4-i 09 .0 4-i 07 .0 4-i 04 .6 4-i 08 .8 4-i 06 .8 4-1 04 .4 4-i 08 .6 4-1 06 .6 4-i 04 .2 o .9 -o .9 o .8 15 4-i 03 .0 + 1 02 .7 4-i 02 .4 + 1 02 .2 4"! O2 ,O 4-i oi .8 + i oi .6 o .8 3 4" I OO . 1 4-o 59 .8 4-o 59 .5 + 59 .3 4-o 59 .1 4-o 58 .9 4-o 58 .7 -0.8 45 4-o 57 .0 4-0 56.7 TO 56 .5 4"O 56 .2 4-o 56 .0 4-0 55 -8 4-o 55 -5 o .7 3 5 3 30 4"0 53 .7 -j-o 50 .1 + o 4 6 . 4 4-o 53 .4 + o 49.8 + 46 .0 4-0 53 .1 4-0 49 -5 + o 45 -7 4-o 49 .2 4-0 45 .5 4-o 49 .0 4-0 45 .2 +0 48 .8 4-o 45 .0 -|-o 52 . i 4-0 4 8 .5 4-o 44 .7 '7 o .6 o .6 3 45 4-o 42 .4 4o 42 .1 + o 41 .8 4o 41 .5 4-0 41 .3 + 41 .0 4-o 40 .7 -5 4 oo 4-0 38 .3 -t-o 38 .0 + o 37 -6 +o 37 -4 4-o 37 .1 +o 36 .8 4-0 36 .5 o .5 4 1 5 4o 34 .0 4-0 33 -6 4-o 33 .3 4-o 33 .0 4-o 32 .8 4-o 32 .5 4-0 32 .1 o .4 4 3 4-o 29 .6 4"O 29 .2 + o 28 .9 -4-0 28 .5 4-o 28 .3 4-0 28 .0 4-0 27 .6 o .4 4 45 4-o 25 .0 4o 24 .6 + o 24 .3 4-0 24 .0 +0 23 .7 4-o 23 .4 4o 23 .0 o .3 5 oo 5 15 5 3 4-0 20 .4 + 15 .6 + o io .8 + 20 .0 j-o 15.3 + 10 .4 4o 19 .7 4-o 14 .9 4o io . i 4-o 19 .4 + 14 .6 4-0 00 .9 4-o 19 .1 j-o 14.3 4-0 09 .6 4-o 18 .8 4-o 18 .4 4-0 14 .o4o 13 .6 +0 09 .2 +o 08 .8 .2 O .2 .1 5 45 -j-o 06 .0 4-0 05 .6 4-0 05 .3 4-o 05 .0 4-o 04 .7 4-o 04 .4 4-o 04 .0 o .0 6 oo + 01 .2 4"0 oo .8 + o oo .5 4"O OO .2 o oo ,i o oo .5 o oo .9 .0 6 15 o 03 .6 o 04 .0 o 04 .4 o 04 .7 o 05 .0 o 05 .3 o 05 .7 4-0 .1 6 30 o 08 .4 o 08 .8 09 .2 o 09 .5 09 .2 o io .1 o io .4 -j-o.i 6 45 7 oo o 17 .9 o 18 .3 o 18 .6 o 18 .9 19 .2 o 19 .6 o 19 .9 4-0.3 7 15 O 22 .5 O 22 .9 23 .2 o 23 .6 o 23 .8 24 .2 o 24 .6 4-0.4 7 3 o 27 .0 o 27 .4 o 27 .7 o 28 .0 o 28 .3 o 28 .6 o 29 .0 4-0.4 7 45 o 31 .4 31 .8 o 32 .1 o 32 .4 o 32 .7 o 33 .0 o 33 -3 4-0.5 8 oo o 35 .6 o 36 .0 o 36 .3 -o 36 .6 o 36 .9 o 37 .2 o 37 -5 4-o.s 8 15 o 39 '7 o 40 . i o 40 .4 o 40 .7 o 41 .0 O 41 .2 o 41 .6 + o .6 8 30 o 43 .6 o 44 .0 o 44 -3 o 44 .6 -o 44 .8 o 45-i o 45 .4 4-o .6 8 45 o 47 -3 o 47 .7 o 48 .0 o 48 .3 -o 48 .5 o 48 .8 o 49 .0 4-0.7 9 oo o 50 .8 O 51 . 2 o 5 l -5 -o 51 .7 o 51 .9 o 52 .1 o 52 .4 -j-o .7 9 15 o 54 .1 54 -5 o 54 -7 o 55 -o o 55 -2 55 -5 o 55 -7 4-0 .8 9 30 o 57 .2 57 -5 -o 57 -8 o 58 .0 o 58 .2 -o 58 -5 -o 58 .7 4-o .8 9 45 I 00 .0 i oo .3 i oo . 6 i oo .8 I OI .O I 01 .2 i oi .4 -t-o .8 10 OO i 02 .5 i 02 .8 i 03 . i i 03 .3 I 03 .4 i 03 .6 i 03 .9 4-0 .9 10 15 i 04 .7 i 05 .0 i 05 .3 i 05.5 I 05 .7 I 05 .9 i 06 .1 4-o .9 10 30 i 06 .7 1 07 .0 I 07 . 2 i 07 .5 i 07 .6 I 07 .9 i 08 .0 4-o .9 io 45 -i 08.4 -I 08.7 -I 08.9 I 09 .2 i 09 .3 I 09 .5 I 09 .7 4-o .9 II 00 i 09 .8 I IO . I i io .5 i io .6 I 10 .9 I II .0 4-i -o ii 15 i 10 .8 I II .1 -i ii .4 i ii .6 -i ii .8 I 12 .0 1 12 .1 4-1 .0 ii 30 i ii .6 I II .9 I 12 .2 I 12 .4 -i 12.5 I 12 .8 1 12 .9 4-1 -o " 45 I 12 .1 -I 12.4 I 12 .6 I 12 .9 i 13 .0 I I 3 .2 i 13 -3 4-i .0 12 00 i 12.3 I 12 .6 I 12 .8 i 13 -o -I I 3 .21 1 3 . 3 -i 13-4 4-i .0 309- TABLES. 321 309. CORRECTION FOR ERROR OF RUN OF A MICROMETER The tabular value is the correction to the forward reading. The sign of the correction is the same as that of the difference Backward Forward. Use this table with such a micrometer as is described in 189, and no other. Forw'd Read- ] 3. - F Forw'c Read- ing. _// _ _// _ ing. 2 .O 3-o o' 15" o".o o".o o".o o".i o".i o".i o".i o".i 0".2 o' 15" o 30 o".o .1 .1 .1 .1 .1 .2 O .2 .2 o -3 o 30 o 45 .1 O . I .1 .2 .2 .2 .2 o -3 o -3 o .4 45 I OO O . I .1 O .2 O .2 O .2 o -3 o -3 o .4 o .4 .6 I 00 I 15 o".o O . I .2 .2 O .2 o -3 o .4 o .4 o .4 o .5 o .8 I 15 I 30 . .1 .2 .2 o -3 o .4 o .4 o -5 o -5 o .6 o .9 I 30 i 45 o . .1 O .2 o -3 o .4 o .4 o -5 o .6 o .6 o .7 I .0 i 45 2 00 o . O .2 .2 o -3 o .4 o -5 o .6 o .6 o .7 o .8 .2 2 OO 2 15 . O .2 o -3 o .4 o .4 o -5 o .6 o .7 o .8 o .9 4 2 15 2 30 . .2 o -3 o .4 o -5 o .6 o .7 o .8 o .9 i .0 5 2 30 2 45 o . .2 o -3 o .4 o .6 o .7 o .8 o .9 .0 .i .6 2 45 3 oo o . O .2 o .4 o -5 o .6 o .7 o .8 .0 .1 .2 .8 3 oo 3 15 . o -3 o .4 o -5 o .6 o .8 o .9 .0 .2 3 2 .0 3 15 3 30 . o -3 o .4 o .6 o .7 o .8 .0 .1 3 4 2 .1 3 30 3 45 .2 o -3 o .4 o .6 o .8 o .9 .0 .2 4 5 2 .2 3 45 4 oo O .2 o -3 o -5 o .6 o .8 I .0 .1 3 4 .6 2 .4 4 oo 4 15 O .2 o -3 o .5 o .7 o .8 I .0 .2 4 5 7 2 .6 4 15 4 30 .2 o .4 o -5 o .7 o .9 i .1 3 4 .6 .8 2 -7 4 30 4 45 .2 o .4 o .6 o .8 I .0 i .1 3 I -5 7 9 2 .8 4 45 5 oo O .2 o .4 o .6 o .8 I .0 I .2 4 I .6 .8 2 .0 3 -o 5 oo 322 GEODETIC ASTRONOMY. 310. 310. Correction for Hour-angle before or after Upper AZIMUTH OF POLARIS COMPUTED FOR DECLINATION 88 46'. i' Increase in Declination of Polaris. Culmination. Lat. 30. Lat. 31. Lat. 32. Lat. 33. Lat. 34. Lat. 35. Lat. 30. o h 15 o 05' 40" o 05' 43" o 05' 47" o 05' 51" o 05' 55" o 06' oo" s" o 30 o ii 18 o n 25 o ii 33 o ii 41 o ii 49 II 58 9 o 45 I 00 o 10 53 22 23 22 38 o 22 53 o 23 09 o 23 26 *7 53 o 23 44 18 i 15 o 27 48 o 28 06 o 28 25 o 28 45 o 29 06 o 29 28 23 i 30 o 33 05 o 33 26 o 33 49 o 34 13 o 34 38 o 35 04 27 i 45 o 38 13 o 38 38 o 39 04 o 39 32 o 40 oo o 40 3 3 1 2 00 o 43 12 o 43 40 o 44 09 o 44 40 o 45 12 o 45 46 35 2 15 o 47 58 o 48 29 o 49 02 o 49 36 50 12 o 50 50 39 2 3 o 52 32 o 53 06 o 53 42 o 54 19 54 59 55 4 43 2 45 o 56 52 o 57 29 o 58 07 o 58 48 o 59 30 i oo 15 46 3 oo 3 15 I 00 58 i 04 47 i oi 37 i 05 28 I 02 l8 I 06 12 i 03 oi i 06 58 i 03 46 i 07 46 1 4 34 i 08 36 50 3 3 3 45 i " 33 I 12 18 I 09 48 I 13 06 i 13 56 i 14 49 i 15 45 -1 4 oo i 14 28 ! I5 I5 i 16 05 i 16 57 i 17 52 i 18 50 61 4 15 i 17 04 I I 7 52 i 18 44 * 19 37 i 20 34 i 21 34 -63 4 30 i 19 19 I 2O 09 I 21 02 i 21 57 i 22 55 1 23 7 64 4 45 I 21 I 4 I 22 05 i 22 59 i 23 55 i 24 55 i 25 57 66 5 oo I 22 48 I 23 40 1 24 35 i 25 32 i 26 32 i 27 36 68 5 15 i 24 oo i 24 53 i 25 48 i 26 46 i 27 47 i 28 51 -69 5 30 1 24 51 i 25 44 i 26 40 i 27 38 i 28 39 i 29 44 -69 5 45 I 25 20 i 26 13 i 27 09 i 28 07 t 29 09 i 30 14 70 6 oo I 25 27 i 26 19 i 27 15 i 28 14 i 29 15 i 30 20 70 6 15 I 2 5 12 i 26 04 i 26 59 i 27 57 t 28 59 i 30 03 -69 6 30 6 45 i 24 34 i 23 36 i 25 27 i 24 27 I 26 21 I 25 21 i 27 19 i 26 18 i 28 19 i 27 17 i 29 23 i 28 20 68 67 7 oo I 22 l6 i 23 06 i 23 59 1 24 55 i 25 53 i 26 55 66 7 X 5 i 20 35 I 21 2 5 I 22 l6 i 23 10 I 24 08 i 25 08 -65 7 30 i 18 34 I 19 22 I 20 12 I 21 05 I 22 OO i 22 59 64 7 45 i 16 13 i 16 59 I I 7 48 i 18 39 i 19 33 i 20 29 62 8 oo i 13 33 i 14 17 1 J 5 4 i ^5 53 i 16 45 i 17 39 60 8 15 8 70 i 10 34 i ii 16 I 12 01 I 12 48 i *3 37 i 14 29 57 8 45 I O7 17 i 03 43 i 07 57 I 04 22 I 05 02 i 05 44 i 06 29 i 07 15 54 5 1 9 oo o 59 54 i oo 30 i oi 07 i oi 47 I 02 29 I 03 12 -48 9 IS 55 49 o 56 23 o 56 58 o 57 34 o 58 13 o 58 54 45 9 30 9 45 o 51 31 o 46 59 o 52 oi o 47 27 S 2 34 o 47 57 o 53 08 o 48 28 o 53 43 o 49 oo o 54 21 o 49 34 42 -38 10 00 o 42 16 o 42 42 o 43 08 o 43 3 6 o 44 05 o 44 35 34 10 15 o 37 23 o 37 45 o 38 08 o 38 33 38 59 o 39 26 3 10 30 32 20 o 32 39 3 2 59 o 33 20 o 33 43 o 34 06 26 10 45 o 27 09 o 27 25 o 27 42 o 28 oo o 28 18 o 28 38 22 II OO O 21 51 O 22 04 22 l8 22 32 o 22 47 o 23 03 T8 " 15 o 16 28 o 16 38 o 16 48 o 16 59 o 17 10 O 17 22 13 II 30 II 01 II 08 o ii 14 O II 22 o ii 29 o ii 37 9 " 45 Elongation: Azimuth.... o 05 31 I 25 27 o 05 34 I 26 20 o 05 38 i 27 16 o 05 42 I 28 14 o 05 45 i 29 16 o 05 49 I 30 20 4 -69 h. nt. s h. nt. s. h. nt. s. h. nt. s. h. nt. s. h. nt. s. s. Hour-angle. 5 57 09 5 57 02 5 56 55 5 56 48 5 56 40 5 5 6 33 ~\~ 2 TABLES. 323 Hour-angle before or after Upper Culmination. AZIMUTH OF POLARIS COMPUTED FOR DECLINATION 88 46'. Correction for i' Increase in Declination of Polaris. Lat. 35. Lat. 36. Lat. 37. Lat. 38. Lat. 39. Lat. 40. Lat. 40. O h TS m o 06' oo" o 06' 05" o 06' 10" o 06' 15" o 06' 20" ,0 06' 26" 5" o 30 II 58 12 08 12 l8 12 28 o 12 39 o 12 50 10 o 45 o 17 53 o 18 07 l8 22 o 18 38 o 18 54 o 19 ii 16 i i5 o 23 44 o 29 28 o 29 51 o 30 15 o 3 4 1 o 25 04 o 25 27 o 31 08 p 31 36 26 i 30 o 35 04 o 35 3i o 41 02 o 36 oo o 41 35 o 36 31 o 37 02 o 37 36 3 1 ,6 2 OO o 40 30 o 45 46 O 46 22 o 47 oo o 47 39 o 48 21 o 49 04 3 40 2 15 o 50 50 o 51 29 o 52 ii o 52 55 o 53 4* o 54 29 45 2 30 o 55 40 o 56 23 o 57 09 o 57 57 o 58 47 p 59 4 49 2 45 i oo 15 I 01 02 i oi 51 i 02 43 1 3 37 1 4 34 53 3 15 i 04 34 i 08 36 I 05 24 I 09 29 i 06 17 i 10 25 I 07 12 I II 24 i 08 10 i 12 25 i 09 12 i 13 30 -S, fll 3 3 3 45 i IS 45 i 16 43 i 14 14 i 17 44 i 18 49 i 19 57 I 21 08 3 66 4 oo i 18 50 i 19 50 i 20 54 I 22 OI i 23 ii I 24 25 -69 4 15 i 21 34 I 22 36 i 23 42 I 24 51 i 26 03 I 27 20 72 4 30 i 23 57 i 25 oi i 26 08 I 27 19 i 28 33 I 29 52 74 4 45 i 25 57 I 27 03 I 28 12 I 29 24 i 30 40 i 32 oo 75 5 oo i 27 36 I 28 42 I 29 52 I 31 06 i 32 23 i 33 44 76 5 T 5 i 28 51 i 2 9 59 I 31 09 I 32 24 * 33 42 i 35 4 77 5 3 i 29 44 i 30 52 I 32 3 i 33 18 i 34 37 * 35 59 -78 5 45 i 3<> 14 I 3 1 21 i 32 33 i 33 48 i 35 07 i 36 30 -78 6 oo I 30 20 i 31 27 i 32 39 i 33 54 i 35 13 i 36 35 -78 6 15 I 3 3 i 31 10 i 32 21 i 33 3 6 i 34 54 i 36 16 -78 6 30 I 29 23 i 30 30 i 31 40 i 32 54 i 34 " i 35 32 77 6 45 I 28 2O i 29 26 i 3 35 i 3i 48 t 33 04 1 34 24 76 7 oo i 26 55 i 27 59 i 29 07 i 30 18 3i 33 i 32 52 75 7 15 i 25 08 i 26 ii i 27 17 i 28 26 i 29 39 i 30 56 73 7 30 i 22 59 i 24 oo i 25 04 I 26 12 i 27 23 i 28 38 72 7 45 I 20 29 I 21 28 I 22 3 I 2 3 36 i 24 45 i 25 57 -69 8 oo 1 17 39 i 18 36 i 19 36 i 20 39 i 21 45 i 22 54 66 8 15 i 14 29 i 15 24 I 16 21 I 17 22 i 18 25 i 19 31 64 8 30 I II 01 i " 53 I 12 48 ' 13 45 i 14 45 i 15 48 61 8 45 i 07 15 i 08 04 I 08 56 i 09 50 i 10 47 i ii 47 -58 9 oo I 03 12 i 03 58 i 04 47 i 05 38 i 06 31 i 07 27 54 9 *5 o 58 54 o 59 37 I OO 22 i oi 09 i oi 59 I 02 51 5 9 3 o 54 21 o 55 oo o 55 42 o 56 25 o 57 " o 57 59 -46 9 45 49 34 o 50 10 o 50 48 o 51 27 o 52 09 o 52 53 42 IO OO o 44 35 o 45 08 o 45 42 o 46 17 o 46 54 o 47 34 -38 10 15 o 39 26 39 54 o 40 24 o 40 55 o 41 28 o 42 03 34 10 30 o 34 06 o 34 30 o 34 57 o 35 24 o 35 52 O 36 22 29 10 45 o 28 38 o 28 59 29 20 o 29 43 o 30 07 o 30 32 24 II 00 o 23 03 o 23 19 o 23 37 o 23 55 o 24 14 o 24 35 20 11 !5 17 22 o 17 35 o 17 48 o 1 8 02 o 18 16 o 18 31 15 II 30 o ii 37 o ii 46 " 54 O 12 04 12 13 12 23 10 Elongation: Azimuth.... o 05 49 I 30 20 o 05 53 i 31 28 o 05 58 i 32 40 o 06 02 33 55 o 06 07 i 35 M 06 12 I 3 6 36 5 -78 h. nt. s. h, nt. s. h. nt. s. h. nt. s. h. nt. s. h. nt. t. s. Hour-angle. 5 56 33 5 56 25 5 56 17 5 56 09 5 56 oo 5 55 5 2 + 3 324 GEODETIC ASTRONOMY. 310. Hour-angle before or after Upper Culmination. AZIMUTH OF POLARIS COMPUTED FOR DECLINATION 88 46'. Correctionfor i' Increase in Declination of Polaris. Lat. 40 Lat. 41. Lat. 42. Lat. 43. Lat. 44 Lat. 45 Lat. 40. O h I 5 m o 06' 26' o 06' 32" 06' 39" o 06' 45' o 06' 52' o 07' oo" ~~~ 5 o 30 12 50 o 13 03 13 is o 13 29 o 13 43 o 13 58 10 o 45 I 9 II o 19 30 19 48 o 20 08 o 20 29 20 52 16 1 00 o 25 27 o 25 51 26 16 o 26 43 o 27 io o 27 40 21 I 15 o 31 36 o 32 05 32 36 o 33 09 33 44 o 34 21 26 I 30 37 36 o 38 ii 38 48 o 39 27 o 40 09 o 40 52 31 i 45 43 26 o 44 07 44 50 o 45 35 o 46 22 o 47 12 -36 a oo 49 04 o 49 50 50 39 o 51 29 o 52 23 53 19 40 2 15 54 29 o 55 20 56 14 o 57 10 o 58 io o 59 12 45 2 3 59 40 i oo 35 oi 34 I 02 36 I 03 41 i 04 49 49 2 45 4 34 i 05 34 06 38 i 07 44 i 08 54 I 10 08 53 3 oo 09 12 i io 16 ii 24 1 12 35 i 13 50 i 15 09 57 3 15 I I 3 3 i 14 38 15 So i 17 06 i 18 25 i 19 49 -60 3 3 I 17 29 i 18 41 9 57 I 21 l6 i 22 39 i 24 08 63 3 45 I 21 08 I 22 2 3 i 23 42 I 25 04 I 26 32 i 28 04 66 4 oo I 2 4 25 i 25 43 i 27 05 i 28 31 i 30 oi I 3i 37 -69 4 15 I 27 2O i 28 40 i 3 4 1 3i 33 i 33 07 i 34 45 72 4 30 I 2 9 S 2 i 3 r 14 i 32 41 i 34 12 i 35 48 i 37 29 74 4 45 i 32 oo i 33 24 i 34 53 t 36 25 i 38 04 i 39 47 75 5 oo i 33 44 1 35 I0 i 3 6 40 i 38 14 1 39 54 i 4i 38 -76 5 i5 1 35 04 i 36 30 I 38 02 i 39 37 i 41 18 i 43 04 77 5 30 i 35 59 i 37 26 I 38 5 8 i 40 34 i 42 16 i 44 02 -78 5 45 i 36 30 i 37 57 i 39 29 i 41 05 i 42 47 i 44 34 -78 6 oo 1 36 35 I 38 02 1 39 34 i 41 io i 42 51 i 44 S 8 -78 6 15 i 36 16 i 37 43 i 39 M i 40 49 i 42 30 i 44 16 -78 6 30 i 35 32 i 36 58 i 38 28 i 40 03 i 41 42 i 43 27 77 6 45 i 34 24 i 35 48 i 37 17 i 38 50 i 40 28 i 42 12 -76 7 oo i 32 52 i 34 IS i 35 42 * 37 13 i 38 49 i 40 31 75 7 i5 i 30 56 i 32 17 i 33 42 i 35 " i 36 45 i 38 24 73 7 3 28 38 i 29 56 i 31 19 i 32 46 i 34 i7 i 35 53 72 7 45 i 25 57 i 27 13 i 28 33 i 29 56 i 3i 25 i 32 58 -69 8 oo 22 54 i 24 07 i 25 24 i 26 45 I 28 10 i 29 40 66 8 15 19 3 1 i 20 41 i 21 55 i 23 12 ' 24 33 i 25 59 64 8 30 15 48 i 16 55 i 18 05 i 19 18 i 20 35 i 21 57 61 8 45 ii 47 i 12 49 i 13 55 i i5 05 i 16 18 i 17 35 -58 9 oo 07 27 i 08 26 t 09 28 i io 33 i ii 41 i 12 54 54 9 15 i 02 51 i 3 45 i 04 43 i 05 43 i 06 47 i 7 54 50 9 3 o 57 59 o 58 49 o 59 42 I 00 3 8 i oi 37 I 02 38 -46 9 45 f > 52 53 o 53 39 o 54 27 o 55 18 o 56 ii o 57 07 42 io oo 47 34 o 48 15 o 48 58 o 49 44 o 50 32 o 51 22 -38 10 15 io 30 o 42 03 36 22 o 42 39 o 36 53 o 43 18 o 37 26 o 43 58 o 38 or o 44 40 o 38 3 8 o 45 25 o 39 16 34 29 1 45 o 30 32 o 30 58 o 31 26 3 1 55 o 32 26 o 32 58 24 It 00 o 24 35 o 24 56 o 25 18 o 25 42 o 26 06 o 26 32 20 IT 15 o 18 31 o 18 47 o 19 04 19 22 o 19 40 o 20 oo IS II 3 12 23 o 12 34 o 12 45 o 12 57 o 13 09 o 13 23 IO ii 45 O 06 12 o 06 18 o 06 23 o 06 29 o 06 36 o 06 42 5 Elongation: Azimuth. .. . I 36 3 6 i 38 03 1 39 35 I 41 II 42 53 44 40 -78 h m s h. tn s t. W J" ft ttl S '. fft S t . ftt , S . J Hour-angle. 5 55 52 5 55 43 5 55 34 5 55 24 55 M 55 4 + 3 TABLES. 325 Hour-angle before or after Upper Culmination. AZIMUTH OF POLARIS COMPUTED FOR DECLINATION 88 46'. Correction for i' Increase in Declination of Polaris. Lat. 45 . Lat. 46. Lat. 47. Lat. 4 8. Lat. 49. Lat. 50. Lat. 50. o* i 5 m o 07' oo" o 07' 08" o 07' 16' o 07' 25" 07' 34' o 07' 44' 6" 3<> o 13 58 o 14 13 o 14 30 o 14 48 15 06 o 15 25 T 3 45 20 52 21 15 o 21 40 O 22 06 22 33 23 02 19 oo o 27 40 28 II o 28 44 o 29 18 29 55 3o 33 25 *5 o 34 21 o 34 59 o 35 40 o 36 23 37 08 37 56 32 30 o 40 52 o 41 38 o 42 26 o 43 17 44 ii 45 08 -38 45 o 47 12 o 48 05 o 49 01 49 59 51 02 52 07 43 oo o 53 *9 54 '9 o 55 22 o 56 28 57 38 58 52 49 15 o 59 12 I 00 l8 I OI 28 i 02 41 3 59 05 21 54 30 i 04 49 i 06 01 I 07 17 i 08 38 10 03 II 32 59 2 45 I 10 08 I II 26 I 12 48 1 !4 15 15 47 17 2 4 -64 3 oo i 15 09 i 16 32 i 18 oo 1 J 9 33 21 II 22 54 68 3 15 i 19 49 I 21 17 I 22 50 i 24 29 26 I 3 28 02 72 3 3<> i 24 08 I 25 40 i 27 18 i 29 02 30 51 32 46 76 3 45 i 28 04 I 29 41 1 3 1 23 i 33 ii 35 5 37 6 80 4 oo i 3i 37 i 33 17 i 35 03 i 36 55 38 54 40 59 83 4 IS i 34 45 i 36 29 i 38 18 i 40 14 42 16 44 *5 86 4 30 i 37 29 i 39 15 i 41 08 i 43 06 45 " 47 2 4 88 4 45 i 39 47 i 4i 35 i 43 30 1 45 3 1 47 39 49 54 90 5 oo t 41 38 i 43 29 1 45 25 i 47 28 49 38 Si 55 91 5 15 i 43 4 1 44 55 i 46 53 1 48 57 51 08 53 27 92 5 30 i 44 02 i 45 54 1 47 53 i 49 58 52 10 54 3 93 5 45 i 44 34 i 46 26 i 48 25 i 50 30 52 43 55 03 94 6 oo i 44 38 i 46 31 i 48 29 1 So 34 5* 46 55 06 93 6 15 i 44 16 i 46 08 i 48 05 i 50 10 52 21 54 40 93 6 to i 43 2 7 i 45 18 i 47 14 i 49 17 I 5 I 2 7 i 53 44 92 6 45 I 42 12 i 44 01 i 45 56 1 47 56 I 5" 4 I 52 20 91 7 oo i 40 31 i 42 18 i 44 10 i 46 09 I 48 M I 50 27 -89 7 15 i 38 24 i 40 09 i 4i 59 1 43 54 i 45 57 I 48 06 87 7 3 i 35 53 i 37 35 i 39 21 i 41 14 i 43 13 i 45 19 -85 7 45 : 32 58 i 34 36 i 36 19 i 38 08 i 40 03 i 42 05 82 8 oo i 29 40 i 3i M i 32 53 1 34 38 i 36 29 i 38 26 79 8 15 i 25 59 i 27 29 i 29 04 1 30 44 i 32 30 i 34 22 -76 8 30 i 21 57 i 23 23 i 24 53 i 26 28 i 28 09 i yg 55 72 8 45 i 17 35 i 18 56 I 20 21 1 21 51 i 23 26 i 25 07 68 9 oo i 12 54 i 14 10 I 15 30 i 16 54 i 18 23 i 19 57 64 9 15 i 07 54 i 09 05 i 10 19 i ii 38 i 13 01 i 14 28 59 9 3 I 02 38 i 03 44 I 04 52 i 06 04 I 07 21 i 08 41 55 9 45 o 57 07 o 58 07 o 59 09 i oo 15 I OI 24 I 02 38 50 TO 00 51 22 o 52 16 o 53 I2 54 " o 55 X 3 o 56 19 45 10 15 o 45 25 46 12 o 47 01 o 47 53 o 48 49 o 49 47 40 10 30 10 45 II 00 o 39 16 o 32 58 o 26 32 39 57 o 33 32 o 27 oo o 40 40 o 34 08 o 27 28 o 41 25 o 34 46 o 27 59 o 42 12 o 35- 26 o 28 31 o 43 02 o 36 08 o 29 05 34 29 2 | II 15 o 20 oo o 20 20 20 42 21 05 O 21 29 o 21 55 18 II 30 11 45 o 13 23 o 06 42 o 13 36 o 06 49 o 13 51 o 06 56 o 14 06 o 07 04 14 22 07 12 o 14 39 07 21 6 Elongation: Azimuth i 44 40 i 46 32 I 48 31 i 50 3 6 I S 2 48 i 55 08 93 h. m. S. h. m. s. h. nt. s. i. nt. s. h. m. s. h. nt. s. s. Hour-angle. 5 55 4 5 54 53 5 54 42 5 54 3i 5 54 20 5 54 07 + 5 326 GEODETIC ASTRONOMY. 310. Correctionfor Hour-angle before or after Upper AZIMUTH OF POLARIS COMPUTED FOR DECLINATION 88 46'. i' Increase in Declination of Polaris. Culmination. Lat. 50. Lat. 51. Lat. 52. Lat. 53. Lat. 54. Lat. 55. Lat. 50. "T o 07' 44" o'o 7 ' 5 4< o 08' 05' o 16 08 o 08' 17' o 08' 29" o 08' 42' 6" 45 o 23 02 o 23 33 o 24 06 o 24 41 o 25 18 25 57 19 00 3 33 o 31 14 o 31 58 o 32 44 o 33 33 o 34 25 25 15 o 37 56 o 38 47 o 39 40 o 40 38 o 41 38 o 42 43 32 3 o 45 08 46 08 o 47 12 48 20 o 49 32 o 50 49 -38 45 o 5* 07 53 J 7 o 54 3i o 55 49 o 57 12 o 58 41 43 oo o 58 52 OO II i 01 34 i 03 03 1 04 37 i 06 16 49 15 I 05 21 06 48 i 08 21 i 09 59 i ii 43 1 J 3 33 54 3 I II 3 2 13 08 i 14 48 i 16 35 i 18 29 i 20 30 59 2 45 I 17 24 19 07 i 20 55 i 22 51 i 24 54 i 27 04 -64 3 oo i 22 54 24 44 i 26 41 i 28 44 i 3 55 * 33 15 68 3 15 3 3 I 28 02 I 32 46 29 59 34 49 I 32 02 I 3 6 5 8 i 34 i| i 39 16 i 36 32 i 41 42 i 39 oo i 44 18 72 -76 3 45 i 37 06 39 4 I 4 I 2 9 43 52 i 46 25 i 49 07 80 4 oo i 40 59 43 12 i 45 32 48 01 i So 39 53 27 -83 4 15 i 44 25 46 42 i 49 07 51 40 1 54 23 57 16 -86 4 3 i 47 24 49 44 i 5 *3 54 50 i 57 37 oo 35 88 4 45 1 49 54 52 17 i 54 49 57 29 2 OO 2O 03 21 90 5 oo * 5^ 55 54 21 i 5 6 54 59 37 2 02 31 5 35 91 5 i5 * 53 27 55 54 i 58 29 01 15 2 04 10 07 16 92 5 3 1 54 3 56 58 i 59 34 02 20 2 05 16 08 23 93 5 45 6 oo 6 15 * 55 03 i 55 06 i 54 4 57 3* 57 34 i 57 06 2 00 08 2 00 10 i 59 4i 02 53 02 56 02 26 2 05 50 2 5 5 2 2 05 21 08 58 08 58 08 26 94 93 93 6 45 7 oo 1 53 44 I 52 20 I 50 27 i 54 42 i 52 47 i 57 14 55 IS i 59 54 i 57 52 2 02 44 2 oo 39 2 05 45 2 03 36 92 91 89 7 15 1 48 06 i 5 23 52 48 i 55 21 I 58 4 2 oo 57 -87 7 30 i 45 19 i 47 32 49 52 i 52 21 i 54 59 1 57 47 -85 7 45 i 42 05 i 44 13 46 29 i 48 53 i 51 26 i 54 08 82 8 oo i 38 26 i 40 29 4 2 40 i 44 58 1 47 25 i 50 01 79 8 15 i 34 22 I 36 20 38 25 i 40 38 i 42 58 I 45 27 76 8 30 1 29 55 I 3 I 48 33 47 i 35 52 i 38 06 i 40 28 72 8 45 i 25 07 i 26 53 28 45 1 30 44 1 32 5 i 35 04 68 9 oo '9 57 i 21 37 23 22 i 25 13 i 27 ii i 29 17 -64 9 *5 i 14 28 i 16 01 17 38 I 19 22 I 21 12 i 23 08 59 9 30 9 45 i 08 41 I 02 38 10 06 03 55 ii 36 05 17 I I| 12 i 06 44 1 *4 53 i 08 16 i 16 40 i 09 53 55 So 10 00 o 56 19 o 57 28 o 58 42 I 00 00 i 01 23 I 02 50 45 10 15 o 49 47 o 50 48 o 5i 53 o 53 02 54 IS o S5 32 40 10 30 10 45 o 43 02 o 36 08 3 43 5 6 o .36 52 o 44 S 2 o 37 39 o 45 5 1 o 38 29 46 54 o 39 22 o 48 01 o 40 18 34 29 It 00 o 29 05 o 29 41 o 30 18 o 30 5 8 o 31 41 o 32 26 23 ii 15 o 21 55 22 22 O 22 50 23 20 o 23 52 o 24 26 18 it 30 o 14 39 14 57 o 15 16 o 15 37 o 15 58 o 16 21 12 Elongation : Azimuth o 07 21 i 55 08 o 07 30 i 57 36 o 07 39 2 OO 13 o 07 49 2 02 59 o 08 oo 2 05 55 o 08 ii ', 09 02 - 6 93 ft tn s I Wl S h. tn s t. Wl S. t tn s t tn s Hour-angle. 5 54 >7 5 53 54 5 53 4 5 53 27 5 53 S 52 57 + 5 TABLES. 327 Correction for Hour-angle before or after AZIMUTH OF POLARIS COMPUTED FOR DECLINATION 88 46'. i' Increase in Declination of Polaris. Upper Culmination. Lat. 55. Lat. 56. Lat. 57. Lat. 58. Lat. 59. Lat. 60. Lat. 60. o" is" o 08' 42" o 08' 56" o* 09' 12" o 09' 28" o 09' 45" io' 03" 8" o 30 o 17 22 o 17 50 l3 20 o 18 53 o 19 27 20 04 on cH 17 o 45 I 00 o 25 57 o 34 25 o 20 39 o 35 21 36 20 o 37 23 o 38 31 29 5 39 44 25 33 I 15 o 42 43 o 43 52 o 45 06 o 46 24 o 47 48 49 19 I 30 o 50 49 o 52 ii o 53 39 o 55 12 o 56 52 58 40 49 1 45 2 00 o 58 41 i 06 16 i oo 16 i 08 03 i oi 50 i 09 57 i 03 44 i ii 58 i 05 40 i 14 08 07 44 16 28 -35 2 15 i 13 33 1 *5 3 1 i 17 37 i 19 52 I 22 l6 i 24 51 71 2 30 I 20 30 i 22 39 i 24 56 i 27 24 i 30 oi 32 So - 78 2 45 I 2 7 04 i 29 23 i 31 52 i 34 3i i 37 21 40 23 T At oft 80 3 3 J 5 1 33 X 5 i 39 oo 1 35 43 i 4i 37 i 44 25 i 47 25 i 5 37 47 2O i 54 3 - 09 94 3 30 i 44 18 1 47 03 i 50 oo i 53 08 i 56 30 2 00 07 99 3 45 4 oo 1 53 27 i 56 26 1 55 4 1 59 37 2 03 OI 2 06 40 2 05 37 2 io 34 108 4 15 i 57 16 2 00 21 2 03 38 2 07 09 2 io 54 2 14 55 in 4 30 4 45 2 oo 35 2 03 21 2 03 44 2 06 34 2 07 00 2 IO OO 2 IO 42 2 13 40 2 I 4 32 2 17 35 2 18 39 2 21 47 -114 116 5 oo 2 05 35 2 08 51 2 12 20 2 16 03 2 2O O2 2 24 17 118 5 IS 2 07 16 2 io 34 2 14 05 2 17 50 2 21 51 2 26 09 119 5 30 2 08 23 2 08 58 2 II 42 2 15 I 4 2 19 01 2 23 04 2 2 1 QQ 2 2 7 2| 2 27 t;8 120 I2O 6 oo 2 08 58 2 12 17 2 15 49 2 19 35 * * o jy 2 23 37 * */ 5 2 27 56 120 6 15 2 08 26 2 ii 44 2 15 14 2 18 59 2 22 59 2 2 7 15 II 9 6 30 2 07 22 2 io 37 2 14 05 2 17 47 2 21 44 2 25 57 118 6 45 2 05 45 2 08 57 2 12 21 2 l6 00 2 19 53 2 2 4 3 116 7 oo 2 03 36 2 06 44 2 10 05 2 13 39 2 17 27 2 21 3 2 114 7 IS 2 oo 57 2 04 00 2 07 16 2 io 45 2 14 27 2 18 26 in 7 3 i 57 47 2 oo 45 2 03 55 2 07 18 2 io 54 2 14 46 108 7 45 i 54 08 i 57 oo 2 00 04 2 03 20 2 06 49 2 10 32 104 8 oo i 50 oi i 52 47 1 55 43 I 5 8 52 2 02 12 2 05 47 IOO 8 15 i 45 27 i 48 06 i So 54 i 53 54 i 57 06 2 00 32 - 96 8 30 i 40 28 i 42 58 1 45 39 i 48 30 i 51 32 i 54 47 91 8 45 1 35 04 i 37 26 1 39 57 i 42 39 i 45 3i i 48 35 86 9 oo i 29 17 i 31 30 i 33 5i i 36 23 i 39 05 1 4i 57 80 9 15 9 30 i 23 08 i 16 40 I 25 12 i 18 34 i 27 24 I 20 36 i 29 44 i 22 45 i 32 14 i 25 03 * 34 55 i 27 30 75 - 69 9 45 * 09 53 1 IT 37 I 13 28 i 15 25 i 17 31 i 19 45 63 10 00 I 02 50 i 04 23 I 06 03 i 07 48 i 09 41 i ii 41 56 io 15 o 55 32 o 56 54 O 58 22 o 59 55 i oi 34 i 03 20 50 io 30 o 48 oi o 49 12 o 50 27 o 51 48 o 53 14 o 54 45 43 io 45 o 40 18 o 41 18 42 21 o 43 28 o 44 40 o 45 57 - 36 II 00 o 32 26 o 33 M o 34 05 o 34 59 o 35 57 o 36 59 29 II 15 o 24 26 25 02 o 25 41 20 21 o 27 05 o 27 51 22 ii 3 o 16 21 o 16 45 o 17 io o 17 38 o 18 07 o 18 38 14 45 Elongation : o 08 ii o 08 23 o 08 36 o 08 50 o 09 04 o 09 20 7 Azimuth... 2 09 O2 2 12 21 2 15 54 2 19 4 2 23 43 2 28 O2 I2O Jt 9tt L jfl f k. nt. s h. m. s h nt s h in s - Hour-angle 5 52 57 5 52 41 5 52 24 5 52 06 5 5 1 47 5 51 27 -f 7 328 GEODETIC ASTRONOMY. 3'I- 311. NOTATION AND PRINCIPAL WORKING FORMULAE. The following general notation is used throughout the book. The special notation involved in each working for- mula will be found below each group of formulae. GENERAL NOTATION. a and 3 = the apparent right ascension and declination, respectively, at the time of the observation under consideration. a m and d m = the mean right ascension and declination, re- spectively. a l87S and tf l875 (with a year as a subscript) = the mean right ascension and declination, respectively, at the beginning of the fictitious year indicated. <* and # = the values of a m and $ m at the beginning of the fictitious year during which the observation under consideration was made. }* and X = proper motions, per year, in right ascension and declination, respectively. A = altitude. C = zenith distance. = astronomical latitude of the station of observa- tion. t = hour-angle, measured eastward or westward from the upper branch of the meridian as the case may be, but always considered positive, and never exceeding 180 (or I2 h ). 3 IX - NOTATION AND WORKING FORMULAE. 329 z = the azimuth of a star, measured to the eastward or westward from north as the case may be, but always considered positive and never exceeding 180. d = value, in arc, of one division of a level. WORKING FORMULA, WITH THEIR SPECIAL NOTATION, AND WITH REFERENCES TO CORRESPONDING PORTIONS OF THE TEXT AND TO THE TABLES. To convert mean solar to sidereal time. See 23 and the table of 290. To convert sidereal to mean time. See 24 and the tables of 290, 291. To interpolate along a chord. ,Vi-V. or Fj = Ft (F 9 F t ) * _ y ; . . . . (i) * the first form being used when the interpolation is made forward from the value F lt and the second when it is made backward from the value F 9 . F 7 is the required interpolated value of the function corresponding to the value V t of the independent variable. V^ and F a are the adjacent stated values of the independent variable to which correspond the given values F^ and F t of the function. See 30. * The number assigned to each formula corresponds to that used in the body of the text. 33O GEODETIC ASTRONOMY. 311. To interpolate along a tangent. FI is the required interpolated value of the function corre- sponding to the value F/ of the independent variable. F 1 and F l are, respectively, the nearest given value of the independ- ent variable, and the corresponding value of the function. f J is the given first differential coefficient corresponding to V,. See 31. To interpolate along a parabola. If the first differential coefficients are given, KdF\ ^) 1 or according to whether the interpolation is made forward from V l or backward from F 2 . Fj is the required interpolated value of the function corresponding to the value F/ of the inde- pendent variable. F, and F, are the adjacent stated values of the independent variable to which correspond the given (dF\ values F l and /% of the function, and the given values \-Jyf and \-jp>) of the first differential coefficient. See 33. If the first differential coefficients are not given, gS 11 - NOTATION AND WORKING FORMULAE. 331 or ~ V, + D)\ [ V, - F,]. (4) is the general formula which is applicable even when the successive differences between V lf F a , F s , . . . are not all the same, and (40) is the formula for the special case in which those differences are all the same. F[ is the required interpolated value of the function cor- responding to the value F/ of the independent variable. F lt FV and F s are three successive given values of the function corresponding, respectively, to the values F p F,, F, of the independent variable, F a being the stated value of the variable nearest to which lies the value F/. In (4*) D = (F Q - FO = (F 3 - F a ) = . . . , and (26) number of observed lines For the process of finding the equatorial intervals see *8 in which T e ' is the observed time of transit across the mean line corrected for inclination of the horizontal axis and for diurnal aberration. B cos C sec d is tabulated in 299. b = fi /,-, /, being the pivot inequality derived as indicated in 94, 115. fi=\(w+uS)-(e + S)\ . . . (29) if the level divisions are numbered from the middle toward each end, w and w' being the west end readings of the bubble 3 IX ' NOTATION AND WORKING FORMULA. 335 before and after reversal of the striding-level, and e and e' the corresponding east end readings. /*={(,+ ,)_ (w 1 + e')\ . . . (30) if the level divisions are numbered continuously from one end to the other, the primed letters indicating the readings taken with the zero of the level to the westward. k is tabulated in 315. T c = 77+ Aa + Cc, in which T c is the reading of the timepiece when a star crosses the meridian, and Aa and Cc are the corrections for azimuth error and collimation error, respectively. A = sin sec # and C = sec d are tabulated in 299. To compute the azimuth and collimation errors (a and c) from the observations, WITHOUT THE USE OF LEAST SQUARES ( 101-106), use the formulae (a- T ;\ v -(a- T C '} E \Ot l c 6 )time stars \ a *c ^-^) azimuth star / x a ~ ~^~~ ~A~ > (40 '- 1 time stars ri azimuth star to derive a and c, by successive approximations, as indicated in 101-106. The clock correction as determined by each observation is AT C = a T. To compute the azimuth and collimation errors (a and c), and AT C , from the observations, BY LEAST SQUARES. ( 107-110.) GEODETIC ASTRONOMY. 311. The observation equations are of the form A w a w +Cc-(a- 27) = o, . . (45) and AT C + A E a E + Cc - (a - T c ') = o, . . (46) for the observations made with illumination west and east, respectively. The normal equations are SA w a w + 2A s a B + 2Gc - 2(a - T e ') = o; + 2A w Cc-2A u Aa-T c ')=o; ^A* E a E + 2A B Cc - 2A (a - T c ') = o', -2C(a-T c ') = o. The solution of these four equations gives the values of j and c. The probable error of a single observation is / 2v* = 0.674* / , V . - n n in which the v's are the residuals of the observation equations, n is the number of observations, and n u is the number of unknowns (and of normal equations). The probable error of the computed AT C is e = e V Q, in which Q is a quantity obtained as follows: In equation (47) write Q in the place of AT C , i in the place of 2(a T/), and o in the place of the other absolute terms, and then solve for Q. For two modifications of this method of computing, which may be used if considered advisable, see 109, no. For the form of computation if unequal weights (depend- 3 TI - NOTATION AND WORKING FORMULA. 337 ing upon the declination of the star and the number of lines of the reticle observed upon) are assigned to the separate observations, see 111-113. TO COMPUTE THE LATITUDE FROM OBSERVATIONS MADE WITH A ZENITH TELESCOPE. ( 146-157.) The latitude from a single pair of stars is - M')~ + d -\(n + n') - (s - '.. (57) In (57) the primed letters correspond to the northern star of the pair; M is the micrometer reading expressed in turns; r is the angular value of one turn ; n and s are the north-end and south-end readings, respectively, of the level, for the northern star, and n' and s 1 for the southern star; R is the refraction, and m the reduction to the meridian of a star observed off the meridian. The level correction as given above is for a level tube which carries a graduation of which the numbering increases each way from the middle. If the level-tube graduation is numbered continuously from one end to the other the level correction becomes ~S(X + s ') ~ ( n + s )\- ^ See X 48.) The term %(R R') is tabulated in 304. (See 150.) The term is tabulated in 305. (See 151.) 338 GEODETIC ASTRONOMY. 31 1- To combine the separate values of and to compute the probable errors. ( 154-157.) The probable error of a single observation is /v^s-a^H,. ..... (6 4 ) in which the J's are the differences obtained by subtracting the mean result for each pair from the result on each separate night from that pair; [JJ] is the sum of the squares of the J's; n is the total number of observations; and/ is the total number of pairs observed. The probable error of the mean result from any one pair is (o.45 5 ) in which the v's are the residuals obtained by subtracting the indiscriminate mean result for the station from the mean result from each pair; and [vv\ is the sum of the squares of the 7/s. The probable error of the mean of the two declinations of the stars of a pair is t, = / - e', ...... (68) in which 3 XI - NOTATION AND WORKING FORMULA. 339 n lt n^ #,,... are the numbers of times that pair No. I, pair No. 2, pair No. 3, . . . , respectively, are observed. The proper weights w lt w t , w if . . . for the mean results from the separate pairs are proportional to n s The most probable value, , for the latitude of the station is ' 3 .-. O0] ,. = 7^T (7) in which

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