A "BROKEN" TRANSIT. Length of axis, 34 inches. Approximate cost, $1200. A TEXT-BOOK OF FIELD ASTRONOMY s. GEO Director of the Wa$JH>\irn Observatory, Professor of Astronomy in the University of Wisconsin* SECOND EDITION, REVISED AND ENLARGED. FIRST THOUSAND.^ " ?, NEW YORK : JOHN WILEY & SONS. LONDON : CHAPMAN & HALL, LIMITED. 1912 Astron. Dept. Copyright, 1902, 1908, BY GEORGE C. COMSTOCK. THE SCIENTIFIC PRESS ROBERT DRUMMOND AND COMPANY BROOKLYN, N. Y. PREFACE TO THE SECOND EDITION. THE author has embraced the opportunity afforded by a second edition of this work to introduce into it certain changes which, without modifying the general plan and scope, will, it is hoped, increase its usefulness. For the most part these changes are made in the direc- tion of increased simplicity or of increased precision, and are most conspicuously shown in 19, 23, 3,2, 37 and 38. The tables at the end of the book have been considerably increased, both in extent and precision, and now suffice for approximate as well as rough deter- minations of time, latitude and azimuth without the use of an almanac. June i, 1908. March i, 1912. 670910 PREFACE TO THE FIRST EDITION. THE present work is not designed for professional students of astronomy, but for another and larger class found in technical colleges. For many years it has been the author's duty to teach to students of engineering the elements of practical astronomy, and the experience thus acquired has gradually produced the unconventional views that find expression in the present text and which, to the author's mind, are justified by the following considerations : In the engineering curriculum, work in astronomy is a part of a course of technical and professional training of students who have no purpose to become astronomers. Under these circumstances it seems the duty of the instructor to select for presentation those parts of astronomical practice most closely related to the work of the future engineer and, with reference to the narrow limits of time allotted the subject, to keep in the back- ground many collateral matters that are of primary in- terest and importance to the student of astronomy as a science. The parts of astronomical practice most pertinent to vi PREFACE. engineering instruction seem to the author to be (a) Train- ing in the art of numerical computation ; (6) Training in the accurate use of such typical instruments of precision as the sextant and the theodolite, with special refer- ence to the elimination of their errors from the results of observation; (c) Determinations of time, latitude, and azimuth, with portable instruments, as furnishing sub- ject-matter through which a and b may be conveniently realized. If this work is to be done during the single semester usually allowed for the subject, the time given to its theoretical side, spherical astronomy, must be reduced to the minimum amount compatible with the student's intelligent use of his apparatus and formulas, and in the present work this pruning of the theoretical side has been carried to an extent that would be unpardon- able in the training of an astronomer, but which appears necessary and proper in this case. Since many engineering students acquire from the mathematical curriculum little or no knowledge of spherical trigonometry and its numerical applications, the first chapter of the work is devoted to a brief presen- tation of the elements of this subject with special refer- ence to its astronomical uses and to the student's acqui- sition of good habits in the conduct of numerical work. The astronomical problems presented in the following chapters are those that have been indicated by experience as best adapted to the author's own pupils, and while many of the methods given for their solution are not contained in the current text-books, in every case these are either methods in use in the best geodetic surveys, PREFACE. vii or such as have been repeatedly tested with students and found well suited to their use. These methods are classi- fied in the text as rough, approximate, and precise, with respect to their precision and the corresponding amount of time and labor required for their application, and the student is advised not to use the refined and laborious methods when only a rough result is required. As a rule, in the development of formulae no attempt has been made to deal with the general case when the solution of a particular case would suffice for the prob- lem in hand; e.g., the earth's compression is ignored in treating of the effect of parallax, since its influence is vanishingly small in the great majority of cases that the student will ever encounter, and cases in which this influence is of sensible amount should be avoided by the instructor. A more serious omission, but one required by the general plan of the work, is found in the theory of the transit instrument, Chapter IX, where broken transits, thread intervals, curvature of a star's apparent path, flexure, etc., are passed by without treatment or even suggestion. They are not required for the begin- nings of work with a transit instrument, and therefore constitute a part of more advanced study than is here contemplated. As a partial guide to such study there is given upon a subsequent page a list of references to works that may be consulted with profit by the student who seeks a more complete knowledge of the processes of practical astronomy. The adopted notation follows, with only slight devia- tions, that of Chauvenet, to whose elaborate treatise vill PREFACE. upon Spherical and Practical Astronomy the author is under obligations that are common to every present-day writer upon those subjects. His thanks are also due to many of his former pupils, and in particular to Dr. 'S. D. Townley and Dr. Joel Stebbins, who have read and criticised portions of his manuscript. It is also a pleasure to acknowledge here the comment and criticism that has come to the author from practis- ing engineers who have made use of the work. In part these suggestions are incorporated in the present text, and in no case have they been rejected without careful consideration. TABLE OF SYMBOLS. THE following table contains a brief explanation of the principal symbols employed in the text, with references to the page at which they are respectively defined. There are omitted from the table a considerable number of symbols employed only in immediate connection with their definition. Mathematical. 2 p. 154 Summation symbol. [ ] ii The enclosed number is a logarithm. Coordinates, etc. h p. 26 Altitude. e 25 Zenith distance. Complement of h. A 26 Azimuth, reckoned from south. a , d 65 Azimuth, reckoned from north. t 26 Hour angle. 3 26 Declination. a 26 Right ascension. 29 Latitude. A 42 Longitude. 5 52 Sun's semi-diameter. P 53 Horizontal parallax. R, R' 58 Refraction, etc. p 70 Polar distance, = oo 5. Time. 6 p. 30 Sidereal time. M 41 Mean solar time. T 44 Time shown by a chronometer, whether right or wrong, AT 44 Chronometer correction. p 45 Chronometer rate. E 40 Equation of time. V 40 Date of conjunction, mean sun with vernal equinox. X TABLE OF SYMBOLS. Q p. 42 Sidereal time of mean noon, as given in almanac. Q v 43 Sidereal time of mean noon reduced to the local merid- ian. Rough and Approximate Determinations. a p. 71 Tabular difference of azimuth, Polaris and north pole. 6 71 Tabular difference of altitude, Polaris and north pole. F,f 71 Factors to transform a and b into their true local values, H 87 A horizontal angle. y, D 72 Auxiliaries for finding hour angle of Polaris. U 91 An approximate value of the chronometer correction, AT, referred to sidereal time. A 9 An approximate azimuth of a mark. x, y 91 Corrections to transform U and A 9 into true values. dA C 92 A coefficient, equals -T-. r 92 An hour angle. Instruments. R, r p. 118 Circle or micrometer readings. d 100 Value of half a level division. k 159 Value of one revolution of a micrometer. f 118 Deviation of a vertical axis from the true vertical. 6', 6" 114 Components of f parallel and perpendicular to hori- zontal axis. * 118 Deviation, from 90, of angle between axes of theodo- lite. w 1 20 Complement of spherical angle at zenith between hori- zontal axis and line of sight. a 171 Deviation of horizontal axis from true east and west. b 171 Deviation of horizontal axis from true leveL c 171 Collimation constant. A,B,C 173 Mayer's transit factors. C m 162 Calibration correction to micrometer-screw. Precise Determinations. L p. 144 Correction to equal altitudes. (Time.) AR Q 148 Correction to equal altitudes. (Azimuth.) g, k 149 Auxiliaries used in computation of azimuths. s 1 60 Auxiliary used in computation of differential refrac- tion. / 83 Auxiliary used in computation of reduction to merid- ian. a 189 Auxiliary collimation coefficient. TABLE OF CONTENTS. CHAPTER I. PAGB INTRODUCTORY i Spherical trigonometry. Approximate formulas. Numeri- cal computations. Logarithmic tables. Limits of accuracy. CHAPTER II. COORDINATES 22 Fundamental concepts. Definitions. Notation. Table of coordinates. Transformation of coordinates. CHAPTER III. TIME 35 Three time systems. Longitude. Conversion of time. Chronometer corrections. The almanac. CHAPTER IV. CORRECTIONS TO COORDINATES 49 Dip of horizon. Refraction. Semi- diameter. Parallax. Diurnal aberration. s CHAPTER V. ROUGH DETERMINATIONS rg Latitude from meridian altitude. Time and azimuth from single altitude. Meridian transits for time. Orientation and latitude by Polaris. xi xii TABLE OF CONTENTS. CHAPTER VI. PACK APPROXIMATE DETERMINATIONS 79 Circum-meridian altitudes for latitude. Time from single altitude. Azimuth observations at elongation. Time and azimuth from two stars. CHAPTER VII. INSTRUMENTS 99 The spirit-level. Value of half a level division. Theory of the theodolite. Repetition of angles. The sextant. Chro- nometers. CHAPTER VIII. ACCURATE DETERMINATIONS. , 141 Time by equal altitudes. Precise azimuth with theodolite. Zenith-telescope latitudes. CHAPTER IX. THE TRANSIT INSTRUMENT 168 Preliminary adjustments. Theory of the transit. Ordinary method for time determinations. Personal equation. Methods and accuracy of observation. Time determination with reversal on each star. Azimuth of terrestrial mark. BIBLIOGRAPHY 196 INTRODUCTION TO TABLES 197 DIFFERENTIAL COEFFICIENTS 207 TABLES . . .208 FIELD ASTRONOMY;;;,; -, ; i CHAPTER I. INTRODUCTORY. i. Spherical Trigonometry. Any three points on the surface of a sphere determine a spherical triangle, whose sides are the arcs of great circles joining these points, and whose angles are the spherical angles included be- tween these arcs; e.g., on the surface of the earth, assumed to be spherical in shape, the north pole, the city of St. Louis, and the borough of Greenwich, England, are three points making a spherical triangle, two of whose sides are the arcs of meridians joining St. Louis and Green- wich to the pole ; the third side being the arc of a great circle connecting St. Louis and Greenwich, and measur- ing by its length the distance of one place from the other. The spherical angle at the pole between the two meridians is the longitude of St. Louis, while the angle at St. Louis between its meridian and the third side of the triangle represents the direction of Greenwich from St. Louis, a certain number of degrees east of north. The particular number of degrees in this angle is to be found by solving 2 FIELD ASTRONOMY. the triangle, i.e., determining the magnitude of its un- known parts by means of the known parts, and in this case we may suppose these known parts to be the differ- ence of longitude between the two places, and the distance of each place from the north pole, i.e., the complement of its latitude. The foi-mulae required for the- solution of a spherical "triangle afe' best derived by the methods of analytical geometry, and in Fig. i we assume a spherical triangle, FIG. i. ABC, situated on the surface of a sphere whose centre is at 0, and we adopt as the origin of a system of rect- angular coordinates, in which the axis OX passes through the vertex, A, of the triangle, OY lies in the plane AOB, and OZ is perpendicular to that plane. From the vertex C let fall upon the plane OA B the perpendicular CP, and from P draw PS perpendicular to OX and join the points C, 5, thus obtaining the right-angled plane triangle CPS. INTRODUCTORY. 3 The lines 05, SP, PC are respectively the x, y, and z coordinates of the point C, and OC, which we shall repre- sent by the symbol r, is the radius of the sphere. It is evident from the construction that the points 0, S, A, and C all lie in the same plane. Also, 0, 5, A, B, and P lie in another plane, and the angle between these two planes is measured both by the spherical angle BAG and by the plane angle CSP, and these angles must there- fore be equal each to the other. We may now express the coordinates of the point C in terms of the sides, a, b, c, and angles, A y B, C, of the spherical triangle as follows: OS = x = r cos b, SP=y = r sin b cos A, (i) PC = z = r sin b sin A. If the axis of x, instead of passing through A, had been made to pass through B, as is shown by the broken line OX' , the axis of Z remaining unchanged, we should have had for the coordinates of C in this system, x f = r cos a, y = r sin a cos B, (2) z' = r sin a sin B. For the sake of simplicity each angle of the triangle ABC has been made less than 90, and the point P, therefore, falls between the axes OX, OX' , thus giving y and y r opposite signs, as shown above. It is evident from the figure that the relations between # x> ' j y> y' > are those furnished by the formulae for the 4 FIELD ASTRONOMY. transformation of coordinates in a plane, when the origin remains unchanged and the axes are revolved through an angle, which in this case is measured by the side c of the spherical triangle. We have, therefore, *'=z, y f y cos c x sin c, (3) x* = y sin c + x cos c ; and introducing into these equations the values of the coordinates above determined and dividing through by r, we obtain the following relations among the sides and angles of the triangle : sin a sin B = sin b sin A , sin a cos B = cos b sin c sin b cos c cos A, (4) cos a = cos b cos c + sin b sin c cos A . These are the fundamental equations of spherical trigonometry and hold true not only for the particular triangle for which they have been derived, but for every spherical triangle, whatever its shape or size. 2. Numerical Applications of Equations 4. We proceed to apply these equations to the logarithmic solution of the triangle above described, premising that in this solu- tion the signs of all the trigonometric functions must be carefully heeded, since upon them depend the quadrants, first, second, third, or fourth, in which the unknown parts of the triangle are to be found. In this connection we shall reserve the signs + and for natural numbers and place after a logarithm the letter n whenever the number corresponding to the logarithm is negative. The student INTRODUCTORY. 5 should accustom himself to this practice, since it is the one in general use. The assumed data of the problem are : Angular distance, Greenwich to Pole. ... b =3 8. 5, Angular distance, St. Louis to Pole c = 5i.4 ; Spherical angle at North Pole A =go.4. and these data we treat as follows : Logarithms, sin A =0.000 sin 6=9.794 cos A =7.8447* cos =9.795 sm b cos A =7.638?* cos 6=9.894 sin =9.893 SOLUTION. Numbers. cos b sin c =+0.613 sin b cos c cos A = 0.003 cos b cos c= +0.489 sin b sin c cos A = 0.003 log cos a = 9.686 Logarithms. sin a sin B 9.794 9.852 sin a cosB =9.789 sin = 9.942 a = 6i.o* In the solution printed above, the student should examine the orderly manner of the arrangement. Each number is labelled to show what it is, and from these labels we see that the first column contains the logarithms of the several trigonometric functions that appear in the second members of Equations 4. The second column contains natural numbers representing the values of the several terms contained in these second members. These are obtained by adding the proper logarithms shown in the first column, and looking out the corresponding num- bers in the tables. An expert computer will do this work "in his head" without writing down a figure that is not shown in the printed solution. At the bottom of the second column is given log cos a, obtained by looking out the logarithm of the sum of the two numbers that stand just above it. This sum being positive shows that the side a lies in either the first or 6 FIELD ASTRONOMY. fourth quadrant, but it alone cannot decide between these two possibilities. We must now have recourse to the third column, which gives the logarithms of the products, sin a sin B and sin a cos B, as derived from the first and second columns, and indicates that these products are positive quantities, since no n is appended to either of the logarithms. The products being positive, the factors sin a, sin B, and cos B must all have like signs, and assum- ing, temporarily, that sin a is a positive quantity we find that B must lie in the first quadrant, since sin B and cos B are positive numbers. To obtain its numerical value we divide sin a sin B by sin a cos B (subtract mentally the corresponding logarithms) and find, as the result of the division, log tan B =0.00 5. This furnishes the value of B given in the solution, and fixes as the direction of Greenwich from St. Louis, N. 4 5. 3 E. Now, looking up in the logarithmic tables the value of sin B (log sin B =9.852), and dividing it out from sin a sin B, we obtain the value, 9.942, given in the solu- tion for sin a. This number might equally have been obtained by looking up in the tables the values of cos B ( = 9.847) and dividing it out from sin a cos B, and with reference to this double possibility the label for the line between sin a sin B and sin a cos B is omitted, it being understood that either sin B or cos B, whichever is the greater of the two, will be entered here and used in the proper manner to obtain sin a. The value of log cos a is given in the middle column, and both sin a and cos a being positive numbers, a is to be taken in the first quad- rant. The agreement between the numerical values of a INTRODUCTORY. 7 furnished by the sine and cosine, is a check upon the accu- racy of the computation, and an asterisk or check-mark is placed after the value of a to show that this check has been applied and found satisfactory. In determining the quadrant of J5, sin a was assumed to be a positive number. It might equally well have been assumed a negative number, which would have made sin B and cos B both negative, and would have furnished as the solution of the triangle B = 225. 6, a^2gg.i. This is also a correct result, for if we travel from St. Louis in the direction N. 2 2 5. 6 E., over an arc of a great circle 299. i long, we shall find Greenwich at the end of it. The first solution represents the least distance, the second solution the greatest distance, on the surface of the sphere, between the two points, and as a matter of convenience it is customary to use the first solution and to assume that sin a is a positive number. 3. Analytical Applications of Equations 4. Equations 4 suffice for the solution of any triangle in which there are given two sides and the included angle, but they are not immediately applicable when other parts of the tri- angle are the data of the problem, e.g., when the three sides are given and the angles are required. A large part of spherical trigonometry, therefore, consists in purely analytical transformations of these equations into forms adapted to different data. For the particular case above suggested, a, b, and c given, we find from the last of Equations 4 cos a cos b cos c cos A = - , (5) 8 FIELD ASTRONOMY. by means of which the angle A may be computed with the given data, and similar equations may be written for the other angles. By transformations more tedious than difficult, and involving the introduction of two auxiliary quantities, denned below by Equations 6, we may change Equation 5 into a form more convenient for computation when all three of the angles are to be determined (see any treatise on spherical trigonometry for the analytical processes involved) . As a result of these transformations we have the following auxiliaries, . (6) which determine the angles through the relations cot \A = k sin a) cot %B=k sin (s-b), (7) cot \C = k sin (s c). Right-angled Spherical Triangles. Since Equations 4 hold true for all spherical triangles, we may apply them to the special case of a triangle right-angled at A, i.e., one in which the angle A equals 90. We shall then have sin A = i, cos A=o, and with these special values we obtain by substitution the following equations, which should be compared with the corresponding formulae of plane trigonometry : INTRODUCTORY. From the first equation, sin B = Sm sin a' From first and second equations, tan B = sin c (8) From second and third equations, cos B = JH1 tana From the third equation, cos a = cos b cos c. These equations together with those derived in the preceding sections, while far from covering the whole field of spherical trigonometry, will be found sufficient for the purposes of this work. 4. Approximate Formulae. In an important class of cases all the preceding formulae may be greatly simplified for numerical use as follows : It is shown, in treatises on the differential calculus, that the trigonometric functions may be developed in series; e.g., x 3 x 5 sin 00=00 -r-\ etc., 6 120 X 2X^ tan* = x + + + etc., (9) O 3 x* x 4 cos x = i f- etc., 2 24 where x is expressed in radians (one radian = 5 7. 3, = 3437'-75> =206264".%). When the angle x does not exceed a few minutes of arc, x radians is a small fraction, and its powers, x 1 , x 3 , etc., are still smaller quantities, so that in these series we may suppress all terms save the first, or all terms save the first two, and the error pro- duced by neglecting these terms of a higher order, as 10 FIELD ASTRONOMY. they are called, is approximately measured by the first term thus neglected. For illustration we assume x = i, and turning this into radians find the results shown in the following short table : Radians. Arc. #=0.0174533+= i ^=0.0001523+= 3 i". 4 2 (IQ) ^x 3 =0.0000009 += o".i8 $x 4 =0.0000000 += o".oo It appears from the value of # 3 , here given, that if we are prepared to tolerate in our work an error of one part in a million we may, for an arc of i, substitute the arc itself in place of its sine, in any formula where the latter occurs ; and similarly (from the value of |# 2 ) we may sub- stitute unity in place of the cosine of an arc of i , if we are willing to admit an error of one part in seven thousand. Expressed in arc these errors are as shown in the table, o".2 and 3 1 ".4 respectively, and with reference to these numbers we may establish the approximate relations: the square of a degree equals a minute; the cube of a degree equals a second ; and find readily, from these rela- tions, the square and cube of any small arc, and thus decide whether, in a given case, these quantities may or may not be neglected. For example: if 00 = 2, we find x 2 = 4 f , # 3 = 8", and for any work in which the data can be depended upon to the nearest minute only, we may assume sin % =x, but we cannot assume cos % = i without sacrificing some of the accuracy contained in the data. It must be constantly borne in mind that the series given above are expressed in radians, and that when applied numerically, x and its powers must be trans- INTRODUCTORY. 11 formed from arc into radians by dividing by the appro- priate factors given above; e.g., The divisor given above is numerically equal to the reciprocal of the sine of i", and in place of the preceding equation it is customary to write x (radians) =x" sin i". (12) As these numerical factors are of frequent use, we record here their values : log sin i" =4.6855749 10, 206264. 8 = [5. 3144251]. (I3) Observe the peculiar notation of the last line. The brackets indicate that the number placed within them is a logarithm, and the equation asserts that this bracketed number is the logarithm of the first member of the equa- tion. This use of the brackets is very common and should be remembered. We may apply to the equations of spherical trigo- nometry the principles here developed, and assuming that the sides, a, b, c, do not much exceed i, i.e., that for a triangle on the surface of the earth the vertices of the angles are not more than sixty or seventy miles apart, we shall find that Equations 8 become b c b = - cos = -, a a c 12 FIELD ASTRONOMY. These are the formulae of plane trigonometry, and indicate that small spherical triangles may be treated as if they were plane. The use of these approximate relations is not limited to the solution of triangles, but they may be applied to the trigonometric functions of any small angle wherever found, and we shall have frequent occasion to use them in the following pages. 5. Numerical Computations. Engineer and astronomer alike should acquire the art of rapid and correct compu- tation, and as a means to that end there will be found on subsequent pages examples of numerical work which should be studied with reference to their arrangement and the order in which the several processes were executed. Often the order in which this work was done is not the order in which the numbers appear upon the printed page, although their arrangement upon the page always follows exactly the original computation, and in no case is to be regarded as a mere summary of results, picked out and rearranged after the actual ciphering had been performed. For illustration we revert to the example of 2, and note that sin A is the first number written in the solution and sin b stands second. But the second number actually written down in the computation was cos A, instead of sin b, for, having found the place in which to look up sin A, it is more convenient and more economical to look up cos A at once, while the tables are open at the right place, rather than to turn away for something else and then have again to find the page and place corresponding to the angle A. Having finished INTRODUCTORY. 13 with the required functions of A , sin b was next looked out and was followed by cos b, although this required the computer to skip two intervening lines of the com- putation and, temporarily, to leave them blank. The general principle here observed is: When a table is open at a given place, look up, before leaving it, all that is to be taken from that place. In order to do this it is necessary to block out the computation in advance, and this was done in the case under consideration, every label, from the initial sin A of the first column to the concluding a of the last column, being written in its appropriate place before a number was set down or the logarithmic table opened. The form of computation thus prearranged is called a schedule, and it is to be strongly urged upon the student as a measure of econ- omy and good practice, that he should draft, at the beginning of each computation, a complete schedule, in which every number to be employed shall be assigned the place most convenient for its use. In general the beginner will not be able to do this without assistance from an instructor, or from models suitably chosen, and for the purposes of the present work the numerous exam- ples contained in the text may be taken as such models. Some cardinal points in the arrangement of a good schedule are as follows: (A) Make it short but complete. Do as much of the work "in your head" as can be done without unduly burdening the mind, and write upon paper only the things that are necessary. But all things that are to be written should have places assigned them in the schedule. 14 FIELD ASTRONOMY. No side computations, upon another piece of paper, should be allowed, and the entire work should be so arranged and labelled that a stranger can follow it and tell what has been done. (B) When the same quantity is to be used several times in a computation (sin b appears as a factor in three different terms of the preceding example) the schedule should be so arranged that the number need be written only once, e.g., since sin b is to be multiplied by both sin A and cos A it is placed between these numbers in the schedule, and for a similar reason the product sin b cos A is placed between cos c and sin c. In adding the logarithms to form the product sin b sin c cos A , cover cos b with a pencil or penholder and the addition will be as easily made as if the intervening number were not present. (C) Frequently, several similar computations are to be made with slightly different data, e.g., it may be re- quired to find the direction and distance of half a dozen American cities from Greenwich. A single schedule should then be prepared and the several computations should be carried on simultaneously, in parallel columns, all placed opposite the same schedule; e.g., look out sin A and cos A for all six places before proceeding to find sin b for any of them, etc. In this particular case sin b and cos 6, depending on the latitude of Greenwich, are the same for all the solutions, and instead of writing their values in each column, they should be written upon the edge of a slip of paper and moved along from column to column as needed. As a memorandum for future INTRODUCTORY. 15 reference they should also be written in one column of the computation. Practise this device whenever the same number is to be used in several different places. See 36 and 40 for examples of two computations de- pending upon a single schedule. 6. The Trigonometric Functions. There is opened to the inexperienced computer an abundant opportunity for error in looking out from the tables the trigonometric functions of angles not lying in the first quadrant. The best mode of guarding against such errors is the acqui- sition of fixed habits of procedure, so that the same thing shall always be done in the same way, and to this end the following simple rules may be adopted : (1) For any odd-numbered quadrant, first, third, etc. Reduce the given angle to the first quadrant by casting out the nines from its tens and hundreds of degrees (add these digits together and repeat the addition until the sum is reduced to a single digit, less than nine), and look up the required function of the reduced arc. (2) For any even-numbered quadrant, second, fourth, etc. Reduce the angle to the first quadrant, as above, and look out the function complementary to the one given. The algebraic sign of the function is, of course, in all cases determined by the quadrant in which the original angle falls. See the following applications of these rules: Quadrant. Required. Equivalent. Process. 2d, Even 005144 29' = sin 54 29' 1+4=5 3 d, Odd tan 264 33' = + tan 84 33' 2+6=8 4 th,Even sin 316 51' = -00546 51' 3 + I= 4 5th, Odd cot 414 1 8' = + cot 54 18' 4 + 1 =5 6th, Even tan 499 49' = cot 49 49' Reject the 9 etc. etc. etc. etc. 16 FIELD ASTRONOMY We may readily formulate a corresponding rule for the converse process, of passing from the function to the angle, as follows : (1) When the arc lies in an odd quadrant. Look out, in the first quadrant, the angle that corresponds to the given function and add to it the required even multiple of 90, i.e., o or 180. (2) When the arc lies in an even quadrant. Change the name of the function (for cos read sin, for tan read cot, etc.). Look out, in the first quadrant, the corre- sponding angle and add to it the required odd multiple of 90, i.e., 90 or 270. See the following examples, in which we represent the required angle by z and suppose that there is given the numerical value of its tangent, e.g., log tan 2 = 9.654. The process of looking out in the several quadrants the angle corresponding to this tangent is as follows : Quadrant. Use. Angle. Add. Result. 2d, Even cot 65 45' 90 155 45' 3 d, Odd tan 24 15' 1 80 204 15' 4th, Even cot 65 45' 270 335 45' etc. etc. etc. etc. etc. The degrees, minutes, and seconds of the required angle should be obtained from the table, the multiple of 90 added to them, and the final result written down in its proper place, without writing the intermediate steps. 7. Determination of Angles. In this connection the student will do well to examine the beginning of a table of logarithmic trigonometric functions and observe how difficult it is to interpolate accurately the value of log sin z or log tan z, corresponding to a small angle, e.g., INTRODUCTORY. 17 = o 33' 17". The difficulty comes from the rapid varia- tion of the function, large and changing tabular differ- ences. On the other hand, log cos z changes slowly and may be readily and accurately interpolated. If we take the converse case and suppose the logarithmic function to be given and the corresponding angle required, we shall obtain the opposite result. The angle will be accu- rately determined by the sine or tangent and very poorly determined by the cosine, e.g., log cos o 33' 17" = 9.99998 and every angle between o 29' and o 36' has this cosine, thus leaving a possible error of several minutes in the value of the angle determined from this function, while the log sin, if correctly given to five decimal places, will determine the same angle within a small fraction of a second. In the interest of precision an angle should always be determined from a function that changes rapidly (large tabular differences), while a quantity that is to be found from a given angle is best determined through a function that changes slowly. In the example of 2, sin a might have been determined through sin B or cos B, and the former was used for this purpose because it varied the more slowly. In cases of this kind, and they are very common, use the function that stands on the right-hand side of the page, in the tables, and subtract it from the larger of the two numbers, sin a sin B or sin a cos B, and it will be then unnecessary to consider whether it is sine or cosine that is employed. The angle a, in this example, was determined through its tangent (log tan a = log sin a log cos a), since the 18 FIELD ASTRONOMY. tangent always varies more rapidly than either sine or cosine and should generally be preferred for this purpose. After obtaining a, its sine and cosine were looked out from the tables and compared with the numbers obtained in the solution, for the sake of the "check" thus fur- nished upon the accuracy of the numerical work. In subsequent pages other checks will be shown, and these should be applied to test the accuracy of numerical work whenever they are available. The mental strain accom- panying a long computation is, under the best of circum- stances, considerable, and a check properly satisfied serves to relieve this tension and facilitate the subse- quent work. 8. Accuracy of Logarithmic Computation. The exam- ple of 2 was solved with logarithms extending only to three places of decimals, and corresponding to this use of a three-place table the results are given to the nearest tenth of a degree. If it were required to obtain results correct to the nearest minute or nearest second, a greater number of decimals must be employed (four-, five-, or six- place tables). The labor of using these tables increases very rapidly as the number of decimals is increased, and a compromise is always to be made between extra labor on the one hand and limited accuracy on the other. As the choice of a proper number of decimal places is usually an embarrassing one for the beginner, there is given below for his guidance a formula intended to repre- sent, at least approximately, the limit of error to be ex- pected in the results of computation on account of the inherent imperfections of logarithms (neglected deci- INTRODUCTORY. 19 mals, etc."). The actual error may fall considerably short of this limit or may overstep it a little. It is evident that the limit will be greater for a long compu- tation than for a short one, and if we measure the length of a computation by the number, n, of logarithms that enter into it and represent by m the number of decimal places to which these logarithms are carried, there may be derived from the theory of probabilities the following expression, in minutes of arc, for the limit of probable error: Limit = 2800' . Vn . io~ m . Applying this formula to the example of 2 we may put n = i6, m = 3, and find 10' as the limit of unavoid- able error; corresponding well with the one-tenth of a degree to which the results were carried. If the data were given to the nearest minute and it were required to preserve this degree of accuracy in the results, we should write, and solving, find ^ = 4.0, i.e., a four-place table is re- quired for this purpose. Let the student verify by means of the above equa- tions the following precepts: To obtain Use Tenths of degrees Three-place tables. Minutes Four-place tables. Seconds Five- or six-place tables. Tenths of seconds Seven-place tables. 30 FIELD ASTRONOMY. If the results are to be expressed in linear instead of angular measure, the limit of error must be represented as a fractional part of the quantity, x, that is to be determined, and corresponding to this case we have Limit = o . 8 # . Vn . io~ M . Corollary. Do not attempt to obtain from a table more than it is capable of furnishing; e.g., do not inter- polate hundredths of a degree in the example of 2, and in connection with linear quantities do not, as a rule, interpolate more than three significant figures from a three-place table, four from a four-place table, etc. 9. Logarithmic Tables. There exists a great variety of logarithmic tables of different degrees of accuracy, from three to ten places of decimals, and having deter- mined the number of decimal places required in a given computation, the choice among the corresponding tables is largely a matter of personal taste. The beginner, however, will do well to observe the following rules foj distinguishing good tables from bad ones: (A) Wherever the tabular differences exceed 10, a good table should furnish proportional parts, PP, in the margin of each page, so that the logarithms may be interpolated ''in the head." (B) The tables should be accompanied by tables of addition and subtraction logarithms. For an explana- tion of these, their purpose and use, the student is re- ferred to the tables themselves, but we note here that by their aid the example of 2 might have been much INTRODUCTORY. 21 more conveniently solved, as is illustrated in a similar problem in 15. The most generally useful tables are those of five decimal places, but computers find it to their advantage to have and use at least one table of each kind, from three to six or seven places. In the examples solved in the present work the following tables have been used : Three-place, Johnson. New York. Four-place, Slichter. New York. Gauss. Berlin. Five-place, Albrecht. Berlin. Six-place, Albrecht's Bremiker. Berlin. As a very useful supplement to the logarithmic tables a slide-rule and the extended multiplication tables of Crelle and Zimmermann are highly esteemed and where extensive computations are to be made much advantage may be derived through the use of computing machines, of which several types are to be had. CHAPTER II. COORDINATES. 10. Fundamental Concepts. For most purposes of practical astronomy the stars may be considered as attached to the sky, i.e., to the blue vault of the heavens, which is technically called the celestial sphere, and is re- garded as of indefinitely great radius but having the earth at its centre, so that a plane passing through any terrestrial point intersects this sphere in a great circle, and parallel planes passing through any two terrestrial points intersect the sphere in the same great circle. If the axis about which the earth rotates be produced in each direction, it will intersect the celestial sphere in two points, called respectively the north and south poles. If a plumb-line be suspended at any place, P, on the earth's surface, and be produced in both directions, it will intersect the celestial sphere, above and below, in the zenith and nadir of the place. The direction thus determined by the plumb-line is called the vertical of the place. The figure (shape) of the earth is such that the vertical of any place, when produced downward, intersects the rotation axis, and a plane may therefore be passed through 22 COORDINATES. 23 this axis and the vertical. This plane, by its intersection with the celestial sphere, produces a great circle which passes through the poles, the zenith and nadir, and is called the meridian of the place, P. A plane passed through P perpendicular to the direc- tion of the vertical produces by its intersection with the sphere the horizon of P. Any plane passing through the vertical is called a vertical plane and produces by its inter- section with the sphere, a vertical circle. That vertical circle whose plane is perpendicular to the meridian is called the prime vertical. With exception of the poles, all of the terms above denned depend upon the direction of the vertical, and as this direction varies from place to place upon the earth's surface each such place has its own meridian, horizon, zenith, etc., while the poles of the celestial sphere are the same for all places. A plane passed through the centre of the earth per- pendicular to the rotation axis produces by its inter- section with the earth's surface the terrestrial equator, and by its intersection with the celestial sphere it pro- duces the celestial equator. Owing to the motion of the earth in its orbit we see anything within the orbit from different points of view at different seasons of the year, and by the earth's motion the sun is thus made to describe an apparent path among the stars, making the complete circuit of the sky in a year. This path is a great circle intersecting the celestial equator in two points diametrically opposite to each other, and that one of these points through which the 24 FIELD ASTRONOMY. sun passes on or about March 22 of each year, is called the vernal equinox. ii. Systems of Coordinates. Most of the problems of practical astronomy require us to deal with the appar- ent positions and motions of the heavenly bodies as seen projected against the sky, and for this purpose there are employed several systems of coordinates based upon the concepts above denned, and three of these systems we proceed to consider. These are all systems of polar coordinates having the following characteristics in common: 1 i ) The origin of each system is at the centre of the celestial sphere. (2) Each system has a fundamental plane and con- sists of an angle measured in the fundamental plane; an angle measured perpendicular to the fundamental plane; and a radius vector. The first of these angles is frequently called the horizontal coordinate, and the second the vertical coordinate, of the system. Latitudes and longitudes on the earth furnish such a system of coordinates. The longitude of St. Louis (horizontal coordinate) is measured by an angle lying in the plane of the equator, which is the fundamental plane of this system. The latitude of St. Louis (vertical coordinate) is measured by an angle lying in a plane perpendicular to the equator, and the radius vector of St. Louis is its distance from the centre of the earth, which latter point is taken as the origin of coordinates. (3) In each system the horizontal coordinate is measured from a fixed direction in the fundamental COORDINATES. 25 plane, called the prime radius, through 360. The ver- tical coordinate is measured on each side of the funda- mental plane from o to 90. (4) Those vertical coordinates are called positive that lie upon the same side of the fundamental plane with the zenith of an observer in the northern hemi- sphere of the earth. Those that lie upon the opposite side of the fundamental plane are negative. (5) It is frequently convenient to measure a vertical coordinate from the positive half of a line perpendicular to the fundamental plane instead of from the funda- mental plane itself, (e.g., in 2 we take as the vertical coordinate of St. Louis its distance from the pole instead of from the equator) . In such cases this coordinate is always positive and is included between the limits o and 1 80. If h represent any vertical coordinate meas- ured in the manner first described and z be the corre- sponding coordinate measured in the second way, we shall obviously have the relation, z = go h. The several systems of astronomical coordinates differ among themselves in the following respects : (a) Different fundamental planes for the systems. (b) Different positions of the prime radii in the fun- damental planes. (c) Different directions in which the horizontal co- ordinates increase. The data which completely define each system of coordinates are given in the following table together with the names of the several coordinates, the letters by which these are usually represented, and the point 26 FIELD ASTRONOMY. of the heavens, called the pole of the system, toward which the positive half of the normal to the fundamental plane is directed. The terms east and west are used in this table with their common meaning, to indicate the direction toward which the horizontal coordinate increases. The letters associated with the several co- ordinates are conventional symbols that should be committed to memory. SYSTEMS OF COORDINATES. System I. II. III. Horizon Equator Prime radius points toward Meridian Meridian Vernal Equinox Horizontal coordinates increase West West East Zenith North Pole North Pole Name of horizontal coordinate.. . Name of vertical coordinate Azimuth A Altitude=^ Hour angle=^ Declination=8 Right Ascension = a Declination=5 EXERCISES. Let the student define in his own language the several quantities above represented by the letters A , t, a, k, and d. 1 . What is the azimuth of the north pole ? 2. What do the hour angle and altitude of the zenith respectively equal ? 3. What are the azimuths of the prime vertical? 4. What are the declinations of the points in which the horizon cuts the prime vertical? 5 . Does d in the second system differ in any way from d in the third system ? The directions of the prime radius as above defined for systems I and II, are ambiguous, since the meridian cuts the fundamental plane of each of those systems in two points. Either of these points may be used to determine the direction of the prime radius, but in general that one is to be employed which lies south of the zenith. COORDINATES. 27 Let the student show the relation between the coordinates furnished in System I by adopting each of the possible positions for the prime radius. 12. Uses of the Three Systems. It is well to consider here, very briefly, the reasons for using more than one system of coordinates, and the relative advantages and disadvantages of these systems. The coordinates of System I are well adapted to observation with portable instruments, e.g., an engineer's transit, since the horizon is more easily identified with such an instrument than is any other reference plane, and the circles of the instrument may be made to read, directly, altitudes and azimuths. The horizon has been defined by reference to the direction of a plumb-line, but in practice a spirit-level, or the level surface of a liquid at rest, are more frequently used to determine its position. System I possesses the disadvantage that, through the earth's rotation about its axis, both the altitude and azimuth of a star are constantly changing in a compli- cated manner, and in this respect System II possesses a marked advantage. Since the normal to its funda- mental plane coincides with the earth's axis, rotation about this axis has no effect upon the vertical coordi- nates, decimations, which remain unchanged, while the horizontal coordinates, hour angles, increase uniformly with the time, 15 per hour, and are therefore easily taken into account and measured by means of a clock. Suppose a watch to have its dial divided into twenty- four hours, instead of the customary twelve. If this watch be held with its dial parallel to the plane of the equator, the hour hand, in its' motion around 28 FIELD ASTRONOMY. the dial, will follow and keep up with the sun as it moves across the sky. If the watch be turned in its own plane until the hour hand points toward the sun, the time indicated upon the dial by this hand will be approximately the hour angle of the sun, and the zero of the dial will point toward the meridian, i.e., south. Let the student compare the ideal case above considered with the following rough rule sometimes given for determining the direction of the meridian by means of a watch with an ordinary, twelve-hour, dial: Hold the watch with its dial as nearly parallel to the plane of the equator as can be estimated. (See 13 for the position of this plane.) Revolve the watch in this plane until the hour hand points toward the sun, and the south half of the meridian will then cut the dial mid- way between the hour hand and the figure XII. A further advantage is gained in the third system of coordinates, since here the prime radius shares in the apparent rotation of the celestial sphere about the earth's axis, and both the horizontal and vertical coor- dinates are therefore unaffected by this motion. In- struments have been devised for the measurement of the coordinates in each of these systems, but we shall be mainly concerned with those that relate to the first system, and shall consider System III as employed chiefly to furnish a set of coordinates independent of the earth's rotation and of the particular place upon the earth at which the observer chances to be. These features make it suited to furnish a permanent record of a star's position in the sky, and it is so used in the American Ephemeris (see 21) and other nautical alma- nacs, where there may be found, tabulated, the right ascensions and declinations of the sun, moon, planets, and several hundred of the brighter stars. 13. Relations between the Systems of Coordinates. A problem of frequent recurrence is the transformation of the coordinates of a star from one system to another; COORDINATES. indeed most of the problems of spherical astronomy are, analytically, nothing more than cases of such trans- formation, and as an introduction to these problems we shall examine the relative positions of the funda- mental planes and prime radii of the several systems. The plane of the equator intersects the plane of the horizon in the east and west line, and the angle between the two planes is called the colatitude, since it is the complement of the geographical latitude of the place EQUATOR n \ / ! HORIZOM i /\ \ / \ \ X / \ / X N / FIG. 2. to which the horizon belongs. The latitude is commonly defined as the angular distance of any place from the equator, but more precisely the latitude is the angle which the vertical of the given place makes with the plane of the equator. From Fig. 2, which represents a meridian section of the earth with the several lines 30 FIELD ASTRONOMY. and planes passed through its centre, it is apparent that the latitude, 0, equals the declination of the zenith and also equals the altitude of the pole. The angular distance of the zenith from the pole is equal to the co- latitude, 90 0. The second and third systems of coordinates have the same fundamental plane, and their relation to each other is therefore determined by the angle, 0, between their prime radii. Since one of these prime radii is directed toward a fixed point of the heavens, while the other lies in a meridian of the rotating earth, it is evident that the angle 6 is continuously and uniformly variable, at the rate of 360 in twenty-four hours. Methods of determining, for any instant, the value of this angle, which is called the sidereal time, will be given hereafter. For the present we note that may be regarded as the horizontal coordinate of the vernal equinox in the second system, or as the horizontal coordinate of the meridian in the third system, and correspondingly we may define the sidereal time as either the hour angle of the vernal equinox or the right ascension of the meridian. 14. Transformation of Coordinates. The transforma- tion of coordinates from the first to the second system is conveniently made by means of the ''astronomical triangle," i.e., the spherical triangle formed by the zenith, the pole, and the star whose coordinates are to be transformed. In Fig. 3 this triangle is marked by the letters PZS, P indicating the position of the celestial pole; Z, the observer's zenith; and 5, the apparent place of the star as seen against the sky. Imag- COORDINATES. 31 ine the triangle projected against the sky and the three points to be visible in their true positions. PZ is an arc of a great circle passing through the pole and zenith and must therefore be a part of the observer's celestial meridian, and in Fig. 2 we have already seen that this arc of the meridian is equal in length to the complement of the latitude, 90 0. The broken line FIG. 3. The Astronomical Triangle. HH f in the figure, is an arc of a great circle, every part of which is 90 distant from Z. But the great circle 90 distant from the zenith is the horizon, -and the arc HS that measures the distance of 5 from HH' must be the star's altitude, h, and the side SZ of the astronom- ical triangle, being the complement of this arc, is equal to go h. In like manner, EE' t drawn 90 distant from P, is an arc of the celestial equator; the a-c SE, that measures the distance of 5 from EE f , is the star's declination, d, and the side PS of the triangle equals go-d. 32 FIELD ASTRONOMY. The star's hour angle, i.e., horizontal coordinate lying in the equator, is measured by the arc EE', and this arc, by a theorem of spherical geometry, is numer- ically equal to the spherical angle, t, included between EP and E'P, which is therefore the star's hour angle. In like manner the spherical angle SZE' is shown to be equal to the star's azimuth, A, and the angle SZP of the astronomical triangle is equal to 180 A. The third angle of the triangle, marked q in the figure, is called the parallactic angle. To apply to the astronomical triangle the fundamental formulae of spherical trigonometry derived in i, we replace the general symbols used in Equations 4 by the particular values which they have in the astronomical triangle, as follows : a = o-/* A=t and introducing these values into Equations 4 we obtain the required formulae for transforming altitudes and azimuths into declinations and hour angles, as follows: cos h sin A = + cos d sin t, cos h cos A = cos sin d + sin cos d cos t, (14) sin h = + sin sin d + cos cos d cos /. The transformation formulae between the second and third systems are much simpler. In Fig. 3 if V repre- sent the position of the vernal equinox, we shall have the arc VE', or the corresponding spherical angle at P, equal to the sidereal time, 0, since the sidereal time is the COORDINATES. 33 hour angle of the vernal equinox. Similarly the arc VE and its corresponding angle at P are equal to the right ascension of the star, and from the figure we then obtain the required relations, 0, d = d ..... (i 4 a) The transformation between the first and third systems is best made through the second system; i.e., by using both groups of formulae 14 and i4a. 15. Problem in Transformation of Coordinates. At the sidereal time i3 h 22 49^3 the altitude and azimuth of a star were measured at a place in latitude 43 4' 36", as follows: h = 6 1 19' 36", A =253 9' 42". Required the right ascension and declination of the star. The required transformation formulae may be obtained from the astronomical triangle in the same manner as Equations 14, and are: cos d sin t = + cos h sin A , cos d cos t= + cos h cos A sin + sin h cos 0, sin d = cos h cos A cos + sin h sin a = 6-t. Note that 0, which occurs only in the last equation, is expressed in hours, minutes, and seconds of time, and that it is customary to express both a and t in these units, 15 = i h , etc. A convenient form for the numerical operations involved in solving these equations is given below, and the student should trace it through, verifying each num- ber and ascertaining why the work is arranged as it is. 31 FIELD ASTRONOMY. Compare and contrast this solution with the one con- tained in 2. The difference of arrangement is largely due to the introduction here of addition and subtraction logarithms. These are indicated in the schedule by the words Add, Subtract, and it is to be especially borne in mind that the addition indicated by the word Add requires a subtraction logarithm when one of the given terms is itself a negative quantity, etc. The schedule shows the algebraic operation required by the formula, but the arithmetical character of the operation is altered by the presence of an odd number of negative signs. SOLUTION. sin A 9 .9809772 sin h cos < 9 .80676 cos h 9.68108 cos h cos A sin 8.97740*1 cos A 9 .46191*1 Add 0.75974 sin'( 9.83441 cos d cos t 9-737 J 4 sin h 9.94318 (Seep. 6) 9.88380 cos h cos A 9.1429971 cos d sin t 9 . 6620571 COS 9.86358 t 3 1 9 5 5' 44" cos $ 9- 8 5334 sin h sin 9-77759 t (time) 2i h i9 m 42 8 . 9 cos h cos A cos ^ 9 .0065771 6 I3 h 22 m 49'.3 Subtract o .06797 a i6 h 3 m 6'. 4 sin 6 9- 8 455 6 8 44 29' 12"* The declination, d, is obtained both from sin d and cos d, and the agreement of the two values is a * check ' upon the accuracy of the work. The * indicates that this check has been applied. CHAPTER III. TIME. 1 6. In astronomical practice, time is measured by watches and' clocks that differ in no essential respect from those in common use, but in addition to the com- mon system of time reckoning, astronomers employ several others, of which we shall have to consider the following : Sidereal Time, already referred to in 13. True Solar Time, which is frequently called Appar- ent Solar Time. Mean Solar Time, which is the common system of every-day life. These three systems possess the following features in common: In each system that common phrase ''the time of day" means the hour angle of a particular point in the heavens, which we shall call the zero point of the system. The unit of time is called a day and is, in every case, the interval between consecutive returns of the zero point to a given meridian; i.e., consecutive transits' of a given meridian past the same zero point. This unit is subdivided into aliquot parts called hours, minutes, and seconds. Each day begins at the instant when the zero point is on the meridian; i.e., on the upper half of 35 36 FIELD ASTRONOMY. the meridian (noon) in astronomical practice, on the lower half of the meridian (midnight) in civil affairs. In astronomical practice the hours from the beginning of the day are reckoned consecutively, from o to 24; in civil practice from o to 12, and then repeated to 12 again, with the distinguishing symbols A.M. and P.M. In con- sequence of the different epochs at which the day begins, the astronomical date in the A.M. hours is one day be- hind the civil date; e.g., Civil Time May 10, 5 h A.M. equals Astronomical Time May 9, iy h . In the P.M. hours the dates agree. Since an hour angle must be reckoned from a deter- minate meridian, this meridian must be specified in order to make "the time" a determinate quantity, and this specification of the meridian should be included in the name assigned to the time; e.g., Local Time denotes the hour angle of the zero point reckoned from the observer's own (local) meridian. Greenwich Time is the hour angle of the zero point reckoned from the meridian of Green- wich. Standard Time is the hour angle of the zero point reckoned from some meridian assumed as standard; e.g., in the United States and Canada the meridians 75, 90, 105, and 120 west of Greenwich are called standard, and Eastern, Central, Mountain, and Pacific Standard Times are hour angles reckoned from these meridians. A like practice is followed in the use of the term Noon; e.g., Washington Noon is the instant at which the zero point is in the act of crossing the meridian of Washington. TIME. 37 17. Longitude and Time. We have introduced above a reference to the time at different meridians, and we have now to note that since ' ' the time" is defined as an hour angle, it is evident that the number of hours, minutes, and seconds expressing either time or hour angle will depend upon the meridian from which the latter is meas- ured. The difference between the hour angles reckoned from two different meridians will equal the angle between the meridians, i.e., their difference of longitude, so that if T f and T" represent the times of any event (whether sidereal, mean solar, or true solar time) referred to two different meridians whose difference of longitude is A, we shall have T r T" = A. It is customary in astronom- ical practice to express differences of longitude in hours rather than in degrees, since both members of the pre- ceding equation should be given in terms of the same units. By transposition of one term in the preceding equa- tion we obtain r = r" + A, ...... (16) and this extremely simple equation indicates that any given time referred to the second meridian may be re- duced to the corresponding time of the first meridian by addition of the difference of longitude, where this dif- ference, A, is to be counted a positive quantity when the second meridian is west of the first. A very common blunder is to omit this reduction to the prime meridian when interpolating from the almanac (see 21). Beware of it, and note that the hour and minute for which a quart- 38 FIELD ASTRONOMY. tity is required to be interpolated are usually given in the time of some meridian other than that of Greenwich or Washington, for which the almanac is constructed, and must therefore be reduced to one of these standard meridians, by addition of the longitude, before they can serve as the argument for the tabular quantity sought. 18. The Three Time Systems. The several time sys- tems differ one from another chiefly in respect of their zero points, and these we have now to consider. Sidereal Time. As already indicated, the zero point of this system is the vernal equinox, and since this is a point of the heavens whose position with respect to the fixed stars changes very slowly, it measures well their diurnal motion. In colloquial language, ' ' the stars run on sidereal time, ' ' and this system is chiefly used in con- nection with their apparent diurnal motion. Solar Time. As their names indicate, both True Solar Time and Mean Solar Time have zero points that depend upon the sun, and before drawing any distinction between the two systems we recall that, owing to the earth's annual motion in its orbit, the sun's position among the stars changes from day to day (we see it from different standpoints). While this change in the sun's position is not an altogether uniform one and takes place in a plane inclined to that of the earth's rotation (ecliptic, and equator), its net result is that in each year the sun makes one entire circuit of the sky, so that any given meridian of the earth, in the course of a year, makes one less transit over the sun than over a star, or over the vernal equinox. The number of solar days in a year is TIME. 39 therefore one less than the number of sidereal days; e.g., for the epoch 1900, (according to Harkness,) One (tropical) year = 366. 242 197 sidereal days = 365.242197 solar days. . (17) It appears from this relation that a sidereal unit of time (day, hour, minute) must be shorter than the corre- sponding solar unit, a relation that we shall have to con- sider hereafter. Apparent, or True, Solar Time. This system has for its zero point the centre of the sun, and the hour angle of the sun's centre at any moment is, therefore, the true solar time. This system is very convenient for use. in connection with observations of the sun, but owing to the irregularities in the sun's motion, above noted, apparent solar days, hours, etc., are of variable length, a day in December being nearly a minute longer than one in September. When time is to be kept by an accurately constructed clock or watch such irregularities are intoler- able, and for the sake of clocks and watches there is employed for most purposes the third system, viz., Mean Solar Time. In this system the days and other units are of uniform length and equal, respectively, to the mean length of the corresponding units of apparent solar time. The zero point of the system is an imaginary- body, called the mean sun, that is supposed to move uniformly along the equator, keeping as nearly in the same right ascension with the true sun as is consistent with perfect uniformity of motion. The mean solar time at any moment is the hour angle of the mean sun 40 FIELD ASTRONOMY. and, numerically, it differs from the corresponding true solar time by the difference between the hour angles, or right ascensions, of the true and mean suns. This difference is called the equation of time (the ' ' sun fast ' ' and ' ' sun slow " of the common almanacs and calendars), and its value for each day of the year, at Greenwich noon and at Washington noon, is given in the American Ephemeris (see 21) and other almanacs. To change local solar time from one system to the other we have therefore to interpolate the equation of time from the almanac, with the argument the given local time, reduced to the Greenwich or Washington meridian by addition of the longitude, and apply this difference with its proper sign to the given local time. For exam- ple, let it be required to find for the meridian of Denver, and for the date May 10, 1905, the local apparent solar time corresponding to the mean solar time, M, given below. The course of the computation is as follows : A, Denver west of Greenwich 6 h 59 47*. 6 i M , Denver Mean Solar Time 3 5 10.5 a Greenwich Mean Solar Time 10 4 58 .1 J + 2 Equation of Time +3 44 . 2 3 Denver Apparent Solar Time 3 8 54.7 2 + 3 19. Relation of Sideral to Meaa Solar Time. Once in each year the mean sun passes through the vernal equinox and at this instant, which we shall represent by V, mean solar time and sidereal time agree. At any other moment during the year sidereal time will be greater than mean solar time by an amount, Q, equal to the right ascension of the mean sun at that moment, TIME. 41 and the conversion of the one kind of time into the other is therefore solely a matter of finding the corre- sponding value of Q, i.e., = M+Q, (18) where and M stand respectively for the corresponding sidereal and mean solar time. Since Q increases by exactly 24 h in one mean solar (tropical) year we may find its rate of increase, 5, by dividing 24 h by the number of days in such year (365.2422) and obtain thus log 5 = 0.595781, minutes of time per day, 5 = 236.555, seconds of time per day, and since the motion of the mean sun is entirely uni- form we shall have at any time, T, the value of Q given by the relation The quantity V varies slightly from year to year and its value, expressed in days reckoned from the first Greenwich Mean Noon of each year from 1905 to 1930, is given in Table 2. The instant, T, must be similarly ex- pressed for use in Eq. 19. Use the second column of Table 3 to convert ordinary dates into days from the beginning of the year. For a rough computation "in the head" it is of ten convenient to use 5=4 m (i-^r) V=March 22.7 (Greenwich) (20) 42 FIELD ASTRONOMY. The Nautical Almanac gives for each day of the year the value of Q at Greenwich Mean Noon (in the last column of page II ,of the monthly calendar), and also gives at the end of the almanac, Table III, a table of proportional parts by which to facilitate the inter- polation of Q for intermediate times. If M represents any local mean solar time which it is desired to convert into sidereal time and A be the longitude of the local meridian west of Greenwich, then will M + X be the corresponding Greenwich time and with this argument we obtain immediately from Table III the correction that added to the tabular value of Q for the given date furnishes its value at the given moment. For illustration of each process above given for finding Q let it be required to find the sidereal time corresponding to Boston mean solar time p h i9 m 2 6 s . 6 7 on August 4, 1905, assuming as the longitude of Boston 4 h 44 m i5 s west of Greenwich. First Method. Second Method. h. m. s. h. m. s. M 9 19 26.67 M 9 19 26.67 A 4 44 15 A 4 44 15 M + A 14 3 42 M + A 14 3 42 T 217.586 Greenwich, Q 8 49 31.43 V 82.691 Table III 2 18.60 log(r-y) 2.129996 Local, Q 8 51 50.03 log S 0.595781 6 18 ii 16.70 S(T-V) 531.835 minutes Q 8 51 50.10 e 18 ii 16.77 The difference in the results here found for M is due to neglected figures in the fourth decimal place of V and T and illustrates the superior precision as well as con- venience of the second method. The latter should TIME. 43 always be employed when an almanac for the current year is available, but the first method may render good service in the absence of such an almanac. The converse problem of transforming sidereal time into mean solar time is best treated as follows : Let M\ denote an assumed mean solar time not very different from the true value of M corresponding to the given and compute, as above, the corresponding values of Q and which we represent by Q\ and Q\. If Q\ proves to be different from the given 0, Qi will differ from the true Q by the amount that the mean sun changes its right ascension in the interval 00 l , i.e., the true value of Q will be 0i), (21) when 6 6 1 is expressed in sidereal minutes. When values of Q are taken from the Almanac it is customary to adopt the local mean noon as the value of M\ since in this case the value of M becomes M=(0-Qi)-oM6 4 (tf-Qi). (22) \ The Qi thus defined is obtained by adding to the Almanac value of Q a correction taken from Table III with the observer's longitude as argument, and the last term of the expression is taken from a table of multiples of o s . 1 64 given at the end of the Almanac as Table II. In the absence of the Almanac it is better to take the assumed Mi as near as may be to the true value of M. Using the Almanac we reconvert the found above into the corresponding mean solar time as follows : 44 FIELD ASTRONOMY. Greenwich, Q Table III, X 8 49 3i-43 46.70 Local Noon, Q l 8 50 18.13 6 18 ii 16.70 0-Qi 9 20 58.57 Table II, 0-^ i 31.90 M 9 19 26.67 Without the aid of an Almanac, we may assume Mi = 9 h and by a process entirely similar to the First Method employed above find the Qi shown in the following schedule and determine from it the required M as follows ; M l d Q M 900 8 51 46.80 17 51 46.80 18 ii 16.70 8 51 49-99 9 19 26.71 20. Chronometer Corrections. As already indicated, in actual practice the measurement of time is made by clocks or chronometers. A chronometer does not differ essentially from an ordinary watch, and like the latter is designed to show upon its face, at each moment, the mean solar time (or sidereal time) of some definite meridian, e.g., the merid- ian 90 west of Greenwich. Since the time indicated by such an instrument is seldom correct, the error of the timepiece must usually be taken into account, and in astronomical practice this is done through the equation, T, or M = T + JT, (24) where T is the time shown by the chronometer (or watch) and AT is the correction of the chronometer, TIME. 45 i.e., the quantity which must be added, algebraically, to the watch time in order to obtain true time of the given meridian. When the chronometer is too slow T is less than the true time at any moment, and AT is there- fore positive in this case and negative when the chro- nometer is too fast. While the symbol AT always repre- sents a chronometer correction its numerical value in a given case depends upon the particular use required, i.e., whether the chronometer time is to be reduced to sidereal, or solar, local, or standard time. In the two Equations 24, therefore, AT represents quite different quantities, since 6 and M are usually different one from the other, and in every case a special memorandum must be made showing whether the given AT relates to sidereal, mean solar, or apparent solar time. If the chronometer gains or loses, it is said to have a rate and AT will then change from day to day. If we assume a uniform rate, the relation between T and becomes = T + AT 9 + P (T-TJ, (25) where the subscript denotes the particular value of AT belonging to the chronometer time T , and p is the rate of the chronometer per day or per hour, positive when the chronometer is losing time. The interval TT must be expressed in the same unit as that for which p is given, hours for an hourly rate, etc. A similar equa- tion represents the relation between T and M, but p and AT will be numerically different from the values required in Equation 25. 46 FIELD ASTRONOMY. A sidereal chronometer, i.e., one intended to keep sidereal time, differs from a mean solar chronometer only in the more rapid motion of its mechanism, and is in fact an ordinary timepiece for which p= 3 m 56 S 5 per day. Similarly a watch may be regarded as a sidereal timepiece for which /?= + io s per hour. A sidereal chronometer is most convenient for use in obser- vations of stars, since their diurnal motion in hour angle is proportional to the lapse of sidereal time, but these observations may perfectly well be made with a watch or other mean solar timepiece, provided this is treated as a sidereal chronometer with a large rate; e.g., if p denote the hourly rate of the watch relative to mean solar time, its hourly rate upon sidereal time will be p' = p-\- io s . Use Equation 25 in connection with this value of p f to determine from minute to minute the varying value of JT. 21. The Almanac. The American Ephemeris and Nautical Almanac is an annual volume issued by the U. S. Navy Department for the use of navigators, astron- omers, and others concerned with astronomical data. These data are for the most part quantities that vary from day to day and whose numerical values are given at convenient intervals of Greenwich or Washington boicti time, e.g., the E and Q of the preceding sections, and the right ascensions and declinations of the sun, moon, planets, and principal fixed stars. The varia- tions of these quantities are due to many causes, orbital motion, precession, nutation, aberration, etc., that, in general, lie beyond the scope of the present work, but TIME. 47 we shall have frequent occasion, as in 18 and 19, to take from the almanac numerical values of the quanti- ties above indicated, and these values are to be inter- polated for some particular instant of time, usually that of an observation in connection with which they are required, as logarithms are interpolated to correspond to some particular value of the argument of the table. Since quantities are tabulated in the almanac for selected instants of Greenwich or Washington time, the time used as the argument for their interpolation must be referred to one of these meridians (see 17). For a detailed account of the way in which the almanac is to be used, consult the explanations given at the end of each volume, under the title, Use of the Tables. In addition to those explanations it should be noted that under the heading Fixed Stars, pages 304-399, there are given three separate tables, from the last of which, bearing the subtitle Apparent Places for the Upper Transit at Washington, accurate coordinates of most of the stars may be obtained for use in the reduction of observations. For the remaining stars, five in number and all very near the celestial pole, special provision of this kind is made in the second table, which bears the subtitle Circumpolar Stars. Look here for the coor- dinates of Polaris. The first table, under the subtitle Mean Places, etc., gives in very compact form, for all stars contained in the other two tables, right ascensions and declinations, together with their Annual Variations, that may be consulted with advantage when only approx- 48 FIELD ASTRONOMY. imate values of these quantities are required, e.g., in the preliminary selection of stars suitable to be observed. In this connection the second column of the table of Mean Places, entitled Magnitude, deserves especial notice, since it furnishes an index to the brightness of the stars, which is an important element in deciding upon their availability for a given instrument. The brightness of each star is represented by a number adapted, upon an arbitrary scale, to that brightness, so that a very bright star is represented by the number o, one at the limit of naked eye visibility by 6, and intermediate degrees of brightness are represented by the intermediate numbers, carried to tenths of a mag- nitude. Polaris is of the magnitude 2.2, and is a con- spicuous object in even a very small telescope, provided the telescope is properly focussed. In the telescope of an engineer's transit, stars of the magnitude 4.0 or even 5.0 may be readily observed, while with a sextant, under ordinary conditions, the third magnitude may be taken as the limit of availability. CHAPTER IV. CORRECTIONS TO OBSERVED COORDINATES. IT has already been pointed out that the problems of spherical astronomy are in great part cases of the transformation of coordinates between systems having a common origin but different axes, and it should be noted that the observed data for these transformations frequently require some correction before they can be introduced into the equations furnished by the astro- nomical triangle. Aside from errors arising from de- fective adjustment or other purely instrumental causes, the observed coordinates of a celestial body may require any or all of the following corrections. 22. Dip of the Horizon. This correction is required when an altitude is to be derived from a measurement of the angle of elevation of a body above the sea horizon. Owing to the spherical shape of the earth the visible sea horizon always lies below the plane of the observer's true horizon, and the amount of this depression might easily be determined from the geometrical conditions involved, were it not that the rays of light coming to the observer from near the horizon are bent by the at- mosphere (refraction), in a manner that does not admit 49 50 FIELD ASTRONOMY. of accurate estimation in any given case, although its average amount is fairly well known. We therefore abstain from any formal investigation of this correc- tion, and expressing by e, in feet, the observer's elevation above the water, we adopt as a sufficient approximation to the observed amount of the depression, either of the following formulae, ^-"sArW', >" = [i.7738]V*. (26) The values of D given by these equations are expressed in minutes and seconds, respectively, but owing to varia- tions in the amount of the refraction the numerical values furnished in a given case may be in error by several per cent. As a correction D must always be so applied as to diminish the observed elevation above the horizon. Note that if the depression of the visible horizon be measured with a theodolite or other suitable instru- ment, Equation 26 will furnish an approximate value of the elevation of the instrument above the water. 23. Refraction. In general the apparent direction of a star is not its true direction from the observer, since the light by which he sees it has been bent from its original course in passing through the earth's atmos- phere. The resulting displacement of the star from its true position is called refraction, and, like the similar effect noted in the previous section, its analytical treat- ment presents mathematical and physical problems whose solution must be sought in more advanced works than the present. Some of the results of that solution CORRECTIONS TO OBSERVED COORDINATES. 51 which we shall have occasion to use hereafter are as follows: Save at very low altitudes, less than 10, the refraction does not sensibly change the azimuth of a star, but its whole effect is to increase the altitude, so that every star appears nearer to the zenith than it would appear if there were no refraction. The amount of this displacement depends chiefly upon the star's distance from the zenith, but is also dependent in some measure upon the temperature of the air and its barometric pressure. If we represent by h' the star's altitude as measured, by h the corresponding true altitude, by t the tempera- ture, in degrees Fahrenheit, of the air surrounding the observer, and by B the barometric pressure in inches, the amount of the refraction in seconds of arc may be obtained from the following formula within half a second for all altitudes greater than 20, ( 27) For altitudes less than 20 this formula gives results that are too great and as the star approaches the hori- zon the error increases rapidly, as may be seen from the following brief table of corrections required to reduce the numbers given by Eq. 2 7 to the true values of the refraction. h Corr. 5 10 15 20 25 -68" - 9 2 + The readings of a mercurial barometer, B', do not 52 FIELD ASTRONOMY. furnish immediately the barometric pressure, B, but require a "reduction to the freezing-point," i.e., a cor- rection to reduce the reading to what it would be if the mercury were at the normal temperature assumed in the theory of the barometer. This reduction may be ob- tained with sufficient accuracy from the equation ( 2 8) 10 000 where T is the temperature of the mercury, in degrees Fahr., and the barometer reading and its resulting cor- rection are expressed in inches. 24. Semi- diameter. Observations of the sun or other body presenting a sensible disk are usually made by pointing the instrument at the edgo of the body, techni- cally called the limb, and the resulting altitude or azimuth is that of the limb observed, while the data furnished by the almanac relate to the centre of the body. The semi-diameters of the sun, moon, and planets, i.e., the angles subtended at the earth by their respective radii, are given in the almanac at convenient intervals of time, and the interpolated values of these quantities may be used to pass from the observed coordinates of the limb to those of the centre of the body, e.g., the sun. In the case of the altitude or zenith distance we have the very simple relation h'=h"S, (29) where 5 denotes the semi-diameter and h" and h f are, re- spectively, the observed and the corrected altitude. The CORRECTIONS TO OBSERVED COORDINATES. 53 sign of 5 depends upon whether the lower or the upper limb was observed. In the case of an azimuth the relation is more complicated. From the right-angled spherical triangle formed by the zenith, the sun's centre and that point of the limb at which the latter is tangent to a vertical circle (see Fig. 4) we obtain, sin z sin (A r - A") = sin 5, (30) which determines the correction, A f A" , for difference of azimuth between centre and limb. Since 5 does not much exceed 15', we may in most cases assume the arcs to be propor- tional to their sines and simplify this rigorous equation to the form, A'=A"S seek, (31) in which the positive sign is to be used for the following and the nega- tive for the preceding limb. 25. Parallax. In the reduction of astronomical observations it is usually necessary to combine the cd coordinates, azimuth, alti- ., FIG. 4. Semi-diameter. tude, etc., with data obtained from the almanac, e.g., the right ascension and declina- tion of the body observed. But the origin to which these latter coordinates are referred is the centre of the earth, while the origin for the observed coordinates is 5 FIELD ASTRONOMY. at the eye of the observer, and before combining these heterogeneous data we must reduce them to a common origin, for which we select that used in the almanac. In Fig. 5 let C represent the centre of the earth, the observer's position, and P the observed body, at the respective distances p and r from C. Neglecting the FIG. 5. Parallax. earth's compression, i.e., its slight deviation from a truly spherical form, the line OC is the observer's vertical and, therefore, OPC is a vertical plane and marks out upon the celestial sphere a vertical circle, against which the body P will appear projected whether seen from or C. Its azimuth will, therefore, be the same for the two origins and requires no reduction to the centre of the earth. The altitude, however, does require such a reduction, and to determine its amount we let OH in Fig. 5 repre- sent the plane of the observer's horizon and obtain as CORRECTIONS TO OBSERVED COORDINATES. 55 the observed altitude of P the angle there marked h f . As seen from the centre of the earth the altitude of P will be measured by the angle PIH, marked h in the figure, and from principles of elementary geometry we have This last angle is called the parallax in altitude, and rep- resenting it by P we find from the triangle OPC p sin (90 + h')=r sin P. Since r is always much greater than p, P must be a small angle and, applying the principles of 4, we may write, in place of the preceding equation, P=h-h' = 206265 cos h', (32) which is the required correction to reduce an observed altitude to the corresponding coordinate referred to an origin at the centre of the earth. For the fixed stars this correction is absolutely insen- sible, less than o".oooi, on account of their great distance from the earth. For the sun and planets it amounts to a few seconds of arc, and in its computation the value of the coefficient, 206265 , should be taken from the almanac, where it is given for each of these bodies and is called their horizontal parallax, since it is the amount of the parallax in altitude when the body is in the horizon, h r = o. For the sun it is usually sufficient to assume 56 FIELD ASTRONOMY. 8". 8 as a constant value for its horizontal parallax. The moon's parallax is much greater, about i, and the simple analysis given above neglects some factors that are of sensible magnitude in this case, although for ordi- nary purposes they may be ignored in connection with every other celestial body. Since the effect of parallax is to make the body appear farther from the zenith than it really is, the corrections for parallax and refraction will always have opposite signs. 26. As an example of such corrections and their proper sequence we give the following reduction of an angle observed between the sun's upper limb and the sea horizon, as seen by an observer at an elevation of 63 feet above the water. Date Oct. 6, 1907 Refraction Const. 2.989 Temperature 44 Fahr. B 1.466 Barometer 29.26 in. colog (456 + 7.301 B'-B .04 cot h' 0.282 B 29.22 logR 2.038 Observed Angle 27 43' 20" R 109" h" 27 33 39 Dip of Horizon Semi -Diameter 16 9 e 63 V" 27 17 37 Constant 1-774 Parallax Const. 0.944 log \/e 0.900 cos h'" 9-949 Dip 472" = Parallax 8" Dip 7' 52" h 27 17 45 V 27 35 28 . All computations such as the above may conveni- ently be made with a slide rule. In accordance with general custom the symbol log is printed in the above schedule only when necessary to CORRECTIONS TO OBSERVED COORDINATES. 57 avoid misunderstanding, as at the bottom of the first column. Usually the figures themselves indicate whether they are logarithms or natural numbers; e.g., the several numbers marked Const, are clearly the logarithms of constant coefficients. For similar reasons of convenience the 10 that strictly should be placed after a logarithm whose characteristic has been increased by 10 is usually left to the imagination. 27. Diurnal Aberration. There is a very small cor- rection to observed data, arising from the fact that the observer himself is not at rest relative to the stars, but is always in rapid motion toward the east point of his horizon, carried along by the earth in its diurnal rotation. This correction is so small that it may usually be omitted and we therefore abstain from an analytical investigation of its effect, such as may be found in the larger treatises upon spherical astronomy, and note as a result of that investigation that all stars when near the meridian are displaced toward the east point of the horizon through an angular distance equal to o".32 cos 0, where de- notes the observer's latitude. As a result of this dis- placement each star comes a little later to the meridian than it otherwise would come and since the rate of motion of a star when measured in arc of a great circle is propor- tional to the cosine of its declination, the amount of this retardation, expressed in time, is o s .o2i cos sec d. See the theory of the transit instrument for an example of the application of this correction, and see also the de- termination of precise azimuths for another case in which diurnal aberration is to be taken into account. CHAPTER V. ROUGH DETERMINATIONS OF TIME, LATITUDE, AND AZIMUTH. 28. General Considerations. For the purposes of field astronomy, which are the only ones contemplated in the present work, the most important astronomical problems relate to the determination of time, latitude, and azimuth. A time determination implies the making and reduc- ing of astronomical observations which suffice to furnish the correction, AT, of a chronometer or other timepiece, and for this purpose we obtain from 15 and 20 the relations a + t = 6 = T + JT, (33) where a and / represent the right ascension and hour angle of any star at the chronometer time T. The student should particularly note that the chronometer is not supposed to be correctly set ; T is the time shown by the chronometer regardless of whether that time be right or wrong, since the AT fully compensates for any error of this kind. In the case of the sun we have, from the rela- tion between mean and apparent solar time, (34) where E denotes the equation of time at the instant T. 58 ROUGH DETERMINATIONS OF TIME, ETC. 59 Since a and E may be obtained from the almanac, any observation which determines the hour angle of a celestial body at the observed time T will suffice to determine 4T, and such an observation when properly reduced constitutes a time determination. An azimuth determination may be required either for fixing the true azimuth of the line joining two terres- trial points, or for determining the relation of a particular instrument to the meridian; e.g., to determine the read- ing, K, to which the azimuth circle of a theodolite must be set, in order that the line of sight shall point due south. A theodolite is said to be oriented when its verniers have been set to read the true azimuth of the object toward which the line of sight is directed, i.e., when K = o. By a latitude determination we mean any set of ob- servations from which a knowledge of the observer's latitude may be obtained. For each of these determinations, time, azimuth, latitude, many methods have been devised and these differ greatly among themselves with respect to the instrumental equipment and expenditure of time and labor which they require, and with respect to the corre- sponding degree of accuracy furnished in their results. In any given case a choice must be made among these methods with reference to the required precision of the results and also with reference to convenience and economy in obtaining it. To facilitate this choice the methods to be presented in the following pages are classified as: (A) Rough Determinations; in which there may be 60 FIELD ASTRONOMY. permitted in the final result an error amounting to two minutes of arc or one tenth of a minute of time. (B) Approximate Determinations; in which the final errors ought not to exceed 15" and i s respectively. (C) Accurate Determinations; in which the required precision is limited only by the capacity of the instru- ment and of the observer. In the case of a sextant this limit may be placed at 2" or 3", and for a good engineer's transit at i". We proceed first to consider that class of observations whose advantage consists in economy of time and labor, viz., rough determinations. 29. Latitude. A determination of any one of the quantities time, latitude, or azimuth is greatly facilitated by a knowledge of one or both of the others, and if all three are unknown, the simplest mode of procedure is to observe the Pole Star as set forth in 32. But this commonly requires observations by night, which may be inconvenient, and by day the sun is the object most readily available. From the astronomical triangle, or from Equations 14, it is apparent that when the sun is on the meridian, i.e., when t = o, its altitude is a maximum, and if this maximum altitude be measured with a sextant or theod- olite it will furnish a latitude determination through the equation, true for noon only, = 8 + z = 9 o + d-h, (35) which may be obtained by inspection from Fig. 2, or analytically from the last of Equations 14. With the instrument employed, follow the sun's motion in altitude ROUGH DETERMINATIONS OF TIME, ETC. 61 until it begins to dimmish, and take the greatest reading obtained as corresponding to the maximum altitude. This reading, or the altitude, h', derived from it, will require correction for instrumental errors, semi-diameter, etc., as shown in Chapter IV, but the application of these corrections may be abbreviated by interpolating, in minutes of arc, the combined correction for refraction and parallax from Table I, at the end of the book. This gives, with the observed altitude as argument under the heading R, the amount of the refraction cor- responding to an average condition of the atmosphere, viz., Bar. 29.0 in., Temp. 50 Fahr., while under the heading R' is given the combined effect of refraction and parallax for the sun. Use these tables for the reduction of any altitude of sun or star when the re- quired precision is not greater than 10". This table may be adapted even to extreme atmospheric condi- tions by increasing R or R' by one per cent for each 0.3 inch that the barometric pressure exceeds 29 inches. Similarly diminish R by one per cent for each 5 that the temperature exceeds 50 Fahr. The following latitude determination was made by measuring with a sextant and artificial horizon (59) the maximum double altitude of the sun's lower edge (limb) upon a date, Dec. 19, 1898, at which the sun's declination, as furnished by the almanac, was, d = 23 26'.o. The reduction of the observation is a? follows : 62 FIELD ASTRONOMY. Sextant Reading 57 44' 30" Instrumental Corr. i 50 Corr'd Sextant 57 42 40 .h' 28 51.3 Ref.-Par., R' - 1.6 Semi-diameter +16.3 h 296.0 90 + 8 66 34.0 Latitude, (f> 37 28.0 Make a determination of your own latitude by a simi- lar method. 30. Time and Azimuth from an Observed Altitude. If the latitude be thus observed at noon, time may be determined with a sextant, and both time and azimuth may be determined with a theodolite by measuring an altitude of the sun when at a considerable distance from the meridian. Any error in the assurned latitude or in the measured altitude will affect both the azimuth and time resulting from the observation, but the influence of these errors may be greatly reduced by observing the sun when near the prime vertical and ; for azimuth, when at a low altitude, e.g., 10 or 15. When a theodolite is used, there should be at least two observations of altitude made, one Circle R. and the other Circle L., in order to eliminate instrumental errors (see 50). Observe the edges of the sun, not its center, and correct the results for semi-diameter '( 24); but if the instrument has stadia threads less than 30' apart, avoid this correction as follows: Point the telescope at the sun so that the two horizontal threads cut off equal segments from the upper and lower PLATE I. \Tofacep. 62.] An American Engineer's Transit. Diameter of Horizontal Circle 7 inches. Approximate Cost $350. ~ i ROUGH DETERMINATIONS OF TIME, ETC. 63 edges of the sun, and by turning the slow-motion screw in altitude, keep these segments of equal area as the sun drifts across the field of view, until it reaches a position in which the vertical thread bisects each segment. Record this time to the nearest second, and also record the readings of the four verniers of the instrument. Before reversing the instrument to obtain the second observation, read and record both its levels (azimuth and altitude levels, 48 and 50), and after reversing bring the bubbles back, by means of the levelling-screws, to the position thus recorded. This process eliminates errors of level. The better class of engineer's transits are usually provided with shade-glasses to moderate the intensity of the sun's light and permit it to be viewed through the telescope. But these glasses are by no means neces- sary, since an image of the sun and the threads of the instrument may be projected upon a piece of cardboard and be there seen and observed quite as accurately as in the telescope. Pull the eyepiece out, away from the threads, until the latter can no longer be seen dis- tinctly with the eye ; then allow the sun to shine through the telescope upon the cardboard held behind the eye- piece, and shift the cardboard toward and from the instrument until a position is found in which the projected images of the threads appear sharp and distinct. Then turn the focussing-screw until the edge of the sun's image also appears well defined and the projected images will be ready for observation. To diminish as far as possible the effect of errors in the measured altitude 64 FIELD ASTRONOMY. and the latitude assumed in the reduction, observe the sun when it is as near as may be due east or west, but its altitude should not be less than 10. Reduction of the Observations. Observations con- ducted as in the following record will furnish as data the true altitude of the sun's centre and the horizontal angle, H, between the sun's centre and some fixed object whose azimuth is to be determined, both quantities cor- responding to one and the same time, T, taken from some watch or other time piece. These data must be supplemented by the latitude, <, supposed to be inde- pendently known, and by the sun's declination, d, and equation of time, E, to be obtained from the almanac. Reference to Fig. 3 will show that d, h and (j> are respec- tively the complements of the three sides of the astro- nomical triangle, and putting 9 o - = a 9 o 90 We find that the solution of the triangle is given by Eqs. 6 and 7, when in place of the general symbols there employed for the angles of the triangle we substitute the particular values which they have in this case, i.e., sin 5 sin (s a) sin (s b) sin (s c)' cot %a Q = k sin (s d), cot \t =k sin (s-6), (36) cot Ag = k sin (s c] . ROUGH DETERMINATIONS OF TIME, ETC. 65 The symbol ao here introduced to represent the angle of the triangle opposite the side 90 d, is readily seen from Fig. 3 to be the sun's azimuth reckoned from north toward west, and if we desire to retain the cus- tomary mode of reckoning azimuths, from the south, we must write A = i8o a . If to A there be added the observed horizontal angle, H, we shall of course find the desired azimuth of the terrestrial point, viz., A M =i8o + JJ-a . (37) The angle t is the sun's hour angle at the moment of observation, T, and we have therefore, T + 4T=t + E (38) from which to determine the chronometer correction. The third angle of the triangle, q, is introduced only for the sake of the following control which it furnishes upon the reduction. Multiplying together the last three equa- tions of (36) we find, cot \a Q cot \t cot \q = k sin s, (39) a relation that must be satisfied within one or two units of the last decimal place if the numerical work of re- duction has been correctly performed. Another useful "check" thai ih^uld be applied midway in the computa- tion is (5 -a) + (5 - 6) + (s - c) =s. (40) The ambiguous sign of k, Eq. 36, covers the two cases of observations made when the sun is west or east 66 FIELD ASTRONOMY. of the meridian. For an A.M. observation use the minus sign, and find a , t and q, all negative quantities. The following illustrative record and reduction relate to observations made at a place whose latitude and longitude are respectively 43 4.6', and 5 h 58"* west of Greenwich. ALTITUDE AND AZIMUTH OF SUN'S CENTRE. At Station A. April 16, 1897. Engineer's Transit, B. Watch No. 6. Observer, C. Object. Circle. Watch. Vertical Circle. Horizontal Circle. Ver. A. Ver. B. Ver. A. Ver. B. Sta. B Sun Sun Sta. B R. R. L. L. h m s 7 54 10 259 14 o 79 45 o 187 54 10 54 i 13 5 45 54 i 4 12 33 4 15 28 26 26 o 25 57 20 25 5 57 o REDUCTION. a 10 28'. sin (s a) 9.4300 do 100 3' -4 Vert. Circ. 26 11.5 sin (s b) 9-7157 i8o + jH" 288 24.7 R'. Table IB -1.8 sin (s c) 9.8726 AM 187 54-3 h 26 9.7 Sum 9.0183 $ 43 4-6 sin 5 9.9982 t (arc) 63 50'. o k 2 0.9799 * (time) 1 5m 20* a 79 32.o k 0.4899 E 25 b 63 50-3 t+E 4 T 4 55 c 46 55-4 cot \a 9.9199 T 4 J 3 55 25 190 17.7 cot \t o. 2056 jr + 1 S 95 8.8 cot \q 0.3625 sa 15 36-8 Sum 0.4880 s-b sin s 0.4881* 5C 48 13.4 Sum 95 8.7* The true azimuth of Station B was known to be 187 54' 10", and a comparison of the watch with standard time furnished as the true value of AT, + 57 seconds. The differences between these results and those found in the preceding solution furnish a fair idea of the pre- cision to be expected in such work. ROUGH DETERMINATIONS OF TIME, ETC. 67 When time only is to be determined from an observa- tion of the sun the measurement of the angle H is unnecessary, and the reduction of the altitude obser- vation may be made by the method of 36. Where azimuth only is required both H and h must be ob- served, and the sun's azimuth maybe computed some- what more conveniently than above, from the third of Eq. 15, written in the form, sin < sin h - sin $ Cos A = - -, (41) cos (j> cos h but as no " check" relation accompanies this formula the longer computation given above will often be prefer- able. 31. Time by Meridian Transits. If astronomical ob- servations are to be made for any considerable length of time at a given station, as at a university, it will be convenient for many purposes to determine the azimuth of a permanently marked line, at one end of which an instrument can be set up and oriented. If a theodolite be thus mounted and its line of sight brought into the meridian, a time determination may be very simply made by observing the chronometer time of transit of the sun's preceding and following limbs past the vertical thread of the instrument. Since the thread is by sup- position in the meridian, the hour angle of the sun at the mean of the observed times, T, is zero and we have JT = a -T (Sidereal), or 4T=ET (Mean solar). (42) 68 FIELD ASTRONOMY. If the azimuth of the line is well determined, this method may rank as an approximate rather than a rough determination, since under ordinary circumstances there must be an error of nearly 2' in the orientation of the instrument, to produce an error of 6 s in the chro- nometer correction. In any case the instrument must be carefully levelled, particularly in the east and west direction, and in the following example the readings of the striding level are employed as a control upon this adjustment. Observe the slight variation of method here intro- duced in order to obtain in place of a single observation two observations, one Circle R., and one Circle L. TRANSITS OF SUN FOR TIME DETERMINATION. At Station A. April 17, 1897. Theodolite, F. Watch No. 6. Observer, G. Instrument oriented on Station B. Circle. Ver. A. Limb. Watch. Striding level. Reduction. / h. m. s. T = n h 57 m 37 s R. 359 30 Pr. 11 55 34 1.4 12.6 i2 b + = n 59 23 L. 180 30 Fol. ii 59 40 14.0 0.5 jr= +i 4 6 By a comparison with standard time the true AT referred to the local meridian was found to be +i m 47*. The telescope of an engineer's transit is usually capable of showing a first-magnitude star by daylight whenever the sky is clear and blue, and such a star is equally available with the sun for a determination of either latitude or time. In the spring and summer Sirius, by reason of its great brilliancy, is a peculiarly favorable object for such observations (see Table 5 ROUGH DETERMINATIONS OF TIME, ETC. 69 for the approximate right ascension and decimation of this and other stars). Even the brightest of these stars is not a conspicuous object by daylight, and is most readily found by placing the telescope in the meridian and at the proper zenith distance, z= , and await- ing its arrival in the field of view. A very slight error of focus in the telescope will render the star invisible, and this adjustment should therefore be carefully made upon a distant terrestrial object before setting the tele- scope for the star. 32. Orientation by Polaris. If a rough determination of time, latitude, or azimuth is to be made by night, or if a theodolite is to be oriented as a preparation for other work, observations of the Pole Star by the following method will be found especially convenient, since no almanac is required and no instrumental equipment other than an engineer's transit and a watch. Even the error of the watch need not be known in advance. If Polaris were exactly at the pole of the heavens the instrument might be oriented by pointing directly upon the star and setting the verniers of the horizontal circle to read 180; and simultaneously the latitude might be determined by measuring the star's altitude, since in this case.

sin p cos t, (43) are applied to Polaris we find that p and a are small quantities rarely exceeding ior 2, whose third powers may be neglected without producing error exceeding i" or 2", and to this degree of accuracy we may use in their place the simpler forms ao= p sec h sin t cos *-^ 2 sin 2 J tan <. (44) The quantities given in Table 4 are respectively the terms of these equations, a = p sec h sin t bo=p cos t when p is put equal to 70'. 5 and h is equal to 40. To adapt these tabular quantities to other values of p and h let us introduce the auxiliary quantities, F=p + *]Q r .$ f=sech-r- sec 40 ROUGH DETERMINATIONS OF TIME, ETC. 71 and substitute for the star's true altitude, h, its apparent altitude, h' ', corrected for the amount of the refraction, R, and in termi of thes3 find from Eq. 44, &'-< = Fbo+R-^p 2 sin 2 t tan 0. (47) The last term in this expression is very small, on account of the factor p 2 , and if we combine its average value with the refraction, R, we shall find that the result agrees very closely with the combined effect of refrac- tion and the sun's parallax, given in Table I B , under the heading R'. Eq. 47 may be used to compute the true azimuth of Polaris and the difference between its apparent altitude and that of the pole, h f 8 2 10 20 2 10 12 46 Obsd. T 2 ioh gr\ AT + 3 T-i IO D+Y 2 25 t 12 28 Obsd. /4 4i 55' Fbn i ii R' + r 43 5 C + 11 Ff 1.041 180 ii Light-Polaris 7 46 Az. of Light 187 57 76 FIELD ASTRONOMY. RECORD OF THE OBSERVATION. Tuesday, May 19, 1908. At Station Z. Object. Circle. Watch. Az. Circle. Vert. Circle. Polaris L. b. m. s. 10 o / 1 80 14 / 41 19 42 9 0.13642 App't h Refraction 27 24 18 1 55 28 39 19 i 49 sec 6 0.02620 h 27 22 23 28 37 30 tan (f> 9 .97082 sin h 9.66255 9 .68040 tan 6 9-55401 sec sec 8 sin h 9.82517 9.84302 sec sec d o. 16262 tan tan 6 9-52483 9-52483 Subtract 9 .99862 0.03367 h. m. s. cos t 9-52345 9-55850 a 14 ii 6.1 t 4 42 0.5 4 35 9-o a+t 9 2 9 5- 6 9 35 57-i Adopted JT= -6oV8 (Local sidereal). In place of the laborious process of separately reduc- ing each observed altitude we may usually treat the mean of the sextant readings and the mean of the ob- served times as if they constituted a single observation. When the observed body is near the prime vertical the time interval covered by a set of observations which it is purposed to unite into a mean result may extend to 15 or 20 minutes without sensible error, but the error 86 FIELD ASTRONOMY. of the process increases rapidly with increasing distance from the prime vertical, and the time interval must be correspondingly diminished. In the preceding example of a time determination from sextant altitudes, the sextant was set accurately to a set of readings differing by a uniform interval of 30', and the times noted at which the observed body came to the corresponding altitudes. In the reduction ad- vantage is taken of this circumstance by computing the value of a + t for the first and last observations only, and interpolating the intermediate values. Observe that the columns a + t and AT , although placed near the beginning of the reduction, are really the last to be filled out. 37. Azimuth Observations at Elongation. An excel- lent approximate determination of the azimuth of a terrestrial mark may be made by measuring, with a theodolite, the horizontal angle between the mark and a circumpolar star at the time of its elongation, i.e., its maximum digression from the meridian. It may be seen by inspection that at the instant of elongation the astronomical triangle, Fig. 3, is right- angled at the star, and we obtain from it, sin A e = cos d sec 0, (58) cos t e = cot d tan 0, where the subscript e shows that the quantities thus marked relate to elongation only, and A e is measured from the north toward either east or west as the case may require. APPROXIMATE DETERMINATIONS. 87 The time of elongation is then given by Sidereal. Q e =at e ~ Elon S atlon - Mean solar. If H denote the measured angle between the star and the mark, positive when the mark is east of the star, we shall have as the azimuth of the mark A M =iSo+A e +H. (60) Equations 58, 59, 60 leave nothing to be desired on the score of simplicity and the only observation required in connection with them is the measurement of the angle H at the time O e or Me. This is frequently done by pointing the instrument upon a star a little before the computed time, M e , and following the star's motion in azimuth until it ceases to move away from the meridian. The verniers are then read and immediately thereafter a single pointing upon the azimuth mark is made. This method is convenient when the time is very imperfectly known but, in general, a better pro- cedure is to determine the time to the nearest minute, by the method of 33, or otherwise, and to make at least two pointings upon both mark and star, one Circle R and one Circle L, to eliminate instrumental errors, since this elimination is of far more consequence than any precise agreement between the time of elongation and the actual time of observation. Indeed the error introduced by failure to measure the angle when the star is exactly at elongation, may be completely remedied as follows : In Fig. 6 let SE' represent the vertical circle passing through a circumpolar star, 5, 88 FIELD ASTRONOMY. which is T minutes away from its elongation, and from the pole, P, as a center describe a circle tangent to SE'. A star at the point of tangency, E f , clearly has the same azimuth as 5 and being at the elongation in its own circle this azimuth may be found through Eq. 58, using for 6 the declination of 5 increased by the distance between the two circles of the figure. Calling the radii of these circles p and p f respectively, we find from the spherical triangle PSE' t FIG. 6. tan p' = tan p cos T, which may be transformed by substituting for cos T its T 2 . value i into the relation 2 x : p = T 2 : n, (61) where x is the required difference of radii of the circles and n is a constant number whose value depends upon the units employed for the other quantities. When p is expressed in minutes and x in seconds of arc we find # = 1745, when T is given in minutes of mean solar time, n = 1755, when T is given in minutes of sidereal time. APPROXIMATE DETERMINATION. 89 Use a slide rule to compute the value of # from Eq. 61 and find the azimuth of 5" from the equation, sin ^4 T =cos (8+#) sec d>. (62) Since the solution here given is only approximate it must not be applied to a star too far away from its elongation and as a rough guide to the permissible limit we note that an error of i" in x need not be feared when the product tpx< 10,000 cot <, T being expressed in minutes of time, p in degrees, and x in seconds of arc. Within the limit thus fixed the mean of several consec- utive observations of a star may be reduced as a single observation by substituting in place of T 2 in Eq. 61 the mean of the T 2 s corresponding to the several observations. The following illustrative observations at elongation were made with an engineer's transit, using the method of repetitions ( 53), two pointings in the set, with a reversal of the instrument between them. After reversal the bubble of the azimuth level was brought back to the initial position, as is shown by the level readings, and the instrumental errors may be considered as well eliminated. Wednesday, May 14, 1902. At Station A. Inst. No. 306. Obsr. C. Object. Circle. Point- ing. Watch. Horizontal Circle. Levels. Ver. A. Ver. B. dUrs. Min... Az. mark . . . L. R. I 2 8h 44m 8 49 4 39' 5" 251 24 o 39' 5" 24 o W. E. 9.8 12.2 12.3 10. o + 0.15 90 FIELD ASTRONOMY. The measured angle between star and mark is # = $(251 24' o"- 4 39' 5") = 123 22' 27". 5 , and the further reduction of the observation is as follows : t cotd tan< te a 43 4' 37"- 86 36 44 .8 8.7723 9.9708 5 h 47 m 18" 18 4 i 3 27 36 8 49 7 * + o .8 d+x 86 36' 45". 6 cos (+#) 8.77148 sec (f> 0.13642 AT 184 38 23 . 2 H 123 22 27 . 5 AM 308 o 51 The recorded times of observations are from a watch approximately regulated to Central Standard Time and supposed to be about two minutes slower than local time. Allowing for this error the values of T for the two pointings are 2 m and +3 Respectively, and we have therefore x \ 2 03 = -': 1745 from which to determine oo. On account of the small values of r, tenths of seconds are here retained in the computation in order to illustrate the method, but the result for A M can pre- tend to no such degree of accuracy. A frequent source of substantial error in similar determinations lies in the assumed latitude, and unless this is well known the elongation method should be either avoided altogether or supplemented by observing the elongation of another star on the opposite side of the pole. In the above case an error of i' in the assumed would vitiate AM to the extent of 4". 5, but this error could be in great part eliminated by observing also the western elongation of 51 H. Cephei which occurs PLATE II. An American Theodolite. Diameter of Horizontal Circle 8 inches. Approximate Cost $400. [To face p. 90.] APPROXIMATE DETERMINATIONS. 91 about half an hour after that of d Urs. Min. Adopt the mean of the resulting values of A M . It is always ad- vantageous to make the mean of the pointings coincide as nearly, as may be with the computed time of elonga- tion in order to eliminate any effect arising from error of the time piece employed. 38. Time and Azimuth from Two Stars. The methods of 32 may be refined so as to furnish the required azimuth, latitude or time with any desired degree of accuracy, but such refinement naturally is at the cost of increased complexity and increased labor in their application, e.g., in the observations we may so use the instrument, Circle R and Circle L, as to eliminate its errors of collimation, level, etc. ; we may resort to the method of repetitions, 53, to secure increased accuracy in the horizontal angle between mark and star, etc. Corresponding to the increased precision thus obtained there is required a more rigorous method of reduction for the observed data and the mathematical theory of such a method is as follows : Let M denote a terrestrial mark between which and some known star there has been measured the horizontal angle, H, corresponding to the time, T, as indicated by some watch or other time piece, and let h denote the altitude of this star measured simultaneously with H. See the following example for an illustration of the method of observing : Let us put The Chronometer Correction (Sidereal) M =Ao+y=True azimuth of the mark 92 FIELD ASTRONOMY. where U and AQ are provisional values of AT and AM obtained from the orientation method, 32, or other- wise, and x, y, are the unknown corrections required to transform U and AQ into AT and A M . To determine the values of oc and y we note that the true azimuth of any star observed may be expressed in the form, A=A Q H+y=Ai + y, (63) where AI is employed as an abbreviation for the meas- ured approximate azimuth, A Q H. Through the first of Eq. 12 we may also compute the star's azimuth A fxom the observed time T, and the known h, d and a, thus sin A =cos d sec h sin (T + x) (64) where r = T-f / a is introduced as an abbreviation for the second member of the above equation. Let us further put as abbrevia- tions : sin A'=cos d sec h sin T C'=cos d sec h cos r (65) and expanding sin (T+X) in Eq. 64 find sin A = sin A' cos x+ C' sin x. (66) APPROXIMATE DETERMINATIONS. 93 When % and sin A' are small quantities whose cubes may be neglected, we may put cos # = i, without sensible error, and find by transposition and development of sin A sin A', 2 sin \(A-A') cos \(A+A')=C sin #. (67) Again neglecting terms of the order # 3 this expression becomes A=A'+C'secA'-*, (68) which when subtracted from Eq. 63 furnishes the relation y-C'sec A'-x-A'-Ai. (69) The quantities y and A' AI of this equation should be expressed in seconds of arc, but the chronometer correction, x, will be required in seconds of time, and combining with sec A' the numerical factor 15 required for this purpose, we call the product G and put the coefficient of x in the form C'G. Values of log G are given in the following short table. THE G FACTOR. A' logG A' -1.1761+ i8oio 1 .1762 I 2 .1764 2 3 -1767 3 4 -1.1772+ 4 The symbols + , placed adjacent to log G indicate the essential sign of G, + for a northern, and for a 94 FIELD ASTRONOMY. southern star, and they must be heeded in computing the coefficient of x. Every observed star will furnish an equation of the type (Eq. 69), and two stars will therefore suffice for the determination of both y and x. It is a matter of consequence, however, to choose wisely the particular stars that are to be observed. One of them should always be near the pole and Polaris should usually be chosen, wherever it may be in its diurnal path. The other star should be on the opposite side of the zenith from Polaris, and because of the assumptions made in Eq. 66 it should be observed only when near the merid- ian ; e.g., when # is as great as 3 the star's azimuth should not exceed 3. We now collect and arrange our formulae as follows : r=T+U-a C=G sec h cos d cos T sin A'=sec h cos d sin T y+Cx=A'Ai (2 Equations) 4T=U+x. Example. The application of the method is illus- trated by the following record and reduction of obser- vations made with the instrument shown in Plate I, and an ordinary watch. APPROXIMATE DETERMINATIONS. 95 Tuesday, April 23, 1907. At Station E. Azimuth of Asylum Light. Observer. C. Object. Circle. Watch. Horizontal Circle. Vertical Circle. Levels. Ver. A Ver.B Ver. I. Ver. II. Light R. R. R. L. L. L. h m s 90 2 16.5 5 ii 10 14 14 8.5 9 17 187 54 o 358 14 15 i79 5 40 359 8 30 181 41 20 7 54 3 53 45 14 15 5 3 25 40 45 54 10 Azimuth Level W. E. 25-7 9-3 9-i 25.6 d Crateris. . Polaris. . . . Polaris d Crateris.. Light 3 2 40 45 42 9 30 42 3 15 32 38 45 39 IS 7 o 3 45 39 o + 0.15 Altitude Level N. S. 2.O 8.O 8.0 2.2 O.I REDUCTION. Star. Polaris. 5 Crateris. U + 2h 6m O s at x h 241*1 4 6s nh 14111 428.3 X + 25.1 T 9 7 42 9 8 12 .5 AT 2 6 25.1 T+U ii 13 42 II 14 12 .5 Q 2 4 0.3 T 9 48 56 o o 29 .8 AT + 2 24-8 T (arc) 147 14' o" - o 7' 27" 8 88 48 31 14 16 41 J 4i 179 7 i 359 57 39 A, 187 54' 6" h 42 6 32 39 y + 59 G i . 1762 i . 17617* A* 187 55 5 COST 9-9 2 47w o .0000 sec h o. 1296 0.0747 cos S sin? 8.3179 9-7334 9.9864 7-3359" Equations. C 9.5484** I.2372W y- 0.35*=+ 50" sin A' 8.1809 7-397CW y-i7.27*=-374 A' 179 7' 5i" 359 5i' 25" The instrument was approximately oriented before beginning the observations, a precaution that it is always well to take, since we may then assume as provisional 96 FIELD ASTRONOMY. azimuths, A , A\, the mean readings of the horizontal circle corresponding to mark and star respectively. Sim- ilarly, taking the mean of the observed times for each star and comparing the time for d Crateris with the right ascension of that star,, we find with sufficient approximation, 7 = +2 h 6 m . The instrument was re- levelled when reversed by bringing the. bubble back to the position it occupied at g h s m and the bubble readings in the last column show that in the mean the errors of level are so small that their effect may be ignored. See 42 and 50. The two values of. AT above derived relate respect- ively to sidereal and to mean solar time. A com- parison of the watch with the standard clock of the Washburn Observatory made immediately after this observation gave as its true correction, AT = + 2 m 24 s .6. In the reduction, the coordinates of the stars and the quantity Q have been taken from the almanac. For comparison let the student derive their values from the tables at the end of this book and Eq. 19. The altitudes, h, that appear in these formulae need not be observed with great precision, an error of i' or even 2' having in general no appreciable effect upon the values of x and y. Indeed the observations of alti- tude may be entirely omitted and adequate values of h computed from the orientation tables for Polaris, and from the declinations and hour angles for the southern star. If, however, these altitudes are accurately ob- served their values, corrected for refraction and in- APPROXIMATE DETERMINATIONS. 97 strumental error, may be made to yield an excellent determination of the latitude. We may use for their reduction, in the case of the southern star, the methods f 34, 35 and for Polaris the following equivalent of Eq. 44. 6=h~-p cos (r+x)+C" C" = (4(180 -AS))* sin 2 h. In the last equation the value of AI must be ex- pressed in degrees and the resulting value of C" will then be given in seconds of arc, e.g., if ^1 = 178 48' h = 45, we find = 4X1-2 sin9o = 23. It may be noted that when the chronometer correc- tion, AT, is sufficiently well known we may put x=o, omit the observation of the southern star and find AM from the equation furnished by the Polaris obser- vation alone. Similarly, if the azimuth of the mark is given, we may put y= o, omit the observation of Polaris and determine AT from an observation of a southern star, or, by daylight, from an observation of the sun when near the meridian. Compare with the method of 31. When applied to a star this procedure will often be of advantage in connection with precise deter- minations of azimuth. Errors of Levelling. A major source of error in results obtained as above is found in imperfect levelling of the instrument, and it should be avoided by reading the azimuth level ( 50) at some time during the first half ASTRONOMY. of the set of observations and relevelling the instru- ment after reversal. Bring the .bubble back to the exact place that it occupied in the tube when first read and record the second readings. Whatever error affected the first half of the set will now be balanced by an equal and opposite error in the second half. It may happen that the recorded level readings, before and after reversal, do not agree, and in this case the angle, b', that the vertical axis makes with the plane of the meridian may be found from the recorded level readings ( 42, 43) and a corresponding correction applied to the time and azimuth found through the vitiated values of x and y. From the spherical triangle formed by the pole, the zenith and the point of the sky towards which the vertical axis is directed we find, Ax= +b' sec < Ay= + 6' tan <, which are the required corrections for level error. Count & as positive when, on the whole, the east end of the azimuth level is higher than the west end. As- suming the value of a level division to be 15" (d=f. 5) we find from the observations on p. 95, b'= +i", Ax= + 1 ".3 = +o s .i, Jy= + 1", which are quantities quite inappreciable for the present purpose. STUDENTS' OBSERVATORY INSTRUMENTS. IN the several determinations thus far considered we have for the most part assumed that the data fur- nished by the instruments employed were free from purely instrumental errors, and in approximate work this may usually be done if due care has been bestowed upon the adjustments. But where a higher degree of precision is required it becomes necessary tb study the instrument employed, as being in itself a source of errors that need to be eliminated, and we must turn therefore to a more detailed consideration of some of the instru- ments used in field astronomy before taking up the class of methods called accurate. 42. The Spirit-level. The spirit-level is used in astronomical practice to measure small deviations of a line or surface from a vertical or horizontal position, and incidentally to adjust a part of an instrument to such a position. It consists essentially of a glass tube bent or ground into an arc of a circle of large radius and so mounted that the plane of this circle is approximately vertical. The tube being nearly filled with ether and its ends hermetically sealed, the small -volume of air or 99 100 FIELD ASTRONOMY. vapor that remains in the tube is collected into a bubble which always stands at the highest point of the circle, so that a line drawn from its middle point through the cen- tre of curvature of the tube is vertical. The upper surface of the tube is usually provided with a scale of equal parts, and the position of the bubble in the tube is determined by the readings of its ends upon this scale. The angle subtended at the centre of curvature of the tube by the space between two consecutive lines is called the value of a division of the level, and this value, which will be represented by 2d, is required for transforming the indications of the level into seconds of arc. Note that d represents one half the value of a level division. Let such a level be supposed attached to a theodolite, the inclination of whose vertical axis to the true vertical is to be determined from readings of the bubble. We are here concerned with angular measurements, e.g., the angle that the axis makes with the true vertical; the angle moved over by the level bubble, as seen from the centre of curvature of the tube, when the instrument is turned from one position to another; etc., and as the simplest method of dealing with these angles we shall imagine the whole apparatus projected radially upon the celestial sphere, so that the arc joining the points in which any two projected lines meet the sphere, meas- ures the angle between these lines. This method of analysis by projecting the parts of an instrument upon the sphere is in common use, and the student should acquire a clear conception of the simple case to which it is here first applied. INSTRUMENTS. 101 To determine the relation of the bubble readings to the required inclination, we imagine the axis of the in- strument and the plane of the level extended until they meet the celestial sphere, as in Fig. 7, which represents L FlG. 7. Theory of the Spirit-level. a small part of the sphere adjacent to the zenith, Z. In this figure V is the point in which the produced axis meets the sphere, and LB is the trace of the plane of the level tube upon the sphere. The projection of; thfe, n^i^ die of the bubble upon the sphere must be at^ the, point, in LB nearest to Z, and found, therefore, by ^ letting iall a perpendicular from Z upon LB. If, now, the theod- olite be turned 180 in azimuth, i.e., rotated about V as a pivot, the level tube will be revolved about V as a centre, into the position L'B' ', and the point B will fall at B', but the middle of the bubble will stand at B" instead of B', since this is now the point nearest to the zenith. From elementary geometrical considerations, VM = %B'B", where B'B" is the space moved over by the level bubble when the instrument is turned from 102 FIELD ASTRONOMY. one position to the other, and VM is the projection upon the plane of the level tube of the arc, VZ, that measures the angle between the axis of rotation and the true vertical. Calling this projection b and repre- senting by a', b', a" , b" the scale readings of the ends of the bubble in the two positions, we have ( , It is customary to record the several readings in the form, (Symbols used above.) (Actual observations.) N S N S a 1 b' l6 -4 39-6 a" b" 32.2 9.1 7-35 The letters N and S denote the north and south ends of the level tube, or some equivalent system of distinguish- ing between them. 43. Discussion of the Level Readings. The student Should no\Y note that : ; ^K^f 1 ^ , coefficient of d in Equation 79 is the mean of the diagonal differences in the square array formed by the four numbers tabulated in the preceding example. This example represents the manner in which level read- ings should be recorded, and the mean of the diagonal differences, 7.35, written below the line, should be worked out and entered with the record. (b) If the bubble readings have been correctly taken and there is no change in the length of the bubble during the observation, these differences must be equal, one INSTRUMENTS. 103 to the other, thus furnishing a check upon the accuracy of the level readings, which should always be applied immediately after recording them. If the temperature is changing rapidly, the length of the bubble may be changed and the check impaired without necessarily diminishing the accuracy with which b is determined. (c) If the greatest of the four numbers stands in the column marked N, the north end of the level tube is on the whole higher than the south end, and the vertical axis is tipped toward the south. Determine the sign of b in this manner. (d) The zero of a level graduation is sometimes placed at one end of the scale and sometimes in the middle, but the method of record and reduction given above applies to both cases. (e) It is apparent from the figure that the point of the level tube midway between B f and B" marks that radius of the level tube which is most nearly parallel with the rotation axis of the instrument. Since this radius ought to pass through the middle point of the scale, and does so pass when the level is in adjustment, we have as the error of adjustment of a level numbered continuously from one end to the other, (80) where s represents the total number of divisions in the level scale. In the example given above 5 = 50 and e =0.7 division. The essential element in the determination of b is 104 FIELD ASTRONOMY. the reversal of the level with the resulting displace- ment of the bubble, and it is a matter of indifference whether this displacement is produced by revolving the level about a vertical axis to which it is attached, as in the case considered above, or by picking the level up bodily from a plane or line upon which it stands, turning it end for end and replacing it in the reversed position, as is done in measuring the inclination of an approximately horizontal axis. Let the student show that the inclination of this axis to the plane of the horizon may be obtained from the bubble readings exactly as the inclination of the vertical axis was deter- mined above. The greatest of the four readings is adjacent to the high end of the axis. Determine in this way the inclination of the horizontal axis of a theodolite. A fine level is an exceedingly sensitive instrument and requires great care in its use. Unless unusually well supported its readings may be vitiated by the observer passing from one side of it to the other, or even by shift- ing his weight from one foot to the other. Therefore observe the following precepts: 1 . Keep away from the level as much as possible. 2. Don't allow the sun to shine upon it. 3. Don't hold a source of heat, e.g., a lamp or your own hand, near a level longer than is strictly necessary. 4. If the level has a chamber with reserve supply of air at one end of the tube, use it to regulate the length of the bubble, keeping this always about one half as long as the scale. 5. Make the inclinations that are to be measured INSTRUMENTS. 105 as small as possible, in order to avoid any considerable run of the bubble and the resulting effect of possible irregularities in the level tube. 44. Value of a Level Division. The value of a level division is most conveniently determined by measuring with a micrometer, or finely graduated circle, the vertical angle through which its tube must be tipped in order to cause the bubble to run past a given number of divi- sions of the scale. If the necessary apparatus for such a determination is not at hand, the following method will furnish equally good results and requires only an engineer's transit, to which the level must be attached with its plane approximately vertical. Let the instrument be firmly set up but very much out of level, e.g., with its vertical axis making an angle, f, with the true vertical amounting to 2, more or less. See p. 109 for a method of determining the exact value of this angle, which will be required in the reduction of the observations. If the transit is now turned slowly about its vertical axis (azimuth motion), the level -bubble will run back and forth in its tube, and two positions of the instrument may be found at which the bubble will come to the middle of its scale. We shall designate the readings of the azimuth circle corresponding to these two positions by A l and A 2 . Any slight turning of the instrument either way from A j^ or A 2 will cause a corresponding slight motion of the bubble, and to determine the relation of the bubble readings to the corresponding circle readings we resort to Fig. 8, which represents a portion of the celestial 106 FIELD ASTRONOMY. sphere adjacent to the zenith, Z. V is the point in which the deflected axis of the instrument meets the sphere, and SV is the trace upon the sphere of the plane of the level-tube, which is assumed to have been adjusted approximately parallel to the vertical axis of the transit. Small errors in this adjustment are of no consequence. FIG. 8. Determination of d. As the instrument is turned in azimuth, carrying the level-tube with it, the arc SV must revolve about V as a pivot, and the amount of its rotation will be measured by the successive readings of the azimuth circle. It may be seen readily that the angle T of the figure corre- sponding to any particular circle reading, A, is given by the equation A,)-^, (82) t -f A 2 ) being the circle reading at which SV coincides with VZ. INSTRUMENTS. 107 Since a level-bubble always stands at the highest point of its tube, the point nearest the zenith, we may find the point in the figure corresponding to the middle of the level -bubble by drawing from Z an arc of a great circle perpendicular to 5 V, and the intersection, 5, will be the required point. In the right-angled spherical triangle SVZ thus formed we have the relation, tan p = tan ? cos T, (83) in which p measures the distance of the middle of the bubble from the fixed point V. To find the effect upon p of any small variation in r, i.e., to find how far the bubble will run when the instrument is turned slightly in azimuth, we differentiate this equation and obtain dp= tan f cos 2 p sin r dr, (84) and substituting in place of these differentials, small finite increments of the respective quantities, we obtain 2 d(b'-b") =tan r cos 2 p sin r (A' -A"), (85) where d represents the value of half a level division and b' and b" are the scale readings of the middle of the bubble, corresponding to the circle readings A' and A" . Equation 85 may be used to determine the value of d, but whenever ordinal y care is bestowed upon the ad- justment of the level, i.e., to make the radius passing through the middle point of the scale parallel to the ver- tical axis of the theodolite (Equation 80), the readings Aj and A 2 will be so nearly 180 apart that we may put 7 = 90, cos = i, for all positions of the bubble within 108 FIELD ASTRONUMY. the limits of its scale, and thus obtain in place of Equa- tion 85 the simpler relation ~ In this equation A' A" and b' b" are to be derived from the readings of the horizontal circle and level, respectively, and in making observations for their deter- mination it is well to bring the bubble as near as may be to one end of the tube and set the circle to read the nearest integral 10'. Then turn the instrument to each successive 10' or 20' reading, and record the readings of the bubble until the former has traversed the entire length of its scale, after which repeat the operation in the inverse order, using the same circle settings as before. With reference to the direction of the bubble's motion these two series will be designated as Forward and Backward. Having completed these observations, turn the instrument to the second position in which the bubble plays, e.g., from A^ to A 2 , and make a similar double set of readings. The readings obtained at any two settings of the instrument will determine values of A' A" and b' b" , and therefore a value of d, but it is advisable to secure a considerable number of these determinations, ranging over the whole length of the level -tube, in order to test its uniformity. Supposing such a series to have been made, the manner of forming the differences b' b" illustrated below, may be followed with advantage, i.e., subtract the first b from the first one following the INSTRUMENTS. 109 middle of the set, the second b from the second one after the middle, etc. The angle 7- of Equation 86 should be determined at the time of deflecting the axis, as follows : After having carefully levelled the instrument, take a reading of the vertical circle when the line of sight is directed toward a fixed mark, that we may call P. By means of the levelling screws deflect the axis exactly toward or from P through some convenient angle, e.g., i if the vertical circle reads to seconds, 3 if it reads only to minutes, and again point upon P and read the circle. The difference of the two readings is the value of ?. To make sure that the deflection of the axis is made in the proper direction, by means of the levelling screws make the reading of the azimuth level (see 50) the same after deflection that it was before deflection, and there will then be no component of deflection perpendicular to the direction P. 45. Example. We have the following example of the record and reduction of the first half of a complete set of observations for the determination of d. In the reduction we note that the divisor 2(6' b") of Equation 86 is equivalent to 26' zb" , and since b is the scale read- ing of the middle of the bubble, 26 is equal to the sum of the readings of the ends of the bubble. The column headed 26 is found in this way from the mean of the two sets of bubble readings opposite each circle reading. The regular progression of the numbers in the column 2(6' 6") suggests a level-tube of variable curvature, but the amount of data is not sufficient to decide this with certainty. More observations are needed. 110 FIELD ASTRONOMY. Friday, Dec. 7, 1894. Alidade Level of Universal Instrument, No. 2598. Readings to Mark. Mic. I. Mic. II. Mean. Axis Vertical. . 180 26' 45" 26' 45 i8o26'45' Axis Deflected 179 27 o 27 6 179 27 3 Azimuth Circle. 291 o'o" IO O 20 O 30 o 40 o 50 o 2 b 2 (b'-b") 26.45 30.0 38.5 42.4 46.75 12 .05 12 .40 I2 -75 12 .40 y = o 59 42 Bubble. Forward. Backward. 26.3 0.2 26.0 0.4 28.1 2.3 27.7 1.9 30.2 4.2 29.7 3.9 32.3 6.4 32.1 6.2 34.0 8.1 34.2 8.5 36. 2 10. 2 36 . 5 10.6 A' -A" log tan f log(A'-A") colog 2 (6'-6") 30' = 1800" 8. 239 3- 255 8. 907 log d d o. 401 2. 52 46. Inequality of Pivots. When a spirit-level is used to determine the inclination of a line, such as the hori- zontal axis of a transit, its readings and the resulting inclination will be vitiated by any inequality which may exist in the diameter of the pivots upon which it rests. To test for such an inequality let the instrument be firmly mounted and the inclination, 6', be measured with the level as shown in 42 ; then lift the axis out of the wyes, turn it end for end, and replace it so that what was the east pivot shall now rest in the west wye. Again measure the inclination, b" , and repeat the level- ings and reversals several times, so that any systematic difference which may exist between b f and b" shall be well determined. We now put V>\ (87) INSTRUMENTS. Ill where i is the correction for inequality of pivots, and find for the true inclination of the axis in the two posi- tions, b, = V-i, b 2 = b" + i. (88) The correction i should be carefully determined and applied to all measured values of the inclination. 47. The Theodolite. This instrument, which is also called engineer's transit, altazimuth, universal instru- ment, etc., is one with whose general appearance and construction the student is supposed sufficiently familiar to recognize its close relationship with the coordinates of System I, altitude and azimuth. Trace out this rela- tionship in Plates I, II, and III, which represent different types of this instrument. The line of sight (telescope) is a radius vector of undetermined length; the hori- zontal and vertical circles measure azimuths and alti- tudes, or zenith distances, and in an ideally perfect instrument the readings of these circles should be the true azimuth and altitude of the line of sight, or at most should differ from these only by a constant index correction. It may readily be seen that among the conditions which must be satisfied in the construction of such an instrument are the following: (1) The axes must be perpendicular to each other. (2) The line of sight must be perpendicular to the horizontal axis. (3) The vertical axis must be truly vertical. Owing to unavoidable imperfections of mechanical 112 FIELD ASTRONOMY. work it is not probable that any one of these conditions is exactly fulfilled in any given instrument, and they are therefore to be regarded as so many sources of error, whose effects may be made small by careful adjustment, but whose complete elimination must be sought in some other way; e.g., if the vertical axis is not truly vertical, we may determine as follows the means for correcting the effect of this error upon the measurement of altitudes. 48. Zenith Distances. In Fig. 9 let HZ be the direc- FIG. 9. Measurement of Zenith Distances. tion of the vertical; 5, a point whose zenith distance, ZHS, is to be determined; HV, the projection of the vertical axis of the instrument upon the plane HZS\ INSTRUMENTS. 113 and let r' denote the reading of the vertical circle when the line of sight is directed toward S. After reading r' let the instrument be turned about the vertical axis through an angle of 180, bringing the line of sight into the position HS', and then let the telescope be turned about the horizontal axis until the line of sight again points at the object 5, and let r" be the reading of the vertical circle in this position. If the circle is numbered in quad- rants, as is very common in small instruments, r' and r" will be approximately the same number but with a graduation extending from o to 360, as is here sup- posed, they will be widely different. From the figure, the angle SHS' is measured by the difference of these circle readings, r' r" , and since VHS = VHS', we have for the angular distance of the point 5 from V the equation, r"). (89) When two pointings of the telescope are made as above, the instrument is said to be reversed between them, and it is customary to designate its two positions as Circle Right and Circle Left, respectively, the reference being to the vertical circle of the instrument, which faces to the observer's right in the one position and to his left in the other. The student should note that the angle z" is here determined quite independently of the adjustment of the verniers, which may be intended to read altitudes, zenith distances, or anything else, since the reversal eliminates all question of adjustment from the difference r' r" , and is made for this purpose. 114 FIELD ASTRONOMY. The true zenith distance of 5 is, however, not z" but the angle, z"+ b", and b" may be determined, as in 42, from readings of the spirit-level, LL, attached to the instrument in such a way that its plane is parallel to the line of sight, HS. Such a level, i.e., one whose tube is perpendicular to the horizontal axis of the instrument, will be called the altitude level of the instrument. A convenient method of taking into account the readings of the level-bubble by applying them directly to the circle readings instead of to the measured angles, is as follows: Let w represent the reading of that point of the level scale through which passes that radius of the level which is parallel to the vertical axis, HV, Cn Q in the figure, and let n denote the position of the middle of the bubble corresponding to the circle reading /; i.e., since the bubble always stands at the highest point of its tube, n is the point exactly above the centre of curvature, C. It is evident from the figure that b" = (n-n Q )2d = (a + b- 2 w )d, (90) where d is the value of half a level division, and a and b are the actual scale readings of the ends of the bubble. If the instrument had been, from the first, perfectly levelled we should not have obtained r r as the reading to the point 5, but in place of r' a number either greater or less than it by the amount b" ; and if, therefore, we apply to r' and r" level corrections determined by the equation above given for 6", we shall reduce the read- PLATE III. A German Universal Instrument. Length of Horizontal Axis 12 inches. Approximate Cost $400. \To face p. 114.] INSTRUMENTS. 115 ings to what they would have been for a perfectly levelled instrument, and therefore obtain the zenith distance of 5 immediately from the half difference of the corrected readings. Since any constant term which appears in the level correction will be eliminated from this difference of the corrected readings, / r", we may substitute in Equation 90, in place of 2W , any constant number what- ever, e.g., zero, but it is usually convenient to take as this number 5, the total number of divisions included in the level scale, since in the long run this will make the level corrections small. Making this substitution, we have finally, Level Correction = (a + b S)d, (91) where the ambiguous sign depends upon the direction in which the numbers increase along the level scale, and may be determined, once for all, for a given instru- ment as follows: Two readings of the vertical circle of a certain instrument were taken to the same object, but with the instrument thrown out of level in such a way that the bubble stood at quite different parts of the scale in the two observations ; e.g. : Observation. Bubble. Circle. Level Corr. Corrected r. a b First 2.0 25.8 91 9' 8' + i8"-7 91 9' 26". 7 Second 7.9 31.9 91 9 40 12.5 91 9 27. 5 The numerical values of the quantities above marked Level Corr. were computed from Equation (91) with an assumed value of d = 2 ff .6, and since the effect of these corrections must be to bring the corrected circle readings into agreement, it is evident that the + sign must be used for the first observation and the sign for the second. The whole number of divisions in the level scale being 35, the formula for this instrument becomes, 6"=+2".6 [35- (a + 6)]. 116 FIELD ASTRONOMY. A similar formula may be obtained for every instrument, and a table should be constructed from it, which with the argument a+b will show the value of b" for any given position of the bubble. Part of such a table is given below, and from it the level correction correspond- ing to any ordinary position of the bubble may be determined by inspection. a + b b" a + b 30 +i3"-o- 40 31 +10 .4- 39 32 +7-8- 38 33 +5 .2- 37 34 + 2 .6 36 35 +o .o- 35 In the second observation given above we have a + b = 7. 9 + 31.9=39.8, and corresponding to this number we find, by interpolation from the table, b" = -12". 5. The level formulae thus derived show that if the bubble be brought to the same place in the tube, same values of a and b, both Circle R. and Circle L.,the level correction will be eliminated from the difference r' r", and may therefore be neglected. To obtain the maxi- mum precision however, the level should always be read and a correction applied to each circle reading, but even when this is done it is good practice to touch up the levelling screws after each reversal and bring the bubble back as near as may be to its first position, without, how- ever, spending too much time in obtaining an accurate agreement. In some instruments the level and verniers are at- tached to a frame (alidade) which admits of rotation about the horizontal axis without disturbing the direc- tion of the line of sight. For such an instrument it may be shown that if, before reading the vernier, the frame be turned until the bubble stands at the middle of *he scale, the resulting vernier readings are equiva- INSTRUMENTS. 117 lent to the corrected circle readings derived above, and therefore require no further correction for level error. This mechanical device, although convenient for some purposes, is of inferior accuracy. 49. Effect of Errors of Adjustment. A geometrical investigation similar to the above may be made to show the effect of each source of instrumental error, but we shall find it more convenient to develop the combined effect of these errors through an analysis based upon Fig. 10, which represents a part of the celestial sphere, FIG. io. Theory of the Theodolite. where Z is the zenith, V is the point in which the vertical axis of the instrument, when produced, cuts the sphere,, H is the point of the sphere determined by the prolonga- tion of the horizontal axis, and 5 is a star or other object whose azimuth and altitude are to be determined from readings of the horizontal and vertical circles of the instrument. The angles measured by means of these 118 FIELD ASTRONOMY. circles lie in the planes of the circles, but just as the azimuth of a point is measured either by an arc of the horizon or by the corresponding spherical angle at the zenith, so the data furnished by the vernier readings may be regarded as spherical angles having their ver- tices respectively at V and H. Thus if r represent the reading of the vertical circle when the line of sight is directed toward 5, and r is the reading when this line is directed toward some point in the arc HV, the differ- ence, r r , measures the spherical angle VHS. Simi- larly, for the horizontal circle, by rotating the instrument about its vertical axis, H may be moved from its present position, corresponding to the reading R, into a new position falling upon the arc VM, and if R t be the circle reading in this position, we shall find that R R l equals the spherical angle HVM. From these spherical angles, determined by the circle readings, it is required to find the true direction, MZS = A f , and the true zenith dis- tance, ZS = 2, of the star 5. It is evident from the figure that the arc VH = 90 i, measures the angle between the vertical and the horizontal axis of the instrument, and that i is therefore the error of adjustment of the axes, corresponding to Condition i, 47. Similarly, HS = 90 + c measures the angle be- tween the horizontal axis and the line of sight, and c is the error in the adjustment corresponding to Condition 2. Also, VZ = ? is the error of level of the instrument, i.e., deviation of the vertical axis from the true vertical, corresponding to Condition 3. The arc HZ = go b measures the angle that the horizontal axis makes with INSTRUMENTS. 119 the true vertical, and b is therefore the level error of this axis. Note that as the instrument is turned into differ- ent positions by rotation about the axes V and H, the quantities 7-, i, and c remain unchanged and are there- fore called instrumental constants, since they define the condition of the instrument with respect to its several adjustments. The level error, b, is sometimes included among these constants, but is not strictly one of them, since its value changes as the instrument is turned in azimuth. We shall suppose the instrument to be so well ad- justed that none of the instrumental constants exceeds 2', and H will then be so near the pole of the great cir- cle VZM that we may assume without sensible error HVM=HZM and, replacing these quantities by their equivalents, obtain or A^R-(R l + go)-w. (92) The azimuth of 5, reckoned from the true meridian instead of from the arc VM, differs from A' only by the substitution of another constant, the index correc- tion of the horizontal circle, in place of 7? t + 90 ; and as this index correction must in any case be separately determined (see 38), we may replace the constant term 7^ + 90 by R , the index correction referred to the true meridian, and we shall then have for the true azimuth of 5, A-R-R.-W. (93) 120 FIELD ASTRONOMY. The auxiliary quantity w has thus far been defined only by means of Fig. 10, where the spherical angle HZS is labelled 90+ w. To determine the value of w in terms of the instrumental constants, we have from the triangle HZS by means of Equations 4, the relation, sin c = sin b cos z cos b sin z sin w, which, since b and c are small quantities, is equivalent to c b sin z tan z 9 or, replacing the zenith distance, z, by the star's alti- tude, h, w = c sec h + b tan h. (94*) Since neither i nor 7- enters into this equation, the effect of these errors must be taken into account through 6, the inclination of the horizontal axis. This is to be deter- mined with a spirit-level, and each circle reading, R t must be corrected for the particular inclination of the axis that corresponds to R. The factor tan h becomes zero for an object in the horizon, and for this special case the effect upon the azimuth readings of an error of level is zero. On the other hand, when the object to be observed is at a considerable elevation, e.g., the Pole Star in an azimuth determination, the factor, tan h, becomes large and the effect of level error is magnified. It is in fact one of the chief sources of error in such deter- minations. 50. Determination of Errors of Adjustment. The error above represented by c is called the collimation, and INSTRUMENTS. 121 its effect is usually to be eliminated through a reversal of the instrument. Since the angular distance of 5 from one end of the horizontal axis is 90 + c, its distance from the other end must be 90 c, and as in the rever- sal these ends change places the effect of c must have one sign Circle R., and the opposite sign Circle L., and will therefore be eliminated from the mean of observa- tions taken in both positions. In precisely the same way it may be shown that the effect of i, error of adjustment of the axes, is eliminated from the mean of observations taken in the two posi- tions, and wherever any considerable precision is required in azimuth observations or in the measurement of hori- zontal angles, the observer should not fail to make an equal number of pointings in each position of the instru- ment to secure this elimination of errors. In the triangle HVZ the angle HVZ=HZM is very nearly equal to 90 + ^4', and assuming this equality we find from this triangle, sin b = sin i cos ? cos i sin 7- sin A', (95) which is equivalent to, b=i-rsinA'. (96) The quantity 7- sin A', which we shall represent here- after by the symbol &', and which corresponds to the arc ZI of Fig. 10, is that component of the level error of the vertical axis, 7-, which lies at right angles to the line of sight and which may therefore be determined from the readings of a level parallel to the horizontal 122 FIELD ASTRONOMY. axis of the instrument. Such a level is called the azi- muth level, and if resting upon the axis and capable of reversal (striding-level) , it is most conveniently used to determine the level error of this axis, b. If fastened to the frame of the instrument and incapable of reversal, it may be used to determine, from bubble readings taken Circle R. and Circle L., the value of b f for the vertical axis, and corresponding to these two cases we shall have the following expressions of the level corrections to be applied to readings of the horizontal circle : Striding-level, 6 tan h. Both Circle R. and Circle L. Fixed Level, b' tan h. Mean of Circle R. and L. If, as is usual, the graduation of the circle increases from left to right, b and b' are to be considered essentially positive when the high end of the horizontal axis has an azimuth 90 greater than the object 5. The student should not fail to note in connection with the use of a fixed azimuth level that if the bubble is brought to the same scale reading, Circle R. and Circle L., b' will be zero and the level error will be eliminated from the mean result. A reversal furnishes a convenient method for deter- mining or adjusting the collimation. For this purpose let R f and R" be readings of the horizontal circle corre- sponding to observations of a fixed mark in or very near the horizon, made in the two positions of the instrument ; then, from Equations 92 and 94, 2C = R'-R". (97) INSTRUMENTS. 123 To determine the error of adjustment of the axes, i, let the inclinations of the horizontal axis, b v b 2 , be meas- ured in two positions of the instrument differing 180 in azimuth, i.e., when Vernier A reads o and when it reads 180. We shall then have, from Equation 96, b i =i j- sin A', b 2 =i- r sin (A' + 180) =i+ r sin A', from which we obtain immediately 2 i = bi + b 2 . (98) If we call the inclinations b v b 2 positive when the circle end of the axis is too high, a positive value of i will indi- cate that the same end is too high, i.e., it makes too small an angle with the upward extension of the vertical axis. The value of 7-, which will seldom be required, may be found from four values of b determined at intervals of 90 in azimuth. 51. Additional Theorems. By an analysis similar to that employed above, it may be shown from the trian- gles HSZ, HZV, of Fig. 10, that the errors b, c, and i have no appreciable influence upon observations of altitude or zenith distance. Indeed, it may be seen without formal analysis that when c, b, and i are small quantities, H is so nearly the pole of the circles ZS, VS, that these arcs are measured by the corresponding angles at H, i.e., by the readings of the vertical circle unconnected for instrumental error. Since the error corresponding to 7- is taken into account in the approximate analysis FIELD ASTRONOMY. of 48, we may adopt as definitive the results there obtained. The correction b" there determined is the arc VI of Fig. 10, i.e., it is the projection of 7- upon the line of sight, VS. The demonstration of the following theorems, which are of some consequence in the use of a theodolite, is left to the student. i. If, as is quite common in engineer's transits, the vertical circle is graduated into quadrants instead of from o to 360, observations of altitude should be made in the way already indicated, but in their reduction we shall have, in place of the formula for 2", the substi- tute, , (99) i.e., the mean of the readings gives directly the instru- mental altitude. 2. The altitude level of such an instrument usually has the zero of its scale placed at the middle of the tube, and when such is the case readings of that end of the bubble nearest the objective end of the telescope should be marked o, and those of the end nearest the eyepiece should be called e\ the formula for level correction then becomes, b" = (o-e)d. (100) 3. A theodolite maybe reversed by lifting the tele- scope from its supports, turning the axis end for end, and replacing it in the wyes in the changed position. This mode of reversal eliminates errors of level and colli- mation quite as well as does the one above described, INSTRUMENTS. 125 and also eliminates the inequality of pivots from the determination of b. It is therefore to be preferrecl when it can be conveniently practised. 52. Errors Arising from the Circle Readings. Numerous errors of a class not considered above, creep into the results of observation through the circle readings, which may be vitiated in greater or less degree by: (a) Defective graduation of the circle itself. (b) The plane of the circle not being normal to the rotation axis. (c) The circle not being truly centred upon the axis. (d) The spaces on the vernier being too large or too small relative to those on the circle. (e) Error of focussing (runs) in the reading micro- scopes, etc. etc. etc. The detailed study of these sources of error lies be- yond the scope of the present work, but we note that in great part their effects may be eliminated by taking the mean of a considerable number of observations in which the circle readings are symmetrically distributed through- out the whole 360 of the graduation. Thus if an angle of 120 between objects A and B is measured three times and the circle turned 120 after each measurement so as to obtain the following system of readings : To A. To B. P-A. Observation i o o' o" 120 o' o" 120 o' o" 2 I2O O O 240 O O I2O O O " 3 240 o o 360 o o 120 o o 126 FIELD ASTRONOMY. whatever graduation errors may affect the particular reading 120 o' oo" will be eliminated from the mean value of B A, since this reading enters into that mean once with a plus sign and once with a minus sign. If the required angle is small, e.g., i, it will not be con- venient to carry out the above programme of reading around the entire circle, but the elimination of errors may still be made by shifting the circle so that the readings to object A may be symmetrically distributed through the entire circumference, e.g., every 60 or every 30. For an instrument provided with two ver- niers or microscopes it will suffice to distribute the read- ings of each vernier over an arc of 180. 53. The Method of Repetitions. A peculiar method of measuring horizontal angles may be adopted with advantage if, as is often the case, the instrument is provided with two motions in azimuth called, respectively, upper and lower, one of which produces a change in the vernier readings, while in the other, verniers and circle remain firmly clamped together and turn simultaneously, without change in the circle reading. Reverting to 52, we may note that the circle readings 120, 240, there recorded, are quite unnecessary since, if the first reading, o, be subtracted from the last one, 360, and the result divided by 3, we shall have as the value of the angle i2oo'o", which is precisely the same as the mean of the three values of B A, and is all that that mean can furnish. This process is called the method of repetitions and consists, essentially, in making a series of pointings upon INSTRUMENTS. 127 two objects between which an angle is to be measured, turning always from A to B upon the upper motion of the instrument and from B to A upon the lower motion, so that the vernier reading in the latter turning is not changed. A series of such pointings is called a set and the verniers need be read only for the first and last pointings of the set. If the initial and final readings be represented by R' and R" ', and n be the number of point- ings to each object contained in the set, we shall have, as shown above, T)f D// Angle = - . (101) It is often advantageous to reverse the instrument at the middle of a set, turning on the lower motion, and thus secure an additional elimination of instrumental errors. The advantages of the method of repetitions are a saving of labor through the diminished number of vernier readings and, where the verniers are comparatively coarse, an increase of accuracy through the introduction of the divisor n into the value of the angle. The pre- cision of a small instrument, such as an engineer's transit, may be considerably increased in this way, but for the larger instruments, provided with micrometer micro- scopes, experience shows that the best results are to be obtained by reading the microscopes after every pointing. Where a horizontal angle between objects at very- different altitudes is to be measured by the method of repetitions, as in an azimuth determination, an addi- tional source of error requires careful attention, viz., the effect of a lack of parallelism between the axes corre- 128 FIELD ASTRONOMY. spending to the upper and lower motions of the instru- ment. To eliminate this error we proceed in the follow- ing manner: The axis of the lower motion should be made as nearly vertical as possible, and whatever may be the error of the upper axis it will produce no effect upon the final result if the number of repetitions is so chosen that the set extends through 360; for in the successive turnings about the lower motion the upper axis has been made to describe a complete cone about the lower axis, and any error which may have been caused by a deflection to the east in one part of the set is bal- anced by the opposite error, caused by a deflection to the west, in another part, etc. If the angle to be meas- ured is so small that the set cannot be made to extend through 360, the following observing programme will also eliminate the error of the axis : Measure a set of any desired number of repetitions. When it is completed leave the instrument clamped at the last vernier read- ing, reverse about the lower motion and repeat the set in the opposite direction, i.e., beginning with the object last sighted upon and with approximately the vernier reading last obtained. The level correction to the circle readings should be derived in the ordinary way, 50, from readings of a level taken when the instrument is reversed about the lower axis. 54. Precepts for the Use of a Theodolite. The ex- perience of the principal geodetic surveys indicates that the following precepts should be observed in all precise work with a theodolite : INSTRUMENTS. 129 (1) An equal number of measurements should be made in each position of the instrument, Circle R. and Circle L. (2) An equal number should be taken in each direc- tion, i.e., the line of sight turned from right to left and from left to right. (3) The position of the circle should be so shifted from time to time that the readings to each object are sym- metrically distributed throughout the 360. (4) The observations should be made as rapidly as the observer can work without undue haste. 55. The Sextant. A sextant consists essentially of two mirrors and a graduated arc of a circle, about 60, for measuring the angle between the planes of the mir- rors. The peculiar value of the instrument lies in the fact that it is light and portable, requires no fixed support, and may therefore be used for the measurement of angles at sea as well as on shore, and in any plane, ver- tical, horizontal, or inclined. For the purpose of de- scription and analysis we suppose the sextant to be placed upon a table, with the plane of its arc horizontal, and we shall use the terms altitude, azimuth, etc., with reference to this special position of the instrument. The conclusions drawn from this consideration of the in- strument apply equally when it is used in any other plane. The essential parts of a sextant are indicated in Fig. ii which should be compared with Plate IV. At the centre of the arc is a vertical axis carrying a vernier-arm, V, and also supporting one of the mirrors called the index-glass, I, whose plane is vertical, passes 130 FIELD ASTRONOMY. nearly through the axis and rotates with the vernier- arm as the latter is turned in azimuth. At one side of the sextant frame is the other mirror, H, called the hori- zon-glass, with its plane vertical and fixed parallel to that radius of the graduated arc which is numbered o. Only the lower half of the horizon-glass is silvered, the upper half is left transparent. A telescope, T, is mounted on the side of the frame opposite to the horizon-glass and has its line of sight directed toward the latter. From Fig. n it may be seen that an observer looking FIG. ii. Elements of a Sextant. into the telescope and through the unsilvered upper half of the horizon -glass will see that part of the horizon toward which the telescope is directed, and will also see superposed upon it a view of another part of the horizon reflected from the index -glass to the silvered half of the horizon -glass, and from this again reflected into the tele- scope. This part of the horizon is said to be seen | INSTRUMENTS. 131 reflected, while the part seen through the horizon-glass is observed direct. Any reflected image which is super- posed upon a direct image is said to be in contact with the latter, and we shall represent these images as seen in the telescope, by / and H respectively. By turning the index -glass in azimuth, different parts of. the horizon may be reflected into the telescope, and since the rays of light incident upon and reflected from the mirror make equal angles with its surface, it is apparent that for every i that the mirror is turned, the azimuth of the point reflected into the telescope will be changed by 2. There may be found by trial a set- ting of the index -glass at which both a direct and a re- flected image of the same object may be seen simul- taneously and may be made to pass one over the other as the vernier-arm is slightly turned. Let R denote the vernier reading when these images are brought into contact, and let R be the reading at which any other object, /, is brought into contact with the H just ob- served; then it appears from the above that the differ- ence of azimuth between / and H is twice the angle included between R Q and R. On account of this mul- tiplier, 2, each half -degree of the sextant arc is numbered as if it were a whole degree, and we have, therefore, for the difference of azimuth, PI-I = R-R Q . ( I02 ) The R which appears in this equation is called the index correction, and it should be observed that, owing to the angle subtended at the object H by the space 132 FIELD ASTRONOMY. separating the index and horizon glasses, the reading R Q will depend upon the distance of H from the instrument. If it is near at hand, less than three miles, R Q should be determined as above and the axis of rotation of the index -glass should be centred over the point at which it is desired the vertex of the measured angle should fall. If the objects are very remote, all question of the exact position of the vertex is eliminated, and a mode of deter- mining the index correction given hereafter will be found more convenient than the above. It may now be seen that the following conditions must be satisfied in order that the Equation 102, given above, shall furnish the true value of the angle between H and /: (a) The rotation axis and the plane of the index -glass must be perpendicular to the plane of the graduated arc. If they are not perpendicular, this arc cannot accurately measure the amount of rotation of the mirror. (b) The horizon-glass must be perpendicular to the plane of the arc. If it is not perpendicular, the direct and reflected images of H cannot be brought into con- tact,. but one will pass above or below the other as the vernier-arm is turned. (c) The objects H and / must lie in the sextant hori- zon, for otherwise the difference of their azimuths would not be the true angle between them. The sextant horizon must here be understood to mean the plane of the graduated arc, and this condition will be satisfied if the sextant is so held during the observation that this plane passes through the objects H and 7. INSTRUMENTS. 133 56. Adjustments of the Sextant. (A) The Index -glass. Take the telescope out from its support, set it on end at any part of the arc, and turn the index -glass until its plane passes a little to one side of the telescope. By holding the eye a little to the right of the line joining the index -glass to the telescope a reflected image of the telescope may be seen simultaneously with a direct view of it, and these two images should be parallel, provided the telescope stands normal to the plane of the arc. Any error in this last condition may be eliminated by turning the telescope 180 about its own axis and repeat- ing the test. No ad jus ting -screws are provided for the index-glass, but it may be adjusted, if necessary, by re- moving it from its frame and filing down the bearing- points against which it is held. (B) The Horizon- glass. Bring the direct and re- flected images of a distant object into contact if possible. If this cannot be done, bring them near together and tilt the horizon -glass by means of its adjusting screws until by turning the vernier -arm the images can be made to coincide. (C) The Telescope. To enable the observer to make the plane of the sextant pass through the objects H and / it is customary to place in the eyepiece of the telescope a pair of coarse threads which should be set parallel to the plane of the sextant. By means of its adjusting screws the telescope should be tilted up or down until the line of sight passing midway between these threads is parallel to the plane of the sextant. If the objects H and / are brought midway between these 134 FIELD ASTRONOMY. threads when contact between them is made, they will lie in the plane of the sextant as required. To deter- mine if the telescope is properly tilted, select two well- defined objects about 120 apart, and bring them into contact when the sextant is so held that they are both seen in the upper part of the field of view. Then shift the position of the sextant plane so as to bring the objects to the lower part of the field and note whether they remain in contact or appear separated; if they are appreciably separated the telescope requires further adjustment. 57. Outstanding Errors of the Sextant. The methods of adjustment above described are only approximate, and the readings of the instrument will be affected by what- ever error remains in the adjustment. In general the effect of these errors will be small for small angles, but will increase rapidly with the magnitude of the angle measured, and the adjustments should be made correct to within 10' if the resulting errors for an angle of 90 are to be insensible. However carefully these adjustments are made there will remain a source of error which cannot be removed by adjustment, but whose effect must be determined and applied as a correction to the readings if the maxi- mum attainable precision of the instrument is required. It is assumed above that the centre of the graduated arc falls exactly at the centre of motion of the index -glass, but the maker is seldom able to secure this exact agree- ment, and without it the readings of the vernier are not an accurate measure of the amount of rotation of the INSTRUMENTS. 135 mirror. The effect of this error, which is called eccen- tricity, combined with the effect of all other outstanding errors of the instrument, is best determined by carefully measuring with it a set of known angles of different magnitudes, from o to the largest one possible, and treat- ing the difference between the measured value and the true value of each angle as a correction to the corre- sponding reading of the sextant. These corrections may be plotted as ordinates with the sextant readings as abscissas and a curve drawn, from which intermediate values of the correction may be read. The length of the arc joining two stars whose right ascensions and declinations are given, may be computed and used as a known angle for this purpose, provided the effect of refraction in altering this distance is duly taken into account; or if a distant part of the horizon can be seen, a set of angles may be measured with a good theodolite for comparison with the sextant results. 58. Index Correction. Since the value of the index correction for very distant objects is constant so long as the adjustments of the sextant remain unchanged, it may be determined from special observations made for this purpose, but the determination should be fre- quently repeated since the adjustment is easily disturbed. Let a shade -glass be placed over the eye end of the tele- scope and the direct and reflected images of the sun brought into contact, externally tangent to each other, in each of the two possible positions, H first right, then left of /. The mean of the corresponding sextant read- ings will be the required value of R Q . Since the index 136 FIELD ASTRONOMY. correction enters into the value of every measured angle, it should be carefully determined from several settings, as in the following example: DOUBLE DIAMETER OF SUN FOR INDEX CORRECTION. Caroline Island, April 22, 1883. Observer, W. U. On Arc. Off Arc. Reduction. o 26' o" 359 22' 5" #0 = 359 53' 54" 26 o 220 i= +66 26 5 22 o 26 o 21 40 45= i 4 o 25 40 21 50 5= 0160 25 40 21 50 Almanac = o 15 56.4 360 25' 54" In place of subtracting R Q from each subsequent reading of the instrument it is in this case more con- venient to employ the quantity ^ = 360 R as a correc- tion to be added. The readings ' ' Off Arc " were taken on the supplementary arc to the right of the o, and the student should note that the resulting R falls off the arc. Referring to the position of R Q , the sign of the index correction may be determined from the sailor's rule: When it (R Q ) is on it's off (i subtr active] , and when it (R ) is off it's on (i additive}. The values of the sun's semi- diameter furnished by the observations and given in the almanac are shown above for comparison. 59. Artificial Horizon. Altitudes may be measured with a sextant either from the natural (sea) horizon or from an artificial horizon, one form of which is a shallow box containing mercury, covered by a glass roof to pro- tect it from wind. The reflecting surface of the liquid is improved by adding to it a little tin-foil and removing INSTRUMENTS. 137 the resulting scum (oxide) with the edge of a card. The reflected image of the sun or star lies as much below the true horizon as the real object is above it, and if the angle between the two is measured with the sextant it gives at once the double altitude of the body, subject to correction for index error, etc. See 29 for an ex- ample. 60. Precepts for the Use of a Sextant. i. Keep your fingers off the graduation. It tarnishes readily. 2. Focus the telescope with great care so as to secure sharply defined images. 3. Make the direct and reflected images equally bright, by moving the telescope to or from the plane of the sextant with the adjusting-screw provided for this purpose. 4. Bring the images into contact midway between the guide -threads. 5. Don't try to hold the images still in the field of view. Give the reflected image a regular oscillating motion by twisting the wrist, and note its relation to the direct image as it swings by. 6. In observing the sun take an equal number of observations on each limb (edge). 7. Take an equal number of observations in each position of the horizon roof, direct and reversed. 8. Determine the index correction as carefully as the angle which you wish to measure. 9. Whenever possible use a shade -glass over the eye- piece instead of those attached to the sextant frame. 10. Work as rapidly as you can without hurrying. 138 FIELD ASTRONOMY. 61. Chronometers. This section will be confined to a consideration of the proper care and use of timepieces. For an account of their mechanical construction see the article Watches in the Encyclopaedia Britannica. A chronometer is a large and finely constructed watch, whose face, hands, and train (wheels) are to be considered as a mechanical device for automatically counting and registering the vibrations of a steel helix, called the balance-spring. In most chronometers this spring makes one complete vibration every half second, producing a beat (tick) of the chronometer and a forward movement of the seconds hand through o s .5. This spring may vibrate too slow or too fast, thus producing a rate of the chronometer, and it is practically convenient that this rate should be small, but the real test of excel- lence in a timepiece is not the magnitude of its rate, but its uniformity of rate from day to day. In order that the rate of a chronometer shall remain constant, every precaution must be taken against dis- turbing the balance -spring, and most of the following precepts for the treatment of a chronometer have reference to this condition. Of the various mechanical disturbances to which it is subject, experience shows that a quick rotary motion about the axis of the balance -spring is the most injurious. According to the chronometer makers a single quick motion of this kind through half a turn and back may change the chronometer correction several seconds and so disturb the rate that it will not resume its normal value for hours or even days. A chronometer is usually supported in gimbals and INSTRUMENTS. 139 should be allowed to swing freely in them when at rest, in order that it may assume a vertical position; but when carried about, the gimbals should be locked since the oscillations that would otherwise be imparted to the balance -spring are more injurious to the rate than the isolated shocks that it may receive when firmly held in one position. A chronometer should be kept in a dry place, not exposed to magnetic influences. If possible it should always rest in the same azimuth, e.g., the zero of the dial always pointing north. It should be wound at regular intervals, and its temperature should be kept as nearly uniform as possible. The average chronometer runs best at a temperature near 70 Fahr. 62. Comparison of Chronometers. A problem of fre- quent recurrence is the comparison of one chronometer with another, e.g., in order to determine the correction of one from the known value of AT for the other. This comparison consists in noting the time indicated by one chronometer at a given time shown by the other, and presents little difficulty when no greater accuracy than the nearest half -second is required. If the comparison is to be made correct to the nearest o s .i, the method of coincident beats may be employed if one of the chro- nometers keeps sidereal and the other solar time. Since sidereal time gains 236 seconds per day uporj mean solar time and the chronometers beat half -seconds, there will be 472 epochs during a day, at which thfe chronometers beat in unison, i.e., a coincidence of the beats occurs every three minutes throughout the day, and if the comparison be made at one of these coinci- 140 FIELD ASTRONOMY. dences by noting by each chronometer its indicated time when the beats are coincident, no fractions of a second need be determined and the comparison can be made correct within one or two hundredths of a second. This mode of comparison is illustrated in the follow- ing example of the comparison of two mean -time clocks, M and F, with each other by comparing each with a sidereal clock designated H . OBSERVED TIMES OF COINCIDENT BEATS. M ip h 30 49 s F i9 b 34 m 55 s H 10 42 i H 10 46 o The interval between the coincidences, as measured by H* is 3 m 59 s (sidereal), and this interval reduced to mean solar units and added to M, or subtracted from F, gives a comparison between the mean-time clocks as follows : M i 9 h 34 473.35 i 9 h 3 o m 49 *.oo F 19 34 55 .00 19 30 56 .65, either form showing that F was 7 s . 6 5 faster than M. Every observer should acquire the ability to ' ' carry the beat" of a chronometer, i.e., to listen to and count the beats while attending to something else, since nearly all observations in which it is required to note the time of an event, e.g., the transit of a star over a thread, require this ability unless special mechanical devices, such as a chronograph, are employed. (See 79.) CHAPTER VIII. ACCURATE DETERMINATIONS. 63. General Principles. Where a high degree of precision is desired in the results of observation, the purely instrumental sources of error that have been examined in the preceding chapter must be eliminated by the methods there shown, or by others equivalent to them. But this alone is not sufficient, and we note, for example, that an instrument taken from a warm place and set up in a cold one undergoes a process of cooling and contraction of its parts that, while in prog- ress, renders the errors of adjustment variable quantities, whose effects cannot be represented by the formulae derived for the case of "instrumental constants." We have therefore as a rule to be carefully observed when precision is required: Let the instrument be set up and levelled in the place where it is to be used, at least half an hour before observations are commenced. Let the sur- roundings of the instrument during this period be as nearly as possible like those under which the observa- tions are to be made, i.e., shutters open, lamps lighted, etc. As a corollary to this rule we have the further pre- cept that during the progress of the observations the 141 142 FIELD ASTRONOMY. observer and his lamp should be kept away from the instrument as much as possible. There is large room for the display of good judgment in the selection of stars to be observed for a given pur- pose, such as the determination of time or azimuth, and precepts bearing upon this choice, both with reference to the precision of the observations themselves and to the elimination of errors in the right ascensions and declinations of the stars as furnished by the almanac, are given in the following sections. Whenever observations are to be made upon a con- siderable number of different stars, as in determinations of time and latitude, an observing list should be prepared in advance, giving the names and magnitudes of the stars, arranged in the order in which they are to be observed, and giving also such data as may be required for finding them with the given instrument, e.g., their right ascen- sions, declinations, zenith distances, etc. Also, a form should be prepared in which to record the observations, each figure that is to be written down as a part of the record having its proper place allotted it. This place must be filled up before the observation is complete, and the presence of an unfilled space in the form is to be considered as a reminder that something remains to be done. 64. Time by Equal Altitudes. The best method of determining time involves the use of a transit instru- ment (see Chapter IX), but an excellent time determina- tion may be made with a theodolite, zenith telescope, or sextant by the method of equal altitudes, as follows : ACCURATE DETERMINATIONS. 143 We note the chronometer time, 7\, at which a star west of the meridian reaxhes the zenith distance z l and the time, T 2 , at which another star, east of the meridian, reaches a zenith distance, z 2 , which differs as little as possible from z r In sextant observing it is customary to assume that if the sextant is set to the same reading in the two observations we shall have z l =z 2 . For an instrument of the other type (theodolite) the telescope must be left firmly clamped in altitude as it is turned from one object to the other, and any slight change in the altitude of the line of sight must be carefully deter- mined from readings of the altitude level of the instru- ment. If the bubble changes its position in the level- tube when the latter is turned from the first to the second star, it should be brought back to its original place by the levelling screws of the instrument, but the angle between the telescope and level-tube must not be altered. If the instrument is provided with an azimuth circle, it will be well to note its readings, ^ and R 2 , correspond- ing to the observed T l and T 2 . For the reduction of the observations we take from the formulae for transformation of coordinates, { 14, the equations cos z t = sin sin d 1 + cos cos d 1 cos ^ , cos z 2 = sin sin 6 2 ~r cos

o . 1364 t (arc) 72 38' cosec t o .0203 cosec A o .0032 a 7 4 D i. .8860 logL 3 9.4406^ A.T. C cot t 9-4952 1/70 TJ \ * *rO O 8-? 14 tan 5 9.09331* L, + 75-37 ^ o *** logL, 1.8772 2 98 logL 3 0.4745** L* 0.28 ACCURATE DETERMINATIONS. .147 For the sake of illustration the reductions in the pre- ceding example are carried to hundredths of a second of time, but this is a quantity quite inappreciable in the telescope of an engineer's transit, and with such an instru- ment, or with a sextant, it will usually be sufficient to carry the reductions to tenths of seconds only. Corre- sponding to this degree of accuracy the difference of declination of the stars may be as great as two or three degrees without the introduction of sensible error into the results by reason of the approximate character of the reduction formulag. The difference should not ex- ceed one half of this amount if hundredths of seconds are to be taken into account. 66. Observing List. Without transgressing these rather narrow limits for d l d 2 , a considerable number of suitable pairs of stars may be selected from the almanac, as is illustrated by the short observing list given below, and such a list should be prepared for the particular time and place at which observations are to be made. At least one of the stars in each pair should be a bright one, easily recognized and found with the telescope by sighting over its tube. The second star of the pair, even though much fainter, may be readily found by the method given below. In the selection of pairs of stars care should be taken to secure those that are as near as may be to the prime vertical at the time when their altitudes are equal, since the motion in altitude is then most rapid and most accurately observed. The analytical expression for this condition is tan $(,+,) =tan ^ cos (i cos A, (us) and differentiating this, treating and z as constants, we find as the change of azimuth of the second star produced by a small change of declination, cos d dd =cos 9" sin z sin A dA =cos 9" cos d sin t dA , (i 16) from which, dA= , . 4 . (117) cos 9 sin t Let R lt R 2 , R represent respectively the readings of the horizontal circle when the telescope is directed to the western star, to the eastern star, and to the meridian; we shall then have A^R.-R,, A 2 =R 2 -R,, (n8) and substituting in Equation 114 these relations together with the approximate values, we obtain Ri+R 2 = 2R ^ j- . 2 -, ,* r . (120) cos 9 sin i( a 2~ a i) The last term in this expression, computed for 9" =43, i s tabulated in the observing list under the heading AR, and by means of it and the reading RI to the first star of a pair, the reading of the horizontal circle, ^ 2 . mav be found at which the instrument should beset and the arrival of the second star in the field awaited at a time as much after the computed 6 as the first observed time was earlier than Q. For convenience' sake orient the instrument and make jR =o. As a control upon the sign of AR, note that the star that has the larger declination must be the farther from the south point. ACCURATE DETERMINATIONS. 149 TIME BY EQUAL ALTITUDES. Partial Observing List for < =43. Stars. Mag. R. A. Dec. 0. J*. a Orionis O h. m. C, co / 7 2 ^ h. m. / a Serpentis 2 . 7 1C 7Q 6 4< 10 44 o 54 a Can. Min a Serpentis ft Geminonim -5 2.3 I 2 7 34 15 39 7 30 5 2 9 6 45 28 16 ii 34 + 2 fi Herculis 7 e 17 4"? 2 7 47 12 43 -o 41 a Leonis I . ? IO ? 12 28 a Ophiuchi 2 2 1 7 3O 12 ^8 13 46 + o 16 ct Aouilce O O 1 46 8 ?6 o Leonis * 8 9-?6 I O 21 14 44 -2 28 67. Precise Azimuths. The azimuth of a terrestrial line, e.g., the line joining the centre of a theodolite to a distant mark, may be determined by measuring the difference of azimuth, D, between the mark and a star at an observed time, T. From the observed time and the right ascension of the star its hour angle, t, may be derived, and from Equations 14 we obtain, by division and introduction of the auxiliary quantities, the relation tan A = gsin (121) (122) I k COS t' from which the true azimuth 01 the star at the time of observation is readily computed. The azimuth of the mark is then A'=A+D, (123) where D is assumed to be measured from the star toward the east. 150 FIELD ASTRONOMY. The precision of A' depends equally upon D and A, and through A it depends upon the assumed latitude, declination, right ascension, and chronometer correction that are employed in the computation. The observa- tions should therefore be planned with reference to elim- inating whatever minute error may exist in any of these data, and to overcoming, by the methods indicated below and in 54, the effect of instrumental errors upon the measured angle D. Errors in the Assumed Data. The effect of these errors may be greatly diminished by selecting for observation a star very near the pole of the heavens, since the factor g is thus made small, and such a star, e.g. Polaris, should always be chosen. If the chronometer correction is well determined, the observations may be made at any con- venient hour, whether near elongation or not. As a guide to the required precision in JT we note that for observa- tions of Polaris within the limits of the United States an error of 2 s in the time will in no case produce in the computed azimuth an error greater than i". If the highest precision is required, the star should be observed at two points of its diurnal path which are diametrically opposite to each other, i.e., there should be two groups of observations separated by an interval of twelve hours, or some odd multiple of twelve hours. Errors in 0, d, and a will then be almost perfectly elim- inated, and there will also be eliminated any systematic personal error of observation depending upon the direc- tion of the star's apparent motion, such as is sometimes found to exist in the work of even the best observers. ACCURATE DETERMINATIONS. 151 A similar but less complete elimination of errors may be obtained from observations made at a single epoch if these are equally divided between stars upon opposite sides of the pole and equidistant from it. Examples of pairs of stars which approximately fulfil this condition are Polaris and 6 Ursae Minoris; 51 H. Cephei and d Ursae Minoris. The angle D may be measured with either a repeating: or a non-repeating (direction) instrument, and the student should observe the following respects in which their use differs: For a repeating instrument the azimuth level should be used to determine the inclination of the verti- cal axis corresponding to the lower motion of the instru- ment. For a non-repeating instrument the inclination to be determined is that of the horizontal axis. In both cases the bubble readings are to be taken when the line of sight is directed toward the star and also when it is turned toward the mark, unless the latter has a zenith distance of 90, in which case erroneous levelling will not affect the readings to it. With any type of instrument the horizontal circle is to be turned in its own plane from time to time during the observations, so that the vernier or microscope read- ings shall be symmetrically distributed throughout the entire 360 of the graduation; e.g., for an instrument with two microscopes let one ninth of the total number of observations be made with the circle reading to the mark approximately o, another ninth with the circle reading 20, 40, 60, etc. But see 53 for the peculiar 152 FIELD ASTRONOMY. manner in which the circle settings should be changed in the case of a repeating instrument. Level Corrections. The correction for level error is to be applied to each circle reading as shown in 50, but for observations made by the method of repetitions the level correction, b' tan h, there given for the reading to the star, must be multiplied by n, the number of point- ings in a set, since the difference of the corrected read- ings to star and mark is to be divided by n in order to obtain the measured angle. It will usually be expedient to arrange the form of record of the observations so that the level corrections may be applied and the angles worked out in the record book. 68. Reduction of the Observations. After the hour angles have been formed from the relation t = T + 4T a, and the constants g and k computed with the known decli- nation and latitude, the computation of A presents no difficulties, but it may be considerably abbreviated through the use of Albrecht's Tables (reproduced in Appendix VII, Annual Report U. S. Coast and Geodetic Survey, 1897-98), which with the argument log x give the logarithm of - . Calling this last factor F, and i x putting k cos t = x, Equation 122 assumes the very simple form tan A=gFsin*. (124) In the absence of special tables for F its value may be readily obtained from an ordinary table of addition and subtraction logarithms as follows: Representing ACCURATE DETERMINATIONS. 153 by A and B the argument and function in such a table,* i.e., A=logx, B = log(i+x), we have, whenever cos t is negative, A = log (k cos t), log F= B. When cos t is positive, we use the development , etc., and interpolating from the table of addition logarithms the values of B v B 2 , B v etc., corresponding to the argu- ments x, x 2 , x 4 , we find logF = B l + B 2 + B 4 + etc. (125) For observations of Polaris made within the limits of the United States it will never be necessary to use more than the first two terms of this series, e.g., corresponding to this case the greatest possible value of k cos t furnishes log x and the several values of B given below: log* 8.39386 BI 0.0106248 log* 2 6.7877 2 2663 log* 4 3-575 #4 2 logF 0.0108913 In ordinary practice the value of log F will be required to only six places of decimals, and B^ +B 2 furnishes this degree of precision, Where the highest degree of precision is sought, it is customary in the reduction of the observations to com" pute for each observed time the corresponding value ol A, but this process may be very greatly abridged by treating the mean of a considerable number of observa- tions as a single observation made at T , the mean of the recorded times. The azimuth A computed from T will not correspond exactly to the observations, but the correction required on this account is readily obtained. * Do not confound this use of A with its wholly different meaning in Equation 124. 154 FIELD ASTRONOMY. We may develop by Taylor's Formula the relation be* tween azimuth and time in the form, A -A+ an equation which obtains for each observed T and its corresponding A. If we take the mean of these several equations and note that the mean of the (TT )s is necessarily zero, since T is the mean of the Ts, we find for the average of the set, where the last term of the expression is the required correction to reduce A Q to the mean of the observed azimuths. For the numerical application of this formula we need to introduce a convenient expression for /"(A ), and there must also be a numerical factor such that the value of the term shall be given in seconds of arc when TT is expressed in minutes of time. This factor, combined with the coefficient J which appears in the equation, is readily shown to be, /6oXi5\* 206265 r^7f7 x r- 2 - =* [0-2930]. The differential coefficient, f"(A Q ), does not admit of an expression that is both simple and rigorous,* but, with entire accuracy at the pole and approximately for any star near the pole, we may write /"(A )=sinA , * See p. 207 for the complete expression for/" (A ). ACCURATE DETERMINATIONS. 155 and combining these several expressions we find as the correction to the computed azimuth, A , 4A =+[0.2930] sin A^Z(T-T Q Y t (127) where n is the number of Ts included in the mean, T m and the differences, T T , are to be expressed in minutes of time. AA Q must always be so applied as to bring the computed A nearer to the meridian. See 85 for the extremely small effect of diurnal aberration upon azimuth determinations. 69. Example. Precise Azimuth. The example on p. 156 represents a determination of azimuth made with an engineer's transit, using the method of repetitions, four pointings in a set, and combining two sets in such a way as to eliminate the effect of lack of parallelism of the axes of the instrument, see 53. The graduation errors are not here eliminated, and other sets with read- ings symmetrically distributed about the circle are re- quired for this purpose. 70. Precise Latitudes. Zenith Telescope Method. In Fig. 12 let V represent any point on the meridian, 5^ POLE FIG. 12. Zenith Telescope Latitudes. and 5 2 the points, on opposite sides of V, at which two stars, of declination d 1 and d 2 respectively, cross the meridian in their diurnal motion, and let z l and z 2 denote 156 FIELD ASTRONOMY. PRECISE AZIMUTH DETERMINATION. At Station M. Monday, May i, 1899. Instrument No. 386. Chronometer, S. Observer, C. Chronometer JT ' = 2' n 39 s .7. 4 tan h.d = io".'j. Object. Circle n. L L 4 R R 4 Chronometer. Horizontal Circle. Vert. Circle. Levels. Angle. Ver. A. Ver. B. Mark Polaris. . Polaris . . Mark. . . . h. m. s. 12 25 .. 12 27 28 30 59 33 28 36 24 o / // 150 31 2 5 117 21 40 117 22 O 150 10 o / // 31 15 21 35 21 30 9 40 / O O W. E. 8.5 15.0 12.7 n .0 o / // 150 31 20 = -26 117 21 38 2.4 4i 5 4i 43 6.3 17.4 18.0 5.8 )i28 19 )33 10 8 12 32 4.8 12 40 7 45 28 47 38 49 40 8 17 32 117 21 45 + 6 !5 9 5 + Q-55 12 52 .. )i82 53 )3 2 47 59 12 45 43.2 8 12 REDUCTION. h. m. s. h. m. s. t a sec cotd tan < 43 4 37 88 46 12.5 I 21 28.O o. 13641 8.33180 9 .97082 T+AT-a t cos t k cos t sin t 12 29 25.1 11 7 5 7 i i665 9 'i6" 9 . 98870^ 8 . 29132*1. 9-35249 12 43 3-5 11 21 35.5 1 70 23' 52" 9 . 99388^ 8 . 2965011 9 . 22221 8.46821 g 8.46821 8.46821 8.30262 F--B, 9-99I59 9.99149 tan A 7 .81229 7 .68191 (T TO) 20, 1,2, 18 -o, o, 4, 16 A 179 37 4i 179 43 28 [2(T T ) i .009 i . CK )7 4A o o sin A Q 7.812^ 7.6* \2K Const. o.293tt o.2( >3 D 8 17 32 8 12 logJA 9.114 9 .o ; 12 A A, + o".i 3 +o' '.12 A' 187 55 13 187 55 28 The corrections AA^ computed above are too small to be taken into account in observations of this character, but with a larger instru- ment or when the star is near elongation they become of sensible magnitude. ACCURATE DETERMINATIONS. 157 the arcs VS 1 and VS 2 . Denoting by 0" the declination of V, we have from the figure and by subtraction, + 1 -* 2 ). (128) Since V, by supposition, is any point of the meridian, we may now define it as the projection upon the merid- ian, of the point in which the vertical axis of a theod- olite or other similar instrument meets the celestial sphere, and we may represent by b" the zenith distance of V, reckoned positive when the zenith lies between V and the pole. Since the latitude is equal to the declina- tion of the zenith, we shall have In the practice of American government surveys all precise determinations of latitude are based upon this equation and are made with an instrument, the zenith telescope, especially designed for the micrometric measurement of small differences of zenith distance, the z t z 2 of the equation. But Equation 129 may be applied with any instrument capable of measuring alti- tudes theodolite, sextant, etc. and in general it will furnish better results than other modes of using the instrument, since if the stars are so selected that z l differs but little from z 2 , any constant errors which may be present in the instrumental work will be very nearly the same for the two stars, and will be approximately eliminated from the difference z l z 2 . We shall here 158 FIELD ASTRONOMY. develop the zenith-telescope method with reference to its use with an engineer's transit provided with a gradi- enter and an altitude level, which latter may be its striding-level properly fastened to the alidade at right angles to the horizontal axis. With very small modi- fications the resulting formulas will be applicable to the zenith telescope as usually constructed. The first step in the application of the method is the selection of an observing programme, consisting of a number of pairs of stars whose right ascensions and declinations, for each pair, satisfy the conditions a 2 -a 1<2 o m , 2 + 1 - 2 0 an d observe it precisely as before. If the level-bubble changes its position in the tube as the instrument is turned from the first to the second star, it should be brought back to its initial position by means of the levelling screws. The readings of the level in the two positions deter- mine the average value of b" , and if R l and R 2 represent the respective gradienter readings and k is the angle moved over by the line of sight when the gradienter is turned through one complete revolution, we shall have, z 1 -z 2 =k(R l -R 2 ). (131) 71. Minor Corrections. Before introducing this value into the expression for 20 we proceed to examine some matters that require further explanation, viz. : Level Error. The small term 26" arises from a devia- tion of the vertical axis of the instrument from the true vertical Its amount and sign are to be determined from readings of an altitude level, as shown in 42, Make this error small by turning the levelling screws, if necessary, so that the bubble readings shall be the same for the second star as for the first. Refraction. The effect of refraction upon the latitude observations is most readily determined by substituting, 160 FIELD ASTRONOMY. in place of the true declinations of the stars, their appar- ent declinations as affected by the refraction. This dis- places each star toward the zenith by the amount, (23) 982" B tan z\ (132) and since for the southern star this displacement in- creases, while for the northern star it diminishes, the declination, we shall have as the sum of the apparent declinations, (tan z, - tan z t ) , which is equivalent to, 982 [ B cos z l cos z 2 456 + tj 0. (133) The following table gives the value of the bracketed coefficient in this equation, computed with the argument z=^(z l +z 2 ), for an aver- age condition of the atmosphere, barometer 29.00 inches, temperature 50 Fahr. In all ordinary cases the correction for refraction may be found with sufficient accuracy by multiplying the tabular number, 5, by the difference of the zenith distances of the two stars, expressed in degrees, r=s(z l -z 2 ). (134) Since 5 is a positive number, the correction thus found will always have the same sign as the term z l z 2 , measured with the gradienter. REFRACTION COEFFICIENTS. s j z j IO 20 3 40 - T'? 4 ' 7 50 55 60 65 70 3-'o 4 " 6 3-9 J 84 29 5 2.4 7 75 14-5 The use of the table is illustrated in the following short example taken from the data of 73: z Z l -Z 2 s r 5-9 5-6 '"5 14.0 ACCURATE DETERMINATIONS. 161 Reduction to the Meridian. It is sometimes conve- nient or necessary to observe a star at some other instant than that of its meridian passage, and for this purpose the instrument may be turned out of the meridian, set at an azimuth that we will represent by a', and the ob- servation made precisely as before. It is evident that this is equivalent to observing on the meridian a star whose meridian altitude is equal to the altitude of the given star at the moment of observation, and whose declination, therefore, differs from that of the latter star by the reduction to the meridian corresponding to the azimuth a', (Equation 55). In the reduction of the ob- servation we have therefore to substitute in place of the star's true declination, d, a corrected declination, d", given by the relation d" = df(a')*, / = [7-9407] cos cos h sec d, (135) where a' is to be expressed in minutes of arc and the upper sign applies to a star between the zenith and pole, the lower sign to all other cases. For the sake of increased precision it will frequently be advantageous to make several gradienter pointings upon a star in different azimuths, during the two or three minutes nat precede and follow its culmination, and, having first oriented the instiumciit, to determine from readings of the horizontal circle the corresponding azimuths required in the reduction. 72. Errors of the Screw. In Equation 131 it is tacitly assumed that the angle moved over by the line of sight when the gradienter is turned from one reading 162 FIELD ASTRONOMY. to another is strictly proportional to the amount of turning of the screw. This is, however, an ideal con- dition seldom realized in fact, and if the capabilities of the instrument are to be fully utilized the errors of the gradienter must be investigated, and a set of cor- rections, C, determined, such that the angle moved through by the line of sight when the gradienter is turned from the reading R^ to R 2 may be strictly proportional to the difference of the corrected readings, R = R"). (137) This calibration of the screw may be made as follows : Let some fixed vertical angle, e.g., the difference of ele- vation of two terrestrial points, be measured upon con- secutive parts of the gradienter screw, from the begin- ning to the end of its run, so that, calling this angle v, we shall have, (137*) The second reading of the screw in the first measurement of v must be the same as the first reading in the second measurement, etc., and to secure this the gradienter should not be touched after the second pointing, R lt has been made, but the telescope should be undamped, set back by hand, approximately, upon the first point ACCURATE DETERMINATIONS. 163 and the accurate pointing completed by means of the levelling screws. From^the mean of the preceding equations we obtain i-f^v^. which contains the three arbitrary quantities k, C m , C , and is the only equation that these quantities are re- quired to satisfy. We may therefore impose two addi- tional relations among them, and, as convenient ones for the present purpose, we assume C m = C = c, where c is a constant whose value we shall, for the present, leave ni undetermined. Representing by p the value of T cor- K responding to these assumptions, and introducing it into Equations 137, we find the fol- lowing results : C = +c CX# + p) -R, +c, C 2 = (R + 2 p) -R 2 +c, R*+c, (140) The corrections thus derived from the readings, R, may be plotted in a curve, from which values of C for all intermediate readings may be obtained. The par- ticular value assigned to c will have no influence upon 164 FIELD ASTRONOMY. the shape of this curve, but will determine its position with respect to the axis of x, and we may assign to c with advantage a value that will make the entire curve lie above the #-axis, i.e., one that will make all the values of C positive quantities. The following example represents the record and reduction of a set of readings made for the investigation of the errors of the gradienter of an engineer's transit. The quantities in the column R are those directly ob- served; the column m gives the serial number corre- sponding to that used in the above analysis. Thursday, June 7, 1900. Gradienter of Instrument No. 386. Observer, P. m R /?+// (R^+ntfi-R c o .027 0.027 0.000 +31 I 2.037 2 .025 .012 19 2 4.045 4.023 .022 9 3 6.047 6 .022 -.025 6 4 8.048 8 .020 .028 3 5 10 .049 10 .018 -.031 6 12.047 12 .Ol6 -.031 7 14-045 14.014 .031 8 16.035 16.013 .022 9 9 18 .022 18 .on .on 20 10 20 .009 20.009 . 000 + 3i In the reduction of the above we use ^ = ^0(20.009 0.027) = 1.9982. The last column, expressed in thousandths of a revolu- tion, is obtained by adding to the numbers of the pre- ceding column the assumed constant, c +0.031. A considerable number of such determinations should be made and the mean of the several results adopted as definitive corrections to the gradienter readings. Similar ACCURATE DETERMINATIONS. 165 corrections must always be applied where a high degree of precision is required in the use of a gradienter or other similar micrometer, e.g., the eyepiece micrometer of a zenith telescope or transit, and particular care should be given to them in determinations of k, the value of one revolution of the gradienter screw. In a similar manner the gradienter should be exam- ined for periodic errors, i.e., errors peculiar to a particular part of a turn and which repeat themselves whenever the same part of the head, as the o, comes under the index, regardless of the number of whole revolutions at which the screw stands. 73. Gradienter Latitudes. Example. We may now write the equation for zenith-telescope latitudes in the form, (141) through which a value of the latitude may be derived from each pair of stars observed, if k is known. This value of a revolution of the screw, k, may be determined by measuring with the gradienter a known angle, such as the difference of declination of two stars, or it may be treated as an unknown quantity whose value is to be derived from the latitude observations themselves. In the latter case at least two pairs of stars, preferably ten to twenty pairs, must be observed for the deter- mination of the two unknowns, and k, and these should be so selected that in one pair the sum of the declinations is greater than 20 and in the other pair is less than 20. The following example represents the observation 166 FIELD ASTRONOMY. and reduction of a single pair of stars made with the instrument shown in Plate I, whose errors are investi- gated in 72. The gradienter readings as directly observed are given in the column marked R, and in the following column there are given the corrections to these readings as interpolated from the table at p. 164. The instrument having been oriented by the method of 32, the readings of the horizontal circle, in the column H.C., furnish immediately the azimuths, a' , required for computation of the reductions to the meridian, which are here represented by the letter M. The stars' merid- ian altitudes, h , that are also needed for the compu- tation of these reductions, may be obtained with suffi- cient accuracy from the declinations and the known approximate latitude of the place, 43. The value of a revolution of the gradienter, k, was known to be about 20' 30", and this value together with the observed dif- ference of the gradienter readings determines z l z 2 with sufficient precision to permit the refraction correction to be interpolated from the table at p. 160. The level correction, 26" = 7", is negative since the level read- ings show that the vertical axis of the instrument pointed north of the zenith, i.e., in too great a latitude. The declinations of the stars are taken from the American Ephemeris, but in the case of Polaris, which was observed at its transit over the lower half of the meridian, sub polo, the almanac declination is subtracted from 1 80 in order to obtain the distance of the star from the upper half of the equator, which is the quantity used in the analysis and required in the reduction. ACCURATE DETERMINATIONS. 167 Monday, May 20, 1901. At Azimuth Stake. Instrument No. 389. Observer, C. Star. Level. N. S. R Corr. H. C. s M Remarks. o / o i n it Polaris, S.P. a Virginis 6.2 9.2 8.0 7.5 0.302 16 . 704 + 2 9 + 13 J 79 55 358 21 91 13 18 -io 38 57 -6 -5 1 Level, d = 3" -1.25 Reduction to Meridian. Star Polaris a Virginis VtV a' 5' 99' cos 9.863 9.863 26" COS UQ 9.872 9.906 sec d i .671 o .008 Ref'n Const. 7.941 7.941 (a') 2 1.398 3-992 1?' -/?" logM -745 1.710 M -5-6 -51.2 2< = 8o33' 3 8o 33' 24" ~ 7 ! + 14 i 1 6 . 386 rev. )' 3i" + i6. 3 86fe Each observed pair of stars furnishes an equation similar to the above, involving and k as unknown quantities, for which definitive values are to be obtained from a solution of all the available equations. For illus- tration we select a single additional pair of stars and its resulting equation, viz., 20 = 91 9' 52" 14.671^, and combining it with the one derived above we obtain, & = i22 9 ". 4 , = 43 4' 38"- This value of agrees within i" with the known latitude of the place of observation and represents about the limit of accuracy attainable with an engineer's transit. With the zenith telescope, used in essentially the same manner as above, a precision of about o".i is attained. See Appendix 7, Report of the U. S. Coast and Geodetic Survey for the Year 1897-98, for an exposi- tion of the methods employed with such an instrument. CHAPTER IX. THE TRANSIT INSTRUMENT. 74. General Principles. Adjustments of the Instru- ment. If the celestial meridian were a visible line drawn across the heavens, the local sidereal time corresponding to this meridian might be determined by observing the chronometer time, T, at which a star of known right ascension, a, crossed this line. We should then have for the correction of the timepiece employed, AT = a-T. The transit instrument, different forms of which are shown in the Frontispiece and in Plate VI, is a substitute for the visible meridian above supposed. Its essential parts are illustrated by the telescope and standards of a large theodolite firmly mounted, with the horizontal axis of ro- tation perpendicular to the plane of the meridian, i.e., east and west, and level. The telescope is usually provided with several vertical threads (an odd number of them), each of which, as seen by the observer, is projected against the sky as a background, and each of which, when the telescope is turned about the rotation axis, traces upon the sky, by virtue of this rotation, a circle whose plane is perpendicular to the axis. Also, one or more horizontal threads are usually introduced to mark the middle points of the transit threads. 168 PLATE VI. A Straight Transit Instrument. Length of Telescope 30 inches. Approximate Cost $1000. \Tofacep. 168.] THE TRANSIT INSTRUMENT. 169 A transit instrument is said to be perfectly adjusted when the circle thus traced upon the sky by its middle vertical thread coincides with the local meridian, and for such an instrument it is evident that the time of a star's transit over this thread may be substituted for the time of its transit over the visible meridian above supposed, and the chronometer correction, AT, will then be fur- nished by the equation printed above. But in general it cannot be assumed that these adjustments are perfect, and we must consider them as so many possible sources of error whose effects must be in some way eliminated from the results of observation. Optical Adjustments. We assume that great care has been given to the optical adjustment of the instru- ment, so that both the transit threads and the star are sharply defined and distinctly seen. For this purpose the eyepiece should first be so set that the threads appear black and distinct, and threads and eyepiece should then be moved in or out together until a star, preferably a double star, presents a clear image without trace of fuzziness, projecting rays, or stray light. This last ad- justment may be a little more accurately made by cov- ering the upper half of the telescope objective with card- board or paper and making an accurate pointing of the horizontal thread upon a circumpolar star near cul- mination. Having made a satisfactory pointing, quickly shift the card so as to cover the lower half of the objective and leave free the upper part, when, if the threads are not properly adjusted with respect to the objective, there will be a slight vertical displacement of the star 170 FIELD ASTRONOMY. with respect to the thread, and this must be corrected by further adjustment. Vertically of Threads. To make the threads perpen- dicular to the rotation axis, point the telescope at a ter- restrial mark, and turning the telescope in altitude with the slow-motion screw, note whether the mark in its apparent motion up and down the field of view runs exactly along the thread. Any outstanding error in this adjustment may be removed by slightly rotating in its own plane the collar which carries the threads ; but a small error here may be rendered harmless by always pointing the telescope, at the times of observation, so that the stars cross the same part of the field, e.g., be- tween the parallel horizontal threads. The principal errors of adjustment that remain to be considered in connection with the use of a transit instrument are three in number, viz. : The azimuth error, a, is the angular amount by which the rotation axis deviates to the south of due west. The level error, b, is the angle of elevation of the rotation axis above the western horizon. The collimation error, c, is the amount by which the angle between the line of sight and the west half of the rotation axis exceeds 90. The line of sight as here used means the imaginary line passed through the Optical centre of the objective and the mid- dle transit thread, or through the mean of a group of transit threads. 75. Theory of the Instrument. To determine the rela- tion of these several instrumental errors to the time, T, at which a star will pass behind a given transit thread THE TRANSIT INSTRUMENT. 171 we have recourse to Fig. 13, which represents a projec- tion of the celestial sphere upon the plane of the horizon. Z is the projection of the zenith, P of the pole, H of the point in which the rotation axis, produced toward the west, intersects the celestial sphere, and 5 is the projec- tion of a star observed at the moment of its transit over FIG. 13. The Transit Instrument. a thread whose angular distance from H is measured by the arc 90 + c. From the definitions given above, c represents the collimation of the particular thread in question, and similarly b and a, in the figure, are the level and azimuth errors above defined. The symbol T of the figure represents the hour angle of the star, reckoned toward the east, d is the star's declination, 90 K is the arc HM, and A is the distance of the star from the meridian measured along HS. This latter arc 172 FIELD ASTRONOMY. must not be confounded with the diurnal path of the star; the one is an arc of a great circle defined by the points H and 5, while the other is a small circle having its pole at P. Note that in all cases the symbols here defined represent the actual magnitudes of the arcs and angles on the sphere, and not of their projections on the plane of the horizon. From the spherical triangle PMS we obtain the rela- tion, cos d sin T = sin A sin $, (143) and from the triangle ZHM we find, sin K = sin b cos + cos b sin sin a. (144) These equations may be greatly simplified by substi- tuting arcs in place of sines whenever the quantities a and b are so small that their cubes and higher powers may be neglected, and we shall therefore assume that we have to deal with an approximately adjusted instru- ment, in which neither of these quantities much exceeds 10'. On this supposition the point H is so nearly the pole of the meridian, PZM, that we may put sin $ = i and " = d, where denotes the latitude of the place of observation, and our equations now take the form, T = A sec d, Prom the figure we have the relation, THE TRANSIT INSTRUMENT. 173 and eliminating A and K between these equations we find, T = sin (0 d) sec 5. a + cos (05) sec d . b + sec 5. c. (146) Since T is an east hour angle, we have also, in terms of the observed time, the chronometer correction, and the star's right ascension, r + J7 = a-r, (147) from which we obtain, by the elimination of T, Mayer's equation of the transit instrument, a T = JT + sin (05) sec d . a + cos (0-5) sec 5 . 6 + sec 5 . c, (148) or, as it is usually written, a-T = 4T + Aa + Bb + Cc, (149) where the capital letters are introduced as abbreviations for the coefficients given above, i.e., A =sin (0-5) sec 5, B = cos (0-5) sec 5, C = sec 5. (150) Since a, T, and AT are expressed in time (hours, minutes, and seconds), it is customary in connection with this equation to express a, b, and c in seconds of time. 76. Discussion of Mayer's Equation. The coefficients A, B, C are called transit factors, and when many obser- vations are to be made in the same latitude, 0, as at an observatory, it is customary to tabulate their values with the declination as argument, and to interpolate from these tables the values of the factors corresponding to the particular stars observed. In the U. S. Coast and Geodetic Survey Report for the year 1880 there may be 174 FIELD ASTRONOMY. found extensive tables of this kind for different latitudes covering the whole extent of the United States. In the use of such tables the following distinction must be carefully observed: Every star whose distance from the pole is less than the latitude remains continu- ously above the horizon throughout the twenty-four hours, and during this period crosses the meridian twice, once above the pole, e.g., between the pole and zenith, and once below the pole, e.g., between the pole and the northern horizon. The latter transit is usually desig- nated sub polo, and from Fig. 13, where 5' represents the star 5 near its transit sub polo, it may be seen that its coordinates at this transit will be obtained by sub- stituting in place of the a and 90 d, corresponding to 5, i2 h + a and (90 d). When these values are introduced into Mayer's equation it becomes, for stars observed sub polo, i2* + a-T = 4T + A'a + B'b + C'c, (151) where the new transit factors have the following values : A' =sin (0+5) sec d, B f =cos (0+ d) sec d, C' = -sec5. (I52) As an exercise in analysis the student may show that the transit factors for a star above and below the pole are connected by the relations, A + A' -2 sin 0, + ' = 2cos0, C + C'=o. (153) Use these equations to derive A' , B', C' from the tabulated values of A, B, and C. From a consideration of the trigonometric functions THE TRANSIT INSTRUMENT. 175 that enter into the transit factors the algebraic signs of these factors are found to be as follows for places in the northern hemisphere : Factor. ABC South of Zenith + + + - Zenith to Pole - + + - Below Pole + - + Note that in every case the transit factors for a given star have opposite signs above and below the pole, and compare with this statement the fact that stars on oppo- site sides of the pole move in opposite directions, east to west above pole and west to east below pole. Query. The above relations of sign are for a place in north latitude. How must they be changed to adapt them to a place south of the terrestrial equator ? In explanation of the double set of signs given above for C, we recall what was shown in 50, that a reversal of the instrument changes the sign of the collimation constant, c\ i.e., 90 c is substituted for the 90 + of Fig. 13, by lifting the axis out of the wyes and replacing it, turned end for end. It is customary to ignore this change of sign in c, and to represent its effect in Mayer's equation by changing the algebraic sign of C when the instrument is reversed; e.g., For Circle W C( + c) = ( + Qc For Circle E C(-c) = (-C)c The collimation constant, c, may be either positive or negative, depending upon the adjustment of the instru- ment ; but it retains the same sign in both positions of the 176 FIELD ASTRONOMY. circle, while the collimation factor, C, is positive (above pole) when the circle end of the axis points west, negative when it points east. 77. Choice of Stars. In the right-hand member of Mayer's equation, as printed on page 173, there are in- volved four unknown quantities, AT, a, b, and c, one of which, 6, the inclination of the axis to the plane of the horizon, is always to be determined by some mechanical method, e.g., the use of a spirit-level. The collimation constant, c, may also be determined mechanically (see 84), but for the present we shall assume that this has not been done and that the instrumental constants a and c, as well as the clock correction AT, are to be deter- mined from observations of stars. Since there are three quantities to be thus determined, there must be at least three observations, and it is practically convenient to make four the minimum number instead of three; ob- serving two stars Circle E. and two Circle W. for the sake of a good determination of the collimation, c, through the reversal of the instrument. The stars thus chosen should not all lie on the same side of the zenith, but should be distributed on both sides, so as to make the sum of their azimuth factors as small as possible. When I A =o, the effect of the azimuth error, a, is completely eliminated, and a nearly complete elimination may usually be obtained by care in the selection of stars. In the example of 78 this condition is approximately satisfied by the four stars marked a', b' ', d' ', e' ', and the student after tracing through the reduction there given, should note that if the azimuth star, i H. Draco., were THE TRANSIT INSTRUMENT. 177 dropped and the azimuth error entirely ignored, the resulting value of AT would be substantially the same as is obtained when the azimuth error is taken into account. In this case, therefore, an accurate deter- mination of a is of little consequence. 78. Example. Ordinary Determination of Time. The following example, taken from the time service of the Washburn Observatory, = 43 4' 37", illustrates the record and reduction of a set of transit observations. In addition to the date and the measured inclination, 6, of the horizontal axis, given in the column of Constants for the two positions of the instrument, Circle W. and Circle E., the observed data are contained in the three columns marked, at the foot, with Roman numerals, I, II, III. The observed times of transit given in III are each the mean of the observed times of transit of the given star over 15 threads, and in the reduction the collimation constant, c, is assumed to refer to the mean of these threads instead of to the middle thread. Note that this particular convention with regard to c can be adopted only when each star is observed over precisely the same set of threads as every other star. The failure to observe a single star at its transit over one of the threads will require either the rejection of the transits of other stars observed at this thread, or a determination of thread intervals and a ' ' reduction to the mean thread" for which reference may be made to Appendix 7, U. S. Coast and Geodetic Survey, Annual Report for 1897-98. The remaining columns are marked with Arabic nu- merals, showing the order in which they are reached in 178 FIELD ASTRONOMY. the computation. Of these columns i and 2 are obtained from the almanac (in this case the Berliner Astronomisches Jahrbuch, plus the corrections given in Astronomische Nachrichten, No. 3508). The declinations are taken to the nearest minute only, while the right ascensions are accurately interpolated for the instant of the star's transit over the local meridian, i.e., 0.3 day after their transit over the meridian for which the almanac is constructed. The third star, being observed sub polo (and before mid- night), was observed half a day before its transit over the local upper meridian, and its right ascension is there- fore interpolated for an instant 0.2 day before its transit over the Berlin meridian. The transit factors contained in columns 4, 5, and 6 were inter- polated from tables of such factors, and the products contained in columns 7 and 8 were next filled in by the use of Crelle's multiplication tables. It may be noted that the effect of diurnal aberration shown in column 7 has already been found ( 27) to be o s .o2i cos (f> sec 5, which, for the given latitude, is equal to o s .oi5 C,and the collimation factor C was employed in computing the correction. These corrections were next added mentally to the numbers contained in III, and the resulting times subtracted from the right ascensions in i, thus giving the absolute terms of the equations numbered 9. The first members of these equations, 3, 4, 6, are obviously derived from Mayer's equa- tion. We have now five equations involving only three unknown quan- tities and presenting, therefore, a case for the application of the Method of Least Squares. A rigorous solution by that method furnishes the following values of the quantities sought: AT = +2 m 57 s .oio, a= +08.858, c=+o s .g66. But such a solution is rather laborious, and a simple method of obtain- ing approximately accurate results is indicated under the heading Solution, where the symbols at the left indicate the manner in which the successive equations are derived. Equation k' is derived from *' by dividing through by the coefficient of c, and /' is similarly de- rived from h r , using the coefficient of AT ', as divisor and substituting in place of c its value given by k' . Equation m f is obtained from c' by substituting in place of AT and c their values as given in k f and /'. THE TRANSIT INSTRUMENT. 179 The value of a furnished by this equation when substituted in k' and /' gives definitive values of AT and c, all of which are entered in the column of Constants. By means of these values of a and c, columns 1 2 and 1 3 are filled up and the sum of the corrections contained in columns 7, 8, 12, 13, is entered in 14 and added to the corresponding numbers in III, thus furnishing the corrected times contained in 15. Only the seconds are entered here, since the minutes remain unchanged. The indi- vidual values of the clock correction contained in 16 are now obtained by subtracting 15 from i, and their agreement, one with another, is a check upon the accuracy of the entire work, both observations and computations. For the sake of this check it is better to proceed as is here done than to rely upon the value of AT furnished by the solu- tion of the equations. The numerical work here shown is greatly facilitated by the use of a slide-rule or an extended multiplication table such as that of Crelle. It may readily be seen from the course of the above solution that the collimation, c, is obtained from the four observations marked a', 6', d f , e' , while the azimuth, a, is furnished by the third observation. A star near the pole, like i H. Draco., is introduced into the observ- ing programme solely to. determine a, and with refer- ence to this use it is called an azimuth star, while the others are known as clock stars, since it is they that deter- mine the value of AT. As there is always a possibility of disturbing the azimuth, i.e., changing a in the act of reversing the instrument, there should, in all strictness, be two values of a determined, one for Circle W. as well as the one above found from the observation of a star Circle E. ; but in the present case it may readily be seen that there was no such disturbance, since the value of a for Circle E. brings into perfect agreement the values of AT furnished by the two stars observed Circle W., although their azimuth factors are widely different. Whenever necessary, introduce into the solution two azimuths, one for each position of the instrument, as 180 FIELD ASTRONOMY. Constants. i U Tj- \O i 4 ._Transits there is indicated the position of the star by Eye and Ear ' with respect to the thread at 26" and at 27 s , as noted 182 FIELD ASTRONOMY. and temporarily remembered by the observer. From the manner in which the thread divides the space between the two star images it is evident that the actual transit over the thread occurred at 26.4 s , and it should be so recorded. The fraction of a second depends upon the observer's estimation (an estimate of space seen in the telescope and not time as counted by the ear), and a skilled observer should be able to follow a star in its progress across the field of view, observing and recording to the nearest tenth of a second the times of transit over as many threads as may be desired, without taking the eye from the telescope during the process. He should, while watching the star, give no heed to the hour and minute, but concentrate attention upon the seconds and fractions of a second, until the transit over the last thread has been recorded; then, still counting seconds, let him look back at the face of the chronometer and note if the time there shown by the seconds hand agrees with his count. This is called checking the beat, and if it checks properly, the minute and hour corresponding to the last observation should be written down as a part of the record. 80. Precision of the Results. By the method above outlined a skilled observer may, from the mean of several threads, determine the time of a star's transit within very small limits of error; e.g., there is found for the probable error of a transit of a single star over the mean of from 10 to 15 threads, some o s .o2 or o s .o3. But this apparent precision is in some degree fallacious, for most observers possess individual peculiarities, called personal equation, THE TRANSIT INSTRUMENT. 183 by which they tend to observe all stars either too soon or too late, by a nearly constant amount, and the proba- ble error of a transit based upon the agreement of indi- vidual results, one with another, furnishes no indication of the presence or magnitude of this constant personal error. Closely related to the precision attainable in esti- mating the times of transit of a star over the threads of an instrument, is the degree of accordance to be expected among the values of AT furnished by the several stars composing a set, such as that of the illustrative example of 78. The range of values there exhibited, while smaller than is to be expected from a beginner, may be regarded as fairly typical of the results to be obtained by an experienced observer provided- with . a good instru- ment. See in this connection the example of 82, where the results show an even closer but by no means abnor- mal agreement. 81. Personal Equation. The personal equation, al- though a real and oftentimes a considerable source of error, is, however, of small consequence save where the observations of different persons are to be combined, one with another, as in a determination of longitude. In such cases, however, the problem of personal equation must be met and seriously dealt with, and various devices have been employed for this purpose; e.g. : (i) An exchange of observers at the middle of the work in question, so that its first half may be affected with the personal error in one direction and the second half in the opposite direc- tion, thus eliminating this influence from the mean. 184 FIELD ASTRONOMY. (2) The determination of the exact amount of the per- sonal equation for each observer, by means of so-called personal-equation machines, is sometimes attempted; but at present the best device for the elimination of personal equation seems to be : (3) The Repsold Transit Micrometer, an apparatus in whose use the methods of observing above set forth, 79, are completely aban- doned, and as a substitute for them the observer, while looking into the telescope, seeks to keep the image of a star, as it moves across the field, constantly covered by a micrometer thread, which he manipulates with his fin- gers and which is so connected with a chronograph as to give an automatic record of the star transits. The experience of the Prussian Geodetic Institute indicates that in this mode of observing, personal differences between observers are nearly annihilated. 82. Reversal of the Instrument upon Each Star. A method of using a transit instrument introduced into general practice in connection with the transit microm- eter, but which may be equally well applied with the ordinary chronographic or eye-and-ear methods, con- sists in noting the time of transit of a star over a group of threads placed at some little distance from the centre of the field, then, after quickly reversing the instrument, to observe the same star again on the same threads in their new position. It is obvious that the effect of collimation is thus completely eliminated from the mean of the observations on each thread and therefore from the general mean of the observed times. This elimina- tion, while an important advantage of this mode of ob- THE TRANSIT INSTRUMENT. 185 serving, is far from being the only one, and a considerable number of sources of error which have not been con- sidered above, but which are dealt with at length in the larger treatises, such as Chauvenet, Spherical and Practi- cal Astronomy, are equally eliminated by the reversal; e.g., inequality of pivots, flexure, thread intervals, and the disturbance of the spirit-level incident to reversing it upon the axis. When the telescope is reversed upon every star a hanging level may be allowed to remain upon the axis without ever being reversed, since the level readings in the two positions of the axis then give its mean inclination, which is the datum required for the reduction of the star transits. Whenever it can be employed the method of reversal upon every star is to be preferred to the older method illustrated in the preceding example, but it requires special facilities for quick reversal of the instrument without disturbing its azimuth, and these are not always present. The following is an example of the record and reduc- tion of such a series of observations, made with the same instrument and arranged in nearly the same manner as the example on p. 180. Each star was observed on five threads in each position of the instrument, and a value of the level constant, 6, was determined for each star from readings of the hanging level, taken immediately before or after the observed transits in each position of the circle, the level remaining unre versed upon the axis during the entire set of observations. 186 FIELD ASTRONOMY. O $ I 00 oo" CtJ 0) 00 . o\oo oo oo loo ^ - M 00 . ONOO O *t 3 CO M Tf 10 W M M CO * . \O to cs ^ "" O w M co .... 00 + + + 1 * ' . too 3" M M tH o w 00 O CO ON CO 04 CO VO \5 i> X ^ 0\ t>. 00 THE TRANSIT INSTRUMENT. 187 83. Determination of Azimuth with a Transit Instru- ment. Let the line of sight of a transit instrument be supposed directed accurately upon some terrestrial mark, and the telescope then turned up to the sky and the time of a star's transit over the line of sight observed. From this observed time the hour angle of the star may be derived, and this hour angle, in connection with the known declination and latitude, will determine the star's azimuth at the instant of observation. If there are no instrumental errors present, this computed azimuth will be the true azimuth of the terrestrial mark at which the line of sight was originally directed. This simple method of determining azimuth requires some modifications on account of instrumental errors, but when these are duly taken into account and a proper selection of stars and mark is made, the method ranks as the best of all known ones for azimuth determination. The star to be observed should be very near the pole, usually Polaris, and if the chronometer correction, AT, is accurately known, the observation may be made at any convenient time, e.g., the time at which the star stands directly above a mark already established. If the chronometer correction is not well determined, the observation should be made when the star is near elonga- tion, since the effect upon the computed azimuth of an error in the assumed AT is then a minimum. But this latter procedure requires the establishment of a special mark whose azimuth shall be very approximately equal to that of the star when at elongation, and it will often be more convenient to determine the time with 188 FIELD ASTRONOMY. the required accuracy, e.g., one tenth of a second, and thus obtain more freedom, in the choice of a mark. A transit instrument of the better class is usually provided with an eyepiece micrometer, i.e., one or more threads parallel to the fixed transit threads, but capable of being moved to and fro in the field of view by a screw whose axis is parallel to the rotation axis of the instrument. This screw is provided with a grad- uated head whose readings indicate the successive posi- tions of the thread and measure the amount of its motion between consecutive pointings upon the star and mark. When such a micrometer is present, transits of the star may be observed over its threads as long as the star remains within the field of view, and many comparisons between star and mark may be substituted for the single one above supposed. The instrument should be re- versed at least once during these observations, and the inclination of its axis, b, must be carefully determined since, as will appear later, the level error has an important effect upon the azimuth. 84. Theory of the Method. To derive from the micrometer readings upon star and mark the difference of their respective azimuths we have recourse to Fig. 15, which represents a projection of the celestial sphere upon the plane of the horizon. 5 and M represent respectively the star and the mark, Z is the zenith, and the spherical angle SZM is the required difference of azimuth. Let H be the point in which the rotation axis of the instru- ment, produced toward the west, meets the celestial sphere, and the arcs 90 - b, 90 -f c, will then have the THE TRANSIT INSTRUMENT. 189 same significance as in Fig. 13. The spherical angles HZS and HZM are represented by the symbols 90 FIG. 15. Azimuth with a Transit Instrument. and 90 -f w', and the zenith distance of the star, ZS, by z. From the triangle H ZS we obtain, cos (90 + c) =cos (90 -b) cos z + sin (90 b) sin z cos (90 + w), (154) which, when b and c do not much exceed 10', is equivalent to, c + b cos z = w sin z. (J55) In this equation we substitute 90 h in place of z and put sec h = i -f 0", and it becomes, w = c + c VA 82.691 Time corr. + A + oh 53 m .3 1910, VY 82.904 Stand, time i 30 .0 Greenwich t. 2 23 -3 VA-VY -r0.2I3, will be found i' too small. In the notation of the differential calculus this relation is represented by the equation d(j>/dh = -i, a relation which in this simple form holds only when the body is in the meridian. In Eqs. 36, 37, 38, if h be measured i' too great the computed azimuth, A, and time, T, will be to some extent vitiated, but the amount of error thus produced is not immediately evident. The ratio of the resulting errors, dT, dA, to the dh that pro- duced them is, however, given by the differential coeffi- cients -TTj -jTj and the analytical equivalents of these expressions as well as others of frequent use, are given on p. 207. By a proper use of these expressions a cor- rection for small errors, e.g., o' to 5', whose presence in the data is detected only after a long reduction has been made, may be readily computed without the necessity for a second reduction of the observations; thus, in the example solved in 30, if it be desired to sub- stitute in place of the latitude there employed an im- proved value o'.4 greater, we may find the resulting change in A M to be dA JA = +o'. = -o / .sec cot t= -o'. TABLES. 205 using for sec < and cot / the numerical values furnished by the original reduction. An equally important use of these differential coeffi- cients is that they furnish criteria by which to distin- guish between favorable and unfavorable conditions for observations of a given type. Thus if it be desired to determine time from an observed altitude of a star, the conditions should be so chosen that any small error in the observed altitude shall have a minimum effect upon the concluded T, or the hour angle, t, from which T is determined. We find from the first of the differential coefficients on p. 207. dt__ i dh cos (f> sin A' which for any given value of the latitude, , has its minimum value when sin A = i, i.e., the observation should be made upon a star near the prime vertical. Let the student show from the fourth of the differential coefficients that in the determination last considered a small error in the adopted latitude will also have its minimum effect upon the resulting chronometer cor- rection if the observation is made upon a body due east or due west. DIFFERENTIAL COEFFICIENTS. 88. dh = cos (f) sin A .................... (f> and d constant =_cos^. ......................... d

s^.- COS (f) /COS d>\ 2 . -fsm d sin A { 7 ) sm 2 A