Chemistry Library QB 145 D72i 1900 THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES in m A TREATISE ON PRACTICAL ASTRONOMY, AS APPLIED TO GEODESY AND NAVIGATION. C. L. DOOLITTLE, Professor of Mathematics and Astronomy, Lehigh University FOURTH AND REVISED EDITION. SECOND THOUSAND. NEW YORK JOHN WILEY & SONS LONDON CHAPMAN & HALL, LTD. 1900. COPYRIGHT, 1885, BY C. L. DOOLITTLE. Braunworth, Munn & Barber Printers and Binders Brooklyn, N. Y. Chemist^ PREFACE. THE following work is designed as a text-book for univer- sities and technical schools, and as a manual for the field astronomer. The author has not sought after originality, but has attempted to present in a systematic form the most approved methods in actual use at the present time. Each subject is developed as fully as the necessities of the case are likely to require; but as the work is designed to be a practical one, those methods and developments which have merely a theoretical or historic interest have been ex- cluded. Very complete numerical examples are given illustrative of all the prominent subjects treated. These have been selected with care from records of work actually performed, and will show what may be expected in circumstances ordi- narily favorable. Such auxiliary tables as are applicable only to special prob- lems will be found in the body of the work; those which have a wider application are printed at the end of the volume. The universal employment of the method of Least Squares in work of this kind has led to the publication of an introduc- tion to the subject for the benefit of those readers who are not already familiar with it. This introduction develops the method with special reference to the requirements of 6 45 IV PREFACE. this particular class of work, and it has not been the design to make it exhaustive. For the materials employed original papers and memoirs have been consulted whenever practicable. The illustrative examples have been drawn largely from the reports of the Coast and other government surveys. For most of the exam- ples of sextant work, as well as for many valuable' sugges- tions, the author is indebted to his friend and former col- league Prof. Lewis Boss. Much assistance has also been derived from the excellent works of Chauvenet, Brunnow, and Sawitsch. Fully appreciating the difficulty of eliminating all mis- takes from a work of this character, the author can only hope that this one may not prove to be disfigured by an undue number of such blemishes. C. L. DOOLITTLE. BETHLEHEM, PA., May 20, 1885. CONTENTS. INTRODUCTION TO THE METHOD OF LEAST SQUARES. PACK Errors to which observations are liable i Axioms 2 The law of distribution of error 3 The curve of probability 5 Determination of the law of error 6 Condition of maximum probability 1 1 The measure of precision 12 The probable error 13 The mean error 15 The mean of the errors 17 Precision of the arithmetical mean 18 Determination of probable error of arithmetical mean 20 Probable error of the sum or difference of two or more quantities 22 Principle of weights 23 Probable error when observations have different weights 26 Comparison of theory with observation 29 Indirect observations 32 Equations of condition Normal equations 35 Observations of unequal weight 36 Arrangement of computation 37 Computation of coefficients by a table of squares 41 Solution of normal equations 43 Proof- formulae 47 Weights and probable errors of the unknown quantises 54 Mean errors of the unknown quantities 65 vi CONTENTS. INTERPOLATION. PACK Notation 7 1 General formulae of interpolation 7 2 Arguments near beginning of table 78 Arguments near end of table 82 Interpolation into the middle 84 Proof of computation.. , 85 Differential coefficients 86 The ephemeris Lunar distances 92 PRACTICAL ASTRONOMY. CHAPTER I. THE CELESTIAL SPHERE TRANSFORMATION OF CO-ORDINATES* Spherical co-ordinates 100 The horizon Altitude Azimuth 102 The equator Declination Hour- angle Right ascension 103 The ecliptic Longitude Latitude 104 Having altitude and azimuth, to find declination and hour-angle 107 Having declination and hour-angle, to find altitude and azimuth 112 To find hour-angle of star in the horizon 114 To find distance between two stars ' 115 CHAPTER II. PARALLAX REFRACTION DIP OF THE HORIZON. Definitions 120 To find equatorial horizontal parallax 120 Parallax at any zenith distance 121 Form and dimensions of the ear-th 122 Reduction of the latitude 124 CONTENTS. vii PAGE Determination of the earth's radius 127 Parallax in zenith distance and azimuth I3 1 Parallax in right ascension and declination 142 Refraction J 53 Descartes' laws 154 Bessel's formula for refraction 155 Refraction in right ascension and declination 157 Dip of the horizon 160 CHAPTER III. TIME. Sidereal time 163 Solar time 164 Inequality of solar days 164 Equation of time 166 Sidereal and mean solar unit 168 To convert mean solar into sidereal time 170 To convert sidereal into mean solar time 172 CHAPTER IV. ANGULAR MEASUREMENTS THE SEXTANT THE CHRONOMETER AND CLOCK. The vernier 174 The reading microscope The micrometer 176 Eccentricity of graduated circles 180 The sextant 183 The prismatic sextant 186 Adjustments of the sextant 188 Method of observing 190 Index error 194 Eccentricity of the sextant 196 The chronometer 207 Comparison of chronometers 208 The clock 209 The chronograph 211 VJii CONTENTS. CHAPTER V. DETERMINATION OF TIME AND LATITUDE METHODS ADAPTED TO THE USE OF THE SEXTANT. PACK Determination of time By a single altitude of the sun 215 By a single altitude of a star 220 Conditions favorable to accuracy 222 Differential formulae 223 Equal altitudes of a star 228 Equal altitudes of the sun 230 Latitude 2 33 By the zenith distance of a star on the meridian 233 By a circumpolar star observed at both upper and lower culmination. . 235 By the altitude of a star observed in any position 236 By circummeridian altitudes 238 Gauss' method of reducing circummeridian altitudes of the sun 247 Correction for rate of chronometer 250 Latitude by Polaris 256 Correction of altitudes for second differences in time 260 Probable error of sextant observation 265 CHAPTER VI. THE TRANSIT INSTRUMENT. Description of instrument 269 Value of level 276 Adjustments of instrument 279 Methods of observing 283 Theory of the transit 284 Diurnal aberration 289 Equatorial intervals of threads 291 Reduction of imperfect transits 294 Determination of constants: The level constant 295 Inequality of pivots 296 The collimation constant 302 The azimuth constant 305 Personal equation 316 Probable error and weight of transit observations 318 Application of the method of least squares 322 CONTENTS. IX PAGK Correction for flexure 335 The transit instrument out of the meridian 338 Transits of the sun, moon, and planets 331, Correction to moon's defective limb 343 The transit instrument in the prime vertical 348 Mathematical theory 352 Errors in the data 356 Reduction to middle or mean thread , 356 Application of least squares to prime vertical transits 372 CHAPTER VII. DETERMINATION OF LONGITUDE. By transportation of chronometers 379 By the electric telegraph 388 By the moon 398 By lunar distances 400 By moon culminations 413 By occul tat ions of stars 423 Prediction of an occultation 435 Graphic process of prediction 443 Computation of longitude 444 Correction for refraction and elevation above sea-level 460 Observations of different weights < 474 CHAPTER VIII. THE ZENITH TKI.ESCOPE. Description of instrument 478 Adjustment of instrument 481 The observing list 484 Directions for observing .... 48" Value of micrometer screw 488 Value of micrometer when level is not known 493 General formulae for latitude 501 The corrections for micrometer, level, and refraction 502 Reduction to the meridian : 504 Combination of individual values of the latitude 507 Ya'ue uf micrometer from latitude observations 509 X CONTENTS. CHAPTER IX. DETERMINATION OF AZIMUTH. PAGE The theodolite .- 521 The signal 523 Selection of stars Method of observing v . 524 Errors of collimation and level 525 Azimuth by a circumpolar star near elongation 526 Correction for diurnal aberration 530 Circumpolar stars at any hour-angle 535 Correction for second differences in the time 537 Conditions favorable to accuracy 542 Azimuth when time is unknown 543 Azimuth determined by transit instrument 546 Circumpolar star at any hour-angle 552 CHAPTER X. PRECESSION NUTATION ABERRATION PROPER MOTION. Secular and periodic changes 559 Mean, apparent, and true place of a star 560 Precession 560 Struve and Peters' constants 563 Bessel and Leverrier's constants 564 Precession in longitude and latitude 564 Precession in right ascension and declination 571 Proper motion 578 Expansion into series 583 S'ar catalogues and mean places of stars .'. 590 Nutation 598 Aberration 603 Reduction to apparent place 609 The fictitious year 617 The Tabula Regiomontance 620 Conversion of mean solar into sidereal time 623 LIST OF TABLES . . 626 INTRODUCTION TO THE METHOD OF LEAST SQUARES. 1. When a quantity is determined by observation, the re- suit can never be regarded otherwise than as an approxima- tion to the true value. If a number of measurements of the same quantity are made with extreme care, no two of the values obtained will probably agree exactly ; at the same time none of them will differ very widely from the true one. There is a limit to the precision of the most refined instru- ment, even when used by the most skilful observer, and therefore the determination of a quantity depending on in- strumental measurement, however carefully made, must be imperfect. It becomes then a problem of great .practical importance to determine how the mass of data resulting from observation shall be combined so as to give the best possible value of the quantity sought. The theory of probabilities furnishes the basis for such an investigation.* 2. Observations are liable to errors of three kinds : First. Constant errors, or those which affect all observa- * The reader is supposed to be familiar with the theory of probability as de- veloped in the ordinary text-books on algebra. See, for instance, Davies Bourdon, edition of 1874, p. 322, or Olney's University Algebra, p. 294. 2 LEAST SQUARES. 3. tions of a given series alike. These may result from a variety of causes, such as errors in the instruments used, personal error of the observer, errors in the constants of re- fraction, parallax, etc., used in the reduction of observations. A proper investigation will generally show the magnitude of such errors, and consequently the necessary corrections at least the more important ones. We shall suppose the data to which our discussion applies freed from such errors, as their investigation does not come within the scope of this subject. Second. Mistakes, such as recording the wrong degree in measuring an angle, or the wrong hour in the clock reading. When such errors are large they are not likely to give much trouble, as their true nature appears at once. When they are small they may prove embarrassing. The present discussion does not apply to them, and we shall suppose that no undis- covered mistakes have been made. Third. Errors which are purely accidental. It is to these that our present investigation applies. At first sight it might seem that such purely accidental errors were entirely outside the sphere of mathematical in- vestigation, but we shall see that they follow a very definite (aw, and that theory is verified in an exceedingly satisfactory manner by observation. 3. We shall assume as the basis of our investigation the following axioms : I. If we have a series of direct measurements of a quantity, all made with equal care, the most probable value of the quantity will be obtained by taking the arithmetical mean of the individual measurements. II. Plus and minus errors will occur with equal frequency. III. Small errors will occur with greater frequency than large ones. 4- DISTRIBUTION OF ERRORS. 3 Various attempts have been made to prove the first of these as a proposition. All such proofs are more or less unsatisfactory, and for elementary purposes it is more ex- pedient to assume its truth at once. The "most probable value" there mentioned must- be understood as the value which most nearly represents the given data, and from the evidence furnished by this series of observations alone it is the best attainable approximation to the true value. The principles are supposed in all cases to be applied to a large number of observations; the larger the number the more closely will the results correspond to the laws assumed. The Law of Distribution of Error. 4. Let x be a quantity whose value is to be determined by observation either directly or indirectly. Let MV M^ MV . . . M m be the individual values obtained. Then regarding M^ as a determination of the unknown quantity x, its error will be (M l x). Similarly, (M, x], (M 3 x), . . . (M m x] will be the errors of the other ob- served values. Let us write (M, ~x} = A, (M, -x) = A... (M m - x) = A m . (i) Let j, = the probability of the occurrence of the error A l ; y^ = the probability of the occurrence of the error A^ y m = the probability of the occurrence of the error //,, Then our second and third axioms assume a law as existing such that the probability of a given error occurring will be LEAST SQUARES. % 4. a function of the magnitude of the error itself. We shall therefore have the equation y = vW, ........ (2) in which A represents any error, and y the probability of its occurring. If this reasoning seems obscure, a different application of the same logic may possibly assist in comprehending it. Suppose we have a large number of tickets in a lottery- wheel. Let a definite proportion of them be numbered i, a certain other proportion respectively 2, 3, etc. Then the probability of drawing any given number from the wheel will be a function of the number itself viz.: Suppose i ticket in every 55 numbered I 2 tickets in every 55 numbered 2 3 tickets in every 55 numbered 3 10 tickets in every 55 numbered 10. Then every ticket would have one of the numbers, i, 2, 3, 4, 5, 6, 7, 8, 9, 10, and The probability of drawing a i would be ^; The probability of drawing a 2 would be ^-; The probability of drawing a 10 would be ^g-. Or if k represents any one of the numbers from i to 10 in- clusive, the probability of drawing a k will be - =/(), or y f(fy 1S tne equation which represents the probability of drawing a k. If now we were ignorant of the relations existing between the successive numbers i, 2, 3, etc., and the relative number 5- CURVE OF PROBABILITY. 5 of tickets so marked, we could, by drawing a sufficiently large number of tickets from the wheel, determine it, at least ap- proximately. In this case we have to determine the proba- bility of a given event occurring, viz., that of drawing a ticket marked with any given number k. In the above prob- lem we have also to discuss the probability of a certain event occurring, viz., that of the appearance of any given error A in any one of our observations taken at random. The Curve of Probability. 5. In the equation y = (A -f- i), y-i = (p(A -\- 2), and / 3 = q>(A -f- 3). If now the limits between which the errors of our series he extend to 10", we see that the probability _y, will differ but little from y 3 , and the sum of all the probabilities j', -f-Js -J- y 3 will differ but little from jy, or 8 Ay = (A?) -\- log (M m - x] ~ --- = d log y(J/, .r) ^ log d(M~- x] d(M~- This equation gives the means of determining x as soon as the form of the function q> is known, and this can best be determined bv considering a particular case. As this func- tion is strictly general, if we have once determined its form in a special case the result will be applicable to all cases. We have assumed as an axiom that in the case of direct measurement of the quantity sought the most probable value will be the arithmetical mean of the individual measurements. This principle will furnish the basis for investigating the form of the function (p. In case of direct measurement we have for the unknown quantity 6. DETERMINATION OF THE LAW OF ERRORS. Q I which may be written (M l -x} + (M,-x)+...+(M m -x)=o. . (8) Equation (6) may be written ("-*) +<".-> Comparing equations (8) and (9), we see that since the quantities (M l x}, (M z x), etc., are independent of each other, these equations can only be satisfied when the coeffi- cients of (M l x), (M^ x), etc., in (9) are respectively equal to the same constant quantity. We have therefore ;- *) _ = . d\ no- m( M r\ = k. (10) (M, - x) d(M, -~xj ~ (M, - x)d(M,-^} d log- cp( M m x) (M m x) d(M m x) Writing for (M x] in general A, we have d log cp( A) = kAdA, and, by integration, log tp(A) = %kA* -\- log c, c being the constant of integration, or (p(A) = ce&* ........ (11) From axiom III. it appears that as A increases this quan- tity must diminish, and this requires the exponent of e to be 10 LEAST SQUARES. 7. negative. As J 2 cannot be negative, it follows that k must be so. Writing therefore \k = If, our equation becomes (12) 7. Let us now consider the constant of integration c. This may be determined by substituting the value of 9>(J) in (4), giving us a special form of the integral known as the gamma function. For the purpose of integrating the expression, place nA = /. Then dA = , and we have As / in this expression is involved only in the quadratic form, we evidently have -*dt = 2e-^dt = 2A (in which we write the integral equal to A for convenience). In the definite integral jf e~ fl dt the value will be the same if we write another symbol instead of t. Therefore Multiplying both members of this equation by f e~ fl dt, we have 8. CONDITION OF MAXIM I'M PROBABILITY. I I In the second member of this equation write v = tu, dv tdu. Then /^-* a d+) ' "" )/ '-*-*! (.3) In this equation the constant h will require further con- sideration ; hut if we assign any arbitrary value, as unity, to h we can readily construct the locus of the equation. It will at once appear that the general form will be that shown on page 5. Condition of Maximum Probability. 8. Substituting in equation 5) the values of etc., from (13), it becomes 12 LEAST SQUARES. 9. From this equation we see that P will increase in value as the exponent of e diminishes, or P will be a maximum when A* -f- A* -{-... -J- A^ is a minimum, thus giving us the im- portant principle The most probable value of the unknown quantity is that which makes the sum of the squares of the residual errors a minimum. From this principle comes the name Method of Least Squares. The Measure of Precision. 9. Let us now consider the constant h. Substituting in equation (3) the value of (A"}, (A") of the value A", etc., and in all + J, + 4 + . . . dm. = 2mcp(A'}A' -f + 2mcp(d'")A r " + etc. From the definition of the mean error e we shall have _ 2mcp(A')A" -f 2tn(p(A"}A"* -f 2w and substituting in J udv = uv J vdu, we find which readily gives e* = -y- a (20) 13- THE MEAN OF THE ERRORS. I/ Substituting the value of h from (19), we have = I.4826r; ) r = .6745, 1 ....... (2I) From these r is readily computed when we know e, and vice versa. 7/fc Mean of the Errors. 13. Another quantity which is much used as an auxiliary for computing r is The Mean of the Errors. This must not be confused with the mean error. It is thus defined : The Mean of the Errors is the arithmetical mean of the differ- ent errors all taken with the positive sign. Let ?/ = the mean of the errors. Then to determine the relation between // and r we proceed in a manner similar to that followed in the previous section. As before, let d', A" , A'" , etc. = the individual errors. '\ J Combining equations (27) and (24), we readily find . = ,.2533 -4=; r .= 0.8453 Ll J . ^ ( r,7 o> ' (28) In these expressions [+ z/] represents the sum of the residuals all taken with the positive sign. These simple formulas (27) and (28) are of great practical value. When the number of observations is not large the values given by (27) will be a little more accurate than those * From what precedes we see that this assumption would be rigorously true if the number of observations were infinite. 22 LEAST SQUARES. 1 6. by (28), but when the number is large (28) will be sufficiently accurate for practical purposes, and the facility with which they are applied is something in their favor. Probable Error of the Sum or Difference of Two or More Observed Quantities. 16. Let us next suppose the unknown quantity x, instead of being directly observed, to be the sum or difference of two or more quantities whose values are obtained by direct measurement ; viz. : Let x y, y in which y, and y^ are independent of each other and whose values are directly observed. Let the individual errors of observation be For 7,, J/, J/', . . . A\ Forj,, 4', 4",... A. The errors of the individual determinations of x will then be (j/ J/), (j," z/;o, . . . (4 m A m ); and if is the mean error of a determination of x, we shall have m? = (J/ j a y 4. (j/' 4") + . . . + (4 * j,*)'. Expanding and making use of the symbol for summation, we' = [J,J,] 2[J,J a ] + [J 9 J 9 ], Let f, and e t = the mean errors of a measurement of y l and /, respectively. Then since, for reasons before explained, I/- PRINCIPLE OF WEIGHTS. 2 j the middle term ([J^]) may be regarded as vanishing in comparison with [A^J an< 3 \_^^^\, we shall have m? me* -f- w^ 2 , or * = V^~+~C (29) In a manner precisely similar we may extend the method to the sum or difference of any number of observed quanti- ties, so that in general if we have x = y^ y^ . . . _ y mt the mean errors being respectively , e,, . . . m , we shall have = *V + < J + V + + *' = nl. (30) Suppose next that we have x = a^y^ a^y^ . . . a m y m , in which a v a v . . . cr m are constants. If, as before, e,, f a , . . . f m are the mean errors of y )> y m , then the mean errors of a t y t , a^y v . . . a m y m will be respectively a v a t e v . . . a m m , and the mean error of x 2 C = vtv]. . (31) Principle of Weights. 17. In the foregoing we have assumed all the observations considered to be equally trustworthy, or, as it is expressed technically, of equal weight. As will readily be seen, we shall frequently have occasion to combine observations of different weights. It is therefore important to ascertain how to treat them, so that each shall have its proper influ- ence in determining the result. Confining our discussion for the present to the case of a directly observed quantity, the most elementary form of the 24 LEAST SQUARES. 1 7- problem will be that where the quantities combined are them- selves the arithmetical means of several observations of the weight unity. Thus, suppose the quantity x to be deter- mined from m such observations ; the most probable value of x' will then be From a second, third, etc., series of m", m'", etc., observa- tions we have respectively _ Combining all these individual values, we have for the most probable value of .*- ' + '" + + The value of x will not be affected if we multiply both nu- merator and denominator of this fraction by any constant a ; viz., _ am' x' -J- am"x" -J- am'"x'" -}-... / \# ;' -j- am" -\- am'" -j- . . . ' ' \3 ) I/. PRINCIPLE OF WEIGHTS. 2$ in which we may regard am', am" , etc., as the respective weights of x', x", etc. a may be integral or fractional. From this we see that the weights are simply relative quan- tities and are in no case to be regarded as absolute. From the foregoing we have the following practical rule : When observations are to be combined to which different weights are to be ascribed, the most probable value of the unknown quantity will be obtained by multiplying each observation by its weight, and dividing the sum of the products by the sum of the weights. It is clear that the difference of weights may result from a variety of causes other than the simple one considered above ; as, for instance, one series of observations may be made with a more accurate instrument than another, or by a more skilled observer. Thus, for example, it may be the case that ten measurements made by one observer will have as much value as twenty made by another. If the weight of an observation of the first series be unity, one of the second would only be entitled to a weight of one half ; or more gen- erally, Letting/ = the weight of an observation of the sepond series, Then 2/ = the weight of an observation of the first series. If then we have a series .* x n x v etc., of observations of the weights A A A etc " an ^ consequently A+A+A + .-- as the most probable value of x, it is evident that, whatever may have been the cause of this difference of weight, we may consider each value x t , x^ etc., as derived from /,,/,, etc., in- dividual observations of the weight unity. Let 2 6 LEAST SQUARES. 1 8- _ t he mean error of an observation of the weight unity ; etc., the mean errors of # x v etc. The whole number of observations being equal to /, + A _|_^ s _j_ . . . [^] observations of the weight unity or of the mean error e, we have for the mean error of*, from (25), nfl (34) 77^ Probable Error when Observations have Different Weights. 18. The mean taken according to weights, as in equation (32) or (32)*, is sometimes called the General Mean. In order to derive the formula for the probable error in this case, let, as before, 8 be the error of the general mean x n \ viz., x x = 6. Then, the notation being as before, we have J i = v t S, A^ = v^ 3, 4, = v t 6, etc. The error A l belongs to x l and therefore appears /, times ; The error A^ belongs to x^ and therefore appears / 2 times; Therefore [/^J] = [pvu] - 2\_pv\d + [p]d\ For the same reason as in previous cases \_pv\ may be dis- regarded as being inappreciable in comparison with the other terms, when we have 1 8. OBSERVATIONS OF DIFFERENT WEIGHTS. Substituting for d the mean error of x from (34), we- have = \pvv\ + 6*. Now, as *! is equivalent to/ : observations of weight, unity, there will be the equivalent of /, errors equal to J, ; and s l being the mean error of x^ we shall have Whence from (33), Similarly, And m being the whole number of quantities, or observa- tions, Xv x t , etc., we have Our equation therefore becomes zc a = [/^ -j- **, from which and from (34), and from (21), (35) (m - I)' J ; in these formulae is the number of individual observations, or quantities, * x v etc., and must not be mistaken for the sum of the weights. It will be evident upon a careful comparison of these ex- pressions with the formulae (27) that we should have reached 28 LEAST SQUARES. IQ. the same result by multiplying each quantity * x etc., by the square root of its weight, and then proceeding exactly as we have previously done with observations of equal weight. We have therefore established the following rule which we may apply in combining observations of different weights : First reduce all observations to a common unit of weight by multiplying each by the square root of its weight, then combine them precisely as if they had originally been of equal weight. For examples of the application of the formulae see pages 515 and 516. General Remarks. 19. We have hitherto considered only those cases where the unknown quantity is derived in the simplest manner from observation, viz., by direct measurement or by the sum or difference of directly measured quantities. Before proceeding to the more complex cases a few general remarks may not be out of place. Equation (13), which represents the law of distribution of error, and on which the subsequent discussion is based, rests upon two hypotheses neither of which is ever fully realized in practice, viz., that the number of observations is infinite, and that they are entirely free from constant errors, i.e., errors which affect all alike. The formulas deduced when applied to the cases which actually arise can give us only approximate results, although they will be the best attainable approximations from the given data. This is particularly to be borne in mind when the number of observations is small. The probable errors in such cases are apt to be entirely illu- sory, and in general are only reliable when the number of observations is large enough to exhibit approximately the law of distribution of error derived from the hypothesis of an infinite series of observations. 20. COMPARISON WITH OBSERVATION. 29 The second hypothesis mentioned above, viz., that con- stant errors do not exist in our data, can never be fully realized, and this fact is often the source of great annoyance and un- certainty in combining observations taken under different conditions. Such errors arise from a variety of causes, some easy to investigate and others not at all so. It is of very frequent occurrence that a result derived from a single series of observations will give a small probable error, and yet differ widely from that derived from a second series to all appear- ances equally good. It sometimes happens that computers who are puzzled by such occurrences attribute the difficulty to faults in the method, the truth being that they are due to the presence of a class of errors with which the method does not profess to deal. The remedy for this difficulty is to vary as much as pos- sible the conditions under which the observations are made, and in a manner calculated to eliminate as far as possible those constant errors which cannot be investigated. Comparison of Theory witJi Observation, 20. The test of theory is its agreement with observed facts. We may in this manner test the truth of the law which we have derived for the distribution of errors. We have the probability that an error falls between the limits a expressed by the equation In accordance with the theory of probabilities, / here is a fraction which expresses the ratio of the number of errors 3 o LEAST SQUARES. 2O. between a to the whole number. If then the number of observations is m, the number of errors between a will be To test the law expressed by this formula we have only to compute the probable error of the series of observations under consideration by (27) or (28), and then h by (19). The value of the integral will then be obtained from Table I., and we shall be in possession of everything necessary for comparing the number of errors between any two limits as indicated by this formula with the number shown by the series of observa- tions. Many such comparisons have been made, and always with satisfactory results, when the number of observations compared has been large. A perfect agreement is of course not to be looked for, as our formula has been derived on the theory of an infinite number of observations ; and further, we are not in possession of the true errors for comparison with the formula, but the residuals instead, which will always differ from the errors unless we are in possession of the absolutely true value of the unknown quantity- As an illustration of the above the following tabular state- ment gives the result of a comparison with theory of the errors of the observed right ascensions of Sirius and Altair. The example is given by Bessel in the Fundament a Astrono- mice. In a series of 470 observations by Brad lev the probable error of a single observation wns found to be r = o".2637, whence h = 1.80865. Therefore for the number of errors less than ".i the argument of Table I. will be t = hA = .180865. With this argument we find for the integral .20188, which multiplied by 470, the entire number of errors, gives 95 as 20. COMPARISON WITH OBSERVATION. the number of errors less than " .\. In a manner similar to this the following results' were found : Between No. of Errors by Theory. No. of Errors by Experience. o" o and o". I 95 94 o.i and o".2 89 88 o '.2 and o .3 78 73 o".3 and o .4 64 58 o".4 and o .5 50 5i o".5 and o' .6 36 36 o".6 and o .7 24 26 o".7 and o' .8 15 14 o".8 and o' .9 9 10 o".g and i' .0 5 7 over i' .0 5 8 This agreement is very satisfactory, but here, as in other similar examples, the larger errors occur a little more frequently than theory would indicate. This is probably due to the fact that (unconsciously, per- haps) every observer will occasionally let an observation pass which is not up to the average standard of accuracy. Small mistakes will sometimes occur, also, which are not of sufficient magnitude to attract attention. A consideration of the matter has led to attempts on the part of Peirce of Harvard College and Stone of England to establish criteria for the rejection of such doubtful observations. On the other hand it has been proposed to overcome the difficulty by determining a system of weights which should give those observations which show large discrepancies less influence than those showing small ones. This branch of the subject, however, is beyond the scope of the present work. It is an exceedingly delicate matter to deal with, and from its nature is probably incapable of a mathematical treatment which shall be entirely satisfactory. Every computer occasionally feels compelled to reject 3 2 LEA S T SQ UA RES. 21. observations. This should always be done with extreme cau- tion. As for the criteria for this purpose hitherto proposed, probably the most that can be said in their favor is that their use insures a uniformity in the matter, thus leaving nothing to the individual caprice of the computer. Indirect Observations. 21. We have now investigated the simplest case of the determination of the unknown quantity by observation, viz., that when the quantity to be determined is measured directly. In the more general form of the problem the unknown quantities are connected with the observed quantities by an equation of the form f(x,y,z, . . .) -M, M being given by observation, and x,y,z, etc., being the un- known quantities. This general form includes the case which we have previously investigated, where there was only one unknown quantity. Each observation furnishes an equation of this form ; therefore a number of observations equal to that of the unknown quantities will completely determine their value. This would leave nothing to be desired if the observations were perfect ; but owing to the errors to which they are liable, the values of x, y, z, etc., will be more reliable the greater the number of observations on which they depend. If now we have four unknown quantities, x, y, 2, and w, four observa- tions will give us four equations from which the values of the unknown quantities may be determined. If we have more than four equations, we may determine values of the unknown quantities by combining any four of them. As the equations depend on observations more or less erroneous, we. should thus obtain a variety of values for x, y, z, and w, all of them probably in error to some extent. 21. IN DIRE C T OBSER VA TIONS. 3 3 The problem then is this : Of all possible systems of values of the unknown quantities, to find that which most accurately represents all of the observations. We shall confine ourselves to the consideration of linear equations; and as the problems in which we shall be more particularly interested do not give rise to equations of more than four unknown quantities, we shall limit our discussion to that number. It will be obvious, however, that it can be extended to any number. Suppose we have the following system of equations : a \ x ~\~ b^y -\- c^z -\- d^w n^, a** + b, y + cjs + d,w = n 3 ; } - - (3 6 ) in which x, y, z, and w are unknown quantities, a, b, c, d, etc., are coefficients given by theory, and ;/ a , etc., are quantities given by observation. If now the data were perfect we should obtain the same values of x, y, 2, and w by combining any four of these equations. Owing, however, to the errors of observation to which #,, n^ etc., are subject, it is not probable that a substitu- tion of the true values of x, y, z, and w (if we knew them) would exactly satisfy anv one of the equations. Let z',, v a , v 3 , etc., be the residuals obtained by substituting in equations (36) for x, y, z, and w their approximate values such that the following equations will be rigorously satisfied : a^x -\- 6, y -f- c^s -\- d^<,v = , a^x -f- b n y -\- c^z -f- d^w = n^ a 3 x -f- b 3 y + c^z -|- d 3 w = , >,. f (37) 54 LEAST SQUARES. 21. Now the most probable values of our unknown quantities will be those which make the sum of the squares of these residuals a minimum, viz., v>* + < + < + etc. = f(x, y, z, w) (38) must be a minimum. In these equations x,y, z, and w are supposed independent, therefore the differential coefficients with reference to each variable must separately be equal to zero to satisfy the conditions of a minimum. That is, d\yv\ _ ~dx~ ' d\vv\ _ dy o, d_\vv\ _ ~dz~ ~ dw Writing out these expressions in full, we have the following : dv, dv^ dv, \ , i dy dz\ , , dy dy dv. . dv* . dv. v, -r 1 + ^ -H + ^s -r 1 1 dw ' dw ' dw = o . (39) x, y, z, and w being independent, we have from -(37), dv, dv, dv, a a) etc. ; dx ~ a " dx dx dv, dv -> h dv, b etc. ; dy = ' * dy - - * dy dv, _ &s = ~ C dv, dz = - c v etc. ; dv, _ d dw dw ~ " dv, dw = d,, etc. ; 21. INDIRECT OBSERVATIONS. 35 by means of which values equations (39) become ^+w+^t;;;=o;[; (4o) Substituting for v lt v etc., their values from (37), we have for the first of these -= O. afi^y -{- a^c^z -(- a^d^w a^n, ajj + a,d,w - a,n, The second of (40) becomes and similarly for the remaining equations. Using Gauss' symbols of summation, we have therefore {ad\x + \ab~\y + \ac~\z + \ad}w = [an] ; -| \ab~\x + \bb\y -j- \bc\z + [ac]x -\- \bc\y + \cc\z + These are called Normal Equations, and the values of the unknown quantities obtained by solving them will be the system of values which makes the sum of the squares of the residuals v^ v etc., a minimum, and therefore the most prob- able system of values. Equations (36) are called Equations oj 36 LEAST SQUARES. 22. Condition, or Observation Equations. An inspection of (41) gives us the following rule for solving a series of equations of condition : Multiply each equation by the coefficient of x in that equation, then add together the resulting equations for a new equation, then multiply each equation by the coefficient of y in that equation, and, as before, form the sum of the resulting equations. Continue the process zvith the coefficients of each of the unknown quantities. The number of resulting Normal Equations will be equal to that of the unknown quantities, and t/ie values of the unknown quanti- ties deduced therefrom will be the most probable T a lues. It must be borne in mind that this process supposes the number of equations of condition to be greater than that of the unknown quantities. If it is less, this process will give us a number of equations equal to that of the Quantities to be determined, but they will be indeterminate none the less than the original equations were, as can be easily shown. Observations of Unequal Weight. 22. In deriving the normal equations from the equations of condition, we have regarded the latter as of equal weight. In the more general case the weights will be unequal. In the equation a,x + b, y -f- cj& -f- dju if we suppose, as in (33), that /, represents the weight of an observation, viz., of that , is the mean error of and e the mean error of an observation of weight unity, we have Multiplying the above equation by Vp^, we have + b, Vp,y + c, Vp,z + d, Vp,w = , Vp,, (42) 23. ARRANGEMENT OF COMPUTATION. 37 an equation in which the mean error of the absolute term #1 ^A i s f a "d tne we ight unity. In the same manner we multiply each equation by the square root of its weight, thus reducing them all to the same unit of weight, when we pro- ceed precisely as before in forming the normal equations. Computation of the Coefficients. 23. The method of forming the normal equations is now fully explained; the work of computation, however, is some- what laborious, especially when the number of equations of condition is large. It will therefore be important to arrange the work so that the numerous multiplications and additions may be performed with the least liability to error, and so that convenient checks may be applied for insuring accuracy in the results. The multiplications may be performed by logarithms, in which case a four-place table will give the necessary degree of precision, or Crelle's multiplication-table may be employed with advantage.* We shall also show how to perform the multiplications by the use of a table of squares. Convenient proof-formulas may be derived as follows: Let the sum of all the coefficients entering into each equation be formed in succession, and represent them by s with the proper subscript. Thus : a, + b, + c, + d, - a = s, . * Dr. A. L. Crelle's " Rechentafeln vvelche alles multipliciren und dividiren mit Zahlen unter Tausend" (Berlin, 1869). 38 LEAST SQUARES. 24. Multiplying these sums by their respective a, b, c, etc., in succession, and adding the products, we shall have the follow- ing equations for checking the accuracy of the coefficients of the normal equations : + [off] + M + M - [an] = [] ; [W] - [to] = [*] - M - [dd} - \dn\ = [off] + [dff] + [fo] + [W] - [to] = M ; M + M + M + [*] - M - This requires the computation of the additional terms [as], \bs~\, . . . and the agreement must come within the limit of error of the computation. These additional terms will be further useful for checking the accuracy of the solution of the normal equations, as will afterwards appear. 24. If it should happen that the coefficients of one unknown quantity in the equations of condition were much larger than those of another, considerable discrepancies might exist in the agreement of the proof-formulae with the sums of the co- efficients. It will generally be necessary practically to limit the computation to a certain number of decimals, when the products of the large quantities may introduce errors into the last places, where the products of the small quantities introduce none. This difficulty is overcome by substituting for the unknown quantities other quantities which will make the coefficients of the same order of magnitude throughout. This is con- veniently accomplished by selecting the largest coefficient with which an unknown quantity is affected and dividing each of the coefficients of this quantity by it. Thus, let , A y, & be the largest coefficients of the quantities x,y, z, w, respectively, which occur in the equations of condition, and let v be the largest of the series of known quantities t , 25. ARRANGEMENT OF COMPUTATION. 39 n 3 , . . . Then we may place the equations of condition in the following form : (*) + (W + ?) + () = ' ; where the unknown quantities are (ax), (Py\ and the values obtained in solving the equations will be in terms of i'. The equations will be made homogeneous by this pro- cess before beginning the work of forming the normal equa- tions. The sums s lt s t , . . . will be most convenient for the purpose to which they are applied, if they are formed from these homogeneous equations. For the kind of problems which we shall have occasion to solve in the following pages there will seldom be a system- atic difference in the magnitudes of the coefficients of the different unknown quantities of importance enough to render this operation necessary. In cases, however, where there is a marked difference in this respect it will be advisable to incur the slight additional labor involved, and in some cases it becomes a matter of considerable importance. 25. The formation of the normal equations with the accom- panying proof-formulas will therefore require the computa- tion of the following quantities: [aa] [ad] [ac] [ad] [an] [as] ; [bb] [be] [bd] [bn] [bs] ; M [cd] [en] []; [dn][ds^ [nn] [ns] . 40 LEAS 7' SQUARES. 25. The latter will be employed for checking the final compu- tation, as will be shown hereafter. As will be seen, there are twenty of these quantities required in a series of four equa- tions. In general the number will be* - i, where n is the number of unknown quantities. Let a sheet of paper be ruled with a number of vertical columns represented by the above formula. In the first horizontal line will be the symbols of the products written in the columns below, viz., [aa], \_ab~\, . . . and in the last line the sums of the products. If the results are correct the proof- equations (44) must be satisfied. The algebraic signs of the various products will demand special attention, as they form a very fruitful source of error. If the application of the proof-formulas is postponed until the conclusion of this part of the computation, the position of an error is often shown at once, since each sum, with the exception of the sum of the squares, is found in two different proof-equations. If two of the proof-form ulae fail to be satisfied, while the others prove true, the error is in the term common to both ; while if only one equation fails to be satis- fied, the error is in the quadratic term. Before proceeding further it is recommended that the reader refer to the example found on page 329. The num- ber of observation equations is twelve, each of which has been multiplied by the square root of its weight. The num- ber of unknown quantities is three, the coefficients of which have no systematic difference in magnitude of sufficient importance to require the application of the process for rendering them homogeneous. The formation of the normal equations is found on page 330. The number of * It is the sum of a series of terms in arithmjtical progression minus i; num- ber of terms = (n -\- 2); first term = i; last term = (-)- 2). 26. ARRANGEMENT OF COMPUTATION. 41 unknown quantities being three, we require by the formula just given fourteen columns. It will be observed that the proof-formulae are perfectly verified, as they should be in this case, no decimal terms having been neglected. Computation of the Coefficients by a Table of Squares. 26. By whatever method the multiplications are performed a table of squares will be found very convenient for the quadratic terms. Terms of the form [ab] may also be com- puted with such a table, as will appear from the following. We have a,b } = {(<*, -f- b^ - a? - b?\\ The quadratic terms [aa], [bb], . . . will be computed in any case, so there will only be required in addition the terms of the form [(a -[- &f~\. In case of four unknown quantities we shall require the following quadratic terms : [(a + Vf\ \(a + ;) 2 ] [(a + d}^ [(a - )] ; (46) [dd~] \(d n [ss] []. The last two will be employed in checking this and the sub- sequent computation. Thus for the case of four unknown quantities we have sixteen terms of the above form, or, in general, < + -H + *) + ,. LEAST SQUARES. 26. The equations having been multiplied by the square roots of their respective weights, and the coefficients made homo- geneous if necessary, the computation will be carried out as shown in the following scheme : [*] [*] [(<* + *)] In order to derive a convenient proof-formula we square both members of equations (43) and add [>] + 3 {[aa] + + + f + []} - (47) For an example of the application of the above method the reader will turn to page 334, where the normal equations are computed from the equations of condition before re- ferred to. This method possesses some advantages over that by direct multiplication: the most important of these is in the fact that the liability to error in algebraic signs is for the most part avoided. Care being taken in forming the sums (a -f- b\ (a -(- c\ etc., no further attention need be given to the algebraic signs until the coefficients of the normal equa- tions are completed. 27. SOLUTION OF NORMAL EQUATIONS. 43 Solution of the Normal Equations. 27. In the solution of the normal equations the work should be arranged so that it may be conveniently reviewed for detecting errors in case such exist, and so that proof-formulae may be applied at the various stages of progress. The order in which the unknown quantities are determined is generally indifferent except in the case where the nature of the problem is such that one or more of them cannot be determined with accuracy from the equations. We may know in advance that we have a case of this kind, or it may be discovered in solving the equations. It will be shown hereafter that the weight of any unknown quantity will be determined by arranging the solution in such a way that this quantity is determined first. The weight will then be represented by its coefficient in the last equation from which the others have been eliminated. If now this coefficient is very small it shows that this quantity cannot be well determined without additional data, and the solution must then be arranged so that the uncertainty in this quantity will have the least effect on the others. In case a preliminary computation shows that the weight of any unknown quantity is very small, the elimination will be repeated in such a way that this quantity is first determined. The values of the others will then be expressed in terms of this one. If then at any time additional data become available for determining this quantity, or if it is known from any other source, the other quantities become known also. As such cases will seldom occur in the problems with which we shall have to deal, it will not be necessary to enter more fully into the matter at present. 28. In the elimination it will be convenient to employ the method of substitution, using a form of notation proposed by 44 LEAST SQUARES. 28. Gauss. In developing the formulas, we shall suppose as before the number of unknown quantities to be four. It will be a simple matter to extend or abridge them in case of a greater or less number. The equations to be solved are \aa~\x + \ab~\y + \_ac\z -f \ad~\w = [an] n \ab\x + \bb\y + \bc\z + \bd~\w = \bn~\ , I \ac\x + \bc\y -f \cc\z + \cd~\w = \cn\ ; f ' \_ad^x+ \bd~\y + \cd]z + \_dd~\w = \_dn\.\ From the first of these we have x= \an\ _ \ab\ M \ad\ , \aa\ [oaf [aa] [aa] which value being substituted in the remaining three equa- tions, we shall have x eliminated. The first of the resulting equations will be and similarly for the remaining two. Let us now write f ~Vl - \bb i] ; \bd} - f^[^] = \bd i] ; } }- (49) 28. SOLUTION OF NORMAL EQUATIONS. 45 and for the coefficients of the second equation, M (49) Similarly for the third, Mrwi _ r^ Our three equations then become \bb \\y + [^ i\z + [^i]w = [^ i \bc \\y + \cc i> + [^ i]w = [ i] ; . . . (50) \bd\\y + [^i> + [^i]w = \dn i In these the same symmetry of notation is preserved as in the normal equations, and it can easily be shown that the terms \bb i], \cc i], and \dd i], which have the quadratic form, will always be positive. From the first of (50) we have r^i] \bd\ \bdi\ y - ~ This is to be substituted in the second and third, and the fol- lowing auxiliary coefficients computed : 46 LEAST SQUARES. 28. which process gives us the following equations : \cc 2\z + \cd 2\w = [en 2] ; \cd2~\z -\- \dd2\w = \dn2\. From the first of these, Substituting this in the second, and writing \dd2\ - [^2] = \dd& \_dn 2] - we have \dd $\w = [dn 3] ; ...... (54) from which W=[ ....... (55) z, y, and x can now readily be found by substituting succes- sively in (53), (51), and (48). The first equation in each of (41), (50), (52), and (54) are called elimination equations, and are here brought together for convenience of reference : \ad\x + \ab~\y + \ac~\z + \_ad\w = [an] ; ^ \bb \\y + \bc i> + \bd i]w = \bn i] ; I \cc 2\s + \cd2~\w - \cn 2] ; [ \ddi\w = This is all that will be strictly necessary in case the weights and probable errors of the unknown quantities are not re- quired. 29. PROOF-FORMULAE. 47 Proof -Formulas. 29. Convenient proof-formulas for checking the accuracy of the successive auxiliary coefficients may be derived from the summation terms [/], [fa], ... of equations (44). Referring to these formulae, let us write Substituting for [fa] and \as\ their values, this expression may be written in the form Therefore, writing for the quantities in the brackets their values, we have [fa i] = \bb i] + \bc i] + \bd i] - \bn i], a formula by which the accuracy of the coefficients in the second member can be tested, and which requires the addi- tional auxiliary quantity \bs i]. Proceeding in a similar manner, we shall require for check- ing the computation at the end of the first stage of the eli- mination the following auxiliary quantities : [fa I] = [fa] - ; !.= - 48 LEAST SQUARES. 30.. when we shall have the following proof-equations : \bs i] = \bb i] + \bc i] + \bd i] - \bn i] ; \ \cs i] = \bc i] + \cc i] + \fd i] - [en i] ; V . (57) [ds i] = \bd\\ + [fl] -f [>tf i] - [dn i] . ) In the same manner we have, for checking the next step in the operation, \cs 2] = [cs i] - I[fo i] ; [ds 2] = [_ds i] - [^ ] [fo i]: , g , and finally, [_ds 3] = [ds 2] - [ 2] ; [^3]= [^3]- [<*3].. ..... (59) The agreement of these two values of [ds 3] must be within the limits of error of the computation, and it furnishes a very accurate control over the accuracy of the computation up to this point. 30. After the values of x t y, z, w have been determined, a most thorough proof of the accuracy of the entire computa- tion is obtained by means of the residuals, v lt v . . . obtained by substituting these values of x, y, z, w in the equations of condition, (37), p. 33, viz. : -f- c^ -\- d^w n l -f cjs + djv n,= v^, (37) 3- PA' OOF FORMULAE. 49 Multiplying these equations by v^ v v 3 , . . . in order, adding, and writing, in accordance with the notation em- ployed, we have [/;] \av\x \bv\y [cv~\z \dv~\w = \vv\\ but by equations 140), [av] = o, [bv] = o, \cv\ = O, \dv] = o. Therefore \irJ\ = \vv\ (60) Now multiply equations (37) by #., n v n 3 . . . in order, and add, viz. : [] j/27/];tr \bn\y \_cn~\s \ciii\w =[nv] = [vv\. (61) By means of this equation \vv\ may also be computed as soon as x, y, z, w become known. But we have _[*] \aV\ \ac\ \ad-\ S M M r M M Let this value be substituted in (61), and write r -. \an\ r -. r -. ^ "M = also write [^ i], [en i], etc., for their values, when we have [ i] \bn \\y \cn \\z \dn \}w \v~J\. Let the same process be carried on for eliminating y, z, and 50 LEAST SQUARES. ^31. iv in succession from this and the resulting equations. We shall have in all the following auxiliary quantities to com- pute : \nn I] = [] - |gj[>] ; \nn 2] = \nn i] - ^j^bn i] ; Om 3] = [2] - [^|][2]; [ 4 ] = [ 3] - f^]^ 3]- Either of the following equations will then give the value of [mi] \an~\x \bn~\y - \cri\z - \_dri\w \vv~\\ -^ \nn i] \bn \\y \cn i\z [dn i~\w = [vv] ; \ \nn 2] \cn2\z \dn 2\w \vv] ; \- (62) [3] - [dn$\w= \_vv~]', ' \nn 4] = [vv] . Only the last of these will generally be used. 31. The value of [nn4] [vv] can be derived from the summation quantities [ns], [ns i], etc., with very little addi- tional labor. We have [ns] = [an] + \bn\ + \cn\ + \dn\ - []. Let us write [ns i] = [ns] - [ ^M, and substitute in this expression for [its'] and [as] their values, when it may be placed in the following form : 32- ARRANGEMENT OF COMPUTATION. 5 1 or what is the same thing, [ns i] = \bn i] + \cn i] + \_dn i] - \nn i]. Proceeding in a similar manner to form in succession the following auxiliary quantities, we have the series of equations by which the accuracy of the quantities \bn i], \cn i], . . . \nn 4] may be verified : 3] - \ns i] \bn i] -f [en i] -|- \dn i] [ i] ; [5 2] = [r 2] -f- [^ 2] [# 2] ; (63) Only the last of these equations will generally be required. Form of Computation. 32. In computing the various auxiliary quantities which occur in the solution of a series of normal equations, the work should be arranged so that it may be carried through from beginning to end in a systematic manner in order to keep a general oversight of the results at the various stages of prog- ress, and to apply conveniently the proof-formulas. This will be the more important the greater the number of unknown quantities. The following scheme will be found to answer these requirements. It will generally be found expedient to make the computa- tion by the use of logarithms, but in some cases the computer may prefer to perform the multiplications and divisions by the aid of Crelle's table. In the following scheme we have 52 LEAST SQUARES. 3 2 - supposed logarithms used. A sheet of paper is first ruled with vertical columns, the number of which is greater by two than that of the unknowa quantities. In the first horizontal line will be written in order the coefficients which are com- bined with a, viz., [aa~], [at], . . . [an], [as], and immediately below these their logarithms. Attention is directed to this line by means of the letter E in the margin, as it is the first of the elimination equations (56), and will be used for deter- mining x after/, z t and w become known. In the third line are the coefficients \bb\ [be], . . . [fa], so placed that the letters combined with b fall in the same verti- cal column with the same letters combined with a, viz., [be] under [ac], \bd] under [ad], etc. In the fourth line of the first column is now written log p-J the value of which, as well as those of all the quan- tities in this column, must be carefully verified, as an error in this factor may not be detected by the proof-formula. The log ^ ^ is now written on the lower edge of a card and added in succession to the logarithms of [aft], [ac], . . . [as], and as each addition is performed the natural number is taken from the logarithmic table and written in the place in- dicated in the scheme. With a little practice the computer will be able to make this addition mentally, and take from the table the corresponding number without writing down this logarithm. Thus we shall have |>#] written under [//]; t .Sac] written under [bc\; 32. ARRANGEMENT OF COMPUTATION. 53 i.r&i i [Mx] log \bb i] [/;<: log [ log [a log \cd 2] [rfrfl] if* 3 log L^ i] <:*a] log [. 2] U. rf] log [rfrf 3] .0^3, log [*] [ftn] log [** i] " IIP. IV. v {i L X'. Prcof-Equati -[!] Practically only those proof-equations which are distinguished by an accent will ordinarily be employed. The lines marked by an E in the margin give the logarithms of the coefficients of the elimination equations. The logarithms marked * must be carefully verified, since aa wor in one of these may escape detection by the proof-equation. For the application to a numerical example see page 331. 54 LEAST SQUARES. 33. and by subtraction, \bbi\\bci\,\bdi\, \bni\\bsi\. These are the coefficients of the second elimination equation, and will be used for determining y after z and w have become known. The I in the margin refers to the proof-formula by which the values of these quantities will be verified. It will not be necessary to proceed farther with this ex- planation, as a reference to the scheme in connection with the formulae for the auxiliary quantities will show clearly the process. The elimination being completed, the quantities [4] and [nsj] are computed as shown in the scheme, the agreement of which with each other and with [vv], obtained by substituting the values of x,y, z, w in the equations of condition, furnishes a most thorough proof of the accuracy of the entire computation. Weights of 'the Most Probable Values of the Unknown Quantities. 33. In case of a single unknown quantity determined by direct observation, the computation of the weight of the arithmetical mean was found to be very simple. In the case under consideration, where the equations to be solved con- tain several unknown quantities, the difficulty is greatly augmented. In our equations of condition we have supposed the quanti- ties observed to be , etc. We have already shown that if the resulting equations of condition are not of equal weight, they may be made so by multiplying each by the square root of its respective weight. We shall therefore in investi- gating the weights of the unknown quantities assume the weight of each observation to be unity. 33- WEIGHTS OF UNKNOWN QUANTITIES. 55 Let p x ,p u ,p z ,p w , be the weights of x,y,z, and w respectively; e x , s y , z , w , their mean errors. Let be the mean error of an observation. As all of our equations are linear, it is evident that if the elimination of the three unknown quantities x, y, and z be completely carried out, the resulting equation will give w as a linear function of ;/ # 2 , ;/ 3 , etc. Similarly, if x, y, and w be eliminated, we shall have z expressed as a linear function of the same quantities, and so of each of the others. We may therefore write x or 1 i -f a^ n , -f ay* 3 + etc.n jiS+S+Sstf (64) w = S 1 n 1 -f- tfjW, -|- ] ; \aV\x + [M] J + \bc\z ,. \_ac\x + \bc\y + M* Let us now assume the following system of equations : \_ad\Q + [>0] 0' -f \ac-\Q" + 0^]0 //7 = P n /7 \bd-\Q" = o ; I \cd-\Q'" = o ; These equations will be possible, as there are four unknown quantities, Q, Q', Q", and Q'", and four equations for determin- ing their values; further, as the equations are of the first de- gree there will only be one system of values for Q, Q', etc. Now let the normal equations be multiplied by Q, Q', Q", and Q'", in their respective orders, and the resulting equations added. Then in consequence of (67) in the resulting equations the coefficients of x,y, and z will be zero, and that of w unity. Therefore we shall have w = \_aii\Q + \bn~\Q' + \cn\Q" + [_dii\Q" r . . (68) We shall now show that Q" = [#tf], and is therefore the reciprocal of the weight of w. Let us expand the quantities contained in the brackets, equation (68), and compare the results with the last of equations (64). We thus find the following values of d it # 2 , etc.: . . . (6 9 ) J 34- WEIGHTS OF UNKNOWN QUANTITIES. 57 Multiplying each of these by its a and then adding, then multiplying each by its b, c\ and d successively and adding, we have by (67) the following equations : = o + 4A + ' = [cd] = o ; f " " [dd~\ i . J Now let each of (69) be multiplied by its d and the results added. Then by (70) we have *A + *A + *A + = [**] = <2'". Q- E. D. (71) The solution of equations (67) therefore determines the weight of w. In a precisely similar manner the weight of each of the unknown quantities may be determined. Thus, to determine the weight of x, we write for the second mem- ber of the first of (67) unity instead of zero, and write zero for the absolute term of each remaining equation. The re- sulting value of Q will be the reciprocal of the weight of jr. This process is simple enough in theory, but its application is laborious, as we must solve equations (67) separately for the weight of each unknown quantity. This does not involve so great an amount of labor as may at first appear, as much of the computation will already have been performed in the solution of the normal equations. It is easy, however, to derive a process which will generally be much more con- venient. It is as follows : 34. In the solution of equations (41) by successive substitu- tions we found for the final equations in w see (56) We shall now show that the coefficient \dd^\ = -^777, and is therefore the weight of w. 58 LEAST SQUARES. 54- For this purpose let us write equations (41) as follows : \ad\x -f- \ab~\y + \ac~\z + \_ad~\w [an] = A ; \ab\x -f \bb\y + [&]* + [&/]w - \bii\ = B ; -w - [en] = C; - ^ = D. Let us now suppose the equations solved by means of the auxiliaries Q, Q', Q", and Q'", determined from (67), when we shall have w = \an\Q + \bn\Q + \cn\Q" + \dn\Q" + AQ + BQ' + CQ" + DQ>". (72) This will now be the same value of w as before obtained, if we make A==C=D = o. Let us now suppose the equations solved, as before, by substitution. Since in this process no new terms in D are introduced, the coefficient of D will not be changed in the final equation for w, and we shall have [dd$\w = \dn 3] -f D + terms in A, B, and C; from which w = f^3 + ^-- -f terms in A, B, and C. Now it is evident that the coefficients of A, B, C, and D must be the same in this equation as in the value before obtained, equation (72). Therefore Q- E. D. We therefore see that we can obtain the values of the un- known quantities from equations (41), and at the same time their respective weights, by arranging the elimination so that 35- WEIGHTS OF UNKNOWN QUANTITIES. 59 each in succession shall come out last. The coefficient of the unknown quantity in the final equation will be its weight. 35. In solving a system of four equations like the above it is best to proceed as follows: Let zv be determined, as above, by substitution in the order x, y, z. We then have w with its weight from [dd^]w = [dn 3]. Equations (56) then give successively z, y, and x. . Let now the elimination be performed in the opposite order, viz., w, z,y, when we have x with its weight from the equa- tion \aa 3> = [an 3], [aa 3] being the weight of x. This value of x must agree with the former value within the limits of error of the computation, thus furnishing a con- venient check to the accuracy of the computation. For the weight of y and z we need not repeat the elimina- tion, but proceed as follows : Let us suppose the elimination performed in the order x, y, w, z. We shall then have the same auxiliary coefficients as in the first case, as far as those indicated by the numerals i and 2, and equations (52) will be the same as before ; but as the elimination will now be performed in the order w, z, instead of z, w, we write them \dd2~\w + \cd2~\z = [dn2] ; \cd2~\W + [cc 2\z = [en 2] . From the first of these, - fr** 2 3 _ \fd2~\ ~ \dd2~\ \dd2~f* 60 LEAST SQUARES. 35. Substituting- this in the second gives us for the coefficient of.* But we have \dd$\ = \dd*\ - From these two equations we find And in a similar manner, We therefore have the following precepts and formulae for computing the weights in the case of four normal equa- tions : First, perform the elimination in the order x, y, s, w, then p w = \_dd-$\ ; -(73) Second, perform the elimination in the order a/, z,y,x, then p x = [a a 3] ; WEIGHTS OF UNKNOWN QUANTITIES. 6l The formulae for the auxiliary coefficients for the second elimination may be derived from those for the first by simply interchanging the letters a and d and b and c. The process is so simple that it will be unnecessary to write them out in full. Other Expressions for the Weights. 36. When the equations have been solved, as already ex- plained, and the various checks applied, so that the computer is convinced that the results obtained are reliable, it may be undesirable to repeat the elimination merely for determining the weights of the first and second unknown quantities. We may derive convenient expressions for computing tne weights in this case, as follows : Suppose four solutions of the equations to be carried through so that each unknown quantity 7 in turn is first deter- mined, the order of the others remaining the same : we should then have each unknown quantity with its weight completely determined, as we have already seen. The solution of the equations for which we have given the complete formulas is in the order d, c, b, a, where we have written the coefficients instead of the unknown quantities. Tf now we substitute the values of w, z, and y in the third, second, and first of equations (56) in order, we have finally the expression for^r, which will be a fraction with the denominator [aa] \bbi\ \cc2\ In the four solutions which we have supposed made, the un- known quantities last determined will be in succession x,x,x, 62 LEAST SQUARES. 36. y, and the denominators of the expressions for their values will be as follows : \_aa\\bb i\ c \_dd2\ e O \aa\\cci\\_dd2\\bb \bb\ a \cc i] a [dd2\ a \a where the subscripts show which unknown quantity is first determined in each solution. As the elimination is performed by successive substitutions, no new factors being introduced, it follows that these expressions are equal to each other re- spectively. It is evident that when the order of the elimination is changed so that a different quantity is first determined, the order of the others remaining the same as before, the values of the auxiliary coefficients \bb i], [cc2], etc., which do not contain the coefficient of this quantity will remain as before. Suppose, as above, the unknown quantities to be determined in the order d, c, b, a. Now let a second solution be made in the order c, d, b, a; then all of the auxiliary coefficients as far as those designated by the numerals i and 2 will remain as before. In a third solution following the order b, d, c, a, the coefficients designated by the numeral i will have the same values as in the first case ; while in a fourth determina- tion in the order a, d, c, b, they will all differ from the first series of values. Thus indicating by the subscripts only those coefficients which have values different from those given by the first elimination, we have the following equations: \aa\ \Mi] [CC2-] \ddj\ = \ad\ \bb i] \dd*\ [0:3]; M \bb I] \cc2\ \ddj\ = \ad\ \cc i] [dd2\ [W 3 ]; \ad\ \bb I] \cc2\ \ddi\ = [W] [cc i] \dd2\ [aa 3]. 36. WEIGHTS OF UNKNOWN QUANTITIES. We already have the weight of w. The weights of z, y, and x are given by these last equations, viz. : - - (74) In applying these formulae the following additional auxiliary coefficients must be computed : j>]-to-.], fc i] a - M (75) In case of three unknown quantities the formulae become - (76) where [^ i] a has the value given above. 64 LEAST SQUARES. 37. An elegant expression for the weights is obtained by making use of the determinant notation. Thus, referring to the normal equations (41), (2'", the reciprocal of the weight of w, given by equations (67), is the same as the value of w obtained from the above equa- tion by making [ari\ = [bti] \cn\ o and \dn\ = i. Therefore writing A for the complete determinant which forms the denominator of the above expression, D'" for the partial determinant formed by dropping the last horizontal line and last vertical column, D" for the partial determi- nant formed by dropping the third horizontal line and third vertical column, and similarly D' and D for the other two, we have A = 4; (77) A number of other forms may be derived for the weights, all of which involve about the same numerical operations as the above. In certain special cases different forms may be more convenient, but for our immediate purposes it will not be necessary to develop the subject further. It may readily be seen fram what precedes that the rela- tive weights of the unknown quantities may be derived, even when the number of observations does not exceed the num- ber of unknown quantities. No probable errors, however, can be determined in this case. 3- MEAN ERRORS OF UNKNOWN QUANTJ7^IES. 65 Mean Errors of the Unknown Quantities. 38. For determining the mean and probable error of an unknown quantity nothing further is required except the ex- pression for the mean error of an observation. It is supposed that the equations of condition have been reduced to the common unit of weight by multiplying each equation when necessary by the square root of its weight. The values of x.y, z, and w, as deduced above, are the most probable values as deduced from the given data. When substituted in the equations of condition the residuals ?/,, z' 2 , v 3 , etc., will not be the true errors unless the derived values x, y, z, and w are absolutely the true values, a condi- tion not likely to be realized. Let (x -\- 6x\ (y -\- dy), (z -)- 8z), (w -j- dzv) be the true values ; A^ A v ^ 3 , ... 4 m , the true errors. We shall then have two systems of equations, as follows : a^x -\- b } y -\- c^z -\- d^v n 1 = a,x + b,y + cj + djv , = = v . [ (78) -*,= -2f,K79) Let us multiply each of equations (78) by its v and add the resulting equations. Then by (40) the coefficients of x, j, z, and w will vanish, giving us the relation before derived, 66 LEAST SQUARES. 38. Proceeding in the same manner with (79), we find \vn\ = [vA] ........ (81) Therefore \vA\ = \vu\ ........ (82) In order to obtain an expression for the sum of the squares of the true errors, viz., \_AA\ in terms of the sum of the squares of the residuals [vv], let us first multiply each of equations (78) by its A and add the resulting equations; secondly, let us multiply each of (79) by its A and add in like manner. The results are as follows : \aA~\x + \bA\y +\cA\z + \_dA\w - \nA\ = - \yA~\ = - [vv~] [aA} (x + 8x) + [_bA] (y+6y) + \cA] (z + fcr) + \dA\ (w + dw) \nA\ = - Subtracting the first of these from the second, we obtain = \vv\ - \aA~\Sx - \bl\Sy - [cA]dz - [d^Sw. (83) If we could now assume dx, 8y, 6s, awrl 8w to vanish, we should obtain, since m? = \4A~\ by definition, This will give us a close approximation to the true value of e when m is large. For a more accurate determination of s we must endeavor to find approximate values of (ad~\$x, \bA~\dy, etc. The true values are beyond our reach, but principles already estab- lished give us a means of approximation. Multiplying each of equations (79) by its a, and adding, we have \_ad\x + \aV\ y + \ac~\z + \_ad~\w - [an] \ _ _ f ,-, -f \_ad\Sx + \ab\Sy + \ac\dz + \ad\1w I 38. MEAN ERRORS OF UNKNOWN QUANTITIES. 6? Comparing this with (41), we see that the first line is equal to zero. Multiplying each equation of (79) by its b and adding, then in a similar manner by its c and d and adding, we have finally \aa~\dx + [aV]dy + [ac\3z + \_ad~\dw = - [>J] n [oft]6* + \bb~\6y + \bc\6z + [&/]tfw = - \bA\ [ [ac\Sx + [bc}Sy -f [><;] [z/z/], and the terms following [w] in (83) must be positive. Let us now perform the indicated multiplication in (86). Confining ourselves to the last equation, since the form is the same for all, we can indicate the result as follows : - \dA-\dw = 4*^,4+ 4V.4 f 4 we shall have -^ (88) 38- MEAN ERRORS OF L\\'KXOW.\' QUANTITIES. "69 With the values of p x , p v , p z , and p w computed by (73), we have finally and the probable errors of ;r, y, z, and w will be obtained by multiplying these respectively by .6745. We have now developed the subject as far as is necessary for our purposes. A complete example of the solution of a series of equations with three unknown quantities, together with the determination of their respective weights and probable errors, will be found in connection with article (191) of this volume. INTERPOLATION. 39. In the Nautical Almanac are given various quantities, such as the right ascension and declination of the sun, moon, and planets, places of fixed stars, etc., which are functions of the time. This is assumed as the independent variable, or argument as it is termed by astronomers. The ephemeris gives a series of values of the function corresponding to equidistant values of the argument. In case of the moon, which moves rapidly, the position is given at intervals of one hour; the place of the sun is given at intervals of twenty -four hours ; while the apparent places of the fixed stars vary so slowly that ten-day intervals are sufficiently small. When any of these quantities are required for a given time, this time will generally fall between two of the dates of the ephe- meris seldom coinciding with one of them ; the required value must then be found by interpolation. Interpolation in general is the process by -which, having given a series of numerical values of any function of a quantity (or argu- ment], the value of the function for any other value of the argu- ment may be deduced without knowing the analytical form of the function. We shall consider the subject more in detail than will be necessary for the simple purpose of using the ephemeris, on account of its importance in other directions. In what follows we shall suppose the values of the function given for equidistant values of the argument, which will always be the case practically. Also the intervals must be 39 INTERPOLATION, GENERAL FORMULA. /I small enough, so that the function will be continuous between consecutive values of the argument. Let w = the interval of the argument. . . . (7--/3W), (T-2w\ (T-w\ (T\ (T+w), (T+2w\ (T+3w), . . . = the values of the argument. The notation for the arguments, functions, and successive differences will be shown by the following scheme : Argu- ist ad 3d 4th 5th ment. Function. Difference. Difference. Difference Difference. Difference. T The notation shows at once where each quantity belongs in the scheme. The first differences are forrned by subtract- ing each function from the quantity immediately following it, the argument being the arithmetical mean of the arguments of the two functions. Similarly the second differences are formed by subtracting each quantity in the column of first differences from the one immediately below it, and so on for the successive orders of differences. It will be observed that the even orders of differences, /", /"', etc., fall in the same horizontal lines with the functions themselves, and have the same arguments, while the odd orders, /', f", etc., fall be- tween those lines. The even differences all have integral argu- ments, and the odd differences fractional arguments. The arithmetical mean of two consecutive differences is indicated by writing it as a function of the intermediate argument. For example : f\ T) = %[f\ T-\w 72 INTEKPOLA TION. 40. 40. Suppose now we set out from the function whose argu- ment is T. Evidently, ) +f"(T Proceeding in this manner, we readily discover the law of the series; viz., the coefficients are those of the binomial formula, and each successive function,/',/", etc., is on the horizontal line drawn under the one which immediately pre- cedes it. Thus we have the general formula nw) =f(T) + nf(T+ $w) + " f"(T+ w) + (90 If we assign integral values to n we obtain the tabular values, viz.,f(T-{- w),f(T-\- 2w\ etc.; but the formula is not used for this purpose, but for interpolating between the tabular values, in which case n is fractional and must be ex- pressed in terms of the interval of argument w as the unit. 41. A more convenient form may be given to this expres- sion (91), as follows : We have (T+ *0 ; 2w) =f iv (T)+ 2f\T+ %w) +f\T) +f vii (T+ 4 ! - INTERPOLATION, GENERAL FORMULA. 73 Substituting these values in (91) and reducing, we readily obtain f(T+ w) =f(T) + nf'(T+ i The law of the series is obvious ; viz., a factor is added to the numerator of each succeeding coefficient alternately after and before the other factors, the last factor of the denomi- nator being the same as the order of differences. The succes- sive differences are taken alternately below and above the horizontal line drawn immediately below the function from which we set out. Formula (92) will be used for interpolating forward. For interpolating backward a better form may be derived by writing ior f\T -\- \w], f'"(T -\- \w], . . . their values in terms Changing ;/ at the same time into n, since the formula is to be used for interpolating backwards,, we readily find f(T- nw} =f(T) - nf(T- *,) + *!L=J->f(T) - i- 2 t 1.2.3.4 74 INTERPOLA TION. 42. 42. In applying (92) and (93) it will be more convenient to write them as follows : =f(T] + n f'(T+&) + F- f"(T) (92)j -rnu] = f(T) - n 5 (93), In (92), and (93), each difference is used to correct the one of the next lower order immediately preceding it, and the quanti- ties to be multiplied will generally be small. In interpolating a value of the function corresponding to a value of the argu- ment between Tand (T + w), we use (92), and set out from f(T}. If the argument is between (T -}- %w) and (T-\-w), we use (93), and set out from f(T-\- w). When the interpolation is carried to any given order of differences, as the fifth, it is a little more accurate to take the arithmetical mean of the last differences, which fall immedi- ately above and below the horizontal line drawn in the vicinity of the required function. Thus the last term of (92), and (93), would bef v (T\ 43. For the quantities tabulated in the American Ephe- meris it will only be necessary to carry the interpolation to second differences ; but for computing ephemerides or tables 44- INTERPOLA TION, EXAMPLE. 75 of any continuous function, much labor is saved by comput- ing the quantity directly for a comparatively few dates and supplying the intermediate values by interpolation. If the function is of such a character that some order of differences, as the third, fourth, or any other, vanishes, this gives exact values for the interpolated quantities, and in fact the process may then be used for computing values of the function for any value whatever of the argument. It is on this principle that "tabulating engines" are constructed. 44. As an example of the application of (90), (92),, and (93),, we take from the American Ephemeris the following values of the moon's right ascension for intervals of 12 hours: 1883, July /=* /' /" /'" /* /" 3 d, o h 5' 4*5' i5 S .'6S 29 39.05 I2 h 6 14 54.73 - 27.08 29 11.97 6.91 4th, o h 6 44 6.70 33.99 + 2.01 28 37.98 4.90 .06 I2 h 7 12 44.68 - 38.89 -f- 1.95 27 59.09 - 2.95 - .01 5th, o h 7 40 43-77 -41-84 + 1.94 27 17.25 i.oi .16 I2 h 8 8 1.02 -42.85 + 1.78 26 34.40 -f .77 - .33 6th, o h 8 34 35.42 - 42.08 + 1.45 25 52.32 +2.22 -.33 I2 h 9 O 27.74 39.86 -+- 1. 12 25 12.46 +3-34 7th, o h 9 25 40.20 - 36.52 24 35-94 i2 h 9 50 16.14 76 INTERPOLA T1OX. 44. Example i. As an example of the application of (92),, let us interpolate the moon's right ascension for 1883, July 5th, 4 1 '- Since the interval of the argument w is here I2 h , we have in this case nw = 4 h , or n -fa = |. Setting out from July 5th, o h , we have ATC/) = .01 f v = .040 /" = + I-94Q Corrected, f iv = -f- 1.900 VU*+--- - 792 f" I.OIO Corrected, f" = - 1.802 ^y- 1 !/'" + =- .801 / ;/ 41.840 Corrected,/" = 42.641 << /" + -=+ 14-214 Corrected,/' =27 m 3i 8 .464 f= a = 1883, July 5th, 4 11 , a = 7 h 49 u '54 8 .26 This value agrees exactly with that found in the American Ephemeris for 1883 (see page 115). 44- INTERPOLATION, EXAMPLE. Example 2. Let us now apply (93), to determine the moon's right ascension, July 5th, 2o h . Here we set out from July 6. As before, n = %,f v (T) = .33. +v 5 f f iv Corrected,/' 11 154 = + 1-450 = + 1.604 . . = + .668 = + -770 = + 1-438 = - -639 = 42.080 4 r Corrected, /'" ^ r Corrected, /" tnl J ^ = - 42.7 J 9 T \ " tr\ Corrected,/' =26 n '2o s .i6o /= = 8^^4^ 1883, July 5th, 20 h a 8 h 25 m 48 9 .70 The algebraic signs of the various corrections are deter- mined without difficulty, as follows: If a horizontal line be drawn in the table of functions and differences (p. 75) in the vicinity of the given argument (in the first of the above examples immediately below 5 d o h ), the successive differences required will fall alternately below and above this line. 78 INTEKPOLA TlOtf. 45- Beginning with/" 1 ' we determine the correction to/" 1 ", which is to be applied so as to bring the value nearer to that imme- diately below the line. In this case/"'" = -j- 1.94; that which immediately follows is + 1.78 ; therefore the correction must be subtracted from 1.94, giving the corrected f iv = 1.90. The value of f" is i.oi ; the value immediately above the line is 2.95. The first must be corrected so as to bring it nearer the latter, giving in this case the corrected f'"= 1.802, and so on for each difference in succession. That is, When the quantity is the horizontal line > the correction so as to bring it in the direction of the one in the same vertical column immediately it. Special Cases. 45. Whenever (92), or (93), can be applied, nothing more will be necessary ; they require, however, a knowledge of the value of the function for several dates both before and after those between which the interpolation is made. It is sometimes necessary to interpolate between values of the function near the beginning or end of the table: as, for in- stance, we might require from the tabular values of the moon's right ascension, given on page 75, to determine the value between the dates July 3d, o h , and 3d, I2 h , or between 7th, o h , and 7th, I2 h . In either of these cases the series of differences terminates with f'\ so the above formulae will only give the value to first differences inclusive. We shall consider the two cases separately. 46. First. For arguments near the beginning of the table. As before, calling the arguments between which it is re- quired to interpolate the function, T and T -f- w, we may apply formula (91), setting out from f(T). 46. INTERPOLA TION, SPECIAL CASES. 79 If the argument for which the value of the function is re- quired is nearer T-\- w than T, it will be a little simpler to set out from T-}- w and interpolate backwards. In this case the formula requires the following modification: Changing n into n, we have f(T- nw)=f(T)-nf'(T + *w) + - f"( T + w) _ n(n +0( + 2) ( + 3) (n + 4)^,- , 5 , 1.2.3.4.5 From the manner of forming the successive functions, we have ' (T+\w-)=f(T-%w)+f"(T) f"(T + iw) = /'"( 7*+tH-/"( r + / - ( T + 2 w) = /iv( 7- + /' ( 7- + f w) = / Substituting these values in the above and reducing, we have RT- nw} = f(T] - nf'(T- &) + ( f"(T) (g^jQ^^-f^l) ( + 2) 1.2.3.4 _ (H - I> ( +0( + 2) (n + 1.2.3.4.5 80 1XTERPOLA TIOX. 46. For greater convenience in the application, (91) and (941 may now be written as follows : S(T+ w) = f(T) + n /'(7-+ia>) + - L f"(T+ w) f(T-nw) =./!T) + | -f'(T~^) + "-=^- \f"(T) . . (95) , Example 3. Required the moon's right ascension, 1883, July 3d, 4 h . Referring to the series of values (Art. 44), we have for this case uw = 4'' ; /. n = . f v = - -06 H -r = = + -044 f _j_ 2.010 Corrected,/' 1 ' = -j- 2.054 Corrected, y" 7 = - 8.279 46- INTERPOLATION, SPECIAL CASES. 8 1 ^ !/-...=+ 4 . 599 /" = -JT-o ;8_ Corrected,/'' = 22.481 ^T^ {/"= + 7494 /' =29-39^.050 Corrected,/' =29 m 46 s .544 {/'...= 9 m 55 s -5i5 /= or = 5"45 m i5 s .68o 1883, July 3d, 4", or = 5 h 55 m uM95 Example 4. Required the moon's right ascension, 1883, July 3d, 8 h . In this case \ve use formula (95),, since the argument is nearer I2 h than o h . n \. - = -06 f"> = + 2.01 Corrected,/'" = + 2.05 Corrected,/"' =+ 8.082 / 7/ = 27.080 Corrected,/" = 23.488 82 INTERPOLA TION. 4/. n I " = + 7-829 Corrected,/' = 29'"3i s .22i { /'...= 9'" 50/407 / = a = 6 h 14'" 54 s . 730 1883, July 3d, 8 h , = 6" 5'" 4 S . 3 2 3 47. Second. Arguments near the end of the table. Proceeding in a manner precisely similar to that of the previous article, we readily obtain the formulae f(T -f nw) = f(T) -_ i .2.3 _ |w) . (97) 3) ( _ 4} (T _^ ( } I .2.3.4.5 I second i ^ tnese a ppli es ^ or interpolating in the 47- INTERPOLATION, SPECIAL CASES. 83 j ^c^eaS j . direction in which the argument . The above may be written as follows : f(T+nw) =f(T) + n {/'(T + Ja,) + ^=-' \f"(T) f"(T- - m) = f(T)+n -f'(T- *,) + - f"(T- w) + " | -/'"( r-f ,) + ^ | /( T - zw) . (9 8,) Example 5. Required the moon's right ascension, 1883, July ;th, 4 h . /" = ~ 36-52 ; /' = 24 35.94; /= Substituting in (98) as above, we find Example 6. Required the moon's right ascension, 1883, July ;th, 8 h . By substituting the numerical values in formula (98^ we find for this case a = 9" 4 2 m 7 8 . 9 7. It will be observed that in the application of formulae (95), (95)i' (98), and (98), the algebraic signs of the various correc- 84 I. \TERPOLATIOX. tions may be determined in a manner entirely similar to that explained in connection with formulas (92), and (93),. (See Art. 44-) Interpolation into the Middle. 48. When the function is to be interpolated for a value of the argument half way between two consecutive dates of the table, this is called interpolation into the middle. For this case either 192), or (93), may be used, but a more convenient formula is obtained as follows. Write in place of n in (92) : yr T + to) = A T) + \f\ T + to) + ^, */"( T) to) Then in (93) let = J, and set out from ( T + of) : S(T+ to) =f(T+ w) - i/' Taking the mean of these equations, obscrvinjr in the result- ing equation that the coefficients of the odd differences, /',/'", etc., vanish, and writing ' )} =-[f(T+ 49- PROOF OF COMPUTATION. 85 \ - $f"(T+*w)+TfoF(T+tw) - roW~ l \ T + i^) + . . . (99) ') }}} (99), Example 7. Let it be required to determine the moon's right ascension, 1883, July 5th, 6 h . We must interpolate into the middle between July 5th, o h , and July 5th, 12''. /* = -f 1.860 - A/" - - -349 /" = - 42.345 Corrected,/'' 42.694 -\\f- .. = + 5-337 Therefore 1883, July 5th, 6 h ,tf=7 h 54 m 27 8 .73 Proof of Computation. 49. The method of differences furnishes a very convenient check on the accuracy of a computation, when, for a series of values of an argument succeeding each other at regular intervals, a series of values of any function have been com- puted. Suppose an erroneous value of one of these quanti- ties, f(T) -(- x, has been obtained, x being the error. The functions, with the respective differences, would then be as follows : (7->+6, r+I, - , 86 INTERPOLATION. 50. Thus the errors in the function has increased to >x in the fourth difference, the greatest deviation being in the horizon- tal line where the erroneous value of the function is found. Suppose, for example, an error of 5 s had been made in computing one of the values of the moon's right ascension given in Art. 44. The scheme of differences would then be as follows: July f=a f f" f" /"*' n. m s. 3d, o" 545 15.68 29 39-05 I2 h 6 14 54-73 27.08 2 9 11.97 - 1.91 4th, O h 644 6.70 - 28.99 - 17-99 28 42.98 - 19.90 I2 h 7 12 49.68 - 48.89 + 3I-95 27 54-09 + 12.05 5th, O h 740 43-77 - 36.84 1 8.06 27 17.25 - 6.01 I2 h 8 8 1.02 - 42.85 26 34-40 6th, O h 834 35-42 We see at once without going further than second differ- ences that the value for July 4th, I2 h , is erroneous. Differential Coefficients. 50. When we have a series of numerical values of a func- tion, corresponding to equidistant values of the argument, we may compute the numerical values of the differential co- efficients from the tabular differences as follows: Either form of the interpolation formula is arranged according to ascending powers of n. The function f(T-\- nw) expanded by Taylor's formula, and the differential coefficients, com- pared with the coefficients of the different powers of n in the above expansions, give at once values of these quantities. 5- DlFl-EREXTIAL COEFFICIENTS. 87 The most rapid convergence, and consequently the best formulae, will be obtained by introducing into formula (92) the arithmetical means of the odd differences situated above and below the horizontal line drawn through the function irom which we set out, using the notation for the arithmeti- cal mean given on page 71. From the manner of forming the differences we readily see /' (T + &>) = f (T) + if"(T) ; f"(T+&) = f'"(T} + if(T). These values being substituted in (92), we readily derive , - / m , 1.2.3 .- (n + 2) (n + i) n (n - i) (n - 2) 1.2.3.4-5 Arranging this according to ascending powers of n, it be- comes 88 INTERPOLA TION. 5 I. Expanding the function by Taylor's formula, dj nW d*f ,M "TtfT 4 i.2.3.4~ t ~^7" 1.2.3.4.5^ Comparing- the coefficients of like powers of n in these two series, we have the following values for the differential co- efficients : 51. Formulas (ioi) will not apply to values of the function near the beginning or end of the table. We obtain formulae for these special cases by comparing formulas (91) and (97), respectively arranged according to ascending powers of n with Taylor's formula. We thus obtain For arguments near the beginning of table : %w) - \f"( T+w) + */"'( T+ f 51- DIFFERENTIAL COEFFICIENTS. For arguments near end of table : J Example 8. Let it be required to compute the numerical values of the differential coefficients of the moon's right da d*oi ascension with respect to the time, -Jr f ~dT i ' ' * or l88 3> July 5th, o h . In substituting the numerical values in (ioi), u>,f',f" . . . must all be expressed in the same unit. It will be convenient to express them in seconds. From the numerical values given on page 75 we have = ~ - 000 458; = - - 000 20 - Therefore - = + .038391 ; = ~ -000972. This value of - may be regarded as the fractional part of 90 INTERPOLATION. 52. a second which the moon's right ascension increases in one second of time at the instant July 5th, o' 1 . In the hourly ephemeris of the moon given in the Nautical Almanac there is given in connection with the moon's right ascension the "difference for one minute," which is simply the value of the differential coefficient multiplied by 60 ; i.e., we may sup- pose the a in -p^ to be expressed in seconds, and the T in minutes. Thus we have for the example above the " differ- ence for one minute" = 2 S .3O346. So in connection with the solar ephemeris there is given the sun's hourly motion in right ascension, which is the value of -^multiplied by 60x60. The hourly motion in declination is expressed in seconds of arc. . 52. By means of these differential coefficients as given in the ephemeris, the second differences are taken into account in the interpolation in a very simple manner*, for we have to second differences inclusive W = f ' (T + ^ ~ * ///(r) ; The difference of these expressions is and f(T+nw} = Thus we have only to correct the value of the first differen- tial coefficient by adding to it algebraically the product of 53- DIFFERENTIAL COEFFICIENTS. 9! the difference of two consecutive values by one half the in- terval n. We then use the corrected differential coefficient, as we should do if the first differences were constant. Example 9. Required the sun's right ascension and decli- nation, 1883, July 4th, 4 h , Bethlehem mean time. As the longitude of Bethlehem from Washington is 6 m 4O s .2, the corresponding Washington time is 3 h 53 m i9 s .8 = July 4th, 3 h .8888 = July 4.162. From the solar ephemeris for the meridian of Washington we then find : Date. Ci. Hourly Motion. $. Hourly Motion. July 4.0 6 h 53 ra 33 8 .7 9 io s . 3 o7 2252 / 5i // .i - i3".i 9 July 5.0 6 h 57 m 4i s .02 io s .294 22 47' 22".7 - I4".i8 d*a n w -^ . - = .013 x .162 = .00105 Corrected hourly motion = io s -3o6 10.306 X 3 h .889 = 40 s .o8 Required a = &$4 m i3*.Sf. w* -^ . * = .99 X .162 as .080 Corrected hourly motion I3 S .27 13.27 X 3 h .8S9 = 51" 61 Required d = 22 $i' $9".$. 53. If values of the differential coefficients are required for values of the argument between the dates of the table, we may derive the necessary formulas by differentiating the function developed by Taylor's formula (100), viz.: (103) ', f'(T+\w}=A, f"(T) = A". In which A" will be the difference between two consecutive values of A. 54- THE EPHEMERIS. 95 Then 3" -* 22 6 and formula (92), becomes D' D -f- (-^ - g ^"]. Let \A r ^ 2f"J = [z/] = corrected tabular difference ; (2 - /'^j [fi] - Then we may assume - g <2"J = [?] with sufficient accuracy, (105) in which Q" is the difference between two consecutive val- ues of Q. (Q and A are inverse functions one of the other, but the algebraic sign of the correction need give no trouble.) It will be a little more accurate if we take for Q" the arithmetical mean of the differences between Q and both the preceding and following values found in the table. Example 11. Required the distance between the centre of the moon and Fomalhaut, 1883, July 2Oth, ig h 20 5 8 , Gh. M. T. From the ephemeris, July 20th, 15" Q July 20th, i8 h D 32 41' 20" (5 = ^536 Q,, _ _]_ 2II = -4747 July 2oth, 21" Z>3i4i' o" Q = . 4995 ^' = + ^3 Then / = i h 20 m 5 s = i" 3347 [Q] = -4683 J' = o 27' 14". 5 Mean Q" = 230 log t 3-6817 D< = 32 14' 5"- 5 If we had neglected the second differences in this example we should have found A' = o 26' 51", which can only be 96 IN TERP OLA 7 'ION. 55- considered a rough approximation. If the interpolation be extended to third differences, we find A' = 27' 13". 8. This differs from the first value by a quantity which will be of very little importance in practical cases. To Find the Greenwich Time Corresponding to a Given Lunar Distance. 55. First. We may interpolate the time directly from the ephemeris, neglecting- the second differences; then with the time so found as a first approximation we deduce the cor- rected proportional logarithm [<2], and repeat the computa- tion. / being the required quantity, either (104) or (104), give the first approximation, viz., log / = log A' -f PLA, ..... (106) or PLt = PL A' - PL A (106), Then with this value of / we determine the corrected pro- portional logarithm [<2] by (105), and repeat the computation. Example 12. 1883, July 2oth: determine the Gh. M. T. when the distance between the moon's centre and Fomal- haut was 32 14' 5".5. 4536 We find from the ephemeris that on July aoth, i8 h D = 32 41' 20" PL .4747 Given value of D 1 = 32 14' 5". 5 -4995 log A' 3.2134 Therefore A' = 27' 14". 5 PLA = .4747 log t = 3 6881 Approximated = i 1 ' 2i m i6 9 By (105), - _5 being the latitude of the place. Transformation of Co-ordinates. 64. PROBLEM I. Having given the altitude and azimuth of any star, to find the corresponding declination and hour-angle. Let us refer the star's position to a system of rectangular co-ordinates in which the horizon shall be the plane of XY, the positive axis of X being directed to the south point, the positive axis of F to the west point, and the positive axis of Z to the zenith. Then will x, y, z = the rectangular co-ordinates of the star; 4, //, a = the polar co-ordinates of the star; 'A being the distance or radius vector. We then have* x = A cos h cos a; \ y = A cos h sin a\ > ( IIQ ) z = A sin h. J *See Davies' Analytical Geometry, edition cf 1869, p. 302; or any other work on analytical geometry of three dimensions. IOS PRACTICAL ASTRONOMY. 64. Let the star now be referred to the equator as the funda- mental plane, the positive axis of X being directed to the south point of the equator, the positive axis of Fto the west point, and the positive axis of Z to the north pole. Let now x' , y', z' be the rectangular co-ordinates; A, d, t be the polar co-ordinates. We then have x' = A cos S cos /; \ y' A cos $ sin /; v (i 1 1) z' = A sin 3. } The problem now requires these values of x ' , y r , and z' to be expressed in terms of x, j, and z. We observe that the axes of Fare the same in both systems; that the axes of X and Z make the angle 90 q> with those of X' and Z'. We therefore require the formulas for transformation of co- ordinates from one rectangular system to another having the same origin, viz.: x' = x cos (90 9?) -f z sin (90 ) + z cos (90 cp)\ or x' = x sin cp -f- z cos cp; \ y' = y; [ . (112) z' x cos q> -j- z sin (p. ) Substituting in (112) the values of x,y, and z from (no), and oix',y', and z' from (in), dropping at the same time the factor A which is common to every term, we have cos 3 cos t = cos h cos a sin cp -f- sin h cos -(- sin h sin (p. } 64. TRANSFORMATION OF CO-ORDINATES. 1 09 These equations express the required relation, but they are not in convenient form for logarithmic computation; be- sides, the required quantities d and t are given in terms of their sines and cosines. It is always best, when practicable, to determine an angle in terms of its tangent. The tangent varies rapidly for all angles great or small, and consequently if a small error from any cause exists in the tangent it will have but little effect on the value of the angle. On the other hand, if the value of the angle is near 90 or 270 and is given in terms of its sine, this function will vary slowly with the angle, and a small error in the sine will produce a large error in the angle. The same is true of the cosine for angles near o or 1 80. If the angle is near 90 or 270 it may be determined with accuracy from its cosine, or if near o or 180 it may be accurately determined from its sine. In any case it can be determined with accuracy from its tangent. For the purpose of effecting the required transformation in (113), let us introduce the auxiliary equations sin h n cos A'; ) (HA) cos h cos a = n sin N. ) This will be possible, for we have the two arbitrary quan- tities n and N, and the two equations (114) for determining them. Substituting these values in (113), we have cos S cos t = n sin N sin q> 4- cos N cos

N)- t \ cos S sin t cos h sin a; I (115) sin S = n sin A 7 " cos

an d when tan is ; third, for the species of t let us examine the equation cos 8 sin t = cos h sin a. Cos & and cos h will always be -|- therefore the species of t will be the same as that of a. As an example of the application of these formulas, take the following: Latitude of Sayre Observatory = (p = 40 36' 23".9; Sun's altitude = h = 47 15' i8".3; Azimuth = a = 80 23' 4".47; Required tf and t. The computation is as follows :

N} = 9.7929304 tan t = .0252996 cos t = 9.8364670 cos = 9.8364670 tan S = 9.6293974 cos d = 9.9637894 9.8002564 sin N _ cos h cos a ~( IT]v) = 9-2542495 (proof) cog s CQS - f = 9-2542496 112 PRACTICAL ASTRONOMY. 65. 65. PROBLEM II. Having given the declination and hour- angle of any star, to determine the altitude and azimuth. This is the converse of the preceding problem. In this case we require the values of x, y, z in terms of the values of x' , y', z' . Our formulas (112) for transformation then become x = x' sin q> z' cos <>; \ y=y f ; [ . . . . (120) z = x' cos cp -\- z sin q> . ) Substituting in these the values of x, y, z, x',_y', z' , from (i 10) and (i 1 1), dropping at the same time the common factor J, we have cos h cos a = cos 8 cos /sin q> sin 8 cos

, 90 S =1 p, and 90 h = z. The angles are t, i So a, and q, the angle at the star, called the parallactic angle. When any three of these quan- tities are given, the determination of any other part is merely a question of trigonometry. 1 1 4 PR A C TIC A L AS '2 'A' ONOMY COROLLARY. To find the hour-angle oj a star when in the horizon, or at the time of rising or setting. When the star is in the horizon the altitude, h, is zero, and the last of equations (121) becomes cos S cos t cos fp -f- sin 8 sin q> = o, sin 3 sin (p or cos*= ---^V 7^= - tan tf tan ?>. . . (122) From this equation we may determine / ; but, as before re- marked, it is better to determine the angle from its tangent. For this purpose first add both members of (122) to unity, then subtract both members from unity, and we have cos 8 cos q> sin 3 sin cp i -4- cos t -- cos d cos (p cos d cos cp -4- sin 3 sin cp I cos /= - ^-f- -- -; cos o cos cp COS (cp + (?) or 2 cos \t - ^-- ~~ ; cos tp cos 3 ' . cos Q S) 2 sm *' = ' Dividing the second of these by the first and extracting the square root, At the time of rising the lower sign will be used ; at the time of setting, the upper. This formula may be used to compute the time of sunrise and sunset at any place whose latitude is known. For example, let it be required to com- pute the apparent time of sunrise at Bethlehem on the morn- ing of July 4th, 1 88 1. 67. ANGULAR DISTANCE BETWEEN TWO STARS. 11$ From the Nautical Almanac, page 329, we find for the sun's decimation = - 9 23' i6".7 /T = - 89 42' i".8g ^= 89 52' 55". 5 tanA'= .7677470 0560^= .0062374 cos^'-Jr) = 7.7182360 cos d = 7.3134726 Il8 PRACTICAL ASTRONOMY. 6/. Applying formulas (IV), to the solution of the same prob- lem, we have the following: a' a = 85 52' 37". 35 cos = 8.8567115 cos = 8.8567115 6 = 22 50' 2l"-9 COt = .3755415 COS = 9.9645407 JV = 9 41' 14". 8 tan = 9.2322530 8.8212522 d' = 9 23' i6".7 Ar+S' = o 17' 58".! B = 66 48' 40". 8 d= 8 9 5 2'55".5 tan (a' a) = 1.1421632 sin N = 9.2260154 cos (IV + d') = 9.9999940 cot (AT + 5') = 2.2817621 factor = 9.2260214 tan B = 0.3681846 cos .#=: 9.5952317 cos = 9.5952317 tan d = 2.6865304 sin = 9.9999991 9.5952308 proof 9.2260214 sin N = 9>226 214 CHAPTER II. PARALLAX. REFRACTION. DIP OF THE HORIZON. 68. The same star may be observed from points on the surface of the earth separated from each other by several thousand miles. If the distance to the star is so great that the diameter of the earth is inappreciable in comparison, it will appear in the same part of the heavens from whatever part of the earth it is seen. If, however, the diameter of the earth bears an appreciable ratio to the distance of the object, then when the observer's position changes there will be an apparent change in the place of the star. This difference in position is called parallax. It is customary in dealing with bodies which have an ap- preciable parallax to reduce all positions to the earth's cen- tre. Thus the places of the sun, moon, and planets, which we find given in the ephemeris, are the places as they would appear to an observer at the centre of the earth. This which we are considering is \\iQdiurnalparallax. With the subject of annual parallax, which depends upon the position of the earth in its orbit, we have at present nothing to do. It may be remarked that on account of the great distances of the fixed stars their diurnal parallax is in all cases inappreciable. It is only necessary to consider it in connection with the bodies of the solar system. I2O PRACTICAL ASTRONOMY. Definitions. 69. THE GEOCENTRIC POSITION of a body is its position as seen from the earth's centre. THE APPARENT* or OBSERVED POSITION is its place as seen from a point on the earth's surface. THE PARALLAX is the difference between the geocentric and the observed place. It may also be defined as the angle at the bod)' formed by two lines drawn to the centre of the earth and the place of observation respectively. THE HORIZONTAL PARALLAX is the parallax when the star is seen in the horizon. THE EQUATORIAL HORIZONTAL PARALLAX is the parallax when seen in the horizon from a point on the eartlis equator. It may also be defined as the angle at the body subtended by the equatorial radius of the earth. 70. PROBLEM I. To find the equatorial horizontal parallax of a star at a given distance from the earth's centre. Let n = the equatorial horizontal parallax = PSC; a = the equatorial radius of the earth = PC; A star's distance from the earth's centre = SC. Then from the figure we have SHITTY ~; .... . ; . . (125) *The terms apparent place and true place are to be considered simply as relative terms. When dealing with parallax we speak of the true place as the place when corrected for parallax. So when speaking of refraction the appar- ent place is the place affected by refraction, and the true place is the place cor- rected for refraction, but it may still require corrections for parallax and a va- riety of other things. When dealing with the places of the fixed stars we use the term apparent place in a still different sense, as we shall see hereafter. PARALLAX. 121 s being the place of the star,/ a point on the surface of the earth, and c being the centre. For astronomical purposes the mean distance of the earth from the sun is regarded as the unit of measure. Then for the sun we have A = i; sin n = a , (126) 71. PROBLEM II. To find the parallax of a star at any zenitJi distance, the earth being regarded as a sphere. In the figure, s represents the place of the star, z the zenith, E the centre of the earth,/ a point on the surface. Let z' = the observed zenith distance; z = geocentric zenith dis- tance; / = parallax = PSE; a = radius of earth = PE\ A = distance of star = SE. From the triangle SEP we have A : a sin z' : sin p. FIG. 6. From which or, from (125), sin / = j- sin z'; sin = sin it sin (127) (128) / and it will generally be very small ; hence for most pur- poses we may write p = n sin z' . (129) 122 PRACTICAL ASTRONOMY. 72. The foregoing solution is only an approximation, the earth not being a sphere as we have there regarded it. For many purposes this is sufficiently exact, while for others, particu- larly where the moon is considered, it is not so. A more rigorous solution requires us to consider the true form of the earth. Form and Dimensions of the Earth. 72. The earth is in form approximately an ellipsoid of rev- olution, the deviations from the exact geometrical figure be- ing so small as to be inappreciable for our purposes. The dimensions of the ellipsoid as given by Bessel are as follows: Equatorial radius A = 3962.8025 miles; Polar radius B = 3949.5557 miles; Eccentricity of meridian e = .08169683; log e 8.9122052. Many other determinations of these quantities have been made, differing more or less from the above, but these are still in more general use than any others. Definitions. 73. THE GEOGRAPHICAL LATITUDE of a point on the earth's surface is the angle made with the plane of the equator by a normal to the surface at this point. THE GEOCENTRIC LATITUDE is the angle formed with the plane of the equator by a line joining the point with the earth's centre. THE ASTRONOMICAL LATITUDE is the angle formed with the plane of the equator by a plumb-line at the given point. 73- THE DEDUCTION OF THE LATITUDE. I2 3 If the earth were a true ellipsoid and perfectly homoge- neous, the geographical and astronomical latitude would always be the same. Practically, however, the plumb-line frequently deviates from the normal by very appreciable amounts. This deviation is always small, but in mountainous countries, as the Alps and Caucasus, deviations have been observed as great as 29". Unless otherwise stated, when speaking of latitude the astronomical latitude is to be under- stood. We shall also assume for present purposes that it coincides in value with the geographical latitude. Let the annexed figure z represent a section cut from the earth's surface by a plane passing through its axis. This section will be an ellipse. Let K be any point on the surf ace, E) P and P the north and south poles respectively. Then HH' will represent the horizon of the point K. FIG. 7 . Let p = CK = radius of the earth for latitude KO'E'\ (p KO'E' = geographical latitude of point K; ' may be readily computed for any given value of g>') in the form of a series. For this purpose we make use of Moivre's formulae, viz.:* * As some readers may not be familiar with these very useful formulae, we give their derivation. Developing = e* by Maclaurin's formula, we have r"= i+--f -f -4- ^ . etc.;. . . (a) ^ I~l.t^ 1.2.3^1.2.3.4 also, cos .r = i - + - a . etc - ....... (*) sin JT = JT ---- 1 -- . etc 1.2.3 1.2.3.4.5 74- THE REDUCTION OF THE LATITUDE. 12$ 2 COS X = 2 V i sin x = V^l tan * = (135) Writing tan q>' = p tan ?> where / = ~j. substituting for tan -f- 2 I/ i . \q* sin 4^ -f- 2 V i -J^ 3 sin 6^>, etc., or q>' (p = q sin 2 i".i6 sin 49?. . (138) To Determine p. 75. x and y being the co-ordinates of the point K, we have ..... (139) (130) tan q>' = = - tan cp ...... (134) Combining (130) and (134), eliminating j, we have or x\\ + tan ^ tan . . ^^ = ^ V cos ^' cos (>' - ^)' l The computation of p from (140) is very simple, but it may be rendered much more so by developing p, or log p 128 PRACTICAL ASTRONOMY. ;6. into a series. For this purpose we shall regard A the equatorial radius as unity, when we have secy = I + ^ tan V = cos>4-| 4 sin> p ~ i + tan

' ~ & ' ~~J? i + - tan 3 (p cos" cp -f sm> B' B* Let us write . - = i - g ; - -, = i - e\ we have Taking the logarithms of both members, 2 log p = log (i g' 1 sinV) log (i e* sin 2 ^)- Developing the second member by the logarithmic formula, 2 lo + \M(? g'} sin> (e* g*) sin>, etc. Substituting for e, g, and M their values, J/being the mod- ulus of the common system of logarithms = .43429448, we readily find log p .00143968 sin j (p .00001438 sin 4

= - -[**- 2 + ~~\: sn = - ^- Therefore log p = [^ + |/? + -| r + etc.]; But 4T 1 -f- = ^* v=r ^ + e- 2 * i/:r ^ 2 cos X x* + = ^4*^- i _|_ ^- 4*^^ 2 cos X* 4- = ^^^=^ -f*- ^^^ = 2 COS 130 PRACTICAL ASTRONOMY. /8. Substituting these values with the numerical values of a, ft, and y as given in (141), and we find log p 9.9992747 -f- .0007271 cos 2cp .0000018 cos 4^. (142) 77. We therefore have for computing (tp tp') and log p, (V) q> q>' [2.839258] sin log p = 9.999 2747 + [6.861594] cos 2(f> + [4.25527] c6s 4 (H3) in #' = // sin s sin a p sin (cptp 1 ) sin (0 #); !- = ^ cosz pcos( cp') cos a-, /sm z' sin (a' a) = p sin TT sin (^ q>') sin a. p sin n sin (cp qj\ sm z Then (149) become /sin / cos (' a) = sin # (i m cos #); /sin z' sin (V a) = m sin * sin #; and by division, m sin # tan (a a] = ' i m cos a (150) and (151) determine the parallax in azimuth. To determine (z' z) we proceed as follows : Multiply the first of (149) by cos \(a' a), and the second by sin %(a! a) ; add, and divide the result by cos \(a! a). A simple reduction then gives .^ cos Ua' -4- a) /sin z = sm z p sm TI sin (cp (p) 7^-7 (. (152) COS '2\# 1) Let us write , cos \(a! -f- a) / ,\ .. cos %(a! + a) or tan y = tan (g> tp ) - , . , v . . (153) 134 PRACTICAL ASTRONOMY. (152) then becomes f sin z =. sin z p sin 7t cos (9? q>'} . Writing now n = p- CQS ^ , . (i55) and we have /sin (z' z] n sin (z y)\ /cos (z' z) = i n cos (z y)\ n sin (s v\ ta "( g '-') = .-. cos (,-Vy ' ' ' (I56) (155) and (156) now determine the parallax in zenith distance, and the problem is completely solved. 80. Formulae (150), (151), (155), and (156) may be placed in a form more convenient for logarithmic computation, as fol- lows : Write p sin 7t sin (a? a/) cos a . sin 5 = m cos a = - ~- ~- ( ! 57) 80. PARALLAX IN AZIMUTH AND ZENITH DISTANCE. 135 Then sin 3 tan a = tan a -^^ = tan a sin 3- cos 3 & 2 sn 3 cos + sn' sin 5 tan a ^ sin 3)* sin 5 -cosj3-f sinj3 = tan tan cos sn cos sn tan ^3- therefore tan (a' a) = tan a tan 5 tan (45 -f- 13). . (158) In a similar manner writing p sin TT cos (a) ') cos (^ v) sin3' = cos(^-r)-- cosy "' we find sin 5' tan (g ^) = tan 3' tan (45 + ^ tan (* - y). (160) For computing y we have tan ^ 136 PRACTICAL ASTRONOMY. 80. Therefore tan y = tan (9> 9') C cos a sin a tan ^(a' a)~\. By Maclaurin's formula we have tan x = x -\- ^x*, etc. Therefore if we neglect terms of the third and higher orders in y, (tp (p f ) [cos a sin a \(a! #)]. . (161) , m sin a From tan (a a) -- ' i m cos a we have, by neglecting terms of the higher orders, . p sin n(

') sin a (a' a) m sin a -- : - - --- . sin z Substituting this in (161), we have P sin n sinW^ 90* sin i" y=(g>- (VI), tan (z' z) = tan \_z (q> (p'}~\ tan 3' tan (45+ ^-3'). ) As an example of the application of (VI) let us take the following: 1881, July 4th, 9 h , mean Bethlehem time, the geocentric position of the moon was as follows: Zenith distance = 2 = 65 40' 46".$; Azimuth = a = 48 19' 49".8. Required the parallax in azimuth and zenith distance for Bethlehem. We have found for the latitude of Bethlehem (Art. 77) cp q>' =. 1 1' 22 // .I9; log p = 9.9993875- From the Nautical Almanac, page 113, n = 56' 2o".4. Our computation is now as follows: 1 38 PRACTICAL ASTRONOMY. 81. a = 48 19' 49". 8

') = 2.8339053 logy 2.6566178 tan a = 0.0506037 tan 5 = 5.5964625 tan (45 -f- iS) = .0000171 tan (a 1 a) = 5.6470833 (a 1 - a) = 9". 152 tan 3' = 7.8307540 tan (45 + IS') = 0029412 tan (z y) = .3423734 tan (z' z) = 8.1760686 (z' - z) =51' 33". 58 We thus have for the apparent place Take the following example of application of (VI), a' = 48 19' 5g".o z' = 66 32' 20". I moon 45 "- 5 9.9993875 .2138035 9 - 79959 3 Equatorialhorizontalpar- ) _ ,, ,/ g allax, f (p cp' = n' 22". 19 5' = 35' 24". 29 45 + $y = 45 17' 42". 15 -gm 44' 3". 13 tan (z' z) = 8.1077169 82. PARALLAX IN AZIMUTH AND ZENITH DISTANCE. I 39 Case Second. 82. To compute the parallax in azimuth and zenith dis- tance, having given the observed azimuth and zenith distance. To obtain the expression for (2' z) we multiply the first of (154) by cos z' and the second by sin z', and subtract. We thus have . .. (I64) For (a' a) we multiply the first of (148) by sin a', the second by cos a' and subtract, recollecting that cos (a a)=i, sin (a a) = o. We thus find p sin n sin (cp cp'} sin a' sin (V a) = - . . . . (165) sm z We thus have for the parallax in zenith distance and azi- muth, having given the apparent zenith distance and azimuth, y = (cp - (p'} cos a; 1 . p sin n cos ((p (p') sin (z' y) \ ~^^V~ " ; \ (vii) p sin 7t sin (cp cp'} sin a' sin (a a) = --- = - . sin z } To compute y we may substitute a' for a without appre- ciable error. To compute (a! a) we must first obtain z by applying the correction (*' z] to the observed zenith distance. In the meridian, a = a'= o, whence y = cp cp' , a' a = o, and (VII) become sin 0' - z] = p sin n sin \_z' (cp ?/)] . (VII), For all bodies except the moon (VII) may be greatly sim- plified, as follows: 140 PRACTICAL ASTRONOMY. 82. (z' z\ (a 1 a], and n being very small, we may write the arcs in place of their sines. (

q>' = n' 22". 19 log (

') = 2.8339053 cos (p qJ = n' 22". 19 logp = 9.9993875 sin TT = 8.2138035 sin [>' - (0j - - y = rectangular co-ordinates ) of the observer's position referred to the eai th's p, cp f , 6 = polar co-ordinates centre. Here p is, as before, the line joining the observer's position with the centre of the earth, and (170) ^' sin 6' = A sin d p sin ?/. ) 144 PRACTICAL ASTRONOMY. 84. As before, let us divide through by /^, and write / = -- ; sin n = . J A A Then 8' cos a' =. cos d cos a p sin n cos cp' cos <9 ; (170 /cos tf' cos ar' = cos 8 cos a- p sin ?r cos " \ , . - r- cos ^(a' or) The last of (171) is \( 1 77) /sin d' sin 6 p sin n sin 9'. Let us write tan o/ cos \(a' of] tan v = 1./ / i x s^ .... (178) cos [(<*' -j- or) 0] Then (177) become J sin ' = sin d p sin zr sin sin n sin - 5) - Si " " 7 r) = ^n (5 - r ) tan 3' tan (45 + W (183) Equations (175), (176), (178), (182), and (183) give the com- plete solution of the problem. We thus have for computing the parallax in right ascen- sion and declination, having given the geocentric right ascen- sion and declination, the following formulae: 84- PARALLAX IN RT. ASCENSION AND DECLINATION. 147 . _ p sin n cos (p' cos (0 a) COS 8 tan (a a') = tan (9 a) tan 3 tan (45 -f 3); tan sin TT sin cp' cos (v #) sin x = = -; sin y tan (d d') = tan (y 6) tan $' tan (45 + ^). J In the meridian, a = a' = 6. Therefore y = cp', and the above become sin B 7 = p sin TT cos (cp' #); ) cnr\ tan (tf tf') = tan (cp r 6) tan 3' tan (45 + A3'). ) ^ ^ l Application of Formula (IX). Required the parallax of the moon in right ascension and declination, 1881, July 4th, g h , Bethlehem mean time, as seen from Bethlehem. Converting g h mean time into sidereal time by the method to be explained hereafter (p. 170), we have Bethlehem sidereal time = Q = I5 h 52 5o'.2 From the Nautical Almanac, p. 114, we find a = i2 h 57 io".56 S = - 11 3'48"-4 Astronomical latitude of Bethlehem = cp 40 36' 23". 9 q> q> = n' 22". 2 Geocentric latitude of Bethlehem = cp' = 40 25' i' .7 Nautical Almanac, p. 113, equatorial horizontal parallax = it = 56' 20". 4 6 - a = 2 h 55 3Q-.64 = 43 54' 54"-6 148 PRACTICAL ASTRONOMY. S 4 . cos (8 a) = 9.8575542 sec S = .0081471 cos q> = 9.8815812 tan cp' = 9.9302268 log p = 9.9993875 cos |( - a') = 9.9999957 sin Tt = 8.2145238 sec [i(a -f- a') 0] = .1443121 sin 8) = .0904399 tan 3' = 8.0128043 tan (45 -f 43) = .0044726 tan (d 8') 8.1077168 8 - d' = 44'3"-i3 Fourth. 85. Required the parallax in right ascension and declina- tion, having given the apparent right ascension and declina- tion. Multiply the first of (r/i) by sin a ', the second by cos a'; subtract and reduce. The result is sin -*= p sin TT cos (p' sin (0 a'} (184) To obtain d 6' we make use of (1/9). Multiply the first by cos 6', the second by sin #'; subtract and reduce. We thus have sn _ = We have therefore the following formulae for the parallax in right ascension and declination, having given the appar- ent co-ordinates: PRACTICAL ASTROXOMY. p sin TT cos = 40 25' i".? cos (5 a') = 9.46459 tan . (189) sin a sin B = . sin b sin A. ) Applying these formulae to the triangle formed by the zenith, the pole, and the star, we have cos cos Also, sin tf=sin

sin z cos a; cos y=sin q> sin ^+ cos 9 cos z cos a 'i sin q cos sin a. cos ,sr=sin (p sin d-\- cos (p cos d cos /; sin .3 cos <7=sin q> cos d cos cp sin d cos /; sin 2 sin q cos (p sin /. FIG. 9. 158 PRACTICAL ASTRONOMY. 87. Now differentiating the first of (190), regarding 8 and .sonly as variables, cos ddS = (sin cp sin z -\- cos

= 6s".82 f* in feet (195) This formula would give us the true value of the correc- tion if there were no refraction, the effect of which is to di- minish D. The refraction very near the horizon is always a somewhat uncertain quantity, but for a mean state of the air the dip corrected for refraction will be found by multi- plying the value given by (195) by the factor .9216, or D" = 58' / .82 Vx in feet. .... (196) An approximate value sometimes used by navigators is obtained by taking the square root of the number of feet above the water and calling the result minutes. Thus if the eye is 25 feet above the water, this process would give for the dip 5'; formula (196) gives 4' 54". The dip must be subtracted from the observed altitude to obtain the true altitude. CHAPTER III. TIME. 89. For astronomical purposes the day is considered as beginning at noon instead of at midnight; the hours are reckoned from zero to twenty-four, instead of from zero to twelve as in civil time. Thus, July 4th, 9 h A.M., civil reckon- ing, would be July 3, 2i h , astronomically.* In all operations of practical astronomy the time when an observation is made is a very important element. There are various methods of reckoning time, of which three are in common use, viz., mean solar, apparent solar, and sidereal time. Before entering upon the relations between these different kinds of time, some preliminary considerations are necessary. 90. The transit, culmination, or meridian passage of a heaven- ly body at any place is its passage across the meridian of that place. Every meridian is bisected at the poles; and as a star in the course of its apparent diurnal revolution crosses both branches, it is necessary to distinguish between the upper culmination and lower culmination. The Upper Culmination of a heavenly body is its passage over that branch of the meridian which contains the ob- server's zenith. The Lozvcr Culmination is the passage over that branch which contains the observer's nadir. Any star whose north-polar distance does not exceed the * The prime meridian conference which assembled at Washington October ist, 1884, recommended the adoption of a universal day for astronomical and other scientific purposes, to begin at Greenwich mean midnight and to be reckoned from o h to 24*". The Astronomer Royal of England has adopted the suggestion for the Greenwich Observatory. Whether it will be generally adopted remains to be seen. TIME. 163 north latitude of the place of observation is constantly above the horizon, and may be observed at both upper and lower culmination. Any star whose south-polar distance does not exceed the north latitude of the place of observation is al- ways below the horizon, and therefore cannot be observed at all.* Stars between these limits can be observed at upper culmination only. The rotation of the earth on its axis being uniform, it follows that the intervals of time between the successive transits of a point on the equator over either branch of the meridian will be of equal length. Such an interval is a si- dereal day, and the point with the transit of which the side- real day is regarded as beginning is the vernal equinox. A SIDEREAL DAY is the interval between two successive transits of the vernal equinox over the upper branch of the meridian. THE SIDEREAL TIME at any meridian is the hour-angle of the vernal equinox at that meridian. The right ascensions being reckoned from the vernal equi- nox, it follows that a star whose right ascension is a will culminate at a hours, sidereal time. Therefore the sidereal time at any meridian is equal to the right ascension of that meridian. In the figure let EE' be the equator, P the pole, PM the meridian of any place, PN the hour-circle of any star S, T the vernal equinox. Then from our definitions, MPN = hour-angle of star 5 t ; = right ascension of star S a = the sidereal time at the meridian PM = &. * If the latitude of the place of the observer is south, obviously these con- ditions will be reversed. 164 PRACTICAL ASTRONOMY. 91. Therefore = a -\- t (197) Thus, if we have by any method determined the hour- angle of a star, this equation gives the sidereal time: a, the right ascension, being taken from the ephemeris, or from a star catalogue. The interval between two successive transits of the sun over the upper branch of the meridian is an APPARENT SOLAR DAY. The hour-angle of the sun at any meridian is the APPARENT TIME at that meridian. Owing to the annual revolution of the earth, the sun's right ascension is constantly increasing ; therefore it follows that the solar day will be longer than the sidereal day. Thus in one year the sun moves through 24 hours of right ascension. In one year there are, according to Bessel, 365.24222 mean solar days; therefore in one day the sun's 24 h right ascension increases -? - = 3 $6*.$$$. In one hour 305.24222 one twenty-fourth of this amount = 9 S .8565. These figures represent the mean or average rate of change. The actual change, however, is not uniform, and in conse- quence the apparent solar days are not of equal length. This want of uniformity results from two causes, which will now be explained. First Inequality of the Solar Day. 91. The apparent orbit of the sun about the earth is an el- lipse with the earth in one of the foci. Let the ellipse, Fig. 12, represent this apparent orbit. When the sun is at A the right ascension is increasing more rapidly than when it is at A'; therefore in the first case it will have a larger arc to pass over between two successive meridian passages than in the second. This inequality alone being considered, the 92- INEQUALITY OF SOLAR DA VS. I6 5 length of the solar day will be a maximum when the sun is in perigee, and a minimum when it is in apogee. We may imagine a fictitious sun to move in the ecliptic in such a way that the an- gular distances AEP, PEP^ PEP,', etc., described in equal times, shall be equal. Let both start together A | from A on January ist, moving in , N the direction of the arrow. On Jan- uary 2d the true sun will be in ad- vance of the fictitious sun, and will FIG. 12. continue so until June 3Oth, when they will be together at A'. Therefore from January ist to June soth the fictitious sun, having the smaller right ascension, will always pass the meridian in advance of the true sun. From A' to A the fictitious sun will be in advance of the true sun, and will con- sequently pass the meridian later, until they both reach A, when they will again be together, January ist. Second Inequality of the Solar Day. 92. The figure represents a projection of the sphere on the plane of the equinoctial colure. P is the north pole, P' the south pole, T 0^= the equa- tor, v =c:V3 the ecliptic. Now the fictitious sun before con- sidered moves in the ecliptic describing the equal arcs fA, AB, BC, etc., in equal times. Let the hour-circles PAP, PBP', etc., be drawn; then the distances pa, ab, be, etc., intercepted on the equator, will not be equal, but the distance TO = TO, both being quadrants. 1 66 PRACTICAL AS 7'KONO M Y. 92. We may now suppose a second fictitious sun to move in the equator in such a way as to complete the circuit of the equator in the same time that the first completes the circuit of the ecliptic. Let both start from the vernal equinox u = 1.00273791. Then /# = //< = / + 7 oO !) = / +.00273791/0; /0 = = 7 * ~ By the use of these formulae the process is very simple. It is rendered still more so by the use of tables II and III of the appendix to the Nautical Almanac. Table II gives the quantity ( i j/^, with the argument /^, and table III gives (ft i)/o, with the argument 70. One or two examples will illustrate their use. Example i. Given the mean solar interval / = 4 h 40 30". Find the corresponding sidereal interval. /o = 4 b 4o m 30 s .ooo Table III gives for 4 h 40 + 45 s -997 Table III gives for 30* -j- .082 7* = 4 h 41 i6 s .o79 Example 2. Given the sidereal interval 7^ = 4 h 4i m i6 s .O79. Find the corresponding mean solar interval. 7* = 4 h 41 i6 s .o79 Table II gives for 4 h 4i m - 46 S .O35 Table II gives for i6".o79 -44 70 = 4 h 40 m 3O s .ooo I/O PRACTICAL ASTRONOMY. $94. To Convert the Mean Solar Time at any Meridian into the Cor- responding Sidereal Time. 94. Referring to Fig 11 and formula (197), we see that if S represents the mean sun, then MPN = the mean time = T; = the right ascension of the mean sun = or. Then we have = or + 71 ...... (199) The right ascension of the mean sun, <*, is given in the solar ephemeris of the Nautical Almanac, for Washington mean noon of each day. It is there called the sidereal time of mean noon, which it is readily seen is the right ascension of the mean sun at noon, since at mean noon the mean sun is on the meridian when its right ascension is equal to the sidereal time. If L the longitude of the meridian from which T is reck, oned, then (T-\-L) = the time past Washington mean noon. Let Vc, = sidereal time of mean noon at Washington. Then ao = V + (7+ and 9= 7*+ Fb-f (r-j-ZX/t- I). . . (200) The last term may be taken from table III before used, or we may compute it by the method given in Art. 90. We there found the hourly change in right ascension of the mean sun to be 9".8565. If we express (T -\- L) in hours, we have ao = F + (r+Z) 93.8565. When this operation has frequently to be performed at any meridian other than Washington, it is a little more con- venient to use the sidereal time of mean noon at the merid- ian itself. Let V = the sidereal time of mean noon at meridian whose 94- SIDEREAL AND MEAN SOLAR TIME. I/I longitude is L. Then if we consider L as reckoned towards the west, the Washington time of mean noon at the given meridian will be L, and we shall have V = Fo + L (v - i), or V = FO + 9 s .8s65Z; L being expressed in hours. Formula (200) then becomes \,u - i) ..... (201) Example i. Longitude of Bethlehem = 6 m 4O 9 .3 = h .ni2; Mean solar time, 1 88 1, July 4th, p h oo m oo s . Required the corresponding sidereal time. From the Nautical Almanac, p. 329, we find FO = 6 h 5i m 22 8 .6io .1112 X 9 8 .8565, or from table III, N. A., O - \]L - i -.096 V = 6 h 5i m 2i s .5i4 Mean solar time T= 9" oo m oo s .ooo Table III, O - i)2" + i m 28 9 .;o8 Sidereal time O = i5 h 52 5o 9 .222 Example 2. T = 1881, July 4th, 2i h 7 m 3 S .2, Ann Arbor mean time. Required 0. Longitude of Ann Arbor = -f26 m 43*. : _ ''.4453 Fo = 6 h 5i m 22 3 .6io 4453 X 9 8 .8565, or table III, (/* - i) L + 4 9 389 T= 2i h ; m 3 3 .2oo Table III, (^ - i)r = + 3 m 28M45 Sidereal time @ = 4 h oi m I7 2 PRACTICAL ASTRONOMY. 95- To Convert Sid. real into Mean Solar Time. 95. This process, the converse of the preceding, may be briefly stated as follows: First. Subtract from the given sidereal time the sidereal time of mean noon; we then have the sidereal interval past noon, viz., & V. Second. Convert the sidereal interval (0 V) into the corresponding mean time interval, by subtracting the quan- tity (& - F)(i - -) found in table II, N. A. The formula is as follows: T = (0 - V) - (0 - F)(l - I). . . (202) Example i. Given 1881, July 4th, I5 h 52 5O 8 .222 Bethlehem sidereal time. Required the corresponding mean solar time. = I5 h 52 m 50 S .222 4s before, V = 6 h 51'" 2i s .5i4 V = 9 b oi m 28 s .;o8 Table II, (0 - F)(i - -) i m 28 s .;o8 Mean time T = 9'' oo m oo s . Example 2. Given 1881, July 4th, 4 h i m 58 a .344 Ann Arbor sidereal time. Required the mean solar time. &= 4 h i m 58 s -344 As before, V = 6 h 51"' 26^999 V 2i h io m 3i s -345 Table II, (0 -V}(\- -) 3 28M45 Mean time T 2i h 7 m O3 8 .2 95- SIDEREAL AND MEAN SOLAR TIME. 1/3 It is sometimes necessary to convert mean solar time into sidereal, or vice versa, in reducing old observations made before the publication of the solar ephemeris in the form now employed. Bessel's Tabula Rcgiomontance furnish the data necessary for solving the problem for any date between 1750 and 1850. The method of using these tables for this purpose is fully explained in Art. 362 of this work. CHAPTER IV. ANGULAR MEASUREMENTS. THE SEXTANT. THE CHRO- NOMETER AND CLOCK. 96. The circles of astronomical instruments are graduated continuously from zero to 360. With ordinary field-instru- ments the smallest division is commonly 10', though sometimes less. The large circles of fixed observatories are graduated much finer. Fractional parts of a division are read by means of the vernier, or reading microscope. The edge of the circle on which the division is marked is called the limb. The circle or arm which carries the index is called the alidade. The vernier, also called the nonius, is an arc carried by the alidade, and graduated in the manner described below, for measuring fractional parts of a division. Let AB (Fig. 14) be a portion of the limb of a circle. Each division is supposed to be one _^ degree of the circle. The arc J\Jf CD, carried by the alidade and } graduated as shown, forms a FlG 14> vernier. In this case there are ten divisions on the vernier, cover- ing a space equal to nine divisions of the limb. Each space on the vernier is therefore shorter by -fa of one degree (equals 6') than a space on the limb. In the figure the index coin- cides with the zero-point of the limb; division one of the ver- nier falls behind division one of the limb, 6'; division two of THE VERNIER. 1 75 the vernier falls behind division two of the limb, 2 X 6' = 12', etc., etc. The method of using the vernier will now be clear by re- ferring to Fig. 15. In this ^ R R to case the index falls between 42 and 43 on the limb. The reading of the circle is there- * &~~ l ~ii ' 4 6 ' 43 fore 42 plus a fractional part FlG IS ' of a degree. This fraction is given by the vernier as follows : Looking along the scale until we find a line of the vernier which coincides with a line of the limb, we find this to be the case with the one marked 4. Therefore, following down the vernier scale towards the zero-point, it is evident that Line 3 of the vernier is 6' to the right of 45 of the limb; Line 2 of the vernier is 2 X 6'= 12' to the right of 44 of the limb; Line I of the vernier is 3X6'= 18' to the right of 43 of the limb; Line o of the vernier is 4X 6 / =24 / to the right of 42 of the limb. The reading is therefore 42 24' or 424, the number on the vernier where the line of the latter coincides with a line of the limb, giving the tenths of a degree at once. In general let d = the value of one division of the limb ; d' = the value of one division of the vernier ; n = the number of divisions of the vernier corresponding to n i of the limb. Then (n - i)d = nd', and d d' = -d. (203) dd'is the least reading of the vernier. We have therefore the following very simple rule : I /6 PR A CT1CA LAS TROXOM J ". 97- To find the least reading of a vernier : Divide ike length of one division of the limb by tJie number of spaces of the vernier. For example, suppose the limb graduated to 10', and the number of divisions of the vernier-scale to be 60. Then the least reading of the vernier will be 10' 600" =-7 = 10". 60 oo This is a very common arrangement. In the vernier just described n divisions of the vernier were equal to n I of the limb. Verniers are sometimes made in which n divisions are equal to n + I of the limb. Then (n -\- \)d = nd and d d -d, as before. It is to be observed that in this case the reading of the ver- nier proceeds in a direction opposite to that of the limb. Many different forms of division and arrangement are found in verniers, but they all follow the same general princi- ple, a practical familiarity with which makes the reading of any form of vernier very simple. The Reading Microscope. 97. Instead of the vernier, in very fine instruments the alidade carries a microscope the optical axis of which is per- pendicular to the plane of the circle. This is a compound microscope with a positive eye-piece. In the common focus of the object-lens and eye-piece are the micrometer-threads for reading the circle. The micrometer (Fig. i6a] consists of a frame of brass, across which are stretched two spider-lines. Sometimes these lines make an acute angle with each other, as shown in the figure ; sometimes they are made parallel and quite close together. The plane of the frame is parallel to THE READING MICROSCOPE. 77 the plane of the circle MN, and it is moved parallel to a tan- gent to the circle by the screw G. Attached to the screw and revolving with it is the cylinder FE, graduated, as shown in the figure, for recording the fractional parts of a revolution of the screw. The cylinder is generally graduated into either 60 or i oo parts. Suppose now the distance between two divisions of the circle to be 5', and that five revolutions of the screw are just sufficient to move the cross-threads over this distance: then evidently one revolution moves the threads over i'. If the head is divided into 60 parts, then each divi- sion of the head corresponds to a motion of the cross-threads over i". By making the screw sufficiently fine and increasing the number of divisions of the head, at the same time increasing the power of FIG. i6. THE MICROMETER. FIG. 16. THE READING MICROSCOPE. the microscope, this division of space may be carried to an almost unlimited extent. For the purpose under considera- tion, however, we should soon reach a limit beyond which nothing would be gained by increasing the delicacy of the microscope. For reading the entire, number of revolutions of the screw there is sometimes a scale attached to the outside of the box in which the slide moves. More frequently the scale is inside the box, placed at one side of the field of view. When so placed it consists of a strip of metal in the edge of which I7 PRACTICAL ASTRONOMY. 98. notches are cut ; the distance between two consecutive notches being equal to one revolution of the screw. Every fifth notch is made deeper than the others for facility in counting. Suppose now the cross-threads to stand opposite the centre notch (which is generally distinguished in some manner), and the zero point of the head to be exactly at the index-mark. The point in the field now occupied by the cross-threads is the fixed point to which all angular measurements are re- ferred ; it corresponds exactly to the zero-point of the ver- nier. Suppose, further, the zero-point of the circle to be exactly under the intersection of the threads. Now let the instrument be revolved on its axis through any angle : the number of divisions of the circle which pass by this point of reference will then be the measure of the angle. For the purpose of fixing the idea, let the arrangement be that described above, viz., the circle graduated to 5', and the micrometer reading to single seconds. If now the revolu- tion of the instrument has brought the scale into the position shown in Fig. 17, we see from the position of the threads that the entire angle passed over is between 45 15' and 45 20'. By means of the screw let the cross-threads be moved so as to coincide with division 15'. Then the entire number of revolutions of the screw will I I I I ill I I I I I I I I I I I lve t ^ ie number of minutes to be added M'lNu M' JJ - -" to 45 15', and the fractional part of a 1 46 revolution given by the head will be expressed in seconds. Thus if the whole number of revolutions were two, and the reading of the head 53, the angle would be 45 17' 53". In making the bisection, the screw should always be turned in the same direction, to guard against the effect of slip or lost motion in the screw. If the thread is to be moved in a negative direction it should be moved back beyond the line, and the final bisection made by bringing it up from the other side. 98. When everything is in perfect order a whole number THE READING MICROSCOPE. 1/9 of revolutions of the screw is exactly equal to the distance between two consecutive lines on the circle. This is pro- vided for by an arrangement for changing the focal length of the microscope, and for moving the object-lens nearer to or farther from the plane of the circle. This adjustment is subject to small disturbances, on account of changes of temperature and other causes. The error caused by an im- perfect adjustment is called the error of runs. The correc- tion for runs is found by reading the microscope on two con- secutive divisions of the circle. If this does not correspond to the exact number of revolutions of the screw, the excess or deficiency is to be distributed in the proper proportion to measurements made with the screw. For determining the correction a number of readings should be made in different parts of the circle in order to eliminate from the result the accidental errors of graduation. Some observers in certain kinds of work always read the micrometer on both divisions of the limb between which the zero-point falls. For example, in Fig. 17 the micrometer- thread would be set on both division 15' and 20', thus eliminat- ing from the resulting reading the effect of runs, and to some extent the accidental errors of graduation and of bisection. For insuring greater accuracy two or more microscopes or verniers are used. When there are two they are placed opposite each other, or 180 apart. When there are three or more they are placed at uniform distances around the circle. If the probable error of the reading of one micro- \" scope be i", that of the mean of two will be* ^ = '' '.71 ; \" that of four will be r- = ".5. 1/4 The principal value of two or more microscopes, however, is for eliminating the error of eccentricity. * See Introduction, Art. 14, Eq. (25). I8o PRACTICAL ASTRONOMY. 99- Eccentricity of Graduated Circles. 99. The centre of the alidade seldom coincides exactly with the centre of the graduated circle. This deviation from exact coincidence is called ec- centricity. In order to understand the effect of eccentricity, let C be the centre of the circle ; C', the centre of the alidade ; O, the zero-point of the limb ; a, the point on the limb where it is intersected by a line joining C and C' ; C'n, the direction of the line drawn from the centre of the alidade to the zero-point of the vernier when the telescope is directed to any object. The true position of the object is given by the direction of the line C'n, while the reading of the circle gives the direction Cn, differing from the former by the small angle n'Cn = CnC'. FIG. 18. Let now CnC = p ; CC = e Cn = r Cn = r' Angle OCn = n ; OCa = a. Then C'Cn = n a. From th? triangle C'Cn we have r' sin p e sin (n at) ; r' cos/ = r e cos (n at) sin (n n) from which tan/ = . . . . (204) I cos (n a) 100. ECCENTRICITY OF GRADUATED CIRCLES. iSl The angle / will always be small, and the denominator of (204) differs but little from unity. We may therefore write, without appreciable error, / = - sin (n a). -. f . , . . (205) 100. It is more elegant to expand the above expression into a series in terms of ascending powers of -. Equation (204) is of the form . . sin p a sin x cos p I a cos x ' from which we readily find sin p = a sin (/ -j- x) (206) Now add sin (/ -(- x) to both members of (206) ; then subtract sin (/ -}- r ) from both members ; finally, divide the first expression by the second: sin p + sin (p -\- x) _ (a -\- i) sin (p + x) _ sin / sin (/ -|~ x) (a i) sin (/ -j- x) ' from which tan (/ -|- %x) = tan $x (207) Applying to this the process of development made use of in Art. 74, Eq. (137), we find / = a sin x -\- |a 2 sin zx -\- fa 3 sin $x, etc. Writing for a and x their values and dividing by sin i", in order to express/ in seconds of arc, we find e e l e 3 p = 77 sin ( a) -] sin z(n a) -\ sin 3(na). (208) r sin i 2r* sin i y 3 sin I The first term is identical with (205), and will always give the necessary accu- racy without using the following terms. 101. Besides the eccentricity above considered there is a similar effect due to the play of the axis of the instrument in 182 PRACTICAL ASTRONOMY. IOF. its socket. This is not a determinate quantity like that we have been considering, but when two verniers or microscopes 180 apart are used, the effect of both will be eliminated, as appears from the following: Let n' and n" be the readings of the two microscopes ; n, the true value of the angle. Then from the first microscope = '-{- e" sin (V ). Similarly, n = n" -f- e" sin (n" a). Jn which e" has been written for Now n" differs very Uttle from 180 -(- n', so that no appre- ciable error will be introduced by writing the second of the above equations n = n" -f e" sin [180 + (n' or)] = n" e" sin (n' a}. Therefore n = \(n' -f- n"}, from which the correction for eccentricity is eliminated. In a similar manner it may be shown that the mean of three microscopes will be free from the effect of eccentricity. In case of four, as the mean of each pair 180 apart is free from this error, it follows that the mean of the four will be. The constants e" and a may be determined very readily by taking readings in different parts of the circle; but with a complete circle they will not be required. It is only in the case of the sextant, where we have a limited arc of the circle read by a single vernier, that this becomes a matter of importance. The application to this case will be considered in the proper place. 102. THE SEXTANT. The Sextant. 102. In the determination of time and latitude when ex- treme accuracy is not required, the sextant is one of the most convenient and useful of astronomical instruments. It is light and easy of transportation ; in observing it is simply held in the hand, and consequently entails no loss of time in FIG. 19. THE SEXTANT. mounting and adjusting ; it is therefore especially adapted to the requirements of navigation and exploration. For use on land the sextant is sometimes mounted on a tripod, which adds something to its accuracy. When the instrument is used by a skilful observer, however, the advantage is not great. In most cases where such an arrangement could be made use of the sextant will not be employed at all, but will give place to an instrument of greater precision. 1 84 PRACTICAL ASTRONOMY. The principal features of the sextant may be seen from Fig. 19. The graduated arc is about 60 in extent, hence the name, sextant. This arc of 60 is divided into 120 parts, called degrees for reasons which will soon appear. The arc commonly reads directly to 10', and by means of the vernier to 10". A mirror, C, called the index-glass, is attached to the arm carrying the vernier, and revolves with it about a pivot at the centre. A second mirror, N, is attached to the frame of the instrument, and is called the horizon-glass. Only half of this glass is silvered, viz., that next the plane of the instru- ment an arrangement which makes it possible to see an ob- ject directly through the unsilvered part by means of the telescope, and at the same time the image of the same object, or of a second one, reflected from the silvered part of the mirror. In order to make these images equally distinct an adjusting-screw is provided (not shown in the figure), by which the telescope can be moved nearer to the plane of the instrument or farther from it. Attached to the frame are sev- eral colored glasses, E and F t which may be brought into a position to protect the eye when observing the sun. These are sometimes attached to an axis so that they can be at once reversed, the object being to eliminate any error due to want of parallelism of the surfaces by taking half of a series of measurements in each position. There is also a revolving disk attached to the eye-piece of some instruments containing a number of colored glasses of different shades. Other minor features can best be learned by the inspection of the instru- ment itself. 103. The principle which lies at the foundation of the sex- tant and instruments of like character is the following: If a ray of light suffers two successive reflections in the same plane by two plane mirrors, then the angle between the first and last direction of the ray is double the angle of the mir- rors. In Fig. 20 let M and m be the two mirrors supposed 103- THE SEXTANT. 1 85 perpendicular to the plane of the paper ; let AM be the first direction of a ray of light falling on the mirror M; it will be reflected in the direction Mm, and finally from in in the direc- tion mE. Draw MB parallel to mE* MP perpendicular to M,\ Mp perpendicular to m. The angle between the first and FIG. 20. last direction of the ray is equal to the angle AMB. The angle between the mirrors is equal to PMp. We have now to show that A MB = 2PMp. Consider first the mirror m. The incident ray Mm makes with the normal the angle Mmp' mMp = pMB = pMP + PMB. . . (a) Consider now M. The angle mMP = PMA = AMB + PMB (6) 1 86 PRACTICAL ASTRONOMY. 104, Subtracting (a) from (#), mMP - mMp = AMB - pMP, from which 2pMP = AMB. Q. E. D. If now the angle between two objects is to be measured, the instrument is held so that the plane of the graduated arc passes through both. The telescope is then directed to one of the objects, which is seen through the unsilvered part of the horizon-glass, and the index-arm is revolved until the re- flected image of the second object is brought in contact with the direct image of the firs*. The reading of the limb will then be the required angle ; the graduation before explained, viz., each degree being divided into two, gives the angle between the objects, which is twice that of the mirrors. 104. In the prismatic sextant of Pistor & Martins (Fig. 21) the horizon-glass is replaced by a totally reflecting prism. The arrangement has this advantage, viz., that by its use angles of all sizes from o to 180, and even larger, can be measured, while the common form of sextant is not adapted to the measurement of angles much greater than 120. In using the instrument the prism B interferes with the rays of light which should reach the index-glass, A, when the angle is about 140; but angles of this magnitude may be measured by turning the instrument over and holding it in the reverse position. If, for instance, the double altitude of the sun is being measured, the instrument, will ordinarily be held in the right hand, with the arc below and the tele- scope above. If, however, the double altitude is about 140, the instrument must be held in the left hand, with the tele- scope below and the arc above. In case the head of the ob- server interferes, as will be the case when the angle is near 1 80, the difficult} 7 is overcome by means of the prism E 105- PRISMATIC SEXTANT. REFLECTING CIRCLE. IS 7 placed back of the eye-piece so as to reflect the rays of light coming through the telescope in a direction at right angles to its axis. 105. The arc of the sextant may be extended to an entire circumference, and the index-arm produced so as to carry a FIG. 21. THE PRISMATIC SEXTANT. vernier at each extremity. The instrument then becomes the simple reflecting circle. As previously shown, this arrange- ment possesses the advantage of eliminating the eccentricity, and to some extent the errors of graduation. This instru- ment is used precisely like the sextant. 1 88 PRACTICAL ASTRONOMY. IO/. Other forms of reflecting circles have been made possess- ing advantages in certain directions, but they do not seem to have met with great favor, although they are theoretically much more perfect instruments than the sextant ; practically, however, this superiority is not so great. This is no doubt due in part to the fact that, except in the hands of an obser- ver of more than usual skill, the errors of observation are so great as practically to neutralize their greater theoretical advantages. Adjustments of the Sextant. 106. First Adjustment. THE INDEX-GLASS. The plane of tJie reflecting surface must be perpendicular to the plane of the sextant. To ascertain whether this is the case, place the index near the middle of the arc, then look into the glass so as to see the image of the arc reflected. If the adjustment is perfect, the arc seen directly will be continuous with its reflected image. This adjustment is attended to by the maker and is not liable to derangement; for this reason no provision is com- monly made for correcting a want of perpendicularity. It may be corrected when necessary by removing the glass from its frame and filing down one of the points against which it rests, or by loosening the screws holding the frame to the index-arm and inserting a piece of paper or other thin substance under one side. 107. Second Adjustment. THE HORIZON-GLASS. The plane of this mirror must also be perpendicular to the plane of the sextant. The index-glass must first be in adjustment; if then it is possible to place it in a position parallel to the horizon-glass by moving the index-arm, then the latter will also be per- pendicular to the plane of the sextant. To test this adjust- ADJUSTMENT OF THE SEXTANT. 189 meat proceed as follows : Bring the index near the zero- point apd direct the telescope to a well-defined point a star is best. If then the index-arm be moved slightly one way and then the other the plane of the instrument being verti- calthe reflected image of the object will move up and down through the field. If the adjustment of the two glasses is perfect, the two images may be made to coincide exactly, otherwise the reflected image, instead of passing over the direct, will pass to one side or the other of it. Two small capstan-headed screws are provided for making this adjustment when necessary. A pair of adjusting-screws is also provided for correcting the position of the glass in the opposite direction, viz., to make it parallel to the index-glass when the vernier is at zero. If the direct and reflected image of the star are brought into exact coincidence by means of the tangent-screw, the reading of the vernier, if not zero, is called the index error. The screws just men- tioned are for correcting this error. It will be found better in practice not to attempt this adjustment, but to determine the error and apply the necessary correction to the angles measured," as will be explained hereafter. 108. Tliird Adjustment. Tlie axis of tJie telescope must be parallel to the plane of the instrument. Two parallel threads are placed in the eye-piece to mark approximately the middle of the field: they should be made parallel to the plane of the instrument by revolving the eye- piece. The axis of the telescope will now be the line drawn through the optical centre of the object-glass and a point midway between these lines. To determine whether this line is parallel to the plane of the instrument, select two well-defined objects 100 or more apart, and bring the re- flected image of one in contact with the direct image of the other, making the contact on one of the threads; then move the instrument so as to bring the images on the other thread. igO PRACTICAL ASTRONOMY. 1 09. If the contact still remains perfect, the line is in adjustment ; if any correction is required, there will be found a pair of screws for the purpose on opposite sides of the ring which holds the telescope. The above test will be found difficult to apply, especially if the observer has not a considerable amount of experience in the use of the instrument. One less difficult is the follow- ing : Place the instrument face upward on a table, then lay on the arc two strips of metal or wood, the width of which must be the same and equal to the distance of the axis of the telescope from the plane of the instrument. Now sight across the upper edges of these strips, and have an assistant mark with a pencil on the wall of the room (which should be 15 or 20 feet distant) the place where the sight-line inter- sects it; then, without disturbing anything, look through the telescope, which has been previously directed to this part of the wall and properly focused, and see whether this mark is found in the middle of the field ; if so, then the adjustment is satisfactory. Method of Observing with the Sextant. 109. To Measure the Distance between Two Stars. Direct the telescope to one of the stars, then revolve the instrument about the axis of the telescope until its plane pusses through the other (taking care to have the index-glass on the right side), then move the index-arm until the image of the second star is brought into the field, clamp the instrument and bring the two images into perfect contact by means of the tangent- screw. The reading of the vernier corrected for index error will be the required distance. Unless the two stars are quite near each other it will be expedient to compute the distance approximately before attempting the observation. The in- dex may then be set at the approximate distance, which will I IO. METHOD OF OBSERVING WITH SEXTANT. IQI greatly facilitate finding the two images. A common obser- vation of this character is that of observing the distance of the moon from the sun or a star for determining longitude. In the Nautical Almanac will be found given for every day throughout the year the distance of the moon from the sun, and certain stars and planets, which may be used for this purpose. The index may at once be set at the approximate angle without any preliminary computation. If the distance of the moon from a star is measured, the image of the star is brought into contact with the bright limb of the moon, the contact being made at the point where the great circle join- ing the star with the centre of the moon intersects the limb. To ascertain this point the instrument must be revolved through a small arc back and forth about the axis of the telescope (supposed to be directed to the star); the image of the moon's limb will then pass back and forth across the field, and should appear to pass exactly through the centre of the star's image, which will in general not be reduced to a simple point by the feeble telescope of the sextant. This distance is to be corrected for the moon's semidiam- eter in order to give the distance between the star tfnd the centre of the moon. In measuring the distance' between the moon and sun, the bright limb of the moon is brought in contact with the near- est limb of the sun. The measured distance must then be corrected for the semidiameters of both moon and sun. HO. Measurement of Altitudes. At sea altitudes are meas- ured by bringing the reflected image of the body in contact with the line of the horizon as seen directly through the telescope. In order that the result may be correct the plane of the instrument must be held exactly vertical To accomplish this the instrument is revolved or vibrated slightly about the axis of the telescope, at the same time moving it so as to keep the image in the centre of the field. 192 PRACTICAL ASTRONOMY. IH- The image will appear to describe an arc of a circle, the lowest point of which must be made tangent to the horizon by moving the index-arm. If the sun is observed, the lower limb must be made tangent to the horizon. As the altitude of the sun's centre is required, the reading of the vernier must be corrected for index error, refraction, parallax, and semidiameter. If a star is observed, there will be no correc- tion for semidiameter or parallax. in. For observing altitudes on land the artificial horizon must be used. This is a shallow basin, about 3 inches by 5, for holding mercury. It is provided with a roof formed of two pieces of plate glass set at right angles to each other in a metal frame, for protecting the mercury from agitation by the wind. The surface of the mercury forms a mirror from which the image of the sun or star is reflected ; and as it is perfectly horizontal the reflected image will appear at an angular distance below the horizon equal to the altitude of the body itself above the horizon. If now the image of a star reflected from the mirrors of the sextant is brought into contact with the image reflected from the mercury, the angle which will be measured is evidently twice the altitude of the star. The opposite sides of the glass plates forming the roof to the horizon should be exactly parallel, otherwise the pris- matic form introduces an error into the measured angle. It is possible to derive a formula for the correction necessary to free an observation from this source of error, but it will be better in practice to observe half of a series of altitudes with one side of the roof next the observer and then reverse it, taking the remaining half in the opposite position. The mercury must be freed from the particles of dust and impurities which will generally be found floating on its sur- face. It may be strained through a piece of chamois-skin or through a funnel of paper brought down to a fine point 113- METHOD OF OBSERVING WITH SEXTANT. 193 at the end. Another method is to add a small amount of tin- foil to the mercury, when the amalgam which will be formed will rise to the top and may be drawn to one side with a card, leaving the surface entirely free from specks of any kind. 112. In measuring altitudes for any purpose, a number of measures should be made in quick succession and the mean taken. In this way the accidental errors of contact and reading will be greatly diminished. Thus, in taking the altitude of the sun for determining the time, a series of not less than three alti- tudes should be measured on each limb. Suppose the observations made when the sun is east of the meridian, and the altitudes therefore to be increasing; the readings on the upper limb will be made first, as follows: Set the index on an even division of the limb at a reading 10' or 15' greater than the double altitude of the upper limb. When the two images are then brought into the field they will appear separated, but will be approaching each other. The observer watches until they become tangent, when the time is carefully noted by the chronometer. The index is then moved ahead 10', 15', or 20', and the same pro- cess repeated. A little practice will enable the observer to take the altitudes in this manner at intervals of 10' without difficulty, in which case five readings may be taken which will correspond to an increase of 40' in the double altitude or 20' in the actual altitude. As the sun's diameter is about 32' of arc, the index may now be moved back to the first reading, and five readings on the lower limb taken at the same altitudes as before. In this case the images will overlap and will gradually separate, the time to be noted being that when the two disks are tangent. If the sun is observed west of the meridian, the readings on the lower limb will be made first. The altitudes will of course be decreasing. 113. The beginner will sometimes find difficulty in bringing the two images into the field together. A convenient way of accomplishing this is as follows: Bring the index near the zero-point and direct the telescope to the sun, when two images will be seen; then bring the instrument down towards the mercury horizon, at the same time moving the arm so as to keep the reflected image in the field until the image reflected from the mercury is found, when both will be in the field together. A little practice will make this process very easy. In observing stars care must be taken to avoid bringing the direct image of one star in contact with the reflected image of another. Sometimes a small level is attached to the index-arm to facilitate finding the reflected image, and at the same time for preventing mistakes of the kind just mentioned. It may be shown geometrically that when the two images of any star are brought in contact in the manner we have been describing, the angle formed with the 194 PRACTICAL ASTRONOMY. I 14. horizon by the index-glass will be equal to that formed with the horizon-glass by the axis of the telescope. As both telescope and horizon-glass are fixed to the frame of the instrument, it is therefore a constant angle. If then the level above mentioned is adjusted so that the bubble will play (the plane of the in- strument being vertical) when the index-glass makes this constant angle with the horizon, it may be used for the purpose mentioned. The method of finding the reflected image will then be as follows: Look through the telescope at the image reflected from the mercury; then, holding the instrument in the same position, move the index-arm until the bubble 'plays. If the reflected image is not then in the field also, the reason will be that the plane of the instrument is not vertical. It will be brought into the field by revolving the instrument back and forth about the axis of the telescope. To adjust this level, bring the two images of the sun or a known star into the centre of the field and move the tube until the bubble plays. Errors of the Sextant. 114. Among the various theoretical errors to which sex- tant observations are liable there are two which call for a detailed investigation, viz., index error and eccentricity. To Determine the Index Error. The arc is graduated a short distance backward from the zero-point ; when the reading falls on this side of the zero-point the reading is said to be off arc ; a direct reading being on arc. First Metliod of Determining the Index Error. By a Star. Direct the telescope to a star, and by means of the tangent- screw bring the direct and reflected images into exact co- incidence. The reading of the vernier will then be the index error, and it must be applied as a correction to all angles measured with the instrument. The correction will be -|- when the reading is off arc; The correction will be when the reading is on arc. The mean of several readings should always be taken so as to diminish the effect of errors of observation. US- INDEX ERROR. 195 Example. The following readings were made with a Pistor & Martins sextant for determining the index correction : On arc. 45" 60" 70" 70" 75" 60" 30" 75" 70" 65" Mean of ten readings, i' 2 // .o. The index correction being 7, we have therefore / = i' 2 / '.o. 115. Second Method. By the Sun. Measure the apparent diameter of the sun by bringing the direct and reflected images tangent to each other and read the vernier ; then bring the opposite limbs into the position of tangency and again read the vernier. If the first reading is on arc, the second will be off arc, and vice versa. Let r = the reading on arc ; r' = the reading off arc ; / = the index correction ; 5 = the true diameter of the sun. Then S = r -f /; S = r' - /; from which / = \(r' r) (209) 196 PRACTICAL ASTRONOMY. 1 1 6. When observations are made on the sun for any purpose, the gradual heating up of the instrument sometimes changes the value of the index correction. For this reason some ob- servers determine its value both at the beginning and end of such a series of observations. The following example taken from the Astronomische NachricJiten, Band 23, No. 548, will illustrate this, and at the same time the application of for- mula (209) : FIRST DETERMINATION. SECOND DETERMINATION. On arc. Off arc. On arc. Off arc. 32' 20" 30' 60" 32' $" 31' 15" 20" 60" o" ' 10" 25" 50" o" 20" 20" 50" o" 10" r=$2 / 2i // .2 r' = 30' 55" r 32' \" .2 r' = 31' i3".8 / = -43". i /= - 23".; Eccentricity of the Sextant. Il6. As the arc of the sextant is limited and is read by a single vernier, the effect of eccentricity is not eliminated; it should therefore be investigated. This can only be done by comparing the values of angles measured by it with their known values determined in some other way. The angles between terrestrial objects may be measured with a good theodolite, and the same angles measured with the sextant, or, what is better, stars may be used. In using stars for the purpose we may proceed in either of two ways. First, by measuring the distances between known stars. The right ascensions and declinations of the stars will be taken from the Nautical Almanac (it will be best to use none except Nautical Almanac stars for the purpose). The posi- Il6. ECCENTRICITY OF SEXTANT. iQ/ tions of the stars as they seem to us will differ from those given in the Nautical Almanac by the amount of refraction in a and M)= o.i57i52 n ' = 4". 6 cos a = 9.944463,! cos a = 9.944463,, tan h =. 9.787311 cos h = 9.930768 Proof 9.474505

= 9.93890 cos (p = 9.81679 A" = 23 41 39 tan N 9.64231 sin A 7 " = 9.60407 9.52020 S = 23 23 31 6"4- A" =52 5 10 sec (<5 4~ A 1 ") = .21150 cot = 9.89147 q 48 10 21 tan ^ = .04819^ cos^ = 9.82405 cos q = 9 82405 s = 49 25 46 tan? = .06742 sin z = 9.88059 9 70464 9.81557 Proof 9.81556 From table, mean refraction = 68". I Factor = .960 Therefore r = 65". 4 a PEGASI. T = 20" 26 m 3 s . 6 AT = 22 50 . f J = 20 3 13 .6 cr = 22 58 28 .5 t 2 h 55 m I4 s .g t 43 48' 44" cos t = 9.85830 tan /= g.gSigg n cos t= 9.85830 q> = 49 i 2 .4 cot (p = 9.93890 cos

$"<) By (194) IDROMED^E. a. PEGASI. cos q 9.90479 log r =. I 81889 sin q 9-77508,, dS = 43" .6 log dd = i.72368 d5 = 52". 9 d = 28 23 30 . 8" cos 8 da = 1-59397 #o = i43i 33- ,2 d = 28 24 14 . 4 1 5 cos 8 = 1.16198 8 1432 26. ,1 da = + 3 69 log da = .43199 da = 4 2 . 70 ff = o i 5i 78 6*0 = 22 58 28, 5" d = o" i m 48 8 .09 a 22 h 58' "25 s . 80 cos q 9.82405 logr = 1.81558 sin q 9.87225,, log dS 1.63963,, cos Sda = 1.68783 15 cos d = 1.12043 log da = .56740 These values of the right ascensions and declinations of the stars are the ones to be employed in computing the apparent distance between the two stars by equations (IV)i. 48".09 25 .80 22 s . 29 50' 34 .35 cos (a' a) = 9.983181 tan (a 1 a) = 9.452982 26 .1 cot 8 = .586075 sin N = 9.984762 39 .6 tan A 7 " = .569256 sec (A 7 "-}- 5') = .637698,, 14 -4 tan B = .075442 54 .o cot (A 7 " + 5') = 9 374136,, 5 .9 cos B = 9 808504 Proof .622460 39 .8 tan d = g 565632 43 . cos (a 1 a) = 9.983181 22 .8 cos 8 9.985862 20.5 9-969043 2". 3 = N. cos B = 9.808504 sin d = 9.538079 a' = 24" I 1 a = 22 58 a' a = I h 3* a' a = 15 5o' 8 ss 14 32 N 'as 74 54 8' as 28 24 N + 5 ' = 103 18 B = 49 57 d = 20 li I = 3 n = 20 15 n = 20 15 n ' ss + Proof .622460 The value of A 7 " obtained by the original computation, and which is employed in our equations, is 2". 2. The difference is of no importance here. N is now the absolute term of equation (212). For the coefficients A = sin \n, cos \n, and B = sin 2 J we must employ for n not the above angles, but the angle corresponding to the point on the limb which coincides with the vernier-scale. For example, the first measured angle of the first series is 63 25' 50". The limb I 1 6. ECCENTRICITY OF SEXTANT. 203 was graduated directly to 10'; these intervals were subdivided by the vernier to 10". The zero-point of the vernier falls between 63 20' and 63 30'; then read- ing along the vernier to the point where coincidence takes place, we find this to be at the reading 69 10' of the limb. It is therefore the eccentricity of this point by which our angle is affected, and not that of the point 63 25' -J-. In this way we find the point of contact for each reading of our series as follows : 63 25' 50" Point of contact = 69 10' 15' 50" = 69' oo' 6' 45" = 6 9 45' 62 57' 10" = 70 oo' 48' 10" 70 50' 39' 45" 72 15' 29' 50" 72 10' in = 17 n'| /sin = 9.47075 2l' IO" = 63" 30' /cos = 9.98014 13' 5" = 65' 15' A = 0.2824 log -4 = 9.45089 62" 3' 55" 65 55' B = 0.0874 log B = 8.94150 Mean = n 68 47' Therefore from this series we derive the equation 0.2824* + 0.08747 + z = 9"- 1- By proceeding in a similar manner with each of the eleven angles measured, the following equations of condition are obtained : .0703.* + .00507 + z =- 5.5; .no4jc -|- -01237 + z = + 2- 2 ; .2019* + .04257 + z = - 7-3; .2341* + .05827 + z = - 17.5; .2824* + .08747 + z = 9-1; .3295* + I239V + z = - 18. 5 ; .3586* + . 15157 + z = - 10.5; 3933-* + -19137 + 2 = - i4-o; 3997-* 4- .19967 + * = - 24-0; .4244* + .23577 + z = - 46.2; .4423^ + .26687 4~ z = ~ 28.6. It will be seen that the coefficients of x and 7 are much smaller throughout than those of z, while the absolute terms are relatively large. It would there- fore be a little more systematic to render the equations homogeneous, as ex- 204 PRACTICAL ASTRONOMY. plained in Art. 24, before forming the normal equations. This has not been done, however. The details of the formation of the normal equations (Articles 21 and 25) are as follows : As the number of unknown quantities is three, we rule our sheet (3 + 2H3 + 3) 2 added two columns for the residuals (v) and their squares (vv). These will be filled in after the unknown quantities have been determined. into i = 14 vertical columns (Art. 25), to which we have No. be cc en cs ab an x .0703 .0050 - 5-5 6-5753 .00494 .00035 - .3867 2 .1104 .0123 + 2.2 - 1-0773 .01218 -00136 4 .2429 3 .2019 .0425 7-3 85444 .04074 1-4739 4 .2341 .0582 17 8.7923 .05480 ^362 4 0967 5 .2824 .0874 - 9- 0.4698 .07976 .02468 - 25698 6 3295 239 - 18. 99534 . 0859 .04084 - 6.0957 I 3586 3933 9'3 14- 5.'s846 . 2854 5467 05432 .07522 - 5-5062 9 3997 996 - 24. 5-5993 5974 .07976 - 9-5928 4 2 44 .4423 a - 46. - 28.6 7.8601 30.3091 . 80,4 .100U2 .11801 - J9 6073 - 12 6498 3.2469 1.3742 11. - 179.0 194.6211 1.11971 .51677 - 65.5013 ^ [6c] \cc\ [en] [] [i M (an} No. as bb bn fa V j + .4622 00002 -0275 + -0329 30.25 - 36.16 + I- 2.89 2 - .1189 .00015 + -0271 484 - 2.37 - 6. 3 4 ^7251 00l8l + .3631 53-29 62.37 -|- I 1. 00 4 4-3993 00339 - 1.0185 1-0937 306.25 - 32887 + 9 94.09 5 2.9567 00764 -7953 9I5 1 8281 - 95-28 i. 3.61 6 65746 01536 2.2922 2.4722 342-25 36914 + 3- 10.24 I 4.3068 6.1294 02296 03657 - 15907 2.6782 2.9813 196^ 12611 - 218.18 - 8. - 98 67.24 96.04 9 ii 10.2319 203118 13 457 03982 S - 4-7904 - 10.8893 7-6305 5.1096 11.2806 8.0865 576.00 2i.-4.44 817-96 - 614.38 - 866*84 + 16^6 - 5-2 o 81 275 56 27.04 Iff 25444 3I-9958 w t 54 f - 4930.84 \ns\ 6l #f The correctness of the work up to this point is now verified by substitution in proof-formulae (44). Therefore the normal equations are as follows : 1.1197-r-f- .5i6Sj/4- 3.24693= 65.5013; .5168*+ .2544/4 1-37422 = - 3I-9958; 3.2469.* 4- I -374 2 .)' 4- n.ooooz = 179.0000. I 1 6. ECCENTRICITY OF SEXTANT. 205 For the solution of these equations we make use of the form given in Art. 32. [aa] 1.1197 / = 0.049102 [*] .516* ' = 9-713323 L[6si] 1.6550 / = 0.24638^ / - o 21880 9-4I53 [en] -179.0000 189.9403 [cj] 194.6211 204. 1009 '[54 [.]= 1.5847 9733 [cm] =10.9403 13-7974 [i] = - 9-4798 -12.9485 []= .6.. 4 I = 9-78633 [cm] 2.85^1 [c] + 3-4687 /z = o.66g59 n z = - 4 ".6 73 roof-Formula. i]=[H>i]-j-[6ci l^-^M. [nn] = 4654.34 3831-76 [H=-493o8 4 4"7-43 / I'. [6s vii. r VIII. [j VIII IX. [wi The wo rious stag IX or all of t 'i3 * [i]= 822.58 195-59 [ns i] - 8.3-41 - 183.56 rjg^i =o.66959n >*2] = 626.99 '3-35 [ns 2] - 629.85 - 16.21 rk is checked at the va- es by substitution in any ic above proof-formulae. [* 3 ]= 613-64 [ns 3] - 613.64 The elimination equations (56) are here rewritten for convenience: By substituting in these the coefficients, the logarithms of which are in the horizontal lines marked E in the foregoing scheme, we find y = - 147 -47; = + 23". 12. These values substituted in the equations of condition give the residuals v. For the final proof of the accuracy of the entire computation we have, Eq. (62), [ 3] = \vv\- The agreement, though not exact, is sufficiently close for our purpose, and as close as could be expected when the magnitude of some of the numerical quan- tities involved in the equations is considered. 206 PRACTICAL ASTRONOMY. 116. For determining the weights of x,y, and z we employ equations (76), by means of which we find f Z = .6114; py ~ .006l35; PX .01196. The mean error of an observation we obtain by formula (88), viz., The mean errors of x, y, and z are then given by equations (89): e f e e x = = 80 .21; e y =112 .00; f z = ^7 = n" .22. These quantities multiplied by .6745 give the probable errors. Collecting our results, we have the following values of x, y, z, with their probable errors: x - + 23".! 52".g; y= 147". 5 75"-s; 2 = - 4"- 7 7"-6. We next compute a table of corrections, to be employed with this instrument, by formulae (211) and (210), viz. : 4?" cos (i = JT; 4e" sin a = y; n ri = ^e" sin \n cos (&n a). We find 4 - (

+ (y - tf)| sin j[> - (y - d)] = V cos \\z + (^ + 0')] cos i[* - (

<5) = 26 4 28 ^[z -f- (

= 38 4' o" Barometer 26.05 Longitude L = i h 44"' 41* w. of Washington. Assumed A T = 6 41 7 INDEX CORRECTION. On Arc. Off Arc. 31' 50" 359" 28' 45" 31 30 28 40 31 40 28 40 31' 40" 359 28' 42" Index correction = 7 = u" From the refraction table we find Mean refraction = 59". i Barometer factor = .880 Thermometer = .946 Therefore r = 49". 2 From the American Ephemeris we find p. 248, eq. hor. parallax it 8". 72 p. 327, S = + 18 42' 16". 7 p. 327, equation of time E -j- 6 m 12 s .99 p. 327, semidiameter s = 15' 47". 7 S is interpolated from the ephemeris by the method explained in Art. 52. The ephemeris is given for the meridian of Washington; therefore we require the Washington time of our observation. Time of observation T = 3 h ,37"' 26 8 .3 Approximate correction A T = 641 7 Approximate local time = 20 56 19 Longitude = i 44 41 Washington time, July 28 = 22 41 o = i 1 ' ig" 1 o 9 before noon of July 29 At noon, July 29, S = 18 41' 29". 6 Hourly change July 28 = 35". oo Hourly change July 29 = 35"-77 Therefore the correction to 6 = I h .3i7[ 35.77-H.77X d .O55] + 47"-i At time of observation 8 = 18 42' 16". 7 At noon July 29. eq. of time = + 6'" I2 S .89 Correction for d . 055 = .10 E 6 m 1 2". 99 In taking E from the ephemeris, second differences need not be considered for this purpose, though it has been done in this case. 220 PRACTICAL ASTRONOMY. 125. If a sidereal chronometer had been used we should have had only to convert the mean time t -j- E into sidereal time, when we should have had A T by comparing with the observed time as now. It may be remarked also that in using a sidereal chronometer the observed sidereal time must be converted into mean solar time for the purpose of taking d and E from the ephemeris, since these are given for mean solar time. In reducing such a series as this it is perhaps a little better to reduce the readings on the two limbs separately; the two reductions will then mutually check each other. Of course the altitudes must be corrected for semiciiameter. If a con- siderable number of series have been reduced in this way the observer can see, by comparing results, whether his per- sonal equation is the same for both limbs. 125. By a single altitude of a star. It will be convenient to use a sidereal chronometer when practicable. Let & = the true sidereal time of observation ; = the chronometer time of observation ; A& = the chronometer correction. Then t is computed the same as above ; recollecting that for a star the semidiameter and parallax will be inappreciable, we have z = cp -(h 1 - r); (219) G = + a) = + //; A = (t + a) - @ (220) 125. TIME BY ALTITUDE OF A STAR. 221 Example 2. West Las Animas, Colorado. 1878, July 29.3. Observation of Arcturus for time. Observer B. Sidereal chronometer. Negus 1590. Means E Sextant. 87 40' 30 2O 10 87 oo 87 20' oo" - 18 - 42 Chronometer. i8 h n m 2g 8 .O II 55 -0 12 21 .O 12 46 .5 13 13 -o Latitude q> = 38 4'oo" Longitude L = i b 44 41" w . of Wash . Thermometer 74.o Barometer 25 .91 From ephemeris, or=i4 h io m 8*. 2 5=19 48' 58" A 0.10386 -f- 9.9 .02651 9.72786 -f 20.0 938525 +50.5 I8 U I2 m 20'. 9 sec

- 5) = > = 87 19' oo" 43 39 30 - 46 43 33 44 46 21 16 38 4' o" 19 48 58 18 15' 2" 64 36 18 28 6 14 32 18 9 14 3 7 t = 24 44' 33". 3 49 29 7 sin 2 t = sin |/ = 9.24348 9.62174 -f 27-5 or = 14 10 8 .2 6 = 17 28 4.7 = sidereal time Observed 6 = 18 12 20.9 = chron. reading M = 44'i6 8 .2 = chron. cor. [Eq. (220)]. It will be seen that the numerical work is somewhat less in case of a star than of the sun. In case a mean solar chronometer has been used, the side- real time (t -f a) must be converted into mean solar time by (202), and the resulting value compared with the chronome- ter time. 222 PRACTICAL ASTRONOMY. I2O. Example 3. West Las Animas, Colorado. 1878, July 27 3. Observation of a Corona Borealis for time. Observer B. Mean solar chronometer. Sextant. 95 50' 40 30 ?o 95 10 Negus 1326.. Chronometer. 17*' 3 m i6 8 .o Latitude

= o. 10386 -j- 9.9 sec 8 = .05061 -f- 4.5 sin 5 = 9.65113 -f 25.2 sin D = 9.43137 + 45-0 zA A r = 95 = 47 29 8 44 34 -46 h z S z-}-(cp d) z (g> 8) D = 47 = 42 = 38 = 27 = 10 = 53 = 26 = 15 43' 48" 16 12 4' o" 7 32 56 28 32 40 19 44 36 20 39 52 *, = 24 = 49 32' 43" 5 26 sin 3 \t = 9.23697 sin $t = 9 61848 + 27.7 rt = 15 29 34.1 6 = 18 45 55 .8 = sidereal time. This is now converted into mean solar time by equation (202). V = & 21 15 .7 = sidereal time of mean noon from ephemeris 6 V = 10 24 40 . i i 42 .3 Table II, Appendix to Ephemeris. M. S. time = 10 22 57.8 Chronom. = 17 4 6.2 AT= - 6 41 8.4 126. Conditions most favorable to accuracy in determining time by a single altitude. As our data will always be liable to more or less uncertainty it becomes a matter of great practical importance to so arrange our observations that small errors in the quantities regarded as known shall have the least effect on the computed value of /. * These quantities are written down so that we may employ them in computing the differen- tial formulae when desirable. (See Articles 128-131.) 128. DIFFERENTIAL FORMULAE. 22$ As we require equations (121), we rewrite them here for convenience of ref- erence. cos A cos a = cos 6 cos t sin

sin a From this we see that for a given latitude cp a small error dh in the altitude will produce the least effect when sin a has its greatest value, viz., when the star is on the prime vertical. Also, that for a constant positive error dh the error produced in / will be T when the star is ^ r of the meridian, and may therefore be eliminated by observing both east and west stars. (221) also shows that dt will be least when cos

- (cp - 6)] = igo - 2/i - i(9> - 5). \ First differentiate (224) with respect to zh and \t. We find 2dl sin if _ dl sin S dS dzh dl sin D dD dzh d^t ' dS ' ^z/i ' ~d~\t ' ' lizh ' ~d~\( dS dD I From (225), ^ = ^ = - -. Therefore we nave, writing - vrr~ = ^ sin \t and = J/sin S . . . , d\t a?) dt _ Als'm S -\- Als'm D ~dzl ~ 4^7si^l/ The quantities Al sin S, Al sin D . . . are the rates of change of the loga- rithms for the values of S. D, etc., employed. It requires, therefore, very little time to take these from the tables while computing t, as we have done in the examples in the foregoing pages. Thus, in example i we have found ///sin 5=19.9, which is the change expressed in units of the last decimal place of log sin S produced by a change of i' in 5. In practice the /sin of the angle 5' less than 5 is subtracted from that of the angle 5' greater, and the difference divided by 10. This is a little more accu- rate than to take the difference between consecutive logarithms. 131- DIFFERENTIAL FORMULM, 22$ In our example S = 32 24' /sin 32 19' = 9.72803 /sin 32 29' = 9.73002 Difference for 10' = 199 Difference for i' =// = ig.g In like manner we have found Als'\nD = 54.6 J/sin \t = 28.7 A correction to the assumed value of zh may result from a variety of causes, such as the employment of values of the refraction, parallax, index error, or eccentricity, which are only approximately correct, or from errors in the pre- liminary computation. Suppose the value of zh employed in example i was found to require the cor- rection Azh = i'. Then the resulting correction to the hour-angle would be At = .649 X = 2 s . 596. 130. For the value of ^ we differentiate (224) with respect to t and d, viz., zdl sin It _ dl sec <5 dS dl sin S dS d$ dl sin D dD dS ~ dS ' ~dt ~" ZS ' <7<5 ' ~ ' ' ~~ ' . and t, viz., d cp Zdl sin \t _ dl sin D dD dq> dl sec

^i/ dS ' ~d

dt zAl sec cp + Al sin S Al sin D Therefore - = - -j -. T . dq> zA sin / For our example I we have by this formula 19 8 -j- 19.9 54.6 dt 57-4 = - .260, (228) and a correction of i to the assumed latitude produces a corresponding cor- rection to the time of At .260 = i s .04. Probable Error. 132. By means of formula (226) we may reduce the time of each altitude to the time of the mean altitude for the purpose of comparing the individual meas- urements and computing the probable error. The application to example I will sufficiently explain the process. The mean value of 2/1 is 89 10'. so that each time will be reduced to the time corresponding to this altitude. Further, as one half the readings were made on the lower limb and one half on the upper limb, we must add to the latter and subtract from the former the time required for the sun to move in altitude over an arc equal to the sun's semidiameter, or in double altitude a space equal to the diameter. Thus we have see example i Semidiameter of sun = S = 15' 47". 7; Diameter of sun = 31'. 590. From previous article, - = .649. dzh Therefore reduction for semidiameter The reduction is now as follows: = .649 X 3i'.S9QX6o = Limb. Observed a*. * Correction for Semi- diameter. Observed Time. Reduced Time. V. I'V. Upper 88 50' + 20' + 5''.9 + I m 22".0 3 h 35 m 12 .. Q 3 h 37 m as . g 16 89 o + IO + 26.0 35 39-5 27 -5 + T - 144 89 10 .0 30 35 25 -5 _ 64 89 20 8 9 3 10 - Si -9 36 3-5 36 56.5 26.5 26 .6 I: 4 9 Lower 88 50 + 20 + 5' -9 I 22 .0 3 37 55 -5 25 -4 _ 81 89 o -j- 10 + 26 .0 38 22 . 26 .0 89 10 o o 38 48. 9 8 9 20 89 3 10 20 - 26.0 - 5i -9 39 14 -5 39 4i -o 26.5 3 37 27 .1 i: 4 64 Mean 3" 37 2 6". 3 [] = 4 . 04 134- DIFFERENTIAL FORMULAE. 22/ Then by formulae (27), probable error of single observation = r = '.43; probable error of mean = r = '.14. The reader must not fall into the error of supposing that this quantity repre- sents the actual probable error of a determination of time by this method, since no account is here taken of the relatively large constant errors to which observa- tions of this kind are liable. The subject will be considered more at length hereafter. (See Art. 156.) Corrections for Refraction and Motion in Declination. 133. The refraction of the atmosphere and the sun's motion in declination affect the computed value of At by small quantities, which it may be considered desirable to take into account in a more refined discussion. Correction for Refraction. Since refraction decreases with the altitude, it fol- lows that when the sun's altitude increases by a given quantity 10' for example as measured with the instrument, the actual space passed over is greater than 10' by the difference of refraction for the first and last position. Thus, instead of simply Aih as used in our formula, we should employ Azh + lAr, Ar be- ing the difference between the refraction for altitude h and that for h -f- Ah. For our example we find for the mean altitude of the sun, viz., 44 34', Change in refraction corresponding to 10' altitude = o".3o = zAr. Therefore the correction to At corresponding to Azh = 10' is .649 X -^ = '.013 This must be added to the computed interval, viz., At = 25*.g6 At = 25-.Q73 134. Correction for Sun's Motion in Declination. Since the sun's declination is not constant, but is ever increasing or diminishing, the time required for the altitude to change by a given amount will be slightly modified by this cause. For our example with Aih = 10' we find At = 25". 97. Referring to the example, we have found the hourly motion in declination to be 35".?; there- fore in the interval 25". 97 the change is ".26. By formula (227) we have found for this example = .754. Therefore correction to At = .754 X = + ' 013. Therefore the final value of At corresponding to Aih 10' is 25'.986. 228 PR A CTICAL A STRONOM Y. ' J 3 5 If both limbs are reduced together, as in our example, the reduction for semi- diameter should be corrected for motion in declination, but not for refraction, since both limbs are observed at the same altitude. Determination of Time by Equal Altitudes. 135. By a star observed at equal altitudes east and west of the meridian. Method of observing. When the star is at some distance east of the meridian (the nearer the prime vertical the better), measure with the sextant a series of five or more altitudes in the manner already explained (Arts, in, 112, and 113); then, a short time before the star reaches the same altitude in the west, set the vernier at the reading of the last altitude and observe the same number of alti tudes as before at the same readings. Some observers prefer to take only one reading east and then lay the in- strument where nothing will disturb it until it is time for the west observation. In this way both observations are secured at absolutely the same altitude so far as it depends on the reading of the instrument ; but there is the objection that only one reading can be made, which more than neutral- izes the advantage. No correction for index error, refrac- tion, or parallax is required. Now, as the declination is constant and the altitudes the same, the numerical values of the hour-angle measured east and west of the meridian will be equal. Suppose a sidereal chronometer used. Let & = the chronometer time of the first observation; " = the chronometer time of the second observation; A = the chronometer correction. Then the sidereal time of the star's meridian passage equals its right ascension a. 136. EQUAL ALTITUDES OF A STAR. 22g For the first observation a = 0' -{- AQ -|- t\ For the second observation a = 0"-|- ^/0 /. From which J0 = or (' -(- 0") (229) Example i. 1856, March igth, equal altitudes of Arcturus east and west of the meridian were observed as follows: East of meridian, 0' = ii !l 4'" 51". 5 West of meridian, 0" = 17 21 30.0 i('+ 0") = 14 13 10.75 From ephemeris, ex = 14 9 7.11 Therefore ^0 = 4'" 3 S .64 136. If a mean time chronometer is employed, the sidereal time of the star's culmination (which is equal to the right ascension) must be converted into mean time, and this com- pared with the mean of the observed times as before. Example 2. 1856, March I5th, equal altitudes of Spica were observed as below, the time being noted by a mean time chronometer: Latitude cp = 33 56' Longitude L = i h 13' 56* from Greenwich. CHRONOMETER. SEXTANT. CHRONOMETER. East. Double Alt. West. IO h 20 ra s . 5 104 o' 2 h 40' 38'. 20 p 28 IO 40 IO .5 20 55 20 39 42 T' io h 20 ni 27 8 .83 T" 2'' 40"' io 8 . 17 ") = 12 30 19.0 From ephemeris, a = = 13 17 37 92 Then Art. 95 from ephemeris V = 23 32 53 .22 O - r = 13 44 44.70 Table II, ephemeris, 2 15.12 Mean time = 13 42 29.58 \(T- + T"} = 12 30 19.00 Therefore J T = -f- i 12 10.58 230 PRACTICAL ASTRONOMY. 137. By equal altitudes of the sun. This method is less simple when applied to the sun, for the reason that the sun's declination cannot be considered con- stant for the interval of time between the morning and after- noon observations. The mean of the observed times will not therefore be the time of meridian passage as in case of a star. The correction due to this cause is called the equation of equal altitudes. To determine its value we proceed as follows : Let AS = the hourly change in declination taken from the Nautical Almanac. Then tAd = the total change in S in the time / ; dt = change produced in / by the increment tAd of tf. Then since t = f(6\ t + dt f(S -f tAd} ; and neglecting terms of higher order than the first, 3* = %'*^ (230) To determine -^ we differentiate the last of equations (121) with respect to / and d, viz., dt _ sin (p cos S cos cp sin d cos / _ tan cp tan S dS ~ cos (p cos 8 sin t sin / tan t ' Therefore substituting this value in (230), and dividing by 15, as St is required in seconds of time, we find ftan

Now suppose a mean time chronometer used, and let 7"' and T" = chronometer times of east and west observation. 138. EQUAL ALTITUDES OF THE SUN. 2$l Then will t dt = the hour-angle of the A.M. observation ; t -f- dt = the hour-angle of the P.M. observation ; E = equation of time. Then E = T' -f- AT -{- (t dt} from A.M. observation; E X" -\- AT (t + dt) from P.M. observation. From which AT=E- \k(T + T") - df\. ..... (232) Example 3. 1856, March 5th, at the U. S. Naval Academy the sun was observed east and west of the meridian as follows: East, 7*' = i h 8 m 26'.6 West, T" 8 45 41 .7 Latitude

= 9.9081 tan S = g.oo42n sin t 9.9243 tan / = .1900 = - 4 h 45 m i3'.86 9.9838 *A = 1.1696 *B = 1.1980 log t .5809 log AS = 1.7642 log ^ = 8.8239 log dt 1.1812 138. Equal altitudes of the sun observed in the afternoon of one day and the morning of the day following. In this case the mean of the observed times plus the neces- * See tables of addition and subtraction logarithms. 232 PRACTICAL ASTRONOMY. 138. sary corrections will be the time of the sun's passing the lower branch of the meridian, or midnight. Let t' = the sun's hour-angle, reckoned from the lower branch of the meridian. Then /' = t + 180 ; sin t = sin t' ; tan t = tan /'. Therefore for this case (231) becomes (p tan = 9 9750* tan 5 = 9.4356 8t = 22.2 sin /'= 9.9809 tan t' = .5179,2 I2 h -(- . = it 56 41 .33 9994i = 2 h 4 m 40'.4 A = 1.0764 .# = 1.1114 log t .8528 log JS = 1.6411 = 8.8239 log (- 8t) = 1.3469 140. LATITUDE. 233 139. The chief advantages possessed by the method oi determining time by equal altitudes are the following: the computation is very simple, and no corrections are required for parallax, refraction, semidiameter, or instrumental errors, nor is a knowledge of the latitude required, except very rough- ly, when the sun is employed. The disadvantages are the diffi- culty and oiten impossibility of obtaining the observations at exactly the same altitude, owing to clouds or other hinder- ances ; also, the changes which often take place in the re- fraction between the morning and afternoon. A correction for this last mentioned source of error may be computed by means of a differential formula, but it has not been thought necessary to develop it here. Latitude. 140. We have seen (Art. 63) that the astronomical latitude of any place is equal to the declination of the zenith of that place, or to the elevation of the pole above the horizon. The distinctions between the different kinds of latitude, as denned in Art. 73, must be borne in mind. We are at present only dealing with the astronomical latitudes there defined. It is perhaps unnecessary to state that all formulae derived will be applicable to either north or south latitude, care being taken to use the proper algebraic signs : n r . j| [ latitudes and declinations being First Method. 141. By the zenith distance of a star observed on the meridian. Resuming the last of equations (121), 234 PRACTICAL ASTRONOMY. 141. cos z = sin q> sin d -f- cos

8) ,- z =

= d z\ For lower culmination q> = 180 S z 9 . The mean of which gives

38 53' 38". 8 By the second method we have Third Method. 143. By an altitude of a star observed in any position, tlie time being known. , the sidereal time, is known ; <*, the right ascension, and S, the declination, are taken from the Nautical Almanac. We then have t = & a. This will be given in time, and must be multiplied by 15 to reduce it to arc. We then have sin h = sin cp sin 3 -f cos (p cos 3 cos t ; in which q> is the only unknown quantity. For solving the equation introduce two auxiliaries, */and D, determined by the equations d sin D = sin tf ; ....... (a) d cos D cos d cos / ..... . (a'} The above equation then becomes, by substituting the value of d from (a), cos < /? = sin h sin D cosec $. 143- ALTITUDE OBSERVED AT ANY HOUR-ANGLE. 237 Dividing (a) by (rt')'to determine D, we have the following formulae for determining (p: tan D = tan # sec t ; cos <> D = sin / sin Z> cosec | f V f Z> is taken less than 90, + or according to the algebraic sign of the tangent. (q> D), being determined in terms of the cosine, may bo either -|- or . There will therefore be t\vo values of the latitude which will satisfy the above condi- tions. Practically an approximate value of the latitude will always be known with accuracy sufficient for deciding this ambiguity. Example. On March 4th, 1882, I observed the following double altitudes of Polaris with a Pistor & Martins prismatic sextant and artificial horizon : Sextant. 79 12' o" 10 50 10 30 10 5 9 50 Clock. io h 43 m 4" 43 56 45 2 45 50 47 45 Means 79 10' 39" Index correction / i 2.0 I0 h 4 -m y9-4 40 +1.5 Refraction 5 = t D = - h -

D] being negative, the angle must be in the second or third quadrant. If we had taken it in the third quadrant we should have found (p = 141 +. As q> is never greater than 90, this value is in any case excluded. 144. Effect of Errors in the Data upon the Latitude determined by an Altitude of a Star. Differentiating equation (g), Art. 126, regarding h and (p as variable, and reducing by equation (e), we readily find From this we see that a small error in the measured altitude will have the least effect on the latitude when the star is on the meridian. Again, differentiating the same equation with respect to

= tan a cos tpdt ; ....... (239) from which it appears that the effect upon (p of a small error, dt, in the hour- angle will be least when a is zero or 180. It appears, therefore, that the latitude will be determined with greater accu- racy the nearer the star is to the meridian. When the star is very near the meridian the method which follows will be preferable. Fourth Method. 145. By cir cummer idian altitudes. When the latitude is determined by the altitude of a star observed on the meridian, the accuracy is greater than in any other position, and at the same time the computation is extremely simple. We can, however, only measure one altitude when the star is on the meridian ; and frequently at the time when the observation is made we shall not know the chronometer correction with sufficient accuracy for determining the exact instant when this observation should be taken. If, however, altitudes are measured near the meridian (how near we shall discuss later), the observed altitudes may be reduced to the meridian alti- tude by a simple computation. It will thus be possible to 145- LATITUDE BY CIRCUMMERID1AN ALTITUDES. 239 make a considerable number of measurements instead of rely- ing on one alone. When this method is applied observation is begun if possible a few minutes before culmination, and a series of altitudes measured in quick succession so as to have about the same number on each side of the meridian. Altitudes measured in this manner are called circumme- ridian altitudes. It is not essential, however, that the series should be symmetrical with respect to the meridian ; the method is equally applicable to the reduction of one or more altitudes taken on either side of the meridian if sufficiently near. Let h = any altitude of a star corresponding to the hour- angle / ; // = the altitude when the star is on the meridian ; 2 g = the zenith distance = 90 // = cp 8 Then sin h = sin cp sin d -f- cos cp cos d cos /. Let us write for cos t its value, i 2 sin't^. Then the above equation becomes sin h = cos z cos (

= $ z + Am Bn =F G?. . . . (242) 146. This computation is made very simple by the use of table VIII, where m and n are given with the argument / ex- pressed in time (the last term, Co, is seldom used). As A and B will be constant for the entire series, we shall have, If z a z v z 3 , etc., Zp, are the observed zenith distances, m,, m m^ etc., m^ the corresponding values of m taken from the table, 148- LATITUDE BY CIKCUMMERIDIAN ALTITUDES. MI, M*> n *, e tc., n^, the corresponding values of n, cp = 3 Zi rp Am, Bn,; cp = has been used which proves to be considerably in error, it may be necessary to repeat the computation of A, using for

and d. Thus, referring to the table, we see that if cp = 40 and 5 = o, then / = 4O m ; or, in this case, the error committed in neglecting this term amounts to i" only when the star is 40 from the meridian. If cp = 40 and 5 = 23 about the maximum declination of the sun, then t = 2O m . LIMITING HOUR-ANGLE AT WHICH THE THIRD REDUCTION AMOUNTS TO ONE SECOND. Latitude. Declination same sign as Latitude. Declination different sign from Latitude. 80 70 | 60 50 40 30 20 to" 10 20 30 40 50 60 70' 80 20 3 4 50 60 7 128 118 107 95 82 67 45 fa e 54 37 67" 59 5* 43 S 2 "9 VJ 43 9 16 *9 4 9 40 29 m 32 2! 23 j 12 2<,"' II T I 111 O m Q 99 P* 90 20 38 37 47 i ^8 s6 67 to 89- * 75 47 59 67 75 67- 75 2?' a 16 54 X4 5 4 3S 5' 73 43 59 02 Let us now consider the term /c I \s cos Q) cos 5\ s 2 sin 4 \t . I cot 2 ^- = Bb. . cot 2 sm (

9 9 33 28 20 7 6c, 50. o 3'-' 20 10" 10 20 3" 40 50* fe" 70 80 ,,m 33 ''9 20 27 m . 2I m ,,'m IS" 8 5'" <* f 8" IS" i6 3I m 7* r 17 *3 14 7 6 5 6 t 10 5 9 13 i 16 '5 '9 23 89 40 11 a 37 1 99 37 19 o j j 3 22 '7 " 9 4 33 39 s 149- LATITUDE BY CIRCUMMERIDIAK ALTITUDES. 243 If we are able to choose our own times for observing, we can always make our measurements so near the meridian that these terms may be neglected. As i" is much within the error of an ordinary sextant measurement, the limits may'be extended somewhat beyond those of the table without serious error. We may, in a similar manner, determine for what values of t Co or Bn will have the values o".i, o".oi, or any other value. Lower Culmination. 149. When the star is observed near the meridian at lower culmination, the hour-angles should be reckoned from the lower branch of the meridian. This is equivalent to substi- tuting 1 80 -f- 1 in the formula in place of t. We then have cos z =. sin (p sin d cos (p cos d cos /. Writing, as before, cos / = i 2 sin'^/, this becomes cos z = cos ((p -\- 6) -f- cos

2O 7 27 .5 Star's declination d = 8 32' n''.5 * If the rate of the chronometer is appreciable it must be taken into account For the simplest manner of doing this see Art. 152. I49 LATITUDE BY CIRCUMMERIDIAN ALTITUDES. 245 q> 49 01 . 5 = 8 32 .2 S = 40 28.8 ^ = .9991 B 1.169 cos cp = 9.8168 cos d = 9.9952 cosec 2 = .1876 log A = 9-9996 log A* = 9.9992 cot Z = .0688 = 0.0680 The observations and method of reduction are shown in the following tabular statement, which will be sufficiently explained by reference to formulas (XIII). Sextant. 2/4. h. Chronometer. &'. t. T 99 5' 35" 49 32' 47". 5 20 h jm 35, - 5 m 52'. 5 2 6 10 33 5 2 37 4 50.5 3 7 5 33 32 .5 3 57 3 30.5 4 7 55 33 57 -5 5 5 2 22 .5 5 8 10 34 5 $ 41 - 46.5 6 8 o 34 o 7 52 + 24-5 7 7 50 33 55 8 51 I 23-5 8 7 40 33 50 9 47 2 19-5 9 7 5 33 32 .5 10 41 3 13 -5 10 99 6 55 49 33 27 .5 20 12 O +4 32 .5 Am. ..* Bn* A + Am - Bn. V. vv. i 67". 8 67"-7 ".01 ".OI 49 33' 55"- 2 4-6 21. l6 2 46 .0 46 .0 .01 .01 51 .0 8.8 77-44 3 24 .2 24 .2 56 -7 3.1 9.61 4 II .1 II .1 68 .6 8.8 77-44 5 I .2 I .2 66 .2 6.4 40.96 6 3 3 60 .3 5 25 7 3 -8 3 -8 58 .8 I.O I.OO 8 10 .6 10 .6 60 .6 .3 64 9 20 .4 20 .4 52 .9 6.9 47-61 10 40 -5 40 .5 49 33 68 .0 8.2 67-24 Mean h = 49 33' 59". 8 Index error = \I = I 51 .5 Eccentricity =$E= 10 .1 Refraction r = 47-3 [w] = 343-35 '= 3"-9 ro = I -3 * It is easy to see in advance than the term Bn is inappreciable in this case. J.t is introduced here to illustrate the method. 246 PRACTICAL ASTRONOMY. 150. Corrected altitude = 49 31' io".g Zenith distance z = 40 28 49 .1 Declination d = 8 32 n .4 Resulting latitude q> = 49 i o .5 i".3 If it is not considered necessary to reduce each observa- tion separately, the work is abridged somewhat by the fol- lowing process [see Art. (146)] : Mean of zh = 99 7' 14". 5 Index / = 3 43 .o Eccentricity E = 20 .2 Corrected zh 99 3 n .3 h = 49 31 35 .6 Mean of m = 22".6 = m' Am' = + 22 .6 Refraction 47 .3 Am' = 22".6 Corrected h = 49 31 10 .9 Zenith distance z = 40 28 49 .1 Declination d = 8 32 n .4 Latitude

cos d 9, = , + <*. + /- ^- - . 2 sin 2 if, etc. (250) The peculiarity of the process is in the method by which the small term t . -7- is taken into account. For this pur- pose we determine the value of / corresponding to the maxi- mum value of h by placing ,- equal to zero and solving for t. Take the equation sin h = sin cp sin d -\- cos

cos 3 k is given in table VIII, C. If a star is observed with a mean time chronometer whose rate is ST, the factor Vk will convert the chronometer inter- vals into mean time intervals ; we then require the factor /.<* = 1.00273791 to convert these mean time intervals into sidereal intervals. The formula for computing A will then be , COS Q> COS $ A = ktf -- - , ..... (261) sin 2. where log n .0011874. If the sun is observed with a mean time chronometer the intervals of the chronometer corrected for rate will not correspond exactly to the solar intervals, as these will be apparent time intervals. * See Art. 93. 252 PRACTICAL ASTRONOMY. 152. If we let SE = the increase of the equation of time in one day, then (one apparent solar day) = (one mean solar day) SE, and ST SE = the chronometer rate on ap- parent time, k will then be given by the formula (262) Finally, if the sun is observed with a sidereal chronometer, we must introduce the factor - to convert the sidereal inter- vals into mean time intervals. The log- =9.9988126. The formulae for the four cases are then as follows: k = r _ __ f 8T ~ Sr \*' 86400 86400 , COS CD COS S Star with sidereal chronometer, A k ; sin 2 , cos cp cos S r(XV) Star with mean time chronometer, A = [p. 002375]^ : sin z Sun with sidereal chronometer, A = [9 997625]^' k and k' are. taken from table VIII, C. Example. Determination of latitude by circummeridian altitudes of the sun. 1869, July 24th. Des Moines, Iowa. Observer Harkness. Instruments: Sextant and Mean Time Chronometer. The declination, equation of time, etc., are taken from the ephemeris for the 15-- CIKCL'MMERIDIAN ALTITUDES OF SUN. 253 instant of the sun's meridian passage at Des Moines = i h 6 16* apparent time at Washington. Assumed Latitude cp = 41 35'^ Longitude L = -f- i h 6 m 16* Chronometer correction A T = 6 18 8.9 From ephemeris, d = 19 46' i6".i Ab - 31 -94 Equation of time = -{- 6 m I2 8 .o Semidiameter S 15' 47". 2 Equatorial hor. parallax it = 8 .44 Computation of ,4 and B. ' 290 .3 920 .9 849 .2 737 -5 662 .7 549 -5 .68 :S 37 3 .20 6". i 5 -2 4 -3 3 -3 1:1 682 5 '28".o 20 .7 14 .9 07 53 49 -2 35 -o 52 -7 Mean h O = 68 25' 21". 2 = 67 53' 45". 6 Semidiameter S 15 47 .2 -|- 15 47 .2 Refraction r = 21 .6 21 .8 Parallax / = + 3 .1 + 3-2 Index cor. \I = -J- i 10 .4 -[- I 10 .4 Eccentricity \E -f 14 .8 -j- 14 .8 Corrected h = 68 10' 40". 7 68 10' 3g".4 Mean h = 68 10' 40". o z = 21 49 20 d = 19 46 16 Resulting latitude

cos d cos /, it becomes sin h = sin (h x] cos/ -f- cos (h x] sin/ cos t. (a) Expanding sin (h x) and cos (h x) by Taylor's, and 153- LATITUDE BY POLARIS. 257 sin/ and cos p by Maclaurin's formula, we have, as far as terms of the order/ 4 and x\ sin (hx) = sin h x cos // %x* sin h + i* 3 cos h -(- ^x* sin h\ cos (A ^r)=cos /z -j- x sin // ^ 2 cos h \x* sin sin />=/-/; cos/ - i - i/ 2 + Substituting these values in (a), we readily obtain x =1 p cos / \(x* 2xp cos / -(- / 2 ) tan // cos / + 3.r/ - / cos /) ( tan h. Which contains all terms in /and or, from the first to the fourth orders inclusive, x must now be determined from () by successive approximations. For the first approxima- tion let x = p cos / Substituting this value in the second term of (U) and retain- ing terms of the order/ 2 , we find for the second approxima- tion x = p cos t i/ 2 sin 2 / tan h ..... (d) Substituting this value in the second and third terms of (#) and retaining terms of the order/ 3 , we find the third ap- proximation, viz., x = / cos / / a sin 2 / tan h -(- ^/ 3 cos / sin" /. . (e) Similarly for the fourth and final approximation, x = p cos t ^/ 2 sin 2 / tan h -(- -J/ 3 cos / sin 2 t |/ 4 sin 4 / tan 3 // -\- ^V/ 4 (4 9 sin 2 /) sin a t tan h.(f) 258 PRACTICAL ASTRONOMY. 153. As x and / will be expressed in seconds of arc, the series must be made homogeneous by multiplying/ 2 by sin i'',/ 3 by sin 2 i", and/ 4 by sin 3 i". Then the expression for the latitude is

76 tan 3 //. 153- LATITUDE BY POLARIS. 259 This term will then be only o".oi in latitude 48, and o".i in latitude 67. It may therefore always be neglected when the instrument used is the sextant. Writing v = i/ 3 sin 2 i" cos / sin 3 t, dv forming -,,-, placing it equal to zero, we readily find that v is a maximum when sin 2 / = f . The maximum value of this term will then be o".333. If then we drop this term with those which follow, the error introduced in this way will seldom amount to half a second, and will generally be much smaller as the maxima values of the different terms occur for different values of /. Therefore for determining the latitude by Polaris by sex- tant observation, = _ a _; (p k p cos / -+- [4.384S4]/ 2 sin 2 / tan h. \ Let us apply this method to the example solved in Art. 143. We have given From Nautical Almanac. By Observation. a = i h i5 m 6 s .o h = 39 33' 38". 8 5 = 88 41' 6". 2 &= io h 45 m 7'.4 Therefore/ = 4733". 8 AQ + i .5 Therefore t 142 30' 43". 5 constant log 4.38454 log/ = 3-675210 log/' 7.35042 cos / = 9 899537,, sin 2 / 9.56866 tan h 9.91704 First correction i 2' 36". 2 log = 3.574 747 Second correction -f- 16 .6 log 2d cor. = 1.22066 Therefore (p = 40 36' 31". 6 260 PRACTICAL ASTRONOMY. 154. We find the third correction to be o".24, which makes the value of cp agree exactly with the value before found (Art. 143). Tables have been prepared with the design of abridging this computation, but the direct application of the formula is so simple that tables are of no great advantage, especially if the third and fourth corrections are not required. Correction for Second Differences. 154. When a series of, say, ten altitudes is observed, if the measurements are made in quick succession, so that the arc of the circle in which the apparent motion of the star takes place does not differ appreciably from a straight line, then the mean of the observed altitudes will be the altitude corresponding to the mean of the times. If, however, the deviation from a straight line is ap- preciable, this mean altitude will require a correction which may be obtained as follows: Let ti, ft, / 3 , . . . f u be the times of observation; hi, hi, ti 3 , . . . h^ be the observed altitudes; h = the altitude corresponding to the time /<>; Ati = / ti, from which t a = *, -f- At^\ At* = to - ^ *o = / + ^A>; Then A. = f(t ) hi = /fr) ; A* = / cos 5\ s . a And since cos A = sin z, this equation becomes ~ = - A cos A, + ^ 2 sin 2 /, tan >4 (266) The quantities At\, At?, etc., will be expressed in seconds of time ; they must be reduced to arc by multiplying by 15. Also, I5^/] 2 , etc., must be multiplied by sin i" in order to make formula (264) homogeneous. The last term will there- fore be multiplied by i(i5) 5 sin i", the logarithm of which is 6.73673 10. Therefore formula (264) becomes ' - [6.73673] * " ' " . . ( 2 6 7 ) 262 PRACTICAL ASTRONOMY. 155 As an example, we may apply formula (267) to the observations of Polarig given in Art. 143, where we have ti = 1235.4 At? = 15227.6 t, = 71 .4 J/T = 5098.0 t a = 5 .4 At* = 29.2 / 4 = 42 .6 AT* = 1814.8 /. = - 157 -6 Mean = 9401.5 log = 3-9732 By formula (265), with the data given in Art. 143, 1 constant logarithm = 6. 7367 Correction = o".lo log = 8.9997 We may in a manner precisely similar derive the correction to be applied to the mean of the times, to obtain the time corresponding to the mean of the zenith distances: this may be more convenient in certain cases. The necessity for applying a correction for second differences may generally be avoided by dividing a long series of observations into two or more parts, neither of which shall embrace an interval of time long enough to require such correction. This proceeding has the advantage that in reducing the two halves of the series separately they will mutually check each other. 155. The methods of determining time and latitude which have been given in this chapter are especially adapted to the requirements of the explorer. The observations can generally be obtained more conveniently at night, and both time and latitude will be required. From the observed time the longitude will be obtained, as will be explained more fully hereafter. As we have already shown, the time will be best determined by observing two stars, one east and one west of the meridian, both as near the prime vertical as prac- ticable. The latitude will generally be most conveniently deter- mined in the northern hemisphere by observing Polaris 155- GENERAL REMARKS. 263 north, and another star south, by circummeridian altitudes. Then, with the best attainable approximation to the latitude, the time can be computed by the method of Art. 125. With this value of the time the correct value of the latitude may then be determined by (XIII) and (XVI), and if this differs much from the assumed latitude the time must be recom- puted. In extreme cases it may be necessary to recompute the latitude, but with proper care this need not often occur. As a survey of the line of travel is generally made by means of a compass and odometer (which is a little instru- ment for recording the number of revolutions of a cart- wheel), the observer always knows his position approxi- mately. The same process, essentially, is followed at sea, where the approximate place of the vessel is always known from the " dead reckoning," which is the course as indicated by the compass and log. The methods of this chapter are those which are most con- venient and useful in practice. On land, where the observer has a certain degree of choice as to time of observation and methods, and where the results must have a considerable degree of accuracy to be of any value, it will seldom be de- sirable to employ others. At sea, however, the case is some- what different. It sometimes happens that the determina- tion of the place of the vessel is of the greatest importance when, from cloudy weather or other causes, observations cannot be obtained which are suitable for the employment of the methods of this chapter. Further, a high degree of accuracy is not required for purposes of navigation. Vari- ous methods of determining the place of a vessel are there- fore given in works on navigation, in order that the mariner may be in a position to utilize any data which he may obtain. It can readily be seen that by varying the conditions a great variety of solutions of the problem may be obtained. Some of these are exceedingly elegant from a mathematical 264 PRACTICAL ASTRONOMY. 155. point of view. Such, for instance, is the method given by Gauss for determining both the time and latitude from obser- vation of three stars at the same altitude. Thus if k is the common altitude, #, d', 6" the declinations, /, / -j- A, / -j- A' the hour-angles of the three stars respectively, we have Three equations from which t and g> may be found. Further than this, as there are three equations, we can also determine h from them, so that the altitude need not be measured at all, but only the instant of time observed when each star reaches the altitude //. If, however, the altitude is measured by the instrument, this process shows the error of the instru- ment, thus giving us one equation for determining the eccen- tricity by Art. 116. If three altitudes of the same star are measured, a similar process gives us three equations for determining the latitude, hour-angle, and declination of the star. Also, it is evident that two measured altitudes either of the same star or of different stars will give two equations of the lorni of (268), from which the latitude and hour-angle may be determined.* A variety of cases may also be considered in which the measured quantity is the azimuth of a star, or three different altitudes of the same star and the differences of the azimuths, or the data may be varied in many ways ; but these solu- tions are of little practical value. * For a solution of this problem graphically, see Captain Sumner's New Method of Determining the Place of a Ship at Sea. 156. PROBABLE ERROR OF SEXTANT OBSERVATION. 26 ] Probable Error of Sextant Observations. 156. In all instrumental measurements the error of the result obtained con- sists of two parts: first, that due to the observer; and second, that due to instru- mental and other sources with which the observer has nothing to do. When the instrument employed is the sextant, the latter consists for the most part of the various undetermined errors noticed in Articles 114-117. In any given series of observations these affect all alike, and therefore nothing is gained in this direction by increasing the number of individual measurements. With the first class, however, the case is different. These form the accidental errors of observation, and, as they occur in accordance with the law of least squares, their effect diminishes with an increase in the number of measure- ments. Let A' = the probable error of the mean of a series of observed altitudes; .A"] = the error due to the observer, not including personal equation; Ri = the error due to instrument and causes other than the observer. Then, by Art. 16, A'o = VAY4- AY ......... (269) Thus if the observer could do his part perfectly, he could never diminish the probable error of a single series below /I'j. The values of J? , A",, and R-> for a given instrument and observer may be determined by methods which we have already employed. Thus (Art. 132) we have found for the probable error of the time determined by a series of ten double altitudes of the sun, RI* = ".14. The corresponding error in the double altitude zh is found by the differential formula, viz., and for this case we have found - = .640. azn Therefore Jzh = *' I4 X -^ = 3". 2 = #,". .049 From the latitude observations (Art. 149) we have found 2". 6 = AY'. By a discussion of the ninety individual measurements of altitude employed in the investigation of the eccentricity of the sextant (example, Art. 116). Prof. Boss finds the probable error of a single measurement of double altitude to be 14'', and of the mean of ten measurements 4". 4 = AV From the solu- tion of the equations of condition of the same example we found for the probable 266 PRACTICAL ASTRONOMY. 1^6. error of a single equation R* = 5". 9. Therefore by equation (269) R* = 3". 93. Thus the instrumental probable error is nearly equal to the observer's probable error of a mean of ten measurements. If now we assume the probable error of a single measurement to be 14" as above, we have for the observer's probable error of the mean of m measure- ments, by equation (25), and the total probable error # = i/ h J 5 -45- R* = 14". 5; m = 10, J? = 5.9; m 50, /i' a = 4.4; /? = 7 .4; m = 20, R = 5.0; m = 100, >fo = 42. Thus it appears that with a skilled observer almost nothing is gained by ex tending the number of observations of a given series beyond ten. Instead, therefore, of multiplying observations in the same circumstances, when accuracy is desired, the circumstances must be varied with a view to eliminating the in- strumental errors. Thus for good results a determination of time or latitude should never depend tn a single series, no matter how carefully made or how elaborately the instru- mental errors have been investigated. Latitude should be determined by both Aorth and south observations, giving both equal weight, no matter whether determined from an equal number of measurements or not. In like manner time should be determined from observations both east and west combined with equal weights. (See also Harkness, Washington Observations, 1869, Appendix I, page m.) CHAPTER VI. THE TRANSIT INSTRUMENT. J 57- When the time is required with extreme accuracy, as in a careful determination of longitude, the methods of the preceding chapter are not adapted to the purpose. The instrument used will then be the transit. The common form of transit instrument consists essentially of a telescope attached to an axis perpendicularly. As it revolves with the axis the line of collimation produced to the celestial sphere describes a great circle. The instrument is generally mounted so that this great circle is the meridian, and it is used in connection with the sidereal clock or chro- nometer for determining the instant of a star's transit over the meridian. If our clock is accurately regulated to show side- real time, such an observed transit gives us at once the star's right ascension, the latter being, as we have seen, the same as the sidereal time of culmination. If, however, we observe a star whose right ascension is already known, this process gives us the error of the clock. The field-transit mounted in the meridian, with which we are at present more par- ticularly concerned, is always used for this latter purpose. Theoretically the instrument may be used in any vertical plane. It is sometimes used in the plane of the prime ver- tical for finding the latitude, or in a fixed observatory for finding the declinations of stars. When speaking of the transit instrument simply we understand it to be mounted in the meridian. 268 PR A C 7 '1C A L AS TRONOM Y. I 5 8. FIG. 6. I 58. THE TRANSIT INSTRUMENT. 269 Description of the Instrument. 158. The "transit instrument designed for a fixed observa- tory, where it is permanently mounted, is much larger and more complete than one designed for use in the field, where it must be transported from place to place. The transit- circle of the Washington observatory, for instance, has a telescope of twelve feet focal length, the aperture being eight and one half inches ; it is mounted on massive piers of marble, which rest on a foundation of masonry extending ten feet below the surface of the ground. Figs. 26, 27, 28, and 29 show different forms of the field- transit used by the coast and other government surveys. Fig. 26 is a very common form. The telescope is 26 inches focal length and 2 inches aperture. It is provided with a diagonal eye-piece for observing transits of stars near the zenith, the magnifying power being about 40 diameters. As may be seen from the figure, the frame folds up so that the entire instrument may be packed in a single box of compara- tively small dimensions. The frame rests on three foot- screws by means of which it is levelled, the final adjustment in this direction being made by a fine screw at the right end of the axis, as shown in the figure. At the opposite end is a screw, or pair of screws acting against each other, by means of which the final adjustment in azimuth is made. The two lamps at opposite ends of the axis are for illuminating the field. The axis being perforated, the light enters it, falling on a small mirror at the intersection with the telescope, by which it is reflected down the tube to the eye-piece. The threads of the reticule then appear as dark lines in a bright field. With some instruments there is only one lamp: with two the unequal heating and consequent expansion of the 2/0 PRACTICAL ASTRONOMY. 159. FIG. 27. THE TRANSIT INSTRUMENT. 2/1 two pivots is to a great extent avoided, also the inconvenience of changing the lamp from one side to the other when the instrument is reversed. The two small circles attached to the telescope below the axis are called finding-circles ; they are used for setting the telescope at the proper elevation. They are about 6 inches in diameter. The alidade carries a level, as shown in the figure. The index is generally adjusted so as to read zero when the telescope is horizontal. If then the vernier is set at the meridian altitude of a star 'and the telescope revolved until the bubble stands in the middle of the tube, the star will be seen in the middle of the field when it passes the meridian. One circle could be made to answer every pur- pose, but it would read differently in the two positions of the axis, and this would be likely to prove a fruitful source of annoyance. The instrument is reversed by lifting the axis up out of the supports by hand, turning it around and carefully replacing it. 159. Fig. 27 shows a larger and more complete instru- ment designed for longitude work. The focal length of the telescope is 46 inches, aperture 2f inches. Magnifying powers varying from 80 to 120 diameters are used. A special apparatus is provided for reversing the instrument, which will be understood by reference to the figure. The cam worked by the crank below the frame raises the axis out of its supports, when it is turned around and again low- ered into its place. One of the finders has two levels at- tached, one the ordinary finding-level, the other a much finer one for use in determining latitude, as will be explained hereafter. 160. Fig. 28 is a somewhat common form of transit, one end of the axis being made to take the place of the lower half of the telescope. A reflecting prism is placed at the intersection of the telescope with the axis, which bends the 2/2 PRACTICAL ASTRONOMY, 160. FIG. l6l. THE TRANSIT INSTRUMENT. 273 rays of light at an angle of 90, the eye-piece being at the end of the axis. The instrument shown in the figure may be used as a transit, zenith telescope, or azimuth instrument, and is very convenient for use in positions where it is not practicable to have two or three separate instruments. It has, besides, the advantage that, for stars of all zenith distances, the observer occupies the same position: with the common form of instru- ment the position of the observer is sometimes uncomfort- able, which is prejudicial to accuracy. 161. Fig. 29 shows another form of instrument, made for the Coast Survey by Fauth & Co. of Washington. This form was first proposed by Steinheil (Astronomische Nach- ricktcn, vol. xxix. page 177). Here a separate tube for the telescope is dispensed with entirely, the axis being made to serve this purpose by placing the object-glass at one end and the eye-piece at the other. The reflecting prism is placed in front of the objective, as shown in the figure, and almost in contact with it. The tube is placed horizontally and in the prime vertical. When the reflecting surface of the prism is adjusted at the proper angle, the image of any star may be made to transit across the threads of the reticule, precisely as in the other forms of instruments. The instrument shown in the figure has a focal length of 25 inches, and 2 inches aperture. It is fitted with the appliances necessary to adapt it to use as a zenith telescope. It is very compact and portable, and is therefore particularly adapted for use in a rough country where transportation is difficult. The portable transit instrument is mounted when practi- cable on a pier of brick or stone, set into the ground deep enough to insure stability. Where such a foundation is not available a log sawed off square and firmly planted in the ground answers a very good purpose. The observatory may be a shed made of boards or a canvas tent. 2/4 PRACTICAL ASTRONOMY. l6l. 163. THE TRANSIT INSTRUMENT. The Reticule. 162. This consists of a number of spider-lines arranged as shown in the figure. The middle line is placed as nearly as may be so that a line joining it with the optical centre of the object-glass shall be perpendicular to the axis. In field-instruments a very thin piece of glass ruled with fine lines is often used, and is found more satisfactory in some FIG. 30 . respects than the spider-threads. In the larger instruments intended to be used with the chronograph there are some- times as many as twenty-five lines; in the smaller instruments there are usually five or seven always an odd number. The two horizontal lines are for marking the centre of the field. The instrument should always be set so that the star will pass across the field midway between them. The Level. 163. Every transit instrument is provided with a delicate striding-level. It is supported by two legs, the bottoms of which are V-shaped. The length is such that these V's rest on the pivots of the axis when the level is placed in the posi- tion shown in Figs. 27, 28, and 29. The tube which is nearly filled with alcohol or sulphuric ether is apparently cylindrical, but in reality has a curvature of large radius. The bubble of air which is allowed to remain in the tube will always occupy the highest point, and so any change in the relative elevation of the two ends will cause a change in the position of the bubble. It may therefore be used not only for determining when the axis is horizontal, but, by ascertain- ing the angle corresponding to a motion over one division of 2/6 PRACTICAL ASTRONOMY. 164. the graduated scale, we may by reading the two ends of the bubble determine the small outstanding deviation from per- fect adjustment. The level when so used is a very delicate instrument for angular measurement. 164. To find the value of one division of the level. This is most easily accomplished by the use of a little instrument called a level-trier, which is simply a bar of wood one end of which rests on two pivots, while the other is supported by a micrometer-screw. Let d = the distance between two consecutive threads of the screw ; L = the length of the bar between the points of sup- port ; r = the angle corresponding to one revolution of the screw. Then Suppose the scale of the level to read from the middle in both directions. Call the two ends of the level E. and W. The readings in the direction W. may be considered -f- ; those in the direction E., . Let the level be placed on the bar of the trier, and both ends of the bubble read ; then let the micrometer-screw be turned so as to cause the bubble to move from its first position, and the two ends read again. Let e and w be the readings of the bubble in the first position ; e' and w' be the readings of the bubble in the second position ; d, the value of one division of the level; v, the true angle through which the bar has been moved, as given by the micrometer- screw. i6 4 . VALUE OF ONE DIVISION OF LEVEL. Then |(w e) will be the reading for the middle of the bubble in the first position; e') will be the reading for the middle of the bubble in the second position. from which ..... (274) #' = ^r cos Substituting for x,y, z and x' , y', z' their values, and dropping the common factor A t we have cos n sin m = cos sin -f- sin b cos <^ ; i cos n cos / = cos b cos ; V (275) sin n cos b sin a cos

-f- b sin From these we readilv derive a = m sin tp - n cos _ ( , b = m cos cp -\- n sin See equations (112.) 172. THEORY OF THE TRAXSIT INSTRUMENT. 287 172. Now let T = the east hour-angle of a star when seen on the middle thread ; c = the error of collimation ; plus when the star reaches the thread too soon.* Now let the star when on the middle thread be referred to a system of rectangular co-ordinates, the plane of x, y being the plane of the equator, the axis of x being perpendicular to the rotation axis. Then d the star's declination is the angle formed with the plane of x, y, by the radius vector; T m the angle formed with the axis of x by the pro- jection of the radius vector on the plane of x, y. Then x = A cos d cos (r m);\ y = A cos 8 sin (T m} ; > . . . . (278) z = A sin 8 } y being reckoned towards the east. Let the star be now referred to a new system of co-ordi- nates in which the axis of x coincides with that of the last sys- tem, the axis of y being the rotation axis of the instrument. Then c = the angle formed with the plane of x, s, by the radius vector ; tf, = the angle formed with the axis of x by the pro- jection of the radius vector on the plane of x, z. Then x' = A cos c cos tf, ; \ y' = A sin c ; ! (279) z' = A cos c sin tf r ) * The star is supposed to be observed at upper culmination. 288 PRACTICAL ASTRONOMY. 1/2. In these two systems the axes of x coincide, tne axes of y' and z' make the angle n with those of y and z. Therefore y' = y cos n z sin ; V ..... (280) z' = y sin n -\- z cos n. Combining (278), (279), and (280), we have cos c cos #, = cos tf cos (r m); j sin c = cos # sin (T m) cos n sin # sin n\ V (281) cos sin #, cos tf sin (r m) sin n -\- sin tf cos n. ) With these equations, as with (275), no restrictions have been placed on the quantities involved, and they will serve for computing T when m, n, and c are known. When these quantities are small, as with the instrument adjusted in the meridian, the second of (281) becomes c = (T m) cos 8 n sin 6; from which T = m -\- n tan d -j- c sec 6 ...... (282) This is BesseV s formula for computing the hour-angle of the star when it passes the middle thread of the reticule. In ap- plying it, the unit in which ;, ;/, and c are expressed must be the second of time. If we substitute in (282) the value of m from the second of (277), viz., m = b sec

=".3IQ COS = S .O2I COS G>.(286) \ 186380 . sin i I If the star's declination is tf, the effect upon the star's / I hour-angle being k ', we have, by applying Napier's L^J .first rule for right-angle triangles to the triangle shown FlG * 33 .in ths figure, sin k = sin k' cos tf; or k' = k sec d s .O2i cos (p sec 3. . . (287) As this will cause the star to appear too far east, the ob- served time of culmination will be too late and the correc- tion must be subtracted. The correction for diurnal aberration may be combined with the collimation constant by making c' = c s .02i cos (p (288) As observations are made in both positions of the axis, it is necessary to distinguish between them. This may be done by noting the position of the clamp, whether it is east or west. If then the sign of c is determined for clamp ivest, the alge- 1/4- EQUATORIAL INTERVALS OF THREADS. 2gi braic sign must be changed when the position is -clamp east. It must be remembered that the algebraic sign of the aber- ration does not change when the instrument is reversed; so if this correction has been combined with c, c' will in one case be the sum of the two, and in the other case the difference. Equatorial Intervals of the Threads. 174. When the transit of a star over one of the side threads is observed, we may regard the distance of this thread from the collimation axis as its error of collimation, and proceed with the reduction precisely as in case of the middle thread. It is simpler in practice, however, to determine the angular distances of the side threads from the middle thread, when the times may all be reduced to the time over this thread. This angular distance when expressed in time is evidently the time required for an equatorial star to pass from the side thread to the middle thread. Let i = the equatorial interval for any thread; / = the interval for a star whose declination is tf. Then 2 -(-^ the collimation error for this thread; r -)- / = the hour-angle of a star when seen on this thread. The second of equations (281) may be written sin (r m) = sin c sec n sec S -\- tan n tan #, and for the side thread sin (r -f / m) = sin(* -f- c) sec n sec d -\- tan n tan 3. 2Q2 PRACTICAL ASTRONOMY. ' 174. By subtraction, sin (r -f / m) sin (r *w) = [sin(/ + <^) sin c] sec sec \ sm/> L L sin i L sin z sin 15" #) sec 6 -\- c sec d S .o2i cos

tf) sec ^ c sec d S .O2 1 cos q> sec S. J Subtracting the first of these from the second, we readily find - T) cos rf + i(T' - T) STcos d "3-W - b} cos ( 9 - 6). (305) This formula is applicable to lower culmination by chang- ing 6 into 1 80 6 as usual. In most cases the term in ST will be inappreciable. The Azimuth Constant, a. 186. This can only be determined by observation of stars. Let two stars be observed which differ as widely as possible in declination. 306, PRACTICAL ASTRONOMY. Let Tand T' be the times of observation reduced to the middle (or mean) thread; $ and 6', the declinations of the stars; a and a', their right ascensions. Then equations (304) will apply to these stars, except that in the second we shall have a' and d' in place of a and tf, and the sign of c is not changed. Let us write t = T -\- <$T(T T ) -f- b cos ((p tf) sec 8 -{- c sec d S .O2 1 cos cp sec d t' = r' + 6T(T T ) + ' cos (?> - d') sec d'+ c sec d 7 s .O2i cos (p sec 6'. That is, we place t and /' equal to the sum of the known quantities in the second members of the equations. Equa- tions (304) then become a = t -f A T + a sin (cp d ) sec 6 a' = t' -\- A T + a sin (cp #') sec d'. From which sin (<7> d') sec d' sin (

(tan d -f tan d'}. 1 87. TO DETERMINE n DIRECTLY. 307 This combination is therefore most favorable for the pur- pose. If the rate of the clock and the stability of the instru- ment can be relied on for twelve hours, the same star may be observed both at upper and lower culmination. This will not be practicable, however, with a portable instrument. If two stars are observed at upper culmination, one should be near the pole and the other near the equator. If m and n are required, they may now be computed by (276), or we may proceed as follows. To Determine n Directly. 187. Using the same notation as in the determination of sec 3, t' T + tf r(:T r o ) + <; sec 8 ~ 48 5 o * Example, Art. 179. l8/. REDUCTION OF TRANSIT OBSERVATIONS. 313 log |(7" - T) = 0.90849* log \(b' - b) = 7.74036* cos 8 = 8.35913 cos ((? 5) = 9.82481 sum = 9.26762* 7.56517* Nat. No. .1852 Nat. No. .0037 Therefore c - T '.1889 clamp j ^J j- . In applying the formula of (XVII), the term \(T' T) 8 T cos 5 has been disregarded, as in this case it is inappreciable. It is convenient to combine the correction for diurnal aberration with c. Thus, if we write c = c ".021 cos , we have in this case c = -f- 9 . 173 clamp east, c = '.205 clamp west. The last but one of (XVII) will now give us the azimuth constant a. We have seen that the best result is to be expected when we use the observed transit? of two circumpolar stars, one at upper and the other at lower culmina- tion. We therefore determine this constant from 5 Ursa Minoris and 47 Cephei. Referring to the derivation of the formula for a (Art. 186), we have for t and t' t = T -f- b cos (

d) = 8.307*

6} sec 8 ; B = cos (tp 8} sec d ; C = sec tf ; A T = the clock correction at time T ; dT = the hourly rate ; a = the stars' apparent right ascension. We can always infer from our observations a value of AT % which will be very near the true one, and as the labor of computation will be diminished by making the numerical values of the unknown quantities as small as possible, we may assume an approximate value of this quantity, and determine a correction to this assumed value. Let 5 = the assumed value of the clock correction ; AT. = S + *. Then x is a small unknown correction to 5. Introducing this notation into Mayer's formula, it becomes In which x, 3T, a, and c may be considered unknown quan- tities. Writing / = T + $ -\- Bb S .O2 1 C cos

). C is of course given with the argument S. When many observations are to be reduced at one place, or in the same latitude, a special table is more conveniently computed for the latitude of the place. The only argument will then be d. It will be convenient to make the computation of / directly in the book used for recording the transits. The means of the times over the threads being taken, this will be T, which is written below. In case of incomplete transits, the time over the mean thread is computed as already illustrated, a and d are taken from the Nautical Almanac and written in the same book. The small corrections B .b and s .O2i coscp.C are applied directly to T. Subtracting a from the algebraic sum, we have / 5, in which 3 will be assumed of such value as to make / small. An example follows. PRACTICAL ASTRONOMY. Reduction of Transit Observations made at the Sayre Observatory, 1883, October n. An observing list was first prepared, of which the following is a specimen : STAR. Magnitude. & Setting. // Aquarii 4-7 2O h 46 21 s - 9 25'. 3 140 I '.7 4 80 <;T A 6' 2 Ursae Majoris, s.p. .. 5-0 21 O 5 112 23.5 18 12 .9 Cvgni 3 21 7 57 29 44 8 4 21 10 7 37 3 2 8 2 7 21 15 47 62 54 68 310 Pegasi .... 2 3 21 38 26 Q 2O 3 4 3 21 42 28 48 46 1 79 Draconis 6.3 21 51 25 73 8.9 57 27.5 a Aquarii 3-0 21 59 46 - o 53.3 131 29.7 32 Ursae Majoris, s.p... 6.0 22 9 31 114 18.5 16 17.9 n Aquarii 4-7 22 ig l8 -f- 47.0 129 49.4 The two groups are intended to be observed one in each position of the axis. The right ascension and declination are taken from the mean values of the Nautical Almanac. The column headed " Setting" gives the setting of the finding circle. In this case the circle reads zero when the telescope is directed to the north point of the horizon, the latitude being 40 36' 24"; the circle will read 130 36' 24" when the line of collimation of the telescope lies in the equator. Therefore the setting for any star will be 130 36'. 4 5. Below is the copy of the recorded transits of the above stars as observed on the night of October n, 1883 : Level. E. W. 12. 9.9 9-2 I3-I 12.0 9.9 9-6 13.0 I II III IV V H Aqu 20 Clamp East. arii. 57- 13-9 30. 46.7 47 3-i I II III IV V "Cyg 20 53 14.4 36.3 57-5 19. 40.4 10.70 11.475 2x -j- .16. 3. I.ISa I.2IC -|- .46* = .35. 4. .2oa -|- I.GSC -(- .91-* = -07. 5. .060 -|- i.o-jc -\~ .&$x = -f- .12. 6. .44"* + I.2CW -f- -5 o-*' = + -2O. 7- -53 a i.oif -J- i. oar = .09. 8. . i6a i.i2f -f- .74^ = .19. 9. .6-ja 1.24^ -f- .36^ = .19. 10. .66 i.ooc -[- LOOT = .37. 11. i.i6a -\- i.2ic -\- .$ox = -\~ .05. 12. .64^ i.ooc -\- i.oox = .15. These now have the general form of the equations of condition (36), viz., a\x -j- c\z -(- d\w = n\, there being in this case the three unknown quantities a, c, and x, correspond- ing to the x, z, and w of the general form. The term corresponding to^y has disappeared here, as we have assumed the rate of the clock to be inappreciable for the short time over which the observations extend. We have now to form the normal equations (see Eq. 41). In order that no confusion may arise from the difference of notation, the general form of these equations is here given in full, viz.: \aa\a + \ac~\c + \ad]x = [an] ; \ac~\a + [cc}c + [cd]x = |>] ; We shall give the solution of these equations in full with the various checks on the accuracy of the computation, as an illustration of the method. Practically, however, this part of the work will generally be more or less abridged by ex- perienced computers when the number of unknown quantities does not exceed that of the above equations. We shall require, besides the quantities already indicated, the sums of the coefficients of each equation, viz.: 330 PRACTICAL ASTRONOMY. Also, we compute the quantities [as], [], [ -.03 16 9 i.oioo .0909 .6161 i.oooo JQOe + .6100 .008 + .054 ) -15 .8288 .2128 + .3020! .5476 IMS .2590 .0361 -.066 + .02 4 - .4464 - .231-6 + 1.6864 .1296 &( - .4896 .036! .258- + .07 49 I.OOOO + .6050 -3700 - .0605 1.0300 + 3-4122 I.OOOO .2500 _; 9700 3 + 1.0300 + 1.4100 .7369 .0025 4- .381 .I4 > .04 "3 i.oooo .1500 .7900 I.OOOO f . 1500 + .7900 .0225 + .118. .10 100 + .1958 1.9201 12.6107 7.6754 + 7635 11.9807 4577 + .o 4 oi .0887 12.6107 11.9807 + .040! [cd] CH\ [dd] [4 [nn] [**,] The agreement of the values of [as], [cs], and [ds] proves the accuracy of this part of the computation. The normal equations are then 5.11433 - .2792^ 4- 3.3460* = - .7397; ,2-jgza + 14.6142^ -i- .1958* = 1.9201; 3.34600 + .i 9 58<: + 7.6754* = - .7635. 191. REDUCTION OF TRANSIT OBSERVATIONS. 331 These equations are now to be solved, following the method and notation ex- plained in Art. 28. We shall therefore require the following auxiliary coeffi- cients, viz., ], [cm], \, [dm], [dsi\, [i], [* i], , [rf2], [A 2], [2], [ra]; & i], [j i], etc., being computed for checks on the accuracy of the work. The computation will then be made according to the following scheme: a. " jr. . S. Proof. *<*] 5-"43 log .70879 [ac] .2792 log 9.44592.. ad] 3-346o log 52453 >] - .7397 log 9.86906,1 [as] 8.9208 log 0.95040 8.9208 log w\ 8 ' 737I3 \cc\ 14.6142 g] M - 0152 [*] -1958 ^j[ a rf]-.l8 27 \cn\ 1.9201 S M [cs] 12.6107 MM-. ,8, [cci] 14.599 log 1.16432 [crfi] .3785 log 9-57807 [c* i] 1.8797 log o.2 74 c9 [csi] 13-0977 log 1.11719 13.0978 log -j^j - 8l 574 [dd] 7.6754 [^|[^] 2 .i8 9 i [rf] - .7635 [ = 029 ; = 7= = -043 ; r c = \B C = .017 ; e c = ' = .026 : v^ n, = f e,x = .035 ; 1.09 .6084 1. 0201 1. 0000 i.o8 - .16 1.90 .92 .66 .1664 .6724 - .06 1.61 1-50 -75 - .86| .81 1.3225 .4641 .2116 1.25 i. ii .27 1.96 I. 12! .98 .0400! .1025 .8281 Mi -QI -A i^' \ll 2 .4400 .7225 .3136 - -48 i. 53 .62 .01 .92 1.09 .2809 .0201 1. 0000 1.28 .58 03 - -38 -9. 93 .0256 2544 .5476 1.91 .31 - -48 .88 - i.o. -55 .4489 5376 .1296 -34 1.66 1.03 o -6. ! -37 4356 .0000 1. 0000 + 2.37 - .36 1.66 1.64 '79 1.71 i. if - -8. -45 1.15 1-3456 .4096 .4641 .2500 1 .0000 &r r? 42 &r . . (a + c) 1 *. (a + rf). (a - ). (c + d)*. (c - *)"- (d-n)*. .0081 .025^ 8.2944 3.0270 3.2041 1.1664 3-1684 .6724 7569 .0256 4 0401 3.6100 .'846. 1.1881 .4356 .1225 .004^ 5625 4.9729 , .0036 5625 2.5921 1.2321 2.2500 .0729 -5625 3.8 4 i 7396 I - 2 544 .6561 .9604 0144 3-4596 I .2769 .8281 .0036 3.6864 .9025 5329 .0400 1.2544 -S77& .0144 .4096 3.0976 I. 0000 .1296 .0081 3721 .2304 2.3409 -3844 .0001 .846^ 1.1881 .0361 .1225 i 6^84 3364 .0009 .1444 .8649 .8649 .0361 .1369 1.8496 i .0609 3.6481 .0961 .2004 1.0609 7744 I . 1025 .3969 3025 1.8769 .0025 7-9524 5 .6169 2.7556 1.2321 2.9241 '3450 .2025 .0225 .6241 .1296 2.6896 .624! 7225 1.3225 4577 33-553 '9 .1701 19.4817 7-0514 22.6812 11-2317 9.6601 19.7285 - .5584 12.7897 6.6920 5-5720 1-4794 22 2896 .* 15.0719 3.8402 8.133, 1.5270 .2792 3.3460 739; 1958 1 .9201 -7635 \nn\ l] [* [0 -[*! \cd 1 -\_cn\ -[ 336 PRACTICAL ASTRONOMY. 192. If we write (cp 8} = 3, the terms of Mayer's formula, which give the correction of the observed time of a star's transit for collimation, flexure, and inequality of pivots, may be written as follows : (p cos z /cos z + f) sec 6; . . . (319) in which/ is determined by (297) or (297),, and which we see is involved in the same manner as/". These instruments are generally provided with micro- meters, which may be used for determining / and c at the same time, as follows: In order to make a satisfactory determination, and at the same time to test the accuracy of the assumed law of change expressed by the formula /cos z, a collimating telescope is necessary, mounted in a frame in such a manner that it may be placed vertically over the transit telescope and at dif- ferent zenith distances from zero to 90. The collimation error is then measured, as explained in Articles 182-184, with the telescope pointed at various zenith distances. This measured value will include the term f cos z, which will be zero when z = 90, and a maximum when z o. It will therefore be possible to separate c from/1 It will be advisable to make a considerable number of measurements, from which c and /"can then be derived by the method of least squares. If the resulting values satisfy the equations within the limit of the probable error of meas- urement, the assumed law of change expressed by the for- mula/cos z will be verified. In some cases there is found to be a correction required depending on the temperature. This may be detected by making the measurements for collimation and flexure at different temperatures. If then different values are found varying with the temperature according to any law, the necessary correction may be determined. CORRECTION FOR FLEXURE. 337 In Vol. XXXVII, Memoirs Royal Astronomical Society, Captain Clarke, R.E., gives an example of the investigation of the flexure coefficient with an apparatus of the kind just described. In addition to the movable collimator, another was used which was fixed in the horizon. The collimation measured on this was free from the effect of flexure, so that by taking the difference between the quantity (f cos z -j- c}, measured at a zenith distance z by means of the movable collimator, and the quantity c, measured at the same time with the fixed collimator, a direct measurement of the quantity /cos z was obtained. Twelve measurements made at zenith distances from o to 55 gave the following results: 2 Difference. V * Difference. V z Difference. , 5 2.8o 2.68 + 22 + 33 20 25 2.72 2.98 + 09 - 24 40 45 2. 4 6 1.98 53 10 3-" - 13 30 2.40 + 22 50 2.02 1 08 15 3-04 12 35 2. 9 - 43 55 1.6 9 +o 4 The column headed 5 gives the zenith distance of the upper collimator ; the next column gives the difference between the collimation determined on the upper and lower collimators; and the column headed v gives the residuals. Referring to equation (319), we see that the quantity called "difference" is equal to (/ /) cos z. From the twelve measured values of this quantity it was found that (//) = 3.02 1 .050 expressed in divisions of the micrometer. From level-readings, p == .779 .026 expressed in divisions of the micrometer; therefore /= 3.800. 338 PRACTICAL ASTRONOMY. 194- One division of the micrometer = o".8345 ; therefore / 3".i7i = o s .2ii. 193. The use of such an apparatus as we have described will not generally be practicable in the field. The coefficient /may then be determined ir.om the observed transits by adding to the equations of condition (317) the term cos d The complete equation will then be Aa -f Bf + Cc -f ST(T - T ) -f x + / = o. . (320) a\ f-> c , T, and x being unknown quantities. If ST is known, as it ordinarily will be, the number of un- known quantities will be four. The Transit Instrument out of tJie Meridian. 194. Equations (275) and (281) are strictly general, and are applicable to the reduction of transits with the instrument in any position whatever. We have seen that when the in- strument is so near the meridian that the squares and higher powers of a, b, m, and n may be neglected* these formulas become very simple. Bessel, Hansen, and others have given more general methods of solving the equations intended for use in those cases where the observer in the field cannot af- ford the time for adjusting his instrument accurately in the meridian. When, however, the observer is provided with a good list of stars reduced to apparent place, like that given * That is, we may write a, b, m, and n for sin a, sin b, etc., and unity for cos a, cos b, etc. 195- TRANSITS OF THE SUN, MOON, AND PLANETS. 339 in the American Ephemeris, this adjustment is made so readily, and the labor of reduction is so much less than with the more general methods, that the latter have not found much favor, especially in this country. Therefore, however interesting- some of these may be from a mathematical point of view, we shall not give their development here. Transits of the Sun, Moon, and Planets. 195. In the field, transits of the moon will be observed for the determination of longitude when no better method is available. The sun and occasionally a planet will be observed for time. In case of the sun and moon the method of observing is to note the instant when the limb is tangent to the thread. With the sun the transit of both limbs may be observed; with the moon this will not be practicable except when the transit is observed very near the instant of full moon. In observing a planet, the transits of each limb may be ob- served alternately, or when a chronograph is used both limbs may be observed, as in case of the sun. With any of these bodies, when both limbs are observed, the time of tran- sit of the centre will be the mean of that of the two limbs. It may, however, be desirable to reduce the limbs sepa- rately for the purpose of comparison. When the moon's limb is observed on a side thread, the hour-angle is affected by parallax : the time required to pass from the thread to the meridian is affected by the moon's motion in right ascension. The reduction is as follows: Let 6' and /' be the apparent declination and east hour-angle of the moon's limb when observed on a side thread; <5 and t, the geocentric declination and hour-angle; z and z', the geocentric and apparent zenith distance. 340 PRACTICAL ASTRONOMY. 195. We can reduce the observation by either of the equations (282), (283), or (284). Taking the latter, viz., Mayer's for- mula, we have i being the equatorial interval of the thread. Having- /', / may be determined as follows: In Fig. 37, let P be the pole, Z the zenith, O the geocentric place of the moon at. the instant of observation, O' the ap- parent place. Angle MPO t\ ZO = z\ MPO' = /'; ZO' = *'. From the triangles MZO and M'ZO', sin MO sin M'O' sin MZO = = = = -, . . (322) cm & cii-i " cos '~ + cos S') 15(1 -A) cos ' 34 2 PRACTICAL ASTRONOMY. T 95- The geocentric declination, tf, and the equatorial horizontal parallax, n, are taken from the ephemeris. Then from (XI),, Art. 85, we have with sufficient accuracy for this purpose d' = 6 Ttp sin ( S') a -- R - - = ".035 cos S' cos (

which is to be determined. The process is then analogous to that employed with the instrument mounted in the meridian ; viz., the adjustments are made as accurately as may be, and the corrections to the final result determined for outstanding deviations. As we shall see, the value of the method consists largely in the 350 PRACTICAL ASTRONOMY. 199. facility with which the effect of instrumental errors may be eliminated. It is evident that only those stars can be ob- served on the prime vertical which culminate between the equator and the zenith, that is, whose declinations are be- tween o and (p. Adjustments. 199. It is only necessary to explain the method of placing the instrument in the prime vertical, all the remaining ad- justments being the same as when the instrument is in the meridian. For this purpose a star is selected whose declina- tion is small, and the clock time computed when the star will be on the prime vertical. Triangle PSZ of Fig. 42 gives tan 6 cos/ = tsr?- ..... (335) The clock time of the star's passing the prime vertical will then be (336) When the clock time is that given by this formula, the middle thread of the reticule must be brought on the star by the fine-motion azimuth screw. It will be observed that a knowledge of the latitude is necessary for computing t, but from (335) it appears that when a star is chosen whose declination is nearly o, a small error in the assumed value of

202. PRIME VERTICAL TRANSITS. 353 If the instrument is carefully levelled and adjusted in the prime vertical, we way write cos b = i ; cos a i ; sin b b ; sin a = a ; when the above equations may be written cos n cos m = sin (cp b} ; j cos n sin m = a\ I . . . . (342) sin n = cos (tp b). ) We shall find these formulas useful in subsequent transfor- mations. 202. Let 90 -|- c = the angle between the clamp end of the rotation axis and the object end of the collimation axis; /and tf = the hour-angle and declination of a star observed on the m'ddle thread. Let the star be referred to a system of rectangular axes, the equator being the plane of xy, the axis of x being directed to the point where the hour circle through the north end of the rotation axis intersects the equator. Then the angle formed by the radius vector with the plane of xy will be d, and the angle between the projection of the radius on the plane of xy and the axis of x will be 1 80 + (/ m). x = cos tf cos(/ m)', y = cos tfsin (tm}\ z = sin tf. (343) In the second system, let the axis of x coincide with the rotation axis, the axis of y coinciding with that of the former system. Then the position of the instrument being clamp north, c will be the angle formed by the radius vector and 354 PRACTICAL ASTRONOMY. 203. the plane of yz. Let tf, be the angle formed with the axis of y by the projection of the radius vector on the plane of yz. Then x' = sine; _/ = cos <: cos , ; s' cos^ sin tf,. (344) The angles between the axis of x and x' being ;/, we have x'= xcosn -\- where the values of sin n and cos n given by (342) have been substituted. Let h sin r clamp north; q = the unknown error in determining c. Then (b -}-/) and (b' /) the true inclination of axis for clamp north and south respec- tively ; c -(- q = true value of collimation con- stant. See equation (305). 205- PRIME VERTICAL TRANSITS. 357 Let q>' and " + *' - / - (c + q) The mean is Unless the errors of adjustment are very large the last term of this equation will be inappreciable, so that practi- cally constant errors of collimation and level are eliminated by combining observations on the same star in different positions of the axis. Errors in $ may result either from errors in the clock rate or they may be simplv the unavoidable errors of observation. To ascertain their effect upon cp we differentiate (350) with respect to cp and 5, by which means we derive dtp = isin 2cp tan 5 d$ (nearly). . . . (353) From this equation it appears that an error in 3 will produce the less effect upon cp the smaller 5 is. Also, that the alge- braic sign when the star is east is the opposite of that when it is west. Therefore The effect of a small error in 3 will be eliminated by ob- serving the star both east and west of the meridian. Differentiating (350) with respect to (p and d, we find 358 PRACTICAL ASTRONOMY. 2O6. As the declination cannot be greater than . For (ft greater than 45, d

6. REDUCTION TO MIDDLE THREAD. 359 by reducing each thread separately to the middle (or mean) thread ; second, by applying a correction to the mean of the times over the different threads to reduce it to the time over the mean thread. First. The thread intervals should be determined by meri- dian transits as already explained.* Let i = the equatorial interval of any thread from the middle thread ; / = the corresponding star interval ; / = the hour-angle of the star when on the middle (or mean) thread ; t I = the hour-angle when on the side thread ; c -f- i may be regarded as the collimation error of the side thread. Then, from the first of (346), sin (c -{- i~) = sin n sin d -(- cos n cos 8 cos (t I m)\ sin c = sin n sin $ -\- cos n cos d cos (t m}. Subtracting, we readily find 2 cos (%i -|- c] sin \i = cos n cos 82 sin \t m /) sin /. Since c will be very small, the first term of this may be writ- ten sin i without appreciable error. Then 2 Sin * 7 = co7^coTTslrr(/ -m- i/)' ' (355) From (342) we may write cos n = sin (cp b). Also, (t m) = 3". sin i may be written i. 2 sin i/ = I(\ - ^/ 3 ) = /(cos 7)A. * Art. 174- 360 PRACTICAL ASTRONOMY. 2O/. Therefore (355) may be written without appreciable error, 7 = sin (tp ~b) cos 6 sin (S- /) (cos 7)5 ; (356) and with accuracy sufficient for most cases, ~~ sin q> cos <$ sin (S /)' ' ' ' (357) Log (cos /)* might be tabulated, but it will be required so rarely that it will hardly repay the labor. The value of / required in the second member of the above formulas may be found directly from the observations themselves, by taking the difference of the observed time over the side thread and middle thread. Care must be taken to give the proper algebraic signs to /, /, and S, i and / being plus for north threads and minus for south ones ; S, plus for west, minus for east transits. 207. Second. This method of reduction is due to Bessel, and is more convenient when many stars are to be reduced. Resuming the first of (346), and writing c -f i instead of sin c and / / for /, c -f- / = sin n sin d -{- cos n cos d cos (t I m). (358) Such an equation is given by each thread observed. If j* threads are observed, the mean of the resulting equations will be c + *' = sin n sin 8 -f- cos n cos 62 cos (/ m), (359) where z" is the mean of the equatorial intervals, 2 is the sum- mation sign, / represents the hour-angle corresponding to- any thread. VESSEL'S METHOD OF REDUCTION. 361 Let T = the arithmetical mean of the times observed on the individual threads (supposed corrected for clock error and rate) ; T I = the time over any thread. Then (t m) = (T a m} I t and Now let -2 sin 7. . (360) y I k sin n = Then 2 cos (/ m) = k cos (T a K m\ (362) (359) then becomes c-\- t a = sin TZ sin -{- cos cos tf cos(T a n ^.(363) Now let y cos tf, = k cos 8 ; ) / g v Y sin ^ = sin d. \ ' Then (363) becomes - = sin n sin #,-)- cos cos ^cos (T a u m).($6$) Thus, by computing the auxiliary quantities y, & lt and x, the form of the equation for the mean of the threads is the same as that for the middle thread. Practically y will seldom differ appreciably from unity. 362 PRACTICAL ASTRONOMY. 207. tf, and K may very readily be computed by the aid of tables A and B, page 365. These tables are computed as follows: Since 2f = o (T being the mean of the observed times, and /the difference between T and the time on any thread), (361) may be written k cos K = i 2 sin 5 ; (366) k sin K = -5Y/ sin /). M , From these it appears that k sin n is of the order / 3 , and that k cos only differs from unity by a quantity of the order /*. There will then be no appreciable error in writing 2 1 ,- , (367) -sin/). ' And since, from (364), we have tan tf, = - tan tf, (368) the method of Art. 74 for expanding a function of this form gives 'i \sin 20 . ill Vsin n 2 This becomes, by substituting for its value, l^^li- d, = fi + - !15^_ sin 2 the argument being the difference between each ot> 207- VESSEL'S METHOD OF DEDUCTION. 363 served time respectively and the mean of all, expressed in minutes and seconds of time for convenience. The arith- metical mean of these quantities will be the numerator of the coefficient of sin 2$ in (370). The denominator differs very little from unity. When desirable, this small difference may be corrected by table B, the argument of which is the numer- i sin/ The fourth colum n of table A gives the quantity (/ sin 7), the arithmetical mean of these quantities being equal to H. If y is required, we readily find, from (364), i (i K) cos'tf cos (&, tf) ' The denominator does not differ appreciably from unity, and Therefore y = i cos'tf ^ sin 2 /. . . . . (371) Since this only appears as the divisor of the small quan- tity c -\- /, it will very rarely be required. The quantity z' will vanish when the star is observed over all of the threads, and the equatorial intervals reckoned from the mean of the threads. Having shown how our fundamental equation which ap- plies to the time over the middle thread may be reduced to a like form when the time is the mean of the times over the different threads see equation (365) we may now solve this equation for

X ( ^ I ; ;j" sm d Bessel's Method of Reduction. * - - ~ 2 (I - sin /); sin \" i - 2 sin 3 / tan ' 34" 41^.6 9.67901 9.40037 .13085 - 135 -2 42 .0 9.68098 9.40234 .82862 - 67-4 41 .6 42 .0 9.68485 9 . 4062 r .82679 4 67.1 41.6 9.68673 9.68859 9.40809 9.40995 .12517 . 29898 + *33 -4 + 199.1 41 .6 16 34 41 .6 T = 16 34 41 .71 9.67700 9-39836 2.3TOI5 4- 204 .2 20 25 9.2 9.67902 9.40038 2.13084 4- 135 .2 9-2 9.68097 9-40233 1.82863 4- 67.4 8.9 9.0 9.68484 9 . 40620 1.82680 - 67.1 9 .1 9.68673 9.40809 2.12517 - 133 -4 9.1 9.68858 9.40994 2.29899 - I99-I 20 25 9 .0 T' = 20 25 9 .07 In the above the quantity 3 is computed from the second of (XXa), using for T' and T the lime over the middle thread, and neglecting the rate, which will be less than the probable error of the observation. The "observed /" is found by subtracting the observed time over each thread from the time over the middle thread. The quantities headed " log denominator" are computed 368 PR A CT1CAL ASTROXOM V. 20 9 . by writing the quantity log (sin cp cos S) on the lower edge of a slip of paper and adding it in succession to each of the quantities in the previous, column, b is neglected in the quantity sin (q> b}. The quantities log i l , log i^ etc., are then written in order on the lower edge of another s.ip of paper and the " log denominator" subtracted, giving log/. It would be sufficient to compute the intervals / for one transit only, as they are the same for both; but in a case like the above it is well to compute both as a check on the work. In the above, four- figure logarithms would have been sufficiently accurate. In the same manner the other observations are reduced, the quantities T and T' being those given in the following computation: Latitude from a Lyra;. CLAMP SOUTH. Dec. 23. 7" 20'' 25'" 9 s 07 AT' + I 48 .73 tan d = 9 9028502 7" -1- A T' = 20 26 57 .80 sec S = .0573745 (r + AT')-(T+ AT) = 3 50 27 .53 cos m = oo 3 = i 55 13 .765 tan (f> = 9.9602247 = 28 48' 26". 5 i = 42 22' 50' '.19 7 1 = 16" 34 40 9 .66 JT- = 4- 1 48 .60 mean

67 -544-75

-\- a tan s c' sec^+/= o. . . . (377) Each star observed furnishes one equation of this form for determining- the unknown quantities A Herculis SCygni N. S. s. E. W. E. 2 58.0 4 7-2 25 5 .6 39 40.4 24 13 .2 12 38 II .2 45 23-6 23 23 .6 36 38.8 46 46 .4 35 o.o 48 14.4 - 2 .123 i -353 - I .124 hr a nrirent places of the stars for the date of observation, 1827, June 28th, 34 m , Munich sidereal time, I find to be as follows: STAR. S A Bootis I4. h Q m "JO 8 2O 46 m' m" 40 18 31 8 14 XIII 316 14 I 2 .36 17 34 38 04 44 40 53 .53 46 6 20 56 it Lyrae v Herculis Y Cygni cp Herculis 18 50 7 -75 15 57 27 .45 20 16 4 .61 16 3 21 .83 10, 39 38 .03 43 43 28 .14 46 31 23 .50 39 42 34 .46 45 23 40 .34 44 42 52 86 The values of the equatorial intervals of the threads from the mean thread are as follows : i= +598". 08 ; z a = 6 = 6i2".46. The correction for inequality of pivots is 0.294! divisions of level for circle north. The value of one division of the level is 4". 49. * See Astronomische Nachrichten, vol. ix. p. 413. t Bessel uses as the correction .42 divisions, which is evidently computed by the erroneous formula / = '- ("os7IL* 1 ()S ' )' instead of ^ 97 ^' See Ast - Nach -i vi - P- 2 3 6 - 211. EXAMPLE OF REDUCTION BY LEAST SQUARES. 375 A mean time chronometer was used, the hourly rate on sidereal time being -\- 9". 19 ; the correction at 12 hours chronometer time being 5'' 4 44*. 61. Bessel gives the approximate values of the latitude and the azimuth of the instrument as follows : cpo = 48 8' 40" ; 00 = 0*7' 48". If these quantities are not known with accuracy sufficient for forming the equations of condition, a preliminary reduction of a few of the observations will give them. The values of T, u, and 5j are computed precisely as shown in Art. 210. With this series of observations K in no case exceeds .oi ; it has accordingly been neglected. The computation of T l for each star may now be conveniently arranged as follows: STAR. T , , +4 , a '. T > A. Bootis... W. o h i3 m 33'.68 5 V 28- 3 1 ".8- i. 99 + " 8" ,.. 79 + I7 2' 5 6.8 5 a I.vrae XIl'l "16 E. W o 30 10 .29 4 30 .85 34 4' -'4 8 3I 8.14 56 7 .., -44 64 /Herculis. TT l.yrae. . . r Herculis yCypni. . Herculis E. E \V E W. 5 5 -52! 41 38 .90 9 52 .88 24 40 10 38 7 -52 . 41 .80 46.13 48-39 50-45 9 4I-72 46 20 .70 14 39 .01 29 28.49 42 57 -97 7 34 38 .04 8 5 7 -75, 5 57 27 .45 + o 16 4 .61' 6 3 2' -83'+ 24 '7 46 . 39 . 7 .05-30 564 i .56+19 175 6 .12-41 39 6 .14 +24 54 .8 75 ; 8 4 SCygni.... E. 45 26.325 5i -57 50 17.89 939 38-03 49 20.14 -27 20 .1 As we have an approximate value of the azimuth error, we may write (equa- tion 376) cp a -(- Aq> q>\ b -\- (a a -\- A a) tan z (/o -j- c) sec z = o. i a is zero for all the above stars except * Lyrce and y Cygni. In the observa- tion of it Lyra the transit over the second thread was lost. Therefore for this star to is the mean of the equatorial intervals ii, i 3 , z' 4 z s ; viz., 75".775- Similarly for y Cygni, the fifth thread being missed, i = -f- I53"- II2 5- Writing the sum of the known terms, viz., \_ -f- da tan z c sec z = f. 3/6 PRACTICAL ASTRONOMY. 211. The computation of g>i, tan z, sec z, and /is now arranged as follows : A Bootis. o Lyrse. XIII 316. i Herculis. IT Lyrse. ( s l tan 8, COST, tan ! 46 53' 2s".68 .0286798 9.9804823 .0481975 48 10' 22". 10 38 37' 5"-33 9.9026368 9.8561090 .0465278 48 3'47"-93 44 4 57 >22 9.9951876 9-9466792 .O 4 8so8 4 48 n' 3 5"-47 .0167925 9.9694648 ^ .0473277 9.9806703 9.9333111 -0473592 48 7' 4"-22 tan T 9 48667 9.98655,, 9.72228 9-58947,, 9 77784n 9.82405 9 82498 9.82388 9.82454 9.82452 og tan 2 log sec 2 9.31072 .00890 9-8"53n .07611 9.54616 .02533 9. 4 i 4 oi n .01414 9.602360 .03228 tan z + -2045 .6479 + -3517 - -2594 .4003 secz 1.0207 1.1915 i. 0600 1.0331 1.0771 Level-reading Inequality of pivots - 2.113 "*..., - 2.340 + -294 = I: ^6', 4 o - L798 -294 9-39 + -105 .294 o.8 5 z sec z i' 2l".62 tan 2 - i' 35"-? 1 + 5' 3"-24 - 2' 44"-59 + 2' i". 4 i + 3' 7"-33 L + z' sec 2 J 48 8' 3 8".22 48 8' 4 i".98 48 8' 44 ".48 48 8' 48".8o 48 8' 49".o8 / - + i".78 - i. 9 8 - 4"-48 - 8".8o 9 .o8 z/ Herculis. y Cygni. Herculis. 5 Cygni. , 46 31' 3 i". 34 39 42' 35"-37 45 23' 44"-94 44 42' s6".6o tan 8, .0231351 9 9193426 .0060007 9.9956904 cos TJ 9.9748851 9.8734442 9.9576263 9.9485819 tan <, .0482498 .0458984 0483744 .0471085 48 10' 34". 43 48'. i' 3 ".8 4 48 6' 5 ".o 4 . tan T log tan z log sec z 9.54427 9.82402 9.36829 .01153 9-949 TI n 9 82532 9-77443n . .06579 9.66670 9.82395 9 : Si 9.71340,, 9.82466 9-53806,, 02445 tanz sec z + -2335 1.0269 -5949 1.1636 + -3095 1.0468 .3452 1.0579 Level-reading I 122 - 2 123 1.353 I.I2 4 Inequality of pivots - .294 .294 + .394 + -29 4 i - 6". 36 io".85 4-75 3"-73 z'o sec z + 2 ' 5 8".i6 W-- (Chronograph Magnet with pen on armature Observing Key at Transit Talking and , n Signal RelaS piilli *HL_ J C Uu Talking and Signal Key FIG. 43 224- LONGITUDE BY THE ELECTRIC TELEGRAPH. 393 station the main circuit is broken by the signal key, when the armature of the signal relay breaks the circuit B at both stations, causing a record to be made on the chronograph. In these cases the chronometer is placed directly in the circuit passing to the chronograph, and no provision is made for equalizing the resistance at the two stations. A small difference in the armature time is therefore likely to exist. Chronometer IV. Battery l.Cell /'''--. "~~7Eiip!iiill Chronograph Magnet A V__rr 1 "\ B (mjjjL^^Mth pen on armature H :' Observing Key at Transit 224. IV, VII, and VIII show a more complete arrange ment of circuits. The chronometer is placed in a local cir- cuit A with a weak battery, in order to avoid the injurious effect of a stronger current on the mechanism. When ob- serving transits the arrangement is as shown in IV. The chronometer breaks the circuit A, the chronometer relay breaks the circuit B, making a record on the chronograph. The observer breaks circuit B with the observing key, also producing record on chronograph. VII shows the arrangement for exchanging chronometer signals, being alike at both stations. The chronometer breaks circuit A, when the armature of the chronometer re- lay breaks the main circuit, the armature of relay D break- ing circuit B at both stations. VIII is arranged for arbitrary signals, both stations being the same. The chronometer breaks circuit A, the armature of chronometer relay breaks circuit B, making record on the chronograph. At the sending station the main circuit is broken bv the signal key, when relay D breaks circuit B at both stations. 394 PRACTICAL ASTRONOMY. 22 4 . Arrangement during 1881 FIG. 45. 22$. LONGITUDE BY THE ELECTRIC TELEGRAPH. 39$ By means of the rheostat and galvanometer the electric resistance is kept practical!}' the same at both stations, and therefore a constant difference of armature time avoided. In order to eliminate any small outstanding difference in the action of the two sets of electric apparatus, each set may be used at both stations alternately, the instruments being ex- changed with the observers at the middle of the series. 225. MetJiod of Star Signals. This method of exchanging longitude signals was formerly employed by the Coast Sur- vey. A very full description of the method is given by Chauvenet (Spherical and Practical Astronomy). It is briefly as follows : The difference of longitude between two points, being simply the time required for a star to pass from the meridian of the east to that of the west station, may be measured by a single clock placed in the electric circuit so as to produce a record on the chronographs at both points. This clock may be at either point, or in fact anywhere in the circuit. When a star enters the field of the transit instrument at E, the observer records the transit by tapping his signal key in the usual manner, producing a record on both chrono- graphs. When this star reaches the meridian of W, the ob- server in like manner taps its passage over the threads of his transit instrument, also producing a record at both points. This method is theoretically very perfect; but as it requires a monopoly of the telegraph lines for several hours every night when signals are exchanged, it has proved somewhat impracticable. Example. For the purpose of illustrating this subject I give below the record of a series of longitude signals between Washing- ton, D. C., and Wilkes Barre, Penn., 1881, October 6th. 39 6 PRACTICAL ASTRONOMY. 225. At Washington the instruments employed were the tran- sit circle, sidereal clock, and chronograph of the U. S. Naval Observatory. At Wilkes Barre the instruments were a portable transit and mean time chronometer. At the latter place the following programme was followed: Transits of 16 stars were observed, the instrument being twice reversed; the chronometer was then taken to the telegraph office, 200 feet distant, and the longitude signals exchanged, after which 13 stars were observed with the tran- sit instrument, the axis being reversed once. The 29 equa- tions furnished by the observed transits gave the values of the chronometer correction and rate, also the azimuth and collimation constants of the transit instrument. The following is the method adopted in exchanging sig- nals : At Washington the telegraph key was tapped at intervals of about 15 seconds, making a record on the Washington chronograph, and through the telegraph line a click of the sounder at Wilkes Barre. The observer at the latter place, having his eye on the chronometer, noted the instant of this click and recorded the same. After 10 or 15 such signals had been sent from Washington to Wilkes Barre, a similar series was sent in the opposite direction, the operator at Wilkes Barre tapping the key, producing a click of the sounder at that place and a record on the Washington chronograph. This constitutes a complete series. Two such were ex- changed each night when observations were made. It is obvious that with a chronograph at Wilkes Barre noth- ing need be changed in the above programme. The record would then be made on the chronograph instead of by the observer, and if thought desirable the intervals between the signals could be much shortened. The chronometer at Wilkes Barre being regulated to mean 225. LONGITUDE BY THE ELECTRIC TELEGRAPH. 397 solar time, its correction and rate on sidereal time are some- what large. The values obtained from the observed transits are as follows : At 9 h 39 m chronometer time, AT -j- !3 h 9 3S s .9q3 .024 Hourly rate, -(- 9 .952 Rate per minute, -|- .1659 Similarly for the Washington clock, At 22 h 30'" sidereal time, AT = . 2i s .89i .019 Hourly rate, -)- .0360 The record of the signals with the individual values of the longitude immediately follows : Washington to Wilkes Barre. No. Wilkes B. chronome- AT". Washington AT 1 . Wilkes B. sidereal Washington sidereal Differ- ence of 9. ter. c oc . time. time. longitude. f 9 h 39 m 3'-9 13" 9 m 38'. 94 22 h 44 m 34'.44 - 2i'.88 22 h 4 8 m 52'.84 22 h 44 m I2' 56 4 m 40". 28 .03 2 39 8 .9 38.98 44 49 -5 49 7 .881 44 27 .62 40 .26 .,,i 3 4 39 3-8 39 8.8 39.02 39 -0 45 4 .40 45 19 -38 49 22 .82 44 42 .52 49 37 -86 44 57 .50 40.30 40.36 a 5 40 3.6 39 - I0 45 34-3 49 52 .70' 45 12 .42 40.28 .q r> 40 8 .5 39 - [ 4 45 49 -28 5 7 -64 45 27 -40 40.24 .01 7 40 -3-5 39.18 46 4 -32 50 22 .68 45 42 -44 40.24 .01 b 40 8.8 39-23 46 19 .56 50 38.03 45 57.68 40-35 . I' ' 9 10 4i 3-6 39.27 46 34-66 50 52 .87 46 12.78 40.09 .!>.: .12 - - Mean = 4 40 .253 = A w Wilkes Barre to Washington. I 9" i.M J 3 h 9 m 39'-93 22 h 5o m 32.78 - 2i 8 .88 22*54" 51- 03 22 h 5O m JO 8 . 90 4 m 40- i 09 2 26.1 39 .97 50 4 .72 55 6 .07 50 25 .84 40 .2 3 36.0 39 .99! 50 5 .74 55 IS -99 50 35 -86 40 .3' ,-,,, ^ 51 .1 40 04 51 i .70 55 31 -M 50 50 .82 40.3 .10 5 6.4 40.081 51 28.10 55 46-48 5 6.22 40 2 ! .04 6 20 .7 40.12 5i 4 -52 56 o .82 5 20 .64 40.18 . '4 1 4 35-9 50 .8 40 .16 40 .20 51 5 .62 52 i .66 56 16.06 56 31 .00 5 35 -74 5 50 -78 40.32 40 22 Q 6.1 4 -25 52 28.10 56 46 .35 5 6.22 40.13 .og 10 9 21 .1 13 9 40 .29 22 52 43 .00 - 21 .88 22 57 I .39 22 5 21 .12 4 40-27 .- = 4 40.219 : 398 PRACTICAL ASTRONOMY. 226. Then referring to formulas (391), we have \ = \(\ w -f A e ) = 4"' 40^.236 Wilkes B. east of Wn. JJL = %(\ w A e ) = o .017. In the above the reduction of each signal has been carried out separately, in order to show the precision of the individ- ual values. Practically the labor of reduction may be econ- omized by reducing the means of the recorded times. Thus from the above we have Wn.-Wilkes B. Wilkes B.-Wn. Wilkes Barre chronometer, 9'' 40'" 2i s .2o 9'' 46"' i4 s -53 AT, 13 9 39.13 13 9 40.10 Wilkes B. sidereal time, 22 h 50'" o s -33 22 h 55'" 54 S .63 Washington clock, 22 45 41.95 22 51 36.29 AT, - 21 .88 - 21 .88 Wn. sidereal time, 22 h 45"' 2O S .O7 22h 5 1 "' i4 s -4i Wn. Wilkes B. Wilkes B. Wn. Difference of longitude = 4'" 40^.26 4 4o s .22 A = 4 40 .24 Wilkes B. east of Wn. This value of A is affected by the relative personal equa- tion of the observers at Washington and Wilkes Barre, by the personal equation of the observer at Wilkes Barre in re- cording the signals, and by the difference in armature time at the two stations. (See Articles 220-223.) Longitude Determined by tJie Moon. 226. The preceding methods, in circumstances where they are available, leave little to be desired in facility of application or in accuracy of results. Before the invention of the electric 22/. LONGITUDE DETERMINED BY THE MOON. 399 telegraph the most valuable methods for determining longi- tude were those depending on the moon's motion, chrono- mctric expeditions being generally impracticable. Though the necessity for resorting to these methods is constantly diminishing as the telegraph lines become more widely extended, it will probably never entirely disappear. There are various methods of utilizing the moon's motion for this purpose, the most important of which are the follow- ing: By eclipses of the sun and occultations of stars. By moon culminations. By lunar distances. By measurements of the moon's altitude or azimuth. Some use has also been made of lunar eclipses. All of these methods depend upon the same general prin- ciple, viz.: The moon has a comparatively rapid motion of its own, in consequence of which it makes a revolution about the earth in 27^ days. The elements of its orbit, together with the effects of the various perturbing forces, being known, it is possible to determine the position of the moon at any given instant of time; thus in the American Ephemeris and Nautical Almanac will be found the right ascension and declination of the moon computed several years in advance for every hour of Greenwich time. Sup- pose now at a point whose longitude is required the position of the moon to be determined in any convenient manner by observation; the local time being carefully noted, the ephe- meris above mentioned gives, either directly or through the medium of a more or less extended computation, the Green- wich time corresponding to this position. A comparison of this Greenwich time with the observed local time gives the difference of longitude required. 227. Some of the applications of this principle are capable of giving very good results ; but there is one difficulty inher- 400 PRACTICAL ASTRONOMY. 228. ent in the principle itself which precludes the attainment of an accuracy commensurate with that obtained with the tele graph. The angular velocity of the earth on its axis, which is the measure of time, is twenty-seven times greater than the angular velocity of the moon in its orbit ; it follows, there- fore, that errors of observation in determining the moon's position, or of the ephemeris, will produce errors in the resulting longitude twenty-seven times as great. So if the errors to be anticipated in determining the place of the moon are of the same order as those of determining and comparing the errors of the clocks by the electric telegraph, we might expect to attain to an ultimate degree of precision by the latter method twenty-seven times greater than by the former. Longitude by Lunar Distances. 228. This method is chiefly useful on long sea-voyages, where, in consequence of accumulating errors, the indications of the chronometers become unreliable. The observation consists in measuring with a sextant, or other suitable instrument, the distance of the moon's limb from that of the sun, or from a neighboring star, the time being noted by the chronometer. After this measured dis- tance has received the necessary corrections (to be consid- ered hereafter), the Greenwich time corresponding is taken from the tables of lunar distances of the ephemeris by the methods of Art. 55. The difference between this time and the recorded chronometer time is the error of the chronometer on Greenwich time. An altitude of the sun or a star gives the error on local time ; the difference between the two errors is the difference of longitude. The ephemeris gives the distance, as seen from the centre of the earth, of the moon's centre from the centre of the sun, 229. LONGITUDE BY LUNAR DISTANCES. 40! from the four larger planets, and from certain fixed stars situated approximately in the path of the moon. They are given at intervals of three hours Greenwich mean time. By a series of carefully observed lunar distances on both sides of the moon the chronometer error may generally be ascertained within twenty or thirty seconds. A longitude determined in this way should be considered as liable to an error of five miles, a degree of accuracy which answers the requirements of navigation. 229. We shall consider first the distance of the sun and moon. This distance having been measured and corrected for in- strumental errors, such as index error and eccentricity, the result is the apparent distance between the limbs of the sun and moon as seen from the point of observation. In order to have this comparable with the distances of the ephemeris it must be corrected for the semidiameters, parallaxes, -and refraction of the two bodies. In order to apply the necessary corrections a knowledge of the altitudes at the time of observation is necessary. When there are instruments and observers enough, which will frequently be the case at sea, all of the quantities may be observed simultaneously : the altitude of the sun so ob- served, if that body is sufficiently far from the meridian, may be further utilized for determining the local time. When it is not expedient to make all these measurements at once the observer may measure the altitudes of the sun and moon immediately after measuring the distance between these bodies, the altitudes at the time of that observation being computed by assuming the change in altitude to be proportional to the change of time, an assumption which will not be much in error if the time is short. Finally, the altitudes may be computed by formulae (II), Art. 65, the right ascensions and declinations being taken 402 PRACTICAL ASTRONOMY. from the Nautical Almanac. The apparent altitudes will be derived from these computed values by applying the correc- tion for refraction, table II, and parallax formulas (VI) and (VI),, Art. 81. This supposes the longitude to be approxi- mately known ; otherwise we lack the means of determining the hour-angle /, required in formulae (II): but we shall always be in possession of a value sufficiently accurate for this purpose. If in an extreme case this be not true, we may repeat the computation, using the value ol the longitude ob- tained from the first computation as the assumed approximate value. The corrections necessary to apply to the measured dis- tance may be computed as follows. Correction for Scmidiametcr of Sun and Moon. 230. The following quantities are taken from the epheme- ris: s - the geocentric semidiameter of the moon ; 5 = the geocentric semidiameter of the sun ; n = the equatorial horizontal parallax of the moon ; n= the equatorial horizontal parallax of the sun. The moon being comparatively near the earth, the semi- diameter will vary appreciably with the altitude; there will be no appreciable variation in the case of the sun. The moon's semidiameter varies inversely as the distance. In Fig. 46, MOB = s. Call MAC = s' = apparent semidiameter. s' _ A _sin MAZ_s\n (Z -\- p\ 1 hen = = -. ,>^, - ^ n -^~ - 230. LONGITUDE BY LUNAR DISTANCES. 403 Z being the geocentric zenith distance of the moon, and p the parallax in zenith distance. sin (Z-{-fl)=s'mZcosfl-\-cosZsinfl=sinZ-\-sin,pcosZ, nearly ; from (128), sin/ = sin n sin Z, approximately. Therefore s' = s(i -f sin n cos Z) (39 2 ) The eccentricity of the meridian has been neglected, but the error is inappreciable for this purpose. The correction for semidiameter will be still further modi- fied by refraction. Owing to this cause the apparent disks of the sun and moon are approximately ellipses, the refrac- tion being less for the upper limb than for the centre, which in turn is less than for the lower limb. We therefore require the radius of the ellipse drawn to the point where the curve is intersected by the great circle joining the centres of the sun and moon. 404 PRACTICAL ASTRONOMY. 230. FIG. 47. Regarding the figure of the disk as an ellipse, the conju- gate axis will coincide with the vertical circle passing through the centre, the semi-transverse axis will be equal to s' in case of the moon ; ^, the semi-conjugate axis, is found directly from the refraction table by taking out the refraction for the altitude of the upper and lower limbs respectively and subtracting one half the difference from s '. The angle q formed by the radius s q f with the con- jugate axis is the angle formed with the vertical circle by the great circle joining the centres of the sun and moon ; s a ' being the required semidiameter. To find the angle q. 2 In the triangle, Fig. 48, Z is the zenith ; Mand S, the moon and sun. / VO-H Then sin H=s\nh cos D -f- cos // sin D cos q\ . sin H sin // cos D cos q = -. : cos h sm D F 8 For computing the angle at the sun, h and H will be inter- changed. Then in the ellipse (Fig. 47) \ve have x s q sin q , y s q f COS q 231. LONGITUDE BY L UNAR DISTANCES. 405 Therefore s q ' = -== === (394) Vs'* cosV + F siiiV 231. The values of s q ' computed by (394) for both sun and moon are then to be applied to the measured distance of the limbs of those bodies. We thus have the measured distance of the centres as seen from the place of observation. To obtain the required geocentric distance this must now be corrected for refraction and parallax. Let D', H', and k' = the apparent distance and altitudes of the sun and moon ; D, H, and h = the true geocentric distance and alti- tudes. //and h are obtained by applying to H' and h' the correc- tions for refraction, table II or III, and for parallax formulae (VI) and (VI),, Art. 81. Referring to Fig. 48, cosZ?'=sin//'sin^'+cos/7'cos^'cos.'=:cos(//' h') /. , cosD =sinffsinA +costf cosh cos=cos(/7 - k)coaff cos/i 2sin'f. Multiplying the first of the preceding equations by cos H cos /t, and the second by cos H' cos h', then subtracting to eliminate sin 3 $E, we find cos D=cos (H-h} + [C SZ> '- COS W-W' (396) D is therefore expressed in terms of known quantities. The equation is not, however, in convenient form for numerical 406 PRACTICAL ASTRONOMY. 2 3 2 - computation ; therefore we make the following transforma- tion: cos H cos h i cos D' Let TT, n = T- ; 7^ = cos // ; cos //' cos h' C' C H h = d-, L It may readily be shown that C will never be so small as to give impossible values to D" and d'' '. (396) then reduces to cos D cos D" = cos d cos d" ; from which sin \(D D"} = ^J7j) ^ D ,,l sin \(d d"}\ . (398) and with accuracy sufficient for practical purposes, yT-x (d - d"\ . . (399) As the unknown quantity D is involved in the second member, this equation must be solved by approximation. Writing in the denominator D' -\- D" for D -\- D", we obtain a value of D which will generally be sufficiently near the true one. In case the value found in this way differs very widely from D' , the computation may be repeated, using this value just found in the denominator of (399). 232. In the above we have assumed the angle E (the dif- ference between the azimuth of the sun and moon) to bfc the 232. LONGITUDE BY LUNAR DISTANCES. 40? same for the point of observation as for the centre of the earth. We have seen, however, that the moon has an ap- preciable parallax in azimuth the value of which is given by formulas (VI), Art. 81, or (VII), Art. 82. In order to determine the correction to D due to this quantity, we differentiate the second of (395) with respect to D and , viz., cos H cos h sin E dD = -- da,. . . . (400) remembering that dE = da. da is the parallax in azimuth computed by the formulae above referred to. Formulae (392), (393), (394), (397), (399), (400) now give the true geocentric distance D, corresponding to the measured distance D'. Then by the method explained in Art. 55 we take from the ephemeris the Greenwich time corresponding to this distance; the difference between this time and the observed time will then be the chronometer correction on Greenwich time. If a planet has been used instead of the sun, the same formulas will be used ; but if, as is generally the case, the disk of the planet is bisected by the lirnb of the moon in making the observation, there will be no correction for semi- diameter of planet. The effect of parallax in case of the outer planets will be very small. If the distance of the moon from a star is measured, there will be no correction for semidiameter or parallax of the star. 408 PRACTICAL ASTRONOMY. 233- 233. Formula for Reducing an Observed Lunar Distance to the Geocentric Distance. s' = s(i + sin n cos z) ; sn - 8 cos a q 1 c/3 1 sn For parallax of moon, (VI), Art. 8 1, or (VII), Art. 82. For parallax of sun, (VIII),, Art. 82. cos 77 cos h _ cos //' cos // H' h' = d'\ H - h = d- cos D C cos d' cos d C- (397) COS H COS h Sin E 7 dD = - - . - -=r - da. Sin U Correction for parallax in azimuth. (XXII) These formulae have been written down rigorously, but in practice many abridgments may generally be made in the application. Example. 1856, March gth, 5 h 14 6 s local mean time, the following distance of the nearest limbs and altitudes of the lower limbs of the sun and moon were measured: D' = 44 36' 5 8". 6; //' = 8 56' 23''; k' = 52 34' o". These values are corrected for instrumental errors. Barometer 29.5 inches; Attached thermometer 60; Detached ther. 58; Latitude (p 35; Assumed longitude L = 150 = io h west of Greenwich. 233 LONGITUDE BY LUNAR DISTANCES. 409 From the Nautical Almanac we take the following quantities: Sun. Moon. Right ascension, a 23* 22'" 27 s 2 h n m 47' Declination, d = 4 3' 6" 14 18' 41" . Semidiameter, S 16 8 .o j = 16 23.1 Horizontal parallax, 77 = 8.6 it = 60 I .9 Sidereal time, mean noon, 23'' II" 1 5" From the refraction table we find, for the altitudes above given, Refraction, lower limb, 5' 42^.9 43"-l Approx. altitude of centre, 9 6' 48" 52 49' 40" We now compute the apparent or augmented semidiameter of the moon by the first of (XXII). and then the oblique semidiameter of both sun and moon by the second and third of these formulae. z 37 10' cos 2 = 9.9014 TI i o' i".g sin it = 8.2419 Sum = 8.1433 log (i -(- sin 7t cos z) = .0060 j = 983.1 log = 2.9926 s' 996.8 log = 2.9986 Measured D' 44 36' s8".6 s 16 36 .8 5 = 16 8 .o Approximate D' 45 Sun. D= 45 10' ff= 52 51 h= 9 12 'i-H)= o 45.5 )= 53 36 44 25 cosec= .1550 > A //)= 8 26 sec= 47 9 43 -4 = 8 1217 Then for computing q: Moon. D= 45 10' H= 9 12 h 52 51 i(Z>+A-//)= 44 25 l(D+A-\-ff)= 53 36 cos=g.7734 l(Dh+rf)= o 45 .5 cosec=i.S783 |(Z> /* //)= 8 26 sec= 47 i^= 6 5' tan ^=9.0274 q 12 10 Then from the refraction table we find Refraction upper limb = 5' 24". 8 centre = 5 33 .6 Therefore b 15' 59 .2 \q= 79 56' f=i59 52 = .7507 lower limb = 43"- A centre . = 42 .7 b = 16' 36".4 4io PRACTICAL ASTRONOMY. 233 log b = 2.9819 log 6- = 5.9638 sin 2 q = 8.6476 4.6114 A* = 1.3407 log S" = 2.9859 log S* = 5.9718 ' cos 2 ? = 9.9803 5-9521 B* = 1.3601 5.9715 <:logden. =7.0143 logd. = 2.9857 ac log den. = 7.0014 logd. = 2.9986 log S g 2.9821 log sq = 2.9984 S q = 15' 5g".6 Sq = 16' 36". 4 log b = 2.9984 logl>- = 5.9968 sin 2 ? 9.0736 5.0704 A* = .8720 log s' = 2.9986 log j'* = 5.9972 cos 2 q = 9.9452 5-9424 B* = .9267 5-9971 Obs. Z>'=4436' 58. "6 True D' = 45 9 34. 6 An approximate value of the azimuth of the moon is required for computing the parallax; also of the sun for computing the small correction dD given by the last of (XXII). The formulae for this computation are f tan M = tan 5 tan a = M tan t. cos /' sin (q> M) Converting the mean time of observation into sidereal time (Art. 94), we find 6 = 4" 26'" 3" Sun a = 23 22 27 / = (0 - a) = 5 3 36 * = 75 54' Moon a = 2 h n m 47' t = 2 14 16 t = 33 34' cos=9-3867 Sun. *=.- 4 3' '= 75 54 j*/=-l6 v I2 =35 o >M=i7 59' cosec(') = o sin (z' - y) = 9.78036 sec y o sin (z z) 8.02196 z' - z = 36' 9". 6 // = 53 26' 3". 3 log p = 9.99952 sin Tt = 8.24208 sin (q> (p) = 7.49715 sin a' = 9.95384 cosec z = .22494 sin (a a) = 5.91753 a a = I7".l We now compute (397) ana v399) : II 9 6' 57' .1 COS = 99944799 h = 53 26 3 3 COS = 9.7750603 H 1 = 9 12 22 .2 sec .0056304 ti = 52 50 36 4 sec = .2189667 d = 41 19 6 .2 i = - 43 38 14 .2 *<7 9.9941373 cos D' = 9.8482718 Z?' = 45 9' 34"-6 cos D" = 9.8424091 D" = 45 55 7 -o af = - 44 19 6 .2 COS d' = 9.8595724 COS d" 9.8537097 /" = - 44 26 13 -7 = 44 22 40 .o \(D' + />'') = 45 32 21 .8 d - d" = 427.5 412 PRACTICAL ASTRONOMY. 2 33- First Approximation. Second Approximation. sin \(d + d") - 9. 84472 ra sin #d + a"') = 9. 84472,. log (^ ") = -14647 cosec J(Z? + -D") = 14409 log (Z? - /?") = 2.62213,, log (D - D") = 2.6i 9 75 n D D" 418.9 Z> D" = 6' s6".6 Z> = 45 48' 8". D = 45 48' 10". 4 }(> + />") = 45 5137.5 dD- 3.5 > = 45 48 13 -9 Correction for parallax in azimuth: E = A' - a - 14 26' cos /f = 9.9945 cos A = 9.7751 sin E = 9.3966 cosec D = .1445 log (' a) = 1.2330 logaTZ? = 0.5437 dD = 3". 5 We have now to take from the Nautical Almanac the Greenwich time corre- sponding to this distance by the method explained in Art. 55. For 1856, March gth, we find the following distances of the sun and moon: 12" D = 43 59' 3i" PL ~ .2493 15 45 40 54 -2510 18 47 21 53 .2527 17 We have therefore to interpolate between is h and i8 h . Referring to formula (106), we have J' = i 19". 9 log = 2.6433 PLA = .2510 / = I3 m 4 s log / = 2.8943 Therefore T= 15'' 1 3 m 4" * Correction for 2d difference . I Resulting Greenwich time 15 13 3 Local time of observation 5 14 6 Resulting longitude 9 58 57 The above solution of this problem is only one among many, as it has re- ceived much attention from mathematicians on account of its importance to Taken from table I at the end of the Nautical Almanuc. 235- LONGITUDE BY MOON CULMINATIONS. 413 navigators. The majority of the solutions are only approximate, the design being to reduce the numerical work to a minimum without at the same time sacrificing too much in the way of accuracy. Such methods may be found in any work on navigation, and will be preferred where only an approximate so- lution is required. As may be seen, the solution which we have given may be considerably abridged without a great sacrifice of accuracy. The differences between the oblique and vertical semidiameters of the sun and moon are very small, and the correction for parallax in azimuth is not large. When we remember that the least reading of the sextant is 10", and that measurements of this kind are quite difficult, it will be seen that often little will be lost by neglecting this part of the computation. Longitude by Moon Culminations. 234. The right ascension of the moon may be determined by means of a transit instrument, mounted at the place whose longitude is required, and the local time of observation com- pared with the Greenwich time corresponding to this right ascension, either by taking this time from the ephemeris of the moon, or by means of similar observations made at Greenwich, or some place whose longitude from Greenwich is known. Comparison by means of the Ephemeris. 235. The transit instrument having been adjusted as ac- curatelv as may be, the transit of the moon's bright limb is observed, together with a number of stars suitable for de- termining the errors of the instrument and the clock cor- rection. The corrections necessary to give the moon's right ascension, from the observed time of transit of the limb, are then applied according to formulas (XIX), Art. 195. The last term of the formula may be taken from the table of moon culminations where it is given under the heading u Sidereal time of semicliameter passing meridian." 414 PRACTICAL ASTRONOMY 237. 236. To insure greater accuracy, the moon's right ascen- sion may be derived by comparing the observed time of transit with that of about four stars differing but little from the moon in declination, two culminating before the moon and two after. A list of stars suitable for this purpose was formerly given in the-ephemeris, under the heading " Moon culminating stars," but it has been discontinued since 1882. It is an easy matter for the observer to select suitable stars from the general list of the ephemeris. Let A t = the right ascension of the moon's bright limb at the instant of culmination; A = the right ascension of the moon's centre ; = clock time of observed transit of limb, corrected for all known instrumental errors and for rate; a . 6 right ascension and time of transit respectively of a star, the time being corrected for in- strumental errors and rate of clock ; 5, = sidereal time of semidiameter passing the meri- dian, taken from ephemeris. Then A, - a = - 6- A, = a + (9-6); ..... (401) This quantity A is then the local sidereal time of transit of the moon's centre. 237. We have now to take from the ephemeris of the moon the Greenwich mean time T corresponding to this value A of the moon's right ascension; the mean time T must then be converted into the corresponding Greenwich sidereal time . Then A being the difference of longitude, we have f. . . .... (402) 237. LONGITUDE BY MOON CULMINATIONS. 41$ The time 7* may be interpolated to second differences from the ephemeris, as follows: Let A l the ephemeris value nearest to A; TI = the corresponding time. Then T l -f- t the required time corresponding to A. Let AA =. the difference of right ascension for I minute, taken from the ephemeris ; $A difference between two consecutive values of A A. ft A then equals the change in A A in one hour. Then if / is supposed expressed in seconds, we shall have to second dif- ferences inclusive dA AA .d\A A From which / = ',-; A A + 6 A 7200 and with sufficient accuracy, t SA 60 [A - A } ~] x* 6A Writing *= ' * - then (403) becomes / = x -f- x" ....... (44) 416 PRACTICAL ASTRONOMY. 238. Example. Among the observations of the moon made at Washington I find the following : 1877, May 23d. Observed right ascension of the moon's centre, A = 13'' 28 m 5 S .O2 From ephemeris of the moon, T l = I4 h , A l = 13 27 3 .91 AA = 2.0996 A A l = i i .11 dA = + .0029 6o(^ A,) = 3666.6 log = 3.56426 log A ] A = .32213 x = 29 m 6\4 log x = 3.24213 x" = - .6 log X* = 6.48426 t = 29 5 .8 log ( 6 A) = 7.46240* ac log A A = 9.67787 aclog 7200 = 6.14267 log x" = 9.76720* This is now the Greenwich mean time corresponding to the Washington sidereal time A. In order to compare the two, T t + t must be converted into sidereal time. T,-\-t= i4 h 29 5 3 .8 Table III, Appendix N. A., 2 22 .77 Sidereal time Greenwich M. N. = 4 4 48 .56 Greenwich sidereal time = 18 36 17.1 A = @ - A = 5" 8 m I2M, the required difference of longitude. 238. If the ephemeris were perfect, very little could be done further in the way of perfecting this method. The errors of the ephemeris, however, are not inconsiderable, and in consequence it cannot be used directly as above, except when an approximate value of the longitude is sufficient. For the year 1877 tne average correction to the right ascen- 239- LONGITUDE BY MOON CULMINATIONS. 417 sions of the ephenieris, as derived from 66 observations at Washington, was '.31, which would have produced an error of 8 s in the 'longitude if the observations had been used for that purpose. Either of two different methods may be used for eliminat- ing from the result these errors of the ephcmeris. First. Correction of the epliemcris. This method is due to Prof. Peirce.* The ephemeris is compared with all available observations of the moon made at Greenwich, Washington, and other fixed observatories during the lunation, and in this way a series of corrections to the ephemeris obtained which, as they depend on all available data, are much more reliable than simply the place of the moon observed at any one observatory. Peirce found that for each semilunation the corrections to the right ascension of the ephemeris could be represented by the formula X = A + Bt + Cf\ (405) X being the correction required, t the time reckoned from any assumed epoch (which should be chosen near the mid- dle of the period under consideration for greater conven- ience), and A, B, and C being constants determined from the observations made at Washington, Greenwich, etc. The ephemeris when so corrected is used as already explained. 239. Second. Corresponding observations. The difference in the longitude of anv two points mav be found by com- paring the values of the n'ght ascension of the moon ob- served on the same night at both places. The times of transit of the moon's bright limb and of the comparison stars are observed at both places and the cor- rections applied as already explained to find the right ascen- * Report of U. S. Coast Survey 1854, p. 115 of Appendix. 4 1 8 PR A C 7 '1C A LAS! 'RONOM Y. 2 39. sion of the centre at the instant of transit. It will be a lit- tle better if the same comparison stars are used at both stations. Let Z,, and Z., = the assumed longitudes of the two sta- tions ; * A. = the true difference of longitude ; A! and A t = right ascensions of moon's centre from observations at L l and Z- 2 ; H = variation of right ascension for one hour of longitude, while passing from meri- dian of L, to that of Z, a . Then A, - A, = \H\ _ A, -A, //is taken from the table of moon culminations, where it is given for the instant of transit of the moon's centre over the meridian of Washington. When used as in (406) its value must be interpolated for a longitude midway between Z, and Z 2 . Example. As an example of the determination of longitude by corresponding observations, let us take the transit of the moon, the observations and reduction of which are given in Art. 196. We have there found for 1883. October 15 : Right ascension of moon's first limb, i h 15"' so.o8. Secondflimb, i 18 11.76. At Washington the right ascensions of the limbs were observed as follows : First limb, i 1 ' i6 m 7 S .3S. Second limb, i 18 28 .69. * Reckoned from Washington or Greenwich according as we use the epheme- ris computed for Washington or Greenwich. One of the longitudes, L\ or.ii, must be known with some accuracy. f This is corrected for defective illumination. 239- LONGITUDE BY MOON CULMINATIONS. 419 Taking the mean in each case as the observed right ascension of the centre, we have At = i 17 18 .035. At A l = ly'.iis. From ephemeris, H = 153". 88; A* A l A = = o h .iii2 = 6 m 4o'-3. The difference of longitude between Washington and Bethlehem determined telegraphically is 6 m 4O 5 .2. This close agreement is of course accidental; a deviation of four or five seconds from the true value would not have been sur- prising. If we reduce the observations of the two limbs separately, we find : First limb, A = 6 m 44'. 7. Second limb, A. = 6 36 .o. The mean being the same as above. This is an illustration of the necessity of employing transits of both limbs. Frequently the difference of longitude de- termined separately from transits of each limb will show much wider deviations than this, even when all possible care is taken to avoid error. To illustrate the method of Art. 236 for deriving the moon's right ascension by means of comparison stars, take the following transits of the moon : f Piscium and v Piscium observed at the Sayre observatory, 1883, October 15. Object. Clock Time. /Piscium Moon I Moon II v Piscium I" 1 1"' 5 5 s . 6 7 I 15 55 -55 i 1 8 17 .23 i 35 30 .41 These times are corrected for instrumental errors, and that of the second limb of the moon for defective illu- mination. The clock-rate is inappreciable. /Piscium. f Piscium. f Piscium. v Piscium. 6 i ii m S5'-67 i" 35". 30-41 a h n m so.o6 i" 35- 2 4 ".8 7 e e' 1 >5 55 -55 i 18 17 .23 i IS 55 -55 i 18 17 .23 if 15 49 .94 i 15 50 .01 i 18 ii .69 e -e e'- + 3 59-88 + 6 21.56 - 19 34 -86 -17 13.18 Mean of A n A o' 15 49.98 18 11.66 420 PRACTICAL ASTRONOMY. 240. This method of deriving the moon's right ascension is employed with most advantage when the same comparison stars are used at both places whose dif- ference of longitude is required, as then uncertainties in the places of the stars will produce no appreciable effect on the result. In our example we have preferred to use the value of the moon's right ascension derived in Art. 196, since the value of A T there used was obtained from transits of a number of stars, and thus a result obtained more likely to be reliable than the one above, which depends only on two stars. 240. If the difference in longitude between the two places is more than two hours, the above method requires some modification, as then the third differ- ences in the hourly motion // will be appreciable. The right ascensions A\ and AI are obtained from observation precisely as before ; then the right ascensions are taken from the ephemeris for the time of culmination at the two meridians, using for this purpose the assumed values of the longitude. Let i and a* = values of the right ascension taken from the ephemeris for the assumed longitudes LI and LI ; Aa = correction to the ephemeris. Then &i -\- Aa and cr y -}~ Aa = true values of the right ascension. If then LI and Zi are the true values of the longitude, (a 3 -(- Aa) ((Xi-\- Aa) = tr 2 a\ will be equal to Ay A\. Let Li L! -J- AL = true difference of longitude. Then AL is the correc- tion to the assumed difference of longitude. Let H = (At Ai) (a* a,). Then AL = ~ . . . (407) H being, as above, the hourly change in the moon's right ascension, AL will here be expressed in hours. To reduce to seconds we multiply by 3600, viz., *L = K&S. : -' (408) This process is sufficiently simple in theory, but if the table of moon culmina tions is employed the moon's right ascension must be interpolated to fourth or fifth differences, which will involve considerable labor. By using the hourly ephemeris of the moon the interpolation need only be carried to second differ- ences. In any case we must assume the moon's motion in right ascension given in the ephemeris to be correct. 241. LONGITUDE BY MOON CULMINATIONS. 421 The hourly motion, //, is taken from the ephemeris for the time of observa- tion at the meridian whose longitude is tp be determined. Example. 1883, October 16, the moon's right ascension was determined by meridian observation at Greenwich and Bethlehem as given below. The transit of the second limb was observed, the Bethlehem observations being made and reduced precisely as in the example of Art. 196. At Greenwich, AI = a' 1 6 m 17". 46. At Bethlehem, A* = 2 19 32 .18. From the houny ephemeris of the moon we now take the right ascension of the moon's centre. Since the argument is the Greenwich mean time, we must convert the above values of the right ascension, which are equal to the sidereal times of observation, into the corresponding Greenwich mean solar time, using for the longitude of Bethlehem the best approximation to the true value which we possess. Thus : Local sidereal time A* 2 h 19 32 s . 18 Assumed longitude from Greenwich. 5 131.9 Greenwich sidereal time. ...'. 2 h 6 m 17". 46 7 21 4.08 Sidereal time, mean noon 13 38 38.61 13 38 38.61 Sidereal interval past noon 12 27 38.85 17 42 25.47 Table II, Appendix of Ephemeris. . 2 2 .48 2 54.05 Greenwich mean lime 12 25 36.37 17 39 31.42 For these times we find t*i =. 2 1 ' 6'" I7 s .6i

k, thus making sin $ > I, an impossible value. As the observation of occultations near this limit is not of PREDICTION OF AN OCCULTATION. 435 great value for the determination of longitude, it will not be worth while to make a very close computation to ascertain whether the occultation actually does occur when it is found to be near the limit. 252. The successive steps in preparing to observe the oc- cultation of a Nautical Almanac star at a given place, assum- ing it to be visible at that place,* are therefore as follows: I. We take from the "Elements for the Prediction of Oc- cultations" of the American Ephemeris the Washington mean time of geocentric conjunction T , the Washington hour- angle //, also Y, x',y', and the star's apparent declination d. II. T t and H are reduced to the local time and hour- angle by applying the correction for longitude, A. p sin (p 1 and p cos 25 .00026 .00265 and p sin tp', which will be given by 30 .00036 .00255 the formulae 35 .00048 .00243 40 45 .00060 .00073 .00231 .O02I8 p cos = fcos (p; 50 55 .00085 .ooogS .00206 -00193 , sin

' sin A ; ) ( 4II v T? = p sin ' sin tf cos ^ . ) * & = [9-4192] P cos tp' cos //.; ) ( } rf = [9.4192] p cos ^' sin // sin 3. } In which // = // A /< a. IV. m, M, n, and ^Vare computed by m sin ^/ = x ,; n sin N = x' ,'; m cos M = y //; ;/ cos N y' //'; m sin (M - N} then >p and r by sm //> = - -> -f j '-(423) ^ cos_^ _ mcos(M-N) | ' J Calling the value corresponding to the plus sign r lf and that corresponding to the minus sign r a , we have Time of immersion = T + r, = Tj Time of emersion = T -(- r 2 7],. V. With these values 7", and T^ we now repeat the com- putation for a second approximation to the true values of the time of immersion and emersion. h a in (411) and (419) will become (/* -|- r,) for immersion, and (/z -f- T S ) for emersion. 7", will give us two values of T; one a small value giving a more accurate time for immersion, the other a large value giving an inaccurate time of emersion. In the same way T, 253- PREDICTION OF AN OCCULTATION. 437 gives a small and accurate value of r for emersion and a large inaccurate value for immersion. The values of x and y to be used in this second approxima- tion will be given by the formulas x = x'r lt y = Y -\~ y'r^ for immersion, and x = X'-TV y Y -f- y'r v for emersion. The values of T given above will be expressed in hours. If it is considered desirable to express them in minutes we may use, instead of n, a quantity ', viz., ff 'Ss.'Ses, [8.22 1 8] n. As a check upon the values of the times finally obtained, we compute for these times the values of x, j, +ir ) n- Write p cos ?/ sin // = ; ) (420) [9.4192] p cos ?/ cos (^ +ir ) = $'. f ' Then r = ^ f g/ (430) In the first approximation the r in the value of ,' may be neglected ; or we may assume it equal to -J// , which will gen- erally be a little more accurate. As the average duration of an occupation is about one hour, we may therefore, in ordinary cases, assume as the hour-angle in equations (41 1) and (419) For immersion, k a -\- T O 30; ) / N For emersion, /z -f- r -)- 30'". I The value of r may be taken from Downes's table, given in connection with the subject of occultations in the Ameri- can Ephemeris. 253- PREDICTION OF AN OCCULTATION. 439 Example. Required the time of immersion and emersion of the star a* Libra at Beth, lehem, 1883, September 6th.

= 9.8810 constant log = 9.4192 log | = 9.6970 \og(x' ') = 9 5824 log I' = 9 1787 ' = .1509 *r = i h .3O2 log r c = .1146 x' % = .3823 The computation is now as follows: Immersion. Emersion. Ao = 2 h 43 m 6 ^ = 2 h 43 m .6 fr = ii8.i r = ii8.i - 30 +30 &' = 3 h 3i m -7 Ao' = 4 h 3i m -7 = 52 55' = 67 55' We now compute |, v, %, and rj\ as follows: * We might have used Downes's table above referred to, where we find T O = 74. t Strictly T O should here be reduced to sidereal interval, but the approximation is so rough that it is not important. 440 PRACTICAL ASTRONOMY. 253 sin 5 = 9.4284,2 cos 6 = 9.9838 sin h a ' 9.9018 p cos sin d = 9. 3094^ cos ho = 9.7803 log ? = 9.7828 p cos sin 5 cos ^ ' = 9 0897^ p sin ' sin d sin //o' = 9.2ii2 Check." I = + 0.6064 Nat. No. = .1229 Nat. No. = + .6238 7 = + -7467 x = jrr = /= .4276 =+ .5433 sin M = g.8i97 sin M = 9 2524 cos .fl/ log g = 9.0805 log rf = 8.6304,, - ^) = - -2034 tan M = 9.9441 log m = 9-4327 sin N = 9.9930 sin N = 9.6158 n cos N = 8.8727*1 tan N = .743i log n 9.6228 log ' = 7.8446 in (M - N) = 9.9328 log m = 9.4327 sin if, = 9 9305 ^ = 58 26'.2 M = 221 ig'.2 g' = .1204 7' = .0427 *' = -5332 y = - .1173 x - % = .4128 y 77' = .0746 N = 100 14'. 5 M - N - 121 4.7 cos (AT ^) = 9-7i28 logw = 9.4327 log-, = 2 1554 1.3009, cos i/> = 9.7189 log k 9 4350 Nat. No. = 2o m .oo log- = 2.1554 ft 1.3093 Nat. No. = 20 m .39 Immersion T\ = o .39 Emersion (inaccurate) r a = 4~ 4 -39 r a A = 6 h 25 m .i - 30'" = 4-48 .1 . T I = - o .39 T = 7" i2 m .8i * The comparison with the true value of #, viz., .0741, shows the adopted value of A t ' for 253- PREDICTION OF AN OCCULTATION. 441 Emersion. sin d = 9.4284/7 cos S = 9.9838 sin Ao' = 9.9669 p cos qJ sin <5 = 9 3094 COS Ao' = 9.5751 log | = 9 8479 p cos ' cos h = 9 7561 p cos (f> sin 6 sin // ' = 9.2763, log I' = 8.8753 log T}' - 8.6955,, sin M = 9 8341 m sin yl/ = 9 4087 m cos M = 9 4384,, tan J/ = 9.9703 log m = 9 5746 sin N = 99953 n sin A 7 " =; 9 6611 cos A r = 8.8306,1 I = +0.7045 Nat. No. = .0766 Nat. No. = .6238 if = + .7004 y = X = XT = -\- .9608 = + .4260 (* ) = .2563 O w = - 2 744 = 136 57'. 6 ?' = .0750 77' = .0496 *' = .5332 / = - ."73 ^ - ' = -4582 y - T/' = - .0677 log = 9 6658 log' = 7.8876 - A 7 ') = 9.7946 log w = 9.5746 l s^= -5650 sin t^> = 9 9342 # = 59 I5'c M-N= 38 33'- 3 cos (J/ - A 7 ) = 9.8932 log w = 9.5746 log 2.1124 1.5802 Nat. No. + 38 m .o cos V = 9.7087 log k 9 4350 log = 2.1124 1.2561 Nat. No. i8 m .o Emersion r a = 20 .o Immersion (inaccurate) r, = 56 .o T - A = 6 h 25"'. i r + 30" = i 48 -i r 2 = 20 .01 immersion to be nearly correct. That for emersion, however, is considerably in error. 442 PRACTICAL ASTRONOMY. 254. As a check on the accuracy of these values we now recompute x, y, , and 77, when we find (x S) 2 -f (y - rj)- = .07426; (x - |) 2 + (y - r/) 2 = .07447. We have therefore a very close approximation to the true time of immer- sion, the time for emersion being a little less accurate. A partial recomputa- tion of the latter gives a correction of o"M6, making the final value of T = 7 h 53 m .O3. This latter computation is altogether unnecessary for practi- cal purposes. For computing the position angle P at emersion,* formula (424), we obtain a value which will generally be sufficiently exact by using the last values of .Wand ^ obtained in computing T. In this case we have N = 98 24'; # = 59 15 ; P = JV+0+ 180 = 337 39. If the angle at the vertex V is required, we have, (428) and (425), um<; = lll; V = P - C. r/ + i?r Using the values just derived, viz., I = .7045, ' = .0750, ? = + . 7004, rj' = .0496, r a = - o h .3335, we find C = 43 28'. Therefore V = 294 u'. 254. In predicting the occultations which will be visible at a given place within a given time, the first operation will be to go over the list of occultations of the ephemeris and select those which may be visible. The conditions of possible visibility are: 1. The limiting parallels of the last column must include the latitude of the place. 2. The hour-angle H A, taken without regard to sign, must be less than the semidiurnal arc of the star; in other words, the star must be above the horizon. 3. The sun must be below the horizon, or at least not much above it, at the local mean time (T A), unless the star is bright enough to be seen in the day- time. Remark i. If the place is near one of the limiting parallels of latitude an oc- cultation may or may not occur. If it is desirable to observe such stars as are *This angle is not required for immersion. 255- GRAPHIC PROCESS OF PREDICTION. 443 occulted near the north or south limbs of the moon, such doubtful ones may be included in our list, and the occurrence or non occurrence of an occultation will be shown in the computation of the time of immersion and emersion. As before shown, if the occuhation is not visible at the place under consideration it will be indicated by sin ^ becoming > i in the formula sin ib = -. Remark 2. In most cases we may see by inspection whether condition 2 is fulfilled. For those stars near the limit it may be necessary to compute roughly the hour angle of the star when in the horizon, for which we have cos / = tan d tan cp (122) If then (ff A) is numerically less than t this condition is fulfilled. A small table computed for the latitude of the place, giving t with the argu- ment <5, is convenient in examining this condition and the next. Remark 3. For determining whether the sun is above or below the horizon, we may compute roughly the times of sunrise and sunset by the method given above for the star, or, since it is not required with great accuracy, we may take it from a common almanac. In going over the list of the ephemeris, the computer will write the value of A on the lower edge of a piece of paper, and pausing over each star for which condition i is fulfilled, he will see whether 2 and 3 are also fulfilled. If either fails the computer passes on. In those cases where he is unable to decide by inspection whether either of the two fail, the star will be marked for further examination after the list has been gone over. Where many predictions are to be made for a given place the work may be much reduced by computing tables for the given latitude by means of which the computation of f, 77, |', rf , and r is facilitated. ' The necessary directions for forming and using such tables are given in the American Ephemeris, to which the reader is referred. Graphic Process. 255. If the observer possesses a celestial chart containing the stars whose occultation is to be predicted, the necessary computation may be made by a very simple graphic process. The scale of the chart must be large, and the method will be principally useful in case of clusters like the Pleiades, where a considerable number of stars undergo occultation within a short time. The right ascension and declination of the moon are taken from the ephemeris for intervals of half an hour throughout the time covered by the occultations; the correction for parallax must then be applied. The resulting apparent places of the moon are then laid down on the chart, and a curve being drawn through 444 PRACTICAL ASTRONOMY. 257- the points we have the apparent path of the moon's centre; this line being then properly subdivided between the half-hour points furnishes a graphic time- table of the moon's centre. Each star whose distance from this line is less than the augmented semidiameter* of the moon will suffer occultation. From such a star as a centre, with the moon's augmented semidiameter as a radius, let a circle be drawn ; this circle cuts the path of the moon's centre in two points the position of which on the curve will give the time of immersion and emersion of the star, and the direction of the star from the point of intersection gives the position angle on the moon's limb. Computation of Longitude. 256. It has now been shown how we may predict the time of beginning and ending of an occultation, as seen from a point on the earth's surface whose longitude is known. The fundamental equation which expresses the condition neces- sary for such an occurrence is (415) If now all of the data of the problem were perfectly known, and if no error entered into the observed time of the occul- tation, this equation would be completely satisfied. Since, however, such perfection is not attainable, we may employ the observed time of an occultation for determining the cor- rections to the values of the constants used. The correction which it is the immediate object of this discussion to consider is that of the longitude assumed. In order, however, that this may be obtained with all possible precision, we must endeavor to obtain or eliminate as far as possible the corrections to the other quantities which enter into the equation if the values employed are at all uncertain. 257. Before making the transformation which (415) re- quires in order to adapt it to our purpose, let us examine the quantities entering into each term separately, in order to see * Formula (392). 257- LONGITUDE BY OCCULTATIONS. 445 what may be regarded as definitively known and what quantities may require corrections. k. The moon's semidiameter may be determined from oc- cultations more accurately than in any other way. A cor- rection Ak to the value employed may therefore be intro- duced as one of the unknown quantities of our equation. ,, >/. Referring to the expressions for the value of these quantities, equations (411), we see that they depend upon a and d, the right ascension and declination of the star ; /u, the local sidereal time; p, the earth's radius; and g>', the geocen- tric latitude, a and tf should be so well determined that they may be regarded as absolute, that is, no stars should be used for this purpose whose places are not so well determined as to require no further consideration. /u, the local time, must be accurately determined by the transit instrument (see Chap. VI). The time determined by observation will gen- erally be sidereal. The ephemeris of the moon given in the Nautical Almanac is arranged for mean solar intervals, so that when this is employed it may be necessary to convert the sidereal time into mean solar time, or the reverse in some cases. It will be remembered that this conversion supposes the longitude known. We shall therefore require an ap- proximate value of the longitude, which we shall suppose to be accurate enough so that no appreciable error will result from employing it for the above reduction. If a case should ever occur, which is not likely, where this preliminary value was so erroneous that appreciable errors in the subsequent computation resulted from its employment, then it would be necessary to repeat that part of the computation which was affected by it, using the value of the longitude obtained from the first reduction. In this way we should obtain a second approximation to the true value. (p. The latitude must be well determined by the zenith telescope or other suitable instrument. 44 6 PRACTICAL ASTRONOMY. 258. p depends upon the eccentricity of the earth's meridian passing through the place of observation. A satisfactory determination of this quantity from occultations is not pos- sible, but Bessel introduces a term into the equation depend- ing on the correction to the assumed eccentricity, in order to show its effect on the final result. This term will be re- tained for the sake of completeness, though in the practical application of the formulas it will generally be disregarded. x and y. Equations (409). Besides quantities already con- sidered these contain^, D, and r, the right ascension, declina- tion, and distance of the moon. Corrections to the assumed values of all these quantities will be introduced into the equations. Those to the right ascension and declination can be well determined from an occultation observed at any place whose position is known. In order, however, to determine r, or the moon's parallax on which r depends, observations must be combined which are made at widely different points on the earth's surface, whose difference of longitude has been previously well determined. The correction to the parallax will be retained for completeness. 258. Let us now suppose a series of occultations observed at two points, the longitude of one of which is well deter- mined. The immediate object is to determine the longitude of the second point. If one star only is observed at the second point, we must assume all the quantities entering into the equation to be known with one exception. If we assume the longitude to be the unknown quantity, we- obtain from our data a value of that quantity which is affected by all of the errors of the data. If the star is also observed at the first point, this observation may be employed to correct the tabular right ascension and declination of the moon, and the longitude of the second point determined bv the aid of these corrected values. If more stars are observed sufficiently near together so that the errors may be regarded as constant 259- LONGITUDE B Y OCC UL TATIONS. 447 during- the time elapsed, then the correction to the semi- diameter can be included as an unknown quantity. As we have remarked before, the errors of the parallax cannot be well separated from the longitude. If then the number of occultations observed is greater than that of the unknown quantities which can be well determined, a solution of the re- sulting equations by the method of least squares will give the most probable values of the quantities, expressed in terms of the constants, and of those quantities which cannot be separated from the constants. 259. We now proceed to develop the equation in the form required. The method is that of Bessel. The meridian from which the longitude is reckoned will be called the first meridian. Let / = the local time of an observed occupation mean or sidereal ; w = the west longitude of the place of observa- tion. Then t -\- w = the time at the first meridian. Let r = an arbitrary time ?t the first meridian suffi- ciently near (/ -j- w] so that the change in x and y during the interval (t -\- w T) may be assumed to be proportional to the time. * and y, are the values of x and y at the time r. Let Ax, Ay, Ak, be the corrections required to reduce the values of x, y, and k employed to the true values. These corrections depend on the various outstanding errors above considered. The true values of these quantities, corresponding to the instant of observation, will then be k -f Ak; x n +x'(t+w 448 PRACTICAL ASTRONOMY. 259. x' and y' are as before the changes in x and y in one hour, mean or sidereal according as one or the other is employed. Let Aee the correction to the assumed value of e" ; e be- ing the eccentricity of the meridian. Then and ;? will require the corrections -^ Aee and r- Aee, As these quantities, and //, do not depend upon the longitude, they will be correctly given by equations (411), and require no other corrections. Using the corrected values of x, y, B,, t], and k, equation (415) becomes (k + AkJ - V +/(/ + v> - r} + Ay - -fcr. (430 w is supposed known with precision enough so that the val- ues of x' andy, which change with the time, will be known with sufficient accuracy. Let m sin M = (X Q ); ;/ sin N = x'; \ , ^ m cos M = (jj/o '/); n cos N = y'. ) Equation (431) may then be written = wsin M+n sin --~4ee , (433) 259. LONGITUDE BY OCCULTATIONS. 449 which may be placed in the form .r sin N+by cos If- +[, smOtf-AO-f A* cos JV- A, sin JV- -. ( 434 ) Let us write - A . .. . . , T d(S, sin N-\-r> cos N} A. = Ax sin N+Ay cos N - - ~ / >-Aee ; ,, A ,, , . ,, ^(^cos^V r;sin TV) ^ A = Ax CQsNAy sin N ! -Aee. Then \k + AkJ = [n(t+w -r)-\-m cos (MN) + A] a +[> sin (J/_^)-A7. (436) Let w sin (J/ ^V) = ^ sin ^ ...... (437) Then neglecting terms of the second and higher orders in A' and Ak, (436) may be written as follows: t -+-w T = - cos ib -- cos (M N} -| -- sec n n n ~\ (438) >& w ms\r\(M- We have -cos^ cos (M-N) = - J n n n sin $ a form which is a little more convenient when sin ip is not very small. 45 PRACTICAL ASTRONOMY. 261. Equation (438) then gives m sin (MN+$) Ak , V K- TV - (t T)-] -- sec i--\- -tan ^ -, (439) n sin ^ ' ' n { n n' v and the equation is solved for a;. As will be seen, this value of w is ambiguous, /> being de- termined from (437) in terms of the sine, with nothing to fix the algebraic sign of cos fi. As before, however, equation (423), the sign of cos >/' will be in case of immersion and -f- for emersion. This will always be the case except when the occultation takes place very near the north or south limb of the moon, when there will sometimes be exceptions 10 the rule. Such occultations, however, are worth very little for longitude purposes, and therefore will not require further consideration here. 260. x' andy vary so slowly that the above equation will give a very close approximation to the true result, even when (t -\- TV r) is some hours in duration. It will, how- ever, be best to arrange the computation so that (/ -{- w r) is a small quantity, as the labor is less in dealing with small quantities than with large ones, and there is less liability to error. The unit of time in the small terms of (438) and (439) is one hour. If then w and (t r) are expressed in the usual way in hours, minutes, and seconds, it will be convenient to ex- press these small terms in seconds. If then the time of the ephemeris and of observation are both sidereal or both mean solar, these terms should be multiplied by 3600. If, however, the ephemeris time is mean solar, and that of observation sidereal, we must multiply by 3609.856. 261. Let us now consider more fully the quantities A and A'. These depend upon the corrections to the moon's co-ordi- nates, viz., Ax and Ay, and upon the correction to the eccen- tricity, Aee. These will be considered separately. 26 1 . LONGITUDE B Y OCCUL TA TIONS. 45 I The co-ordinates x and y are variable quantities, and the corrections which they require on account of the inaccuracy of the data, viz., Ax and Jy,'\vill also be variables. It will be ^more convenient for present purposes to express these in terms of quantities which remain constant throughout the entire occupation. We have x = .* -(- n sin N(t -\- w T )',\ y = j/ o -f cos;V(/ + w - T); j ' irom winch we have x sin N-\-y cos yV= -r sin N-\-y cos 7V-j-(^-hw r); ) , x cos N-\-y sin A^= ,r cos N-\-y a sin A^. f The last of these is practically independent of the time, and therefore may be regarded as constant throughout the entire occultation. Let n = x^ cos N -f- y n sin N = x cosN -\-y sin N. Then squaring and adding equations (441), *"+/ = n* + [x t smN-\-y t cvsN+n(t + w r)] 8 . (442) This expression is a minimum when the last term is zero. Let the value of (/ -f- w) corresponding to this minimum be T. Then x^ sin N-\- y a cos N -\- n( T T) o; = x cos ;V-f a sin TV. Therefore = vV +y is the minimum distance of the axis of the cylinder from the centre of the earth, and T is the time at the first meridian corresponding to this minimum. 45 2 PRACTICAL ASTRONOMY. 262. We now have x sin N -\- y cos N = n(t -f- w 7"); Referring no\v to the values of A and A', equation (435), we* have for the part of these quantities depending on x and y For A, Axs\i\N-\- For A', Ax cos 7V+ dy sin N. Differentiating equations (444), we have for these quantities = nAT-\- (t -\- w Ax cos TV -f- -4^ sin N = AH. Therefore that part of the terms (A' tan ip A) due to Ax and nA T-\- AH tan if} (t -f- w T}An. . . (445) The corrections Ax and Ay are by this formula expressed in terms of AT, AH, and An, which will be constant for the same occultation. 262. It remains to consider the effect of an error in the eccentricity, viz., Aee, which is considered here for the sake of completeness, though it might be neglected without se- riously impairing the practical value of the theory. From (134) and (140) we have cos (p sin = ; p sin q>' = . (446) .4/1 ^sm 2 (p I dp sin tpf dee p sn ' dee * dp cos q>' dee ' d 11 */?; dee dp sin ' dee Referring now to the values of and /;, equations (411), we have -7 = sin C/u a): -. r^ -, = o i' \ f ^ /* x/icin fri ' dp cos cp' dp sin 7 = sintfcos(,u a}: -3 : 7 = cos#. dp cos (p dp sin cp Therefore -jr- = -/fy&; ^ = -/3fi t j 0cos8. (447) Referring now to the values of A and A', (435), we have for the terms depending on Aee ForA '_ ^ nsAr _ s . nAr r r . (448) For A', -A** = r ^^( f cos N -\-TI sin A'')-)-^ cos S sin A'' Let us write & sin ^-; 454 PRACTICAL ASTRONOMY. 263. which substitution will give us for (449) + - n - cos '/-]+/? cos tf cos N \ Aee; \ , . H + k sin //]+/? cos d sin JV}^. } l Therefore that part of (\! tan $ A) which depends upon ^ is [i _ 7.) - * tan ^ - * sec V] - ^ . (451) Therefore by (445) and (451) the last three terms of equation (438) or (439) will be as follows: - sec i(> -\ -- tan if) -- = A T -\ -- tan tp AH -4- - sec if> Ak - (t + w - T) . (453) Each term is expressed in seconds of time, and h is the num- ber of seconds in one hour of the kind of time employed in the ephemeris of the moon. If the times employed in the ephemeris and in observation are both sidereal or both mean solar, h = 3600. If the ephemeris time is mean solar and the time of observation sidereal, // = 3609.86. 263. We have now obtained an expression for the small terms of our equation, in which the quantities depending- on the corrections to the moon's place are expressed in terms of quantities which are constant during the time of the occulta- tion. It will be advantageous, however, to express them directly in terms of the corrections to the quantities given in the ephemeris, viz., to the moon's right ascension, declination, and horizontal parallax. 263. LONGITUDE BY OCCULTATIONS. 455 Let A(A or) = the correction to the assumed difference of right ascension of the moon and star; A(D 6) = the correction to the assumed difference of declination ; An = the correction to the assumed parallax. We have, equation (409), cos D sin (A a) sin D cos S cos D sin 8 cos (A a) . X = shT^ = y = ifn^ ' ' < 453) X Y Writing for brevity x = , . sin TT' sm TT' and differentiating, we have AX ATI AY A* Ax = x ; Ay = . y CiriTT tiltn -TT * C1117T J tan ar sin TT 'tan TT These equations in connection with (444) give the following: sin TT It will presently be shown that - = tan 7t n and therefore _ nAT= ^J^L^ Sin rr AT , *v *r } (454) AX cos TV -f- A Y sin 7v JTJ sin TT ;r tan 456 PRACTICAL ASTRONOMY. 264. 264. The value of An will now be more fully considered. We have, equations (432), n sin N = x'\ n cos N = y'. From these, n" = x" -j- y'\ Differentiating, nAn = x' Ax' -\- y' Ay' (455) x' andy, it will be remembered, are the changes in x and y respectively in one hour. Regarding them as the differen- tial coefficients of x and y with respect to the time, we have dx__d L t X \ _ dX dt dfr\sin nl dt sin n dy _ d( Y \ _ dY i , dt ~ dt\s\\\ nl~ dt sin n ~ y ' -JT and -JT depend upon the hourly change of the moon in right ascension and declination, which changes are given with accuracy by the ephemeris. Any correction to the val- ues of x' and y' will therefore depend upon n. We may therefore write A ' A Ax' = A -. = sin n tan ?r' A > A Ay = A - = sin n '. tan n Substituting in equation (455), it becomes /a An y ^tan n Therefore = , the value assumed above. n tan n 265. LONGITUDE BY OCCULTATIONS, 457 265. Returning now to equations (454), we see that An AX AY and tan TT' sin JT sin ?r may be regarded as constant throughout the duration of the occultation, since they are expressed in terms of AT and AH, which are constant, and An and TV, which are practically so. AX AY The values of - - and - - will then result from the differ- sm n sin re entiation of equations (453), viz.: X = cos D sin (A ); Y = sin D cos cos Z? sin # cos (A a); ^/JT = cos D cos (^ a)A(A a) sin Z> sin (A a)AD\ A Y = [cos D cos d + sin Z> sin tf cos (^ a)] AD -\- cos Z> sin S sin (y4 at) A (A a] At the time of conjunction of the sun and moon A be- comes equal to a. Therefore AX cos/?.. JF cos(/? tf) , = -- A(A a); - - = -- i -- '-A(D ^.(456) sin 7t sin TC sin n sin n Therefore taking D and re for the instant of conjunction of the moon and star in right ascension, and regarding A(Aa) and A(D tf) as the corrections to the assumed differences of right ascension and declination at this instant, also writing unity for cos (D d), TC for sin n and tan TT, we have, from (454), (457) . AT= ^smTV-f - cos TV; a} A(D-6} . A7 An An= '- cos TV-f sin TV x ; TT ^ TT ' ATC 4$8 PRACTICAL ASTRONOMY. 266. Substituting these values in (452), and writing for brevity we have -* sec ^4. -tan $ - = - r[sin NCOS DA(A - a)-f- cos NA(D - 5)] _|_ j/[ C os yVcos DA(A ) 4~ sin NA(D 5)] 4~ v sec ^> TtAk 4~ v\ji(t -|- w 7") K \.a.n if>\ATt 6[n(t-\-wT)Ktanip&sec ] '- j ! 4 \x4tt. (459) This equation gives the expression for the last three terms of (438) or (439), in which An and 2/^ are completely sepa- rated from the other corrections. 266. Let us now write I 7 1 -N O = *| - cos if? - cos (M A 7 ) J (t T); y sin A 7 " cos DA(A a) -f- cos = _ cos v cos ^ - sn J(Z> - tf); \ (460) = n(t-\-w T) u tan ?/-; ^ = [!/?/?[(,+,_ r) -K tan ^-4 sec ^ _ ^ cos g cos ( A^)! ff> , |_2 COS^ J J Then equation (438) becomes w = fl vy -\- v tan rf>$ -(- v sec tyitAk -\- vEAit -\- vFAee. (461) This equation is now in a form which is well adapted to the purpose in view. w, y, 5, Ttdk, An, and Aee may in certain cases be treated as unknown quantities, but they can never all be determined at the same time from the same series of equations. vy is a constant, and its value is independent of the longi- tude of the place of observation. In order to make its de- 267. LONGITUDE BY OCCULTATIONS. 459 termination possible, therefore, the occultation should be ob- served at one place at least whose longitude is known. In case such an observation is not available, y may be deter- mined from meridian observations of the moon, if such are available, made on the same night or sufficiently near the same time that A A and AD may be well determined from them. Of course if the ephemeris of the moon were perfect this would be unnecessary, as then AA and AD would be zero. 267. In case simply the immersion or emersion of a star -has been observed at two places, the longitude of one of which is well determined, the power of the data will be exhausted with the determination of w and y. If both the immersions and emersions have been observed, we may also determine nAk and as unknown quantities, but in no case can An be determined from occultations unless w has been previously well determined. Still less can a satisfactory determination oiAee be obtained in this manner. The two last terms may, however, be retained in the solution of the equations in order to show the effect on the resulting longitude of an error in TC or in ec. At the same time it will make it possi- ble to apply the necessary correction to the longitude, if from any source values of these quantities become known more accurate than those assumed in the computation. For the determination oidk from single occultations both immersion and emersion must be observed, but contacts at the bright limb can be observed much less satisfactorily than at the dark limb. The best results are obtained from the occultations of groups of stars like the Pleiades, in which the relative posi- tions of the stars are well determined. The passage of the moon through such a group furnishes a number of equations of condition of the form (461), equal to that of the observed disappearances or reappearances of the stars occulted. As 4^0 PRACTICAL ASTRONOMY. 268. before remarked, observations at the dark limb can be made with much greater accuracy than at the bright limb (except perhaps in case of a few of the brighter stars). If it is thought desirable, therefore, only observations made at the dark limb need be used in the equations, especially so if stars are observed both north and south of the moon's equator. On account of the advantages offered by the Pleiades for this purpose, Prof. Peirce developed the equations in a form especially adapted to this group, for use in the longitude work of the U. S. Coast Survey. The reader who is suffi- ciently interested in the subject may refer to the reports of the U. S. Coast Survey, 1855-56-57-61, in the latter of which is given a numerical example of the application of the method. Correction for Refraction and for Elevation above Mean Sea Level. 268. The fundamental equation which has been used as the basis of our analysis expresses the condition that the point from which the immersion or emersion is observed is situ- ated in the surface of a right cylinder enveloping the moon and star. At the same time it has been supposed to be in the spheroidal surface of the earth. The refraction which the ray suffers in passing through the atmosphere causes the elements of this cylinder to be curved lines instead of right lines; or, more correctly, the surface is not that of a cylinder. Further, it follows from the irregularities of the earth's surface that the point from which the observation is made will not in general be in the surface of the mean ellipsoid. Neither of our surfaces there- fore conforms exactly to the mathematical form assumed. The effect upon the observed time of an occupation will 268. LONGITUDE BY OCCULTATIONS, 461 always be small, but in extreme cases must be taKen into ac- count in an accurate investigation. If we consider a ray of light as it comes to the eye at the instant when the star is apparently in contact with the moon's limb, this ray will form a curved line, the asymptote of which will cut the vertical line of the observer at a point where the contact would be seen at the same instant as that observed if no refraction existed. The effect of refraction will then be taken into account if we substitute this point for the point occupied by the observer. Let h' = the altitude of this fictitious point above the observer's position ; h = the altitude of the observer's position above the mean sea level. Then h + h' = the altitude of the fictitious point above the mean sea level. Let us then suppose the observation to be made from a point at this elevation above the surface of the mean ellipsoid. The necessary transformation will be accomplished by changing p cos q>' and p sin cp' into p cos cp' -f (h -\- h'} cos cp and p sin cp' -\- (h -f h'} sin cp ; or, by formulas 446, p cos cp' [i + (h + h') Vi ee sin 2 cp] , f ,,,"'. IA V i ee sin 2 sinG4 a) _ sin (D S) cos 8 ^(A a) -f- sin (D 4- 8) sin* \(A a). sin 7t ' sin if * These values are given by Peirce, Coast Survey Report 1861, pp. 204, 205. They were com- puted directly from Hansen's tables. When the Nautical Almanac is used the intervals will be /aean solar hours. 466 PRACTICAL ASTRONOMY. The computation is given in full for g Celaeno. log it S sin it 3" 4 h s" ffi 3-5575301 4.6855527 8.2430828 3-5573724 4.6855527 8.2429251 3.5572148 4.6855527 8.2427675 3.5570558 4.6855527 8.2426085 cosec it 1.7569172 1.7570749 I-7572325 1.7573915 A a A - a sin (A - a) j cos D cosec it log* 52 40' 29". 52 53 49 34 .68 -i 9 5 .16 4.6855457 3.6175413,, 9.9602268 1.7569172 . 02023 io n 53 18' s8".26 53 49 34 .69 - o 30 36 .43 4.6855692 3.2639744 n 9.9596679 1.7570749 9.6662S64H 53 57' 3i". 09 53 49 34 .70 + o 7 56 .39 4.6855745 2.6779626 9.9591146 1.7572325 9.0798842 54 36' 8". 03 53 49 34 .72 -|-o 46 33 .31 4.6855616 3-4461191 9.9585670 I.75739I5 9.8476392 X D S D-d D + 8 \(A - a) 1.047686 -0.463753 -(-0.120194 -{-0.704108 24 8' 55". 07 23 46 56 .47 21 58 .60 47 55 5i -54 - 34 32 .58 24 18' 44". 85 23 46 56 .47 o 31 48 .38 4? 5 4i -32 15 l8 .22 24 28' 24". 41 23 46 56 .48 o 41 27 .93 48 15 20 .89 + 3 58 -20 24 37' 53"- 73 23 46 56 .48 o 50 57 -25 48 24 50 .21 + 23 16 .66 sin \(A - a) j sin 2 \(A - a) sin (D + d) Sum I 4.6855676 3-3i65ii3 6.0041578 9.8706018 5.8747596 4-6855735 2.9629467 5 . 2970404 9.8717195 5.1687599 4.6855748 2.3769418 4.1250332 9.8728115 3.9978447 4.6855716 3.1450906 5.6613244 9.8738782 5.5352026 cos' \(A - a) sin (D - 6) | Sum 2 9.9999562 4.6855719 3.1201131 7.8056412 9.9999914 4.6855687 3.2806649 7.9662250 9.9999994 4.6855644 3.3958381 8.0814019 9.9999798 4.6855590 3.4853310 8.1708698 52 - Si Zech* cosec Ti logy 1.9308816 .0050625 1.7569172 9. 5676209 2.7974651 .0006918 1.7570749 9-72399I7 4.0835572 .0000358 1.7572325 9.8386702 2.6356672 .0010038 I.75739I5 9.9292651 y + -369506 -529653 .689716 .849699 ' This is the quantity taken from Zech's addition and subtraction logarithmic table 263. LONGITUDE BY OCCULTATIONS. 467 We thus have values of x and y computed for four consecutive hours, from which we can now compute the values of x and y' to the third order of dif- ferences inclusive by means of formulae (101), (101),, and (ioi), viz. : 3 h 1.047686 4 h - .463753 5 h + .120194 6 h -f .704108 X 1 .583910 5S3948 .583938 .583882 y y' .369506 .160189 .529653 .160105 .689716 .160023 .849699 .159941 For the other stars observed we find Taygeta. X X 1 y y 3 h 1.136840 + 583978 +.191768 .159690 4 h - .552839 .584017 351424 .159623 5 h + .031182 .584018 .511015 .159560 6 h + .615185 .583981 .670546 159503 3 h 1.278300 4 h - .694197 5 h .110057 6 h + .474080 Maja. +.584071 .584128 .584145 .584122 .290289 449353 .608340 .767257 I59I05 .159024 .158951 .158884 Computation of |, rj, and . (C is only required for determining the correction due to refraction.) Formulae (412) are as follows: p sin ' sin (u a); p cos q>' cos (j.i a) b cos B\ rj b sin (B 5); = cos (5 - 5). With the known values of q> for Greenwich and Washington, we obtain p and q> by the use of formulae (V), Art. 77- 468 PRACTICAL ASTRONOMY. The computation is then as follows: 268. Greenwich. Washington. ' 9.8913883 9.7955269 p cos

by the formulae m sin M = JT O I; m cos M = y tj\ wsin N = x'\ n cos N = y' ; sin if> = j sin (M - N). The computation for Celaeno is then as follows: Wash.' time Gh. time Assumed r Greenwich. Washington. 5 h 23 53-.8S 5 h -4 22 h 5I m 19 . 9Q 3 59 21 .74 4 h .o JT *.-3 353765 .284772 -068993 -.463753 -.736861 +.273108 y* V > - J? 753720 .488656 .265064 529653 .469105 .060548 log m sin M sin J/ log m cos J/ 8.8388050 9.4012192 9.4233508 9.4363344 9 9895810 8.7820998 tan >/ M log w 9.4154542 14 35' 22". 8 9.4375858 .6542346 77 29' 5 9 ".o 9.4467534 JC 7 y .583916 .159990 .583948 .160105 log sin ^V sin ,V log w cos TV" 9.7663504 9.9842810 9.2040928 9.7663742 9.9842609 9.2044049 tan N N log M N sin(Af - N) log m ac log / sin^ .5622576 74 40' 38". 3 9 7820694 299 54' 44' '.5 99379i3? 94375858 .5650000 99404993" .5619693 74 40' 3". 4 9.7821133 2 49' 55". 6 8.6938108 9.4467534 .5650000 8.7055642 4/0 PRA CT1CAL A STJtOA'OM Y 3 268. Since the emersions were the phases observed, cos ip is plus; therefore Greenwich. Washington. iff = 299 18' 43". 7 We now compute fl from the formula where 2 54' 35". 5- = h \ cos tp cos (M _n n h = 3600; log h = 3.5563025. COS if) zi Greenwich. Washington. 9.6898123 9.4350000 .2179306 99994397 9.4350000 .2178867 Si Nat. No 2.8990454 792. 58 3.2086289 i6i6 8 .7o cos ip n cos(M - A 7 ) log m log - 13"' 1 2 s . 58 26 m 5b.7o 9.6978174 9.4375858 .2179306 9.9994692 9.4467534 .2178867 St Nat. No. 2.9096363 3.2204118 i66iM6 t T - 6.15 27 m 41 s . 16 5 h 8 m 40*. 01 n, -I3M* 45" 7 m 55 8 -55 In a similar manner we find for the other stars ForTaygeta, For Maja, 1 - 9". 30 +5 h 7 m 55 8 -67; fl 9". 79 -}~5 h 7 m 53 8 -8- We next compute T, H, and v by formulae (443) and (458), viz.: T = r -(x a sin cos N); H xo cos W -\- yt sin N; h * It is not necessary for this purpose to know the value of k with extreme accuracy, since the correction A/t to the assumed value appears as one of the terms of our equation. 268. LONGITUDE BY OCCULTA TIONS. For Celaeno we have 471 sin TV 9. 98428 log JT O cos N 8.97073 log - 6.44270 7t log .TO 9-5487I Zech .83098 log- .21793 cos 4^9 42202 logjj/o sin JV 9.86149 log h 3.55630 log > 9.87721 log H 9.80171 log v .21693 sin N 9. 53299 * -6334 v 1.6479 Zech .43345 log;'o cos 7^9.29923 log(*o sin JV-y cos N) 9 73268 log ~ .21793 9 95o6i Nat. No. .8925 T 4- 5075 We now compute the coefficients for the final equations of the form (461), viz.: v tan rf>, vE = v\ti(t -{- w T) H tan ^], and v sec ^>. Greenwich. Washington. t -\- IV 5-3983 39894 t -\- w T .8908 .5181 log(/+ w - T) log n 9.94978 9.78207 9.78211 Sum 9-73I85 9.49652 log K 9.80171 9 80171 tan #> .25069* 8.70612 Sum .05240* 8 50783 Zech . 16969 .04241 log .22209 Q. 53SQ3;/ log K .21693 .21693 logv .43902 9 75586* vE 2.7480 5700 sec ^ .31019 .00056 log v sec J/> .52712 .21749 log y tan ^ . 46762,1 8 92305 v sec ^> 3.3661 1.6500 v tan #,2.9351 .0838 47 2 PRACTICAL ASTRONOMY. 268. Computing the coefficients for the other two stars in the same way, we ob- tain the following six equations: Celaeno: G. TO = - o h o ra 13". 42 - 1.6487 2.935* -f 3.36617^ + 2. 74 8Au-; [i] -. W. w' = Taygeta: G. TO = - o W. ,' = 5 Maja: G. -M - - o 55.55-1-6487 + .084*41.650^*- .jToAir. [4] | 9 .30 - i.6 4 8y - .598* + i.75 3 irA* + 1.307 Aw; [2] I 55 .67 - i.6 4 8y + 1.048* + I.QSJITA* - i o8 4 A,r. [5] f 9 .79 - 1.648? - 2.328* + 2.8 5 27rA* f 2.4 9 2Air; [3] I 53. .08 - 1.6487 - .062*+ i.6soirA - .4 4 2Ajr. [6] J W. / = 5 If we assume y, 3, An. and TtAk to be the same in all of these equations an assumption which involves no appreciable error we shall have six equa- tions between those quantities and w'. w, the longitude of Greenwich, will be zero. It is evident, however, that for various reasons a direct solution of these equations will not be expedient. In the first place, the large terms involved would render the operation very laborious, and further it will not be possible to separate Art from the remaining quantities without assuming both w and w' to be known. We therefore proceed as follows: Assuming the equations to be of equal weight, we subtract the first from the third, the first from the fifth, and the third from the fifth; then we subtract the second from the fourth, the second from the sixth, and the fourth from the sixth. We then have the following six equations: O = 4.12 -f- 2 337^ l.bllTtAk I.24lJjr. [2] [i] o= 3-63+ .6oy3- .5I47T/7/C- - .256Z/7T; [3] [i] o = .49 1.7303 -f i.oggTr/U 4- .985 Jff; [3] [2] , 0-4. . I2+ .96434- .303*^-- .514^*; [5j- [4] r o = 2.47 .1463 .oooTtdk 4- .I28//7T; [6] [4] o = 2.59 i.no3 .wsTtdk -\- .642Z/7T. [6]~[5] By means of these six equations of condition we now determine the most probable values of 3 and TtAk. The value ofZ/Tf, however, cannot be well de- termined, as we have before remarked. If it were not known a priori that such was the case, it would be shown from the normal equations, which would be practically indeterminate for this quantity. We shall therefore determine 3 and TtAk in terms of Ait in order to show what effect an error in rt will have upon the longitude. By the method of Art. 21 we derive from the above equations the following two normal equations: 11.00563 5.35457T^7/ = 16.0306 -f- 5.9864^^; | ,Q 5-3545^ -)- 4.25747rJ = 8.2287 2.8656^. f 208. LONGITUDE BY OCCULTATIONS. 4/3 From which itAk = , ".2588 -f- .O28g//7T: \ 3 = - i".330i + .5577^*. f ' We now substitute these values in the first, third, and fifth of equations (A), writing zero for w, the longitude of Greenwich, when we find the following values for 1.6487: 1.6487 = 8.645 + 1.209.4*; \ 1.6487 = 8.055 + 1.226^*; 1 (E) 1.6487 = - 5.955 -f 1. 276-4. ) Mean 1.6487 = - 7.552 + 1.237^*; y = - 4". 582 + .751^*. We now substitute these values of TtAk, 3, and y in the second, fourth, and sixth of (A), when we find the following values for the difference of longitude between Greenwich and the observatory on Capitol Hill, Washington: Celaeno w = 5 h 8 m 3'. 42 1.712^*; Taygetaa/ = 5 8 2 .33 i.68i2/*; Maja w = 5 8 1.141.665.4*. Mean w 5 8 2 .30 i. 686.4*. The Capitol Hill observatory is io s 25 east of the Naval Observatory. The longitude of the latter, determined telegraphically, is 5 h 8 m I2 8 .O9 west of Green- wich. Therefore the true value of w is 5'' 8 m i s .84, corresponding very closely with the above value if we neglect Ait altogether. With these values of y and 3 we may now determine the correction to the assumed right ascension and declination of the moon. We have sin jVcos DA(A - a) + cos NA(D -8)=y;) ( - cos JV cos DA(A - a) -f- sin NA(D - d) = 3. f Substituting for the coefficients of A(A a) and A(D S) the mean of the values for the three stars, we have the equations - 5) = - 4582; 8) 1330. From which we find A(A a) = 4". 46; A(D - d) = - 2 .49. Assuming the errors of the star places to be inappreciable, these will represent the errors in the computed right ascension and declination of the moon at a time corresponding to the mean of the times of observation. These corrections 474 PRACTICAL ASTRONOMY. 269. it will be seen are affected by any small outstanding error in the parallax, as they have been derived by assuming Ait = o. In the same way, assuming Ait o and taking for it the mean of the values given above, viz., 3608", we find from the above value of itAk Ak = -j- .0000717. We hav2 -assumed k = .272270. Therefore k = .272342, as shown from these observations. This result from so small a number of occultations has no value, however, as a determination of the moon's semi- diameter. Observations of Different Weights. 269. In the solution of our equations we have supposed all to be of the same weight. Such will not in general be the case. Other things being equal, those occultations will be best for longitude determination which are most nearly central in reference to the moon's disk. When both immer- sion and emersion of the same star are observed, the obser- vation at the dark limb of the moon is entitled to greater weight than that at the bright limb, except, perhaps, in case of the brighter stars. In order to determine the proper manner of treating the equations when different weights are assigned, let us suppose, as in our example, three observations to have been made at one place whose true longitude is w, then for the present, considering only terms in y and $, we shall have three equa- tions of this form : Vpw VpaS - VpO = o; 1 Vp'w - Vp'a'S - \fp'O' o; \ . . (466) 269. LONGITUDE BY OCCULTA TIONS. 4/5 Where O = H ry, and/,/',/" are the respective weights. From these we derive the normal equations = ;1 O. i \_pa\w + [paa]5 [paO] The solution of these equations in the usual manner gives . . u m (468) = \jaO\\. Which gives % with the weight [/##!]. But, as we have seen, this form of solution is inconvenient on account of the large quantities involved. Let us write out in full the values of \_paai] and (469) [paai\ =pa*+p-a'*+ P "a"* - , (P* +P'*' _ //(a - a') 2 -\-pp"(a - a") /+/'+/' ' J Comparing these expressions with our equations of condi- tion (466), we see that the final equation for 3 may be obtained as follows: Before multiplying the equations through by Vp, Vp', and Vp" ', subtract the second from the first, the third from the first, and the third from the 476 PRACTICAL ASTRONOMY. 269. second, then give to the three resulting equations the fol- lowing weights respectively: pp' pp" p'p" p + p' VF' P + P' +7' ; 7T7T7 7 ' We may apply the same reasoning to the equation in which all of the unknown quantities are retained, and may extend it to any number of equations of condition. Thus if the number of equations of condition were four, we find by com- bining them in a like manner, two and two, six equations with weights pp' pp" p"p'" P +/ +/' +/"'/ +/ + P" +/''" " / +/ +/" + /"' It is not possible to give a rule by which the proper weight can be assigned in every case, as it will depend upon a variety of circumstances, such as the skill and experience of the observer, the magnitude of the star, condition of the atmosphere, and various other causes. Evidently, if weights are to be assigned depending upon these circumstances, much must be left to the judgment of the observer and com- puter. If the conditions are otherwise the same in case of two stars, the weights may be assumed proportional to the numerical values of cos ^; that is, proportional to the chord of the moon's disk traversed by the stars a central occulta- tion having the weight unity. If we assign weights to our six equations (A) in accordance with this prin- ciple, we shall have for the weights, taken in order, p = .49; pi = i.oo; p' =.94; /:' = .84; /" = .58; pi" ~ i.oo. The weights of equations (B) will then be in accordance with formulae (470). [2] - [I] .22 9 [ 5 ] - [ 4 ] .296 [3] - [i] .141 [6] - [4] -352 [3] - [2] -271 [6] - [5] .296 LONGITUDE BY OCCULTATIONS. 477 Multiplying the equations by the square roots of the respective weights and proceeding in the usual way, we obtain the following normal equations: 2.76303 i 239i7rJ = 3.7605 -f I.5I29//7T; 1.23913 -f- 1.01747^^ = 1.6907 .6678//7T. From these we find itAk = -f- .00931 3 = - 1.3570 -- .5579^*- Substituting these values in [i], [2], and [3] of equations (A), and taking the mean by weights, we find 1.648^ = 8.l6l -f I.22lJff. Finally, substituting these values of 3, itAk, and y in [4], [5], and [6], we find the following values for w: [4] w' = 5 h 8 m 3 9 .6i i.-jc&Jit; wt. = r.oo. [5] w' = 5 8 2 .43 I.675//7T; wt. = .84. [6] w = 5 8 i .34 i.66o//7r; wt. = i.oo. From these we have w 5 h 8 m 2 S .4.6 CHAPTER VIII. THE ZENITH TELESCOPE. 270. This instrument is used in determining latitude, and is particularly useful when a high degree of accuracy is re- quired, the precision being not inferior to that of the most refined instruments of a fixed observatory, while on account of its great simplicity it is especially adapted to use in the field. We have already developed several methods for determin- ing latitude : those of Chapter V". are very useful, but will not be employed in the field except in cases where an error of five or six seconds in the result is not considered objec- tionable. The prime vertical transit gives results of high precision, but not without the expenditure of much labor. The method by the zenith telescope is superior to the first of these in accuracy, and to the second in facility of applica- tion. On account of these advantages *it has superseded all other methods on the Coast and other government surveys in cases where extreme accuracy is required. The most common form of instrument is shown in Fig. 54. In general appearance, as will be seen, it is a telescope with an altitude and azimuth mounting. The essential character- istics are a very delicate level attached to the tube, like the level of the finding-circles in the transit instrument, and the eye-piece micrometer. The vertical axis is made very long to insure steadiness of motion in azimuth The instrument is used in the meridian like the transit. 2/0. THE ZENITH TELESCOPE. 479 FIG. 54. THE ZKNITH TELESCOPE. 4^0 PRACTICAL ASTRONOMY. In the Coast Survey instrument the aperture of the tele- scope is 3^ inches, focal length 45 inches, length of horizon- tal axis 7 inches, vertical axis 24 inches, diameter of horizon- tal circle 12 inches, vertical circle 6 inches (sometimes this is only a semicircle, the radius being 6 inches). The instru- ment rests on three foot-screws. The lamp at the end of the horizontal axis opposite the telescope illuminates the field; the weight seen at the same end of the axis acts as a counter- poise to the telescope. This weight is connected with the telescope by a bent metallic bar, shown in the figure, in such a way as to prevent to some extent the flexure of the axis. The horizontal circle is read by means of two verniers. The level attached to the vertical circle is generally gradu- ated so that the motion of the bubble over one millimetre corresponds to an angle of one second of arc. The accuracy of the instrument depends in a great degree on the delicacy of this level. In testing an instrument it may generally be assumed that if the level is a good one the performance of the instrument as a whole will be satisfactory. The striding- level shown on the horizontal axis is used for adjusting the instrument, and is not necessarily of so great accuracy. The micrometer* is provided with one or more movable threads, the value of one revolution of the screw being from 45" to 60". The head of the screw is divided into 100 parts, of which tenths may be estimated; thus by estimation T -^ of one revolution may be read, or about o".o$. The entire revolutions are read by means of a comb at one side of the field of view, the distance between two consecutive notches corresponding to one 'revolution. There are three, and sometimes five, vertical threads which may be used for observing transits. A rack and pinion is provided for slid- ing the eye-piece in the direction of the vertical so that the star may always be observed in the middle of the field. * For description of the micrometer see Art. 97. 2/1. ZENITH TELESCOPE. ADJUSTMENTS. 481 The instrument is mounted like the transit on a pier of masonry, or simply a solid wooden post planted three feet in the ground. The dimensions given above are those of a large-sized in- strument; much smaller ones are often used. The transit instrument may be used as a zenith telescope if it is provided with the fine level and micrometer. A special appliance for reversing is convenient, but not essential. As we have seen in the descriptions of the different forms of portable transit instruments, the two are often combined. This arrangement is very advantageous on the ground of economy of first cost and of transportation; at the same time nothing is lost in accuracy and little in convenience. Adjustments. 271. First. The vertical axis must be made truly vertical. In setting up the instrument it will be found advisable to place two of the loot-screws .in an east and west direction, otherwise if it is found necessary to move the screws after the instrument has been brought into the plane of the me- ridian this last adjustment will be disturbed. The axis is brought into the vertical position by the use of the striding-level, which should read the same while the instrument is turned completely around in azimuth. This adjustment will also be tested by means of the more delicate level attached to the telescope. Second. The horizontal axis should be perpendicular to the vertical axis. This may be tested by reversing the striding- level after the vertical axis has been properly adjusted. Third. The line of collimation may be adjusted by direct- ing the telescope to some distant terrestrial mark, then turn- ing the instrument 180 in azimuth by means of the horizontal 4^2 PRACTICAL ASTRONOMY. 271. circle. Allowance must be made for the parallax of the in- strument, unless the mark is so far away that it is not appre- ciable. This is necessary, since the line of collimation is not in the same vertical plane as the axis. Let d = distance of the line of collimation from vertical axis; D = distance of mark; p = correction for parallax. Then This method of adjustment depends entirely on the read- ing of the circle, and is therefore not capable of extreme ac- curacy. If considered desirable, a more accurate adjustment may be made by means of a pair of collimating telescopes* or by the mercury collimator.* The error may also be de- termined by transits of stars observed in both positions of the axis, as explained in connection with the transit instru- ment. If stars are chosen which culminate near the zenith, an error of azimuth will have but little influence on the re- sult. When used as a transit instrument a meridian mark is recommended, consisting of two lamps placed side by side and at a distance apart equal to twice the distance of the vertical from the collimation axis. It is perhaps unnecessary to say that the instrument must be focused and the threads placed truly vertical and hori- zontal respectively, precisely as in the transit instrument. Fourth. The instrument must be brought into the plane of the meridian. For this and other purposes we require the local time, a chronometer or clock being an essential part of * See Art. 168. 2/1. ZENITH TELESCOPE. ADJUSTMENTS. 483 the outfit. The clock correction A T ma)- be determined by the sextant, transit instrument, or by transits observed with the zenith telescope itself. In the latter case the process of bringing the instrument into the meridian will be the same as that already described for the transit. If A TIB known within one second of its true value, that will be sufficient. AT being supposed known, Let a = the right ascension of a star near the pole. Then a AT = the chronometer time of culmination. At this instant, as shown by the chronometer, the middle thread is placed on the star, the horizontal circle being pro- 1 vided with a clamp and tangent-screw for this and similar purposes. The reading of the verniers now shows the true direction of 'the meridian. Two stops arranged for the pur- pose are now clamped to the horizontal circle so that the in- strument may be turned freely in azimuth, but brought to a stop when it reaches the meridian. Care must be taken in turning the instrument in azimuth not to bring it up against these stops with a shock, as this will disturb the adjustment. South stars may be used for adjusting in the meridian, pro- vided they are sufficiently far from the zenith. In any case the adjustment should be tested by trying whether a south star crosses the middle thread at the proper time. The stops should be placed so that in reversing the in- strument in azimuth the object end of the telescope always turns towards the east. The observer can then turn it in azimuth a little, so as to find a star a moment before it enters the field ; then knowing exactly where to look for the star, the eye-piece can be brought to the right place by the rack and pinion, and the micrometer-thread moved to nearly the proper place, so that when the star finally comes into view the bisection can be made with all necessary deliberation. 484 PRACTICAL ASTRONOMY. 2 7 2 All of the above matters having been attended to, the in- strument is ready for regular latitude observation. The Observing List, 272. The stars are observed in pairs, one star culminating north of the zenith and the other south. The difference of zenith distance should not exceed 15' or 20'. Let q>, 8, and 6' respectively the latitude of station and declination of south and north star; z and z' = the zenith distances. Then

7 S. N. 6 9-5 5415 5460 6 6 16 6 36 16 15 36 58 16 40 i 9 16 8 59 N. .S. 9 7-5 5502 5523 5 5 16 21 41 16 24 31 55 30 42 10 6 30 6 50 N. S. 6 40 5545 5624 4-5 7 16 28 17 16 40 4 69 3 28 35 20 3 20 25 N. S. 20 14 5644 5658 6 6 16 43 18 16 44 17 42 28 55 38 6 32 6 38 S. N. 6 35 As will be seen, the selection of a good list of stars involves considerable labor. Where great accuracy is required especial care should be exercised in selecting the stars, and none should be employed whose declinations are not well determined. This part of the subject will be considered more in detail hereafter. Directions for Observing. 273. A suitable list of stars having been prepared, the in- strument adjusted, and the chronometer error determined, the observer sets the vertical circle at the proper reading, the telescope is directed towards that side of the zenith 488 PRACTICAL ASTRONOMY. 274. where the first star will culminate, and the bubble brought to the middle of the level-tube by means of the tangent- screw connected with the horizontal axis. At the time of culmination, as shown by the chronometer, the star is bi- sected by the micrometer-thread, and the micrometer and level are read ; the instrument is then reversed in azimuth and the second star observed in the same way : this forms a complete observation. During the operations described the tangent-screw of the vertical circle must not be touched, but the tangent- screw which moves the telescope, and consequently the level, may be turned after reversing, in the exceptional case where the vertical axis is not well adjusted. If for any reason the bisection is not obtained at the in- stant of culmination, the star may be observed off the meridian and the time of observation recorded, when a correction may be computed to reduce it to the meridian. Several bisections might be made while the star is crossing the field, and the observations reduced to the meridian in a similar manner ; but experience shows that little or nothing is gained in this way. The accuracy with which a bisection can be made by a skilled observer being greater than that of the average de- clinations which will be employed, it is advisable to increase the number of stars observed rather than to multiply obser- vations on the same star under the same circumstances. Determination of Value of Micrometer-screw. 274. This value may be determined most advantageously by means of p. circumpolar star observed near elongation. One of the four close circumpolar stars whose peaces are given in the American Ephemeris will generally be selected for the purpose, viz., 51 Cephei, #, a, or A Ursse Minoris. 274- DETERMINATION OF MICROMETER VALUE. 489 The observations are made as follows: From 15 to 30 min- utes before the star reaches elongation the telescope is pointed to the star, the micrometer-thread being near that end of the screw from which the star is moving. The tele- scope is set at such an elevation that the thread is a little in advance of the star, and the bubble of the level brought into the middle of the tube, without disturbing the position of the telescope. The time of transit of the star over the thread is then observed and the level read. The -thread is then moved forward one revolution (or sometimes only half a revolution) and the transit of the star observed in the new position, and so on throughout the entire length of the screw. It is well to time the work so that the elongation will occur near the middle of the series, though this is not essen- tial. With this in view it may be borne in mind that the time required for Polaris to pass over a space equal to the range of an ordinary zenith telescope micrometer will be about 50, for A Ursse Minoris 70'", -for 51 Cephei 30'". The record of the observations will be kept according to the following or a similar schedule : No. Micrometer. Chronom. Time. Level. N. S. To prepare for the observation, the chronometer time of elongation must be computed. It will facilitate setting the instrument on the star if the azimuth and zenith distance are also computed. 49 PRACTICAL ASTRONOMY. 275. In the triangle formed by the arcs of great circles joining the zenith, the pole, and the star, the angle at the It I star 5 will be a right angle at the time of elonga- tion. Then by Napier's rules, ' z cos d -} sin a = ; cos tp sin k . , (474) cos z -. ; w/ ^ z sin o FlG - 55- cos / tan

> 53"' 56' a = i 12 06 a t 19 18 10 AT = - 2 Chronometer time of elongation a t AT = 19'' i8 m 12" The transit of Polaris was observed over the micrometer-thread at every half turn, beginning with revolution 35 and ending with 5.5 sixty transits in all. In the example I have only used those observed at the even revolutions, as this will be sufficient for illustrating the method of reduction. No. Micrometer. Read in p. Chronome- ter Time. Le N. irel. s. Time from Elonga- tion. Reduction to Vertical. Reduction to Mean State of Level. Correction Level. Reduced Times. T 35 18" 38 4 o.o .8.6 19.1 - 39- 3 2'.o ' + ii 8 8 5 + o.6 i8 h 38 m 52'. 2 34 41 38 .0 18.5 19.1 36 34-0 9 -3 .6 -j- 41 48. 3 33 44 32 -8 18.6 19 2 33 39 -2 7 -3 -j- 44 4- 4 32 47 27.6 18.7 19.2 3 44 -4 S -5 -{- 47 33 S 3" 50 24 .0 19 o 27 48.0 4 -I 50 28 . 6 S3 20.6 24 5i 4 2 .9 S3 23 . 7 2Q 56 '3 -7 19.0 19.1 21 58.3 -}- 56 is- 8 9 1 7 18 59 i-o 19 2 4 .4 19.0 19.2 16 '.6 *;| _ ' 1 : 18 59 " 19 2 5 . o f > 5 o .0 19-5 19. 1 13 i -o 4 -f- .^ 4 59 i 5 7 52 -3 19.2 19.2 10 i .7 .2 7 52 - 2 4 10 49 .0 19.6 19 3 7 2 .0 + -I -j- 10 4 8. 3 a 3 3 4i -9 4 3 -i .0 i * ~ 13 4 1 6 34 5 21 9 29 .0 19.6 I9-S + 110 .0 4. _ 9 28 . 8 ! 9 2 21 .9 20.0 20.2 19.4 19.3 4 -9 7 -3 .0 i ; ~~ 2 21. 5 IS- 8 10 .C 20.3 19.4 9 58.6 .2 -f- 8 9. 9 7 i 3 -9 20. 5 19.5 12 51 -9 4 -f- i . I 2 . 1 g:S 20.6 19-5 is 5 :i 6 .8 1 1' _ '_ ;6 50. 2 4 9 46.0 21 34 .0 i .9 fi- - : 9 42 . 4 3 5 35 -4 20.7 19.5 27 23 .4 3 -9 ii '. 2 35 - 5 8 29 .0 2 .0 19-3 30 17 .0 5 -2 + 1. ,8 21 . 26 O i 25.0 2 .0 19-5 6.9 i 16. 54 8 . 28 29 7 '97 H 7 20 I 3 .6 12 .0 2 .0 19 6 19.7 39 2 .7 42 i .6 ii -3 14 .1 + 1. + 1. - 57 i 19 59 57 3 , 6 20 3 8.6 2 .0 19.8 + 44 56 .6 - 17 .2 + 1. ~ 20 2 49 . 278. DETERMINATION OF MICROMETER VALUE. 497 The first five columns require no explanation. The sixth contains the quanti- ties which we have called r. The "reduction to vertical " is taken from the table Art. 275. The "reduction to mean state of level" is (n s) (w s a ), where ( Jo) = o in this case. The "correction for level" is this quantity d multiplied by The value of one division of the level, d = ".893. 30 cos 6' Therefore this factor equals i 23. The elongation being east, the sign of the level reduction is minus. The " reduction to vertical" and "correction for level" being applied to the observed time, we have the " reduced times" of the last column. We combine these quantities by subtracting No. i from 16, No. 2 from 17, ... No. 15 from 30, thus obtaining a series of values for the time required for the star to pass over a space equal to 15 revolutions of the screw. The mean of these quanti- ties multiplied by = cos S then will give the value of one revolution in seconds of arc. The numerical work is as follows: Nos. Time of 15 Revolutions. * 16- i 43"' 28' 8 3-9 15-21 17-2 43 27 .1 2.2 4.84 18- 3 43 28 .5 3-6 12.96 19-4 43 28 .6 3-7 13.69 205 43 28 .9 4.0 16.00 21-6 43 26.5 1.6 2.56 22-7 43 26 .9 2.0 4.00 23- 8 43 24 .4 5 25 24-9 43 24 .6 3 .09 25 10 43 21 .8 3-i 9.61 2611 43 23.7 1.2 1-44 27 12 43 19 -9 5-o 25.00 28 13 43 20 .2 4-7 22.09 29 14 43 23.1 1.8 3-24 30 15 43 21 .0 3-9 15-21 \yv\ = 146'.! 9 Mean 43'" 24". 93 = 2604'. 93 log = 3-4 I 579 I cos d 8.3772074 log one revolution 1.7930035 One revolution 62". 0874 Correction for refraction .0315 Corrected value 62".os6 49 8 PRACTICAL ASTRONOMY. 2/9- The" correction for differential refraction is computed by the last of formulae (481), viz., r r' = [6.44676] sec 5 z(z z) 6.4468 log (z z) = 1.7930 sec 2 s = .2578 lo = ( m - m} + (/+/') + ( r - r '}. Substituting this value in equation (472), 9 = K* + cos 3 2 sin 2 Aw = - . 2 sin z sin i (482) 90-S Aq> will be plus for a north and minus for a south star. . m f, is taken from table VIII A at the end of this volume, sin r 286. When the star is observed off the line of collimation, the instrument remaining in the meridian. In the figure, PK is the meridian, PS the hour- circle passing through the star. If the* star is observed on the meridian, SAT will be the position of the micrometer-thread. If ob- served off the meridian at S', this thread will have the position S' K' . Let KK' = x. Then PK' = 90 -(<* + *), and, by Napier's second rule, cos / = tan 6 cot (S + x). 506 PRACTICAL ASTRONOMY. 286. This may be placed in the form . tan 8 -4- tan x tan d = (i - 2 sin 2 \f\ - --,- --- . ' i tan d tan x Clearing of fractions and neglecting the small term tan x .2 sin 3 ^, we readily find tan x = sin tf cos 6 2 sin 2 ^/, or, with sufficient accuracy, x = i sin 2d . . (483), sin i" As the apparent zenith distance is diminished for a south star and increased for a north star when observed in this man- ner, the correction to the latitude will always be plus and will be equal to \x. That is, ^ = ^\n26 ...... (483) This method of proceeding will generally be preferred when the observation on the meridian is lost, as when the other method is used the stop must be undamped, and where other stars follow in quick succession a pair may be lost in consequence. If the star cannot be observed before it gets beyond the field of view, the observer will generally prefer to let it go altogether. The computation of Aq> by the above formula is very simple, but a table is added from which the value of x = 2.Aq> may be taken at once. The horizontal argument is the hour- angle of the star, and the vertical argument the declination. COMBINATION OF RESULTS. TABLE C REDUCTION TO MERIDIAN. 507 10S. .ST. 20S. 25*. 30*. 35-r- 40S. 45*. SO*. 55*- 6os. 5 .00 .01 .02 .03 .04 .06 .08 .IO .12 14 17 o 10 .01 .02 .04 .O6 .08 .11 15 .19 23 .28 34 80 15 .01 03 05 .09 .12 17 .22 .28 34 41 49 75 2O .02 .04 .07 .11 .16 .22 .28 *> 44 63 70 25 .02 05 .08 13 .19 .26 34 .42 52 .63 75 65 30 .02 05 .09 15 .21 .29 .38 .48 59 71 .85 60 35 03 .06 . IO .16 23 31 .41 53 .64 77 .92 55 40 03 .06 .11 17 24 33 43 54 67 .81 97 50 45 03 .06 .11 I? 25 33 44 55 .68 .82 .98 45 287. Formula for Computation of Latitude from Observations with the Zenith Telescope.

derived from a single pair, Let f s , &> = the probable errors of the declinations; Then if , is the number of observations on this pair the probable error of the mean will be \/"~> and E$ being the probable error of the resulting latitude. The relative weights are proportional to the reciprocals of the squares of the probable errors; or, since the unit of weight is arbitrary, we may write "- Value of Micrometer from the Latitude Observations. 289. If no special observations have been made for deter- mining the value of the micrometer-screw, it maybe derived from the latitude observations themselves. * Equation (29). 510 PRACTICAL ASTRONOMY. 290. Let R = an assumed value of one revolution as near the true value as possible; AR = the correction required. Then^-j- AR = the true value of one revolution; q>' = the latitude computed with the assumed value of R from all of the observations; tp' -f- Acp = true value of the latitude. Then from (478), -79 5-08 + .23 .1502 29 775S 7765 2 25 .20 625 400 .3200 2-37 52 .10 4.98 + -54 .6911 M- 73-98 [/*] = "5-39 [>*H = I5-9920 * " W = 4i r 59 '5i".56. In this column only the last three figures of p$ are given. 5l6 PRACTICAL ASTRO NOMY. 2 9-- The probable error of

= 48 59' 5i".54 .056. 292. For an illustration of the method of Art. 289, let us form the equations for determining the correction to the adopted value of R and to the above value of cp. We shall have 29 equations of the form (486); the above values of 'v will be the absolute terms. If we refer to the observations given in Art. 290, we have for the first pair \(Af M') 8.99. We have from this pair the equation Acp -\- 8.99 AR = n. This star was observed on two nights, so taking the mean of the values of \(M M 1 ) and multiplying the resulting equation through by the square root of the weight determined for this star, we have the following equation: 1.52^95 -(- i^.^AR = 1.46. Proceeding in a similar manner, we derive the following 29 equations of condi- tion for determining Acp and AR, for which we shall write x and y: I.S2X+ 13.577 = - I-46; 1.58* + 7.367 = .49; 1.90* 4.227 = .38; 1.38*4- 1 1.567 = + .69; 1.84* 18.037 = + -9 8 ; 1.77-r - 18.537 = -Si; i.43x 4- 4.457 = .49; 1.95* - 11.977 = + -82; i.88.r + 10.887 = I -3', 293- LATITUDE AND MICROMETER. 517 2.28* + 18.867 = 57 ; 2.12* 13.727 =r + 25 ; 2.0 9 * + 10.307 = i- 55 i- S3* - 12.517 = 34 i .33-* - .207 = -f 77 j 1.82* + 3.697 SB + 35 ; 1.03* 2.997 = .ig i; 1. 00* 4- 2.887 = + .TO ; 1.47* - 357 = .04 ; 1.51* + 7.077 = - 53 1.42* - 4.697 = + .eg :J .83* + 7-63;' = *- .10 ; 1.54-*- - 8.817 = + 1. 1 1 1.03* 2.817 = -f- 72 j 1.54* + 14.607 = -i- .1)6 ; i.6i* + 7.787 = .14 1.04* 4- 1.217 =r + 1.49 1.53* 4- 3017 = + I? j 1.69* - 4.77}' = 4. 39 j 1.54* - 5-7cy = + .83. Proceeding in the usual manner, we derive from these the two normal equations 73.98*4- 17.657 = 004; 17.65*4- 2732.357 '= 85.80. From these, * = 4- -o7 .054; y = .031 .009. The most probable values of the latitude and micrometer-screw as indicated by this series of observations are therefore tp = 48 59' 5i"- 567 -054; R = 62". 025 .009. In order to have the value of R determined in this way of any value in com- parison with that determined by transits of circumpolar stars, the declinations of the stars employed must be well determined. 293. There are various ways in which the observation of stars in pairs at equal or nearly equal altitudes by means of the zenith telescope may be employed for the determination 5l8 PRACTICAL ASTRONOMY. of latitude and time. As may be seen, the instrument is adapted to the solution of any problem of Spherical Astron- omy which depends upon the observation of two or more bodies at the same altitude. The most favorable condition for latitude determination is when the two stars are on the meridian, one north, the other south, while time is best de- termined by observing two stars on the prime vertical, one east, the other west. On account of the facility with which the latitude is deter- mined in the manner already explained, and the ease with which the instrument may be converted into a transit when it is necessary to employ it for determining the approximate time, other solutions of the problem depending on observa- tions out of the meridian have never met with much favor. Some of these methods are interesting from a theoretical point of view, but for the reasons stated the subject will not be developed further in this connection. CHAPTER IX. DETERMINATION OF AZIMUTH. 294. The AzimutJi of a point on the earth's surface is the angle between the plane of the meridian and the vertical plane which passes through this point and the eye of the observer. Since the vertical plane is determined by the direction of the plumb-line, and this line may deviate from the true normal to the earth's surface, a corresponding deviation in the azimuth must exist. We must therefore distinguish be- tween the Astronomical Azimuth and the Geodetic Azimuth. The Astronomical AzimutJi of a point is the angle between two planes drawn through the plumb-line at the point of observation, the first plane parallel to the earth's axis, and the second passing through the point. The Geodetic Azimuth is the angle between two planes drawn through the normal to the earth's surface at the point of observation, the first plane passing through the earth's axis, and the second through the point. It is with the Astronomical Azimuth only that we are at present concerned. The azimuth may be reckoned from either the north or south point of the horizon. For astro- nomical purposes it is usually reckoned from the south point towards the west from zero to 360. In determining the azimuth of a point on the earth's surface it is more conven- ient to use stars near the north pole of the heavens ; conse- quently for geodetic purposes the azimuth is generally 520 PRACTICAL ASTRONOMY, 295. FIG. S 8a. 295- DETERMINATION OF AZIMUTH. 521 reckoned from the north point. For the sake of uniformity we shall in this chapter always suppose the azimuth reckoned from the north in the direction N., E., S., W. A minus azi- muth will be reckoned from north towards west. Extreme accuracy in the determination of azimuth is re- quired in connection with the geodetic operations of primary triangulation. The principal methods employed in such cases will be given, when it will be shown how they may be abridged where a less degree of accuracy is demanded. There is a variety of these methods, depending on the form of instrument employed and the position of the stars ob- served. The instrument will be either the theodolite, used for measuring horizontal angles, or the astronomical transit. In any case the azimuth of the point is determined by meas- uring instrumental!^ the difference between the azimuth of the point and a star. The azimuth of the star is computed by its known right ascension and declination, and the local time and latitude, which have been previously determined ; from these data we have the azimuth of the point. 295. The Theodolite. Figures 58^ and 58^ show two forms of instruments used on the U. S. Coast Survey. The older form, Fig. 580, has a horizontal circle from 20 to 30 inches in diameter. With the newer instruments, circles from 12 to 20 inches are considered sufficiently large, as such circles can now be graduated much more accurately than formerly ; the instrument can therefore be made more compact and portable, a matter of some importance in the field. The horizontal circle is commonly divided directly to 5', these spaces being subdivided by reading microscopes directly to single seconds, and by estimation to tenths of a second. Two or three microscopes are used. The essential features of the instruments will be understood from the plates without further description. For secondary azimuths a less perfect instrument will often 522 PRA CT2CAL A STRONOMY. 295. PIG. 2 9 6. DETERMINATION 0^ AZIMUTH. 523 be used. For magnetic work or ordinary land-surveying a common surveyor's transit with 5- or 6-inch circle will fre- quently be employed. It is perhaps unnecessary to say that the instrument must be carefully adjusted in everv particular. 296. The Signal. For observing at night an illuminated mark is required. A convenient mark is a square wooden box firmly mounted on a post or other support, the light of FIG. 59. a bull's-eye lantern being thrown through a small hole in the front. The box itself may be painted so as to form a con- venient target for day observation. This mark must be placed far enough from the station so that no change will be required in the sidereal focus of the telescope : about one mile will generally be sufficient. When from any cause a distant mark is not practicable a collimating telescope may be used ; but the greatest care must be exercised in mount- 524 PR A C TIC A L ASTR ONOM Y 298. ing both the instrument and collimator firmly, piers of solid masonry being used for both. 297. Choice of Stars. For first-class azimuths only close circumpolar stars will be used. Preference will be given to the four circumpolar stars whose places are given in the ephemeris, viz., a, $, and A Ursae Minoris, and 51 Cephei. Fig. 59 shows their relative positions, and will assist in finding the smaller ones which are not readily distinguished with the naked eye unless the position is previously known. 298. Method of Observing. A complete series of observa- tions on one star will consist of ten or twelve readings on the mark and about the same number on the star, the instrument being reversed about the middle of the series. The follow- ing order of observation is recommended: ist. 6 readings on the mark. 2d. 6 readings on the star. 3d. Read the level. 4th. Reverse. 5th. Read level. 6th. 6 readings on the star. 7th. 6 readings on the mark. If more than one series is taken it is advisable to change the position of the horizontal circle so as to bring the read- ings in another place, in order to eliminate to some extent the errors of graduation. Readings are sometimes taken on the star directly, and on its image reflected from a basin of mercury. When this is done reading the level may be dispensed with. By the process above described we have a carefully-exe- cuted measurement of the difference in azimuth between the star and mark. It only remains to compute the azimuth of the star, when we shall have the azimuth of the mark. OF COLLIMA TION AND LE VEL. 525 Let m = reading of circle on mark ; s = reading- of circle on star; A = azimuth of mark measured from north towards east ; a = azimuth of star measured from north towards east. Then A = a -\- (m s). (487) Different methods of computing a will be employed, de- pending on the position of the star when observed. Errors of Collimation and Level. 299. The mark and star being at different altitudes above the horizon, the measured difference of azimuth will be affected by an error of collimation, also by a want of parallel- ism between the horizontal axis and the horizon. Other theoretical errors of the instrument we need not consider, since their effect may be made inappreciable by careful adjustment. In the figure let NWSE represent the horizon, z the zenith, s any star, w' the point where the horizontal axis pro- duced pierces the celestial sphere. *b is the inclination, -j- when west end of axis is high ; *c, error of collimation, -|- when thread is east of collima- tion axis ; x, error in reading of horizon- tal circle due to b and c. * This designation is sufficiently general for our purpose, since we shall only have occasion to apply it to stars observed near the pole. 526 PRACTICAL ASTRONOMY. 3OO. Then in the triangle sw'z, sz = z = zenith distance of star; ziv' 90 b ; w's = 90 + c ! w ' zs 9 + x > Therefore sin c = sin b cos z cos b sin z sin x. Or, since c, b, and x will be very small, the above may be written c = b cos z x sin z ; from which * = ~ It will seldom be necessary to apply the correction for collimation, since it may be eliminated by observing in both positions of the axis. If the mark is not in the horizon a similar correction to readings on mark will be required, where, of course, for z we shall have the zenith distance of the mark. Azimuth by a Circumpolar Star near Elongation. 300. When the star is within a short distance of elongation, either east or west, the position is especially favorable, since the motion in azimuth then is very slow. Only one reading can be taken at elongation, but we may apply a correction to the readings near elongation to reduce them to the read- ing at elongation. The azimuth and hour-angle of the star at elongation are 301. AZIMUTH BY A CIRCUMPOLAR STAR. 527 computed by considering the right-angle triangle formed at this instant by the zenith, pole, and star. Let a* and t e be the azimuth and hour- angle at elongation ; \- /l ' a-, d, and 6, the right ascension, declina- tion, and sidereal time. Thenf sin a e = cos o sec q>; cos t e = cot d tan 9?; --Vi3S }** W F Chronometer time of elongation = 8 46. The chronometer correction should be known within about one second, and may be determined by any of the methods previously given; or the theodolite itself may be used for the purpose, either as a transit or by measuring altitudes as with the sextant, provided it has a good vertical circle. 301. The formulae for reducing the readings to elongation will now be developed. Formulae (121) give the values of h and a in terms of d and / for a star at any hour-angle. Recollecting that we now measure the azimuth from the north instead of the south point, these equations are (a) cos h cos a = sin 6 cos

cos /; (>) cos Ji sin a = cos d sin /. * a e , since a plus value of the hour-angle t e corresponds to a minus azimuth. f If many observations of the same star are to be made, it will be convenient to prepare in advance a table of the values of a e and extending over the time during which it is intended to observe. 528 PRACTICAL ASTRONOMY, 3OI. At elongation we have cos 8 sin d cos t e (c\ sm a f = - - = - : cos (p sin cp (d} cos a e = sin 3 sin f e . Multiplying together first (a) and (c), then (b) and (d\ we have (e) cos h cos # sin a e sin d cos 6 sin d cos d cos / cos t e \ (/) cos /* sin a cos # e = sin 6 cos tf sin f sin t e . Add (/) to (e\ cos // sin (a e a) = sin d cos 8 sin S cos tf cos (/ e t). sin tf cos tf From this, sin (a e a) = ---- -. 2 sin 2 (/ 6 /"). The computation will be more convenient if for cos h we substitute its value in terms of a e and <5, viz., cos h = cot a e cot 8; and therefore sin (a e a) = tan # e sin 8 ^ 2 sin 2 \(t e t). (489) We now have an equation which gives the difference between the azimuth at elongation and at any hour-angle /. As this will only be used for stars near elongation, and consequently t e t, a small quantity, it will be convenient to expand it into a series, viz., .. 2 sinH (/,-/) l. *vi . , a e a = tan a e sin" 8 - 7 -~, - - + -(tan a e sin 2 5)* L -- ^ r , - .* (490) * y = sin ~~ ' x = x 4- - - r , -f- etc. 1 6 sin i" In this case (a e a) = sin~ x [tan c e sin 8 5 2 sin 8 ^(/ e /)]. 302. AZIMUTH BY A CIRCUMPOLAR STAR. $2Q When this formula is applied to the close circumpolar stars, sin" 6 differs but little from unity, and the last term will in all practical cases be inappreciable. We have therefore the simple formula _ 2 sin" $(t e t) e tt tail u e ,f .... \49 *) 302. Correction for Inclination of Axis. When the west end of the axis is high the reading of the horizontal circle will be small; therefore the correction will \>e plus, The inclination will be given by the formula derived for transit instrument, (289) : b = [_(w + w'}-(e + e'}\. . . . (492) Or if the level is reversed more than once, b = d -\W-E\ ......... (493) Where Wand E are the means of the readings of the east and west ends respectively. The effect upon the reading of the horizontal circle we have by equation (487)!, viz., x = tan z b tan h. Where h is the altitude of the star. Such a correction must also be applied to the reading on mark when appreciable. 530 PRACTICAL ASTRONOMY. 303- With the circumpolar stars observed at elongation we may write tan

cos a cos h . . (496) For a close circumpolar star this will not differ appre- ciably from da = ".319 cos a. (497) This will be added algebraically to the computed azimuth of the star. 304. Formula for Azimuth by a Circumpolar Star near Elon- gation. sin a e = cos d sec cp\ cos t e = cot tf tan cp\ a e a = tan a e 2 sin 2 \(t e /) sin i" * Level = ~\W E] tan tp- t Aberration = ".319 cos a; A = a e -\-(ms)* level -j- aberration. (XXIV) * m = reading of circle on mark; s reading on star. 304- AZIMUTH BY CIRCUMPOLAR STARS. 533 Example. 1847, October I7th, Polaris was observed near western elongation at Aga- menticus, York County, Maine, with one of the 30 inch theodolites of the Coast Survey, as follows: No. Object Tel. Time by Sidereal Chrono- meter. Azimuth Circle. Level. A B C h. m. s. / d. d. d. d. d. d. v. idiv-o" 9 7 2 Mark. R. 6 30 33 63 55 63 55 39- 7 39- 41. o 39. 7 27. 5 27. o 27. o 28. o 27. 7 26. 5 26. o 24. 3 o d 3 34 63 55 41.0 41.0 29. 8 29. o 26. 4 26. 3 4 D. 37 243 55 26. 2 28. 2 16. 8 17. o 16. 8 13.3 it 6 4 2 243 55 27. o 29. o 19. o 19. o 16. 2 14. o 2 ! Star. D. 6 47 12 127 2 68. o 67. o 61. 5 63. o 64. 5 64. 3 C E. W. 2 49 06 127 2 65.0 65.0 63. 5 63. 2 63. i 60. 5 3 44 62 3 5 38 127 2 62. 8 62. 8 57. o 59. 8 60. o 58. 2 i_ 63 44 4 5 12.5 127 2 58.0 58.0 54-o 52-5 55- 3 53- 5 53-o 52.0 1 64 7 R. 7 oo 5 2 -5 307 2 48. o 49. 2 43- 2 44. 2 45. o 44. 8 8 9 o .5 5 307 2 37 42 48. o 48. 7 49. o 49. o 43- o 44- 7 44- 7 45- 46.8 45.0 47- 9 40- 9 S62 46 10 14.5 37 42 49. 2 50. 5 44.8 44.8 47. 2 46. 2 ., 7 Mark. R. 7 16 63 55 40. o 40. o 23. o 25. o 26. 8 25. 2 u 63 43 8 9 D 63 55 63 55 39- 7 39- 7 Its 23. o 23. o 21. 5 22. 7 25. 7 24. 8 25 o 23. 8 p ii 12 20 243 55 243 55 26. 8 26. 8 26. 7 27. 3 14.5 14.8 15. 2 14. o M- 5 13- 9 So o 2 The horizontal circle was read by means of three microscopes designated A, B, C respectively; the value of one division of the micrometer head cor- responding to one second of arc, subject to the correction for run. The circle being graduated directly to 5', if five revolutions of the screw exactly cover this space there is no correction for run; otherwise it represents the excess or deficiency. For reducing these observations we have: Right ascension of Polaris = a = i h 5 32 .g6 Declination of Polaris = S =88 29' 54".27 Latitude of station =

= 9-973O53 1 cos t e = 8.3915818 t e 88 35' i7".8 4 = 5 h 54 m 2i.2 a = i 5 33 .o = 6 59 54 .2 M = i 51 .8 j Chronometer time of elongation = 7'' i m 46 9 .o In the table which follows, the column marked corrected readings is the mean of the readings of the three microscopes corrected for run when necessary; the remaining columns will be explained by referring to formulae (XXIV). No. Position. Corrected Readings. ,-, '-^-^ *.-. Reduced Readings. Means. r R. 6355'3'"-3 2 3 1 - 1 3 4 \ D. 32 .3 243 55 19 -8 D. 127 42 64 .7 +14-34' 416". - i S ".o 7'4'49".7 2 63 .4 i 40 3'5 ii . 52 .1 3 60 .1 i 8 201 . 7 52 .8 4 55 -2 33-5 179 6 . 48 . I R. 307 42 6 '.S 50-5 + 52 120 . 4 49 . 307 42 46 . Level .23 7 39-5 45 8 6 'o 15 -5 10 . 45 9 7 -i 5 32 i . 45 10 7 -i - 28.5 58 . 2 . 45- 307 42 45 .80 Level .00 I R. 63 55 30 .0 9 -4 9 D. 8 .4 243 55 8 .3 n 8 .7 12 8 -3 Mean of readings on mark = m = 243 55' 24". 86 Mean of readings on star = s = 127 42 48 .03 m s = 116 12 36 .83 Azimuth of star = a e = 2 3 39 .21 Azimuth of mark A = 114 8 57 .62 Diurnal aberration -f- .32 Final value of azimuth, 114 8' 57". 94 35- STAR AT ANY HOUR-ANGLE. 535 From the level readings we have Direct. Reverse. E = 53.50 53-50 W= 53-00 53.50 \{W-E~\ =-.24 ^ = ".97 Azimuth by a Circumpolar Star observed at any Hour-angle. 305. This method differs from the preceding in the manner of computing the azimuth of the star, which may be con- veniently done by either of three methods. First. By the fundamental equations (a) and (b), Art. (301), we readily find sin t ~ cos

cos / sin /' a and / being small, we may expand tan a, sin /, cos / into 536 PRACTICAL ASTRONOMY. 305. series, when the equation becomes, to terms of the third order inclusive, , , 3 = ___ sin t(p - j/ 3 ) _ cos cp(i i/) sin q> cos t(p / 3 )' or cos q>= p sin t-{-ap sin

cos^+^ 3 sin 2 i"[(i+4tanV)cos 2 /-tanV] (5) For Polaris within the limits of the United States the term in/ 3 will not exceed 2", while the terms neglected will not be greater than o".i. For a close circumpolar star observed near culmination this formula may be written The corrections for level reading and aberration will be com- puted by the same formulae as in the previous case. 306. CORRECTION FOR SECOND DIFFERENCES. 537 Correction of the Mean Azimuth for Second Differences. 306. In applying the foregoing method to a series of ten or more readings on a star we may proceed in either of two ways : first, we may reduce each reading separately, com- puting the azimuth of the star for each time of observation ; or second, we may take the mean of the readings and com- pute the azimuth for the mean of the corresponding times, applying to this computed azimuth a small correction for second differences. The first method involves considerable labor, but at the same time the individual values furnish a rough check on the accuracy of the work. When the second method is pre- ferred we may derive the expression for the correction as follows : Let / t t^ . . . t n the observed times ; a^,a. 2 ,a 3 , . . . a n = the corresponding azimuths of the star; " ' ' ' = /.= the mean of the observed times ; n a = the azimuth corresponding to /. Let At, =/,/,; At* = t, - t, ; . . . At n = t n - /. Then we have Jf, + J* 9 -f- . . . -f At n = o. We may now write a n =/(/ n ) = fit. + JO - ^o + 4** + 53^ PRACTICAL ASTRONOMY. 306. The mean of these expressions will be 0. + ^+... + ^ _ d\i At? + At? + . . . -f- J/ n a ~^ 0+ <#'~2 ~ 72 -"' The quantities /// will be expressed in time : multiplying by 15 to reduce to arc, and also multiplying each quantity of -the form (i5^/) 2 by sin i", the term multiplied by -y? will be . (502) Or, if preferred, this term may be computed by table VIII A, for, since the quantities At will be small, we shall have prac- tically sin i" and the above term becomes i ^, 2 sin 2 \At It remains to determine a convenient expression for - r3 -. Differentiating equation (b], Art. 301, with respect to a and t, we find d^a _ tan a /cos 2 / cos 2 a\ ~df~ ^"sln^V c^~a /' ' * ' (S 4) For a close circumpolar star cos 2 ^ differs but little from unity, so that we shall have very nearly (505) * It will be seen that the expression which we have derived for reducing the reading taken near elongation to the reading at elongation is a special case of this same form. 307. CORRECTION FOR SECOND DIFFERENCES. We therefore have for the mean of the azimuths 539 - '- ~ = a - tan a [6.73672] ^24 f, (506) where, as usual, the quantity in brackets is a logarithm, and the quantities At are expressed in seconds of time. Example. 307. 1848, April 5. Observations on Polaris at Dollar Point, Galveston Bay, Texas. Instrument, iS-inch Troughton & Simms theodolite. One division of level = o''.82. (f) = 29 26' 2".6 ; S = 88 29' 57".83 ; Object. Position. Chronometer Azimuth Circle. Level. Time. A B c E. W. Mark. D. 158 50' 55" 65" 50" 129 7i5 R. 20 00 Bi 119 126 Star. D. 9 h 3 m 33'-5 337 8 40 35 20 83 117 4 47 -5 8 55 55 35 R. 6 7.0 98 6.5 8 75 9 45 70 55 55 40 9 24.0 9 65 75 55 10 23.5 20 20 3 10 Mark. D. 158 50 55 65 5 '* 79 120 R. '5 121.5 78 77-5 122 The reduction is now as follows : Object. Position. Reduced Reading. Mean of Readings. Chronometer. A/. A/. Mark. D. 158 50' 5 6". 7 R. 5 1 '3 -3 Star. D. 337 '8 31 .7 18 48 -3 9 h 3 m 33'5 210". 2 136 .2 44,84 18523 18 66 .7 6 70 56.7 R. 19 46 .7 8 6.5 62.8 3944 19 65 .0 9 24 .0 140.3 19684 Mark. D. 158 50 56 .7 R. 51 ii .7 158 5' 4 -6 540 PRACTICAL ASTRONOMY. 307. Formula (506): = 129470 log = 5.1122 Mean of times = g h 7 m 3'. 7 log = 9.2218 n Constant log = 6.7367 tan a = 8.4092,, log correction = 9.4800* Correction = o".3 a = i 4 4.7 t = 8 h 2 m 57 8 .2 = I2044'i8".o The azimuth of the star may now be computed either by equation (498), (499), or (500). We shall compute it by each method for illustration. sin/ Formula (498) is tan a = 5 . cos

= 29 26' 2". 6 S = 88 29 57 .83 a = i 28' n".5 Formulae (499) : tan d = 88 29' 57". 83 q> = 29 26 2 .6 S - 9= 59 3 55 -23 $(8 )= 58 58 o .21 \t = 60 22 9 .O = 28 32 20 .60 a) = 30 o 32 .09 a = I 28 II .5 cos q> = 9.9399792 tan 5 = 1.5817575 Sumi = 1.5217367 * Zech .0032688 log denom. = 1.5250055 sin t = 9.9342512 tan a = 8.4092457 sin 4(5 - = .06002 log a = 3.72357 tan 8

) = .3567 cos 8 1 = 9.4170 Sum = 9-7737 tan 9 (p = 9. 5029 Zech = 9.9372 log factor = 9.4401 52 9 i". 4 i 28' n".4 For computing a single azimuth, as in the present case, formula (498) will be preferred. For other cases, where a larger number of values are required, (499) and (500) will sometimes be found more convenient. For the level correction -[ W ] tan cp = [97.56 102.44] tan q>= 2.00 X tan (p = l".l3. Mean reading on star -f- level correction = 337 19' 25". 3 = s. Mean reading on mark = 158 51 4 .6 = m, Azimuth of star -f- correction for 24 1* -\- aberration = i 28 10 .9 = a. Azimuth of mark = a + (m s) = 180 3 28 .4 = A. The aberration, as before, is given by the formula ".32 cos a. 54 2 PRACTICAL ASTRONOMY. 308. Condition* favorable to Accuracy. 308. Reckoning the. azimuth from the north point equations (121) become, (a) cos h cos a = sin 8 cos

) cos h sin = cos 5 sin t ; (c) sin h sin 5 sin q> + cos <5 cos q> cos Also from the triangle whose vertices are the zenith, pole and star, (d) sin q sin 8 = cos a sin * sin a cos / sin q> ; (<") sin q cos 5 = sin a cos

sin a _ sin q dS cos h cos S cos h Differentiating with respect to (a} and ( cos a. 8 and

-v/ ^v cos

sin ${* cp -}-$} cos i(.sr tp _ , ~s %(z -\- (p + 0) sin j(g -|- y -. o) tan \a -A/ CQS ^ _ ^ _ ^ gin ^ _ ^ _^_ rf y <544 PRACTICAL ASTRONOMY. 310. The azimuth of the star may be computed by either of these formulas, the last being most accurate. As this method will not be employed when extreme accuracy is required this consideration will have less weight than in other cases. When the sun is employed the correction for semidiame- ter is obtained as follows : Let 5 the sun's semidiameter taken from the ephemeris. Then from the right-angle triangle formed by the great circles joining the zenith, cen- tre, and limb of the sun we have, calling the angle at the zenith da, sin 5 = sin z . sin da, S oa = - , . . . ("ii2^ smz' the proper algebraic sign being obvious. If the time is also required, we derive it from the meas- ured altitudes by the method of Articles (124) and (125). Conditions favorable to Accuracy. 310. In order to investigate the effect upon the azimuth of small errors in assumed latitude and zenith distance we resume the fundamental equation sin S = cos z sin

cosec a -\- cot z cot a~\dz ; \ , , d$a = [ tan q> cot a -f- cot z cosec a]d + 5) = 57 3', 44" cos = 9-73538 . |(z -\- q> 5) = 43 8 ii sin = 9.83489 \(z tp S) = 4 14 53 sec = .00120 KZ tp -\- d) 1 8 10 26 cosec = .50598 07745 \a - 47 33' 3".0 tan \a = .03872.5 a = 95 6 7 Hor. circle = 25 56 40 290 50 33 = Reading of circle for north point. * A sidereal chronometer was used. The time is only required for taking S from the ephem- eris and need not be very exact. When a star is used no record of the time is required. 546 PRACTICAL ASTROXOMY. Azimuth by the Transit Instrument. 312. It has already been shown, in connection with the general theory of the transit instrument, how the azimuth of the line of collimation is determined, either by special obser- vations made for this purpose or from a series of transits re- duced by least squares. If now the direction of this line is fixed by a meridian mark, we have the azimuth of the mark. Such a determination, though not of the highest order of ac- curacy, is sufficient for many purposes. When the greatest precision is required, the 'telescope must be provided with an eye-piece micrometer moving a vertical thread. The instrument will generally be mounted either in the meridian or in the vertical plane of a circum- polar star at elongation. 313. AzimutJi by a Close Circumpolar Star near Culmination. The instrument is set up and adjusted as already explained in Articles 166-9. The mark whose azimuth is to be deter- mined must be placed so near the meridian that it may be well observed without changing the azimuth of the instru- ment. In positions where a distant meridian mark is not available a collimating telescope may be used, in which case the firmest possible mounting will be required for both tran- sit and collimator. The observations will be made as follows: A short time before the star's culmination the telescope is directed to the mark and a series of readings taken with the micrometer, both in direct and reverse position of the instrument. The level is then read and a series of transits observed over the micrometer-thread, which is moved forward successively one turn or less. The instrument may be reversed or not at the middle of the series. The level is again read and a series 315- AZIMUTH BY THE TRANSIT INSTRUMENT. $4? of readings on the mark taken. Transits of zenith and equa- torial stars will also be observed for determining- the clock correction. 314. Method of Reduction. The value of one revolution of the micrometer-screw is required. If not previously known this may be derived from the observed transits of the star, by the same method used for determining the equatorial intervals of the transit-threads, viz.: Let / = the interval of time required for the star to pass over the space corresponding to one revolution of the screw. Then, eq. (291), R 1 5/ cos tf Vcos /. ..... (514) Vcos / being taken from table Art. 174 when it differs appreciably from unity. R, the value of one revolution, will be expressed in seconds of arc. The collimation constant may be derived either from the transits of the star, the instrument being- reversed at the middle of the series, or by means of the readings on the mark in the two positions as explained in Art. 182. When the transits of the star are used for the purpose the formula for c is (see Art. 185) -T) cos 8+$(T- T)6Tcos tf + (b'- b) cos (?- It is well to derive c from both the star and mark, the two determinations mutually checking each other. 315. The mean of the observed times must next be re- duced to the time over the line of collimation of the telescope. 548 PR A CTICA L AS TRONOM Y. 3 I 5 . Letr,, r v . . . r m = the successive readings of the micro- meter; A> * f m = chronometer times of observation; r c and t c = micrometer reading and time for line of collimation. Then, from (291),, / c - / = ^^-- --sec 8 Vsec (/ c - /). (515) The factor Vsec (4 /) is taken from the table Art. 174 if it differs appreciably from unity. We thus have T, the chronometer time of transit over the line of collimation. Then, equations (284), (285), (287), AT+ Aa + Bb + C(c s .O2i cos = - .018. AZIMUTH BY THE TRANSIT INSTRUMENT, 551 Mark west of collimation axis 3.042 revolutions = 2' 26". 02 Mean value of a 1 = .39 Azimuth of mark = 2 25 .63 316. If the telescope is not provided with an eye-piece micrometer, the azi- muth-screw at the end of the axis may be employed (see description of instru- ment, Art. 158). The mark in this case must be quite near the meridian, as the range of the screw is small. The method of observing is the same as that de- scribed in the last article. Determination of the Value of the Screw. For this purpose a series of transits of a circumpolar star near culmination will be observed, extending over the en- tire available range of the screw. It will be as well not to extend it to the ex- treme limit in either direction. Let M the micrometer reading at any observed time /; Afo = the micrometer reading at time of culmination t 9 ; J? the value'of one revolution of screw. Then since the screw moves the instrument in azimuth, we have, by (517), where r = t t a . This is a little more accurately written R(M M ) sin i" = sin (15*"), R(M - M.) = -k*5r - Ki5r> J sin i"]; R(M - Mo) = -J-[r - i(i5 sin i)*r] (519) Where the log ${15 sin i") 2 = 0.94518 - 10, and the quantity (15 sin i")*^ may be taken from the table Art. 275. When this correction is appreciable it will be convenient to apply it directly to the observed times, when we shall have these times reduced to what they would have been if the star had moved uniformly in a great circle. The method of combining these reduced times is the same as that illustrated in the preceding article. 552 PRACTICAL ASTRONOMY. 317. EXAMPLE. Ursa; Minoris near lower culmination, February 5, 1869. Chronometer time of lower culmination, 6 h I5 m 48". Micr. Chron. time Time from cu mina- on. Red'n. Red'd time. Time of 3 turns. 21. 19 11: 8. 7-5 7- 5 55 57 57 40 59 2 3- 02 41. 04 21. 06 01. 07 40. 09 2O 9-9 8.1 6.4 4.8 3-1 9-8 8.1 6-5 4-8 + i. 0. 0. 0. 5 55 58. 57 i- 59 4- 6 01 3. 04 i. 06 i. 07 40. 09 20 II 00. 21 to 8 20.5 7.5 I s \> \i- s r 5 Mean 9 62.7 9 59-5 9 55-7 9 56-9 9 57-i 9 5i-7 9 55-8 9 57-07 5-5 5- H 13- IS 57- O.I 0. 15 57- Time of three revolutions, 597*. 07 One revolution = r = ig9 9 .o J? = 198". 2 log = 2.29885 log 15 = 1.17609 log = 8.82216 log R = 2.29710 Star's declination = d = 93 23' 48" Latitude = q> = 30 13 54 The computation of the azimuth of the star at the mean of the observed times, and the determination of the azimuth of the mark from the combination of the readings on star and on mark, will require no further illustration. Azimuth by Circumpolar Star at any Hour-angle. 317. When extreme accuracy is required the instrument must be provided with an eye piece micrometer. The mark, of course, must be near the line of collimation. The method of observing will be the same as with the theodolite, Art. 298, except that the readings are made with the micrometer. If there is no eye-piece micrometer the azimuth-screw may 317. AZIMUTH BY THE TRANSIT INSTRUMENT. 553 be used, in which case the reduction will be precisely the same as that given for the theodolite, formulas (XXIV), Art. 304- When the micrometer is em- ployed the reduction will be as follows : In the figure NESW repre- sents the horizon, Pthe pole, s the star, Z the zenith, /* the mark, CZ the direction of the line of collima- tion, w' the point where the west end of axis pierces the celestial sphere. FIG. 65. Let M 9 = micrometer reading on line of collimation ; M micrometer reading on star; M' micrometer reading on mark ; R = value of one revolution of screw ; b = elevation of west end of axis. Then from the micrometer and level readings we require the expression for the difference in azimuth of s and /*. Let Then from figure, R(M - M.} = m ; R(M r - M.} = m' Then if a = azimuth of star, a' = azimuth of mark, a a' = a l #/ = required difference of azimuth. 554 PRACTICAL ASTRONOMY. 3l8. From triangle w'zs, sin m = sin b cos z cos b sin ,sr sin a t . From triangle w'-sy, sin m' = sin b cos ^' cos b sin #' sin #/. m, m', b, a lt and a/ will always be small quantities ; therefore the above equations may be written m = b cos .3- #, sin z ; ;;?' = b COS ^ rt/ Sin #'. From these equations we obtain * 7' , sin (z 1 z} a r a,' = -T -. 7 -f b -v ' : /-. . (520) sin z sin y sm ^ sin g The micrometer reading is supposed to increase with the azimuth ; if the opposite is the case the signs of in and m' will be changed. b includes the correction for inequality of pivots; also for flexure, if the instrument is of the form shown in Fig. 28. (See Art. 192.) Thus the complete expression for b is b = ~(W- E}+p+f. . , . . (521) P is the correction for inequality of pivots, and /"the flexure. The azimuth of the star being computed by any of the methods before given, we have by (520) the required azimuth of the mark. 318. A Circumpolar Star near Elongation. It will be best when practicable to observe the stars near the time of elori- 319- AZIMUTH BY THE TRANSIT INSTRUMENT. 555 gation. The readings on the star may then be reduced to the reading at elongation as follows : In the figure let s e = position of the star at time T e = elongation ; s = position of the star at time T. Then s e a = x is the correction re- s [ quired to reduce the reading at s to the reading at elongation. From the right-angle triangle sPa, we have cos (f e t) = tan S cot (d + x). From this, by the process given for deriving equation (483) . 2 sin 2 (t e - t} x = sin 2d -- / - (522) On account of the rapidity and accuracy with which the micrometer readings may be made several sets may be taken at one elongation if thought desirable. Example. 319. In Vol. XXXVII, Memoirs Royal Astronomical Society, Captain Clarke gives among others the following observation of Polaris : Station Findlay Seat, 1868, October 23. Position E IV E = 1.30 Latitude cp = 57 34' 50". o M M = 580.19 Declination <5 = 88 36 34 4 M' Mo = 77.01 Right ascension a = i h n m 57-46 Sidereal time = i8 h 4i m 30. n Hour-angle t 17 29 32 .65 / 262 23' 9". 75 Zenith dist. of mark z' = 93 2 We also have One division of level = d = i".8io One division of microm. screw = J? = ".8345 Inequality of pivots p ".650 Flexure /= 3". 171 5 $6 PRACTICAL ASTRONOMY. 319- _ The observations given are the means of a series taken in the following order : ist. Level. 2d. Mark. 3d. Direct telescope to star and read level. 4th. Three readings on star. 5th. Level. 6th. Mark. 7th. Level. The instrument is then reversed and another series taken in the same order. The level reading given is the mean of the four above indicated. We shall first reduce the observation by computing the azimuth of the star at the instant of observation. As both zenith distance and azimuth are required, equations (II), Art. (65), may be employed. These equations are rewritten here for convenience, tan d tan M = cos t cos M tan t; tan h Proof: sin (

cos t e = cot d tan cp Time of elongation T e = a t e We readily find a e 2 35' 39". n T e = i9 h 20 m 43M3 Time of observation T = 18 41 30 .n T e T = t e t = 39 13 02 Then by (522), log ' '" " = 3.47892 zS = 177 13' 8".8 sin = 8.68589 log | = 9.69897 log x = 1.86378 Reduction to elongation = x = 73". 08 Micrometer reading on star m 484 .18 Reading at elongation = m -\- x = 557 .26 m -j- x now takes the place of m in equation (520). When the observation is within a few minutes of elongation we take for z the zenith distance at time of elongation ; but in the present example this will not be admissible. Using for 2 the value derived in the previous reduction, we have m -j- x XT' sin 2 1 / 13 .48 m i 4 .36 sin 2' sin (z' 2) 4 27 sin 2 sin z' a a = 18 22 .XX a = 2 35 39 .XI a' = 2 t? 17 .00 Aberration = O .33 Azimuth of mark = 2 17' i7"-32 CHAPTER X. PRECESSION. NUTATION. ABERRATION. PROPER MOTION. 321. The heavenly bodies which are employed for any of the purposes treated of in the foregoing pages are, first, the sun, moon, and planets; and second, the fixed stars. In solving the problems of practical astronomy, we have in most cases supposed the position of the object observed to be accurately known. The co-ordinates which we have in most cases employed are the right ascension and declina- tion. The motions of the sun, moon, and planets are of a com- plicated character, and the prediction of their places for any given instant belongs to another department of astronomy. When their co-ordinates are required for any of the fore- going purposes they will simply be taken from the American Ephemeris or a similar publication. With the fixed stars the case is different ; their relative positions change very slightly from age to age. In most cases no change at all has been discovered. The apparent co-ordinates of all stars, however, are vary- ing slowly but continuously, owing to two causes which are independent of the star's motion, viz.: first, a shifting of the planes of reference, giving rise to precession and nutation ; and second, an apparent motion of the star, due to the earth's motion combined with the progressive motion of light, called aberration. 322. SECULAR AND PERIODIC CHANGES. 559 Secular and Periodic Changes. 322. The small changes to which many of the quantities employed in astronomical operations are subject are divided into two classes, viz., secular and periodic. Secular changes are those which are progressive in the same direction from year to year, requiring long periods of time secitla to complete a cycle, so that during short periods the changes may be considered as proportional to the time. Periodic changes are those which complete their cycle in a comparatively short time, and where the motion from maxi- mum to minimum, or the reverse, is so rapid that the change cannot be considered proportional to the time, except for very short intervals. T\\Q precession of the equinoxes produces a secular change in the co-ordinates of all stars referred either to the equator or ecliptic. It will be remembered that this is the name given to the slow motion which takes place in the line of intersec- tion of the ecliptic and equator, causing the pole of the equa- tor to describe a circle about the pole of the ecliptic in a period of about 25,000 years. This motion is due to the spheroidal form of the earth, in consequence of which one component of the attractive force of the sun and moon tends to draw the equator into coincidence with the ecliptic. This component of the attraction is not uniform. It is a maximum when the sun and moon are farthest from the plane of the equator, and a minimum when they are in the equator. Nutation. The want of uniformity in the forces producing precession gives rise to small changes of short period which together are called nutation. There are a number of small changes embraced under this head, but the principal one causes the actual pole of the earth's equator to describe a 560 PRACTICAL ASTRONOMY. 324. small ellipse about the mean pole ; the major axis of this ellipse is directed to the pole of the ecliptic and embraces about 1 8" of arc. The length of the conjugate axis is about 14". The period is about 18 years. Mean, Apparent, and True Place of a Star. 323. Suppose the right ascension and declination of a star to be accurately observed with a suitable instrument : the place of the star so determined will be the apparent place. The apparent direction of the star is affected by aberration, the effect of which will be considered more fully hereafter. If we apply to the apparent right ascension and declination the corrections necessary to free them from the effect of aberration, we have the true place. If now we apply to this true place the small periodic cor- rections called nutation, we have as the result the mean place. In catalogues of stars the right ascensions and declinations are given, referred to the mean equator and equinox for the beginning of the year of the catalogue. If then the apparent place of the star is required for any given date, the preces- sion must be applied to reduce the mean place of the cata- logue to the mean place at the given date; the nutation and aberration must then be applied to reduce the mean place to apparent place. The determination of these reductions will be the immediate object of the present chapter. Precession. 324. The change in the position of the equinoxes is due to two causes : first, the action of the sun and moon ; and second, that of the planets. The first gives rise to luni-solar precession, and the second to planetary precession. 3 2 5- PRECESSION. 561 By the processes of physical astronomy it is shown that the attractions of the sun and moon upon the matter accumulated about the earth's equator, which gives it its spheroidal form, produce a slow retrograde motion in the line of intersection of the equator and ecliptic, without changing the angle be- tween these planes. As the celestial longitudes are measured from this line, or rather from one of the points where it pierces the celestial sphere, the effect is a constant increase in the longitudes, with no change in the latitudes. This is luni-solar precession, and is due simply to a motion of the equator. The attractions exerted upon the earth by the other planets of the solar system tend to change the plane in which it re- volves about the sun, without changing the position of the equator; this change is relatively small and tends to diminish the right ascensions without affecting the declinations. The latter is called planetary precession and is due to a mo- tion of the ecliptic. The combined effect of the luni-solar and planetary pre- cession is to produce small secular changes in the right ascen- sions and declinations, also of the longitudes and latitudes of all stars, and in the obliquity of the ecliptic. 325. In order to be able to determine the position of the equator or the ecliptic at any given instant it will be neces- sary to select the positions of those circles at some given epoch as fixed circles to which all motions may be referred. Let these fundamental circles be the mean equator and eclip- tic for 1800.0. In Fig. 67, let AA Q be the mean equator for 1800.0; A' A", the mean equator for 1800 -+- * Let EE and EE' be the mean ecliptic for 1800.0 and 1800 -(- t respectively. Then BD, the part of the fixed ecliptic over which the 562 PRACTICAL ASTRONOMY. 325. point of intersection has moved, is the luni-solar preces- sion in / years = ^'. Let D' be the point on the movable ecliptic which coin- cided with D when the ecliptic had the position EE Q . Then CD' is the general precession for / years = 0,. Since B is the point of the equator which at the instant 1800.0 was at D, BC is the arc of the equator over which FIG. 67. the intersection with the ecliptic has moved in a forward direction. BC is therefore the planetary precession in the interval / years = 5. Let G? O = the mean obliquity of the ecliptic for 1800.0 = A n DE- a?, = the obliquity of the fixed ecliptic for 1800 -f / = A" BE- GO = the mean obliquity of the movable ecliptic for 1800 + / = A"CE\ 7t = the inclination of the mean ecliptic for 1800 -f- / to the fixed ecliptic = EEC* D is the mean equinox of 1800; C is the mean equinox of 1800 -+- t. PRECESSION CONSTANTS. 563 Since longitudes are reckoned in the direction DE Q , E will be the descending node of the movable on the fixed ecliptic. Let n = the longitude of the ascending node of the mov- able on the fixed ecliptic, reckoned from the mean equinox of 1800. Then n = 180 - DE. 326. The determination of the values of the above con- stants, by means of "which the position of the mean ecliptic and equator at any time 1800 -\- t can be determined in reference to the fixed ecliptic and equator of 1800.0, belongs to the department of physical astronomy. Three different series of values have been quite extensively employed, viz., those of Bessel, Struve and Peters, and Leverrier. Bessel's values are given for the mean ecliptic and equinox of 1750, those of Struve and Peters for 1800, and Leverrier's for 1850.0. The values which we shall employ are those of Struve and Peters, being those which are more extensively used at present than either of the others. If, however, it is preferred to use other values, it will be a simple matter to make the necessary changes in the formulae which will be derived. The values are as follows:* - . . (523) 0.000 ^ = 50". 241 1 1 -f o.ooo 1 134/ 2 ; co, = 23 27' 54".22; co, = co -f . ooo 0073 5/'; f co= eo ''.4738^ .OOOOOI4/ 2 ; n = ".4776/ ".ooo 003 5 / 2 ; 3 = o".i5ii9* .000 241 86/ 2 . * Dr. C. A. F. Peters' Numerus Constans Nutationis, p. 66 et 71. f In the American Ephemeris the value of the annual diminution employed is o".4645, instead of ".4738. The difference is so small as to be practically almost inappreciable. 564 PRACTICAL ASTRONOMY. 3 Bessel gives the following values for the epoch 1750:* = 5o".37572/ ', = 50 .21129* j,= 23 28' i8".o ".ooo 1217945? ; .000 1221483? ; .00000984233*" ; &? = 23 28 1 8 .o .48368* .000 00272295*' ; LT = 171 36' \o" 5". 21*; Tt = o // .48892* .000 00307 1 9? ; 3=o .17926* .000 (524) The following are Leverrier's values, the epoch being 1850 : i/} = 5o".36924*' ''.ooo 10881*' ; fa = 50 .23465* -|- .000 U288* 2 ; GO, = 23 27' 3 1 ".83 + .000 00719? ; GO = 23 27 31 .83 - -47593' - ".QOOOOI49* 2 ; n = 173 o' 12" - 8".694 * ; yf = o".4795O* .000 003 1 2 ? ; 5=o .14672* .000 24174*'. (525) Assuming the values ot the above quantities to be known, we may now solve the following problems. 327. Problem First. To find the precession in longitude and latitude for any star between 1800.0 and 1800 -f- /. Let the star be referred to a system of rectangular axes, the fixed ecliptic for 1800 being the plane of XY, the positive axis of X being directed to the ascending node of the ecliptic of 1800 -(- t on the fixed ecliptic, the positive axis of Z being directed to the pole of the fixed ecliptic. Let L and B = the longitude and latitude for 1800. Then x = cos B cos (L IT) ; y = cos j5 sin (L 77) ; s = sin.(a) Next, let the plane of JfFbe the mean ecliptic of 1800 + t, Tabulae Regiomontanae, p. v, Introduction. 327- PRECESSION. 565 the new axis of X coinciding with the old, and the new axis of Z directed to the pole of the ecliptic of 1800 -(- /. Let A and ft = the longitude and latitude for 1800 + /. Then *'=cos ft cos (A 77 ^); / =cos ft sin (A 71- 0,); z'=sm ft.(b} II is the same in both (a) and (b} y being the value for 1800.0. The new axes of F and Z make the angle n with the old. Therefore x'=x\ y'=y cos n -\- z sin n\ z' y sin n + i) = x, n tan B = m, 90 -(L-n) = y, m sin y sin x the above formula becomes tan x = = . I + ; cos _y cos x From this we have sin x = m sin (y x). Adding both members to m sin x, then subtracting both members from m sin x and dividing, m -\- i _ sin x -}- sin (_y x) _ tan _y ~~ ~ m i sin x sin (y x) tan (x ^y) Now write -r- =/; x \y = u; \y = v. tan u = f tan v; and by Moivre's formula, equation (135), 2u V I _ 2t> V -1 _ t *ui +I - p t r=* +l ' 328. PRECESSION. 567 328. Problem Second. To find the precession in longitude and latitude between two given dates 1800 -(-/and 1800 -}-/'. Let A and ft be the longitude and latitude for 1800 + /; A' and ft' be the longitude and latitude for 1800 + /'. Then by (527), A L = ?/\ + -n tan B cos (L - U ); A' L = t\ f + n ! tan B cos (L 77'). Subtracting, A' A = (/// ?/<,)-)- TT' tan cos( 77') 7rtan#cos(X 77). (529) This may be placed in a better form by assuming the auxiliary equations a sin A = (*' + n) sin (77' - 77); ) . . a cos A = (rt f - n) cos \(tt' - 77). [ ' From this we find ~ ' - (/ - i) -- I -}- me P+ I since = m. P l Taking the logarithms of both members of the above and expanding, 2(M + v) y-^n = me * - 1 - i, / rl/:ri + i / rV - \ etc. - ;,- 2 ^^ + ^ r -**=3 _ iw a r -^ etc Or -f- w = w sin 2r/ |? s sin 4z/ -f- iw 3 sin 6v, etc. Writing for u, v, and w their values, we have X _ L - #, = TT tan ^ cos (Z - II) - ITT" tan s ,5 sin z(L - II) \n* tan 3 B sin 3(Z 77), etc. 568 PRACTICAL ASTRONOMY. 328. Combining these with (529), and eliminating n and TT', we find ' - A = (0/ - fc) + A cos /: - - 4tan. (531) Similarly from (528) we have for 1800 + t and 1800 -f- /' /3 B - n sin (L - H ); ft' B = 7i r sin (Z - IT). Subtracting and eliminating n a.id TT' by the auxiliary equa- tions (530), we find ft' - ft = - a sin L - - A. . (532) For the auxiliary quantities a and A we find, from (530), tan A = ~~ - tan (/-/'- //). If we substitute for TT and n' their values from (523), neg- lecting the term in f, and recollecting that ^(U r U) is very small, this equation may be written t> + t n f n ' f + / A = -i- ^7^77- - - 8". 5 05 -^ . (533) A being therefore very small even for large values of / and (', we may write cos A = i in (530), when a = n' - 7i = (/' - ".4776 - (/" - /*) ''.0000035. (534) 329- PRECESSION. 569 In equations (531) and (532) we may write A. ^ for L, .and ft for B. Introducing the auxiliary angle M such that (S35) and substituting in (531), (532), and (534) for ^/, 0' ?r, 77, Struve and Peters' values equation (523) we have finally the following practical formulae for computing the preces- sion in longitude and latitude between any two intervals 1800 4- t and 1800 -f t'\ M = 172 45' 31" 4 / 5o".24i - (f 4 f) 8". 505; + (/' /) [ o".477& (/ 4- /) o".ooo 0035] cos (A. - J/)tan /3; /S' /ff = (/' /)[ o".4776 - (f 4 o".ooo 0035] sin (A - M).. J 329. If we divide the expressions for (A/ A) and (fi r ft] by (/' /), and then make / = /', we shall have the values 7-1 J s> of r and -~, or the expressions for the precession in longi- tude and latitude respectively at the instant /, viz. : M= 172 45' 3i" + 3 -^- = 5o // .24ii 4- o.ooo 2268/; 4- [o // 4776 o.ooo oo/o/] cos (A, M} tan = [o"4776 o.ooo 0070^] sin (A J/). ^(537) These formulae may be used to compute the entire pre- cession between two dates 1800 -\- t and 1800 + /', if we compute the values of the differential coefficients for the middle interval, viz., 1800 -f \(t -f t'\ The result will be accurate to terms of the second order inclusive. 570 PRACTICAL ASTRONOMY. 329. We have developed these formulae (536) and (537) (which are those of Bessel, except that we have employed other constants) for the sake of completeness, although they will not be used in connection with the problems of the present treatise, the co-ordinates commonly employed being the right ascension and declination. Example. The mean longitude and latitude of a Lyra for 1850.0 are as fol- lows: A = 283 12' 48". 12; ft = 61 44 25 .45. Required the mean longitude and latitude for 1884.0. Here t = 50; f = 84; t' t = 34; > + t = 134. Therefore we find, by (536), M 173 8' 23"; A M = no 4 25; A' - A = (t' - t) X 50". 2563 + (t 1 -OX -4771 cos (A M) tan ft; ftf - ft = - (t 1 - t) X .4771 sin (A M). A' - A = 28' i8". 3 6 ftf - ft- - 15". 24 A = 283' 12' 48". 12 ft = 61 44' 25". 45 A' = 283 41' 6". 48 /?' = 6r44' io".2i If we wish to employ (537), we shall have for t the middle of the interval be- tween 1850 and 1884, viz., t = 67. For A in the second member we require the longitude for 1867, which we shall have with all necessary accuracy by adding to the longitude for 1850 the general precession for 17 years and neg- lecting the smaller terms. Calling this value A , we have Ao = 283 12' 48" + 50". 24 X 17 = 283 27' 2"; M = 172 45 31 + 33 .231 X 67 = 173 22 37; AO M = no 4 25; = 50". 2563 -f .4771 cos (A - M) tan ft = 49". 9517; dft ~ = - -477I sin (A - M} - -".4481. 330- PRECESSION. 571 Therefore A.' - A = ~(S - /) = 28' i8".36; agreeing with the values obtained by the other formulae. 330. Problem Third. Given the mean right ascension and declination of a star for the date 1800 -f /, required the right ascension and declination for 1800 -f- 1' '. We first require the values of certain auxiliary constants similar to those employed in solving the corresponding prob- lem for the ecliptic. In Fig. 68 let V, V{ = the fixed ecliptic for 1800; QV, the equator for i8oo + /; QVj the equator for 1800 + t'\ Vyi = the luni.-solar precession in the in- terval (/' /). Therefore Vy{ = if,'- $. Let QV, = 90-^; j2^ 1 / =90+* / ; VtQV*'= e - z, z' , and 6 will be quite small quantities, even when the in- terval (t' t] is considerable. In accordance with our notation, angle QV.V.'i^o &? QV.'V, =00,'. Then in the triangle QVy' the quantities a?/, GO,, and 572 PRACTICAL ASTRONOMY. 331- ip' tp are given by (523); we can therefore determine 2, z' t and 6. By Napier's analogies, we readily find tan Aft = sn The second of these may be written (538) In the first and third the denominator may be written equal to unity. 331. We can now solve our problem, viz., to determine the right ascension and declination for 1800 -}- /', having given those quantities for 1800 + /. In Fig. 68, 5 being any star, Sa = 3, Sa' = 3'. If V^ and VJ represent the position of the mean equinox for 1800+ * an d 1800 + *' respectively, then The planetary precession in the interval / = V^ V^ = 5; The planetary precession in the interval /' = F/F/ ^' The right ascension V^a = oc\ V^Q = 90 z 3; F 2 V = a'; V^Q = 90 + ** - 3'. Considering now the rectangular co-ordinates of the star, 33 J - PRECESSION. 573 the mean equator of 1800 -\-t being the plane of XY, the posi- tive axis of X being directed to the point Q, we have x = cos 8 sin (a -\- z -|- 3); y = cos 6 cos (a-\- z-\- 3); # = sin tf. Similarly for the equator of 1800 -f- /', x' = cos 3' sin (a' z' -f- 3'); / = cos . (539) sin 8' = cos S cos (a -j- z-f- 3) sin + s ' n #cos $ ' We might have derived these equations by applying the formulas of spherical trigonometry to the triangle formed by joining the place of the star with the pole of the equator in the two positions. Thus in Fig. 69, 5 being the star, and P and P the pole of the equator at the time 1800+* and 1800-}-^ respectively, we have the following for the sides and angles of the triangle. Calling the angle at the star C, PP' = 6; PS= ao-tf; P'S=9O -(T; SPP = a + z-\-$ A, say, for convenience; SPP= 180 (a'-z'+$'} = 1 80 - A'. 574 PRACTICAL ASTRONOMY. 332. Another solution of the problem is obtained by applying Gauss' equations to this triangle, viz.: cos !(90+cT) cos sin 1(90 -H') cos lt(A'-C)=s'm sin ^ sin ^'-= sin - sn (540) The auxiliary quantities z, z' , and being computed by (538), either (539) or (540) give the required solution of our problem; these equations being solved in the usual manner. 332. Practically it is more convenient to compute the dif- ferences, (a 1 a) and (d f 6). A formula for (a' at) is conveniently derived from the first and second of (539), which we write as follows: cos d' sin A' = cos tfsin A; cos 6' cos A ' = cos # cos A cos 6 sin 6 sin 6. Multiply the first of these by cos A, the second by sin A, and subtract; then multiply the first by sin A, the second by cos A, and add. We readily find cos 5' sin (A 1 A) = cos 6 sin A sin 6 [tan 5 + cos A tan #]; \ ^^ cos d' cos (A 1 A) = cos d cos 5 cos A sin 6[tan 5 -f- cos A tan |fl]. \ Let p sin 0[tan d -\- cos ^4 tan By the first of Napier's analogies, . . (542) 33 2 - PRECESSION. 575* It will be necessary to make the computation in this com- plete form for circumpolar stars when the interval (/' /) is large. When the star is not too near the pole the computa- tion will be much simpler, as \ve shall see. Example. The mean place of Polaris for 1825.0 is as follows: Right ascension a = o h 58 15'. 32; = 14 33' 49". 8. Declination 5 = 88 22' 3i"-47. Required the precession in right ascension and declination between 1825 and 1900. We have here / = 25, /' = 100. We therefore find, from formulae (523), oa, = 23 27' 54". 22459; rj> = 1259". 43; 2 = 3".628; &V = 23 27 54 .29350; if/ = 5036 .90; 3' = 12 .700. Then by formulae (538), which we may write tan i(z' + z) = cos $(,' + <,) tan \(# $), Itf _ z ) - ^(' - tf) cosec !((/ + ojj), tan \Q = sin \(z' + z) tan K 00 / + >i)- ^ ifj) = 31' 28". 74 tan = 7.9617592 cot = 2.03824 KGJ, -f cj/) = 23 27' 54". 26 cos = 9.9625128 cosec = -3999 1 $(z' + s) = o 28 52 .55 tan= 7.9242720 a?,) = o". 03446 . log = 8.53732 - z) = 9.45 } S i(' - 2 > = 0-97547 z 1 o 29' 2". oo tan K') = 9- 6 375775 = o 28 43 .10 sin 4(2' -f z) = 7.9242567 tan -J0 = 7.5618342 |0 =0I2'32".07 6 =o 25 4 .14 7^ PRACl^ICAL ASTRONOMY. We now compute (a a) and (5' 8) by formulae (5^2), viz. : a = 14 33'49".8 tan |6 = 7.56183 z = 28 43 .10 cos A = 9 98486 3 = 3 -63 -- 333- Sum = 7.54669 = 15 2' 36". 53 Zech = 434 tan S = 1.5472620 sin = 7.8628593 sin A = 9 4142243 log/ = 9.4101647 cos A =9.9848553 log/ = 9.4101647 \(A'A} 2 32' 13". 06 sec = 0.0004259 / cos A = 9.3950200 \(A'-\-A) = 17 34 49 .60 cos = 9.9792268 Zech = .1239697 ^S = 7 5618342 log denominator = 9.8760303 log numerator = 8.8243890 tan (A 1 - A) = 8.9483587 A' A = 5 4' 26". 13 A = 15 2 36 .53 1(8' 8) = o ii 57 .65 tan = 7.5414869 3' - 8 = o 23 55 .30 A = 20 7' 2". 66 (A 1 - A) = 5 4'26".i 3 + (2' +z) = 57 45 -10 -($'-$) = - 9 .07 a' a = 6 2' 2".i6 = o h . 24 8'. 144 333- B_y means of the foregoing formulae we readily find the precession in right ascension and declination, viz., -j- , dd and ;-, at any given instant 1800 -f- * We have (A 1 - A) = ('_)_ (^ 4. *) _|_ (5' - 3). (543) 333- PRECESSION. zf 7 If now we make /' = t in the first of (541), we may make 8' = 3, sin (A' - A} = A' - A, sin 6 = 6, sin A = sin (a+3); also, sin 6 tan %8 will vanish, being an infinitesimal of the second order. Therefore this equation becomes A f - A =6ian6 sin (a -f 3). . . . (544) From (538), the same condition existing, viz., / = /', we have = p- 4:) sin G Combining (543), (544) and (545), writing da, d$, and dd> for (a a), etc., and dividing by dt, da d dt dt -= - - + cos co, - -sin co l tan tf sin (or -f 5). (546) The last of (542) by a similar process gives d8 d'h - = - sin oo, cos (a + 3) ..... (547) Writing m = -j- -f- -rr cos <,-; . = ~ sm * If we draw in the plane of the equator lines to the mean equinox of (1800-)-^ and (1800 -f- t -{- i) years, it will be observed that m represents the angle be- tween them, assuming the rate of change to be uniform during one year. Also, will be the angle between the two lines drawn to the poles of the equator in the two positions. 578 PRACTICAL ASTRONOMY. 334. From the values of '/-, a> lt and equation (523) we have m = 46". 062 3 4- ".ooo 2849/1 1 3 '.0708 2 4- s .ooo 01899/1 n 2o".o6o7 - ".ooo 0863/5 m 4- w sin <* tan S; at - COS at (549) We have written r in place of (a -(- S), no appreciable error resulting from neglecting ~ . These formulae may be employed for computing the pre- cession between any two dates 1800 -f- / and 1800 -f /'. If the values of -j- and -7- are computed for the middle date, viz., at at 1800 + 4(* -f *') tne r 651 - 1 ^ will be accurate to terms of the second order in (/' /) inclusive. We shall return to these formulae hereafter. Proper Motion. 334- When the co-ordinates of a star observed at different dates are reduced to the same epoch by means of the pre- cession formulae, a considerable difference in the values is often found, indicating a motion of the star itself. This change is called proper motion, and may be due either to an actual motion of the star in space or to the motion of the solar system, producing an apparent motion of the star. The observed proper motion is in fact the resultant of the two. For our purposes it is not necessary to attempt to separate these components. The proper motions in most cases are very small, requiring many years to produce an appreciable change in the star's place; but there are a few important ex- ceptions to this rule. jj 334- PROPER MOTION. 579 In investigating the subject, the path of the star is assumed to coincide with a great circle, and the motion to be uniform. It is not probable that either assumption is true, but such deviations as may exist will be very small. In order to determine a star's proper motion, its place must be observed on at least two dates which we may call 1800 -\- t and 1800 + /'. The greater the interval (t' t) the more accurate will be the results, other things being equal. Let a and # = the observed mean right as- cension and declination for 1800 -f /; a -\- Aa and d -(- AS = the values given by reduc- ing the values observed at 1800 + /' to the first date by the application of the pre- cession only. Then Aa and A6 will be the changes in a and 6 due to proper motion in the interval (/' t}. Let /-i and // = the annual proper motion in right ascen- sion and declination respectively. Aa Ad Then yu = -r- V = ~~~ ..... These values will be referred to the mean equator of 1800 -\- t. If we had reduced the co-ordinates for this date to 1800 -f- t' we should have obtained the proper motions referred to the equator of the latter date : Aa' Ad' H=- and Vf- (551) 5SO PRACTICAL ASTRONOMY. 336. These values for stars near the pole may differ very con- siderably from the first. 335. Problem 1. To reduce the right ascension and dec- lination of a star from the epoch 1800 -{- / to 1800 -f t', the proper motion being known. First. Suppose the proper motion given in reference to the mean equator of 1800 -j- t, the solution is as follows: Add to the right ascension for i8oo-|- / the effect of proper motion for the interval (t' /), viz., /<(/' t)\ similarly add to the declination //(/' /). With these values of the right ascension and declination the precession is computed as before by formulas (542). Second. The proper motion being given for the mean equator of 1800 + /'. Reduce the star's place to 1800 -f- 1' by formulas (542), and add to the results n(t' t) and //(/' t} respectively. 336. Problem II. Having given the proper motion in right ascension and declination, referred to the mean equator of 1800 -(- /, to derive the values in reference to the equator of 1800 + t'. Equations (539), giving the values of oc' and <$' in terms of a and 6, are as follows: cos 5' sin (a' z -f S') = cos 5 sin (a -f- z + 5); \ cos S' cos (' - z -j- 3') = cos 5 cos (a -f z -f- 3) cos sin S sin 0; ( (552) sin 5' = cos S cos ( (553) sin d = cos S' cos (a 1 z + 3') sin -j- sin 5' cos 0. ) The proper motion which changes the position of the star itself produces no change in the quantities z, z , -, 5', or 6, as these quantities merely serve to fix the positions of the 337- PROPER MOTION. 581 reference planes. Therefore, proper motion alone being considered, these quantities will be constants, a, a', d, 6' being variable. Differentiating the first two of (552) on this hypothesis, we have cos d' cos (a' z -f S') da' sin 5' sin (a' z -f- 5') d8' = cos d cos (a -f- z -f- 5) da sin d sin (a -\- z -\- 3) are tne tota ^ differential coefficients with respect to both precession and proper motion. If we write d p a, d p $ to indicate a variation due to pre- cession, and d^a, d^d to indicate changes due to proper mo- tion, we have df d p d dja a ~ and similarly for the other coefficients. Equations (549) give us -^- and -~, viz., j = m -(- n sin a tan d: at . .-. 559) ~^- = cos . Differentiating these, we have -^ = - 4- sin 20. + I sin a + mn cos a Itan 8 + * 2 sin za tan 2 8; dt* at 2 L. at - = mn sin a -\ '- cos a 2 sin 2 a tan 8; at* at dja. _ mn* 3 , . 3 dn_ . + [( 2 2 - m* + 3 2 cos 20.) n sin a + (*m ^ + n ^) cos a ] tan S -f- \3tn* cos 2a + 3 - sin 2a tan 2 8 + 2* sin a (i + 2 cos za) tan 3 8; (- tun? sin 2a + 3 - sin 2 a ^tan 8 3 s si n 2 a cos a tan 2 8. (S6o> 34- EXPANSION INTO SERIES. 585 340. Let us now consider proper motion. p, x, M, and }JL have the same significance as before, Articles 334 and 338. a' and <$' = the right ascension and declination at end of time t, proper motion alone being considered. In the triangle formed by the pole and the two positions of the star we have PS = 90 - tf; PS' = 90 - eT; SS' = tp\ S'PS=a'-a; S'SP=X- Therefore sin d r = sin d cos /tf+cos d sin pt cos x\ costf'cosfo'' tf)=costf cos/tf sin #sinp/cos;t; cos d' sm(af'a)= sin pt sin % Also, p sin x = M cos tf; p cos ^ = J*'; P* = (^ cos" Differentiating the first of (561) with respect to 8' and /, we find rt/ cos 8' -j- p sin <$ sin pt -4- cos tf cos p^./> cos . Substituting for p cos ^ its value X, and making * = o, we have dt Differentiating a second and third time and reducing in a 586 PRACTICAL ASTRONOMY. similar manner, we have the following partial differential coefficients with respect to // ^~i , nvsn i nv J = " '" sm ; UFJ = Mjt !sm). (562) In a similar manner, by differentiating the third of (561), making t = o, and reducing, we find ^an ru and //, viz., dpdfiCt . d^a , _ d^ = n cos a tan o -^ -f- sin a sec 2 o ; Substituting for -~- and -^- the values given above, we have ^P^*M (Si / 2 ^ , 2 = yw cos tan o -|- X sm a sec "I (564) ''*"* = nu sin a. at Therefore, from (558), (560), (562), (563), and (564), \-?- \ = m -}- n sin tan d -\- yu ; -r. = COS l_* J 34 1 - EXPANSION INTO SERIES. 587 2 "' x/ sin a + \_7t sin a+(/ " + 2ft) " cos a + 2* sin a( cos a + jn') tan 1 5; I -,-5 I = (' + 2fx) sin a -j- cos a -- /n a sin 28 a sin 8 a tan 5. (565)1 Also we have da ' ,_ *' + -' - (566) Differentiating the first of (560) with respect to //, we find ~dn d^ct d-a M iM** . W-u 1 "* JT + ~7r cos ~~7r ~~ mn sm a'/" tan o ut at \_dt at at J + -js'm a -\- mn cos a sec 2 (5-^- , X ^u* + 2 cos 2 tan o--7r + 2ir sin 2a tan o sec tf 7. jf ' + 6/)* cos 2 o]tan2 -\- [(2 2 + 6/u.' a ) sin a -)- 6V' sin 20. + 4 3 sin o cos 2a] tan 3 S; (567) cos a ~ " sin a 3 sin 2 a cos a 3V' sin 3 a 2 |ii a /u.' sin 2 S - fe/f/i/*' sin a + ^(w + 2 /x) sin 2 a + 3 ^ sin 2 al tan 3 2 < J 3 2 ( cos a -(- ^') sin 2 a tan 2 6. 342. These expressions for the third differential coefficients are too complicated for use in practical computation. A series of tables is given by Argelander* by means of which that part may be readily derived which depends on preces- sion onlv. These tables are convenient when the proper motion is so small that it may be disregarded. They are given for the epoch 1850, and Bessel's constants are employed. If the third differential coefficients are required, they may be obtained very conveniently by computing the values of the second differential coefficients for two dates fifty years before and after the given one and proceeding according to the method of Art. 50. If we make/(r) = --, then/(T w) and f(T-\-w] will ' See Untersuchungen iiber die Eigenbewegungen von 250 Sternen, p. 145. 343- EXPANSION INTO SERIES. 589 be the values for-dates fifty years before and after the date T. Then the first of (101) gives (568, the notation being that of formula (101), and the unit of time being one year. 343. If now we require the precession formulas for any given date, as 1875.0, we obtain them by substituting for m and n the values given by (549). m will generally be ex- pressed in time and n in arc. It will be convenient to give the formulas for the second differential coefficients the fol- lowing form: da . cos tan 8 / 2 tt~] (dm m dn\ , dm Ida \ . Id w\ = b -ih?) +-din\aj -*) + - sin ' b -| --- sin i"(-T- -b"' )sin a sec* S -\- zfifj.' sin i" tan 5; ,,(da , \ . (15)* sin i" , * -- ~ - si da dd m, -J-, and ^ will be expressed in time; n, -r , and p in arc. We then have the following formulas for 1875.0: I - = 3 S .07225 -}- [0.126115] si n a tan I '^J=o.oooo322-[4.6338o](^-//) + [ n ; .(569) . ccs a tan |= [1.302206] cos a-\-n'; u'sin2d 59 PRACTICA-L ASTRONOMY 344. The numerical quantities enclosed in brackets are loga- rithms as usual. A numerical example illustrating the application of the foregoing formulae is given in Art. 347. Star Catalogues and Mean Places of Stars. 344. The various catalogues of stars which are in use may bedivided into two classes, viz., compilations and those derived from original observation. Among the most important of the first class are the British Association Catalogue, Newcomb's Catalogue of 1098 Standard Clock and Zodiacal Stars, Boss' Catalogue of $00 Stars, and Saf- ford's Catalogue. These catalogues are of very different degrees of excellence. The British Association Catalogue (often written B. A. C.) contains the right ascensions and north-polar distances of 8377 stars reduced to the mean equator of January i, 1850. The places of many of these are, however, not well determined, errors of from 5" to 10" in north-polar distance, and of corresponding magnitude in right ascension, not being uncommon. It is a very conven- ient catalogue for use in preliminary work, bnt the co-ordi- nates of the stars should be taken from other authorities when accuracy is required. The places given in Newcomb's and Boss' catalogues, on the other hand, have been derived with great care from all of the more reliable authorities, and are entitled to great confidence. The following are among the most reliable of the other class of catalogues, viz., those derived from original observa- tion: Bradley s Observations reduced by Bessel. Epoch of cata- logue 1755. 344- MEAN PLACES OF STARS. 591 Bradley s Observations reduced by Auwers. Epoch 1755. Piazzi. Precipuarum Stellarum Inerrantium Post/tones Medics. Epoch 1800. Groombridge. A Catalogue of Circumpolar Stars, deduced from the Observations of Stephen Groombridge. Epoch 1810. Struve. Posit iones Medics. Epoch 1830. Argelander. DXL Stellarum Fixarum Positiones Media. Epoch 1830. Airy. First Cambridge Catalogue. Epoch 1830. Robinson. Armagh Catalogue of 5345 Stars. Epoch 1840. Gilliss. Observations made at Santiago, Chili. Epoch 1850. Pulkowa. Catalogue in Vol. I, Fulkowa Observations. Epoch 1845. Greenwich. The various catalogues from observations at the Greenwich observatory. Radclifft-. Several catalogues from observations made at the Radcliffe observatory, Oxford. Washington. .Catalogues derived from observations at the Naval Observatory, Washington, I). C, Besides these there are valuable catalogues published by the observatories of Brussels, Paris, Cambridge, England, Cambridge, U. S., Edinburgh, Vienna, and others. These catalogues give the right ascension and declination (or north-polar distance) of the stars referred to the mean equator of the date of the catalogue. Generally the data for reducing the star to the mean equator of any other date are also given. These are commonly given under the headings precession and secular variation ; the proper motion is some- times given when its value is known. The quantities called precession are simply the values of y- and - ,- for the date of the catalogue, precession only at at 59 2 PRACTICAL ASTRONOMY. 345- being considered. The secular variations are the changes which take place in these quantities in 100 years ; i.e., the d*a d*d values of 100 r^ and 100 -^-. Let pa. = the annual precession in right ascension = -^-; d-a s a = the secular variation = 100 TT ; at ac = the right ascension for epoch 7", the date of the catalogue ; a = the right ascension for epoch T -f- t. Then = , + /!>.+ - >--) (570) The declination will be given by a similar process. If proper motion is given, this must also be included in formula (570). In some catalogues the proper motion is included with the precession, when this is generally given under the heading da dd annual motion, and it corresponds exactly to -j- and -j- given by formulas (565). 345. When a star's place is required with extreme accu- racy it should be sought for in as many original authorities as may be available, and the values of the co-ordinates given by the various catalogues combined by the method of least squares to determine the most probable values of these co- ordinates with the proper motion. There are different methods for working out the details of this process, the fol- lowing being perhaps more frequently employed than any other ; Suppose we require the mean place for 1875.0, together with proper motion. If the star has been well observed at 345- MEAN PLACES OF STARS. 593 epochs separated by a considerable interval, the latter may be determined ; otherwise not. We first derive the approximate right ascension and decli- nation for 1875.0 by reducing to that date the place as given in one or more of the best modern catalogues, using for this purpose the annual motion and secular variation of the cata- logue. For this preliminary place the Greenwich catalogues will generally give a value of the right ascension within f .2 or ".3, and of the declination within 2" or 3" of the truth. , da dS d*a J d*d We then compute accurate values of -j-, -7-, -7-5-, and -TT dt at at at for 1875.0 by formulas (569) ; and if great precision is required, -TT and -^r, as explained in Art. 342. Our assumed co-ordi- nates are then to be corrected by comparing them with the places given in the various catalogues. For this purpose the assumed right ascension and declination are reduced to the date of each catalogue. Let , = the assumed right ascension for 1875.0; a,' = the value of or, reduced to the epoch of catalogue, 1875 - t\ a. 2 = right ascension given by catalogue ; tJL = the annual proper motion. The difference (a, or/), supposing for the present or, to be free from error, will consist of two parts, viz., the error in the assumed value of , and the change due to proper motion in the interval /. Therefore (570 is an equation for determining the proper motion ;/ and the correction to the assumed right ascension x. Each cnialogue will give us an equation of this form ; from these the most 594 PR A C TIC A L ASTR ONOM Y. 346. probable values of x and // are derived by least squares. A similar series of equations will also be obtained for the decli- nation. 346. The above is an outline of the method ; practically it is much complicated by the fact that the different catalogues are of very different degrees of accuracy, and in the same catalogue the weight will depend on the number of observa- tions made on the star. It is impossible to give any infallible rule for the assignment of weights ; practically much must depend on the judgment of the investigator. In general, however, the more recent catalogues are entitled to much greater weight than the older ones. Methods and instru- ments are constantly improving, and in consequence a much higher precision is possible now than was the case a hundred years ago. The old catalogues are, however, indispensable in investigation of proper motion. The following table shows the weights assigned by New- comb to the different authorities employed in deriving the right ascensions of the catalogue referred to above : Number of Observations. ' \* 3 4 5 7 . i.S 20 -"5 3 , " 60 80 zoo Bessel's Bradley v If V j f 1 i i I 1 j 1 I J i t i Piazzi S X \ 1 I | | ' J Struve, 1825 Argelander, 1830 Pond \ ^ I i a 3 3 ?. 4 S 3 f, 3 6 4 6 4 7 s 8 6 Airv, Cambridge, 1830 Gilliss. 1840.. " Airy, Greenwich, 1840 1 J i j j j 6 Pulkowa. 1845 . Radcliffe, 184; I 2 2 3 i 4 i 5 7 2 7 2 8 10 15 ?, 20 3 to 4 25 i 25 ?. 1 Airy, Greenwich, 1850 Pulkowa, 1850 7 8 IS IS 20 2=, 2^ 3 Airy. Greenwich, 1860 . Yarnall, Washington, 1860 u ,, .. hi M . r. Airy, Greenwich. 1864. Engleman. Leipzig. 1866 " v " " - Airy. Greenwich, 1870 * Washington, 1870 MEAN PLACE OF STAR B.A.C. 2786. 595 Boss gives a similar table of weights for the declination equations. See Report of the U. S. Northern Boundary Commission, p. 566. If an approximate value of the proper motion is also known it may be employed in computing the differential coefficients by formulas (569), when we shall have in equation (571), in- stead of jj, the correction to the assumed value of ;*, viz., J/f. Example. 347. For the purpose of illustrating the foregoing formulae and methods let us derive the mean co-ordinates and proper motion of the star B. A. C. 2786* for the epoch 1875.0. The following tabular statement shows the values of the co-ordinates given by the various authorities consulted. It probably explains itself sufficiently. 1 II t If L Catalogue. Is !i- 1 * 2'| Catalogue Right Ascension. !! % C l Catalogue Declination. a go.s c"~ |o!s d-S U ^ z Bradley "^ m g . . 4 59' 22". 6 Piazzi 800 8 7 53-5 8 51 13 .0 Gould's D'Agelet.. 800 783.3 8 7 53 3 1783.3 5 I 22 .0 Weiss' Bessel 825 826.2 8 9 24 .70 1826.2 2 46 34 .0 Argelander 830 8 9 43 -43 8 45 40 -3 Taylor 3 35 8 10 1.96 4 44 46 .86 Armayh 840 830. 8 10 19 .99 1853.3 S 43 43 -3' Brussels 856 856. 8 ii 18 .56 1856.2 i 4 49 -37 858 858. 8 ii 25 .98 1858.1 4 40 26 .8 u 860 860. 8 ii 33 -M ,860. I 40 5 -5 Cape of Good HOJ e. Greenwich 860 860 857. 857. 8 ii 33 -38 8 ii 33.28 1857.1 ,857.7 2 8 4 4 -37 40 4 .12 Radcliffe 860 855. 8 ii 33.29 1856.3 7 40 4 .2 Greenwich 864 863. 8 ii 47.88 ,863.7 39 18 .96 868 868 8 12 2 .53 1868.2 9 38 33 .76 u 869 869. 8 12 6.22 1869.2 i ^8 23 .10 ouy 870 ,87L 6 38 .1 -63 ,871.2 6 38 o .30 872 ,872.2 4 37 49 02 Washington 872 1872.2 3 8 12 I 7 .08 1872.2 3 37 48 -5 We first require an approximate value of the star's place for 1875.0, which we may readily derive from the four catalogues which give the co-ordinates for 1860.0, viz., Brussels, Cape of Good Hope, Greenwich, and Radcliffe. Thus we find 1860 a = 8 h n m 33"-27; d = 27 40' 4"-5- < * This is the number of the star in the British Association catalogue. 59^ PRACTICAL ASTRONOMY. 347. For reducing these to 1875 we take from the Greenwich catalogue the follow- ing quantities : In right ascension, precession = -)- 3".66i; secular variation = .017. In declination, precession = io".8g; secular variation = .44; proper motion ju' = .38, Therefore 1875 a = 8 h n m 33".27 + 15(3.661 7.5 X .00017) = 8" I2 m 28M7, 8 = 27 40' 4". 5 -j- is( io".8g 7.5 X .0044 .38) = 27 37' 15". o We may reasonably expect these to prove very close approximations to the final values. With these values of a and 8, and the above value of ju' , we next da. dS d*a d*8 compute -, -, jyj, an "37T b y (59)- This computation is given in full. Constant^ 0.126115 Constant = 4.63380,; Constant = sin a. = 9.923012 doc. rf tan* = 9.718710 *-*" " 56326 37-*' =: _ log = 9.767837 l S- 5-I9706* log = 5.67348 Nat. No. = 0.58592 Nat. No. .000015 7 Nat. No. =-{-.000047 1 5 m = 3.07225 Constant = 5.98778 Constant = J.J^S-jn ^.-ju= 3.65817 ^+>" = "5 6 326 ~-] r /.i= .56326 log n = 1.302206 cos a 9.73748^ sin a = 9.92301 cos a = 9_737476 tan d = 9.71871 log = 1.039682* log =r 6.00723 log = 7.65014^ Nat. No. = 10.95676 Nat. No. = .000 101 7 Nat. No. = .004468 30 fj.' = .38 Constant = 4.81169 dS dd , d*d = -11.3368 -- + //= i o688i -^-=-.0044212 sin a = 9.92301 sec 2 S = .10510 log = 5.9o86i Nat. No. = .000081 o Constant =-|- .000 032 2 =, .000 166 2 347- MEAN PLACE OF STAR B.A.C. 2786. 597 For determining the third differential coefficients, we find for the dates 1825 nd 1925 respectively: 1825 = .000 164 5; r = .004471 5. 1925 -^ = - -ooo 167 9; = - .004 370 o. We therefore find, by (568), d*a d*d = .000 ooo 034; - = -(- .000 ooi 014. Substituting the above values of the differential coefficients in Maclaurin's formula, and making t minus, since we shall want to apply it to dates previous to 1875, we have a. = <*<, - /[3 9 65817 + /(.ooo 083 i /.ooo ooo 006)]; 5 = 6 > + /[n".336S /(.oo2 211 -{-/.oooooo 17)]. By means of these formulae we next reduce the above assumed right ascen- sion and declination to the epoch of each of the authorities where our star is found. The differences between these computed values and the observed values are given in the following table. The'" correction for //'"there given is applied to those catalogues where the epoch of observation differs considerably from the epoch of the catalogue. For example, GouM's D'Agelet: The mean epoch of observation is 1783; the catalogue places are given for 1800. We have as- sumed /*' = ".38, which in 17 years produces a change in d of 6". 46. This is. in this case, the ''correction for n' ." I 3 fc 8 9 20 AUTHORITY. RIGHT ASCENSION. DECLINATION. C u ' S8>< 753 ft 783 020 3 ;< ' 8 3 S 830 656 858 B6o 86. 860 B6< 864 86S 86,, 1872 b T-: ^ .X 5 .05 - C. u No.Obser- vations. ja s ^ IHk u-a 0- C. " !5 5 7 +.03 -.19 .04 * . Z'. Bradley 18 3 '& is a .2 .OS -6". 46 4- .38 +4 -94 - .69 t:S ?S +I-94 - .82 + -14 - .18 -43 = 3 - .06 - -49 + :8 +2.09 -i.6 7 +2 '.19 - -54 + -42 + .10 .'5 .07 - .19 + -23 - .19 Piazzi Gould's D'Asjelet... Weiss' Bessel Argelander Taylor Armagh. .. . Brussels 8 6 |f 8 1 1} 3 2.0 5 5 a.o ;i 4.0 3.0 x.o tS -.04 +.01 .04 .01 .11 4-.^ +.26 -.06 ts +.02 .03 +.0. -.09 ;sn ;g :E 1860 1860 1860 ,8*,, 1868 1869 ,87,, 1871 1872 1872 s 4 ! a.o .a 3 .8 7 5 2.0 .8 Cape of Good Hope. Greenwich Radcliffe Greenwich Washington 59 S PRACTICAL ASTRONOMY. 348. The weights have been assigned in accordance with the systems of New- comb and Boss for the most part. The quantities are now the absolute terms of the system of equations of condition of the form Vp(Aa tn n) and Vp(A8 tdj.i' = n). From these we derive the following normal equations in the usual manner, with the values of the unknown quantities: 21.250^0: 4.045^ = .304; - 4.045^0: + 1.365/1 = + .055; Aa-= .015 .0197; M = .00005 .00078. 11.750^/5 2.416^(1' = 3.263; 2. 4i6J5 -f- .gS-j/lju' = -\- .615; AS = .301 .122; AH' = .00114 .00420. Applying these corrections to the assumed values of a, S, and /*', we have finally, as the most probable values, a = 8 h I2 m 28M55 .0197; M = '.00005 .00078; S = 27 37' 14". 70 .122; X = ".3811 .0042. Nutation. 348. Nutation has already been defined as the name applied to the periodic part of the precession. The components of the attractive force of the sun and moon, which tend to draw the equator into coincidence with the ecliptic, are not con- stant with respect to either of those bodies. The component has a maximum value when the attracting body is in the plane passing through the earth's axis and perpendicular to the ecliptic, and it is zero when the body is in the plane of 349- NUTATION. 59;) the equator. The orbit of the moon and apparent orbit of the sun are ellipses, so that the distances of these bodies from the earth are constantly changing. The angle between the plane of the moon's orbit and the equator is variable; so in a less degree is that between the equator and ecliptic, or ap- parent orbit of the sun. All of these circumstances produce periodic terms in the movement called precession. It will be seen that the law or laws governing this matter are intricate and difficult to investigate ; their discussion be- longs to the department of Physical Astronomy. Various investigators have given more or less attention to the deter- mination of the constants which enter into the formulce ; the values which are most extensively employed at present are those of Peters. 349. Since nutation is simply a motion of the equator, the ecliptic remaining unchanged, it follows that it will produce no effect upon the latitudes of stars. The longitudes will be changed, also the obliquity of the ecliptic. Let A\ and AGO = the nutation in longitude and obliquity respectively. Then, according to Peters, for 1800.0: A\ = 17". 2405 sin fi-f". 2073 sin 2 Q-". 2041 sin 2 ([-f-".o677sin(([ r')1 - i".2692sin2 -)-". 1279 sin (0 F) ".O2i3sin( +F) ; I 4ca= 9". 2231 cos Q ".0897 cos 2 Q -\- ".o886cos 2 C + ''-55(>9 cos2 | -|- ".0093 cos (0 4~ -O- J Where Q = the mean longitude of the ascending node of the moon's orbit ;* ([ = the moon's true longitude ; = the sun's true longitude ; F = true longitude of the sun's perigee ; r" = true longitude of the moon's perigee. *That is, the point where the moon passes from below the ecliptic to above. 600 PRACTICAL ASTRONOMY. 350. The coefficients of the above formulas vary slowly with the time, so that, according to Peters, the values for 1900 will be A\ = 17".2577 sin +".2073 sin2 Q ".2O4isin2([-f ".0677 sin(([ F') i".26g3 sin 2 -(-".1275 sin (0 F) - ".0213 sin (Q -f- F) ; Joo'=-{- g". 2240 cos Q ".0896005 2 Q -(-''.0885 cos 2 ([-|- ".5506 cos 20 4- ".0092 cos (0+T). (573) The numerical values of AX, and the true obliquity, = co -\- AGO, are given in the ephemeris for every tenth day throughout the year. AX is there called the equation of the equinoxes, and is additive algebraically to the longitude re- ferred to the mean equinox in order to obtain the longitude referred to the true equinox. 350. To determine the nutation in right ascension and declina- tion. Since the terms of the formulas are always small, a sufficiently accurate result will be obtained by neglecting the squares and higher powers of these quantities. In other words, we may employ differential formulas, viz., da ^ da Aa = 77T JA + ,77; - (574) For the values of the differential coefficients we employ the equations obtained by applying the general formulas of trigonometry to the triangle formed by joining the poles of the equator and ecliptic with each other and with the star. * In No. 2387, Astronomische Nachrichten, Oppolzer gives formulae for these quantities carried out so as to include all terms which are appreciable in the fourth decimal place. 350- NUTATION. 6C! In Fig. 72, P is the pole of the ecliptic, P of the equator, 5 any star. p PP' = 6,7, PS = 90 - ft, P'S = 9 - tf, SPP' = 90 *, SP'P = 90 + a. Therefore cos 6 cos a- = cos ft cos A ; cos tf sin <* = cos //sin A cos a? sin ft sin a?; (-(575)^ sin = cos/? sin A sin &?-)- sin/tf cos a?. J FlG Differentiating these equations, considering ft as constant, since it is not affected by nutation, cos 8 sin cdcc. -f- cos cr sin 8d8 = cos /9 sin At/A ; cos 5 cos ; | cos 8dS = cos ft cos A sin oodA. -\- (cos /? sin A cos oo sin ft sin oo]doa. J From the second and third of (575) we derive cos ft sin A = cos d sin a cos GO -j- sin # sin o>. Reducing (576) by this and the first of (575), we have cos, 8 sin ccda -j- cos cc sin 5^/5 = (cos 8 sin nr cos ea -j- sin 5 sin cos 8 cos ov/a sin a sin 5. dt dt dt (586) By means of these equations we have the values of a' a an( j $> _ 8 in terms of the sun's distance and longitude, but they are not in a convenient form for practical application unless we are satisfied with an approximation obtained by 353- A B ERR A TION. 607 regarding the earth's orbit as a circle and the motion uniform. dR , dQ In this case we make -^- = o and -j- = the mean apparent angular velocity of the sun in longitude. 353. The true velocity of the earth in any part of its orbit may be taken into account as follows: The orbit being an ellipse, its polar equation will be i -f *>cos(o - ry a being the semi-major axis, e the eccentricity, and (O F) the angle between the major axis and radius vector measured from the perihelion (O and F having the same significance as in Art. 349). Let F = the area of the ellipse = yea* Vi e*; T = the time of one revolution of the earth = one sidereal year; df = an element of area between two consecutive radii vectores; dt = time required to describe df. Then by Kepler's first law, viz. the areas described by the radius vector are proportional to the times we have F df xa* VT^7 i dQ T = Si' r T~ = 2 R W* (588 > since the element of area df = iV(0 - F) = $lTdO. Therefore -n].. (5$9) 608 PRACTICAL ASTRONOMY. By differentiating (587) we find dR 353- (590) But -yfis equal to the mean angular velocity of the earth in its orbit about the sun ; or, what is the same, the apparent angular velocity of the mean sun about the earth. Calling this velocity n, we have, from (586), (589), and (590), d^ _ Tt ~ ] ic 2 -- sin i" sec 2 S[(i + cos 2 10) sin to. cos 202 cos sin S sin a sin u> cos 6) cos 0] K 2 sin i" tan | [(i + cos 2 o>) cos 2 o - sin 2 o>] cos 2 +2 cos o> sin 2 sin 20 [sin 8 cos o sin T (cos o> sin 6 sin a sin cos aa]. (592) 354- REDUCTION TO APPARENT PLACE. 609 The last two terms in each are constant, or are only suDJect to a slow secular change ; they will therefore be combined with the mean right ascension and declination of the star, and will require no further consideration in this connection! as we are only concerned with the periodic terms. The most commonly received value of the constant is that of Struve, who found from a very carefully executed series of observations at the observatory of Pulkova H = 20".445i. (Recently Nyren finds from a still more ex- haustive investigation 2o".4g2.) For 1875.0 the mean value of the obliquity of the ecliptic is a? = 23 27' 19". Substituting these values in (592), and dropping the con- stant terms, we have finally a' a = 20". 4451 sec 5[sin Q sin a -)- cos cos a cos oo] .0009330 sec s S sin 20. cos 20 -\- .0009295 sec 2 S cos 293 cos (O 4- F)] cos a tan S 20". 4451 cos GO sec S cos a cos 20". 4451 sec 8 sin a sin - h sin Q 4-/&'sin2Q A" sin 2([ + A'" sin [ - r')-^ iv sin2 + // sin (0 - T) - A T| sin (0 + T); ' (54- r/<'4- [r /sin Q 4- z" sin 2fi z"sin2([ 4- '" sin (C T) - |i sin 2 4- z' v sin (0 - T) - / Vl sin (Q + F)l X cos a 4- [9". 2231 cos ft o".o897cos2Q 4- o". 0886 cos 2 C 4-. 5509 cos 2 4-0.0093 cos (Q 4--O] sin a 20". 4451 cos oo cos (tan GO cos d sin a sin 5) 20". 4451 cos a sin 5sin . 6l2 PRACTICAL ASTRONOMY. 355- [t will be observed that the corrections to the mean values of a and 8 consist of terms made up of two classes of factors, the first class independent of the star's place and varying with the time, the other class depending on the star's place and varying so slowly that they may be regarded as constant for a considerable time. Writing them in accordance with Bessel's original notation, *A = Ttsln Q+*" sin 2Q i" sin 2< -{-*"" sin(([ T') z iv sin 20 ] + '* sin (0 D - f l sin (0 + D; B = 9". 2231 cos Q -{-".0897 cos 2fi .0886 cos 2( .5509 cos 2 -.0093 cos (O+T); C = 20". 4451 cos oo cos ; D = 20". 4451 sin ; E = - h sin Q+A'sinaQ *"sin2 +Aa' + BV + Cc' +Dd'. \ A, B, C, D, E being the same for all stars are computed in advance for every day throughout the vear, and the values given in the nautical almanac and the similar publications of other countries; so for our purposes we need only take them from these sources. In some star catalogues a, b, c, d and a', b ', c', d' are given in connection with the star's place. For the purposes of an accurate reduction, however, these become obsolete in a few years, as m, ;/, /*, 6, and GO are all subject to slow secular * See Art. 358. f These are divided by 15, since the right ascension is generally given in time. 356. REDUCTION TO APPARENT PLACE. 613 changes. It will be advisable to recompute them if much time has elapsed. Example. Required the apparent place of a Lyrae, 1884, November 10, for upper transit, Washington. Mean a. = i8 h 33 o".6-j8 Mean S = 38 40' 34".4O >u = .oi79 //' = -f- ".2726 r = 0.863 rV"* = 3*.0724 ) by formulae = 20". 0534 Then log a = o 3039 log b = 78842 \OgC = 8.0884 log^ = 8.9269. N. A. p. 284, log A = 9 .9602 log B = 0.9619 logf = 1.0894 log D = 1.1883 log a' = a .4592 logy = 9-9935 logc' = 9.9809 log**' = 8.9528 logAa = 2641 log Bb = 8.8461 log Cc = 9-1778 logDd = 0.1152,, log A a' = 0. 4 X 94 log .Si' = 0.9574 log Cc' = 1.0703 log Dd' = 0.1411 Mean place a : 1 8" 33" o.678 d = 38 " 40' 34' .40 Aa 1.837 Aa = 2 "3 Bb .070 Bb' 9 .07 Cc 150 Cc II .70 Dd - i -304 Dd' = I .38 E .001 TH . .016 T/J.' - .23 Apparent place a' = iS h 33'" i s -44 3 <5 - 38" 40' 59"-47 356. The above form of reduction is most convenient when a considerable number of apparent places is required, or when the star catalogue gives reliable values of the constants a, l>, c, d, etc. If these quantities are not given and only one or two apparent places are required, a different form may be given to equations (597) which will be more convenient. This transformation, also due to Bessel, is as follows: Write / = in A -\- E\ i = C tan GO; g cos G nA ; h cos H = D\ g sin G = B\ h sin H = C. Then we have it" = tr + r// + / +g sin (G -f a) tan 5 -f- A sin (ff+ a) sec S; ) ( ~ S' = S + ryu'-f-i'oosS-j-^costC-l-a) + A cos (H + a) sin 5. The values of r, /. <7, /f, i og ^ log A, and log i are also given in the epheme- ris for every day of the year. 6 14 PRACTICAL ASTRONOMY. As an example, let these formulae be applied to determine the apparent place of a Lyrae on the date given above. We have a = i8 h 33. o = 38 40'. 6 page 291 of ephemeris, G = i 46 .3 *G-\-a = 2o h ig m .3 H= 2 34 .2 *H-\-a = 21 7 .2 log T V = 8.8239 log ^g = 8.8239 page 291 of ephemeris, log^- = 1.3109 log h = 1.2952 *sin (G + a) = 9.9142,, *sin (H + a) = 9.8373 tan 5 = 9.9033 sec 5 = .1075 log (g) 9-9523 log (A) = .0639* logo- 1.3109 log A = 1.2952 cos (G 4- a) = 9.7570 cos (H -f a) = 9.8610 log (^') = 1.0679 s ' n ^ == 9-7958 page 291 of ephemeris, log i = 0.7273 log (/*') = 0.9520 cos 8 = 9.8925 log (i) = 0.6198 a = i8 h 33 m o s .678 8 = 38 40' 34". 40 /= 2.804 (g') = ri -7 () = - .895 (A') = 8 .95 (A) = - I.I58 (0 rr 4 .17 r/i = .016 rfji' = .23 a' = i8 h 33 m i 8 .445 <5' = 38 40' 59". 45 357. Note. Certain of the small terms which have been neglected in the preceding formulae will sometimes be appreciable for stars near the pole where great accuracy is required. ist. The Precession for Time r. We have only used the term depending on the first power of r. The values of the second differential coefficients are given by equations (565). The numerical values being substituted, the only terms which can be appreciable are A(a' -v a) = -|- s .ooo 003 r* sin a tan d o'.ooo 1491-* cos a tan d j .ooob6sr 2 sin 20- tan 1 8; > (599) 4(8' 8) = -f- .000975^ sin 2 a tan 5. ) 2d. In the formulae for aberration (593). rigorously a, d, Q, and oo are not the mean values of these quantities as there assumed, but the true values. They * A table giving logarithmic sines and cosines with the argument expressed in time is con- venient. If this is not available, (G + o) and (H + a) must be reduced to arc. 357- REDUCTION TO APPARENT PLACE. 615 should therefore be corrected for nutation. The necessary corrections to (a! a) and (5' 5) as given by (593) may be determined by differential formulae. Since (a' a) /[a, 5, 0, a>), and similarly for (5' 5), (6 ' Where Aa, AS, etc., represent the corrections for nutation given by (572) and (579)- Practically the terms in AQ and AGO will never be appreciable, and of the values of Aa and AS we need only retain the following terms: Aa = [6". 865 sin or sin Q + 9". 2235 cos a cos Q] tan 5; ) ,, , AS = 6". 865 cos a sin Q -j- 9".2235 sin a cos Q. ) Differentiating (593) with respect to a and 5, neglecting the smaller terms, ^j = 20". 445 1 sec 5[cos a sin sin a cos cos co\; -j- = 20". 4451 sec (5 tan 5[sin a sin Q -|- cos a cos Q cos GO]; 45' <5) -j = 20 .4451 [sin 5 sin a sin -|- sin 5 cos a cos Q cos GO]', -Tft = 20". 445 1 cos d cos a sin Q -(- 20". 4451 cos [cos d sin a cos GO -|-sin 5 sin a?]. Substituting in (600) and retaining only terms multiplied by tan d or sec d, we find A(a ,_ a)= 2.?^145 Isini tan5sec J +; 6 15 2 +(6 [ -(6 -(6".86s f 9" 2235 cos co) sin 2acos(-[- Q ); .865 cos ) sin 2acos(0 Q); .865 cos a>-9".223s) cos 2a sin(0 Q ); -(6 .865+9".2235Cos 24 h . If, as before assumed, we regard the sidereal day of the fictitious year as beginning when the right ascension of the meridian is i8 h 40, then as long as the right ascension of the sun is less than this quantity it will cross the meridian before the point on the equator having this right ascension, and the day of the fictitious year will be the same as the common date. When the sun's right ascension is equal to i8 h 40 (the sun being on the meridian) the two days begin together, and when it is greater than i8 h 40 the sidereal day of the fictitious year begins before the common day, and therefore one day must be added to the common reckoning for the 622 PRACTICAL ASTRONOMY. 361. date of the fictitious year. Therefore the argument of the table will be in which i = ofrom beginning of the year to where the right ascension of the mean sun equals the sidereal time, after whicli 2=1. The Tabulce Regiomontance then give the following quantities : Table I gives k for the longitude of Paris expressed in hours, minutes, and seconds, and also as a fraction of a day, for every year from 1750 to 1849. Table II gives d, the west longitude from Paris of a num- ber of the principal cities of Europe. (Better values can, however, be found in the ephemens.) Auxiliary table, p. 16, gives ' = - . 24 Table VIII, pp. 17-116 inclusive, gives log A, log B, log C, log D, log T, and E. For C and D table IX may be employed. It requires no special explanation here. Example. Required the logarithms of A, B, C, D, r, for 1825, July i d io h , Greenwich sidereal time. Table I for 1825, k .157 Table II for 1825, d = + .007 Page 1 6, g' = -\- .639 i = .000 Argument = July 1.489 Page 92, table VIII, log A = 9.9224 log B = 0.3026 E = -f- ".05 log T = 9.6975 table IX, log C = .4817 log D = i.3co6 n 362. MEAN SOLAR AND SIDEREAL TIME. 623 The quantities have been interpolated directly from the tables; log C and log D are given more accurately by table IX. If thought desirable, the interpolation may be carried out to second differences, but this will not often be necessary. As an example of a case where i = i let it be required to find the above quantities for 1825, Dec. i d io h , Greenwich sidereal time. As before, k = .157 d = + .007 Table VI, right ascension of g' = .639 Mean sun Dec. i is i6 h 40"', therefore * = i.ooo Argument = Dec. 2.489 With this argument we find log A = 0.0867 ; log B = .4976 ; log C = .7599 ; log D = 1.2772; log r 99631; = +.05. Various forms of tables for star reductions have been pro- posed and employed. Some of these are very useful for special purposes, but it is not necessary to enter into the details of their construction in this connection. 362. Conversion of Mean Solar into Sidereal Time and the con- verse. The solution of this problem for any date after the British and American Nautical Almanacs became available in their present form has been treated with all necessary ful- ness in Articles 94 and 95. For earlier dates other methods must be used. The Tabulce Regiomontancz gives the data necessary for solving the problem for any date between 1750 and 1850. We have shown in Art. 94 that the mean time at any meridian is equal to the true hour-angle of the second mean sun, which moves uniformly in the equator, and whose mean 624 PRACTICAL ASTRONOMY. 362. right ascension is equal to the mean longitude of the first mean sun, which mov 7 es in the ecliptic. Also, the sidereal time is equal to the hour-angle of the true equinox. Therefore in our formula (199) at O must be understood to mean the true right ascension of the second mean sun. This equals the mean right ascension plus the nutation of the vernal equinox in right ascension. The latter is found from the general equations (579), by making a = o, d = o to be A\ cos &?, and is given in the ephemeris as the " equation of the equinoxes in right ascen- sion." It is included in the sidereal time of mean noon given by the ephemeris. When the ephemeris is available it will therefore require no further notice. Table VI of the Tabula Regiomontance gives the right ascen- sion of the second mean sun corrected for the solar nutation of the equinox for every mean noon at the fictitious meridian. The fictitious year always begins with the same right ascen- sion of the mean sun, therefore this table is available for every year. The number taken from this table for any date, which must be the date at the normal meridian, is then cor- rected for lunar nutation in right ascension, which is given by table IV. The result is the sidereal time of mean noon, F , at the normal meridian, which may be used in precisely the same way as the sidereal time of mean noon at Washing- ton. (See Articles 94 and 95.) Or writing the formulae out in full, 0= r+ table VI + table IV + (r+ + O<>- i);(6o6) or V= F B + ( + /) (;i - i) = VI + IV == T + V + T(:i - i). 362. MEAN SOLAR AND SIDEREAL TIME. 625 And for converting sidereal into mean solar time, T= d- (607, The notation being- that of Articles 94 and 95. Example. Given 1825, July i' 1 /' 25", Greenwich mean solar time. Required the corresponding sidereal time. By the first of formulae (606), T = 7'' 25 : " o'.coo Table VI = 6 37 33 .099 Table IV = 1.015 (r+ + 2 7 .927514 77 .98769. .28 .307880 .78 .730010 .28 929734 78 .988174 .29 .318284 79 .736104 .29 .931899 79 .98864, 3 32 .328627 .338908 .349126 .80 .81 .82 .742101 .748003 753811 30 31 32 .934008 .936063 .938065 .80 .81 .82 .989090 989525 .989943 33 34 359279 .369365 3 759524 765143 33 34 .940015 .941914 3 990347 990736 35 36 .379382 389330 .85 .86 .770668 .776100 i 943762 .945562 .86 .88 .991473 .992156 '38 .399206 a .781440 .786687 947313 .949016 00 .92 .992790 993378 39 .418739 .89 .791843 39 .950673 94 993923 .40 .41 428392 .437969 .00 .91 .796908 .801883 .40 952285 ,-953852 .96 .98 .994426 .994892 .42 .447468 ! .92 .806768 .42 95S376 .0 995323 43 44 .456887 .466225 i -93 94 .81156* .8,6271 43 44 956857 958296 .2 .997021 .998137 a 47 .48 49 .475482 .484655 493745 .502750 511668 3 '.98 99 .820891 .825474 .829870 .834232 .838508 49 959695 .961054 9^373 .963654 .964898 2-3 2-4 2.5 3- 3-5 .998857 .999312 999593 999978 999999 .50 .520500 1. 00 .842701 1.50 .966,05 00 I.OOOOOO 628 TABLE II A. MEAN REFRA CTION. Barometer 30 inches. Fahrenheit's Thermometer 50. Apparent Altitude. Mean Refraction. 1 Ovq; I j, oo Mean . .5 Refraction. 1 - 1 Mean . .5 Refraction. ONU) 1 g 1 - r | (4 u rt S | Si M fl 1 J | II l| 04 o'^T 3 1 "' 6 .\ ~ -i 4 ' 3 ' I .0 .0 o3o' 2919' 2438 paf i235' I9io' 2' 4 6".I 2 7 io' i' S3 ".i 4 220' ' 3"-9 79 oo' 80 o 3 o ,0 15 5 20 25 30 5 35 5 4 5 45 5 50 ri 1 * 12 its 635 6 40 1819 14 22 ii 45 952 944 -o 936 .2 8 50 8 55 558 .8 555 -7 12 50 12 55 4 io .4 4 8 .8 19 40 '9 50 24. .6 240 .2 27 40 27 50 50 -7 50 .0 43 20 43 40 44 40 45 o i -7 I .0 059 .6 58 -9 58 .2 57 -6 50 .9 56 .2 055 -6 55 -o 54 -3 53 -7 53 -i 52 -5 050 .6 48 .9 47 .2 45 -5 43 -9 82 o K: si: 87 o 88 o 89 o 90 o 9 5 549 -6 546 .6 3 5 4 5 -6 4 4 .1 2020 20 3 237 -4 236 .0 2 34 .6 28 20 28 40 29 o 47 -7 46 .2 44 -8 9 14 .0 9 7 -o 853 -'4 846 .8 840 .4 8 34 .2 828 .1 822 .1 8 16 .2 8 10 .4 8 4 -8 7 59 -3 9 2 5 9 30 9 35 9 40 9 45 9 So 9 55 5 37 -9 535 -i 5 32 -4 5 29 -6 527 .0 5 24 .3 5 21 .7 3 25 3 3 3 35 3 40 3 45 3 5 3 55 3 59 - 6 358 -I 356 .6 3 55 -2 3 53 -7 352 -3 350 -9 20 50 21 20 | 31 50 2 32 .0 29 40 2 30 .7 30 2 2 9 . 4 3 20 2 28 .1 30 40 2 20 .9 31 2 25 .7 JI 20 2 24 .5 31 40 42 .0 40.6 38 .0 36 -7 35 -5 34 -2 33 -o 31 .8 30 -7 29 .5 28 .4 27 .3 45 40 J. $z 47 47 20 47 40 48 o ,49 5i o 52 o 53 10 5 5 16 .7 10 15 5 ii .7 10 20 5 9 .3 10 25 5 6 .9 4 i 4 20 4 3 4 4 4 5 346 8 344 -2 341 - 6 3 39 -o 336 -5 22 10 22 30 22 40 22 50 2 22 .1 32 20 2 19 .8 33 o 2 ,8 . 7 3 , 20 2 '7 -5 33 40 748 .7 743 -5 io 35 5 2 .3 10 40 5 o .0 5 i 5 20 331 -7 329 -4 23 20 215 -4 2,4 .3 34 4 25 .1 24 -i 55 o 56 o i: 040 .8 39 -3 37 -8 36 .4 35 -o 33 -6 650 6 55 7 33 -5 728 .6 io 50 4 55 .6 io 55 ! 4 53 .4 5 4 55 3 24 .8 23 40 3 22 .6 [23 50 2 12 .2 35 20 35 40 36 o 22 .0 21 .0 r 3 20 .5 24 o 2 10 .2 '5 20 25 30 35 40 45 7 50 S'S Lo- ll: - ! 2S 8 3 o 835' 7 14 -6 7 10 .1 75-7 W:J fi s r 6 4 8 . 9 644 .9 641 ..o 637 .1 633 -3 6 29 .6 625 .9 622 .3 618 .8 6x5.3 II 10 II 15 II 20 II 2 5 II 30 " 35 ii 40 ii 45 ii 50 " 55 12 12 5 12 10 J4 47 -0 |444 -9 1442 .9 : 4 40 .9 438 -9 436 -9 .4 35 -o !433 -I 43' -2 429 -4 427 -5 4 2? .7 4 23 -9 6 30 6 40 6 50 316 .3 3 14 - 2 3 I2 .2 11:1 2 4 20 2 4 30 24 40 24 50 2 8 .2 20:: 2 5 -3 24-4 3640 37 o 37 20 %<: 8 .2 S.1 5 -4 4 -5 62 o 6, o 64 o 65 o 66 o 29 -7 28 .4 2 7 .2 25 -9 7 30 7 40 8 5 o 8 20 jji y^ 255 -8 2 54 .1 25 30 2 5 4 :i s o 26 io 2 i :! 2 .7 i59 -8 158 .9 158 .1 I 57 -2 3840 39 o 39 20 39 40 40 o 40 20 40 40 12 .7 II .9 9 -4 8 .6 7 -3 69 o 70 o 72 o ! 73 o 23 .6 22 .4 *8 017 .8 16 .7 12 20 ,4 20 .4 12 2 5 4 l8 . 7 840 8 50 250 .81 126 40 2 49 -2 |26 50 155 .5 i 54 -7 41 40 6 .2 5 -4 76 o 77.0 14 -5 13 5 6' 8". 5 '.-.. o'n-3 1235,.,., TABLE II B. FACTOR DEPENDING ON BAROMETER. In- ches. B log B 27.5 .917 9.9622 27.6 .920 9.9638 27.7 9 2 3 9 9653 27 8 .927 9.9069 27.9 .930 9.9685 933 9.9700 a&'.i 937 9 9716 28.2 .940 9 9731 28.3 943 9-9747 28 4 947 9.9762 28 5 95 9-9777 ! 28.6 953 9 9792 28. 7 -957 ' 9-98o8 28.8 .960 9 9823 28.9 963 9-9838 29.0 .967 29.1 .970 9:0868 29.2 973 9.9883 j 29-3 977 9 9897 ! 29-4 .980 i 9 9912 =9-5 98? 9 9927 29.6 .987 9.9942 29.7 .900 ! g 9956 29.8 993 9.9971 29.9 997 9.9986 30.0 i .000 .0000 1 30.1 1.003 .0014 30 2 i 007 .0029 30.3 i .010 .0043 30 4 I OI3 .0057 30.5 1.017 .0072 30-6 .0086 30.7 1.023 .0100 308 i 027 .0114 30 9 i 030 .0128 31.0 .0142 TABLE II C. FACTOR DEI- ON ATTAC THI-RMOMI ENDING HED :TER. F - log/ - 30 1.007 .0031 20 I . 006 .0027 - 10 ; 1.005 .0023 o 1.005 .oo>o + 10. + 20 1.004 1.003 .0012 30 1.002 .OOOg 40 I .OOI .0005 50 I .OOO .0000 60 999 9.9996 70 .998 9 9992 80 997 9 - 99 8 9 90 100 .996 .996 9.9985 9.9981 TABLE II D. 629 FACTOR DEPENDING ON DETACHED THERMOMETER. F T log T F r iogr P T *r - 25 72 .0688 15 073 0308 M . .990 9-9958 24 69 ! .0678 16 071 .0298 = ', 9-9949 23 66 .0669 , 17 069 .0289 -- .086 9.9941 22 64 j .0658 18 .0280 - , -985 9-9933 21 61 i .0648 , 19 064 .0271 .983 9.9924 20 5 S .0639 20 062 r..> 981 9.9916 19 56 ' .0629 21 060 .0253 > I 979 9.9908 8 53 ! .0619 22 058 .0244 69 977 9 9899 7. 51 | .0609 ' 23 056 0235 ' ; 975 9.9891 6 48 ; .0599 24 054 .0226 '4 973 9.9883 5 45 -059 25 051 .0217 ' 5 972 9-9875 4 43 .0580 26 049 .0209 66 .970 9 9866 3 40 .0570 27 04? .0200 67 -968. 9.9858 3 8 ; .0 5 CI 28 045 .0191 .966 9-9850 i 35 -0551 29 043 .0 82 964 9.9842 i 33 .0541 ' 30 .041 .0 73 7'J .962 9-9834 30 .0532 31 39 .0 64 71 .961 9.9825 28 .0522 32 .036 . -o 55 -j 959 9.9817 -5 : 0513 33 034 .0 47 73 -957 9 9809 j 23 .0503 i 34 032 -o 38 74 955 9.9801 20 .0494 i 35 1 3 .0484 36 .030 .028 !o 20 75 7< 953 .952 9 9793 9-9785 j 15 475 37 ,3 .0465 38 .026 .024 .0 12 .0 03 77 78 .950 .948 9-9777 9 9769 - n j .0456 39 .0094 7 ' .946 9 976 * 08 .0446 40 .020 .0086 So 945 9 9753 + 106 .0437 41 103 .0428 42 .0 8 .0077 .0068 gi Ba 943 .941 9-9745 9-9737 01 .0418 i 43 .0 4 .0060 83 939 9.9729 ] i 099 .0409 44 .0 2 .0051 84 .938 9.9721 .096 .0400 45 6 094 I .03:50 46 7 .002 .0381 47 008 .006 0043 .0034 II S7 .936 934 933 9 97'3 9-9705 9-9697 8 ! 089 .0372 ; 48 .004 .0017 88 931 99689 9 .087 -0363 49 10 085 0353 5 ii .082 1 .0344 | 51 12 .080 .0:35 : 52 , 3 .078 I .0326 ; 53 14 .076 \ .0317 54 ^998 .996 994 .092 .0009 .0000 9.9992 9.9983 99975 9.9966 89 01 93 93 94 .929 .928 .926 .924 923 .92! 9.9681 99673 9.9665 9 9658 9.9650 9 9642 + 15 ! -073 | -3o8 ! 55 99 9.9958 to .919 9-9634 r = (mean refraction) xX.TXt. 630 TABLE III A. BESSEL'S REFRACTION TABLE. II C r* V C V f% M 11 III log a. Df. A. A. al a'i 1 log a. Dif A. A. GO <^5 <"< <" N Q i 1 5 o 85 o 1020 i 0127 . 1229 I 4 20 75 40 ' -7539 1 84 So .71279 59 I.OI2I .1178 3 3 .75408 *7 .0208 20 .71522 43 1. 0115 .1130 75425 r j. . 0204 , 3 40 71749 2 7 .1082 5 10 7544 1 16 .0200 5 10 72160 99 1.0100 .0992 6 74 i -75543 86 o 75 10 Si 40 -72346 725 '9 .72681 I 1.0096 I . 0092 1.0088 .0951 .0914 0879 7 8 9 73 72 7 1 ! -756'5 75675 .75726 7 2 6a -o 56 o 39 3 30 -72832 1 1.0084 .0846 20 70 -75771 n -Oil 72974 1 2 1 1.0081 0815 21 69 - 75809 .0 01 50 10 83 o 73105 .73229 i 4 g 1.0078 1.0075 .0784 0754 23 68 67 75842 75871 S3 JO .0092 1 .0083 I IO 82 50 73347 1.0073 .0725 24 66 75897 .0075 20 40 73459 2 1.0070 0697 25 65 759'9 22 .0068 30 4 30 73564 73663 9 i 0067 1.0065 0671 0646 27 64 ' 63 -75939 75957 s .0063 .0058 5 10 73757 \ i .0362 0622 28 62 75973 .0054 8 o 82 o 81 50 73345 3928 ''3 i. 0060 1.0358 OD30 0579 2 9 3 61 60 .75988 .7600! T 3 .0049 .0046 2O 40 4007 9 1.0056 559 59 .76012 .0043 30 3 4083 1.0354 0540 32 58 . 76023 1 . 0040 ! 40 4155 68 I .0052 0523 33 57 .76033 .0037 ' 5 IO 4223 1.0050 0508 34 5 .76042 I 81 o 4288 ^ 1.0049 0493 35 55 .76050 .0031 1 20 80 50 4352 4412 60 -< 1.0047 I .0046 0479 0466 36 37 54 53 : 7 .0029 .0027 30 30 4468 5 1.0045 0454 38 52 .76071 .0026 4 ' 20 4521 1.0043 0442 39 5' .76077 . 76082 .0025 10 80 o 74623 1.0041 0420 4 1 49 .76087 .0023 IO 79 So 74670 47 1.0040 0409 4 2 48 .76092 .0020 20 40 747H 44 I 0039 0398 43 76096 .0019 30- 30 74757 43 1.0038 0387 44 46 .76100 .OOIQ 40 20 74799 42 1.0037 0377 45 45 76104 .OOl8 50 10 74^39 4 1.0036 0367 46 44 76107 II 79 74876 ,6 1.0035 0357 47 43 76m 10 78 So 74912 3 1.0054 0347 42 .76114 20 40 74947 35 1-0033 0338 49 76117 30 3 7498i 34 1.0032 0328 50 40 76119 40 20 75 OI 3 3 2 1.0031 03-8 39 76122 50 10 75043 3 1.0030 0308 52 76124 78 o' 75072 29 1.0030 0299 53 37 76126 IO 77 So 29 1.0029 0290 54 36 76128 20 40 75129 26 I .0028 0281 35 76130 3 3 75155 1.0027 0272 56 34 76132 40 751-80 25 1.0027 0264 57 33 76134 50 10 75205 2 5 1.0026 0258 58 32 76136 13 o 77 o 75229 24 1 .0026 0252 59 3 1 76138 76 50 75252 2 3 0246 60 3 76139 20 40 75274 22 0241 65 25 76145 30 30 75295 O2 35 7 20 76149 4 753'6 21 0230 75 15 76152 50 10 75336 2O 0225 80 10 76154 14 o 76 o 75355 18 O22O 85 5 76156 10 75 SO 75373 18 O2l6 90 o" o' 76156 14 20' 75 40' 7539' 0212 TABLE III. A. TABLE III D. 631 SUPPLEMENT. FACTOR DEPENDING ON DETACHED THERMOMETER. Apparen Altitude Apparen Zenith Distance I Logarithm of Refraction. A. x. F. Log y. i F. Logy. F. Logy. 25 + .06773 1 15 +.02969 55 - 00528 -24 .06674 i 16 .02878 -,,, .00612 o 30' I 89 30' 80 o 3.24142 3.16572 .0780 593 .5789 4653 06575 .06476 '\l .02787 .02697 a - 00698 .00780 I 30 88 30 3.09723 .'M' S 3797 21 .06377 ' 19 .02606 59 -.00863 3.03606 .0368 20 .06279 .025.4 - 00946 2 30 87 30 2.98269 0298 '.2624 9 .06181 21 .02426 61 -.01029 3 o 3 30 i; 86 o 2 93'74 2 88555 .0244 .0204 - 8 - 7 .06083 22 .023-16 62 .02247 I 63 -.01195 .01278 4 3 2.805QO .0147 ^1408 - 5 .05790 25 .02068 t 5 85 o 2.76687 .0127 .1229 - 4 .05693 26 .OT 9 79 60 i-. 01443 - 3 05596 27 .01890 67 -.01525 2 II .05500 28 05403 i 29 .01713 1 69 -.01689 TABLE III B. TABLE III c. -'9 05307 30 3' .0 624 .0 536 70 7' 7-.' .01933 ON BAROMETER. ATTACHED THERMOMETER. 7 .05020 33 .0 360 73 i .0201; i .04924 34 o 273 74 02096 .0482? 35 .0 185 .02177 Inches. Log B. P. Log r. I ' 047 4 36 .04640 37 .0098 77 02257 -.02338 27 5 27 6 27-7 27.8 27.9 28.O 2 2 8.3 -.03191 -'02876 -.027^0 -.02564 .02409 .02254 .02099 - 01946 ~ 3 3 60 + .00242 +.00125 + 00086 + .00047 + .00008 .00031 .00070 i i i ++++++++ 04545 3 8 .04451 39 04357 40 .04263 41 .04169 2 .04076 3 .03982 4 03889 5 .03796 6 .03704 7 .03611 8 .00924 .00837 .00750 .00664 .00578 .00406 .00320 .00234 00149 +.00064 7 s 79 Si ga 63 & If in - 02499 :-. 01579 - 02659 :- 02738 .02819 -.0289? -.02978 -.03057 -.03136 | .03216 28.4 28.5 01793 .01640 -,, .00 70 80 .00 8 + 9 + 10 .03519 9 -. 00021 .03427 50 - 00106 8g 9 i-" 03294 -03373 28 7 -.01400 - o, !3 6 90 -.00 25 0333 51 ,-.0019 9> 03452 28 8 28.9 -.01,85 01035 100 . 00264 +'3 + '4 0315: .03060 53 1 .00360 54 .00444 93 94 -.03609 29.0 29.1 .00735 . + 15 + .02969 55 j .00528 I 95 -^03765 29.2 29 3 -00438 log ft log B + log T. 29 4 00790 29.5 29.6 + .t 0005 29.7 -00151 log r = log a + A . log + A . log y + log tan z. 29.8 .00297 29.9 00443 g 30.0 .00588 30.1 .00732* 30.2 .00876 30 3 .OTO2O 30.4 .01163 30 5 .01306 30.6 .01448 30.7 .OI58 9 30.8 .OI 7 3I 30.9 .01871 31.0 .02012 632 TABLE IV. To CONVERT CENTIMETRES INTO INCHES. Centi- metres. 11 (3- 1 Centi- metres. iS bi-g C e BS . ; ! oi ft ii 'Jg WJ5 68.0 26. 772 73-5 28.938 .1 .0394 68.5 26969 74.0 29 '34 .2 .0787 69.0 27., 66 74-5 29-33t 3 ."8i 695 27-363 75-o 29.528 4 ! -'575 7-5 27.756 76.0 29.922 6 . 2362 71.0 27-953 76.5 30.119 7 -2756 7'-5 28 150 77.0 30-3 '6 .8 .3150 72.0 28.347 77-5 30.512 9 -3543 72-5 2^.544 780 30 709 .0 , .3937 73- 28.741 78.5 30.906 | TABLE V. C. F. C. , F. C. F. -32 -2 5 .6 4-3 +37-4 o^o^8 3' -23 -8 4 i 39-2 .20 .36 3 5 i 4' - 3 54 29 2 o 6 42 .8 .40 .73 -28 -18 '. 7 44 -6 50 .90 27 -16 .( 8 46 .4 .6 i .08 25 14 10 50 -o o .8 i .44 24 ii . Si - o .91 .62 33 9 12 53 6 i .0 i .80 7 3 55-4 21 - 5 4 57 -2 20 4 5 ! 59 - IQ 6 60 .8 -.8 o 7 62 .6 17 4- i 8 64 .4 16 4- 3 *9' 66 .2 15 -'4 II 68 .0 69 .8 4- 8 22 71 .6 12 10 23 73 -4 II 12 24 75 -2 10 14 25 77 - 9 '5 26 78 8 - 8 17 27 80 .6 7 28 82 .4 - 6 21 .: 2 9 84 .2 - 5 4 23 24 .! 3 3 1 86 .0 87 .8 3 26 33 89 .6 2 28 .4 33 91 .4 I 3 34 93 -2 + 1 3 2 33 P 95 -o 96 .8 3 35 37 % 98 .6 100 .4 TABLE VI. To CONVERT RKADING OF REAUMUR'S THBKMOMETEK INTO FAHRKNHMT'S. R.| F. R. F. R. F. - 24 25 3 38- 75 0.I 0.225 - 9 -75 5 43 25 675 -75' 6 45 -5 4 .90 5 -25 7 47 -75 5 I2 5 3 - 8 50 .0 .6 35 -io .75 9 52 -25 7 575 - 8 .5 10 54 5 ,8 .80 - 6 25 II 56 -75 9 .025 - 4 .0 12 59 .0 25 - i -75 '3 61 . f f 5 '4 63 2 -75 15 65 5 16 68 . 7 -25 '7 70 . 5 9 -5 18 72 - II -7S 19 74 5 14 .0 ao 77 16 .25 21 79 5 18 .5 29 81 20 .75 23 8? . s 23 o 24 86 . 25 -25 2 5 88 . 5 _ 27 -5 -'' 90 . 29 -75 27 9 2 - 5 32 -o *' '-j 95 -f~ 34 25 29 97 5 36 . 5 3 99 -5 38 -75 3i 101 .75 TABLE VII. 653 To CONVERT HOURS, MINUTKS. AND SECONDS INTO A DECIMAL OF A DAY. Hour. Decimal of Day. Minute. Decimal of Day. Second. Decimal of Day. .041 6667 i .000 6944 i .000 0116 2 083 3333 2 .00, 3889 2 .000 0231 3 . 125 ooo-j 3 .002 0833 3 .000 0347 4 .166 6667 4 .002 7778 4 .000 0463 5 . 2-,8 3333 5 .003 4722 c; .000 0579 6 .250 oooo 6 .004 1667 6 .000 0694 7 .291 6667 7 .004 8611 7 .000 0810 8 333 3333 8 .005 5556 8 .000 0926 9 .375 oooo 9 .006 2JOO 9 .000 1042 .416 6667 .006 9444 .coo 1157 12 .458 3333 .500 oooo 12 .007 6389 008 3333 ii is 58 .541 6667 J 3 .000 0278 13 .000 1505 '4 .583 3333 14 .009 7222 '4 .oco 1620 15 .000 I7>'6 16 '.666 6667 16 on mi 16 .000 l8=;9 18 -78 3333 .750 oooo 18 .on 8056 11 .coo 1968 .000 2083 19 .791 6667 19 .013 1944 '9 .000 2159 -83^ 3333 20 .013 8889 20 .000 2315 23 .875 oooo .916 6667 .958 3333 23 24 .014 5833 .0,5 2778 .015 9722 .016 6667 23 24 .000 3431 .000 25-16 .000 5662 .000 2778 24 .017 3611 .000 2894 26 27 .018 0556 .018 7500 27 .000 3009 .000 3125 28 .019 4444 28 .000 3241 29 .020 1389 29 .000 3356 3 ' '02 ^278 3 .000 3472 3 1 3 1 .000 3588 3 2 .022 2722 32 .000 3704 33 .022 9)67 33 .000 3819 34 023 6111 34 .000 35 024 3056 35 .000 4051 3 6 .000 4167 ' 38 .025 6944 .026 5889 ll .000 282 .000 398 39 .027 0833 39 .000 514 .027 7778 40 .000 630 41 .028 4722 4 1 .000 74, 42 .029 1667 42 .coo 4861 43 .029 8611 43 .000 4.77 41 .030 5556 44 .coo 5003 45 .000 5208 46 .031 0444 46 .coo 5324 47 .0^2 6389 47 .000 5440 49 5 033 3333 .034 0278 .034 7222 48 49 5 .000 5556 .000 5671 .000 5787 5' ' 52 .035 4'6 7 .036 nn 5 1 52 .000 5903 .000 6019 53 54 55 56 S7 .036 8056 .037 5000 .058. 1044 .038 8*89 .039 5833 53 54 P 57 .00, 6,34 .000 6250 .000 6366 .000 6481 .000 6507 58 .040 2778 58 .000 6713 * 59 60 .040 9722 .041 6667 g .000 6829 .000 6944 634 TABLE VIII A. in 2 \t o i J n 2 m 3 > log f* M log m m log i m logm .00 .00 6.73673 7 33879 ".96 3 16 29303 3739 32151 8 !?2 .89509 .90230 .90945 i 7 ".6 7 17 .87 18 .07 1.24727 1.25208 1.25687 4 .01 7.94085 23 . 3499 8 -39 92357 18 -47 1.26636 5 6 7 8 9 .01 o .04 8.13467 8.29303 8.42692 8.54291 8 64521 31 38 45 52 .60 36255 3758i .40174 .41442 8 .52 8 .66 8 : 9 4 9 .08. 93^55 93747 94434 95H5 95791 18 .67 18 .87 19 .07 19 .28 19 .48 1.27107 1-27575 1.28041 1.28504 1.28965 10 13 14 o !o1 o .08 o .09 O .11 8 73673 8.8,951 8.89509 8.96461 .67 s .91 99 .42692 43925 .45140 46338 47519 9 .22 9 -36 9 -so 9 .64 9 -79 .96462 .97127 .97788 .98443 .99094 19 .69 19 .90 20 .11 20 .32 20 .53 1.29423 1.29879 1.30332 1.30783 1.31232 ; 17 18 19 o '16 o !i8 9 08891 9.14497 9.19763 9 24727 .07 '5 23 32 .48685 . .49836 50971 52092 9 -94 10 .09 10 .24 10 .39 .99740 .00381 .01017 01649 20 .74 :3 sa 1.31679 1 1.32123 I 1.32566 1.33006 1-33443 20 21 23 24 o .24 o .26 9.33879 938117 9.42157 9.46018 9-497*5 49 .58 .67 .76 83 54291 55370 56436 57489 5852.) io .09 ii .15 ii 31 .02898 .03517 .04131 04740 05345 21 .82 22 .0 3 22 .25 22 .47 22 .70 1-33878 1-343" 1 1-34743 1-35172 i I-35598 i 21 27 29 -34 o .37 o .40 -43 o .46 9 53261 9.56667 9-599-15 9 63104 9.661=2 94 , .22 59557 60573 .61577 .62570 6.155' ii .47 ii .63 ii .79 ii -95 .05946 -06543 .07136 .07725 .083,0 22 .92 23 -14 23 -37 23 .60 23 .82 1.36022 i 30445 1.36866 1.37285 ; 1.37702 3 3i 32 33 34 o .49 o .56 o .59 o .63 9.090^7 9 7'945 9 74703 9-77376 9.79968 -52 .62 i .72 .82 .64521 .65481 .66431 67370 .68299 12 .27 12 .43 12 .60 12 .76 12 .93 08891 .09468 .10042 . 1061 i .11177 24 .05 24 .28 24 -5 1 24 -74 24 .98 i 38116 1.38529 1 1.38940 I.39348 i 9 P 39 o .67 o .71 o -75 o .79 o .83 9.82486 9-84933 9-873'3 9.89629 9 91886 .92 3 '3 .24 34 .69218 .70127 .71027 .71918 .72800 13 .10 13 .27 13 -44 13 -62 M -79 -"739 .12298 .12853 13404 13952 i i ,'i :?6 140160 1.40563 1.40964 1.41364 1.41761 ! 40 4' 42 43 44 o .87 o .91 o .96 i !o6 9.94085 9 96229 9.98323 o 00366 o>363 ^78 .90 73673 74537 75393 . 76240 .77080 13 .96 M -'3 M -3i M -49 14 .67 .14497 .15038 15576 lice 26 .40 26 .64 26 .88 27 .12 27 -37 1.42157 ' 1-42551 1.42943 1-43333 i i 43722 a 47 48 49 I .10 I -15 i ; 2 6 I .31 04315 .06224 .08092 .09921 .11712 .01 !3 .24 .36 .48 .77911 78734 79550 80358 .S.isS 14 -85 '5 -03 15 .21 15 -39 15 -57 -17169 \&i 18735 .10250 27 .61 27 .86 28 [35 28 .60 1.44109 . 1-44494 1.44877 I-45259 1-45639 5 51 52 53 54 I .36 1$ i -53 i .59 13467 .15187 .16875 .18528 .60 ,i .8 4 .96- .09 .81952 .82738 83517 .84288 85053 15 -76 IS -95 16 .14 16 .32 16 .51 .19762 .20778 21281 21782 28 .85 29 .10 29 -36 29 .61 29 .86 1.46018 1-46395 1.46770 I-47I43 1475'S 55 56 57 58 50 i .65 i .71 1 -77 i .83 i .89 .21745 .83110 .24848 .263,8 27843 34 .46 .60 72 85813 -86564 .87310 . 88049 .88782 17 .08 22280 22775 23267 23756 30 .12 3 .38 30 .64 30 .90 1.47886 1-48255 , 1.48622 i 148988 (So ! i". 9 6 ' 29303 1 7"-5 ' -80509 17" 67 < i 24727 3'"-42 I-497I4 TABLE VIII A. 635 s * 6 m 7 m iQgM m log** m | 1"?'" Mt logn, * 3' -94 32 .20 32 .47 1.49714 1.50076 '.50435 '.50793 I 51150 49 -4' 49 -74 50 .07 50 .40 ..69096 1.69385 1.69673 1.69960 1.70246 7 o".68 71 .07 7 ,\ :S 72 .26 1.84931 ,.85,72 1-85412 1.8565, ,.85890 g6".2o 96 .66 97 .12 97 .58 98 .04 1-98320 '98526 i 98732 ,.98937 6 I 9 32 .74 33 -oi 33 -27 33 -54 33 .81 ; '-51505 ..51859 1.52912 So .73 51 .07 51 .40 5' -74 52 .07 '70531 1.708,5 1.71099 1.71382 1.71663 72 .66 73 .06 73 -46 73 -86 74 -26 i 86129 1.86366 1.86603 1.86840 1.87075 98 .50 98 -97 99 -43 99 -90 loo .37 '-99347 '9955' '99755 1.99958 2.00161 10 12 13 4 34 -09 34 -36 34 -64 34 -9' 35 .'9 1.53260 1.53606 '53952 1.54296 1.54639 52 .41 52 -75 53 -09 53 -43 53 -77 1.7:944 1.72213 1.72502 1.72780 '73057 74 -66 75 -06 75 -47 75 .88 76 .29 1.873,0 ' 87545 1.88244 1 100 .84 j 'ox .31 i 10, .78 102 .25 ma .72 2.00363 2.00565 200766 2.00967 2 01167 i 7 9 35 -46 35 -74 36 .02 36 .30 36 .58 1.54980 '55659 '56332 54 -46 54 -80 55 -'5 55 -50 '73333 1.73608 1.73883 '74I57 , 74429 76 .69 77 .'0 77 5i 77 -93 78 .34 1.88476 1.88708 1.88938 1.89168 1.89398 103 .20 I0 3 .67 HI -'63 105 .10 2.01765 2.0.964 2.02,62 20 21 23 24 36 .87 37 -'5 37 -44 37 -72 1.56667 1.57000 '.57663 "57993 55 -84 56 -'9 56 .55 56 .90 57 -25 1.747"' '74972 1.75242 1-755" 1.75780 78 .75 79 .,6 79 .58 80 .00 80 .42 j 1.89627 i ..89855 ..90083 1.90310 1.90536 05 .58 106 .06 106 .55 107 .03 107 .w 2.02360 202557 2.0*753 2.02950 2.03146 3 27 29 38 .3 38 -59 38 .88 39 -'7 39 -46 1.58321 ..58648 1.58974 '59299 1.59622 57 -60 57 -96 58 .32 58 .68 59 -3 1.76048 1.76314 1.76580 1.76846 I.771IO 80 .84 81 .26 8! .68 82 .10 82 .52 1.90762 1.90987 1.91212 1.91436 1.91660 107 -99 108 .48 ,08 .97 100 .46 2-03341 2 03536 203730 i 203924 , 2.041,8 3 3' S 2 33 34 39 -76 40 .05 40 -35 40 .65 40 .95 1.59345 i 60266 1.60586 i 60904 I 6,222 59 -40 59 -75 60 .11 60 .47 60 .84 1:77636 1.77898 1.78160 1.78420 82 .95 %% 84 .23 84 .66 1.9.883 1.92105 1.92327 1.92548 1.92769 I in .44 i io .93 III .43 III .02 112 .41 2.043,1 | 2 04504 2.04697 2.0 4 ij88 H 37 38 39 4' -25 :I 5 42 -'5 42 .45 1.6,538 t.6l8=i4 1.62.63 1.62481 1.62793 6' -57 6 1 .94 62 .31 62 .68 1.78938 1.79,97 '79454 1.79710 85 .09 85 .52 85 -95 86 .39 86 .82 1.0,2990 ..93209 1.93428 1.93646 1.93864 112 .90 "3 -40 113 .00 114 .40 "4 -9 2.05271 205462 2.05652 2.05843 2 06031 42 43 44 42 -76 43 -6 43 -37 43 .68 43 99 1.63,03 1.634,3 1.63722 1.64029 '64335 63 .05 63 -42 6 3 -79 64 .,6 64 -54 1.79967 1.80221 1.80476 1.80729 1.80982 87 .26 f? -70 88 .14 88.57 89 .01 1.94082 1.94299 '94515 1.94731 1.94946 115 .40 "5 -90 116 .40 ..6 .90 117 .41 2.O622O 2.06409 2.06597 2-06785 2.06972 45 46 49 44 .30 j 44 .61 44 -92 45 -24 45 -55 1.64641 1.64945 1.65248 1.65550 1 65851 64 -91 65 .29 65 .67 66 .05 66 .43 1-81234 1.81486 1.8,736 1.8.986 1.82236 89 -45 89 .89 9 -33 90 .78 91 .23 '95375 '95589 1.95802 1.960,4 "7 -92 "8 -43 1.8 .94 2-07,59 2.07346 207532 2.077,8 2.07903 50 52 53 54 45 -87 46 .18 46 -50 46 82 47 -'4 1 1.66,51 1.66450 1.66748 1.67045 1.67341 66 .81 | 67 -'9 67 .58 67 .96 68 .35 1.82484 '.82732 1.82979 '.83225 1.8347' 91 .68 92 .12 92 -57 93 .02 93 -47 1.962*6 1.96438 1.96649 1.96860 i 97070 '20 .47 120 .98 121 49 122 .01 .21- .53 208088 2.08273 2-08457 208641 208824 56 57 58 59 47 -46 1 47 -79 48 .11 48 .43 48 .76 1,67636 1.67930 68 .73 69 .12 69 .51 69 .90 70 .29 1.83716 1.8,960 i 84204 1.84447 1.84690 93 -92 94 .38 94 -83 95 .29 95 -74 1.97279 1.97488 i 97697 1.97905 1.98112 '23 -05 '23 -57 124 -09 .24 .61 125 -'3 2.09007 2 09190 2 CKJ372 209554 I 209735 60 49 -9 1.69006 70. 68 1-8493' 96 .20 i 1.98320 ,25 .65 2.09917 1 636 TABLE VIII A. 8 9 n IO m ii m m log m m log m Jll log in \ m log 1 3 4 i25".6 5 126 .17 126 .70 127 .22 Tv>7 75 .09917 .10098 10278 .10458 '0637 159" .02 160 '.80 161 .39 2 20146 220307 2.20467 2.20627 2.20787 , 9 6". 3 2 196 .97 197 .63 198 .28 198 .94 2.29296 | 2.29441 2 29586 2 29730 2.29874 237"-54 238 .26 238 .98 239 -70 240 .42 = -574 2.37705 2.37836 2.37967 2 38098 7 8 9 128 .2b 128 .81 129 -34 129 .87 130 .40 .10817 10995 .11174 .11352 .11530 161 .98 162 .58 163 .17 163 .77 164 -37 1 '2-> 14!', 2.21106 2.21264 ^1 199 .60 200 .26 200 .92 201 .59 2.30017 2.30l6l i 2.30304 2.30447 2.30590 241 '.87 242 .60 243 -33 244 .06 2.38229 2.38360 2.38490 2.38619 2.38749 130 .94 131 -47 2.11707 164 .97 '65 -57 2.21739 2.21897 202 .02 20 3 .58 2.30732 2.30874 244 -79 245 -52 2.38879 2.39009 i? H I3 2 -55 < ". .<.; 2.12237 2 12413 166 .77 167 -37 2.22212 2.22369 204 .92 205 .59 2.31158 2. 3 I300 246 .98 247 -72 2.39267 2.39396 15 16 17 18 *9 133 - 6 3 134 -17 134 -7i 135 -25 135 -80 8.12589 2. 127 6 4 2.12939 2.13114 167 .97 168 .58 169 .19 169 .80 170 .41 2.22525 2.22682 2.22838 2.22994 2.23150 206 .26 206 .93 207 .60 208 .2 7 2o8 .94 2.31441 2.31582 2.31723 2.3l86 4 2.32004 248 -45 249 -19 249 -93 250 .67 251 .41 2.39525 2.39654 2.39782 2.39910 2.40038 21 23 24 136 -34 136 .88 !37 -43 137 .98 J 3 8 -53 2.13462 2.13635 2.13809 2.139?3 2.14154 E 172 .24 172 -85 173 -47 2.23304 2.23459 2.23614 2.23768 2.23922 209 .62 210 .98 211 .66 212 .34 2.32(44 2.32284 2.32424 2.32563 2.32703 252 .15 252 .89 253 - 6 3 254 -37 255 -12 2.40166 2.40294 2. 4 4 2I 2.40548 2.40675 3 27 28 29 139 .08 139 .63 140 .18 140 .74 141 .29 2.14326 2.14498 2.14670 2.14841 2.15011 174 .08 174 .70 175 -32 $ 3 2.24076 2.24230 2.24383 2.24689 213 .70 2.4 .38 215 .O? 215 -75 2.32842 2 32980 2.33119 2-33 2 53 255 -87 256 .62 257 -37 258 .12 2 5 8 -8 7 2.40802 2.40929 2-4'055 2.4118! 2.41307 30 31 141 -85 142 .40 2.15182 2-15352 177 .18 177 .80 2.248 4 2 2.24994 216 .44 2-33534 2.33671 259 .62 260 .37 2.4M34 2.41560 33 M3 -52 144 .08 2.15691 2.15860 179 -5 179 .68 2.25297 2.25449 218 .50 219 .19 2-33946 2.34083 261 .88 262 .64 2.418*1 2.41936 P 144 .64 2.16029 i 80 .30 2.25600 219 .88 2.34220 263 .39 2.42061 37 M5 .76, 2.16366 ;! is 2.25902 2-34493 264 .QI 2.42310 39 M6 $ 2.16701 182 .82 2.26202 222 .66 2.34766 266 .44 2.42559 40 4 1 4 2 43 44 H7 -46 3 s 149 .17 149 .74 2.16868 2.17035 2 17202 2.17368 2.17534 i8 3 . 4 6 184 .09 184 .72 185 .35 185 .99 2.26352 2^26651 2.26800 2.26049 223 .36 224 .06 224 .76 225 .46 226 .16 2.34901 2.35037 2.35172 2-35307 2.35442 267 .20 26 7 .96 268 .73 269 .49 270 .26 2.42683 2.42807 2.42931 2.43055 2.43178 9 47 49 ISO .31 150 .08 151 -45 3 2 17700 2.17865 2 18030 2.18194 2.18359 186 .63 187 .27 187 .91 188 .55 189 .19 2.27097 2.27246 2.27394 2.27542 2.27689 226 .86 227 .57 228 .27 228 .98 220 .68 2 35577 2357" 2.35846 e 3^980 236,14 271 .79 272 .56 273 -34 274 .11 2.43302 2.43425 2-4J548 2.43070 2.4^793 5 '53 "> 2 l8r,23 189 .83 8.27836 230 .39 274 -88 2-439'S 53 54 154 -93 155 -5i 2.19013 2.19176 191 .76 2.28130 2. 2277 2.28423 231 .81 232 .52 233 .24 2.j6c, 5 2.36648 2.36781 276 .43 277 .20 277 -98 2 4-4159 2.44281 2.44403 9 11 59 60 156 .09 156 .67 157 -25 '57 -84 153 -43 2.19338 2.19500 2.19662 2.19824 2.19985 193 .06 T 93 -7i 194 .36 195 -01 195 .66 2.28569 2.28715 2.28861 2.29006 2.20I5I 233 -95 2 34 -67 235 -38 236 .10 236 .82 2.30913 2.37046 2.37178 2.37310 2.37442 278 .76 279 -55 280 .33 28l .12 28l .QO 2.44525 2.44646 2.44767 2.44888 2.45009 TABLE VUi A. 637 y 12 m 13 M m 15 m \ogrn fH log m m logm m log m 1 3 4 282". 68 283 .47 284 .26 285 .04 285 .83 2.45130 2.45250 2.45371 2.45491 2.45611 33i"-74 332 -59 333 -44 334 -29 335 -'5 2.52081 2.52192 2.52303 2.52414 2-52525 384" 74 385 -65 386 .56 387 -48 388 4O 2.58516 2.58019 2.58722 2.58825 2 58928 44"-63 442 .62 443 .60 444 -58 J4J -- 2.64506 2-64603 2.64699 2.64795 2.64891 5 6 7 8 9 86 .62 87 .41 8 9 : 89 .79 45731 45850 4597 .46089 .46209 336 .00 336 .86 337 -72 338 .58 339 -44 2.52^35 2 52746 2.52856 2.52907 2.53077 389 .32 39 -24 391 -16 392 .09 393 -oi a.t a 2-59 l 34 2 59236 2-59339 2.59441 44" - 447 -54 448 -53 449 -51 45 -5 2.64987 2.65083 | 2.65179 i 2.65274 2 65370 10 '3 14 290 .58 291 .38 292 .18 292 .98 293 -78 46328 .46446 .46565 .,46684 .46802 340 .30 341 .16 342 .02 342 .88 343 -75 2.53187 2.53297 2.53406 2.53516 2.53025 <>, .04 394 -86 395 -79 396 .72 397 65 2 59543 2 59645 2-59747 2.59849 2 5995 ! 451 -50 452 .49 453 .48 454 .48 455 -47 2.65466 2 65561 2.65656 2.65751 2 61846 Jl 17 18 '9 294 .58 295 -38 296 .18 296 .99 7 .79 2.46920 2.47038 2.47156 2.47 2 74 2.47-302 344 62 345 -49 346 .36 347 .23 2 53735 2.53844 2-5,3953 2.54062 398 .58 399 -52 400 .45 401 .38 2.60052 2.60154 2.60255 2.60357 456 -47 457 -47 458 -47 459 -47 2.65941 2.660^6 2.66I3I 2.66 .,25 21 298 .60 299 .40 2.47509 2. 7626 348 .97 349 -84 2-54279 2.54387 403 .26 404 .20 2.60559 2 60660 461 .47 462 .48 46' 48 2.66414 2.66509 2 6660^ 23 24 3 OI .02 301 .83 2. 7860 2- 7Q77 35i .58 '.- 2.54004 2.54712 400 .08 407 .02 2.60861 2.60961 464 .48 465 -49 2.6669 7 2.66791 3 3 29 302 .64 303 .46 34 -27 305 .09 305 .90 2. 8-. 94 2. 8210 2.48327 2.48443 2.48559 353 -34 354 -22 355 -io 355 .98 2.54820 2.54928 2.55035 2 55143 2.5^5'J 407 .96 408 .90 409 .84 410 .79 4'i -73 2.61062 261162 2.61263 2.61363 2.61463 466 .50 % : 5 5 2 469 -53 470 54 2.66885 2.66979 2.67073 2.67166 2.67260 3 3 1 32 33 306 .72 37 -54 3'* -36 303 -18 2.48675 2.48790 2.48906 2.49021 2 49136 357 -74 358 .62 359 -5i 360 .39 361 .28 3.55358 2.55465 2-55572 2 55679 2.55785 413 .68 413 -63 414 -59 K :S 2.61563 2.61662 2.61762 2.61861 2.61961 47' -55 472 -57 473 08 474 .60 475 -62 2.67353 2.67446 $ig 267726 i 37 38 1" % 3'2 -47 313 -3 314 .12 2.49251 2.49366 2.404SI 2.49596 2.49711 362 .17 363 -07 363 .96 364 .85 365 -75 2.55892 2-55999 2.56105 2.56211 2.56317 :: 8 7 : 419 -35 420 .31 421 .27 2.62060 2.62159 2.62258 2.62357 2.62456 476 .64 477 -65. 47 -67 479 -7 480 .72 2.67818 2.67911 2.68004 2.68007 2.68189 40 3'4 -95 2.49825 366 .64 2.50423 422 .23 2.62555 481 .74 2.68281 42 43 44 316 .61 3'7 -44 3'8 -27 -50053 .50167 .5028l 368 .42 369 -3' 370 -2T 2.56635 2.56740 2.^6846 424 -15 425 " 426 07 2.62752 2.62850 2.62949 483 -79 484 .82 485 -85 2.68558 2.68650 45 46 $ 49 319 .10 319 -94 320 .78 321 .62 322 .45 50394 .50508 .50621 50734 50847 37i " 372 -oi 372 -9' 373 -82 374 -7 2 2.56951 2.57056 2.57'6l 2.57266 2.573:1 427 .04 428 .01 428 .97 429 -93 430 .90 2.63047 2-63145 2.63243 263341 2.63438 43 -5 54 -5 Si:* 2.70109 2.70200 2.70291 2.70381 2.70471 5 6 7 ".2 568 .3 569 .4 570 -5 57' 6 2-75373 2 75458 2-75-43 2.75628 2.75713 635"-9 637 .0 638 .2 639 -4 640 .6 2.80336 2.80416 2.80496 2.80576 2.80656 78"-4 709 .7 710 .9 713 -4 2.85029 i 2.85105 2.8518, 2.85257 2-85333 I 5i 2.70561 2.70651 572 .8 573 -9 2.75798 2.75883 641 -7 642 .9 2.80736 2.80816 7U -6 7i5 -9 2.85409 2.85485 8 9 510 .9 5" -9 2.70830 2.70920 576 .1 577 2 2.76052 2.761,6 645 -3 646 .5 2.80976 2.81056 $:; 719 .6 2.85636 2.85712 10 12 13 J 4 5'3 -o 514 .0 515 ;I 516 -I 5'7 -2 2.71099 2.71188 2.71278 2.71367 578 -4 579 -5 580 .6 581 -7 582 .9 2.76220 2.76304 2 76388 2.76472 , 7 . :-,' 647 .7 648 .9 650 .0 6 5 I .2 652 -4 2.81135 2.81215 2.81295 2.81375 2.81454 720 .9 722 .1 7 2 3 -4 724 .6 725 .9 2.85787 2 85863 2.85938 2.86089 3 . *7 18 J 9 5i8 -3 5'9 -3 52 .4 521 -5 522 .5 2.71456 2.7 J M5 2.71634 2 71723 8.7l8ll 584 -o 585 -I 5 86 .2 587 -4 5 &8 .5 2.766 4 2.76724 2.76808 276892 2 76976 653 -6 654 .8 656 .0 657 -2 6 5 8 .4 2.81533 2.81612 2.81691 2 81770 281849 727 .2 728 .4 729 .7 730 .9 732 2 2.86,64 2.86239 2.86314 286389 286464 20 22 23 2 4 523 .6 524 -7 525 -7 526 .8 527 -9 2.71900 2.71989 2.72077 2.72165 2.72254 589 .6 590 -8 59i 9 593 -o 594 .2 2.77059 2 77'43 2.77226 2.77309 659 .6 660 .3 662 .0 66 3 -.2 664 .4 2.81928 2.82007 2.82086 2.82165 2.82244 733 -5 734 -7 736 .0 737 -3 738 -5 2.-S 2 86689 2 86763 2.86838 3 27 28 29 529 -o 530 .0 53i ' 53 2 .2 533 -3 2.72342 2.72430 2.725.8 2.72006 2.72694 595 -3 SQ 6 -5 597 .6 598 -7 599 -9 2.77476 2-77559 2.77642 2.77724 2.77807 665 .6 666 8 668 .0 669 .2 670 .4 2.82322 2.82401 2.82 4 79 2 82558 2.82636 739 -8 741 .1 742 -3 741 -6 744 -9 2.869,2 2.86987 2.8 7 06, 2.87,36 2.87210 3 3 1 32 33 34 534 -3 535 -4 536 -5 537 -6 538 -7 2^72869 2.72957 2.73044 2.73132 601 .0 602 .1 t 3 " 3 604 .5 605 .6 2 77890 2.77973 2.78056 2.78138 2. 7 S?20 671 .6 672 .8 674 .1 gi :1 2.827,4 2.82792 2.82870 2.82948 2.8jO-6 746 .2 747 -4 748 -7 750 .0 75' -3 2.87284 2 87358 2.87432 2 87506 2.87580 35 36 | 539 -7 . 54 .8 54i -9 543 -o 544 -i 2.73219 2.73306 2-73393 2 73480 2 -73567 606 .8 607 .9 609 ., 610 .2 6x1 .4 2.78302 2.78385 2.78467 2.78549 2.7863. 6 7 8 .9 680 .1 681 .3 682 .6 2 83,82 2.83260 2.83337 2.834,4 753 -8 755 ' 756 .4 757 7 2 87728 | 2 87802 2.87876 2.87949 , 40 41 42 43 44 545 -2 546 .3 547 -4 548 -5 540 -5 2 73654 2.73741 2.73827 2.739I4 2.74001 6,2 .5 6- i 6r6 .0 617 .2 2.78713 2.78795 2.78877 2.78958 2.79040 683 .8 685 .0 686 .2 687 .4 -:-:;-; .7 2-83492 2.83^70 2.83648 2.83725 2.83802 759 -o 760 .2 761 .5 762 .S 764 .1 2.8802:; 2.88096 2.88,70 2.883,7 45 46 47 48 49 550 .6 551 -7 552 .8 553 -9 555 -o 2.74087 2 74173 2.71259 2.74346 2.744^2 619 -5 620 .6 621 .8 623 .0 2.79203 2.79284 2.79366 2 70447 69< : 692 .4 693 .6 694 .8 2.83057 2 84034 2.84UI 765 -4 4 769 .3 770 .6 2.88^90 2.88463 l'.88f?,o 2 88683 50 5i 52 53 54 556 -i 557 -2 558 .3 559 -4 560 -5 2.74518 2.74604 2.74690 2-74775 2.74867 624 .1 625 .3 626 .5 627 .6 6-8 .8 2.79528 2.79609 2.79690 2.79771 2 79852 696 .0 697 3 698 .5 609 .7 701 .0 2 84*34? 2.844,8 2 84495 2.S 4 S7, 77' -9 773 -I 774 -5 775 -7 777 ' 2.86756 2.88828 2.8890, 2.88974 2.89047 s6 561 -7 2.74947 630 .0 2 79933 702 .2 2.84648 778 .4 2.891,9 3 59 563 -9 Si 2.75118 2.75203 2 75288 632 -3 633 -5 634 -7 2.80094 2-80175 2.80255 74 -7 705 .9 707 .1 2.84801 2 84877 2.84953 781 .0 7 7 8 8 3 :l 2.80265 2.89337 2.89409 60 567 .2 :-.7---T 635 -9 2.80336 708 .4 2 85020 ' 784 -9 ' 28948, TABLE VIII A. s 2C m 2 0, 2 m a - 1 m log m m \ogrn nt log m m log, 3 4 7 8 4 ". 9 786 .2 787 -5 788 .8 790 .1 2.89481 2-89554 2.89626 2.89698 -." ,:;.. 86s".3 866 .6 8n8 .0 869 .4 870 .8 2.93717 2.93786 2-93855 2.93923 2.93992 949".6 951 .0 952 -4 953 -8 955 -3 2-97755 2:97886 2.97952 2.98017 io 37 ".8 1039 .3 1040 .8 1042 .3 1043 .8 301613 3.01675 3-01738 IE 1 9 791 .4 792 .7 794 -o 795 -4 796 .7 --- 1842 2.89914 2.89986 2.90058 2.90130 872 .1 . 873 -5 874 .9 876 .3 877 -6 2.94061 2.94129 2.94198 2.94266 2-94335 956 .7 958 .2 959 -6 961 .1 2.98083 2 98.48 2.98214 2.98279 1045 .3 1046 .8 1048 .3 1049 .8 301926 3.0,989 3.02052 3.02114 12 748 .0 799 -3 800 .7 2.90202 290274 2.90346 879 o 880 .4 2.94403 2.94471 2.94540 963 -9 965 .4 966 .9 2.98410 2.98475 2.98540 1052 .8 1054 3 1055 -9 3.02239 3.02302 3.02364 4 803 .3 2 90489 884 .6 2.94676 969 .8 2.98670 1057 .4 1058 .9 3.02426 3.07489 i I 9 804 .6 806 .0 807 .3 808 .6 809 .9 290500 2.90632 2.90703 2.90774 2.90845 886 .0 887 .4 888 .8 890 .2 891 .6 2.94744 294812 2.94880 2.94948 2 05016 971 .2 972 -7 974 -i 975 -5 977 .0 2.98735 2.98800 2.98865 2.08^30 2.98995 1060 .4 1062 .0 1063 .5 1065 .0 1066 .5 3.02551 3.026,3 3.02675 3.02737 3.02799 20 21 2 3 24 S3 8i 3 . 9 815 .2 811 6 2.90917 2.90988 2 91058 2.9,129 893 .0 894 -4 895 -8 897 .2 898 .6 2.95084 2-95152 2.95219 2.95287 2-95355 978 -5 979 -9 9 8i -4 982 .9 984 -4 2.99ofo 2.99125 2.99189 2.99254 2.99319 1068 .1 1069 .6 1071 .1 1072 .6 1074 .2 3.02861 3.02923 3.02985 3.03047 3 03109 25 27 28 29 8,7 .9 8l 9 .2 821 .9 823 .2 2.91271 2.91342 2.91413 291484 2.91555 900 .0 901 .4 902 .8 904 .2 905 .6 2.95422 2.95490 2-95557 2.95625 2.9=692 985 .8 987 -3 988 .8 990 .3 991 .8 2.9938.3 2.99448 2.99512 2.99576 2.99641 1075 .7 1077 .2 I0 7 8 .7 I080 . 3 1081 .8 303171 3.03232 3.03294 3-03356 3-034I7 3 3' 3 2 33 34 824 .6 825 .9 827 -3 828 .6 829 .9 2.91625 2.91696 2.91766 2.91837 2.91907 go-' .0 908 .4 909 .8 911 .2 912 .6 2-95759 2-95827 7.95894 295961 2.96028 993 -2 994 -7 996 .2 997 -6 999 -i 2.09705 2.99769 2.99834 2.99898 2.99962 1083 -3 1084 .8 1086 .4 1087 .9 1089 .5 3-03479 3-03540 3.03602 3.03663 3.03725 35 36 37 38 39, 8 3 I .2 832 .6 833 -9 835 -3 836 .6 291977 2 92048 292II8 2.92I8S 914 .0 9'5 -5 916 .9 918 .3 2.96095 2.96162 2.96229 2 96296 looo .6 1003 .5 1005 .0 ,006 .5 3.00026 3.00090 3.00154 3.00218 3.00282 1091 .0 1092 .6 1094 .1 1095 .7 1097 .2 303787 3.03848 3-03909 3-03970 3 04031 4 4i 42 43 44 838 .0 839 -3 840 .7 842 .0 843 4 2.92328 3.92398 2 92468 2.92538 2.92608 921 .1 922 .5 923 -9 925 -3 926 .8 2.96429 2.96496 2.96563 2 96630 2.96696 1008 .0 1009 .4 IOIU .9 1012 .4 1013 .9 3.00 .-46 300409 3-00473 3.00=137 3.00600 1098 .8 1103 .4 1105 .0 3.04092 3-04I53 ' 3.^4214 304275 304336 4 4 i 47 48 49 844 -7 846 i 847 -5 848 .9 8 5 .2 2.02077 2.97747 2.92817 2.92886 2.92956 929 .6 931 .0 932 .4 933 -8 2 06763 2 96829 2.96896 2.96962 2 97028 a.3 1018 .4 1019 9 IO2I .4 3.00664 3.00728 3.00791 300855 300918 "08 \ 1109 .6 11 .2 Jl .7 3-04397 3-04458 3.04519 3.04580 3 04641 5 Si 52 851 .6 852 .9 854 -3 2.93026 2.93096 2.93164 935 -2 936 .6 938 .1 2.97095 297161 2.97227 iii :1 300981 3.0 04=, 3.0 .08 II .3 ii .8 11 -4. 3.04701 3.04762 304823 54 8S7 -I 2 933>3 94 -9 2 97359 3-0 134 ii .5 3-04944 55 56 H ;-) 858 .4 8=9 .8 86, .1 862 .5 863 .9 2.93372 2.93441 293^10 t$ffi 942 -3 ! 943 .8 94S -2 946 .6 948 .1 2 97425 2.07491 297=57 2 97^3 2.07689 1 3 3 103. .8 ' -1 1036 .3 3.0 298 3-0 361 3 424 30,487 3-OI550 ii .6 1126 '.7 1-28 .3 3.05004 30506.; 3-05'25 3.05'85 3-05246 60 1 8*5 -3 2 937T7 949 -6 2-977S* 1037 8 3 01613 1129 .9 305306 640 TABLE VIII A. 24 m m 2 5 m 2 ,m m \ogrn m \ogrn m log m m log,* 3 4 129" '33 3.05306 3.05366 3 05426 3 05547 22 5 ".9 227 -5 229 .2 230 .8 232 .5 3.08848 3.08906 3-08964 3.09022 3.09079 325"-9 327 -6 329 3 33i -o 3.12^07 3-12363 3-12418 3.12474 1431 .4 M33 -2 M35 -0 1436 .7 3.15580 3-15633 3-15740 7 8 9 "39 1140 114-2 3.05607 3.05667 3.05727 3-05787 3-05847 235 -7 2 37 -3 239 -o 240 .6 3.09195 3.09252 3.09310 336 .1 337 -8 339 -5 34' -2 3 12585 3.12640 3-X2695 1440 -3 1442 .1 1443 .9 1445 6 3-^ ---93 3.15500 3-15953 ! 3.16007 12 13 M "45 "47 1150 1 1 52 3.059^7 3.05966 3.06026 3.06086 3.06146 242 .3 243 -9 245 .6 247 .2 3.09425 3.09482 309540 39597 3-09655 342 -9 344 .6 346 .3 348 .0 349 -7 3.12806 3.12861 3.12916 3.12971 3.13026 M47 -4 1449 .2 1451 .0 1452 -8 '454 -5 3.. 6060 3.16113 ' 3.16166 3.16220 3- '6273 15 16 \l "53 "55 "56 "58 "59 3.06205 3 06265 3.06324 3.06384 3 06444 250 -5 2 5 2 .2 253 -8 255 -5 ?S7 -i 3.09712 3.09769 3 09826 3.09883 3.09941 35' -4 353 -2 354 -9 356 .6 358 .3 3.13081 3.13136 3.13191 3 13246 H56 -3 1458 .1 '463 '.4 3 16326 3 16379 3.16432 316485 3 16538 21 3.06503 3.06562 258 .8 260 .5 3.09998 360 .1 313356 3 "34" 1465 2 1466 9 3.16591 3 16643 2 3 24 1166 1167 3.06681 3.06740 263 .8 265 -5 1: 022! || 3 I.352I 3-I3576 '470 -5 1472 .3 3.16749 3.16802 25 26 1169 1171 1172 "74 .3.06800 3.06859 3.06918 3.06977 3 07036 268 '.8 270 .5 273 -7 3- 0283 1: 0396 3- 453 3- 0510 370 .4 372 -i 373 -9 375 -6 3 13631 3.13686 3-13740 3- '3795 3.13850 1474 .1 M75 -9 '477 -7 1479 -5 1481 .3 3-16855 3 16907 3.16960 3.17066 30 3 1 32 33 34 "77 "79 1180 307095 3-07I54 3.07213 3.07272 3-0733I 275 -4 277 - 1 280 .4 3- 0567 3 0623 3. 0680 3- 0737 3- 0703 377 -3 38 : s 384 .2 3.13904 3-13*59 ? I43 T 3 3.14068 3.14122 1483 .1 1484 .9 ,486 .7 1488 .5 1490 -3 3.17170 3.17223 3-17275 35 36 37 38 39 1.85 1187 1188 1190 191 3.07389 3.07507 3.07566 3.07625 283 .8 285 .5 287 .1 290 .5 3- 0850 3. 0906 3- .963 3. 1076 385 -9 387 -7 389 -4 39 1 -2 392 -9 3 H'77 3.14231 3.14285 3.14^40 3.14^04 1492 .1 M93 -9 '495 -7 1497 -5 1491 -3 3-17380 3-'7433 3 17485 3.17538 3 1759 40 42 43 44 193 '95 196 198 '99 3 07683 3-07742 3.07801 3.07859 3.07018 292 .2 293 .8 295 -5 297 .2 298 .9 3. 1.88 3- 1245 3- 1301 3- '357 396 .4 398 .2 399 -9 401 .7 3.14502 3-'4557 3.14611 3 14665 1501 .1 1502 .9 1504 -7 1506 .5 1508 .4 3.17642 3-17694 3 '7746 3- '7799 3-1785 1 45 201 3 07976 300 .5 3 '413 403 -4 3-14719 1510 .2 3.17903 49 204 206 208 3.08093 3.0815! 3.08210 303 9 305 6 307 .3 I SB 3- '638 406 .9 408 .7 410 .4 3.14827 3.14881 3-H935 15.3 -8 1515 -6 XS 1 ? -4 3.18007 88 50 Si 52 53 54 209 212 214 216 3.08268 3.08326 3.08384 3.08442 3.08501 309 .0 312 .4 3M -i U5 -7 3- '694 3- 1750 3- 1805 3. 1861 3- i9'7 412 .2 4'3 9 4'5 -7 417 .4 419 .2 3.14989 3- '5043 3.15096 3-I5I50 3-15204 1519 .2 1521 .0 1522 , 9 1524 -7 1526 .5 3.18163 3.18215 3.18267 3.18371 55 217 3-08^59 3X9 1 3- T 973 4 EO .9 3-'5258 1528 .3 3.18422 58 59 60 222 224 3 86 75 .3-08791 322 .5 324 .2 3- 2085 3. 2140 3. 2196 424 -4 426 .2 4 2 7 -9 3-15365 3 I54I9 3-I5472 3.15526 1532 .0 '533 -8 '535 -6 3.18=2* 3 '8578 3.18629 3.18681 TABLE VIII A. 6 4 I 5 28 29 n 30- D 3' ! m log*, MM log*. m log m NV log* 1 2 3 4 '537"-5 '539 ; 1541 .1 1542 .9 1544 .8 3.18681 3.18733 3- 8784 3.18836 3.18887 i6 49 ".i 1651 .0 1652 .9 '654 .8 1656 .7 3.21725 3 21775 >2l82 5 3 21875 3.2,924 Sf 1768 '770 1772 3.24665 3-247'3 3.24761 3.24810 3-24858 !88 4 ".o 1890 .1 i8g2 .1 3.27509 3-27556 , 3.27602 j 3.27649 3-27695 ! I 7 8 9 1546 .6 1548 -4 1550 .2 1552 .1 1553 9 3.18939 3.18990 3.19042 3.19093 3-'9M5 16^8 .6 1660 .5 1662 .4 .664 .3 1666 2 3 21974 3.22024 3-22073 3.22123 5.33173 53 '778 a 3-24906 3 24954 3.25002 3.25050 3.25098 1894 .2 l8 9 6 .2 !8 9 8 .2 lOXKJ .3 1902 .3 3.27742 1 3 27788 3-27835 3-2788, 3 27928 n '555 -8 '557 -6 3.19196 1668 .1 3.22222 58 3.25146 IW -3 1906 .4 3-27974 3.28020 12 13 14 '559 -5 1561 .3 I-' ! .- 3 19299 3-I9350 3 19401 I6 7 , .9 '673 .8 '675 -7 3.22371 1790 '792 3.25289 3 25337 1908 .4 IgiO .4 ,912 .5 3.28067 3.28,13 3.28159 ! 3 17 18 '9 1565 .0 1566 .9 1568 .7 '570 .5 '572 4 3- '9452 3-'9W3 3-19554 3.. 9606 3-i9 6 57 1677 .6 679 -5 1681 .4 '683 .3 l68 5 .2 3.82470 3.22519 3-22568 7. 226,8 322667 '794 ,796 1798 1800 1802 3-25385 3-25433 3-25480 3.25528 3-25576 '9'4 -5 1916 .5 1918 .6 1920 .6 ,922 .7 3 28206 3.28252 3-28298 i 328344 328390 20 23 2 4 574 -3 1576 .1 '578 .0 '579 -8 1581 .7 3.19708 3-19759 3.198.0 3.19861 3 19912 l68 7 .2 689 .1 692 .9 1694 .8 3.22 7 l6 3.22766 3.22815 3.220,3 1804 ,806 1808 ,809 i8 3.25624 3.25671 3 257 '9 3.25766 3-258,4 1924 .7 1926 .8 ,928 .8 1930 .9 '93 2 9 3-28437 ' 3.28483 3.28529 3-28 =75 3.28621 25 26 27 29 1583 -5 '585 -3 '587 -2 '589 .1 1590 .9 3.19962 3.20013 3.20064 3 20166 1696 .7 1698 .6 1700 .5 1702 .5 1704 .4 3 22q6 3 3.23012 3 23061 3.231,0 3.23159 18. 3 l8l5 l8lg t82I 3.25802 3.25909 3-25957 3.26004 3 260^1 '934 9 1937 .0 1939 .1 1941 .1 1943 .2 3.28667 3.28713 3-28759 3.28805 3.288 5I 30 3 1 32 33 34 3S 3 '59 6 -5 '598 .3 1600 .2 3 20216 3.20267 3.703,8 3-20369 3 20419 1706 .3 708 .2 1712 .1 T 7 I 4 .0 3.23208 3.'3257 3-23306 3 23355 3-23404 '823 8? 5 .8 1.829 '.8 1831 .8 3.20099 3.26,46 3.26,94 3.26241 3.26288 '945 -2 '947 -3 '949 -3 '95' -4 3 28897 3-28943 3.28988 3.29034 35 1,6 9 39 1602 .1 !60 4 .0 1605 .9 1607 .7 1609 .6 3 20470 3.20520 3-20571 3.20621 3.20672 7'5 -9 7'7 -9 719 -8 721 .7 3-23453 3-2350I 3-2355 3-23590 3.23648 '835 -8 1837 .8 ,839 .8 3-26336 3-26383 3.26430 3.26477 3.26524 '955 -5 '957 -6 ,959 .6 1961 .7 1963 -8 3.29126 3 29172 3-29217 3.29263 3.29309 40 4' 42 43 44 1611 .5 1613 .3 I6l 5 .2 1617 .1 1619 .0 3.20722 3.20772 3.20822 3.20873 3.20924 725 -6 727 -5 729 -5 73' -5 733 -4 3.23697 3-23745 3-23794 3 23843 3 2380, '843 .8 '845 .8 1847 .8 1849 .8 1*51 .8 3.26571 3.266,9 3.26666 3.267.3 3.26760 ,965 .8 1967 .9 1970 .0 1972 .0 '974 -i 3-29354 3.29400 3-29446 3-29491 3.29537 45 46 47 48 49 1620 .8 l6?2 . 7 1624 .6 1626 . 5 1628 3 3-20974 3.21024 3-2 075 3-2 25 3-2 -5 735 -3 737 -2 739 -2 741 .2 743 ' 3.23940 3-23988 3.24037 3.24086 3 24'34 .853 -8 '855 .8 18.7 .8 '859 -8 1861 .9 3.26807 3-26854 3.2690, 3.26948 3.26905 1976 .2 ,978 .2 1980 .3 1982 .4 1984 .5 3.29582 3 29628 3.29673 3.297,9 329764 5' 52 53 I6 3 .2 ,632 .. 1634 .0 '635 -9 1637 .7 32 25 3-2 75 3-2 25 3-2 75 3-2 25 '745 ' 1747 -o 1749 .0 1750 .9 '752 .8 3.24,82 3.2423, 3.24279 3.24328 3 24176 1863 9 1865 .9 ,867 .9 1869 .9 ,87, .0 3. 7042 3. 7088 3- 7J35 3- 7'2 3 7229 1086 .5 1.88 .6 1090 .7 1902 .8 1994 .8 3 298,0 3-29855 3.29900 3-29946 3.29991 55 1639 .6 3-2 75 '754 -8 3-24424 1873 -9 3.27276 19.16 .9 3.30036 57 58 59 1641 .5 '643 -4 '645 -3 1647 .2 3-2 575 3-2 625 3-2 675 '7.-8 .7 1760 .7 1762 .6 3245" 3.24^69 3.246,7 ,878 .0 1880 .0 1882 .0 3-27369 3.27416 3.27462 2005 .3 3.30127 3-30172 3.30217 60 1649 .1 3.21725 1764 .6 1 3 24665 T88 4 -o 3-27509 20->7 .4 3.30262 64? TABLE YIII B. TABLE VIII C. t n lo** , n lo R o'(.oo .00 .00 4-9706 6.1747 6.8791 7-3788 22 m o 3 40 50 2". I 9 2 .25 2 .32 2 -39 2 . 4 6 2 -54 0-3396 1 0.3527 0.3057 0.3786 o-39'5 0.4042 .04 .06 .09 '3 8.0832 8. 3^09 8.5829 8-7875 8 975 9.1360 10 30 40 50 2 !69 2 -93 0.4,68 o 4 2 93 0.4418 0.4541 o 4664 30 3 3 2 3 '36 9.3580 9.4262 9.4917 9 5549 9 6158 20 30 40 50 3 -'8 3 -27 3 -36 3 -45 0.4907 0.5027 0.5 '46 0.5264 0.5382 o 5499 >5 o 30 47 49 54 56 9.6747 9.6939 9.71^8 9-73!6 9 7=;o2 20 3 -'4 3 74 3 84 3 -94 056,5 0-57.5" 0.5845 50 59 9 7686 50 4 -'5 o 6,84 16 o 3 40 BO -67 .69 .72 75 9.7867 9.8047 9 8225 9.8402 9.8576 9 8749 26 o 10 3 40 4 .26 4 -37 4 48 4 .60 4 -72 4 83 o 6296 0.6407 0.65.7 o 6626 0.6735 0.6843 20 30 4 5 .78 .81 .84 .91 95 9.8920 9.9089 9-9257 9 9423 9.9588 9 975 T 27 o 3 4 5 4 -96 5 -08 5 - io 5 60 0.6951 0.7057 0.7,64 0.7269 18 o .98 9 99'3 28 o 5-7,3 o 7,82 20 30 40 50 .06 .09 :',S 0.0231 0.0388 0.0544 o 0698 3 40 5 6 : 5 6 .30 6 .44 0.7787 o 7889 0.7990 o P.-OO 19 o 10 30 40 50 .22 .26 30 35 .40 44 o 0851 0.1153 o 1302 0.1450 0.1597 29 o 20 3 40 50 6 -59 6 .75 6 .90 7 .06 7 .22 7 -38 08,90 0.8290 0-8389 0.8487 o.8 5 8s 0.8682 20 49 0.1742 30 o 7 -55 0.8778 20 30 40 50 .60 65 .70 .76 0.2029 2F 7 o 2311 0.2450 30 50 7 89 8 .06 8 .24 8 .42 0.8970 09065 0.9160 0.9254 21 40 87 03 99 .06 0.2589 0.2726 0.2862 0.2997 0.3131 o 3264 20 30 4 50 8 .60 8 -79 8 .98 9 -'7 9 -37 9 -57 0-9347 o 9440 0-9533 0.9625 0.9716 o 9807 22 ro . 2 .19 0.3396 52" cf 9 -77 o 9^98 L 1 86400. Rate. 9 999 6985 SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OF JOTTX WILEY ,t SOXS, NEW YORK. CIIAPMAX & HALL, LIMITED. ARRANGED UNDER SUBJECTS. Ivescriptive circulars sent on application. Books market! with an asterisk are sold at ,irt prices only. All books are bound in cloth unless otherwise state.l. AGRICULTURE. Armsby's Manual of Cattle-feeding 12mo, $1 1% Downing's Fruits and Fruit-trees of America 8vo, 5 00 Grotenfelt's Principles of Modern Dairy Practice. ( Woll.) . . 12mo, 2 00 Kemp's Landscape Gardening 12mo, 2 50 Maynard's Landscape Gardening as Applied to Home Decora- tion 12mo, 1 50 Stockbi idge's Rocks and Soils 8vo, 2 50 \Vi ill's HandlKK>k for Farmers and Dairymen 16mo, 1 50 ARCHITECTURE. 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(Du Bois.) . .8vo, r> Oil Whitham's Steam-engine Design Svo, 5 00 Wilson's Treatise on Steam-boilers. (Flather.) Itiino, 2 50 Wood's Thermodynamics, Heat Motors, and Refrigerating Machines 8vo, 4 00 13 MECHANICS AND MACHINERY. Barr's Kinematics of Machinery 8vo, 2 50 Bovey's Strength, of Materials and Theory of Structures. .8vo, 7 50 Chordal. Extracts from Letters * 12mo, 2 00 Church's Mechanics of Engineering 8vo, 6 00 Notes and Examples in Mechanics 8vo, 2 00 Compton's First Lessons in Metal- working 12mo, 1 50 Compton and De Groodt. The Speed Lathe 12mo, 1 50 Cromwell's Treatise on Toothed Gearing 12ma, 1 50 " Treatise on Belts and Pulleys 12mo, 1 50 Dana's Text-book of Elementary Mechanics for the Use of Colleges and Schools 12mo, 1 50 Dingey's Machinery Pattern Making 12mo, 2 00 Dredge's Record of the Transportation Exhibits Building of the World's Columbian Exposition of 1893 4to, half mor., 5 00 Du Bois's Elementary Principles of Mechanics: Vol. I. Kinematics 8vo, 3 50 Vol. II. Statics 8vo, 4 00 Vol. III. Kinetics 8vo, 3 50 Du Bois's Mechanics of Engineering. Vol. I Small 4to, 10 00 Durley's Elementary Text-book of the Kinematics of Machines. (In preparation.) Fitzgerald's Boston Machinist 16mo, 1 00 Flather's Dynamometers, and the Measurement of Power. 12mo, 3 00 " Rope Driving 12mo, 2 00 Hall's Car Lubrication 12mo, 1 00 Holly's Art of Saw Filing 18mo, 75 * Johnson's Theoretical Mechanics 12mo, 3 00 Jones's Machine Design: Part I. Kinematics of Machinery 8vo, 1 50 Part II. Form, Strength and Proportions of Parts 8vo, 3 00 Kerr's Power and Power Transmission. (In preparation.) Lanza's Applied Mechanics 8vo, 7 50 MacCord's Kinematics; or, Practical Mechanism 8vo, 5 00 " Velocity Diagrams 8vo, 1 50 Merriman's Text-book on the Mechanics of Materials 8vo, 4 00 * Michie's Elements of Analytical Mechanics 8vo, 4 00 Reagan's Locomotive Mechanism and Engineering 12mo, 2 00 Reid's Course in Mechanical Drawing 8vo, 2 00 " Text-book of Mechanical Drawing and Elementary Machine Design 8vo, 3 00 Richards's Compressed Air 12mo, 1 50 Robinson's Principles of Mechanism 8vo, 3 00 Sinclair's Locomotive-engine Running and Management. .12mo, 2 00 Smith's Press-working of Metals 8vo, 3 00 Thurston's Treatise on Friction and Lost Work in Machin- ery and Mill Work 8vo, 3 00 " Animal as a Machine and Prime Motor, and the Laws of Energetics 12mo, 1 00 Warren's Elements of Machine Construction and Drawing. .8vo, 7 50 Weisbach's Kinematics and the Power of Transmission. (Herrman Klein.) 8vo, 5 00 Machinery of Transmission and Governors. (Herr- (man Klein.) 8vo, 500 Wood's Elements of Analytical Mechanics 8vo, 3 00 " Principles of Elementary Mechanics 12mo, 1 25 " Turbines 8vo, 2- 50 The World's Columbian Exposition of 1893 4to, 1 00 14 METALLURGY. Egleston's Metallurgy of Silver, Gold, and Mercury: Vol. I. Silver 8vo, 7 50 Vol. II. Gold and Mercury 8vo, 7 50 Keep's Cast Iron. (In preparation.) Kunhardt's Practice of Ore Dressing in Lurope 8vo, 1 50 Le Chatelier's High-temperature Measurements. (Boudouard Burgess.) 12mo, 3 00 Metcalf s Steel. A Manual for Steel-users 12mo, 2 00 Thurston's Materials of Engineering. In Three Parts 8vo, 8 00 Part II. Iron and Steel 8vo, 3 50 Part III. A Treatise on Brasses, Bronzes and Other Alloys and Their Constituents. 8vo, 2 50 MINERALOGY. Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 2 50 Boyd's Resources of Southwest Virginia 8vo, 300 Map of Southwest Virginia Pocket-book form, 2 00 Brush's Manual of Determinative Mineralogy. (Penfield.) .8vo, 4 00 Chester's Catalogue of Minerals 8vo, paper, 1 00 Cloth, 1 25 Dictionary of the Names of Minerals 8vo, 3 50 Dana's System of Mineralogy Large 8vo, half leather, 12 50 First Appendix to Dana's New " System of Mineralogy." Large 8vo, 1 00 Text-book of Mineralogy 8vo, 4 00 Minerals and How to Study Them 12mo, 1 50 " Catalogue of American Localities of Minerals . Large 8vo, 1 00 " Manual of Mineralogy and Petrography 12mo, 2 00 Egleston's Catalogue of Minerals and Synonyms 8vo, 2 50 Hussak's The Determination of Rock-forming Minerals. (Smith.) , . . Small 8vo, 2 00 * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests " 8vo, paper, 50 Rosenbusch's Microscopical Physiography of the Rock-making Minerals. (Idding's.) 8vo, 500 * Tillman's Text-book of Important Minerals and Rocks . . 8vo, 2 00 Williams's Manual of Lithology 8vo, 3 00 MINING. Beard's Ventilation of Mines 12mo, 2 50 Boyd's Resources of Southwest Virginia 8vo, 3 00 " Map of Southwest Virginia Pocket-book form, 2 00 Drinker's Tunneling, Explosive Compounds, and Rock Drills 4to, half morocco, 25 00 Eissler's Modern High Explosives 8vo, 4 00 Goodyear's Coal-mines of the Western Coast of the United States 12mo, 2 50 Ihlseng's Manual of Mining 8vo, 4 00 Kunhardt's Practice of Ore Dressing in Europe 8vo, 1 O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 2 00 Sawyer's Accidents in Mines 8vo, 7 0( Walke's Lectures on Explosives 8vo, 4 00 Wilson's Cyanide Processes 12mo, 1 Wilson's Chlorination Process 12mo, 1 50 15 Wilson's Hydraulic and Placer Mining 12mo. 2 00 Wilson's Treatise on Practical and Theoretical Mine Ventila- tion 12mo. 1 25 SANITARY SCIENCE. Folwell's Sewerage. (Designing, Construction and Maintenance.) Svo, 3 00 " Water-supply Engineering Svo. 4 00 Fuertes's Water and Public Health 12mo. 1 50 Water-filtration Works 12mo.. 2 50 Gerhard's Guide to Sanitary House-inspection IGmo, 1 00 Goodrich's Economical Disposal of Towns' Refuse. . .Demy Svo, 3 50 Hazen's Filtration of Public Water-supplies Svo, 3 00 Kiersted's Sewage Disposal 12mo, 1 25 Mason's Water-supply. (Considered Principally from a San- itary Standpoint Svo. 5 00 " Examination of Water. (Chemical and Bacterio- logical.) 12mo, 1 25 Merrimaivs Elements of Sanitary Engineering Svo. 2 Of) Nichols's Water-supply. (Considered Mainly from a Chemical and Sanitary Standpoint.) ( 1883.) Svo, 2 50 Ogden's Sewer Design 12mo, 2 00 Richards's Cost of Food. 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Mounted chart, 1 25 " Fallacy of the Present Theory of Sound 16mo, 1 00 Ricketts's History of Rensselaer Polytechnic Institute, 1824- 1894 Small Svo, 3 00 Rotherham's Emphasised New Testament Large Svo, 2 00 Critical Emphasised New Testament 12mo, 1 50 Steel's Treatise on the Diseases of the Dog Svo, 3 50 Totten's Important Question in Metrology Svo, 2 50 The World's Columbian Exposition of 1893 4to. 1 00 Worcester and Atkinson. Small Hospitals, Establishment and Maintenance, and Suggestions for Hospital Architecture, with Plans for a Small Hospital 12mo, 1 25 HEBREW AND CHALDEE TEXT-BOOKS. Green's Grammar of the Hebrew Language Svo, 3 00 " Elementary Hebrew Grammar 12mo, 1 25 " Hebrew Chrestomathy Svo, 2 00 Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. (Tregelles.) Small 4to, half morocco. 5 00 Letteris's Hebrew Bible Svo, 2 25- 16