ON THE RECURRENCE OF SOLAR ECLIPSES WITH TABLES OF ECLIPSES FROM B.C. 700 TO A.D. 2300 BY SINHJN FHEVVCCHWES pnomsson, U. s. NAVY V SUPERINTENDENT OF THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC I HHOSSTW ‘mam 311003 ‘Ifmmoo em‘ 2 190 ‘LLISIRJIAINQ M19 xhnéw CJHVHHTT 3 WASHINGTON BUREAU OF NAVIGATION, NAVY DEPARTMENT 1879 PREFACE. The following paper presents a new theory of the recurrence of solar eclipses, founded on some hitherto unnoticed properties of the I8-year-eclipse cycle. This ‘ theory has been utilized in’ the formation,of tables whereby the solar eclipses of any class which have occurred during the past twenty-five centuries, ‘or are to occur during the next five centuries, may be determined and approximately computed with great rapidity. The tables are founded on the mean motions and other elements of the sun and ‘moon given in Hansen’s Tables, the mean motion of the moon and of its nodes being corrected to accord _With the results deduced in the author’s Researches on the Motion of the Moon. In the concluding section, the eclipses most remarkable for the duration of total phase are pointed out, and the conditions for their occurrence briefly discussed. A considerable part of the work of constructing the tables has been performed by Mr. John Meier, assistant in this office. - 3 TABLEJ OF CONTENTS. General Theory of the Recurrence of Eclipses - Data for Tables of Eclipses - - - - - - — Recurrence of Remarkable Eclipses - - - - _- Tables of Eclipses B. C. 700 to A. D. 2300 - - Computation of Eclipse of Thales as Example Page. THE RECURRENCE OF SOLAR ECLIPSES. § I . GENERAL THEORY. It has been known from ancient times that eclipses both of the sun and moon generally repeat themselves in a cycle of 18 years and II or 12 days, known as the Saros. This cycle is due to the circumstance that 242 revolutions of the moon rel- atively to either of its nodes require nearly the same period with 19 revolutions of the sun relatively to the same node. The time required for either of these returns is 6 58 5% days. Hence, if we note the relative positions of the sun and moon at any mo- ment, and then count forward through this period, we shall, at the end of it, find them in nearly the same position, both relative to each other and relative to the node. If we start from the centre of an eclipse, when the two bodies are nearly in the same straight line, we shall, at the end of the period, find another eclipse very similar in its character. This relation affords a very simple and easily applied method of finding the series of eclipses which occur during any period of 18 years, from those which occurred during the cycle previous There are, however, two remarkable chance relations connected with the Saros, which, so far as I know, have never been remarked, and without which the period would not have served the purpose of foreseeing eclipses so well as it actually does. The cycle takes account only of the mean motions of the sun and moon. But in con- sequence of the eccentricity of the orbits, the sun may be 2 degrees on either side of its mean place and the moon 5 degrees. The relative position of the two bodies may therefore vary 7 degrees from their mean position at any time; this extreme variation would change the time of an eclipse by half a day and the distance from the node at which it occurred about zdegrees. If the corresponding eclipses in two successive cycles were subject to these independent variations, their circumstances might differ so widely that the recurring eclipse would differ considerably from its predecessor, and might be nearly a day later or earlier than the mean length of the cycle in its recurrence. A partial eclipse might fail entirely to recur, and a total one might become partial at the first_recurrence and then total again at the second one. But, as a matter of fact, the irregularities of this class are reduced almost to nothing by two other remark- able relations. At the end of a Saros, not only are the sun, the moon, and the node found nearly in their original relation, but the mean anomaly of the moon has also the same value to less than 3.. degrees, and the mean anomaly of the sun to some 12 There is no a priori reason that this should be the case: it arises only from degrees. 7 8 RECURRENCE OF SOLAR ECLIPSES. the fact that 18 years is a close multiple, not only of the times of revolution of the sun and moon, but also of the times of revolution of the moon’s node and perigee. The following is a more exact statement of the changes at the end of the Saros. Taking as a period the time required for 22 3 lunations, the changes in the elements at the end of the period will be as follows :— In the argument of latitude, - - - - - - - — 28’.6 In the moon’s mean anomaly, — - - — - - - — — 20.831 In‘the sun’s mean anomaly, - - - - - - - - + Io°.494 In the distance of the lunar perigee from the node, - + 2°. 3 5 3 In the distance of the solar perigee from the node, - — 100.971 In consequence of the minuteness of these changes, not only the mean place of the 1noon, but all its larger inequalities, will return nearly to their original values at the end of the period. This will hold true, not only with respect to the time of the eclipse, but also with respect to its character, since the parallax and semi-diameter of the moon must also return nearly to their original values. If the eclipse is of a remarkable character with respect to duration, the corresponding ones of succeeding cycles will be of the same character. ‘ An interesting illustration of this fact is found in a series of total eclipses now in progress, namely, those of 18 50, 1868, 1886, etc., in which the duration of totality is greater than in any others which l1ave occurred for several centuries. This series will be investigated in the course of the present paper. Owing to the mean retrocession of 28’ from the node in each cycle, the corre- sponding eclipses in successive cycles are subject to a progressive change. A series of such eclipses commences with a very small eclipse near one pole of the earth. Grad- ually increasing for about eleven recurrences, it will become central near the same pole.‘ Forty or more central eclipses will then recur, the central line moving slowly toward the other pole. The series will then become partial, and finally cease entirely. 'l‘he entire duration of the series will be more than a thousand years. A new series commences, on the average, at intervals of thirty years. It follows from this that all eclipses may be divided into sets, theseparate eclipses- of each set being separated by intervals of one 18-year cycle, and extending through sixty or seventy cycles. Moreover, from the elements of the central eclipse of each set, those of any other of the same set may be readily found by applying the changes corresponding to the number of intervals which separate it from the central one. It is now proposed to utilize this circumstance by the formation of a series of tables, by which the approximate elemen‘ts of any solar eclipse between the years B. C. 700 and A. l). 2 300 may be found with a few minutes’ calculation, and by which any such eclipse occurring during this period may be promptly identified. The principles on which the most important of these tables are constructed may be readily compre- hended by a conception of movable conjunction points reached in the following manner. Let us suppose the mean motions, n and 72’, of two bodies, planets for instance, revolving round a common centre, to be so related that i'n—in’:o, RECURRENCE OF SOLAR ECLIPSES. 9 i and i’ being integers. Then, i’ revolutions of the first will require the same period as i revolutions of the second, so that at the end of this period, which we may call P, they will have returned to their original positions. I)uring the period P they will have been in conjunction i — i’ times at the same number of equidistant points of either orbit. Every subsequent mean conjunction will occur at these same points. We shall call them conjunction points, and shall represent their number, i — i’, by 1’. ‘If we suppose these points to be numbered, in the order of longitude, 0, 1, 2 . . . . V — I, and suppose the two bodies to start out from the point 0, the number of revolutions which each body must severally make to reach the point 12 will be found by solving the indeterminate equation i’x—iy.-_-;|-;p. at will then be the entire number of revolutions of the one planet and 3/ that of the other before the required conjunction will occur; that is, the one planet will then have passed over 1/ cc +pinterva1s between the conjunction points, and the other over’ 3’ y + p. The condition that these two quantities shall be in the ratio i: i’ gives the above indeterminate equation. In order to avoid the ambiguous sign, we may suppose n > n’, which will make i> i’. This will make the equation i'x—iy:p. In what precedes, we have supposed the mean motions of the two bodies to be 4 exactly in the ratio of the entire numbers i and i’. This is never the case in nature, if we reckon the mean longitudes from a fixed point of departure; but we may always assign such a uniform progressive motion to this point that the condition shall be ful- filled. Let us put is for the progressive motion required. The mean motions relative to the moving departure point will then be n — is and n’— 1.: respectively. The condi- tion that these shall be in the ratio i: i’, or A "-7? n’—k ‘I7 “Is. gives in’-i’n in’— i’n Ii3Z—."—'.—,'—:--—--'---- i—i V The conjunction points, being fixed relatively to the departure point, will have this same mean motion 1:; that is :— By assigning to the av conjunction points the uniform -mcan motion is, the conjunctions of the two bodies will always take place at these points. ' This conception of movable conjunction points is of great assistance in represent- ing and investigating the relations of the two bodies through many revolutions. For instance, in the case of Jupiter and Saturn, taking i: 5 and i’ : 2, there will be three conjunction points having a direct mean motion of 489" per annum relative to a fixed equinox. Their successive passages through a fixed point occur at intervals of A R——-2 I0 RECURRENCE OF SOLAR ECLIPSES. 88 3 years, and we may consider the great inequality between the two planets as depending on the position of the conjunction points relative to their perihelia. Theoretically, the values of i and i’ may be regarded as entirely arbitrary. But to obtain the advantage of the conception, we take them as nearly as practicable in the ratio of the mean motions. Even with this limitation we have a choice of sys- tems, an increase in the assumed values of i and i’ having the disadvantage of increas- ing the number of points to be considered, and the advantage of diminishing their mean motion. The most advantageous systems will of course be found by devel- oping the ratio of the mean motions as a continued fraction, and taking the successive converging fractions which approach to the ratio. Between two such successive sys- tems the following relation subsists :—' _ The interval between the successive transits of the conjunction points of one system over any one of the next higher, and therefore more slowly moving system, is equal to the time required for the conjunctions to occur at all the points of this latter system. ‘ Commencing with the higher system, and supposing the mean motions n and n’ to be counted from a point of this system, and to be in the ratio j : j’, we shall have j’n—jn’:o. The mean motion of the points of the next lower system relatively to the higher one will then be, in’—i’n_ ls: i—i' ’ thetime required for a. complete revolution of the lower system will be, 3_7_r__36o° (i—i’)_ k_ "" in’—i’n ’ and the intervals between successive passages of its i— i’ points over a fixed point of the other system will be, 2 7r _ 360° 7»? "' 572277;‘ Since n and 33' are in the ratio j : j’, we may put 72 : aj, n’: aj', which will make in’— i’ n : (ij’— i’j) a. But, by the properties of continued fractions, the value of the coefficient of a in this expression is :1: 1. Hence, the sign being indifferent, as expressing only the direction of the motion, the interval between successive passages of the conjunction points becomes 0 360 C! RECURRENCE OF SOLAR ECLIPSES. I I In order that the conjunctions may occur at all points of the higher system, it is necessary that the one planet should make and the other j’ revolutions. The time required for this will be, — 6 O0 _ 360° the same as the interval just found. Let us now apply these methods to the problem now under consideration, that of the recurrence of solar eclipses. Let us put g, the mean anomaly of the moon; g’, that of the sun; 2 an, the distance of the lunar perigee from the node; to’, that of the solar perigee from the moon’s node; T, the number of Julian centuries after 1800. 6 Applying to the elements given by Hansen (Tables de la I/zmle, p I 5) the corrections to the mean longitude and the longitude of the node given in my Researches on the Motion of the Moon, p. 268 and p. 274, the numerical expressions for g, co, g’, and co’ will become :— g : 1 10° 19’ 32”.50 + (I325’ + 715807”.98) T + 45”.58 T’+ 0”.050 T3 as : 192° 7’ 21”.91 +( 16’ + 87551 2”.07) T — 44”.32 T’- 0”.044 T3 g’: 0° 24’ 28”.22 +( I00"«—- 3392”.I8)T-— 0”.56 T’ co’: 246° 13’ 50”.28 ( 5’ + 489088”.09) T — , 6”.52 T'— 0”.007 T3. Epoch, + 1800.0, Jan. 0, Greenwich mean noon. In dealing with a subject of this kind, the entire revolution is a more convenient unit than the angular denominations usually adopted. We therefore transform these angles into revolutions and fractions, with the following results :— g : K30646026 + 1325’.55232097 T + 0’.00003517 T’ + 0’.000000039 T3 ~ 60 = ’-53367431 + I6’-67554944 T — 0'.00003420 T’ — 0’.000000034 T3 g’:’.00113289-}- 99'.99738258 T — 0“.00000043 T’ w’='-68397398 + 5’-37738278 T — 0’.00000503 T’ — 0’.000000005 T3. I2 RECURRENCE OF SOLAR ECLIPSES. In the construction of the present tables we shall use the Julian calendar, it being -more convenient to change the dates from this calendar to the Gregorian than to take account of the complexities of the latter. We shall therefore take, as our fundamental epoch, 1800, Jan. 1, Greenwich mean noon of the Julian calendar, : 1800, Jan. 1 2, Greenwich mean noon of the Gregorian calendar. Transferring to this epoch, the constants of the four principal elements will become, go : 0“.741 96000, 60° = 0’-5391 5294, g’, : 0“.03398624, co’, : 0“.68 5 74068, while the coeflicients of the powers of T will remain unaltered. We shall count the timefrom this epoch in Julian centuries or in equal Julian years of 36 5.2 5 days each. This reckoning of time will hereafter be called a fictitious one to distinguish it from the civil reckoning. The expression for the mean distance of the two bodies from the ascending node of the moon’s orbit, which we shall represent by u and u’, putting — l u : .9 "l" my “I: .9,"l" 59/1 will now be u : 0'.281 1 1 2,94 + 1342’.227870,41 T + 0,96 T’ + 0,005 T3, u’: 0".719726,92 + 105’.374765,36 T - 5,46‘ T’ — 0,005 T3. The comma in these expressions is used to cut off six places of decimals. If we differentiate these expressions with respect to T, and then put T : 0 and T : - ; 5, we have the following expressions for the mean motions from the node at the epochs — 700.0 and + 1800.0 :— Epoch, — 700.0, + 1800.0, Mean motion of u ,: ,u, 1342'.227832 1342“.227870,41 Mean motion of u’,: ,u’, 105’.375028 105’.374765,36. Developing the ratios of these two quantities into a continued fraction, we have, For — 700.0, For + 1800.0, '2/;—‘,:12+v:»-+1 I 5,s:12+~:—+_1w I ‘ ’+Y+3 I ' ’+f+5 I 4+8+£ 1 4+i”+I 3+7 3 §+%+1 'i+l. 2 , through the node. RECURRENCE OF SOLAR ECLIPSES. 13 The several converging fractions, so far as it is worth while to carry them, are :— FOT '—' 700.03 “I12, ‘:23, ‘if, 914;) 671 ; , T etc. . £3 _I_§ 51. 343 .777 4..-.I_2. .7. 4904 For +I800.0. I7 I 1 37 47 _I97 617 324; etc- 2I492, which will give Of these systems the one which offers the greatest advantages is us 22 3 conjunction points, each having (relative to the node) a retrograde motion such that it would, if constant, make a revolution in about I‘4,ooo years. This time, how- ever, varies with the mean motion of the moon and its node. From the formulae for k, already given, we find, A Epoch, — 700.0: k.-_-_—.0007050, Epoch, + 1800.0: k:—.0007338. The distance apart of two consecutive conjunction points is, I‘ I . K : 553, : 0“.004484304 : I °.6I4350; and they pass the node at the following intervals :— At the epoch — 700.0, interval: 6332607 : 785 lunations. At the epoch + 1800.0, interval : 615111 I : 756 lunations. Between these two fundamental epochs there will be 40 passages of conjunction points We next investigate the positions of the conjunction points at the first of these epochs. We note that a conjunction (new moon) occurred 7“.01670 before the first epoch, when w u : u’: 0”.327024 : 73 K — 0’.000330 I We conclude that the node is very near the 73d conjunction point back from that at which the new moon just found occurred, and that this point passed the node about {th of an interval, or 42 years before the epoch. We shall take this as the zero con- junction point, and count the others in the order of longitude. Their successive pas- sages across the ascending node will then occur at the times shown in the left-hand half of the following table. The intervals between consecutive passages, as just shown, will diminish from 6331607 at — 700.0 to-61311 I I at + 1800.0. I4 RECURRENCE OF SOLAR ECLIPSES. Passages of Conjunction Points through Nodes. Conj. Ascend-. Conj. Ascend. Conj. Descend. Conj. Descend. Point. Node. Point. Node. Point. Node. Point. Node. y. y. o y. y. 0 — 704.82 25 865.9 112 — 673.03 ’ 137 896.94 1 — 641.25 26 927.96 113 — 609.50 138 958.94 2 —— 577.74 27 989.92 114 — 546.02 - 139 1020_.87 ; 3 — 514.29 28 1051.82 115 — 482.60 140 1082.74 I 4 — 450.90 29 1113.66 116 — 419.24 141 1144.55 ! 5 — 387.58 30 1175.44 117 -- 355.95 142 1206.30 l 6 - 324.32 31 1237.15 118 — 292.72 143 1267.97 I 7 —- 261.12 32 1298.80 119 — 229.55 144 1329.59 : 8 —- 197.98 ~ 33 1360.39 120 —- 166.44 145 1391.15 9 - 134.91 ' 34 1421.92 121 — 103.40 146 1452.66 10 — 71.90 4 35 1483.39 122 — 40.42 ' 147 1514.09 , 11 — 8.95 36 1544.79 123 22.50 148 1575.46 7 12 53.94 37 1606.13 124 85-36 , 149 1636.77 l 13 116.77 i 38 * 1667.41 125 148.16 150 1698.02 14 179.54 f 39 1728.63 126 210.90 151 1759.20 15 . 242.25 40 1789.78 127 273.58 152 1820.32 1 16 304.90 g 41 1850.87 128 336.20 ‘ 153 1881.38 17‘ 367.49 42 1911.90 129 398.75 154 1942.39 18 430.01 43 1972.87 130 461.24 155 2003.32 19 492.47 44 2033.77 131 523.67 156 2064.19 20 554.87 45 2094.61 132 586.04 157 2125.00 l 21 617.21 E 46 2155.39 133 648.34 158 2185.75 22 679.48 i 47 2216. 11 134 710. 59 159 2246.42 23 741 .69 48 2276. 77 I35 772- 76 160 2307.03 j 24 803.84 49 2337.37 136 834.88 1 161 2367.58 ‘ I At the first of the above epochs the descending node will fall between the 1 1 1th and the 112th conjunction point, and the passages will occur midway between those of the ascending node. These times are shown in the right—haud portion of the table. A new moon occurs at each conjunction point at equal intervals of 223 lunations; and, according to the system adopted, eclipses are classified according to the conjunc- tion point at which they occur, those of each series being separated by intervals of 223 lunations. The middle eclipse of each series will be that which occurs nearest the time when the conjunction passes the node; and we now wish to find when these suc- cessive middle eclipses occur. We have just seen that the sun and moon were together at the 73d conjunction point on the 7th day before — 700.0. We wish to find when they were together at the zero point, which is 150 points farther advanced. Each new moon occurs at an interval of 19 conjunction points past the preceding one; therefore, if 3' be the number of lunations required, we must have n 192' E 150 (mod. 223). This gives :— 2'-._-.137, ori:—86. RECURRENCE OF SOLAR ECLIPSES. I 5 The required conjunctions at the zero point are therefore the I 37th following and the 86th preceding that of — 7003’ — 7“, from which we started. The latter, of course, is , nearest the node. The number of lunations between a conjunction at any point and the first follow- ing conjunction at the next point in order is given by the congruence, I92’; I (mod. 232), the solution of which is, '5 : 47. We shall therefore have a conjunction at point n + I at an interval of 47 lunations after any conjunction at point n, whatever be n. The intervals between consecutive middle eclipses must therefore be of the form, 223 93 + 47; a: being an integer. The mean interval must be the same as that between two pas- sages of the node over a conjunction point; that is, 785 lunations about the epoch — 700.0 and 756 lunations about the epoch + 1800.0. The actual intervals are there fore found by putting avg: 3 and :13 : 4, so that they must be either 716 or 939 lunations. §2. DATA FOR TABLES OF ECLIPSES. When the possible solar eclipses which may have occurred during any period are to be investigated, it is convenient to have tables by which we can at once find the limits of time within which their occurrence is possible. A central eclipse can occur only within eleven or twelve days of ‘the time when the sun passes the moon’s node, and therefore only at the new moon nearest such passage. A partial eclipse may occur at any time within eighteen days of such passage: there may, therefore, be two partial eclipses; one at the new moon preceding, and the other at the new moon fol- lowing, the passage of the sun through the node. Our first problem is, therefore, to find the dates of passage of the sun through the nodes of the moon’s orbit, which gives us at once the middle of what we may call an eclipse season. This is effected by two tables, of which the first gives the dates at which the ascending node has the same longitude that the sun has at the beginning of the fictitious Julian year, and the sec- ond the changes in the times of passage for the 19 years following these dates. The data for the construction of the first table are as follows. Hansen’s longi- tude of the node, corrected, is, A 6 : 33° 16’ 3 1”.I5 — 6962929”.6I T + 8”.I9 T’ + 0”.007 T3, Epoch, + 1800.0, Gregorian calendar. I6 RECURRENCE OF SOLAR ECLIPSES. Reduced to the Julian epoch, 12 days later, it becomes approximately, 9 : 32° 38’.47 — 1 16048’.827 T + o’.136 T’. The sun’s mean longitude at the beginning of the fictitious Julian year is, 291° 44’ + 46’.13 T ;1- 0’.021 T’. The distance of the node from the chosen departure point is, 100° 54’ — 1 16094’.96 T + 0’.1 16 T’. The annual motion and period are, Epoch, — 700.0, m : — 1 16100’.76; Period : 18516045 3 : Epoch, + 1800.0, m : — 1 I6094'.g6; Period : 183160546. The first passage through the departure point after + 1800.0 is at the epoch" 1805.2 147. The 135th passage preceding is at the epoch — 706.460. The times of the intermediate passages are then interpolated from the known periods. Table II gives the days of the fictitious year at which conjunctions of the mean sun with either node occur. The argument is the interval which must elapse after the beginning of the year under examination before the next following conjunction in Table I. The units of the argument are on the left hand, and the tenths on the top I of the table. Eclipses can occur only nea.r one of the two or three epochs found in this table, unless a conjunction has occurred nearthe end of the year preceding or shortly after the beginning of the year following. Table III, on the same page, gives the reduction from the time of mean to that of true conjunction of the sun with the node, which reduction arises from the eccentricity of the earth’s orbit. This table is used only to make more definite the eclipse limits by enabling us to decide whether an eclipse could or could not occur at a given conjunction in cases where the mean values of the argument might leave the question doubtful. Table,IV enables us to find the moon’s mean age at any fictitious Julian date. To the fictitious day of the year we add the value of D corresponding to the century, and that corresponding to the year, and subtract the greatest_multiple of Period. We may also subtract the next greatest. multiple, and thus obtain a negative value of D, counted backward from the next following conjunction. 5 By taking, for the required date, that of conjunction of the mean or true sun with the node, we are enabled to judge whether an eclipse of given character could or could not have occurred at the preceding or following new moon. RECURRENCE OF SOLAR ECLIPSES. 17 Tables V and VI give the approximate arguments for the central eclipse of each series from — 700.0 to -|— 2300.0, a period of thirty centuries. To understand its con- struction we call to mind that, on the system adopted, the moon’s orbit is conceived as divided into 223 equal parts by that number of conjunction points; that this whole system of points has a very slow retrograde motion relative to the moon’s nodes, such that 61 years elapse betweenthe passages of two consecutive points; that all mean new moons occur at some one of, these 223 points; that those at any one point are separated by intervals of 223 lunations, or one Saros or cycle; that if we isolate every 47th lunation, we shall find these isolated lunations to occur at consecutive conjunc- tion points in the order of longitude. When a conjunction point, by the slow motion already described, approaches within about 18° of the node, there will be an eclipse of the sun at every. new moon which occurs at that conjunction point. The series of eclipses will become central within 10° or 12° of the node, and will continue unbroken until the conjunction point has got 18° beyond the node. We shall thushave a series of central eclipses gen- erally between 45 and 350 in number, with about 15 partial eclipses on each side of it. The total number will generally range between 75 and 80. Since the conjunction point moves about 0°.48 between the consecutive eclipses of each series, some one eclipse must occur within 0°.24 of the node- This nearest eclipse we have sought to ilfiltiei as lthe cential eclipse of the serles-; but, in some cases, that chosen is not abso- e y t e nearest. The numbers 1n Tables V and VI correspond to the eclipse of each SGPIBS chosen as the central one. The intervals between the passages of consecutive conjunction points through the node.are about 61 years at the present tlme‘, and were 63.6 years 25 centuries ago. Thls must be the mean interval between consecutive central eclipses. But it has been shown that this interval, expressed in lunations, is necessarily of the form 223 as + 47, m being. an integer, and must be either 716 or 939 lunations, the former being the more frequent value. . From the mean motions already given we derive the following numbers and periods for the two fundamental epochs, —- 700.0 and + 1800.0, which have served as the basis of Tables V and VI. 8 . Epoch, — 7000; + 1800.0. One mean lunation, in days, _- - - - - 29.53059562 29253053344 Length of Saros, in Julian years, - - - - .1 8.0296 3127 18.02962689 Length of Saros, in days, - - - - - - 6585322823 5535321222 Annual motion of mean anomaly, in rev., - 13.2 5 5 50638 13 2 5 5 5232, Motion of mean anom. in one Saros, in rev., - 2 38.991892 3 233,992 I 377 Change of the same, in degrees, - - - - — 2.9188 _ 23304 Centennial motion of conj. points, in rev., - — 0.007050 -' 0-007 333 Centennial motion of conj. points, in degrees, — 2.5380 — 2_541 7 Motion of conj. points in one Saros, in degrees, r 0.45758 _ Q47523 Motion of ®’s mean anomaly in one Saros, in 0 degrees, " " ' ' ‘ " ‘ ' " -' + IO.498O + Motion of ®’s mean long. in one Saros, in deg., + 10.802 5 + 10,3037 AR——3 18 RECURRENCE OF SOLAR ECLIPSES. The necessary explanation of the principal columns in Tables V and VI will next be given. The conjunction point at which the new moon of — 689, January 20th, occurred, is arbitrarily taken as the zero one. The others are counted from it in the order of longitude. The slow retrograde motion of the whole system relatively to the node causes them to cross the node in the same order. In column T is found the fictitious Julian date of the central eclipse of each series, already described. Any one of these dates being found, the next following is derived by adding to it the time either of 716 or 9 39 lunations; such intervals being chosen as would keep the dates near the times of passages of the corresponding con- junction point through the node. A table of these passages has already been given. Near the beginning and end of the table, the regular order has been deviated from, for the reason that it was supposed that there would be no occasion to use the tables for epochs outside the limiting dates, — 700.0 and + 2 300.0, while it was desirable . to be able to compute all the partial or total eclipses within these dates. Many of these eclipses would, however, take place at conjunction points the central eclipse of which might take place several centuries without the limits. Instead of choosing the central eclipse of the series, one occurring near the limiting epoch was chosen in each case. It may also be noted that the years before Christ are reckoned in the usual astro- nomical way; the year immediately preceding the first of the Christian era being con- sidered as zero, the next preceding being — 1, etc. The days are, however, considered as positive; so that if we express any one of these dates in years and fractions, the integer number of years would be one less than in the‘ table. The reckoning of fictitious time throughout the table is that already explained, namely, taking Greenwich mean noon of 1800, January I 2th, as the epoch, we call this epoch 1800.0, and count backward and forward by years of 3653 days each. The days are, therefore, not always reckoned from noon, but from noon, 6 hours, 1 2 hours, or 18 hours, according to the number of the year. A correction is therefore required to reduce to the time of noon, and, since I 582, a still further correction to reduce from the Julian to the Gregorian calendar. These corrections are shown in Table XIII b. The times of mean conjunction correspond to 'Hansen’s mean motion and secular variation, with the corrections given in my Researches on the Motion of the Moon, page 268,* the periodic terms being omitted. .The times of mean conjunction, as given, are generally accurate to one or two units in the last place of decimals, or to 8" or 10" of arc in the relative positions of the sun and moon. Their errors, therefore, fall far within the necessary uncertainty of the lunar theory in past and future centuries. The moon’s mean anomaly, g, has been divided from — 180° to + 180°, for greater convenience in the selection of total eclipses. It is derived from Hansen’s tables, applying the same correction as to the mean longitudes. The sun’s mean anomaly, g’, and the mean longitude, L, do not seem to require any special explanation. The moon’s mean argument of the latitude, uo, has been derived from the differ- ence between the date of each central eclipse and the passage of the conjunction point * Vifashington Observations for 1875, Appendix II. RECURRENCE OF SOLAR ECLIPSES. 19 through the node, and is equal to the motion of the conjunction points during this interval. Table VI, which gives the mean elements for eclipses at the descending node, is constructed on the same principles. Here the argument of latitude, uo— 180°, is of course counted from the descending node. Table VII gives the reduction of the arguments in Tables V and VI for other eclipses of the same series. In the use of the tables for calculating a particular eclipse, it is necessary to find the date of the central eclipse of the series to which the one under consideration belongs, as given in Tables V and VI. This is readily done by the precepts given in the tables. Having found the central eclipse, the elements for e the -required eclipse are deduced by adding the corrections for the number of periods elapsed, as given in Table VII. Owing to the secular changes in the motion of the arguments, these motions are given for three epochs, namely, the year 0, the year 1000, and the year 2000 of our era. For greater facility in the use of the tables, the change in the last place of decimals for intervening centuries is added wherever it is necessary. In using these tables, the number must be taken out for an epoch mid- way between that of the central eclipse and that of the eclipse to be computed. Having found the arguments for the moment of mean conjunction, the next step -is to deduce the elements for the moment of true conjunction. The theory of-this pro. cess has been fully developed by Hansen in his Analyse der ecliptischen Tafeln.* The same authorhas given tables for the approximate computation of eclipse elements which are of direct application to the problem' as here presented. These tables are, however, rather meagre, and can only be used in‘ connection with the author’s tables‘ of the moon. On the other hand, the formulae in the later paper are developed with \ such fullness that it is not necessary to go over them. I shall, therefore, accept Han- setys results, with such modifications as are necessary to make them applicable to the form of tables now proposed. The following are the modified expressions, A general remark, applicable to the tables, is, that the quantities required are given for the moment of true conjunction in ecliptic longitude, but are expressed in terms of so the values of g, g’, etc, at the moment of mean conjunction. (1) Reduction from time of mean conjunction to that of true ecliptic conjunction. 6T : — o“.4089 sin g - - +o“.0161 sin 2g — o“.0004 sin 3 g + Od.I 743 sin y’ + 0“.002 I sin 2 g’ —— o“.oo5 I sin (9 -|-g’) + o“.oo75 sin (9 — 9’) + 0“.0I04 sin 2 u. (2) True argument of latitude, reduced to the ecliptic, for the moment of true con- junction. In the special form of tables adopted, it is necessary to reduce the mean longitudes of the two bodies at the moment of mean conjunction to their true longitudes at the T-lV* erichtc iiber dic Verhandlungen der Koxniglicll-Siichsischen Gesellschaft der Wissenschaften, Bd. XV, Leip- zig, I863, and Bd. IX, I357- 20 RECURRENCE OF SOLAR ECLIPSES. moment of true conjunction. If the expression for the elapsed time between mean and true conjunction is correct, this reduction ought to be the same for both bodies. Han- sen gives its expression for the moon; the corresponding correction‘ for the sun is, 0 a Mean motion during interval + Equation of centre for true conjunction. Putting 6T for the elapsed interval, and g’, for the mean anomaly at the moment of true conjunction, the required reduction will be, no T + I°.922 sin g',+ 0°.020 sin 2 g',, -where we must put 91:9’ + WT. Substituting this value, and developing, the expression will be, n’6T (1_+0.0335 cos y’) + |°.922 sin g’+ 0°.020 sin 2g’. Substituting the value of n6 T just given, we find the following expression for the true ecliptic distance of both bodies, counted from the node, at the moment of true conjunction, a being the mean distance at the moment of mean conjunction :— ‘ a,:u—0°.403 sing :u—.00703 sing +0°.016 sin 2g +.00028 sin 2g + 2°.094 sin g’ 7 + .03655 sin g’ + 0°.027 sin 2 g’ '+ .00047 sin 2 g’ — 0°.0I 2 sin (y +g’) - .0002 I sin (g + g’) + 0°.0I.0 sin 2 a + .0001 7 sin 2 a. (3) Vertical distance of the axis of the moon’s shadow from the centre of the earth at the moment of ecliptic conjunction. The expression for this element is, _ sin ))’s latitude """' sin(7r—7t’) ' Hansen puts it into thelform, B::Pcosu+Qsinu_-:3/2. Its numerical expression will, however, be a little more simple by substituting u,,. the true argument of latitude, for u, the mean argument. Hansen’s expressions for P and Q are nearly as follows, some very small terms being omitted :— P : — .0392 sin g Q : + 5.2207 +.0I16 sin 2g —0.3299 cosg + .2080 sin g’ — 0.0048 cos y’ + .0024 sin 2 g’ + 0.0020 cos 2 g’ - .0073 sin (g + g’) — 0.0060 cos (g +.g') + .0067 sin (g — g’) + 0.0041 cos (g — g’). +.01I8 sin 2a. . RECURRENCE OF SOLAR ECLIPSES. If we suppose 3/, : P, cos u, + Q, sin 11,, We shall have, P, : P cos (a, — u) — Q sin (u, — u), Q, : Q cos (u, — u) + P sin (24, — From the preceding expression for u, We find, cos (u, — u) : 1 — .00036 — .00034 cos 2g’ -1- .00014 cos (g + '9’) — °OOOI4 cos (.9 -29,): while we may suppose sin (u, — u) : u, -9-— u, We then find, Qsin (u,—u):— 0386sing Qcos (uI._{,):Q + .0025 sin 2g + .191 7 sing’ + .0022 sin 2g’ — .0070 sin (g + g’) + .0060 sin (y — g’). P sin (u, — u) : + .0039 P cos (u, — u) _—; P_ — .0002 cos 2g — .0038 cos 2g’ + .0014 cos (9 + g’) —'- .0014 cos (g —g'). 9 P,:—.0006 sing Q: + .0091 sin 2g + .0163 sing’ + .0002 sin 2 g’ — .0003 sin (g + g’) + .0007 sin ([1 — g’) + .0118 sin 2 u. 2! —.0019 —.0018 coszg’ + .0007 cos (g + 9’) — .0007 cos (g -—g’)_ + 5.2227 — 0.3299 cos 9 — 0.0048 cos g’ — 0.0036 cos 2g’ - 0-0°39 008 (11 + .0’) + 0.0020 cos (g -— 5;’), The value of P, is so small that We may suppose cos u,: :1: 1'in multiplying it by this quantity. In a total eclipse, the value of u, can differ from 0° or 180°sby only about 11°, and that of u by only about 16°. We shall, therefore, obtain a result nearly accurate to the third place of decimals by replacing 0.0118 sin 2 u'by 0.022 sin 11,. A sufficiently accurate expression for 3/, will therefore be, y2:P1+QxSinu1; 22 RECURRENCE OF SOLAR ECLIPSES. or, omitting terms which will not change g, by .001, y, : :1: (— .0006 sing + .0091 sin 2g + .0163 sin g’) + (5.245 — 0.330 cos g) sin u,, the upper sign being used at the ascending and the lower at the descending node. An error of a ‘unit in the third place of decimals corresponds toone of about 5’ in the position of the shadow-path on the earth’s surface; the probable error of the shadow- path, on account of the quantities neglected in g,, will therefore not exceed 10 or 15 miles. (4) For the hourly motion of the axis of the shadow along the fundamental plane,* Hansen’s expression is equivalent to V I d 2 sic’,-.:_-€z(—;,—’:+ 0.5410 g.,: : (.05-40+ .0034 cosg) cos u,. + 0.0397 cos g — 0.0010 cos g’ + 0.0006 cos (g + g’) — 0.0004 cos (g — g’). . . d y, 1 d :13, We may, without an error exceeding 0.001, regard W as equal to E6 . —d—t—. (5) The radius of the shadow on the fundamental plane and the angle of the shadow-' cone are given by the formulae, p : + 0.0059 sinf: + 0.004653 — 0.0182 cos g + 0.000078 cos g’. + 0.0004 cos 2 g + 0.0046 cos g’ — 0.0005 cos (g + g’). When p is positive, the eclipse will be annular; when negative, total. The value of p for external contact may be found by increasing the above by 0.5460, which will make the constant term 0.55 19. The same value of sin f may be used in the two cases. . C'z'rcumstances of an Eclipse on the Earth’s Surface.——Our next step is to find the relation of any point on the earth’s surface to the shadow. Several systems of co-ordi- nates may be adopted for this purpose, which vary with the adopted direction of the . axis of X on the fundamental plane. We have the choice of three systems, depending on the following three positions of this axis of X in the fundamental plane :— (1) The intersection of the earth’s equator with the fundamental plane; (2) The intersection of the ecliptic with the same plane; (3) A line in the same plane parallel to the path of the axis of the shadow along it. The firstsystem is that of Bessel, while the second and third have been used by Hansen. 6 " It will be remembered that the fundamental plane in the theory of eclipses passes through the centre of the earth perpendicular to the axis of the shadow. RECURRENCE OF SOLAR ECLIPSES. 2 3 Let us put 0, the sun’s true longitude, or, more exactly, the longitude of the sun as seen from the moon; .9, the obliquity of the ecliptic; a, d, the right ascension and declination of the sun as seen from the moon, for which, in the present case, we may take the geocentric direction of the sun; L a, the angle of the shadow-path along the fundamental plane with the inter- section of the ecliptic with the same plane; h _-: p cos ¢’ p being here the earth’s radius and cp’ the geocentric latitude Ic : p sin cp’ ; of any point on its surface. For the present We shall represent the co-ordinates corresponding to these various systems by subscript numbers. It will be remarked that in all the systems the axis of Z passes through the centre of the earth parallel to the line joining the centres of the moon and sun. c The value of the equation of the centre by which (9 is found may be obtained from Table XXVI. The expression is, G) : L + Equation of centre, For the relations between systems (I) and (2), We determine the angle p from any or all of the equations, A cos d sin 19 : sin 3 cos 6), cos d cos p : cos a, sin d : sin 8 sin Q. The required relations will then be, cc, : 93, cos}? + 3/, sinp, y,-_-_—w,sinp+y,cosp, 2,, -_: 2, ; or, reversing them, as, : x, cosp - y, sinp, y, : cc, sinp + y, cosp, For the use of the third system, it will be sufficiently accurate to suppose that the axis Of 773 makesan angle of 5° 30' With that Of 56,. With a little greater proba- ble accuracy We may determine the angle a by the condition, a : 3: 50,7 cos an this angle being positive at the ascending and negative at the descending node. Then, putting pl :19 "l" a: we shall have, 233 : as, cos p’ + y, sin 19’ : as, cos a + y, sin a, g3-_:—-cc, sinp’+y,cosp’:—x, sin a+g,cosa, Tables XVIII to give the value of y, for the shadow-axis at the moment of conjunction in longitude, when ac, -_-_- 0; and Tables XXI and XXII give the hourly 24 RECURRENCE OF SOLAR ECLIPSES. variation of 50,, from which that of 3/, may be obtained by multiplying by tan at or by cos u, tan (50 42’). The expressions for cc, and 3/, will therefore be, :c,_,:x’,,t, y,:g/°+cv’,,ttana:y,°+g/’2t. Here If is the time after true conjunction, T, expressed in hours as the unit. To refer the shadow-axis to either of the other systems, We shall then have, $3 y,° sin a + x’ 15 sec a, 3/3 3/2° COS 5‘: C l I 90, — 3/2° S111 19 + 2: 2 t (cosp — tan a S111 p), —y2°sinp+x’,tsec acos (p+ a), 312° cosp + as’, t (sinp + tan a cos p), y,° cosp + as’, t sec a sin (19 + a:). The values of the coefficients for :3, and 3/, may be taken from Table XXVIII, where we have put a :—sinp, a’: cosp, b : secacos(p;{: cr), 1)’: secasin(p;{:a); so that the expressions for ac, and y, are, xx:-—a y2°+b xlzti y, : a’ y,° + 12’ av’, t. We now require the corresponding co-ordinates for the point on the earth’s sur- face, expressed by the quantities h and k. If We represent these by E, 17, and 8, Bes- sel’s eclipse formulae give, ,:hsinH, 77,_—_kcosd—ksindcosH, H being the hour-angle of the sun, or, to speak more exactly, the hour-angle of that point of the sphere representing the direction of the su11 as seen from the moon. The other co-ordinates of the place will then be, , : k cosd sinp + k (cosp sin H — sin d sinp cos H), 17, : k cos d cosp — h (sin 1) sin H + sin d cosp cos H), 3 : it cos d sin 12’ + h (cos 1)’ sin H — sin d sin 10’ cos H), 273 : .7: cos d cosp’ — h (sin 1)’ sin H —|— sin d cosp’ cos H). The angle H is expressed in terms of t, as follows :—From the conjullction tables V—VII we have, by the corrections from Tables VIII—XII and by correcting for the fictitious date, the fraction of a day of Greenwich mean time at the moment Of true RECURRENCE OF SOLAR ECLIPSES. 25 ecliptic conjunction. Multiplying this fraction by 360° (Table XIV), we have the hour- angle of the mean sun for the meridian of Greenwich at the moment of conjunction. Let us put Ho, this hour-angle; A, the longitude of the place west from Greenwich; E, the equation of time, to be added to apparent time in order to obtain mean time, expressed in arc. Then, at conjunction, the hour-angle of the mean sun will be H0— A, and that of the apparent sun will be Ho — A — E. The expression for H, as a function of t, will then be H:Ho—l—E+ I5O)(t. We have now all the data for proceeding with the computation of the eclipse in any of the usual ways. The data of the present tables are of such accuracy that we may generally expect to predict the phases of an eclipse, by means of them, within one or two minutes of time, and to determine the shadow-path in total or annular eclipses to coarse fractions of a degree. In fact, supposing the tables perfect to the last place of decimals, the pro- bable error of this path should not exceed two or three tenths of a degree, unless near the north or south pole of the earth; but small errors of theory are possible, leading to larger errors in the shadow-path. The following are the formulae for the computation of the path of a central eclipse. They are applied by computing the longitude and latitude of the point in which the axis of the shadow intersects the earth’s surface at any assumed moment of Green- wich mean time. c Compute the Values of as, and y, for the assumed moment. Table XXVIII is designed to facilitate this computation. If it is desired to take into account the ellipticity of the earth, the neglect of which will introduce a probable error of perhaps 10’ in the point required (which amount, however, is hardly greater than the necessary uncertainty of the results from the preceding tables), we compute p, and d, from the formulae, p, sin d, : sin d, p, cos d, : V I — 6’ cos d: [9.99855] cos d, I _ _y_. _ T y * " 0. The Values of p, and d, may be taken at once from Table XXIX with the argument @:L+Equation of centre. If, however, we neglect the ellipticity, we put d for d , y, for 3/',, and m for m, in the following formulae, which are a continuation of the preceding ones. From I 0 sin C : 3/ ,, 0 cos 0 : ~/I —x,’—y’."’, 26 ' ‘ RECURRENCE or SOLAR ECLIPSES. find c and C. The last quantity may be computed by the auxiliary angle ‘,6’, thus :— sin ,6’ sin y : x,, sin ,6’ cos 3/ : 3/,, 0 cos C : cos /0’. Then, from the equations, cos ¢, sin H : 23, cos cp, cos H : 0 cos (C + d,), sin cp, : 0 sin (0 + d,), tan ¢ .-_—_ [o.oo145] tan go,," find q) and H. The former will be the latitude of the point required, and the latter the local hour-angle of the shadow-axis. The Greenwich hour-angle is found by the formula, H,:Ho—E+I5°t; Ho_being the Greenwich mean time of true conjunction in longitude, expressed in ~ arc; E, the equation of time (Tables XXVI and XXVII); t, the interval of the assumed time after that of true conjunction, expressed in hours The west longitude of the point sought will then be :— l:H —H. I €33. RECURRENCE OF REMARKABLE ECLIPSES. The occurrence of eclipses approaching the maximum length of totality is a sub- ject of astronomical interest. We havealready shown that the successive eclipses at the same conjunction point, occurring at intervals of 18 years, are nearly of the same character. Consequently, if we have at any time an eclipse in which the duration of totality approaches the maximum, we shall have a similar one after a lapse of one period, and the duration will vary but slowly from period to period. We shall there- fore search in our tables, not for single eclipses with long duration of totality, but for series of such eclipses, the distinctive mark of each series being the conjunction point at Wl1icl1 it occurs. The conditions necessary to the greatest duration of totality, considered individ- - ually, are the following :— I. The moon must be near its perigee at the time of conjunction. In other words, its mean anomaly, positive or negative, must be small in absolute value. II. The sun must be near its apogee in order that its semi-diameter may be small; hence its mean anomaly must differ little from 180°. It is, however, to be remarked that a deviation of the sun’s mean anomaly from 180° will produce only about one fourth the effect of an equal deviation of the moon’s anomaly from 0°. RECURRENCE OF SOLAR ECLIPSES. - 27 III. The preceding two conditions give the maximum breadth of shadow on the fundamental plane, passing through the centre of the earth at right angles to the axis of the shadow. But, the shadow being conical, its diameter increases as we approach the moon. The observer should therefore be as near the moon as possible. In other words, at the moment of central eclipse, the sun and moon should be near his zenith. The’ diameter of the shadow at his station will then be nearly one third greater than on the fundamental plane. In order that these conditions may be fulfilled, it is neces- sary that the observer should be within the tropics and. that the conjunction should take place near the node. For, the two bodies being in the zenith, the effect of paral- lax is zero, and the eclipse must be central at the centre of the earth, which can only occur when the conjunction coincides with the node. ' These conditions will be most nearly fulfilled by the central eclipses, the dates of which are given in Tables V and VI. IV. The diurnal motion of the observer must be as great as possible, because by this motion he is carried along in the same direction with the axis of the shadow, and thus the time which he remains within it is increased. This condition is best fulfilled when he is upon the equator. , V. The direction-in which the observer is carried by the diurnal motion must be parallel to the direction of the shadow. This condition demands that the direction of the axis of the shadow shall be near the great circle joining the pole of the earth and the pole of the moon’s orbit. . This condition can be fulfilled only when the sun’s lon- gitude is near 90° or 270° ; in other words, near the times of the two solstices. It is impossible that all the preceding conditions can be simultaneously fulfilled, owing to the obliquity of the ecliptic. The 4th condition can be fulfilled only at the equinoxes, and the 5th only at the solstices. Also, since the sun’s apogee has, during ' each century, a nearly definite longitude (at present about 90°), it is only near 90° of the sun’s longitude that the 2d condition can at present be fulfilled. Informer ages, the case was somewhat different. But the distance of the epoch from these solstices was only about 20° at the beginning of the Christian era. We may see, therefore, that during historic times, and for several centuries to come, the solar eclipses of great- est duration can occur only near the summer solstice. If the eclipse at this time occurs also exactly at the node it will be central in the zenith of the Tropic of Can- cer. Conditions 3d and 5th will therefore be fulfilled, but condition 4th will not. If the latitude of the moon be north at the moment of conjunction, conditions 3d and 4th 1 will both be less favorable. If the latitude be south, condition 4th may be more favor- able, because the shadow will then be thrown further towards the equator, but condi- tion 3d will be less favorable. Condition 4th will be best fulfilled by south latitude of about 24’, or by an argument of latitude of —— 4° or — 5° near the ascending node and +40 or —|— 5° near the descending node. Beyond these points of latitude, both conditions are unfavorable. We conclude, therefore, that when the two first conditions are pro- perly fulfilled, the most favorable eclipses will be the ten or twelve which follow the central one of each series at the ascending node ‘and the tenor twelve which precede it at the descending node. The most favorable will generally fall between the fourth and the seventh from the central eclipse; and the first two conditions require 28 RECURRENCE OF SOLAR ECLIPSES. that we should then have g : 0 and g’ : 0. In order that these conditions should be fulfilled at the sixth eclipse, we should have, near the time of central eclipse, the fol- lowing system of values of g, g’, and L :— Ascending Node. Descending Node. .9 = I 7° 9 = - I 7° g’,: I 20° g’: 240° L _—_‘ 25° L : I 55° An examination of Table V will now enable us to select series of remarkable total eclipses almost by inspection- We see that the moon’s mean anomaly is repeated within 12° or 15° at every third conjunction point. Considering, first, eclipses at the ascending node, we perceive that the moon’s mean anomaly is small at the Ioth, I 3th, I 5th. etc, conjunction points, and that the condition, y : 17°, is most nearly fulfilled at the 16th and 19th -points. The conditions, g’ : 120° and L : 25°, though not ful- filled at either point, are so near fulfillment that there were then two series of total eclipses nearer the maximum duration than any which occurred for several subsequent centuries. The last of the most favorable series were in the years 663, 681, 699, etc. To find other series approaching the maximum of totality, we have to pass over more than a thousand years, until the 42d and 45th conjunction points approach the node. To the 42d conjunction point belong the series of great eclipses of. 18 3 2, I8 50, 1868, 1886, etc., of which the maximum was that of 18 32 or 18 50. The position of the solar perigee is unfavorable at this point; otherwise the duration would have gone on increasing through the next century. But at the 45th conjunction point, the con- ditions are more nearly fulfilled than they have been for at least twenty centuries, and we shall therefore have a series of eclipses approaching within a very few seconds of the maximum duration of totality. These will occur in the years 2150, 2 [68, etc. Passing now to the descending node, we see that in a general way the series occur in the same order. The favorable conjunction points near the present epoch seem to be the 149th, 152d, I 55th, etc. In the first two of these, the sun’s mean ano- ‘ maly is not favorable except when the moon’s is unfavorable. The conditions are better fulfilled at the I 55th conjuction point, the central eclipse of which takes place in the year 2009. The eclipses of maximum duration will occur two or three periods before the central eclipse, namely, in the years I95 5 and 1973. To this series belong the total eclipses of 186 5, 188 3, etc. The successive eclipses of this series will therefore increase in duration for five or six periods to come, when the duration will probably be greater than that of any that have preceded them during the past thousand years, TABLES OF SOLAR EOLIPSES. 13, 0. 700 TO A. D. 2300. 29 TABLE I.-9--Dates at which the Moon’s Asce TABLES OF SOLAR ECLIPSES. 31 nding Node has the same Longitude that the ‘Sun has at the Beginning of the Fictitious Y ear. ' Year, Year. Yea!’-3 Year. Year _.73o_373 —148.321 +434-24-I +III6.8I8 +1749.398 -762.273 -129.716 -502 849 I135 423 1768.664 ._743_ 59 —111.112 521-454 1154.028 1786.609 ._725_054 — 92.507 540-059 II72.633 1805.215 -—766.466 -— 73 962 553-654 I191 239 1823.820 —-687.855 - 55-297 577-269 I209 844 1842.426 —669_.251 — 36-693 595-374 1223-450 1861.031 ...(,~50_545 — 18.088 614-479 1247.055 1379_637 —632.042 + 0-517 633-034 1265.660 1898.242 -—613.437 19-122 651 689 I284 266 1916.848 —594.832 37-727 670-294 . 1302-371 1935.453 —576.228 56-332 633-399 1321 -476 1954.059 _557_(,23 74-936 707-504 1340.080 1972.554 -539,019 93 541 726 109 1358.686 1991.276 _52o_4I4 112.146 744-714 1377.291 2oo9_375 -501 .809 130-751 763-319 1395-396 2028.481 —483,2o5 149-356 781-924 1414.502 204-7_o36 -464.600 167-961 300-529 ~I433.Io7 2065.692 _445_996 186.566 819.135 1451.712 2084.298 _.427_39I 205-171 337-740 1470.318‘ 2102.903 ..493,7g5 223 775 856 345 1488.923 2,,,.5O9 ._39o_,32 242-330 374-950 1507.528 2140.114 ._37,,577 266.985 893 555 1526.134 2,58.72o ~—352.972 279-590 912- 160 1544-739 2177.326 _ '-334-363 208' 195 93°'765 1563-344 2195 .931 -315-763 3‘6‘8°° 949°-37° 1531-950 2214.537 -297. 158 335-404 967-976 1600.555 2233_ I42 —278.5S4_ 354-009 936-531 1619.160 2251.743 -259-949 372°6‘4 1°05 ' I86 1637- 766 2270. 354 -24 -344 391-219 ‘°’3-792 ‘ 1656.371 2288.959 -222. 740 409- 324 1042 - 397 1674 . 977 2307 , 565 -26 .135 423°4’9 ‘°°‘°°°’ I693 582 2326.176 -—185.53o 447-034 1079-607 1712. 187 . 2344_ 776 ‘166-926 +4656-°’9 +'°98‘2" +1730-793 +2363.381 TABLE I. I 1.1g§°1fiZ1‘§§i1§f1f."iZ.“?§1i§° ’v‘1‘3§‘1'i.‘1”1”.‘.§'Z..‘.'?.§’.?.‘I?§.?‘i”.£J.§‘£E‘§i."§‘.f.‘Z?.."I.‘.'i“i.’1‘i'.Z.°1f.?.§‘§.’I.°3‘Q;‘§§’.§‘.§‘Z’.ff1”;‘*£..f'1‘L‘iT§1‘IL'?;.fifi‘§‘.T3;° "-‘1”1‘.f.’.‘.'iI’.f':'.‘.’;‘. 'L‘III{3’°f number will be the day and tenths of a day of the fictitious ulian year at which the mean sun was in conjunction with the node il1(lll(:a1t~0(1nilI£: the second column. Should the number be negative 1t W111 mdlcate days before the beginning of the year, and should it exceed 365.25 it will Indicate a conjunction after the end of the year. _ In general, a central eclipse of the sun can only occur within ten or twelve days of the times thus found, and a partial eclipse within elghteen or twenty days. It is to be remarked, however, that an eclipse may occur when the corresponding conjunction takes place near the 61111 Of the year preceding, or during the first seventeen days of the year followmg. - "“‘*~*——__* 32 TABLES OF SOLAR ECLIPSES. TABLE II.-Days of the Fictitious Year when the Meaen Sun is in Conjunction with either Node. . ' 6-‘D —# _ _' ‘M _—‘I - - — H .““‘!‘-H N A 6W__—_—M’V‘h“ ‘NV >—~ 1:‘ h d .Y-ear. Node. I .0 I .1 .2 I .4 I .5 .6 I .7 I .8 .9 °' “" '°d“‘S I 1 I i of ayear. I I , . I I .1. I a’. .1. d. ‘f .2. I .1. .1. a’. I .1. .1. y, ,1, . Asc. 0.0 1.9 3.7} 5.6. 7.4I 9.3 11.2 13.0] 14.9 _ 16.8 0 CI 0 2 0 Desc. 3 173.3 175.2 177.0: 178.9 180.7 ; 182.6 184.5 186.3 . 188.2 190.1 ' ‘ Asc. I 346.6 348.5 350.3 1 352.2 I 354.0 I 355.9 357.8 359.6 361.5 ' 363.4 0.02 0.4 I Asc. 18.61 20.5 22.3 24.2 1 26.04 27.9 29.7 31.6 33.5 35,4 0.03 0.6 Desc. 191.9 193.8 ‘ 195.6 I 197.5 ; 199.4 201.3 203.1 205.0 206.9 208.7 0-04 0-7 5 A 4 I _’ 1 0.05 0.9 2 sc. 37.3 39.2 41.0. 42.9 9 44., 46.6 48.4 50.3 52.2 54.0 Desc. ‘ 210.6 212.5 214.3 216.2 9 218.0 219.9 221.7 223,6 225.5 227.3 0-06 1-I 3 Asc. 55.9 I 57.8 59.6 : 61.5 ’ 63.3 65.2 67.0 68.9 70.8 72.7 0'0’ 1'3 Desc. 229.2 231.1 232.9 234.8 . 236_.6 238.5 240.3 242,2 244.1 246.0 = 0-08 1-5 4‘ Asc. 74-5 76.4 78.2 I 80.1 82.0 83.9 85.7 87,6 89.5 91.3 ‘ 0'09 1'7 Desc. 247.8 249.7 251.5 I 253.4 I 255.3 257.2 259.0 260.9 262.8 264.6 °-‘° 1-9 5 Asc. 93.2 95.1 96.9 98.8 1. 100.6 102.5 104.3 106.2 108.1 109.9 Desc. 266.5 A 268.4 270.2 . 272.1 * 273.9 275.8 277.6 279,5 281.4 283.2 6 Asc. 111.8 113.7 115.5 117.4 ‘ 119.2 121.1 122.9 . 124.8 126.7 128.5 Desc:_ 285.1 287.0 288.8 290.7 . 292.5 294-4 296-2 298.1 300.0 301.9 ' I 7 Asc. 130.4 132.3 134.1 136.0 ’ 137.9 I 139-3 141-6? 143.5 9 145.4 147.2 D050 303-7. 305-6 307-4 309-3 311.1 313-0 314-9 I 316-3 I 318-7 320-5 ; I I 8 7Asc. 149.0 150.9 152.8 154.7 6 156,5 158.4 160.2 I 162,; 164.0 165.8 Desc. 322.3 324.2 I 326.1 328.0 . 329.8 33!-7 333-5 335.4 337.3 339.1 9 Desé. -5.6 -3.7 -1.9 0.0 1.8 3.7 5.5 7,4 9.3 11.1 Asc. 167.7 169.6 171.4 173.3 I 175,; 177.0 178.8 180.7 182.6 184.4 Desc. 341.0 342.9 344.7 346.6 5 348.4 350.3 352.1 354,9 355.9 357.7 Desc. 13.0. 14.9 16.7 18.6 I 20.4 22.3 24.1 26.0 27.9 29.8 10 Asc. 186.3 I 188.2 190.0 191.9 I 193.7 195.6 197.4 1993 201.2 203.1 Dem 359.6 I 361.5 353.3 365.2 I ‘ 367.0 368-9 376.7 372.5 374.5 376.4 I I ll Desc. 31.6 33.5 35.3 37.2 39,1 41.0 42-3 44.7 46.6 48.4 Asc. 204.9 206.7 208.6 210.5 4 212.4 214.3 216.1 213,9 219.9 221.7 , I . ‘ 12 Desc. 50.2 52.1 54.0; 55.9 . 57.7 59-6 61-4 63.3 65.2 67.01 Asc. 223.6 . 225.5 227.3 I 229.2 . 231.0 232-9 234-7 236.6 238.5‘ 240.3 13 Desc. 68.8 70.7 72.6 74.5 g 76.3 73-2 80-0 81.9 83.8 85.6 Asc. 242.2 I 244.1 245.9 247.8 . 249.7 25!-6 253-4 I 255.3 257-2 259.0 14 Desc. 87.5 89.4 91,2 93,1 9 94,9 96.8 98.6 ‘ 100.5 102.4 104.3 Asc. 260.8 4 262.7 264.5 ; 266.4 I 268.3 270-2 272-0 273.9 - 275.8 277.6 9 I I I I . 15 Desc. 1 .1 I 108.0 109.9‘, 111.8 » 113.6 115.5 117.3 4 119.2 121.1 122.9 . Asc. 279.4 281.3 283.2 I 285.1 ; 286.9 288.8 290.6 i 292.5 I 294.4 296.2 I; I I ; I = I ‘ 16 Desc. 124.7 I 126.6 128.4 I 130.3 132.1 134.0 ! 135.8 137.7 I 139.6 141.5 “ Asc. 298.1 4 300.0 301.8 303.7 ' 305.5 307.4 309.2 I 311.1 313.0 314.8 I I ; I . ‘ ' 1., Desc. 143.4 I 145.3 I 147.1 I 149.0 156.8 152.7 ' 154.5 156.4 ‘ 158.3 166.2 I Asc. 316.7 318.6 320.4 I 322.3 ' 324.1 326.0 | 327.8 I 329,7 I 331.6 333.5 I I I i I I 18 Desc. I 162.0 163.8 I 165.7 ‘ 167.5 I 169.4 171.2 ! 173.1 ; 175.0 5 176.9 178.8 ' Asc 4 335.3 337.2 339-I . 341.0 I 342.8 344.7 346.5 4 348.4 I 350.3 352-1 I I I I 6 I ‘. I 5 TABLE III.—Reduction to Time of True Conjunction of Sun with Node. Days. (1. Days. (1. Days. 2 d. Days. d. Days. (1. Days. (1. Days. d. "Days. ‘ d. 0 -0.4 50 i -1.6 100 -1.8 150 -0.7 200 I + 1.0 250 + 1.9 300 + 1.5 350 +0.1 10 0.7 60 1.8 110 1.7 160 -0.3 210 , 1.2 260 1.9 310 1.2 360 -0.3 20 1.0 70 1.9 120 4 1.5 170 0.0 220 g 1.5 270 1.9 320 1.0 370 -0.5 30 1.2 80 1.9 130 1.2 180 +0.3 230 I ' 1.7 280 1.8 330 0.7 40 -1.5 90 -1.9 14o_ -1.0 190 +0.6 240 + 1 8 290 + 1.7 340 + 0.4 '_._..___________V TABLES OF SOLAR ECLIPSES. TABLI8 IV —To find the Age of the Moon at any Fictitious Julian Date. Y , D, 1? . D. Century. D. Year. D. Year. 13- 331' ear I .-) I d' 0 4 6 T 25 - 10-9 50 17-'~’ 75 i 23'5 “ 80° 6-8 ' 6 21 8 51 28.1 76 4.8 I 15-5 2 ' ., 700 2.5 2 26.3 27 3_1 52 9,4 77 15.7 _. 500 _ 23.3 18.6 29 24,9 54 1.6 79 7.9 - 400 18.9 4 - 300 14.6 —— 200 10.3 5 29_4 30 6.2 _ 55 12.5 80 18.8 - 100 5.9 6 10.3 3; 17.1 56 23.4 81 0.2 0 1.6 2I_7 32 28.0 57 4.8 82 11.1 -+ 100 26.7 3 3_0 33 9.3 58 15.6 83 21.9 200 22.4 9 I3_9 34 20.2 59 26.5 84 3.3 300 18.0. 1 ' 400 13-7 500 9.4 £0 24.8 I.6 60 7.9 14.2 500 5_0 II 6_2 36 12.5 61 18.8 86 25.1 700 O_7 I2 I7_0 37 23.3 62 0.1 87 6.4 300 25.9 13 27_9 38 4.7 63 11.0 88 17.3 900 2I_S I4 9_3 15.6 ‘ 2I.(_) 28.2 1000 17.2 1100 12.8 6 mo rs 4° .2 9:’ I300 4'2 I6 ' I-5 41 18. 67 2:‘0 32 1.: ‘400 29-4 ‘7 l2'4 42 o‘: 68 6.4 93 12.6 150° 25'° 18 23% 43 10.9 69 172 94 245 1600 20.7 4- 44 . ' ' ‘ 1700 16.3 I9 ’ 1800 12.0 I 00 . 20 15.5 43 21.8 70 28.1 95 g 4.9 29 7 7 2’ 26_4 46 3.2 71 9.5 96 3 15.8 000 3'3 . 1 .1 72 20.5 7 ’ 26.7 7.8 47 4 9 . 2100 28.5 29 , . '8 6 43 24.9 73 1.7 98 , 8.0 2200 24.2 23 ' 6 3 74 I2 6 99 I8 9 I9.9 0.0 ' ' ' ——-——.____. .. the The age (D) of the mean _1noon at a century, adding the fictitioiis J multiple next smaller and next grea: iilia iiy time is found by ta» (1 ; cli D86 certain. B Egggvvggrt: 1: 2134.: ; $11 Sclipse possible. If the occurrence (if the eclipse is still t1011btfl1 of which the argument is the d of the present time‘ but it in , . v. 5 Using the corrected values of 1); the’ “nuts mu bi’ . ~1' se certain. i gtczlillise possible. a.v ay of the fictitious y ear, already found from Table II. he used for other centuries by simply increasing the argiinient by one day for each ceiitiiry before the nini-teentli. A: king the sum of the values of 11 date and subtracting the greatest multiple of the period. ter than the sum. The first re1naini]l.er.will _the1i indicate the days which ' 5 he next following. The units of D for a central eclipse will tl mean (1 the second those beiorc t new moon, an D between 1) between I, the limits may be narrowed by D corresponilin Genera Multiples of Period. o»ooo\1c~ u\«tsLp1or-1 I2 13 I4 15 33 g 29. to 59- : 83. 3 118. l xlr-1C--U1 147. 177. 206. 236. 265. 295. U-JCDt~)\llO 324. 354- 383. 413. 443- O-F-\O$-(13 «r to the century and to the year of :1: 85.0 : a central eclipse certain. J: 14“.3 : a central eclipse possible. I)’ it will be better to si1bt1':u°t thi- have elapsed sini-c the preceding ien be, applying to D the further correction taken froni Table I II , This table only holds good within two or three ct-11t1i1'ies 9“.9: a central eclipse certain. 3}: 12“ 4: a central eclipse possible. To find the central eclipse of the series to which the required one belongs, take one of the valiies of I’ c01'1‘espo1Hli11g most iiearly to D in ‘the following table ;_ D 0.51 L0 L4 L9 24 2_9 3,4 3.8 4.3 4.8 5.3- 5.8- 6.2 6.7] 7.2 7.7 i 8.2 ’ 8.6 3 4 -E 5 7 8 9 IO 11 12 ‘ 13 14 l 15 16 _ 17 T T‘? .4 72 1 9° 1°: 3:; :83... :1; ii. I 1 :—::_:,;;; ‘OJ -1:}: 11.5 12.0 12.5 13.0 13.5 13.9 14.45 14.9 15.4 15.8 16.3 16.8 17.3- ! 1) Ti 20 21 22 ,3 25 26 27 28 29 30 31 32 33 34 35 36 T F! 361 379 p 397 4,5 1 433 451 469 487 505 5263 541 539 577 596 614 632 650 ._....__..__..__.._;.. AR-——-5 1'°3D0ndin v 1 f P '11 b . the number of periods of 183 .114 between the date in It is tag 1): 33.2.1 thv;-, if (the value of thus found carries the central date with enough to bring the date within these limits will be necessary. If I) is ‘t’ v, 1 t act (me of the neampsf, va,1ues of T fro_ui the nuinber of the year; “mm the dat)e0:lf l(:(:I,1t:‘l:ltl)erclip:se found in Table V or VI, according to the node, or ii I if iii-gative, add it, and we shall hit pon a date ilitferiiig by a iiiiiltiple of 1-iyear very nearly s. The cor- I‘able V or VI and the date of the eclipse sought. out the limits —— 770 and+236o, a value of P and T small ~s 34 TABLES OF SOLAR ECLIPSES. TABLE V.—Mean Elements of Eclipses at Ascending Node. . ; 1 2 T g g’ -, L uo T g g' L uo C°"j- Fictitious Date M00n’s Sun’s Sun’s M00n’s C°.”J.- Fictitious D_ate _M0011’s Sun’s Sun’s Moon's _ P6i“t- of Central Mean Mean An0- MeanAn0-Mean Lon- .\Iean Arg. P°””- of Central Mean Mean An0— Mean An0- Mean Lon- Mean Arg. Conjunction. maly. maly. gitude. Lat. Conjunction maly. maly. gitude. Lat. ' ” ‘—"”" I y. ‘I. o o C ‘ O y_ d_ O b O O 214 — 724 313.2334 + 68.73 344.67 221.25 ;—14.052 25 866 166.7316 + 49.43 185.37 88.92 — 0.013 215 — 72o 240.1713 — 157.00 272.'63 5 149.27 ‘i -12 536 26 924 126.1349 + 174.22 144.78 49.33 + 0.094 216. — 716 167.1093 — 24.52 200.60 77.30 ——11 021 27 1000 96.3598 — 63.83 114-75 20-57 — 0.269 217 — 712 94.0473 + 108.85 128.54 i 5.30 4 — 9.505 - 28 1058 55.7627 + 60.98 74.18 341.00 — 0.165 218 — 708 20.9853 — 117.77 56.48 - 293.33 g — 7.988 29 “I6 I5-1050 — 174-20 33-03 301-43 - 0.062 . 5 ' 219 — 705 313.1733 + 15.61 344.43 1 221.35 — 6.472 30 1173 339.8183 — 49.37 353.08 261.87 + 0.040 220 — 701 240.1112 + 148.98 272.38 149.37 i — 4-955 3! I249 3l0-0425 + 72.61 323.02 233.10 — 0.333 221 — 697 167.0492 — 77.64 200.33 77.37 ‘ — 3-439 32 I307 269-4450 - 162.55 282.45 193.53 — 0.234 222 — 693 93.9872 + 55.73 128.28 5.38 ‘ — 1.921 33 1305 223-8473 - 37-70 241-37 153-95 — 0-137 0 —. 689 299252 ._ 170,89 50.25 293.40 — 0.403 34 1423 188.2496 + 87.16 201.28 114.37 — 0.042 1 — _‘ 632 345.5815 — 46.27 15.67 5 253.82 — 0.259 35 1481 147.6517 — 147.98 160.75 74.80 + 0.052 2 — 574 '304.9879 + 78.36 335.13 214.28 - — 0.116 36 1539 107.0536 — 23.11 120.20 35.23 + 0.145 3 — 516 264.3940 — 157.00 294.57 174.68 4 + 0.025 37 1615 77.2768 + 98.93 90.17 6.48 — 0.239 4 — 440 234.6227 — 35.27 264.52 145-93 i - 6.291 38 1673 36-6783 — 136-19 49-58 326-90 . -. 0-150 5 — 382 194.0285 + 89.38 223.97 106.35 ' —- 0.156 39 1730 361.3299 — 11.29 9.02 287.33 — 0.062 6 — 324 153.4343 —' 145.96 183.43 66.78 ‘ — 0.019 40 1788 320.7313 -1- 113.60 328.47 247.77 + 0.024 7 — 248 123.6626 —- ‘ 24.20 153.35 38.00 ‘ — 0.345 41 1846 280.1325 -— 121.49 287.88 208.19 + 0.108 8 —- 190 83.0682 + 100.47 112.82 358.43 1 —- 0.210 42 ‘1904 239.5337 4 + 3.42 247.32 168.62 -|_- 0.192 9 — 132 42.4736 — 134.85 72.25 ‘ 318.85 E — 0.078 43 1980 209-7557 + 125.51 217.27 139.87 — 0.204 10 — 74 1.8789 — 10.17 31.73 1 279.30 E + 0.054 44 2038 169.1565 — 109.56 176.68 100.28 — 0.124 ~ I 11 + 1 337.3564 + 111.63 1.67 250.52 E — 0.279 45 2096 128.5573 + 15.37 136.13 60.72 — 0.046 12 59 296.7615 — 123.67 321.08 210.93 1 — 0.151 46 2154 37-9579 + 140.32 95.60 21.15 + 0.031 13 117 256.1664 + 1.04 280.53 171.35 ; — 0.024 47 2212 1 47.3584 —— 94.73 55.03 341.58 + 0..106 14 175 215.5712 + 125.75 239.97 131.78 ; + 0.102 48 2270 : 6.7587 + 30.22 14.43 302.00 + 0.180 15 251 185.7981 — 112.41 209.93 103.03 '5 — 0.239 49 5 2327 ‘ 331.4091 + 155.19 9 333.87 262.43 + 0.252 16 309 145.2026 + 12.31 169.40 63.47 _ — 0.116 50 . 2331, 258.3466 —— 71.42 261.83 190.45 + 1.763 17 367 104.6069 -1- 137.04 128.85 23.90 E + 0.005 51 : 2335 1 185.2842 4- 61.98 189.79 118.47 + 3.275 18 425 64.0112 — 98.21 88.28 344.30 7 + 0.125 52 ; 2339 ’ 112.2218 — 164.62 117.75 46.50 + 4.787 19 501 34.2375 + 23.66 58 23 315.55 — 0 223 53 i 2343 . 39.1593 — 31.22 45.70 334.52 4- 6.298 20 558 358.8915 + 148.41 17.65 275.97 E — 0.100 54 K 2340 ; 33'-3?403 + 102-13 333-65 202-53 + 7-309 1 - = 1 ~21 616 318.2953 — 86.83 337.10 236.40 + 0.008 55 2350 258.2844 —— 124.42 261.60 190.55 + 9.321 22 674 277.6990 + 37.94 I 296.53 196.82 + 0.122 56 ; 2354 185.2220 4- 8.98 189.55 118.57 +10.832 23 750 247.9246 + 159.85 266.47 168.05 ' — 0.233 57 i 2358 112.1596 4- 142.38 117.50 46.58 +12.344 24 808 207 3282 — 75.37 , 225.93 128.50 — 0.123 58 2362 39.0972 — 84.22 45.45 334.59 -1-13.856 1 The 1-iecoml and third 13111111111131 of 'I‘11l1l1-.~1 V and VI give-, with s01111- exceptioiis notvtl 011 page 8, the dates of those mean ncjv moons which occur 11eare1~1t to the 11011014. The t'n111' 1-01111111111 t'u1l0wi11g give the \':1.111e1~1 of the four p1'i111-ipal :‘lI‘gll111e11ts for these dat-es. , The 1111111111.-.1‘ of the 1-011ju111-timi point. at which 1111 eclipse 01-1.-Iirs 111113’ he fmnul froni the reslilts of Tables I, II, and IV, as follows :—P11t t for the f1'9.cti011 of :1 c -ntury which has 1-l:1p.~11-11 since the last precmliiig pa «sage of a m11j11111-timi point through the corresponding node, as found on p. 14, 111111 110 for the 11111111101‘ of that point‘-. 'I‘h1~11. the q11a11tit_v no 4- 151-» 0.641) will be nearly an i11teger, which integc-1* ‘will he the 11111111101‘ of the point sought, and the arg111ne11t- for Table V or VI. 4*-“Rh 9 T C°“j- iFictiti0'us‘_Iulian Date Point! of Central Mean 5 Conjunction. 1 y. . d.» I02 3 — 771 j+322-7073 ‘ I03 — 767 6 249-6452 I04 - 763 - 4176,5832 I05 — 759 1 103.5212 106 — 755; 30-4592 107 - 752 322.6471 I03 - 748E 249-5851 109 — 744 170.5231 I10 — 740 9 103.4611 '- III - 736‘; 30-3990 1 I 1 1 112 - 6.79 355-0555 I13 — 603% 325-2849.- II4 - 545% 284-6908 -1 "5' - 487' 244-0970 116 — 411 214.3254 i I I17 - 3533 173-7314 6 "3 -- 295} 133-1372 119 — 2374 92-5428 120 -.- 161 62.7708 I21 — 103; 22.1762 6122 —- 46 f 346.8314 3 '23 7 + 307 317.0589 I24 88 ‘ 276.4639 0 I25 140 235.8687 '26 204 195-2736 I . i '27. 230 _ 105.5003 . '28 333 124.9047 ‘29 396 . 84.3091 4 ‘3° 454 43.7133 ; 63' 530’ 13.9394 ‘32 58 . ' 133 ' 64:3 338.5936‘ ‘ ’ 297-9972 . I35 7 761 216.8046} 5; Sr’ L 7. no — 180° T M00n’s Sun’s Sun’s M00n’s C°“j- Fictitious Julian Date Mean An0- 3.\'Iean Ano- Mean L0n-!VIean Arg. Pom" 0fCentral Mean maly 4 maly. ; gitude. Lat. Conjunction. ____.__ *———-- ~ -I O I 0 7 o 0 y. 11'. ._ ;79_(,9 1 354.43 230.23 -—13.665 136 337 +137-0300 _ 3723 282.40 7 158.26 i —12.150 I37 895 140.4333 4. 96_99’ 210.36 86.28 -—'10.635 133 953 105.8365 _ 13953; 138.30 14.30 -- 9.119 I39 1011 65-2396 4. 2_3_.7 1 00.25 302.31 — 7.602 I40 1069 24-6425 .1. 136,22 1 354.21 230.33 — 6.086 I41 1144 360.1170 _ '90-,“ 282.17 158.35 — 4.569 142 I202 319.5197 .1. 42:97 I 210.12 86.30 - 3.052 143 1260 278.9222 .1. 176,534 138.08 14.38 — 1.535 In 1336 219.1402 _ 50,23 66.03 302.40 ~ 0.018 I45 I394 208.5485 + 674.33 625-45 262.82 + 0.127 146 1452 167.9500 — 163.95 f 355-37 234.02 — 0.188 147 1510 127.3527 — 39.32 7 314-82 194-43 — 0.046 I48 1568 _86.7546 + 85.32; 27.1.30 154.90 + 0.095 149 1626 46.1564‘ - 152.94 5 244.25 126.13 —- 0.225 I50 1702 16.3792 7 . —‘ 28.29 203-70 80.57 — 0.087 151 1759 341.0306 + 96.37 163-15 47 00 + 0.049 152 ‘1817 300.4319 -— 138.96 ' 122.58 7.42 + 0.184 153 1875 259.8331 — I7-I9 92-53 333-65 ~ 0.144 154 1933 219.2342 + 107-49 51-98 299-07 — 0-012 155 2009 189.4502 * I27-32 “-45 259-50 + 0.119 I56 2067 148.8570 - 6-02 = 34!-37 230-72 — 0.215 157 2125 108.2576 + 118.63 3 300.83 7 191.17 — 0.087 158 2183 67.0582 — 116.61 260.27 151.58 + 0.039 159 2241 27,0536 + 8.11 219.70’ 7 112.02 + 0.164 160 2298 3551,7099 1 + 129-95 '39-67 83.25 - 0.177 101 2302 278.6465 - I05-32 ‘49-I0 : 43-67 — 0.055 162 2306 205.5841 +, '9-42 1 108,45 4.10 + 0.005 163 2310 132.5217 + I4-I-I6 68.00 ‘ 324.52 _~1- 0.184 104 2314 59 4592 — 93-97 37-93.‘ 295-75 — 0I65 165 2317 351.6468 , 4 1 ' ' + 30.79 357.38 256.18 — 0.019 166 2321 278.5844’, + 155.50 ! 316.83 7 210.62 + 0 006‘ 167 2325 205.5219 - 79-66 276-25 177 03 + 0.179 108 2329 132.4595 + 45.1! ‘ 235.68 I 137.47 -1 0.290 169 2333 59.3971 1 TABLES OF SOLAR ECLIPSES. TABLE VI.—-Mean Elements of Eclipses at Descending Node. +13.823 35 g 5'' w L u°—I80° 7 M00n’s Sun’s E Sun’s M00n’s Mean Ano- Mean An0- Mean Lon- Mean Arg. maly 6 maly. I’ gitude. Lat. 6" __ I 1 + 167.03 205.65 108.72 — 0.068 —— 08.17 165.07 69.12 + 0.040 + 50-63 124.53 1 29-57 + o.147f — 178.50 84.00 350.00 + 0.252 - 53-75 - 43-43 1 310.43 + 0.357: + 68.22 13.37 281.05 — 0.011 E -- 160.95 332.82 242.10 + 0.090 — 42.12 4 292.27 202.55 + 0.189 . + 79.88 - 202.17 173.75 — 0.18.67 — 155.27 221.58 134.17 — 0.090 I J ! 1 — 30.41 I 181.03 _ 94.58 + 0.005 + 94.46 140.48 55.02 + 0.098 — 140.67 99.90 15.43 + 0.190; ‘ 15-79 59-33 335.90 + 0.280 + 106.26 29.32 307.13 — 0.106 I " 123-35 5 348.73 267.55 — 0.019; '7 3-95 303.17 227.97 + 0.0064 + 120.96 267.60 188.40 + 0.150 — 114.12 227.03 148.83 + 0.233 + 7.98 } 196.98 120.08 — 0.164 6+ I32'.91 . 156.42 80.50 — 0.085 — 102.15 115.87 40.93 — 0.008 + 22.79 7 75.32 1.37 + 0.008 + 147.74 g 34.73 321.78 + 0-1421 — 87.30 354.18 282.26 + 0.214 -1- 46.10 282.12 210.25 + 1.7275 + 179.50 210.07 138.27 + 3.239% — 47.10 138.02 60.28 + 4.750 ; + -86.29‘ 65.97 354.28 + 6.263 - 140.31 353.92 282.30 + 7.773 7 — 6.91 281.85 210.32 + 9.287 + 126.49 209.80 _138.34 +1o,8o0 — 100.11 137.74 06.35 +12.312 + 33.28} 65.69 354.37 In 'I:able VI the values of no are are °°“3‘d61‘0d as values of no — 180° iv o ' ' ‘ a 3 . u . , . O . g en ‘in the last column If (.01111te11 from the des(,e11d111g 110110, lmt, t-0 111.1ke the 110t.at10n unlform, these , no bemg always counted from the ascencltng node. 36 TABLES 01-‘ SOLAR ECLIPSES. TABLE VII. —-Reduction of Quantities in the Preceding Tables to Corresponding Conjnnctions in Other Cycles. ‘ - Reduction of Reduction of g. ‘ CYC1e5- 1. ‘ Change for— *1 Change for- Year .0. 1000. l 2000. I 1 Year 0. 1000. 2000. V ‘1 ‘E ‘I00;/. 200y.. 300y. 400;/.4500}/.41 ' 100;’. 200;». 3003:. 4ooy, gooy, -:1- . 1 — —— 1 1 . - ' l y. d. l .1. 1 a’. l o 1 o o 1 \ 18 10.8224 10.8217 1 10.8211 1 1 2 2 I 3 I — 2.89 j - 2.86 ; - 2.82 1 0 1 1 2 2 1 . l 1 l I 2 36 21.6447l 21.6435 4 21.6422 1 3 4 5 1 6 5.78 5.72 E 5.64»; 1 . 1. 5 2 3 4 3 54 32.4671 1 32.4652 ; 32 4633 2 4 6 8 9 8 68 1 8.58 ' 8 47 1 2 3 4 6 4 72 _ 43.2895 43.2869 ; 43.2844 3 5 8 10 13 1 11.581 11.43 ' 11.29 1 3 4 6 7 5 90 54.1119 ! 54.10861 54.1055 3‘ 6 10 13 16 . 14.47 14.29‘ 14.12 _ 2 .4 5 7 9 6 108 64.9342 64.9304 64.9265 4 8 11 15 1 19 1 - 17.36 - 17.15 ' - 16.94 2 4 6 9 11 _7 126 75.7566 1 75.7521 . 75.7476 4 9 13 18 1 22 20.26 1 -20.01 19.76 3 5 8 10 12 8 144 86.5790 86.5738 l 86.5687 5 10 15 20 ‘26 23.15 22.87 1 22.58 3 6 8 11 ' 14 9 1 162 97.4013 97.3956 97.3898 6 12 17 23 ' 29 26.05 25.73 1 25.41 1 3 6 '9 13 16 10 180 108.2237 1 108.2173 108.2109 6 13 19 26 32 28.94 28.59 28.23 4 7 11 14 18 4 1 Y | E . 11 1'98 119.0461 119.0390 119.0320 7 14 21 28 35 - 31.83 4 — 31.44 — 31.05 - 4 '11 16 19 12 ' 216 129.8684 ‘129.86o8 129.8531 8 15 23 31 ; 38 1 34.73 1 34-30 33-83 13 17 21 13 234 140.6908 1 140.6825 140.6742 8 17 25 33 1 42 37.62 37.16 36.70 5 9 14 19 "_ 23 I4 252 151.5132 151.5042 ‘ 151.4953 9 18 27 36 45 ; 40.52 40.02 39.52 5 10 15 20. 24 15 270 162.3356 1 162.3259 4 162.3164 10 19 29 38 48 43.41 . 42.88 42.35 5 11 16 22 26 16 288 173.1579 1 173.1477 173.1374 10 21 31 41 51 1 — 46 30 — 45.74 - 45 17 4 6 11 17 1 23 28 17 306 183.9803 183.9694 4 183.9585 11 22 '33 44 55 I 49.20 48.60 47.99 8 6 12 18 i 24 30 18 324 194.8027 194.7911 . 194.7796 12 23 35 46 l‘ 58 l: 52.09 51.45 50.81 1 6 13 19 26 ‘ 31 19 1 342 205.6250 ‘1 205.6129 9 205.6007 12 24 37 49 4 61 1 54.99 i 54.31 53.64 7 13 '20 1 27 33 20 1 360 216.4474 ' 216.4346 1 216.4218 13 26 38 51 ~ 64 4 57.88 57.17 56.46 ‘ 7 14 21 29 35 1 1 ' 3 l * . 1 ' . 4 l ; 1 1 21 378 227.2697 i 227.2563 227.2429 14 27 40 54 67 : - 60 77 ‘ — 60.03 — 59 28 7 15 22 30 37 22 3 396 238.0921 1 238.0781 238.0640 14 28 42 56 71 1 63.67 3 62.89 62.11 8 15 23 i '32 39 1 1 ‘ 1 ] . X 23 414 248.9145 1 248.8998 248.8851 15 29 44 g 59 3 74 66.56 . 65.75 64.93 8 16 :4 24 4 33 ‘ 41 24 432 259.7369 I 259.7215 259.7062 15 31 ‘ 46 62 77 ’ 69.45 68.61 67.75 8 17 25 34 42 25 i 450 270.5593 8 270.5432 270.5273 16 32 5 48 E 64 1 80 ? 72.35 71.46 70.58 A 9 17 ! 26 1 36 44 1 2 1 1 1 . , l » 26 1 468 281.3817 ‘ 281.3650 281.3483 17 33 i 50 66 83 < - 75.24 4- 74.32 — 73.40 9 18 27 » 37 45 _ 27 486 292.2041 j 292.1867 4 292.1694 17 35 52 : 69 86 78.13 77.18 76.22 _ 10 19 i 28 ’ 38 '47 28 504 303.0264 303.0084 302.9905 18 36 E 54 I 72 f 90 f’ 81.03 80.04 79.05 V 10 20 29 39 49 29 E 522 313.8488 1 313.8302 313.8116 18 37 56 E 74 ; 93 1 -83.92 82.90 ‘ 81.87 j 10 20 ' '31 Q 41 51 30 540 324.6712 324.6519 324.6327 19 38 58 8 77 96 4 86.82 85.76 1 84.69 A 11 21 32 E 42 53 1 1 i 1 1 . 2 . 1 4 31 558 335.4935 5 335.4736 335.4538 20 40 1 59 79 4 99 ; — 89.71 4 — 88.62 4 - 87.52 1 11 122 33 l 44 55 32 576 346-3159 V 346.2954 346.2749 20 41 61 82 9 102 92.61 1 91.47 l 90.34 1 11 23 34 4 45 56 33 594 357.1383 1 357.1171 3 357.0960 21 42 _ 63 84 106 95.50 1 94.33 93.16 ' 12 23 35 1 47 58 34 613 2.7107 1 2.6888 2.6671 22 44 1 65 87 109 ; 98.39 1 97.19 95.98 12 24 36 48 60 35 631 13.5330‘: 13.5105 13.4882 22 45 I 67 ~ 112 1 101.29 1 100.05 98.81 1 12 25 37 50 62 . I 90 1 ‘ 1 . 1 1 1 1 E 1 1 ' 4 § f 1 _ I 36 649 24.3554 1 24.3322 ~ 24.3092 23 46 l 69 Q 92 115 2 -104.18 1 -102.91 . -101.63 3 13 25 38 i 51 63 37 667 35.1778 35.1539 35.1303 24 47 i 71 1 95 1 119 g 107.07 105.77 104.45 j 13 26 39 4’; 52 65 Having identified the central eclipse of the series to which the required one belongs, the times and arguments from Table V or VI are to be reduced to the required date, for the number of periods elapsed, by means of Table VII. Here the time is to correspond to the middle of the elapsed interval. If the eclipse examined precedes the central one in time, the signs of all the quantities are to be changed. TABLES 01-‘ SOLAR ECLIPSES. 37 T LE VII Rectnction of Quantities in the Preceding Tables, ete.—C0ntinued. AJ3 --- 9 - . R 1 Reduction of u. Reduction of g’. Reduction of L. 9 9 H 9 9 9 9- _ 9 I _ , ~~ 6"””0””0 I I I (Sh r -— I Cycles. _- - --—ang~¢- 2 ~- N Year 0. 2000. Year 0. 2000. Year 0. 1000. 2000- me} 200),. 300),. 4999,, gooy. o 0 ° O O 0 O . l _. , 6 —- . 0 - 0-473 1 2 2 9 3 1 + 19.59 + 10.49 +_ 10.80 + 10.80 0 4 E2: :3) 470 0 956 I 3 4 ; 6 7 2 29_99 29,99 21.61 21.61 0.92 -94 1.433 2 4 7 9 A H 3 31 49 3‘-43 32'4I * 32°41 I 389 I :81 . 11 3 6 9 E I2 l 15 % .8 . I I-9 - 4 41-99‘ 41-98 43'” , 43.22 I SI 2 1 2 389 4 . 7 11 : I5 , I9 5 52.48 52.47 54.91 E 54-02 2-314 -35 ' 9 - 5 > . 1 13 I 18 22 , __ _. ,8 —- 2.867 4 0 6 4. 92,93 9 .2 62,97 4- 64.82 I -2 64.82 2.777 2 22: 3 345 5 IO I6 , 21 26 7 73.48 ‘ 73.46 75 62 I ,7s.63 3 24° 3- E 3 822 6 I2 [8 I 24 . 30 ' _ 8‘, . 03 3-7 2 - ' I 8 83.98 83.96 86 42 V 0 43 3 :66 4 233 4.300 7 9 13 20 5 27 , 34 ' 9 94 48 94 45 97 23 97 24 4'6 8 ' .4 778 7 .15 22 I 30 1 37 . 10 104.97 104.94 103-03 103-04 4- 2 4'70?’ ‘ . 8 16 025 . 33 ' 41 _ 3,3 .2 8.84 - 5.991 —— 5 I73 —- 5 256 . _ 11 +115.47 +115 44 +11 3 11 I8 27 3 36 45 12 125.96 125.93 129.64 129-65 5-554 5-644 5.734 9 _ , ' - 1 0 140 45 6 017‘ 0'-114 0-211 1° I9" 29 3 39 48 13 136.46 136.43 _ 4 -44 .- A - 6 68 It 2! 3! 42 52 14 145,96 146,92 ‘ 151.24 151.25 6.480 6.584 . 9 H 22 34 45 56 ,5 157,45 ‘ 157,42 162.04 162.06 6.943 7.054 7-I07 _ 3 86 _ O6 _ 2 __ -71,45 12 24 36 ' 48 60 16 +167.95 +167.9I 2172-35 . +172- 7-4 7-5 5 ; 17‘ 178.45 178.41 I33-05 133-07 7-303 7-995 3-123 I3 25' 38 3 5‘ 64 18 188.95 188.90 194.45 ‘ I94 47 . 8 331 8 465 3-600 13 27 4° 3 54 * 68 19 199,44 199,39 205.26 '| 205.27 3.794 8-936 9-078 14 28 43 57 ' 71 20 299,94 299,89 216.06 1 216.08 9.257, 9.406 9-550 15 3° 45 ; 0° 9 74 r 3 ‘ 21 +2-20,44 +22o,38 +226.86 E -I-226.88 — 9.720 " 9-876 "10-034 I6 3! 47 63 78 22 230.94 230.88 237-67 237-03 10-133 10-347 10-512 16 33 49 . 66 82 23 241.43 241.37’ ' 248.47 1 248.49 10.646 10-317 10-939 ' 17 34 52 i 69 86 24 251.93 251.87 259.27 ] 259.29 11.108 11.287 11.467 13 30 54 72 9° 25 262.43 262.36 270.07 : 270.10 11.571 11-753 11-945 I9 37 S6 75 03 26 I +272,92 +272.86 +28o.88 +28o.9o -12.03-4 —I2.228 -12-423 I9 39 ~53 73 97 27 283,42 283,35 291.68 3 291.70 12.497 ‘ 12.698 12.901 20 40 01 31 1°‘ 28 293_92 29335 039243 392,5; 12,969 13.168 13.378 21 42 9 63 3 84 I04 29 304.41 _ 394.34 313.29 7 313-3! 13 422 I3 639 13-856 22 43 65 i 37 103 30 314.91 314,83 324.09 324.12 13.8859 14-109 14-334 22 45 07 I; 90 112 u E ! ' i 31 -1325.41 .+325.33 -1334-89 4 -6334 92 -I .348 -I4-579 -I4-312 23 4° 7° 1 93 “° 3’ 335-9! 335.82 345-69 I 345 72 14.811 15.959 15.290 24 48 72 96 I20 33 346.41 349_32 , 356,59 3 356.52 15.274 15-520 15-703 25 49 74 99 v 124 34 35590 3563, . 7_39 E 7,33 15,736 15.990 16.245 25 51 76 102 g 127 35 7.39 7.31 18.10 ; 18.13 16.199 16.460 16.722 26 52 79 105 131 I . 36 4. 1739 + I7_3o .4. 23,99 E 4.. 28.94 —16.662 —-16.93: -—17.200 27 54 81 108 134 37 28,39 23,29 39,70 ‘ 39.74 17.125 17.401 17.678 _ 28 55 83 111 138 Having identified the central eclipse of the series to W be reduce elapsed i hich the required one belongs, the times and arguments from Table V or VI are to (1 to the required date, for the number of periods elapsed, by means of Table VII. Here the time is to correspond to the middle of the Ilterval. If the eclipse examined precedes the central one in time, the signs of all the quantities are to be changed. ‘ TABLES OF SOLAR ECLIPSES. TABLE VIII, Arg. g.—Fo-r Reduction to Momeni of True New Moon. 6 T = — 04.4089 sing + 08.0161 sin 2g — 04.0004 sin 39. ! I O O O O O O O O 1 0 g 0 10 20 30 40 50 60 70 i 80 9 .1. .1. .1. .1. .1. .1. 4. a’. . 0 .1. o 0 —.0000 + .0657 + —-.1298 + —.1910 + .2473 + —.2975 + — 3402 + -—.3736 + i — 3969 + 10 66 65 63 59 53 46 28 ? I7 1 .0066 .0722 .1361 .1969 .2526 .3021 .3440 .3764 3986 9 i 66 65 62 53 37 27 1 ‘ Is 2 .0132 .0787 .1423 .2027 .2579 .3067 . 3477 .3791 4001 8 66 65 52 45 26 15 3 .0198 .0852 .1486 .2085 .2631 .3112 .3513 .3817 4016 7 5 65 65 61 57 51 44 35 25 ; I3 -.0263 + -0917 + -.1547 + -.2142 + .2682 + -.3!56 + — 3548 + .3842 + ’ - 4029 + .6 66 64 . 62 57 51 43 34 24 i I2 . I : S --0389 + -0951 + --‘(>09 + ''-2'99 4- -2733 4' —-3|99 + --3532 + --3366 + — 4041 + 5 : 66 , 64 61 56 50 42 33 23 _ 11 6 .0395 . 1045 . 1670 ’. 2255 .3783 . 3241 .3615 . 3889 . 4052 -4 65 64 61 55 49 42 32 22‘ 10 7 0460 . 1 109 .1731 .2310 .2832 . 3283 .3647 . 391 1 .4062 3 66 63 60 55 48 40 30 20 9 8 .0526 .1172 .1791 .2365 . .2880 .3323 .3677 .3931 .4071 2 "- 66 63 60 54 48 40 30 ' 20 8 . 9 .0592 .1235 .1851 .2419 .2928 .3303 .3707 .3951 4079 1 65 63 59 54 47 -39 29 I8 6 ; 10 —.0657 + .1298 + —.1910 + —.2473 + .2975 + —.3402 + —.3736 + —.3969 + - 4085 + 0 350° 340° 330° 320° 310° 300° 290° 280° 270° g i E S g 90° 100° 1 10° 1 20° 1 30° : 40° 1 50° 1 60° 1 70° § 0 1 4 . - ~— — _— -_. —- o d. d. d. 1 d. d. d. d. :1. ~ d. 0 0 —-.4085 -1- 4079 +- — 3941 + — 3680 + .3293 + —.2791 + —.2188 + — 1506 + —.0767 + 10 . s . 8 2! : 33 45 65 72 I .4090 .4071 3923 ; 3647 .3248 .2735 2123 I434 -0691 9 3 22 1 47 57 66 72 L 2 .4093 .4062 .3901 3 .3612 .3201 .2678 2057 1362 .0615 ‘8 ? I0 ‘ 23 5 ' 47 67 72 77 3 3 .4096 .4052 3378 fig 3576 .3154 .2620 1990 1290 0538 7 I It 24 2 37 49 ' 67 73 5 4 --4097 + .4041 + - 3854 + T — 3539 + .3105 + - 256! + — I923 + — 1217» + -- 0462 + 6 7 - 0 13° 26 50 60 68 74 77 5 —.4097 + _.4028 -4- — 3828 + 4 ~ 3501 + .3055 + —.2501 + — 1855 4- -.1143 -1- — 0385 -4- 5 ’ I I4 27 1 39 5! 74 77 6 . 4096 . 4014 .3801 * 3462 3004 .24 40 1 786 . 1069 0308 4 2 16 28 3 41 52 75 77 7 -4094 .3998 3773 1 .3121 .2952 .2378 1717 .0991 0231 3 4 16 | 41 52 62 70 75 77 8 .4090 .3982 3743 ; .3380 .2900 .2316 1647 .0919 0154 2 5 ' I9 1 43 54 _ 70 77 9 .4085 .3963 .3712 1 .3337 .2846 .2252 1577 .0843 0077 1 6 I9 32 L 44 55 71 77 .10 —.4079 + 3944 4. _ 3680 + 5 - 3293 + .2791 + —.2188 + — 1506 +' ~.0767 + —.00o0 + 0 I .. i L 260° 250° 240° E 230° 220° 210° 200° 190° 180° g TABLES OF SOLAR ECLIPSES. TABLE IX,» Arg, q’.—For Reduction to Moment of True New Moon. 39 «ST -_= + od.17.;3 sin g‘ + 05.0021 sin fig’. g 0° 100 20° 30° 40° ‘50° 60° 700 800 ; + 4- a’. d. :1. d. . a’. d. 4. d. .. .ooo03! — +.o31o -— .o6o93‘o-— +.oS9o26 — +.II4O2 — .1356 — +.1528 — .1651 — +,;-724 _. 10 I '°°-°" -°34‘ -0639 0916 .1165 S -1375” I542“ 166110 1723 4 9 32 3° . 29 27 22 . . . A 2 .oo6 ‘9 ‘4 33° .o3713° .0668“ .o94326 . 118723 . 1394 .1556 1670 , 1732 4 3 3 -0093 0401 .05 5 ’ ‘9 ‘4 9 3 4 + O 3! I . 9 2.9 .o96925 .1210” 1413}? .l57O13 1679 ,1-735 7 - 124 — +.o432 _ _ _ _ _ , 2 32 30 072527 + 099425 + . I232” . 143017 - + . 158313 — 1687 7 —. + .1737 _. 6 » 3 5 + .0156 -— , 5 __ __ A 6 . 8 3! + 04 23° .0752” +.1ot9’6 ‘ '+-.I253” ‘ 444718 e- +. 1596 — 1694 6 -- +4740 _ 5 .0! 7 0 ' 8 , *3 2 7 021730 49229 07 I27 . 104524 . I275“ . 1465“ , moan .700 _ I 742 4 ' 0521 0808 . I069 _ 1296 _ I481 I620 7 ’ 3 -02483‘ .o5S13° 083628 I 24 .. °° *5 I0 I707 6 .1743 o 3 3! 29 . 27 o .1316 .1497 .1630 [7[3 .1743 2 3 9 0279 .0580 .0863 . 1117 . 1.33620 :5 n 6 , t , I 3: - -1513 .164! 1719 1 39 37 23 3° - 744 I I 30 +.o3xo — +.o6o9 — 0390 _. + ".0 _ 15 1 to 5 , ' 35°° 340° 330° 320° ° o . I 3” 30° 290 280° 270° K. 5 I I 5" 90° 100° 1 30° 0 o . E I 20 '30 I40‘) 1 50° I60‘ 170° ___l o (1. Q d (1 d “_.—_§—~ —- b ---+—__-._..- ' - . an 4. ° +4743 1 - 3 +.l7lO _ 1625 — 4-.1492 -— +4314 .. "00 _ 0:2 4'' d‘ ° 1 I742 7 12 17 2° 2 ' 5_ ‘ .0583 — + _()296 _ to j .1 .l- 25 29 3! 2 I7 I I I 5 xx 475:6 29420 ‘O7? ‘OB27 -0554 0265 9 4 2 ‘ ‘ n ‘ . . 24 27 27 29 3 3 I2 18 53 0300 -0527 0236 8 ‘ 739 . 1689 I 590 I 4 ’° 24 26 ,3 3 ° 4! ° I254 1029 .0774 O ' 29 4 + .1736 7 *9 . :6 ,, . - 499 L 0207 7 9 ‘ -— . + .1682 — 1578 ._ + , 1425 _ +_,233 _ mos“ + .127 . 29 3., I 8 x4_ ‘8 22 24 —. .074.” '- .0470” —— + 0177 __ 6 , 29 f 5 ~+-.1732 + '6 3 ‘ - 74 - I564 — . — _ ‘ 5 W293 ‘66 9 + 1407” +4211“ .098I2s— +.o72o2 — ,o4_” '_. _,__o,48 _ 5 5 7 X72 4 ° 5.0 "55' ~I39oI_9 .n9o cqss .o693 7 .0412” 011830 4 ’ . 22 5 . 165; I536 .137! ‘H68 .093!” 066627 29 ,9 8 I720 16 6 9 ‘4 :9 33 ‘ 28 -0383 0089 3 ‘ _ . 4 I 1522 .1352 .1145 .090‘ 0638 4 28 3° 9 .1716 16 t ‘4 I9 22 ‘,7 ’ 2 '°355 0059 2 . « 35 I508 . 1333 . H23 0378 of 7 3° 3° 10 + I to ,5 I » . in ,o3g5 0°29 ' 7‘° " “I-.1625 - I492; _ +.!3I49_ +.IIoo93 _ ‘O8S226— + 0‘8 28' 29 ,9 I "“ \~ __~_ - ' 5 3 " 0295 - + .0009 — 0 260° 250° 2 9 .-I . 40 o ¢ 0 O 230 220 gm 200 I90. I80, g’ ‘-666 40 TABLE X.-—Arg. g+g’. - «ST = -04.0051 sin (g+g'). g+g' 61 0' (1. . 0 0 0.0000 360 1° ," 09 +- 350 20 ‘E 17 340 30 ‘2s 330 40 i 33 320 50 § 39 310 60 ' 44 300 I 70 48 290 80 ' 50 280 90 51 270 100 50 260 I10 48 250 120 44 240 ‘3°-- -39 230 I40’ 33 220 150 25 210 160 17 200 170 - 09 4- 190 180 0.0000" 180 g+g'- The sum of the quantities from Tables VIII to XII, inclusive, being applied to the time T of mean conjunc- tion of the sun and moon in longitude, will give the Greenwich fictitious mean time of true conjunction in longitude. To reduce the fictitious time thus found to the ordinary calendar, a correction from the table following is to be applied. The correction for bissext-ile years presupposes that January 1 is counted as the zero day of the year. TABLE XI.—Arg. g—:q'. TABLES OF SOLAR ECLIPSES. l_ :__ i {X- «ST = + 04.0075 sin (g —-g’), * Eg—gC 61 ° d. o 0 0.0000 360 10 4- 13 -— i 350 20 26 E 340 ‘ 30 37 E 330 40 48 E 320 S0 57 310 V 60 65 i 300 i 70 70 E 290 ‘ 80 74 j 280 90 75 E 270 100 74 E 260 I10 70 250 I20 65 i 240 . 130 57 E 230 ‘ _ I40 48 ! 220 ; 15° 37 210 I 160 26 200 170 +- 13 -— 190 , 180 0.0000 180 I g'+-gfl » i TABLE XII.—Arg. u. 6T = + 0d.0104 sin 2 u. ,,',e, I l i 5 l I-O C I-I F! 9-1 N r-4 9) I-I 4|- I-It-I O\tn n-an-1 ®\l in C ~o0o~10~un.s.~.0::o ) 6T. a’. 0.0000 04 07 11 I4 18 4- 22 25 29. 32 .36 39 42 46 49 52 55 58 61 + 0.0064 TAB LE XIII a. 5 4 I 1 In order that it may correspond to the civil count of days, it must be increased by 1. TABLE XIII b.——F0r Reducing Fictitious Julian Dates to those of the Ordinary Calendar. Calendar and Limiting Dates. Julian calendar - Bissextile Years. Gregorian calendar, 1582 to 1700, February - Gregorian calendar, 1700, March, to 1800, February - - Gregorian calendar, 1800, March, to 1900, February - - Gregorian calendar, 1900, March, to 2100, February - - - Gregorian calendar, 2100, March, to 2200, I~‘ebruary - - - Gregorian calendar, 2200, March, to 2300, February - - - J. 4- (moo nxoo 11.00 nzoo ryoo npoo -k 15.00 For the further expression of the time in days and hours, Tables XIII a and XIV are added. { Day of the Year to Day of the I Month. 3 Common Bissext, ‘ Year. Year.. ‘ , Jan. 0 0 — 1 10 10 9 I 20 20 19 5 Feb. 0 31 30 i 10 41 40 E 20 I 51 50 ' Mar. 0 59 z I0 : 59 ; 20 : 79' April 0 I 90 .10 , 100 g 20 110 3 hlay 0 E 120 23; 3.3 {June 0! 151 ; 10 161 20 171 July 0 181 10 191 20 201 Aug. 0. "212 1 10 222 20 232 Sept 0 243 10 223 20 2 3 Oct 0 273 I 10 283 i 20 293 1 Nov 0 304 10 314 ’4, 29 324 | Dec 0 334 I0 344 ‘ 20 354 Year 1 Year 2 Year 3 after Bis. after Bis. aft-er Bis. d. ' d. J. + 0.25 + 0.50 + 0.75 10.25 10.50 10.75 11.25 11.50 11.75 12.25 12.50 12.75 13.25 13.50 13.75 - 14.25 14.50 14.75 + 15-25 + 15-50 + 15-75 TABLES OF SOLAR ECLIPSES. 41 TABLE XIV.—For Changing -Decimals of a Day to Time and Arc. .,M_2_,_k3 ,3_ _ _, 7 7 1 T T 3 ? - - E FOP 170- For I““‘- . f For 3 For ._T , T- Time. Arc. T Time. Arc 100 10°00 I i Time ‘ Arc. Time Arc ¥ Time. 5 Arc. I Time. ! Arc. 2 ' e 7 7“ —-7 — ‘- ,d- /1. 7/2. .1‘. ° ’ m .1‘. 5 ° ' 5- I ’_ I1. ! /z m. .1‘ ° I m. .1‘. ° 1 . .1-, r , 0.01 0 14 24 . 3 36. 0 8.64 7 0 -2.16 0.09 I 0.02 0.51 I 12 14 24 183 36 7 20.64’ 1 50,16 f 4-4! 1,10 L , 1 : °°°2 0 28 48 7 12 0 17.281 0 4-,32 0 I7 I 004 0-52! 12 28 48 187 12 7 29.28‘ 1 52.32, 4-49 1,12 6 0.03 0 43 12 10 48 0 25.92 I 0 6.48 0 26 1 0 06 0.53 E 12 43 12 .‘ 190 48, 7 37,92 1 54,43 1 4.58 1,14 _ 0'04 0 57 36 I4 24 0 34-56 1 0 3-64 0-35 ' 0 09 0-54 E 12 57 36 5 194 24 7 46.56 1 56.64 E 4-67 1.17 1 0.05 1 12 0 18 0 0 43 20 , 0 10.80 0 43 0 11 0.55 7 13 12 0 1 193 0 7 55,20 1 53,36 ' 4.75 1,19 0.06 1 26 24 21 36 0 51.84 ! 0 12.96 I 0.52 1 0 13 0.56 i 13 26 24 , 201 36,1 8 3.84 _ 2 0,96 : 4.84 1,21 0.07 I 40 48 25 12 - 1 0,43‘ 0 15.12 0.60 o 15 0.57 13 40 48 205 12 , 8 12,48 1 2 3,12 ; 4.92 1,23 . 0.08 1 55 12 28 48 1 9 12’ 0 17.28 0.69 0 17 0.58 I 13 55 12 { 203 43 . 3 21,12 ‘ 2 5,23 5.01 1,25 °-09 2 9 36 32 24 1 17 76 0 19-44 0 78 , o 19 0-59 14 9 36 i 212 24] 8 29.76 6 2 7.44 , 5-10 1.27 ‘ °"° 9 24 0 36 0 1 26.40 0 21.60 0.86 6 0.22 0.60 14 24 0- 1 216 0 8 38.40 i 2 9,60 1 5.18 1,30 1 ? . 1 0.11 2 38 24 39 36 1 35.04 0 23.76 0.95 1 0.24. 0.61 14 38 24 , 219 36 j 3 47,04 2 11,76 , 5.27 1,32 0.12 2 52 48 43 12 1 43.68 0 25.92 1.04 ; 0.26 0.62 14 52 48 7 223 12 1 8 55.68 2 13,92 7 5.36 1,34 ' 0'13 3 7 ‘2 46 48 1 52-32 0 23-08 V 1.12 1 0-23 0-63 15 7 12 E 226 48? 9 4.32‘ 2 16.08 5 5-44 1.36 1 °-‘4 3 21 36 50 24 2 .96 0 30.24 1.21 0.30 0.64_ 15 21 36 E 230 24§ 9 12,96, ,2 13,24 7 5.53 1,33 . 0.15 3 36 0 54 0 2 9.60 0 32.40 1.30 0.32 0.65 15 36 0 7 234 0 P 9 21.60 ; 2 20.40 7 5.62 1,40 ' I l I ~ 0-I0 3 S0 24 57 36 2 13-24 0 34-56 1.38 I 0-35 0-66 I5 50 24 ! 237 36: 9 30.24 f 2 22.56 g 5.70 1,43 0-17 4 4 48 61 I2 2 26.88 0 36.72 1 47 0.37 0.67 16 4 48 1 241 12; 9 38.88 ; 2 24.72 6 5.79 1,45 , 0.18 4 19 12 64 48 2 35.52 0 38.88 1.56 0.39 0.68 16 I9 12 5 244 48: 9 47.52 2 26.88 3 5.88 1,47. ? 0.69 4 33 36 68 24 2 ‘”"6' ° 4‘°°4 ‘-54 1 °-4‘ 0-69 16 33 36 E 248 24 9 56.16 1 2 29.04 5.96 1.49 7 0.20 4 48 0 72 0 2 52.80 0 43.20 1 73 f 0.43 0.70 16 48 0 3 252 0, 10 4,30 2 31,20 ; 6,05 I SI ? i ’ « - ; ' 0.21 5 2 24 75 36 3 1.44 0 45.36 1.81 1 0.45 0.71 7 I7 2 24 ii 255 36. 10 13,44 1 2 33,36 ' 6,13 1,33 0.22 , 5 16 48 79 12 E 3 10.08. 0 47.52 1.90 0.48 0.72 17 16 48 E 259 12 10 22.08 _ 2 35.52 : 6,22 1_;6 0.23 5 31 12 82 -48 j 3 18.72 0 49.68 7 1 99 0.50 0.73 17 31 12 1 262 436 10 30_.2 f 2 37,63 6 31 I 8 0.24 5 45 36 86 24 i 3 27.367 0 51.84 : .07 0.52 0.74 17 45 36 1 266 24- 10 39.36 ‘ 2 39,34 ' 6,39 1.30 0'25 6 0 0 90 0 1 3 36 001 0 00 i 2 16 0-54 0 * 18 ' ' 7 - 7 54- 7 -75 7 0 0 , 270 0 »10 48.00 2 42.00 6.48 1,62 :.:: Z :3 :2 Z: 5: 6 3 94.631 0 52.16 7 2.25 0.56 0.76 1 18 14 2, 1 273 36 ,0 56,64 2 44,16 Z 6,57 1,64 . 0 8 , 3 53.2 0 5 .32 7 2.33 0.58 0.771 18 23 43 7 277 12 11 5,23 2 46,32 3 6,65 ,,66 '2 6 43 I2 I00 48 4 1°92 1 0-43 i 2 42 0-6° 0-73* 13 43 12 280 48 11 13.92 2 48 48 6 6 74 1 63 . 0.2 6 ' ' . ' , , ' ' O 3: 7 5: 36 10: 24 4 10.56 1 2.64 ? 2.51 0.63 0.79 I 18 57 36 284 24 11 22.56 2 50.64 § 6.83 1,71 - 0 10 , _ 0 4 ‘9'2° ‘ 4-8° _ 2-59 °-55 0-30 I9 12 0 _ 288 0 11 31.20 2 52.80 , 6.91 1,73 : 0.31 1 , ; , O 32 7 26 2: 111 36 4 27.84 1 6.96 2.68 0.67 0.81 19 26 24 , 291 36 '11 39.84 2 54,96 , 7.00 1,75 ‘ 0,33 7 4° 4 I‘5 I2 4 36-48 ‘ 9-12 2-70 0-69 0-32 19 40 48 j 295 12 711 48.48 2 57.12 5 7.08 1,77 0,34 8 55 I2 I38 48 ' 4 45 I2 I I‘-28 2-35 0-71 0-83 I9 55 12 ' 298 48 -11 57.12 2 59.28 7.17 1,79 L O 35 8 2: 36 I22 24 I 4 53 76 I ‘3 44 2-94 0-73 0-34 20 9 36 302 24 12 5.76 3 1.44 7.26 1,31 1 ' 0 126 0 : 5 2.40 1 15.60 3.02 0.76 0.85 20 2 0 06 0 12 1 0 60 7 3 E I 4 3 7 4-4 3 3- -34 I. 4 - 0.36 8 33 1 0,37 8 52 2: 129 36 5 II-04 1 17-76 3.11 0.78 0-36 20 38 24 309 36 :12 23.04 3 5.76 7.43 1,36 6.38 9 4 Sm 12 SW68 ' “’'9’* 3-’° 0-30 °-37 20 52 48 « 313 12 I 12 31-68 3 7-92 7-52 1.88 O 39 7 12 7136 48 5 28.32 1 22.08 3.28 0,32 0.88 21 7 12 316 43 12 40,32 3 10,03 , 7,60 1_9o ~ ' 9 21 ' 3 : 0,40 9 36 3° J 140 24 5 30-90 1 24-24 I 3.37 0.84 0.89 21 21 36 320 24: 12 48.96 3 12.24 ; 7.69 1,9, 0 I44 0 5 45-60 I 26-401 3 40 0 86 0 90 21 36 o 324 0‘ I2 57 60 3 14 40 7 78 1 , K. o . 0 - . . . .94 °-4‘ 9 50 F i 1 0,42 10 24 ! 147 36 5 54.24 1 28.56 7 3.54 0.89 0.91 21 50 24 ,327 36; 13 6.24 3 16.56 5 7.86 1,97 (M3 ,0 ,4 48 ‘5‘ ‘Z 4, 6 2 33 I 30 72 i 3-63 0-91 0-92 22 4 48 331 12 g 13 14-88 3 18-72 6 7-95 1.99 O 44 IO 39 ;: ,15g 48 7 6 11 52 1 32.88 g 3.72 0.93 0.93 22 19 12 1 334‘ 48 i1; 23.52 3 20,33 8,04 2,01 ~ 615 24 . 6 20.16 1 35.041 .8 . . 1 2 6 ; 3 .6 . 8. 0 45 10 48 0 F162 0 g 6 28 80 1 37 20 ' 3 8: Z Z: 6 3: 2: :8 30 1 :32 26 6:: :6 80 3 :3 :4 8 6: 2.03 1 ' ' i - - - . , - 3 5- 0 7 - 2.05 0. 6 T 3 6 i ' 7 ’ 2 0:7 2 24 1“’5 3,6 . 037-44 I 39-361 3-97 0-99 0-96 23 2 24 ’ 345 36;13 49-44 1127-36 8.29 2.07 . I , _ 1 . 0 48 II 48 .169 12 , 6 46.08 1 41.52 4.06 1.02 0.97 23 16 48 349 124 13 58.08 3 29,52 : 3,33 2,10 0 49 It 3' ‘2 :‘72 48 1 6 54.72 1 43.68 4.15 1.04 0.98 23 31 12 352 48% 14 6.72 3 31.68 i 8.47 2,12 0,50 ,2 45 35 7176 24 7 3-36 1 45-84 4-23 1-06 0-99 23 45 36 356 24:14 15-36 333-84 8.55 2.14 ;__h______ 0 0 ,180 0 7 12.00 1 48.00 1 4.32 1.08 1.00 24 0 0 360 01 14 24.00 3 36.00‘ 8.64 2.16 42 TABLES OF_ SOLAR ECLIPSES. TABLE XV, Arg. g.— Values 80] u, — u... u; — u0 = - 0°.4o3 sing + 0°.o16 sin 2g. g 0° _ 10° 20° 30° 40° 50° 60° 70° 80° 0 —.ooo+ -.064+ -—.128+ —.188+ —.243+ —.293+ --.335+ —.368+ —.391+ 10 I -006 -071 -I34 .193 .249 .297 .339 .371 .393 9 2 -013 -077 . I40 . I99 . 254 . 302 -343 . 374 . 395 8 3 .019 .084 .146 .205 .259 .306 .346 .376 .396 7 4 — .026+5 —.o9o+ —.152+ —.211+ - .264+ —.311+ -.349+ —.379+ _.397+ 5 5 -'-032+ --096+ —.158+ —.216+ —.269+ — 315+ -.353+ —.-.381+ 8--399+ 5 6 .039 . I03 .164 .222 .274 .319 .356 .383 .400 4 7 -045 - 109 -170 .227 .279 .323 .359 .386 .401 3 3 -052 .115 .176 .233 .283 .327 .362 .388 .402 2. 9 -053 I2! .182 .238 .288 .331 .365 .390 .402 1 10 —.064+ —.128+ -.188+ —.243+ —.293+ —.335+ —.368+ —.391+ —.403+ 0 350° 340° 330° 320° 310° 300° 290° 280° 270° g‘ V g 90° 100° 110° 120° 130° 140° 150° 160° 170° I 0 —.403+ —.402+ —.389+ —.363+ -.324+ —.27s+ -—.215+ —.148+ _.075+ 10 I E .403 .402 .387 .360 .320 .269 .209 .141 .003 9 2 .404 .401 .385 .356 .316 .264 .202 .134 _06_0 3 3 .404 .400 .383 .352 .311 .258 .196 .127 .053 7 4 —.404+ —.399+ —.380+ —-.348+ —.306+ —.252+ —.189+ —.I20+ —.045+ 6 5 ---404+ --397+ -.377+ --344+ -.3°I+ -.246+ —.182+ —.112+ —-.038+ 5 6 .404 .396 .375 .340 .296 .240 .176 .105 .030 4 7 .404 .394 .372 .336 .291 .234 .169 .097 .023 3 8 .403 . 393 . 369 . 332 . 286 . 228 . 162 .090 .015 2 9 .403 .391 .366 .328 .281 .222 .155 .033 _003 I 30 --.402+ -.339+ -.363+ -.324+ --275+ -.2l5+ —.I48+ —.075+ —.ooo+ 0 260° 250° 240° 230° 220° 210° 200° 190° 130° 3. TABLES OF s0LAR ECLIPSES. TABLE XVI.-.-—Arg. g’. 43 241_—- 240 =‘+ 2°.094 sing’. + 0°.o27 sin 2g’. \ I 1 i 2.} g’ 0° 10° 20° 30° 40° 50° 60° 70° 80° . 0 0.000 +0.373 - +o.734 - +1.070 — +1.373 8 -e +1.631 . - +1.837 +1.985 — +2.072 — 10 37 35 32 2 23 I7 22 s I +0.038 — 0.410 0.769 1.102 1.401 1.654 1.854 1.997 2.077 9 37 34 32 27 22 I7 I0 4 2 0.075 0.446 0.803 1.134 1.428 1.676 1.871 2.007 2.081 8 37 37 35 °27 22 17 It 4 ‘ 3 0.112 0.483 0.838 1.165 1.455 1.698 1.888 2.018 2.085 7 36 34 31 27 22 :5 9 3 4 +0 150 — +0.519 ._ +0.872 —- +1.196 _ +1 482 — +1.720 —- +1.903 — +2.027 — +2.088 — 6 37 36 34 30 2! Is 9 3 5 +0 187 — +0.555 &- +0.906 — +1.226 — +1 508 - +1.741 - +1.918 +2.036 8 -‘ +2.091 — 5 37 3 33 30 25 20 rs 2 6 0.224 0.591 0.939 1.256 1.533 1.761 1.933 2.044 8 2.093 4 — 33 30 25 20 14 1 7 0.262 0.627 0.972 1.286 1.558 1.781 1.917 2.052 2.094 3 37 33 29 25 I9 23 7 0 _ 8 0.299 0.663 1.005 1.315 1.583 1.800 1.960 2.059 2.094 2 37 35 ‘ 33 29 24 I9 1 1 9 0.336 0.698 1.038 1.344 1.607 1.819 , 1.973 3 2.066 7 2.095 1 37 36 .32 29 24 18 - 12 6 1 10 +0.373 — +0.734 — +1.070 — +1.373 — +1.631 — +1._837 - +1.985 — +2.072 — +2.094 — 0 350° 340° 330° 320° 310° 300° 290° 280° 270° 3'' g’ 90° 100° 1 10° 1 20° 130° 140° 1 50° 1 60° 1 70° 0 +2.094 - +2.053 8 — +1 950“ — ‘.+1.790 — +1.578 4 — +1.319 8 — +1.024 - +0.699 — +0.354 5 — 10 I 29 2 2 32 34 3 I 2.0 3 2.045 1.937 1.771 1.554 1.291 0.992 0.665 0.319 9 2 2 14 20 25 28 32 34 35 2.091 2.037 1.923 1.751 1.529 1.263 0.960 0.631 0.284 8 3 rs 20‘ 25 29 3! 34 35 3 2-088 2.028 1.908 1.731 1.504 1.234 0.929 0.597 0.249 1 3 9 rs 20 2s 29 32 34 . 36 4 +9-035 — +2.019 +1.893 —— +1.711 —. +1.479 — +1.2o5 —. +0.897 — +0.563 -— +0.213 -— 6 4 to I6 2: 25 29 33 9 35 3s 5 +2-“$1 - +2.0o9 +1.877 ~ +1.69° - +1.454 -- +1.176 — +0.864 +0. 528 — +0-.178 — 5 6 4 go 16 22 26 30 32 34 36 3-077 1.999 1.861 1.668 1.428 1.146 0.832 0.494 0. 142 4 5 12 I7 22 27 3° 33 35 35 7 3-072 1.987 1.844 1.646 1.401 1.116 0.799 0.459 0.107 3 3 5 u 17 22 27 30 33 35 2.066 1.976 1.827 1.624 1.374 1.086 0.766 0.424 0.071 2 5 13 28 23 27 3! 34 34 35 9 2-060 1.963 1.809 1.601 1.347 1.055 0.732 0.390 + 0.036 — 1 7 1 1 2 28 1 3 36 36 10 +2 3 9 3 3 3 -053 — +_1.950 +1.79° +1.578 — +1.319 — +1.024 — +0.699 - +0.354 — 0.000 0 260° ° 240°‘ 230° 220° 210° 200° 190° 180° 3'' The sum of the three numbers from Tables XV-XVII is the reduction from’ the mean argument of latitude at mean conjunction to true argument at true ecliptic conjunction, measured on the ecliptic. ' TABLES OF SOLAR ECLIPSES. TABLE XVII —Arg. (g + 9'). 241- no L-. —.0°.012 s_in (g +g'). I g+g I 5 + g ’ I o 0.000 360 90 — .012 + 270 10 -- .002 + 350 100 .012 260 20 ‘ .004 ' 340 110 .011 250 E 1 f , 30 .006 - 330 _ 120 .010 240 40 . 008 320 1 30 _ . 009 230 50 .009 310 140 .008 220 60 .010 _ 300 150 .006 ' 210 70 .011 ‘ 290 ' 160 .004 ' 200' 80 . .012 280 I70 - .002 + 190‘ 90 -— .012 + 3 270 180 0.000 180 = I § g+g' g+{ F or Values of y, at the Moment of Ecliptic Conjunction (y,°). TABLE XVIII —Arg. g. i TABLE XIX.-Am 9’- - ..._...._ .-_..._.._._._._. __._. § y9° = *.—— .0006 sin g + .0091 sin 2g (near ascending node). yg° = + .0163 sin g’ (near ascending node). y,° = + .0006 sing — .0091 sin 2g(nea1' descending node). : yg° = —- .0163 sing’ (near descending node). - It I i * , i g‘ g 8' I l 5’ g" ' I 5 l - _ 4' -— - 1 g 5 o 0 | 0 l . 0 0 o O 0 i .000 I 360 90 - .001 + 270 . , 180 0 .000 180 360 10 + .003 — ; 350 100 .004 E 260 , 170 10 I + .003 — ; 190 350 20 I . 006 E 340 1 10 . 006 3 250 ' i 160 20 ’ . 006 200 340 ; % : s 30 '+ .008 — ; 330 120 — .008g+ 9 240 150 30 + .008 -- - 210 330 40 .009 E 320 130 .010 I 230 I 3 140 40 ; .011 220 I 320 50 . .008 310 140 .009 J 220 i 130 50 I .012 230 310 I ' 4 60 + .007 - 300 150 — .008 + 3 210 . 120 60 3 + .014 — 240 300 70 .005 ago 160 .006 f 200 ' 110 70 E .015 250 290 so + _oo2 _ 280 170 — .003 + 190 f 1 100 80 .016 ‘ 260. 280 90 — .001 + 270 180 .000 I 180 Q 90 90 « i + .016 — 270 270. g ‘ 3' f In Tables XVIII and XIX, the numbers have the sign given with them near the ascending node, and the opposite sign near the descending node. * The algebraic sum of the numbers taken from the three tables, XVIII to XX, is the value of .1/2, the ordinate of the point in which the axis of the shadow intersects the fundamental plane, at the moment of true ecliptic conjunction. This ordinate is reckoned in a direction perpen-, dicular to the ecliptic. In Table XX, the algebraic sign of the numbers is the same as that of sin 11,. Near the descending node -141 diifers little from 180°; hence near the ascending node the number from Table XX has the same sign as 11;; near the descending node the opposite sign of u,— 180°. 4 ____— 'I‘ABLES OF SOLAR ECLIPSES. ’ 45 TABLE XX.—H0r. Arg., g; Vertical Arg., 81,. 99° z: + (5.245 — 0.330 cos g) sin ul. 0° 10° 1 20° 30° 40° 50° 5 60° 70° 3 80° 90° I f I u I g 6 360° 350° 340° 330° I 320° 310“ I 300° 290° I 280° 3 279° « _ _ _ 1 __ ,,,‘__ J ; E I _ ? .000 86 0.000 86 0.000 86 0.000 6 I 0.000 0.000 3 0.000 0.000 ' 0.000 0.000 1 8 I 8 88 . 8 ‘ .086 0.086 I 0.086 0.086 I 0.087 7 0.088 9 0.089 9 0.090 90 0.091 91 I 0.092 92 1 86 86 87 88 I 88 88 e 89 99 99 9, ; 2 -172 0-I72 ; 0-I73 0-I74 1 0.175 0.176 f 0.178 , 0.180 0.181 f 0.183 ; 85 35 9 85 - 86 I 86 87 I 88 89 91 92 3 .257 0.257 1 0.258 0.260 - 0.261 0.263 0 266 0 26 0 2 2 ¥ 86 86 86 86 5 87 88 5 ' 88 " 9 89 ? ' 7 99 T 0'2” 9: ' 4 -343 0-343 F 0-344 0.346 ‘ 0.348 0.351 5 0.354 0.3580 9. 0.362 0.366 8s 86 I 86, 86 . 87 88 I 89 39 9.-, ‘ 9, 5 -428 8 0.429 8 0.430 86 0.432 0.435 0.439 I 0.443 . 0.447 I 0.452 _: 0.457 S 5 I 86 87 87 . 88 89 I 90 91 .513 86 0.514 86 I 0.516 85 0.518 86 I 0.522 87 0.526 87 0,531 88 0,530 8 I 0,542 0_543 1 I . I 1 0-599 0-600 I 0-601 0-604 ’ 0.609 0.613 5 0.619 0.625 9 r 0.632 90 -‘ 0.639 9 68 85 68 85 ; 86 86 86 87 I 88 89 . 99 9. 0. 4 85 0. 5 85 1 0.687 85 0.690 86 0.695 86 0.700 87 0.707 88 0.714 89 0.722 f 0.730 0-769 84 0.770 84 J 0.772 8 0.776 8 0.781 86 0-787 8 I 9.795 0.803 1‘ 0.812 9°. 0.821 9‘ 1 5 s 8 88 3 8 » 0.853 0.854 » I 0.857 0.861 0.867 0.874 7 0,332 7 0_39; ; 0.90, 9 . O_9” 9° 0 38 85 3 85 1 85 6 85 I 86 6 86 I 87 88 2 89 § 9., -9 0-9 9 2 0-942 0-94 0- 53 0. 0 . 6 . : . J 84 84 I 84 . 85 I 9 85 9 86 I 0 9 9 87 0 979 88 0 99° 89 1.001 89 I I 5 i .022 84 1.023 84 1.026 84 1.031 85 I 1.038 85 1.046 86 I 1.056 8 1.067 8 ‘ 1.079 1,090 f .106 1.107 1.110 1.116 ’ 1.123 1.132 1.143 7 1.155 8 ; L16 88. x 8 9° 8 83 83 I 84 34 I 35 85 I 86 I 87 I 7 88 I .10 89 -1 9 8 1.190 8 1.194 1.200 1.208 1.217 I 1,229 L242 : L255 : 1.269 272 3 1 27 3 1 2 83 8 84 3 84 85 I 86 86 ' 88 E 39 . 83 . 3 83 . 77 83 1.2 4 8 I 1.292 8 1.302 8 I 1.315 1.328 1 1-343 Q 1.358 -355 1-356 : 1-360 1.367 3 3 1 376 4 1 387 5 i 1 00 85 87 ’ 87 E 88 . . 8 . , . ’ ; 437 82 I 43 82 I I 443 82 1 450 83 1.460 83 1.471 84 I 1.485 85 1.501 85 - 1.517 86 I 1.534 8 3}-52° ‘-525 I-533 I1 1-543 1-555 A 1-570 1.58 I 1.693 1 1.621 ’ ,:"T_'—":"“:—. i‘ ,:_:~;;; ~:;_- —;-_~_-»—_ 2. ..+_: .... ......_.._-.... .- _ 1...: .._I.__*___, .__V, ‘,_ ,_>.__ ____.V_V_;, 5V_ ___5_I __T__ ,_HV_‘: :f: V‘. ”;~v_I7nWV_ 9-“: 1’. n';—‘“;‘I; ‘ -. 5 90: 100: I 110° 120° 130° 140° 150° 150° 170:1 180.. (2 0 260 ' O 0 ° ° ° 0 0 7 250 240 230 220 210 200 190 180° I « I “- ‘(*9 " ‘_”'°—' " """ ’ ""‘ “‘ — —~--———-— —. - . - -000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0 000 .092 92 0.092 92 I 0.093 93 O 094 94 0 095 95 O 096 96 O 096 96 O 097 97 O 097 97 0 097 97 2 .183 91 0 185 93 i 0 187 94 0 18 95 96 96 97 97 ‘ 98 98 2 - 2 I - . 9 0.191 0.192 0.193 0.194 0.195 0.195 I 3 -275 9 0 277 9 ' 0 280 93 0 28 94 0 286 95 8 95 8 96 97 96 97 1° 9! . - I - - 3 . 0.2 7 0.2 9 0.291 0.291 0.292 I 4 _366 O 0 93 I O 94 94 8 95 96 97 97 98 97 5 -37 1 -374 0-377 0-3 I 0.383 0.386 0.388 -0.389 0.389 - 5 9‘ 93 93 95 95 96 96 96 96 97 '457 0-462 0-467 0-472 0-476 0.479 0.482 0.484 0.485 0.486 91 6 92 93 93 94 95 96 97 97 97 .548 ét 0.554 0-560 0.565 0.570 0.574 6 0.578 96 0.581 0.582 0.583 92 - 93 94 , 95 9 6 97 96 : .639 9! 0.646 92 0.653 93 0.659 94 0.665 95 0.670 95 0.674 96 0.677 :6 0.679 96 0.679 .730 9! 0.738 9! 0.746 92 0.753 93 0.760 94 0.765 95 -0.770 5 0.773 96 0.775 96 0.776 :I: . 9 9 -821 90 0.829 . 0.838 0.846 0.854 0.860 0.865 I 0.869 0.871 0.872 92 92 94 94 94 .95 96 96 96 '9“ 90 0.921 91 0.930 92 0.940 92 0.948 93 0.954 0.960 0.965 0.967 96 0.968 96 95 9s 95 .001 89 1.012 1.022 1.032 1.041 1.049 1.055 1.060 1.063 1.064 9° 92 93 94 94 95 95 9s 95 ~°90 1.102 1.114 1.125 1.135 1.143 1.150 1.155 1.158 1.159 .180 9° 1 I 91 I _ 91 2 93 94 94 9s 95 95 89 . 93 90 I 1.205 91 1.217 92 1.228 92 1.237 93 1.244 9‘ I 1.250 9 1.253 95 1.254 , 1 " 1 4 95 269 89 1.283 8 I 1.296 1-309 1-320 1.330 1.338 1.344 1.348 1-349 .358 1 9 I n 91 91 92 93 93 94 94 94 88 I-372 I 1.587 1.400 _ 1.412 1.423 1.431 1.438 1.442 1.443 446 6 39 9° - 91 92 92 92 93 93 94 88 1:4 I 8 ) 1.477 1.491 1.504 1.515 1.523 1.531 1.535 1.537 L534 1 0 9 I 66 89 8 91 6 92 60 92 6 94 93 93 - 93 87 -55 88 1.5 1.5 2 1.59 1. 7 1. 17 1.624 1.628 1.630 1962! 89 90 90 92 92 93 93 ' 93 1-633 1.655 1.672 1.686 1.699 1.709 1.717 1.721 1.723 TABLE XXI.—For Hourly Motion of Axis of Shadow. TABLES OF _ SOLAR ECLIPSES. x'g = + 0.5410 + 0.0397 cos g. I g 0° 3 10*’ 20° 30° 40° 50° 60° 70° 80° 1 ‘e 0 .5807 \ .5801 4. .5783 .5754 .5714 .5665 .5608 . 5546 .5479 10 1 . 5807 . 5800 1 . 5781 .5750 , .5710 . 5660 .5602 .5539 .5472 9 2 .5807 .5798 9 .5778 .5747 § .5705 .5654 .5596 .5533 .5465 8 3 .5806 A .5797 .5775 .5743 4 .5700 .5649 .5590 .5526 .5458 7 4 .5806 § .5795 A .5773 .5739 § .5696 .5643 .5584 .5519 .5452 6 ' 1 I 5 .5805 i .5793 4 .5770 .5735 j .5691 .5638 .5578 .5513 .5445 1 5 6 .5805 E .5792 , .5767 .5731 % .5686 .5632 .5571 .5506 .5438 4 7 .5804 i .5790 6 .5764 .5727 ' E .5681 .5626 .5565 .5499 .5431 3 8 . 5803 E . 5788 . 5761 . 5723 E . 5676 . 5620 . 5559 . 5492 . 5424 2 9 .5802 .5785 .5757 .5719 1 .5670 .5614 .5552 .5486 .5417 1 I 10 .5801 .5783 .5754 .5714 f . 5665 .5608 .5546 .5479 .5410 0 I 1 i i‘ 350° 4 340° 330° 320° 310° 300° 290° 280° 270° g ° g 90° 1 oo° 1 10° ; 1 20° 1 30° 1 40° 1 50° 160° 1 70° ! 0 .5410 .5341 A .5274 3 .5212 .5155 .5106 .5066 .5037 .5019 10 l 1 .5403 .5334 . 5268 . 5206 . 5150 . 5101 .5063 .5035 .5018 9 2 .5396 .5328 7 .5261 f .5200 .5144 .5097 .5059 .5032 .5017 8 .5389 .5321 A .5255 .5194 .5139 .5093 .5056 .5030 .5016 7 4 .5382 .5314 ‘ .5249 .5188 .5134 .5089 .5053 .5028 .5015 6 5 .5375 .5307 .5242 § .5182 .5129 .5085 .5050 .5027 ‘.5015 5 6 .5368 .5301 .5236 .5177 .5124 .5081 .5047 .5025 .5014 4 7 . 5362 .5294 % .5230 .5171 .5119 .5077 -5045 -5023 - 5014 3 8 .5355 .5287 j .5224 .5166 . 5114 .5073 .5042 .5022 .5013 2 9 .5348 .5281 3 .5218 5160 .5110 . 5070 .5039 .5020 .5013 1 10 .5341 .5274 ' . 5212 5155 .5106 .5066 .5037 .5019 .5013 0 260° 250° 240° 230° 220° 210° 200° 190° 180° g i 47 TABLES OF SOLAR ECLIPSES. TABLE XXII.—For Hourly Motion of Axis of Shadow. — 0d.O0I0 cos g’ + 0d.0006 cos (g + g’) — 06.0004 cos (g —g’). 0000 oooooooooooo 0000 OXOOOOOOOOO o000oo0oooooooo0oo0oo 6543 9 00000000 000 3333 m32m.nm%mnM..mmmm.u.muummumwsvowwwmmo g 2 2 O V ‘I! II I (‘VI r? t ‘ ‘Ill IIIIIIIC ‘'1 4 .Il 1.7 I III‘! .46: I I1} LIIIIIIII We IIHI 8642o24689mnmmm..um98642024689m..u22 0 II .. ____ ____ + + __.______ m _o 222 9 _.m ._.,_.._.m _Iwzossosmummmmum9?o4..o.....:?.£..u..E.. 0 I: _ + + ________ my 00 mmmmnm97S3II35790....2222I09753113579012 o _m ___________+ ................ 1.... m +________ 2 oo MMUU nm9753II4689...22332I09753...2.4689...2 o 0.. . __+ +_______ n .o0 HMMJUMJMHm8642025790I233321086420257901 o “M ________.___ + .................. ..... o . + ______ M .003? nmmmumumu9863I[468wuu7mJm.mUmn9863I.I468mu o0 ,1. _____________+ +______ we 04 3 I.-. {.1 . ll ..l||,I- TIE- |:r! .. I112 Ivllllltt . _ _ 1:1 rfoevuul. . _o0_ Hwm...M.m.wM.Umm9742O35oo9...23444320974203589... o _m ______.______ .................. .. o 7?. J- . + + ___ _. u _o0 Hm..Um.m.....,.wM...wU97530257912344.4431975303579... o I _____________.+ 11111111: .. o I I + _____ % am mmU..A._.m...U..m...m.........m863I246802344443I0863124680 o 1111111111 1. ¢X I . +_____ 2 o muUm.Jm.M.m.Unm864II468o...334433I0864II468o o 0 9 IIIIIIIIII IL —0/ . +__.__ 2 o0 .n.u.Hm..m.....A._...mo..m.BHmoo641146801234.433100064113680 o + +_____ 2 oo QHMUUUUMH9863....I3689...233332I9863113689 .. + +__.__ 2 oo 9mHMmmmHm9753II3579mUmmmmflm9753113579 o 6 __________.____+ +_____ m 3 00 9mnumun..m986A...20246790...I2II098642024679 o s ________ 11:11:: o _____ + + _____ N om. o%.n_u.%%...i..%.%98753II246789m..uUmm98653II2467oo om ______0+ +______ 3 oo 899m99o0765320I3567899m998765320235678 co 3 . ____________ + + ______ Mu o0 oooo9988~/6532013456780099oo87,05320l3nu.5.07oo o0 2 ___________ +. + _______ ..... o0 0083007654320I.3450788oo88765432013456738 o0 1 __._m______ + + ____.___ u Xe oooo7./6543IOI345677oooo8776543IOI.345677ooo0 .o% 0 _ _________+ +_________ 3 O000000OOOOOOOOOOOOOOOOO0000000000000 100000000 .3 ms I234567%wmmmwwwmmmwmmmmwmmmmmmmmwmmm IIIIIIIIII22222222223333333 when the argument g is at the bottom of the page, or is negative, g’ is to be sought for at the right. of tlfhelalgebreic sum of the numbers from Tables XXI and XXII is the hourly va.ria.ti0n of the co-ordinate 2;; of the point in which the axis 0 8 ladow lntersects the fundamental plane. TABLES OF SOLAR ECLIPSES. V TABLE XXIII.—F0r Radius of Shadow on Fundamental Plane. 1: .0059 — .0182 cosg + .0004 cos 2g. E , 1 g 0° i 10° 20° 30° ' 40° 50° 60° - 70° 80° i _, _. . E . 0 -.0119 —.0117 ——.0109 —.0097 —.0080 —.0059 —.0o34 —.0006 .0024 10 1 119 116 108 ‘ 95 78 '. 56 31 — 03 27 9 2 119 1 15 107 94 76 54 29 00 30 8 V 119 i 115 ' 106 . 92 74 52 26 + 03 33 7 4 119 1 114 105 90 72 49 23 05 36 6 I 5 -—.0118 i «—.0113 —.o103 —.0o89 —.0070 —.0047 —.0020 -,-.0008 .0039 5 6 118 I 1 12 102 87 67 44 I8 II 42 4 7 118 E 112 101 85 ° 65 42 15 1 46 3 8 117 ' 111 100 83 63 39 12 17 49 2 9 117 110 098 82 61 37 09 20 ’ 52 1 to I17 ‘I09 997 i so 59 34 96 24 55 o _ I _ 9 350° 340° 330° 320° 310° 300° 290° 280° 270° g g 90° 100° 1 10° 1 20° 1 30° 140° 1 50° 1 60° 1 70° 0 +-0055 +.o087 +-.0118 +.0148 +.0175 : +.0199 +.0219 ‘-I-.0233 +.0242 10 I S3 90 121 151 178 201 220 23 243 9 3 6‘ 93 _ 124 154 180 203 222 235 243 8 3 64 96 129 157 183 205 224 236 243 7 4 68 99 9 I3! 159 185 207 225 237 244 6 5 + .0071 + .0103 + -0135 + .0162 + .0188 + .0209 + .0226 + .0238 + .0244 5 6 76 106 136 165‘ 190 211 228 239 244 4 7 79 109 V 139 I67 192 213 229 240 245 3 3 31 I I2 142 I70 195 215 231 241 245 2 9 34 I15 145 I73 I97 217 232 241 245 1 10 87 118 1 148 175 199 219 233 242 245 0 % 260° 250° 240° 230° 220° 210° 2oo° 190° 180° g I For radius of penumbra add 0.5460. TABLES OF SOLAR ECLIPSES. TABLE XXIV —For Radius of Shadow on Fundamental Plane. 1: + 08.0046 cos g’ - 08.0005 cos (g + g’). g 0° 10° 20° 30 ‘ 40° 50° 60° 70° 80° V 90° 100° 110° 120° 130° 140° 150° 160° I 170° 180° E 3’ . ’ ’ 0° +41 +41 +41 +42 +42 +43 +43 +44 +45 +46 +47 +48 +48 +49 +50 +50 +51 +51 +51 360° 10° 40 41 41 42 42 43 44 44 45 46 47 48 49 49 49 50 50 3 so 50 350° 20° 39 39 39 40 41 42 42 43 44 45 46 46 47 48 48 48 48 3 48 I 48 340° 30° 36 36 37 37 33 39 40 41 . 41 42 43 44 44 45 45 45 45 4 45 44 330° 40 31 32 33 34 35 35 36 37 38 38 39 40 40 40 40 40 40 : 40 ; 39 320° 50° 26 27 28 29 3° 30 31 32 33 33 34 35 34 35 35 34 34 i 33 33 310° 60° 21 21 22 23 1 24 , 25 25 26 27 27 28 28 28 28 '28 27 27 } 26 25 300° 70° 14 15 16 17 I7 18 19 20 20 20 21 21 21 20 20 _19 19 18 1 17 290° 80° + 7 8 9 10 10 11 12 12 13 13 13 13 13 12 12 11 11 10, + 9 230° 90° 0+1 +2 +3 +3 +4 +4 +5 +5 +5 +5 +5 +4 +4 +3 +2 +2§+1f 0 270° 100° -7 -6-5-5 -4 -4 -3 -3 -3 -3 -3 -4 -4 -5 -5-6 -7;-85°-9 260° 110° -14 -13 -12 -12 -II ‘-1; -11 -11 -11 -11 -11 -12 -13 -13 -14 -15 -16! -17 -I7 250° 120° -21 -20 -19 -19 -13 -18 -18 -18 -18 -19 -19 -20 -21 -21 -22 -23 -24 f -25 -25 240° 130° -26 -26 -25 -25 -25 -25 --24 -25 -25 -26 -26 -27 -28 -29 -30 -30 -31 i -32‘ -33 230° 140° -31 -31 -30 -30 -3° -30 -31 -31 -31 -32 -33 -34 -34 -35 -36 -37 -38 7 -38 -39 220° 150° -36 -35 -35 -.-35 "35 -35 -36 -36 -37 -37 -33 -39 -10 -4: -42 -42 -43 -44 -44 210° 160° -38 -38 -38 -38 -33 -39 -39 -40 -41 -41 -42 --43 -44 -45 -46 -46 -47 g -48 -48 200° 170° -40 -41 -40 -41 “4‘- -41 -42 -43 -43 -44 -45 -46 -47 -48 -49 __,9 ._50 ‘ -50 -50 190° 180° -41 -‘41 -41 -42 -42 -43 -43 -44 -45 -46 -47 -48 -49 -49 -50 -50 -51 -51 -51 180° 190° -40 -4! -41 -42 -42 -43 -44 -44 -45 -'40 -47 -48 -49 -49 -50 -50 -50 -50 5 -50 170° 200° -38 -39 -39 -40 -41 : -42 -42 -43 -44 -45 -46 -46 -47 -47 -48 -48 -43 -48 -43 160° 210° -36 -36 -37 -37 -38 : -39 -40 -41 -41 -42 -43 -44 -44 -45 -45 -45 -45 3 -45 -44 150° 220° -31 -32 -33 -34 7 -34 1 -35 -36 -37 -38 -38 -39 -39 -40 -40 -40 -40 -40 F -40 -39 140° 230° -26 -27 -23 -29 i -30 E -30 -31 -32 -33 -33 -34 -34 -35 -35 -35 -34 -34 -33 -33 130° 240° -21 -21 -22 -23 -24 . -25 -25 -26 -27 -27 -28 -28 -28 -28 -28 -27 -27 . -26 -25 120° 250° -14 -15 -16 -17 -17 -18 -19 -20 -20 -20 -21 -21 -21 -20 -20 -19 -19 -18 -117 110° 260° - 7 - 8 — 9 -1o -10 -11 -12 -12 -13 -13 -13 -13 -13 -12 -12 -11 -11 i -10 -’ 9 100° 270° 0-1-2 -3 -4 -4 -4-5-5-5-5-5-4-4 -3 -2 -2!-1 0 90° ‘280° +7 +6 +5 +5 +4 +4 +3 +3 +3 +3 +3 +4 +4 +5 +5 +6 +7|+8 +9 80° 290° 14 13 12 12 11 11 11 11 11 11 11 12 13 13 14 15 16 ‘ 17 17 70° 300° 21 20 19 19 18 18 18 18 18 19 19 20 21 21 22 23‘ 24 ’ 25 _25 60° 310° 26 26 25 25 25 25 25 25 25 26 26 27 28 29 30 30 31 ’ 32 I «33 50° 320° 31 31 3° 30 3° 30 31 31 31 32 33 33 34 35 35 37 33 ' 33 39 4°° 33°° 30 35 35 35 35 35 36 36 36 37 . 38 39 40 41 42 42 43 44 44 30° 34°° 39~ 38 38 38 38 39 39 40 41 41 42 43 44 45 46 46 47 48 48 20° 35°° 40 40 40 41 41 41 42 43 43 44 45 46 47 48 48 49 50 50 50 10° 3°°° +41 +41 +41 +42 +42 +43 +43 +44 +45 +46 +47 +48 +48 +49 +50 +50 +51 +51 4 +51 0° : 1' 350° 350° 340° 330° 320° : 310° 300° 290° 280° 270° 260° 250° 240° 230° 220° 210° 200° 190° 180° g‘ i . The algebraic sum of the numbers from Tables XXIII and XXIV is the radius of the shadow-cone on the fundamental plane. If this radius is negative, it indicates a total eclipse; if positive, an annular one. To find the radius of the penumbra, the sum of the numbers is to be increased by 0.5460. AR-——7 f§C> -TABLES -OF SOLAR ECLIPSES. TABLE XXV.—Angle of Shadow Cone. sinf = 0.004653 + 0.000078 cos g’.’ g’ sinf log sinf g’ sinf .l0g sin} ~ 0 360 0.004731 7.6750 90 270 0.004653 7.6677 10 350 .004730 7.6749 100 260 .004640 7.6665 20 340 . 004726 7 . 6745 1 10 250 . 004626 7 . 6652 30 330 .004720 7.6739 120 240 .004614 7.6641 40 320 .004713 7.6733 130 230 .004603 7.6630 50 310 .004703 7.6724 140 220 .o04593 , 7.6621 , 60 300 .004692 7.6714 150 210 - .004586 7.6614 70 290 .004680 7.6702 160 200 .004580 '7.6609 ' 80 ' 280 7.004666 7.6689 170 190 .004576 7.6605 n 90 270 .004653 7.6677 180 180 0.004575 7.6604 _ TABLE XXVI, Arg. g’.-S'un’s Equation of the Centre, or Reduction from Mean to True Longitude. g’ Year 0. 2000. ' g’ Year 0. I 2000. 0 + 0.00 - + 0.00 — 360 90 + 2.01 + 1.91 ‘270 0.18 0.17 355 95 1.99 1.89 265 10 0.36 0.34 350 100 1.97 1.88 260 15 0-53 0-50 345 I05 1-93 1-84 255 ‘ 20 0.70 0.67 340 110 1.87 1.79 250 25 0.86 0.82 335 115 1.80 1.72 245 6 30 + 1.02 — + 0.97 - 330 120 + 1.72 + 1.64 240 35 1-17 1.12 325 125 1.62 1.55 235 40 1-31 1.25 320 130 51-52 1.45 230 45 1.44 1.37 315 135 1-40 1-33 225 50 1.56 1.49 310 149 1.27 1.21 220 55 1.66 1.59 305 145 1.13 1.08 .215 60 + 1.76 — ‘+ 1.68 - 300 150 + 0.98 + 0.94 210 55 1.84 1._75 295 155 0.83 0.80 205 70 1.90 1.81 290 160 0.67 0.64 200 75 1.95 1.86 285 165 0.51 0.49 195 80 1.98 1.89 280 170 0.34 0.33 190 85 2.00 1.90 275 175 0.17 0.16 185 90 + 2.01 — + 1.91 — 270 180 + 0.00 + 0.00 180 Year 0. 2000. ' g’ Year 0. 2000. g’ Table xxv $2..‘ the angle of the shadow eonejutl its logarithm. _ Table XXVI gives the sun’: equation of the centre. By applying this quantity to L, the sun’s mean longitude, we obtain G), its true longitude. TABLES OF SOLAR ECLIPSES. TABLE XXVII.—Rednction from ®’s Longitude to’ ®’s Right Ascension. 51 G) Year 0. 2000. G ‘ Year 0. { 2000. 0 180 0.00 0.00 180 360 45 225 P 2.52 4- 9 2.46 4- 135 315 1 181 0.08 0.08 179 359 46 226 3 2.52 I 2.47 134 314 2 182 0.17 0.17 178 358 47 227 I 2.52 E 2.47 133 313 3 183 0.25 0.25 177 357 48 228 2.52 : 2.46 132 312 4 184 0.34 0.33 176 356 49 229 2.51 : 2.46 131 311 5 185 0.42 + 0.41 + 175 355 50 230 2,59 4. ‘ 2..15‘+ 139 310 6 I86 0-50 6-49 I74 354 51 23I 2.49 2-43 I29 369 7 137 0-58 0-57 I73 353 52 232 2.47 2.42 128 308 8 188 0.66 0.65 172 352 53 233 '2,45 2.40 127 307 9 139 0-75 0-73 171 351 ' 54 234 2.43 2.38 126 306 10 190 0.83 4- 0.81 4- 170 350 55 235 2,49 4. 2,35 4. 125 395 11 191 0-91 0-39 109 349 . 56 236 2.37 2.32 124 .304 12 I92 0-99 0-90 103 343 57 237 2.34 2.29 123 303 I3 I93 1-06 1-04 I57 347 53 233 2.31 2.26 122 302 ‘4 194 1-14 I-12 I56 345 59 239 2.27 2.22 121 .301 ‘5 ‘‘95 1-3‘ 3' ‘-19 4- ‘ I55 345 00 240 2.23 4- 2.18 4- . 120 300 16 196 1.29 1.26 164 344 61 241 2,19 2.14 119 299 I7 I97 I-36 I-33 I63 343 62 242 2.14 2.09 118 298 18 198 1.43 1.40 162 342 63 243 ,_99 2_95 1,7 297 I9 I99 1-56 I-47 I6! 34I 64 244 2.94 , 2.99 115 299 20 200 1.57 4- 1.53 4- 160 340 65 245 1,99 4. 4 1,94 4. 1,5 295 21 201 1.63 1,69 159 339 66 246 1.93 . ,_89 I14 294 . 22 202 9 1 . 70 1.66 I58 338 67 247 1.87 I -83 1 13 293 23 203 1 1.76 1.72 157 337 68 218 1 1.81 ? I-77 I12 292 24 204 9 1-32 1-73 '50 330 69 249 Q 1,74 ; 1.70 111 291 25 205 3 1.88 4- 1.84 4- 155 335 70 250 1 7,93 4. L 1,94 9_ 1,0 290 . 25 206 I-93 I-89 I54 334, 7I 251 1 1.61 I 1.57 109 289 27 207 | I-99 - I-94 I53 333 72 252 ‘ 1.54 1.59 108 288 23 203 j 2-04 I-99 I52 332 73 253 1.46 1.43 107 287 29 209 2.09 2.04 15! 33! 74 254 1.39 1.35 106 286 3° 210 2-I4 '+ 2-09 + I50 336 75 255 - 1.31 + 1.28 + I05 285 3‘ "‘ — . 2-13 2-13 149 329 70 256 1.23 1.20 104 284 32 212 2.22 2.17 148 328 77 257 1.15 1.12 103 283 33 213 ' 2.26 2.21 147 327 78 258 1.07 1.04 102 282 34 214 2.30 2.25 146 326 79 259 0.99 0.96 101 281 35 2‘5 2.33 4- 2.28 -1 145 9 325 80 260 0.90 4- 0.88 4- 100 280 36 216 2.37 2.31 144 _324 81 261 0.81 0.79 99 279 37 217 2.40 2.34 143 323 82 262 0.72 0.71 98 278 33 218 2.42 2.36 142 322 -83 263 0.64 0.62 97 277 39 9'9 2.44 2.39 141 321 84 264 0.55 0.53 96 276 40 220 2.46 4- 2.41 4- 140 320 85 265 0.46 4- 0.45 -1 95 275 4‘ 231 2.48 2.43 139 319 86 266 0.37 0.36 94 274 42 222 2.50 2.44 138 318 87 267 0.28 0.27 93 273 43 223 2.51 3,45 137 317 88 268 0.18 0.18 92 272 44 224 2.51 2.46 136 316 89 269 0.09 3 0.09 9! 27! ‘5 225 - 2.52 4- 2.46 4- 135 315 , 90 270 0.00 “A 0.00 90 270 1 - 3 Year 0. i 2000. 0 Yea!’ 0- 3000- Q 9 Table XXVII gives, with a1°gument Q, a. quantigr which, W011-0n added to the equation of the centre (Table XXVI), will be the equation of 9 ‘"001 E1 Oxpressed in degrees and hundredth.-3. . I 52 TABLES OF SOLAR ECLIPSES. TABLE XXVIII —Coe17icients for Besselicm C0-ordinates of Shadow Axis. : 9 G i ‘ At Ascending Node. At Descending Node. Sun's True a log a log a’ j L011gitud0- t logb log 5' 4 5' 10gb log 5' 3' ‘ I 7‘-‘“'“* I ; o o _ o o o 360 — .3981+ —9 6000+ ! 9.9625 9.9440 +9.6871— + .-4865- 9.9803 +9.49o9—- + .3o97- 180 180 1 1 359 9 . 3980 9-5999 1 25 40 9.6871 1 .4865 03 9. 4908 . 3096 181 179 2 358 *_ . 3979 9 . 5998 1 26 40 9 . 6869 ' . 4863 04 9 . 4906 . 3095 182 178 3 357 2 -3976 9-5995 ’ 26 41 9-6867 -4861 04 9-4902 4 -3092 183 177 4 ‘ 356 1 -3973 9.5991 27 42 9-6864 1 -4858 04 9-4897 -3088 184 175 5 ; 355 t — -3953+ —9-5986+ 1 9-9628 9-9443 +9-6860- 1 + -4853- 9-9805 +9-4891- I + -3083- 185 175 6 i 354 - 3963 9. 5980 29 44 , 9. 6855 ' .4848 06 9.4882 ‘ . 3078 186 174 7 ! 353 1 -3956 9-5973 i 30 46 9-6850 1 -4841 07 9-4872 1 -3071 187 173 8 352 -3943 9.5964 4 32 48 9.6843 1 4834 08 9.4861 ’ .3063 188 172 9 ; 351 ‘ -3940 9-5955 ; 34 50 9-6836 [ -4826 09 9-4848 1 -3054 189 171 10 350 F — -3930+ -9-5944+ 5 9-9636 9 9453 +9-6827— 1 + 4816- 9-9811 +9-4834- $ + -3044- 190 170 11 1 349 i -3919 9.5932 1 33 _55 9.5313 1 4806 13 9.4313 , .3932 191 159 12 i 348 t -3907 9-5919 i 40 59 9-6808 1 -4795 14 9-4800 1 -3020 192 168 13 4. 347 i -3895 9-5905 43 62 9-6797 1 -4783 16 9-4781 7 -3007 193 157 14 { I 346 -; -3881 9-5889 E 46 65 9-6785 4769 18 9-4760 ! -2992 194 166 15 1 345 4 -7 -3366+ -9-5873+ 9-9649 9-9469 +9-6772- 1 + -4755- 9-9820 +9-4738- + :2977— I95 155 16 1 344 ; -3850 9-5855 1 52 73 9-6758 ‘ -4740 23 9-4713 2 -2960 196 164 ? 17 1 343 E -3633 9-5836 3 55 77 9-6743 I 4724 25 9-4688 ! -2943 197 163, 18 ? 342 3 -3315 9.5815 5 58 82 9.6727 3 4706 28 9.4660 3 .2924 I98 162 19 341 .3796 9.5795 62 87 9,6710 .4688 31 9.4631 .2904 I99 161 20 340 I — .3776+ —9.5771+ 9.9666 9.9492 +9.6692—- i + .4669— 9.9833 +9.4599- g + .2884— 200 160 21 1 339 1 -3755 9.5745 70 97 9.5573 ; .4543 35 9.4555 .2352 201 159 22 3 338 ; -3733 9-5720 74 9-9502 9.6653 | -4627 39 9-4531 -2839 262 158 23 I 337 -3710 9-5693 78 cs 9.5532 E -4605 43 9-4494 .2315 ’°3 157 . 24 336 3 .3685 9.-5665 83 14 9.6610 1 .4582 46 9.4455 .2790 204 156 E 25 g 335 ; — -3660+ -9-5635+ 9-9688 9-9520 +9_.6587— 1 + -4557- 9-9849 +9-4414- + -2763- 205 155 $ 26 1 334 I -3634 9-5604 92 26 9-6563 1 -4532 53 9-4371 -2736 206 154 § 27 1 333 -3606 9-5571 97 32 9-6538 3 4506 56 9-4326 -2707 207 153 5 28 -' 332 . -3578 9-5537 ' 9-9702 39 9-6511 - -4478 60 9-4278 .2573 293 152 ! 29 331 f -3549 9.5501 03 46 9-6433. -4450 64 9-4228 .2647 209 151 30 330 i - -3518+ —9.5463+ - 9-9713 9-9553 +9.6455— + .4.-420- 9.9868 +9.4176— + .2616—— 310 150 Q 31 : 329 -E 3486 9-5424 19 60 9-6425 4390 72 9-4121 -2583 211 149 i 32 1 328 5 .3454 9.5383 24 67 9.6393 .4358 76 9.4064 .2549 212 148 33 327 ; .3420 9.5341 V 30 75 9.6361 .4326 80 9.4004 V .2514 213 147 34 5 326 .3385 9.5296 , 36 82 9.6327 .4292 84 9.3941 .2478 214 146 35 , 325 E — 3349+ —9.525o+ 9.9742 9.9590 -§-9.6292-— + .4258— 9.9888 +9.3876— § + .2441-— 215 145 36 3 324 -3313 9.5202 * 48 98 9.6258 4222 92 9.3808 5 .2403 216 144 37 323 3275 9.5153 54 9.9606 9.6218 .4186 96 9.3736 .2364 217 143 33 7 322 3236 9-5100 60 14 9.6178 .4148 9.9991 9.3662 .2324 218 142 39 321 -3'96 9-S046 66 22 9.6138 4109 05 9.3584 .2282 219 141 E 40 f 320 — 3155+ -9-4990+ 9-9772 9.9631 +9-6095“ + .4070- 9-9909 +9-3502- + .2240“ 220 140 i 41 319 -3113 9.4931 79 39 9.6052 .4029 14 9.3417 .2196 221 139 42 1 313 -3069 9-437! 85 48 9.6006 .3987 18 9.3328 .2152 222 138 43 317 -3025 9-4303 92 57 9.5960 .3944 23 9.3232 -2106 223 137, 44 316 -2930 9-4742 93 65 9.5911 .3900 27 9.3138 .2060 224 136 45 315 ; — 2934+ -9-4674+ . 9-9305 9.9674 +9.5861- + .3855- 9.9931 +9-3037~ + .2012-— 225 135 1 log b log 5' 5' log 5 log 6' 6' Snn,s True 0 108 0 108 0' Longitude. At Descending Node. 1 1 At Ascending Node. Table XX VIII gives the coefficients by which to express the co-ordinates, 1:1 and 3/1, of the shadow axis on the fundamental plane. These correspond to the co-ordinates .1: and 3/ of the Besselian theory of eclipses and of the American Ephemeris. The expressions are :— at1:ay°.;+b1:’9t, 3ll=“'.'I°2 4‘ b'27_’~2 ‘- 310, having been obtained from Tables XVIII to XX, and 2’, from Tables XXI and XXII. TABLES OF SOLAR ECLIPSES. TABLE XXVIII.—Coefiicients for Besselicm Co-ordinates of Shadow Axz's—C0ntinued 53 At Ascending Node. At Descending Node. When the argument (9 is found on the right, the headings of the columns are to be sought at the bottom of the page. I G I I Sun's True a log a log a’ ’ --__._. Longitude. log (3 log 6’ _ 2' I log 6 log 5’ 6’ ° ° ) o o 45 315 “ -2934+ -9-4674+ 9.9805 9.9674 +9.5861—- + .3855— 9.9931 -1'-9.3037— .2012— 225 135 46 314 .2886 9.4604 11 83 9.5809 .3809 36 9.2930 .1963 226 3 134‘ 47 313 .2838 9.4530 18 92 9.5755 .3762 40 9.2819 .1914 227 133 43 312 .2789 9.4454 24 9.9701 9.5699 .3714 44 9.2702 .1863 228 132 49 311 .2738 9.4375 31 10 9.5641 .3665 43 9.2580 .1811 229 i 131 50 310 — .2687-P -9.-1292+ 9.9837 9.9719 +9.5581— + .3615-— 4 9.9952 +9.2452— .1759— 230 130 5’ 309 -2634 9-4207 44 28 9.5520 - .3564 ’ 56 9.2317 .1705 231 129 S2 308 -2531 9-4113 50 37 9.5456 .3512 61 9.2175 .1650 232 3 128 53 307 -2527 9-4026 57 46 9-5390 -3459 65 9,2026 -I594 233¥ I27 54 306 -2472 9-3930 63 55 9.5322 .3406 9 58 9.1869 .1533 234,: 126 55 305 “ -2415+ -9-3330+ 9-9869 9-9764 +9-5252- + -3351- ;‘ 9-9972 +9-1703- -1480- 235 E 125 56 304 -2353 9-3726 76 73 9.5179 ‘.3295 I 76 9.1528 .1422 236 124 57 303 -2300 9.3618 82 82 9.5103 .3238 80 9.1342 .1362 237 ‘ 123 53 302 -2241 9.3505 88 91 9.5025 .3181 83 9.1145 .1302 238 4 122 59 301 -2181 9-3387 -9894 99 9-4944 -3122 87 9-0936 -1241 2395 121 60 300 — .2120+ —9.3264+ 9-9900 9-9303 +9-4361- + -3063- 9-9990 +9.07l3- .1178— 240’ 120 61 299 -2°59 9-3136 06 17 9-4774 -3002 93 9.0475 .1115 241 ! 119 62 293 -1996 9-3002 12 26 9-4635 -2941 96 9.0219 .1052 9 242 118 63 297 -1933 9.2862 17 34 9.4591 .2879 99 3,9944 .9937 243 ; 117 64 296 -1369 9.2716 23 43 9.4496 .2816 0.0002 3.9647 .9922 244 H5 65 295 '- -1304+ -9-2562+ 9.9928 9.9851 +9.4397— + .2752— . 0.0004 +8.9324— .0856—- 245 115 66 294 - ‘ 738 - 9- 2401 33 59 9- 4293 - 2633 07 8 . 8970 .0789 246 1 14 67 293 - 1672 9- 2232 38 67 , 9-4137 -2622 09 8. 8581 .0721 247 113 63 292 -1605 9-2054 43 75 9.4076 .2556 11 8.8150 .0653 243 112 69 291 -1537 9-1366 48 82 9.3961 .2489 13 8.7666 .9534 249 1 1;; 7° 290 - -1468+ -9-1668+ 9-9953 9-9890 +9-3840- + -2422- 0-0014 +8.7u5— .0515— 250 110 71 289 -1399 9-1458 57 98 9-3717 -2353 16 8.6478 .0444 I 25: 109 72 238 - 1329 9- 1236 61 9.9905 9-3533 -2235 17 8. 5724 .0374 252 108 73 287 .1259 9.0999 65 12 9.3454 .2215 18 8.4804 .0302 253 107 74 286 . 1 188 9.0747 69 19 9. 3314 .2145 19 8. 3627 .0230 254 106 75 285 -— .1116+ —9.0477+ 9.9973 9.9925 +9.3168— + .2074— 0.0019 +8.1992— .0158— g 255 105 76 284 .1044 9.0187 76 32 9.3016 .2003 20 7.9313 .0085 256 ’ 104 77 283 .0971 8.9874 80 38 9.2858 .1931 20 +7.0828— .0012— ; 257 g 103 78 282 .0898 8.9535 83 45 9.2692 . 1859 20 -7. 7889+ .0062+ 258 102 79 281 .0825 8.9165 85 50 9.2519 .1786 20 —8.1316 .0135 259 101 30 280 — .o751+ -8.8759+ 9.9988 9.9956 +9.2337- + .1713— 0.0019 --8.3220 .0210+ 260 100 8‘ 279 .0677 8.8307 90 62 9.2146 .1639 18 —8.4542+ .0285 261 99 32 278 .0603 8. 7802 92 67 9. 1946 . 1565 17 -8. 5557 .0360 262 98 33 277 .0528 8.7228 94 72 9.1735 .1491 16 —8.6381 .0435 263 97 :4 276 .0453 8.6563 96 77 9.1512 .1416 14 —8.7075 .0510 264 96 82 :75 - .0378+ —8.5775+ 9.9997 9.9981 +9.1276- -1- .1341— 0.0013 —8.7674+ .0585+ 265 95 87 274 .0303 8.4809 98 85 9.1025 .1266 11 —8.8202 .0661 266 94 88 27: .0227 8.3562 99 89 9.0758 .1191 08 —8.8673 .0737 267 93 8 7 .0151 8.1804 99 93 9.0474 .1115 06 -8.9098 .0812 268 92 9 271 .0076 7. 8797 0.0000 97 9-0169 . 1040 03 —8.9485 .0888 269 91 9° 270 — .000o+ — oo + 0.0000 0.0000 +8.9841-— + .0964— 0.0000 —8.9841+ .0964+ 270 90 log 5 log 6' 6' log 13 log 6' 6' Sun-S -I-we a log a log a’ Longitude. At Descending Node. At Ascending Node. G) 54, TABLES OF SOLAR ECLIPSES. TABLE XXIX.—Szm’s Declination, etc. E i Q ' i = I I I Q a7 a'1 7*); i Q E J ' :11 I3: 1 i ‘ 0 0 0 0 _ o 0 0 0? .5 0 9 9 0 I30 -F 0-00- E-F 0-00- E 1-0033 ; I30 360 45 I35 i-+16.35- -+16.40- 1.0031 225 315 1 179 0.40 . 0.40 g .0033 181 359 46 134 5 16.64 16.69 .0031 226 314 2 178 0.80 _ 0.80 ; .0033 7 182 358 47 133 ’ 16_93 16.98 .9931 227 313 3 177 1-19 I 1-19 : -0033 1 I33 3 357 43 132 I 17.21 . 17.26 .0030 228 312 : 1:: ;.:::3- :.;:::_1 .:::::: 1:: 3:: :2 ::;1.::::2- .:::::- .223: :32 2:; 6 174 _ 2.38 i 2.39 .0033 186 354 51 ” I29 ~ 18.02 18.08 .0030 231 309 7 173 ! 2.78 ; 2.79 I .0033 ! 137 353 52 128 § 18.28 13-343 .0030 232 308 8 172 E 3-17 4 3-13 4- .0033 ‘ 188 352 53 127 2 18.54 18.60 .0030 233 307 9 171 3-57 1 3-53 V .0033 , . 189 351 54 126 I 18.79 18.85 .0030 234 7306 I0 170 ;-F 3-96" ‘F 3-97- ; 1.0033 3 190 350 55 125 T -219.03-— -FI9.09-— 1.0030 235 305 It 169 1 4-35 4-35 .0033 ‘ 191, 349 56 124 i 19.27 19.33 .0030 236 304 I2 168 i 4.75 1 4-77 5 .0033 . 192 348 57 123 E 19.50 19.56 ' .0030 237 303 I3 167 g 5-14 1 5-16 ; .0033 193 347 53 122 f 19.73 E 19.79 .0030 233 302 14 166 § 5.52 1 5-54 . .0033 ; 194 346 59 121 g 19.95 3 20.01 .0030 239 391 15 165 -F 5.91-— 3 -+ 5.93- j 1_oo33 . I95 345 60 120 '-+20.17- : 4-20.23- 1.0029 . 240 300 16 164 E 6.30 6.32 2 _9933 7 I96 344 61 119 ; 20.38 ; 20.44 ‘ .0029 241 299 17 163 E 6-63 5.70 3 .0033 ' 197 343 62 118 § 20.58 E 20-64 .0029 242 298 18 162 7 7-07 7-09 f .0033 193 342 63 117 L 20.78 ; 20.84 .0029 243 297 19 161 7 7.45 7.48 E .0033 7 199 341 64 116 E 20.97 : 21.03 .0029 244 296 20 160 E-+ 7 33" ‘F 7-36- § 1.0033 3 299 340 65 115 1 4-21.15—— ?-+21.21- 1.0029 245 295 21 159 i 3-20 3.23 1 .0033 1 291 339 66 114 f 21.33 E 21.40 .0029 246 294 22 158 E 3-53 3-51 : .0033 g 292 338 67 113 1 21.50 5 21.57 .0029 247 293 23 157 : 3-95 3-93 L .0033 ‘ 293 337 68 112 5 21.66 E 21.73 19929 248 292 24 156 ’ 9.32 9.35 = .0033 1 294 336 69 111_1 21.82 1 21.89 9.9929 249 291 25 155 .-F 9-59" .'* 9-72- 1 1.0033 I 295 335 70 110 i-+21.97- i 4-22.04- 1.0028 250 290 26 154 10-05 10.03 1 .0033 ' 296 334 71 109 E 22.11 22.18‘ .0028 251 289 27 153 1 10-41 i 10-44 E .0032 ! 297 333 72 108 l 22.25 22.32 .0028 252 288 28 I52 10-77 1 _ ‘O-3° § -0032 1 208 332 73 107 i 22-33 22-45 .0028 253 287 29 151 [ '‘-‘3 } “-‘7 1 -0032 E 209 331 74 I05 ‘ 22-50 22-57 .0028 254 286 30 150 ‘;+II-43- i+H-52- . 1.0032 3 210 330 75 105 +22.61— +22.68— 1.0028 255 285 31 I49 1 “°33 1, “'87 -0°32 3 211 329 76 104 4 22-72 22-79 . .0028 256 284 32 143 ; 12.18 . 12.22 _ ,9932 ; 2,2 328 77 103 ‘ 22.82 22.89 .0028 257 283 33 147 - 12.52 § 12.56 .9932 § 213 327 78 102 22.92 22.99 .0028 258 282 34 146 1 12.86 V 12.90 , ,9932 3 2,4 326 79 101 23.00 23.07 .0025 259 231 35 145 E-+13.20-—” f-+13.24- ‘ 1,9932 { 2,5 325 80 100 -+23.08- -k23.15- 1.0028 260 280 36 144 95 13.53 1 13.57 .0032 3 216 324 81 99 23.15 ; 23.22 .0028 261 279 37 143 g 13.86 ; 13.90 .9932 1 2,7 323 82 98 23.22 ! 23 29 .0028 262 273 33 142 3 14-19 14-23 .0032 213 322 33 97 1 23-27 j 23-34 .0028 263 277 39 141 7 14.51 7 14.56 ; .oo31 219 321 84 96 7 23-32 ; 23.39 .0028 264 276 40 140 f-+I1-33- ' -FI4-33- 5 1.0031 f 220 320 35 95 ’'F23-36-- ; 4-23.43—- 1.0028 265 275 41 I39 1 15.14 A 15-19 .0031 2 221 319 86 94 23-40 23.47 .0028 266 274 42 138 4 I 5.45 7 15 - 50 i .0031 ; 222 318 87 93 23.43 ; 23.50 .0028 267 273 43 137 3 15.75 1 15.80 1 .0031 7 223 .317 88 92 1 23-44 . 23.51 .0028 268 272 44 136 E 16.05 I 16.10 .0031 f 224 316 89 91 E 23-45 1 23.52 .0028 269 271 45 135 ;.+16,35—— 7 -+16.40-— 1.0031 E 225 315 90 90 E-F23.46- 7 4-23.53-— i 1.0028_ 279 279 ‘ 1 I 1 3 I. 7 d 111 ‘ pl G (1 E 111 pl (9 Table XXIX gives, with argument 6), the value of the sun’s declination, d, that of d1, the reduced declination, and that of P.‘ for computing 1 the central line on the earth's surface. TABLES OF SOLAR ECLIPSES. 55 central only in the southern hemisphere. eclipse of Thales. Table V, c. p. 4, T — 440? + 234“.6227 Table VII, — 5123', — I44? — 864.5816 Arguments for date, — 5845’ + I48d.O4ll Table VIII, + 4.0786 Table IX, , — “.0015 Table X, —- ‘.0010 Table XI, + ‘.0016 Table XII, + ‘.0012 _ I48“.I2oo Red. for calendar, 04.00 Ho, True conj., May 28, 2“ 52‘“.8 T; in arc, 43°.2o — E, +2°.28 H; at conj., 45°.48 By Table XXVIII :- xi = — .0624 + .555ot y; = + .2832 + .I796t Asc. Desc. As an example of the use‘ of the Tables, we shall examine what eclipses of the sun were visible during Table In’ E 6:2’ 29 I the year B. C. 584. From Table I, we find the argument of Table II to be 7Y.772. From Table II, the 144.8 318.: times of conjunction of the mean sun with the node are found to be 1445.8 for ascending and 3184.1 for 173,9 347,2 descending node. The values of D show that there were two central eclipses, of which the second was Multiple’ l77'2 3544 We therefore consider only the first one, which is the celebrated 1;’ " 3 ' 3 - 7' 2 . 7 :1: T, + I26 :1: 18 Year of central eclipse, - 458 i 18 3’ 5" L “ J’ 9 -3'9 — 35°.27 .264°.52 I45°.93 .- o°.294 Table XVIII, — .004 Table XXI, .5798 + 23°.29 — 83°.99 — 86°.42 + 3°.673 Table XIX, 0 Table XXII, + .0008 __ 110.98 + 1800.53 + 590.5! _+_ 30.379 Table XX, 2}’ O.294' Table XXVI, — °.o2 Table XV, + °.o77 f9 = + (3,290 x’,=, 0.5806 G): 59°.49 Table XVI, —- °.o2o Table XXIII, — 0.0115 Table XXV, Table XXVII, — Table XVII, - °.oo2 Table XXIV, - 4t 5;nf_—_- .oo4575 Eq. Cent. = E, — 2°.28 u; = + 3°.434 1’ = -— .0156 log 7.6604 Track of Central Eclipse. 1 I 4’: i J”: H H1 ' Long. Lat. I*‘.35 -1- .6868 + .5270 67°.4 65°.8 I°.6 E. + 41°.r I".4o .7146 .5360 7I°.8 66°.5 5°.3 41°,3 I".4s -7423 -5450 76°.7 67°.3 9°.4 4o°.8 1"-50 - 7701 . 5540 82°.4 68°.o r4°.4 39°.r I“.55 . 7978 . 5631 89°.8 68°.8 2I°.O 37°.2 1*‘. 57 .8089 . 5668 93°.5 69°.r 24°.4 36°.o, 1h.59 , .8200 .5704 roo°.6 69°.4 3o°.6 33°.6 r“.59r8 .8211 ’ .5708 ro3°.6 69.°4 34°.2 32°.5 The last point of the shadow-path is between 4° and 6° south of the region within which the celebrated battle must have been fought, which was supposed to have been stopped by this eclipse. This large deviation is due to the corrections which have been applied to Hansen’s mean longitude of the moon. If these corrections are well founded, the sun set upon the combatants about nine tenths eclipsed. l y... N. C§%’$’'l '‘