Page:1902 Encyclopædia Britannica - Volume 27 - CHI-ELD.pdf/679

 E C L IPSE plane. It will be noticed that, in the first case, the eclipse appears annular to an observer at the point where the shadow axis intersects the fundamental plane, and in the second case total. But since an observer on the surface of the earth is nearer to the moon than the fundamental plane is, it may happen that an eclipse which is annular on the plane will be total to the observer on the earth’s surface. Hansen’s method differs from that of Bessel in that the ecliptic is taken as the fundamental plane instead of the equator. The axis of X on the fundamental plane is parallel to the plane of the ecliptic; that of Y perpendicular to it. The other elements are nearly the same in the two theories. As to their relative advantages, it may be remarked that Hansen’s co-ordinates follow most simply from the data of the tables, and are necessarily used in eclipse tables, but that the subsequent computation is simpler by Bessel’s method. Its use, in case Hansen’s form is used at first, requires merely the transformation of co-ordinates from the ecliptic to the equator. The eclipse tables of Oppolzer and Newcomb give the data for computing the echptic Y-co-ordinate at the moment T of true conjunction of the sun and moon in longitude. This coordinate is called B by Oppolzer and y°2 by Newcomb, and is expressed by Oppolzer in the form B =_p sin P. The tables also give, for the moment T, the value of x12 or AL, the hourly motion of the X-co-ordinate. Oppolzer gives separately data for AB, the hourly motion of B, while Newcomb determines the motion of the shadow-axis from x12 and the angle, assumed constant, which the path makes with the X-axis. An approximate value of the sun’s longitude (L or 0) and of the angle between the shadowaxis and the equator (8 [Delta] or d) is also found. Oppolzer uses these elements in the subsequent computation, while Newcomb transforms x2 and y2 into xx and yv the Besselian co-ordinates, which are to be used in the subsequent work. Several problems are involved in the complete computation of an eclipse from the elements. First, from the values of the latter at a given moment to determine the point, if any, at which the shadow-axis intersects the surface of the earth, and the respective outlines of the umbra and penumbra on that surface. Within the umbral curve the eclipse is annular or total outside of it and within the penumbral curve the eclipse will be partial at the given moment. The penumbral line is marked from hour to hour on the maps given annually in the American Ephemeris. Second, a series of positions of the central point through the course of an eclipse gives us the path of the central point along the surface of the earth, and the envelopes of the penumbral and umbral curves just described are boundaries within which a total, annular, or partial eclipse will be visible. In particular, we have a certain definite point on the earth’s surface on which the edge of the shadow first impinges; this impingement necessarily takes place at sunrise. Then passing from this point, we have a series of points on the surface at which the elements of the shadow-cone are in succession tangent to the earth’s surface. At all these points the eclipse begins at sunrise until a certain limit is reached, after which, following the successive elements, it ends at sunrise. At the limiting point the rim of the moon merely grazes that of the sun at sunrise, so that we may say that the eclipse both begins and ends at that time. Of course the points we have described are also found at the ending of the eclipse. There is a certain moment at which the shadowaxis leaves the earth at a certain point, and a series of moments when, the elements of the penumbral cone being tangent to the earth’s surface, the eclipse is ending at sunset. Three cases may arise in studying the passage of the outlines of the shadow over the earth. It may be that

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all the elements of the penumbral cone intersect the earth. In this case we shall have both a northern and southern limit of partial eclipse. In the second case there will be no limit on the one side except that of the eclipse beginning or ending at sunrise or sunset. Or it may happen, as the third case, that the shadow-axis does not intersect the earth at all; the eclipse will then not be annular or total at any point, but at most only partial. The third problem is, from the same data, to find the circumstances of an eclipse at a given place—especially the times of beginning and ending, or the relative positions of the sun and moon at a given moment. Reference to the formulae for all these problems will be given in the bibliography of the subject. There are two well-marked periods in which eclipses recur at nearly the same distance from a node, of the moon’s orbit, one of 223 lunations, the other of 358. At the end of the latter period the eclipse lapses. recurs at the opposite node, and at the end of two periods, at the same node. The length of this period is 10,571'95 days, or 29 Julian years less 20'30 days. Hence 18 periods make 521 years, so that at the end of this time the eclipse recurs on the same day of the year. In the mean, the time of recurrence is so nearly at the same distance from the node that we find each central eclipse visible at our time to be one of an unbroken series extending from the earliest historic times to the present, at intervals equal to the length of the period. For example, starting from the eclipse of Nineveh, 763 b.c., June 15, recorded on the Assyrian tablets, we find eclipses on May 27, 734 b.c., May 7, 705 b.c., and so on in an unbroken series to 1843, 1872, and 1901, the last being the 93rd of the series. Those at the ends of 18 periods occurred on June 15, O.S., of each of the years 763, 242 b.c., a.d. 280, 801, 1322, and 1843. As the lunar perigee moves through 242°'4 in a period, the eclipses will vary from total to annular, but at the end of 3 periods the perigee is only 7°T in advance of its original position relative to the node. Hence in a series including every third eclipse the eclipses will be of the same character through a thousand years or more. Thus the eclipses of 1467, 1554, 1640, 1727, 1814, 1901, 1988, &c., are total. The length of the other period, called the Saros, is 6585J days, or 18 years and 11 or 12 days. The fact that eclipses recur at the end of this period has been known from ancient times. Owing to the fractional excess of J of a day in the period, each recurring eclipse takes place about 120“ farther west in longitude than the preceding one of the series, and is therefore not generally visible in the same region. During the course of a Saros there are 223 lunations and 19 returns of the sun to each of the moon’s nodes. The clearest idea of the law of recurrence thus arising may be gained by the conception of conjunction-points of the moon and sun in the following way :— Imagine a circle, having the earth in its centre, to be situated in the mean plane of the moon’s orbit, and to be fixed to the node of the orbit so as to make one revolution in its plane around its centre in the same period as the revolution of the node (about IS^'b). Imagine also that, as the sun and moon revolve, we mark on this circle the points in the direction of which the mean conjunctions of the two bodies occur. These conjunction-points will fall at various points of the circle until the end of the Saros. Then after 223 conjunctions the 224th will fall very near the first. The deviation will be somewhat less than half a degree. To make it fall exactly upon the first we have only to give our circle, conceived as bearing the conjunction-points, a motion equal to this deviation, or, speaking; more exactly, a uniform retrograde revolution through 0°-476 in one period. The amount of this motion was