Eclipse

Eclipse (Gr. ἔκλειψις, falling out of place, failing), the complete or partial obscuration of one heavenly body by the shadow of another, or of the disk of the sun by the interposition of the moon; then called an eclipse of the sun. Eclipses are of three classes: those of the sun, as just defined; those of the moon, produced by its passage through the shadow of the earth, and those of the satellites of other planets, produced by their passage through the shadow of their primary. Jupiter (q.v.) is the only planet of whose satellites the eclipses can be observed, unless under very rare circumstances.

The geometrical conditions of an eclipse of the sun or moon are shown in fig. 1, which represents the earth E as casting its shadow towards C, and the moon M between the earth and sun as throwing its shadow towards some part of the earth and eclipsing the sun. The dark conical regions are those within which the sun is entirely hidden from sight. This portion of the shadow is called the umbra. Around the umbra is an enveloping shaded cone with its vertices directly towards the sun. To an observer within this region the sun is partly hidden from view. As the apparent path of the moon may pass to the north or south of the line joining the earth and sun, the axis of its shadow may pass to the north or south of the earth, and not meet it at all. An eclipse of the sun is called central when the shadow axis strikes any part of the earth; partial when only the penumbra falls upon the earth. It is evident that an eclipse can be seen as central only at those points of the earth’s surface over which the axis of the shadow passes.

Fig. 2.

Fig. 3.

A central eclipse is total when the umbra actually reaches the earth; annular when it does not. These two cases are shown in figs. 2 and 3. In the first of these the sun is entirely hidden within the region uu′. In fig. 3 within the region aa’ the apparent diameter of the sun is slightly greater than that of the moon, and at the moment of greatest eclipse a narrow ring of sunlight is seen surrounding the dark body of the moon.

We shall treat the subject in the following sections:—

I. Phenomena of Eclipses of the Sun and conclusions derived from their observation.

II. Eclipses of the Moon.

III. The Laws and Cycles of recurrences of Eclipses of the Sun and Moon.

IV. Chronological list of remarkable eclipses of the Sun, past and future, to the end of the 20th century.

V. Description of the methods of computing eclipses.

I. Phenomena of Eclipses of the Sun.

While an eclipse of the sun, whether partial, annular or total, is in progress, no striking phenomena are to be noted until, in the case of total eclipses, the moment of the total phase approaches. It will, however, be noticed that as the moon advances on the solar disk the sharply defined and ragged edge of the moon’s disk contrasts strongly with the soft and uniform outline of the sun’s limb. As the total phase approaches, the phenomenon known as shadow bands may sometimes be seen. These consist of seeming vague and rapidly moving wave-like alternations of light and shade flitting over any white surface illuminated by the sun’s rays immediately before and after the total phase. They are probably due to a flickering of the light from the thin crescent, produced by the undulations of the air, in the same way that the twinkling of the stars is produced. The rapid progressive motion sometimes assigned to them may be regarded as the natural result of an optical illusion. A few seconds before the commencement of the total phase the red light of the chromosphere becomes visible, and will be seen most distinctly as continuations of the solar crescent at its two ends. Owing to the inequalities of the lunar surface, the diminution of the solar crescent does not go on with perfect uniformity, but, just before the last moment, what remains of it is generally broken up into separate portions of light, which, magnified and diffused by the irradiation of the telescope, present the phenomenon long celebrated under the name of “Baily’s beads.” These were so called because minutely and vividly described by Francis Baily as he observed them during the annular eclipse of May 15, 1836, when he compared them to a string of bright beads, irregular in size and distance from each other. The disappearance of the last bead is commonly taken as the beginning of totality. An arc of the chromosphere will then be visible for a few seconds at and on each side of the point of disappearance, the length and duration of which will depend on the apparent diameter of the moon as compared with that of the sun, being greater in length and longer seen as the excess of diameter of the moon is less. The red prominences may now generally be seen here and there around the whole disk of the moon, while the effulgence of soft light called the corona surrounds it on all sides. Before the invention of the spectroscope, observers of total eclipses could do little more than describe in detail the varying phenomena presented by the prominences and the corona. Drawings of the latter showed it to have the appearance of rays surrounding the dark disk of the moon, quite similar to the glory depicted by the old painters around the head of a saint. The discrepancies between the outlines as thus pictured, not only at different times, but by different observers at the same time and place, are such as to show that little reliance can be placed on the details represented by hand drawings.

During the eclipse of July 8, 1842, the shadow of the moon passed from Perpignan, France, through Milan and Vienna, over Russia and Central Asia, to the Pacific Ocean. Very detailed physical observations were made, but none which need be specially mentioned in the present connexion.

The eclipse of July 28, 1851, was total in Scandinavia and Russia. It was observed in the former region by many astronomers, among them Sir George B. Airy and W.R. Dawes. It was specially noteworthy for the first attempt to photograph such a phenomenon. A daguerreotype clearly showing the protuberances was taken by Berkowski at the Observatory of Königsberg. An attempt by G.A. Majocchi to daguerreotype the corona was a failure. Photographs of the eclipse of July 18, 1860, were taken by Padre Angelo Secchi and Warren De La Rue, which showed the prominences well, and proved that they were progressively obscured by the edge of the advancing moon. It was thus shown that they were solar appendages, and did not belong to the moon, as had sometimes been supposed. The corona was barely visible on De La Rue’s plates, but those of Secchi showed it, with its rifts and the bases of the tall coronal wings, to about 15’ from the sun’s limb. The sketches taken at this eclipse proved that the corona extended in some regions 1° from the sun’s limb. As the sensitiveness of photographic plates has increased, they have gradually been wholly relied upon for information respecting the corona, so that at the present time naked-eye descriptions are regarded as of little or no scientific value. Owing to the great contrast between the brilliancy of the coronal light at its base and its increasing faintness as it extends farther from the sun, no one photograph will bring out all the corona. An exposure of one or two seconds is ample to show the details of inner corona to the best advantage, while longer exposures give greater extent of the brighter portions. The most extended streamers are very little brighter than the sky, and must be photographed with long exposures.

The first application of the spectroscope to the phenomenon was made during the total solar eclipse of August 18, 1868, by P.J.C. Janssen and other observers in India. By them was made the capital discovery that the red solar prominences give a spectrum of bright lines, and are therefore immense masses of incandescent gases, chiefly hydrogen and the vapours of calcium and helium. Janssen also found that this bright-line spectrum could be followed after the eclipse was over, and, in fact, could be observed at any time when the air was sufficiently transparent. By one of those remarkable coincidences which frequently occur in the history of science, this last discovery was made independently by Sir Norman Lockyer in England before the news of Janssen’s success had reached him. It was afterwards found that, by giving great dispersing power to the spectroscope, the prominences could be observed in a wide slit, in their true form. At this eclipse the spectrum of the corona was also observed, and was supposed to be continuous, while polariscopic observation by Lieutenant Campbell showed it polarized in planes passing through the sun’s centre. The conclusion from these two observations was that the light was composed, at least in great part, of reflected sunlight.

At the total eclipse of August 7, 1869, it was independently found by Professors C.A. Young of Princeton and W. Harkness of Washington that the continuous spectrum of the corona was crossed by a bright line in the green, which was long supposed to be coincident with 1474 of Kirchhoff’s scale. This coincidence is, however, now found not to be real, and the line cannot be identified with that of any terrestrial substance. The name “coronium” has therefore been given to the supposed gas which forms it. It is now known that 1474 is a double line, one component of which is produced by iron, while the other is of unknown origin. The wave-length of the principal component is 5317, while that of the coronal line was found at the eclipses of 1896 and 1898 to be 5303.

The eclipse of December 28, 1870, passed over the south-western corner of Spain, Gibraltar, Oran and Sicily. It is memorable for the discovery by Young of the “reversing layer” of the solar atmosphere. This term is now applied to a shallow stratum resting immediately upon the photosphere, the absorption of which produces the principal dark lines of the solar spectrum, but which, being incandescent, gives a spectrum of bright lines by its own light when the light of the sun is cut off. This layer is much thinner than the chromosphere, and may be considered to form the base of the latter. Owing to its thinness, the phenomenon of the reversed bright lines is almost instantaneous in its nature, and can be observed for a period exceeding one or two seconds only near the edge of the shadow-path, where the moon advances but little beyond the solar limb. Near the central line it is little more than a flash, thus giving rise to the term “flash-spectrum.” Young also at this eclipse saw bright hydrogen lines when his spectroscope was directed to the centre of the dark disk of the moon. This can only be attributed to the reflection of the light of the prominences and chromosphere from the atmosphere between us and the moon. The coronal light as observed in the spectroscope may thus be regarded as a mixture of true coronal light with chromospheric light reflected from the air, and it is therefore probable that the H and K (calcium) lines of the coronal spectrum are not true coronal lines, but chromospheric.

At the eclipse of December 12, 1871, visible in India and Australia, Janssen observed, as he supposed, some of the dark lines of the solar spectrum in the continuous spectrum of the corona, especially D, b and G. This would show that an important part of the coronal light is due to reflected sunshine. This feature of the spectrum, however, is doubtful in the most recent photographs under the best conditions. At this eclipse the remarkable observation was also made by Colonel John Herschel and Colonel J.F. Tennant that the characteristic line of the coronal spectrum is as bright in the dark rifts of the corona as elsewhere. This would show that the gas coronium does not form the streamers of the corona, but is spherical in form and distributed uniformly about the sun. Photographs were also taken on wet plates by a party in Java and by the parties of Lord Lindsay (at Baikul, India) and of Colonel Tennant (at Dodabetta). The Baikul and Dodabetta photographs were of small size (moon’s diameter = 3⁄10 in.), but of excellent definition. A searching study was made of them by A. C Ranyard and W.H. Wesley (Memoirs R.A.S. vol. xli., 1879), and for the first time a satisfactory representation of the corona was obtained. The drawings in the volume quoted show its polar rays, wings, interlacing filaments and rifts as they are now known to be, as well as the forms and details of the prominences.

The eclipse of April 16, 1874, was observed in South Africa by E.J. Stone, H.M. astronomer at the Cape, who traced the coronal line about 30’ (430,000 m.) from the sun’s limb. The visual corona was seen to extend in places some 90′ from the limb.

The eclipse of April 6, 1875, was observed in Siam by Sir J. Norman Lockyer and Professor Arthur Schuster. Their photographs showed the calcium and hydrogen lines in the prominence spectrum.

The eclipse of July 29, 1878, was observed by many astronomers in the United States along a line extending from Wyoming to Texas. A number of the stations were at high altitudes (up to 14,000 ft.), and the sky was generally very clear. The visible corona extended on both sides of the sun along the ecliptic for immense distances—at least twelve lunar diameters, about eleven million miles. Photographs taken by the parties of Professors A. Hall and W. Harkness gave the details of the inner corona and of the polar rays, showing the filamentous character of the corona, especially at its base in the polar regions. A photograph taken by the party of Professor E.S. Holden showed the outer corona to a distance of 50′ from the moon’s limb. The bright-line spectrum of the corona was excessively faint and, as the solar activity (measured by sun-spot frequency) was near a minimum, it was concluded that the brilliancy of the coronium line varied in the sun-spot period, a conclusion which subsequent eclipse observations seem to have verified. It is not yet certain that the other coronal spectrum lines vary in the same way.

The eclipse of May 17, 1882, was observed in Egypt. On the photographs of the corona the image of a bright comet was found, the first instance of the sort. (A faint comet was found on the plates of the Lick Observatory eclipse expedition to Chile in 1893.) The slitless spectroscope showed the green line (coronium) and D3 (helium) in the coronal spectrum.

The eclipse of May 6, 1883, was observed from a small coral atoll in the South Pacific Ocean by parties from America, England, France, Austria and Italy. A thorough search was made by Holden (with a 6 in. telescope) for an intra-Mercurial planet, without success, during an unusually long totality (5 m. 23 s.). J. Palisa also searched for such a planet. Janssen again reported the presence of dark lines in the coronal spectrum. “White” prominences were seen by P. Tacchini.

The eclipse of August 29, 1886, was observed in the West Indies. The English photographs of the corona, taken with a slitless spectroscope, show the hydrogen lines as well as K and f. Tacchini devoted his attention to the spectra of the prominences, and showed that their upper portions contained no hydrogen lines, but only the H and K lines of calcium. He also observed a very extensive “white” prominence. It was shown on the photographs of the corona, but could not be seen in the Hα line with the spectroscope. It has been suggested by Professor G.E. Hale that the colour of a “white” prominence may be due to the fact that the H and K lines (calcium) are of their normal intensity, while the less refrangible prominence lines are, from some unknown cause, comparatively faint. It is known that the intensity of such lines does, in fact, vary, though it is not yet certain that the “white” prominences are produced in this way. The subject is one demanding further observation. High prominences are generally “white” at their summits, “red” at their bases. The Harvard College Observatory photographs show the corona out to 90′ from the moon’s limb, though no detail is visible beyond 60′. W.H. Pickering made a series of photographic photometric measures of the corona, some of which are given below, together with results deduced by Holden from the eclipses of January and December 1889:—

	August 1886.	January 1889.	December 1889.
Intrinsic actinic brilliancy of the brightest parts of the corona	0.031	0.079	0.029
Do. of the polar rays	· ·	0.053	0.016
Do. of the sky near the sun	0.0007	0.0050	0.0009
Ratio of intrinsic brilliancy of the brightest parts of the corona   to that of the sky (actinic)	44 to 1	16 to 1	32 to 1
Magnitude of the faintest star shown on the eclipse negatives	· ·	2.3	· ·

The eclipse of August 19, 1887, was total in Japan and Russia, but cloudy weather prevented successful observations except in Siberia and eastern Russia.

The eclipse of January 1, 1889, was observed in California and Nevada by many American astronomers. The photographs of the corona, especially those by Charoppin and E.E. Barnard, show a wealth of detail. Those of Barnard, of the Lick Observatory party, were studied by Holden, and exhibited the fact that rays, like the “polar-rays,” extended all round the sun, instead of being confined to the polar regions only. The outer corona was registered out to 100′ from the moon’s limb on Charoppin’s negatives, to 130′ on those of Lowden and Ireland. On other plates the outline of the moon is visible projected on the corona before totality began. The spectrum of the corona showed few bright lines besides those of coronium and hydrogen.

The eclipse of December 22, 1889, was observed in Cayenne, S. America, by a party from the Lick Observatory under rather unfavourable conditions. Expeditions sent to Africa were baffled by cloudy weather. Father Stephen Joseph Perry observed at Salute Islands, French Guiana, and obtained some photographs of value. The effort cost him his life, for he died of malarial fever five days after the eclipse.

The eclipse of April 16, 1893, was observed by British and French parties in Africa and Brazil, and by Professor J.M. Schaeberle of the Lick Observatory in Chile. The Chile photographs of the corona were taken with a lens of 40 ft. focus, and are extremely fine. They show a faint comet near the sun. No great extensions to the corona were shown on any of the negatives, or seen visually, though they were specially looked for by British parties. The neighbourhood of the sun was carefully examined by G. Bigourdan without finding any planet. The spectrum of the corona was the usual one. The following lines were photographed in slitless spectroscopes, and undoubtedly belong to the corona: W. L. 3987; 4086; 4217; 4231; 4240; 4280; 4486; 5303 (the last number is the wave-length of the green coronium line). All of these have been seen in slit spectroscopes also. It is possible that two lines observed by Young in 1869, namely, W. L. (Ångstrom) 5450 and 5570, should be added to the list of undoubted coronal lines. It is not likely that helium or hydrogen or calcium vapour forms part of the corona. The wave-lengths of some 700 lines belonging to the chromosphere and prominences were determined by the British parties.

The eclipse of August 9, 1896, was total in Norway, Novaya Zemlya and Japan. The day was very unfavourable as to weather, but good photographs of the corona were obtained by Russian parties in Siberia and Lapland. Shackelton, in Novaya Zemlya, with a prismatic camera obtained a photograph of the reversing-layer at the beginning of totality. This photograph completely confirms Young’s discovery, and shows the prominent Fraunhofer lines bright, the bright lines of the chromosphere spectrum being especially conspicuous.

At the solar eclipse of January 22, 1898, the shadow of the moon traversed India from the western coast to the Himalaya. The duration of totality was about 2 m. The eclipse was very fully observed, more than 100 negatives of the corona being secured. The equatorial extension of the visible corona was short and faint, and the invisible (spectroscopic) corona was also very faint. The spectrum of the reversing-layer was successfully photographed; one set of negatives shows the polarization of one of the longest streamers of the corona, and proves the presence of dust particles reflecting solar light. The bright-line spectrum of hydrogen in the chromosphere was followed to the thirtieth point of the series, and the wave-lengths were shown to agree closely with Balmer’s formula (see Spectroscopy). The wave-length of coronium was found to be 5303 (not 5317 as previously supposed), and the brightness of the corona was measured. E.W. Maunder made the curious observation of coronal matter enveloping a prominence in the form of a hood.

Observations of the eclipse of May 28, 1900, were favoured in a remarkable degree by the absence of clouds. The photographs of the corona obtained by W.W. Campbell extended four diameters of the sun on the west side. The sun’s edge was photographed with an objective-prism spectrograph composed of two 60° prisms in front of a telescope of 2 in. aperture and 60 in. focus. A fine photograph, 6 in. long, of the bright- and dark-line spectra of the sun’s edge at the end of totality was thus obtained. It shows 600 bright lines sharply in focus besides the dark-line spectrum, to which the bright lines gave way as the sun reappeared. The coronal material radiating the green light was found to be markedly heaped up in the sun-spot regions. No dark lines were found in the spectrum of the inner corona. G.E. Hale and E.B. Frost also photographed the combined bright- and dark-line spectra of the solar cusps at the instants before and after totality. On one photograph showing no dark lines 70 bright lines could be measured between 4070 and 4340. On another were 70 bright lines between Hb and Hs. On a third were 266 bright lines between 4026 and 4381, and some dark lines. These lines show a marked dissimilarity from the solar spectrum.

The eclipse of May 18, 1901, was observable in Mauritius with 3½ minutes of totality, and in Sumatra with 6½ minutes. Unfortunately there was cloudy weather in Sumatra, which at some stations prevented observations entirely and at others neutralized the advantages promised by the long duration of totality. Thus spectroscopic observations for the detection of motion of the corona, for which the long totality gave a special opportunity, failed owing to cloud; and the search for intra-Mercurial planets had only a negative result, though stars down to magnitude 8.8 were photographed on the plates. But though no particular step in advance was taken, successful records of the eclipse were obtained, which will enable comparison to be made with other eclipses and will contribute their share to the discussion of the whole series. These include photographs of the corona, showing that it was of the sun-spot minimum type, and available for measures of its brightness; photographs of the spectra of the chromosphere and corona which are of the same general character as those obtained at previous eclipses; photographs showing the polarization of the corona, available for quantitative measures of polarization at different points. Photographs of the spectrum of the outer corona taken by the Lick Observatory party show a strong Fraunhofer dark-line spectrum, consistent with the view that the light is reflected sunlight. At Mauritius there was no cloud, but the definition was poor. Successful photographs of the corona were obtained for comparison with those taken in Sumatra one and a half hours later, but nothing of great interest was revealed by the comparison.

The eclipse of August 30, 1905, offered a duration of 3½ minutes in Spain, the track running from Labrador through Spain to North Africa, and affording excellent opportunities for observers, who flocked to the central line in great numbers. Unfortunately it was cloudy in Labrador, so that the special advantages of the long line of possible stations were lost. Exceptionally good weather conditions were enjoyed in Algeria and Tunisia, and full advantage was taken of them by H.F. Newall, C. Trépied and others at Guelma, by the party from Greenwich and G. Bigourdan at Sfax. That G. Newall’s spectroscopic photographs for rotation of the corona again gave no result is a clear indication of the faintness of the corona at 3′ from the limb; but F.W. Dyson at Sfax obtained two new lines at 5536 and 5117 in the spectrum of the corona; and a very large number of photographs of the corona (including many in polarized light on several different plans), of its spectrum, and of the spectrum of the chromosphere, were obtained by the various parties, which will afford copious material for discussion. Newall also obtained a polarized spectrum of the corona. Altogether no less than eighty stations were occupied. There were English, American, Russian and German observers in Egypt; English and French in Algeria and Tunisia; English in Majorca; observers of almost all nationalities in Spain; and English and American in Labrador. In Egypt the weather was bright, though the sun was low; in Majorca and Spain there were local clouds. Consequently many observations, in addition to those in Labrador, were lost, notably the special spectroscopic observations undertaken by Evershed on the northern limit of totality, and the observations of radiation undertaken by H.L. Callendar. A search for intra-Mercurial planets was conducted on an elaborate plan, with similar batteries of telescopes, in Egypt, Spain and Labrador, by three parties from the Lick Observatory, but the examination of the plates showed nothing noteworthy. Pending discussion of the greater part of the material, some interesting preliminary results were published in 1906 by the French observers. C.E.H. Bourget and Montangerand conclude that there is a marked division of the chromosphere into two regions or shells, a lower or “reversing-layer,” extending only 1″ from the limb, and a chromospheric layer extending to 3″ or 4″; and that the coronal light contains less blue and violet, but more green and yellow, than sunlight; while Fabry, by visual methods, obtained measures of the total and intrinsic intensity of the light from the corona closely confirming recent photographic observations, finding the total brightness about equal to that of the full moon, and the intrinsic brightness at 5′ from the limb about one quarter of that of the full moon.

II. Eclipses of the Moon.

The physical phenomena attending eclipses of the moon are no longer of a high order of interest either to the layman or scientific observer. A brief statement of them and their causes will therefore be sufficient. An observer watching such an eclipse from the moon would see the earth, which has nearly four times the apparent diameter of the sun, impinging on the sun’s disk and slowly hiding it. The phenomenon would be quite similar to that of an eclipse of the sun seen from the earth, until the sun was completely covered. During the progress of this partial eclipse the moon would be passing into the earth’s penumbra. As the moment of total obscuration approached, a red band of light would rapidly form in the neighbourhood of the disappearing limb of the sun, and gradually extend around the earth. This would arise from the refraction of the sun’s light by the earth’s atmosphere, and the absorption of its blue rays. When the light of the sun was completely hidden, a reddish ring of great brilliancy would, owing to this cause, surround the entire dark body of the earth during the period of the total eclipse.

The aspect of the moon, as seen from the earth, corresponds to this view from the moon. The fading of the moon’s light, due to its entrance into the penumbra, is scarcely noticeable without direct photometric determination until near the beginning of the total phase. Then, as the limb of the moon approaches the earth’s shadow, it begins to darken. When only a small portion has entered into the shadow, that portion is completely hidden. But, as the total phase approaches, the part of the moon’s disk immersed in the penumbra becomes visible by a reddish coppery light—that of the sun refracted through the lower parts of the earth’s atmosphere. The brightness of this illumination is different in different eclipses, a circumstance which may be attributed to the greater or less degree of cloudiness in those regions of the earth’s atmosphere through which the light of the sun passes in order to reach the moon. Its colour is due to absorption in passing through the earth’s atmosphere.

III. Laws and Cycles of Recurrences of Eclipses of the Sun and Moon.

It has been known since remote antiquity that eclipses occur in cycles. These cycles are known now to be determined principally by the motion of the moon’s node and the relations between the revolutions of the earth round the sun and the moon round the earth.

Fig. 4.

Owing to the inclination of the moon’s orbit to the plane of the ecliptic, an eclipse of the sun can occur only when the conjunction of the sun and moon takes place within about 16° of one of the nodes of the moon’s orbit. The Eclipse seasons. eclipse can be total only within about 11° of the node. An eclipse of the moon can occur only when the line sun-moon-earth makes an angle less than about 11° with the line of nodes; and the eclipse can be total only within about 8° of the node, the average limiting distances varying 1° or 2° according to the circumstances. These conditions being understood, the cycles of recurrence of eclipses of either kind can be worked out geometrically from the mean motions of the sun, moon, node and perigee by the aid of geometric conceptions shown in their simplest form in fig. 4. Here E is the earth, at the centre of a circle representing the mean orbit of the moon around it. MN is the line of nodes which is moving in the retrograde direction from N towards S1, at a rate of about 19.3° in a year, making a complete revolution in 18.6 years. Let the sun at the moment of some new moon be in the line ES1, continued. If the angle NES1 is less than 16° there will probably be an eclipse of the sun, which may be central if the angle is less than 11°. Let the next new moon take place in the line ES2 a month later. The mean value of the angle S1ES2 is about 29°; but as the node N has moved towards S1 about 1.4° during the interval, the sum of the angles NES1 and NES2 will be somewhat greater than S1ES2 by about 1.6°. The result is that if these two angles are nearly equal there may be two small partial eclipses of the sun, after which no more can occur until, by the annual revolution of the earth, the direction of the sun approaches the opposite line of nodes EM, nearly six months later. The result is that there are in the course of any one year two “eclipse seasons” each of about one month in duration, in which at least one eclipse of the sun, or possibly two small partial eclipses, may occur. One eclipse of the moon will generally, but not always, occur during a season.

Owing to the retrograde motion of the node the direction ES of the sun returns to the node at the end of about 347 days, so that a third eclipse season may commence before the end of a year. In this way there is a possible but very rare maximum of five eclipses of the sun in a year. Owing to the motion of the line of nodes each eclipse season occurs about 19 days earlier in the year than it did the year before. Another conclusion from the greater eclipse limit for the sun than for the moon is that in the long run eclipses of the sun, as regards the earth generally, occur oftener than those of the moon. But as any eclipse of the sun is visible only from a limited region of the earth’s surface, while one of the moon may be seen from an entire hemisphere, more eclipses of the moon are visible at any one place than of the sun.

If, starting with a conjunction along some line ES1, we mark by radial lines from E the successive conjunctions year after year, we shall find that at the end of 18 years and about 11 days the 223rd conjunction will fall once more very near the line ES1, the angle NES1 being about 24′ greater than before. Successive eclipses will then occur very nearly in the same order as they did 18 years and 11 days before. This period of recurrence has been known from remote antiquity and is called the Saros. What is most remarkable in this period is that in addition to the distance from the node being nearly the same as before, the longitude of the sun increases by only 11° and the distance of the moon from its perigee has changed less than 3°. The result of this approach to coincidence is that the recurring eclipse will generally be of the same kind—total, annular or partial—through a number of successive periods.

To see the law of recurrence of corresponding eclipses in the successive periods let us suppose the line of conjunction ES1 to be that at which there is a very small eclipse, visible only in high northern or southern latitudes. At the end of 18 years 11 days a second eclipse will occur along a line nearly half a degree nearer EN, the line of nodes. The successive eclipses will occur at the same interval through about ten periods, or 180 years, when the line of conjunction will pass within 11° of EN. Then the eclipse will be central, whether annular or total depending on circumstances: in the first one the central lines will pass only over the polar regions; but in successive eclipses of the series it will pass nearer and nearer to the equator until the conjunction line coincides with the node. The path of centrality will then cross in the equatorial region. During 22 or 23 more recurrences the path will continually approach to the opposite pole and finally leave the earth entirely. The entire number of central eclipses in any one series will generally be about forty-five. Then a series of continually diminishing partial eclipses will go on for about ten periods more. The whole series of eclipses will therefore extend through about sixty-five periods; and interval of time of about twelve hundred years.

Another remarkable eclipse period recurs at the end of 358 lunations. At the end of this period the line of mean conjunction ES1 falls so near its former position relative to the node that we find each central eclipse visible in 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. The recurring eclipses in this period do not, however, have the remarkable similarity of those belonging to the Saros, but may differ to any extent, owing to the different positions of the line of conjunction with respect to the moon’s perigee. Moreover, they recur alternately at the ascending and descending node. The length of the period is 10,571.95 days, or 29 Julian years less 20.3 days. Hence 18 periods make 521 years, so that at the end of this time each eclipse recurs on or about the same day of the year. As an example of this series, starting from the eclipse of Nineveh, June 15, 763 B.C., 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 the 521-year intervals 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 7.1° 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.

IV. Chronological Lists of Eclipses of the Sun.

The following is a brief chronological enumeration of those total eclipses of the sun which are of interest, either from their historic celebrity or the nature of the conclusions Notable eclipses. derived from them. In numbering the years before the Christian era the astronomical nomenclature is used, in which the number of the year is one less than that used by the chronologists. The Chinese eclipses are passed over, owing to the generally doubtful character of the records pertaining to them.

We give below, in tabular form, a list of the principal total eclipses during the 19th and 20th centuries, omitting a few visible only in the extreme polar regions, and some others of which the duration is very short. The first column gives the civil date of the point on the earth’s surface at which the eclipse is central at noon. The next two columns give the position of this point to the nearest degree. The fourth column shows the Greenwich astronomical time of conjunction in longitude. The next column gives the duration of the total phase at the noon-point; this is sometimes 0.1′ less than the absolutely greatest duration at any point. Next is given the node near which the eclipse occurs; and then the number in the Saros. Corresponding eclipses at intervals of 18 y. 11 d. have the same number, and occur near the same node of the noon, which is indicated in the next column.

Date at Noon-Point.		Point where Central at Noon.		Greenwich M.T. of conjunction in Longitude.			Duration of Totality.	Node	Series.	Regions Swept by Shadow.
		Lat.	Long.	d.	h.	m.	m.
1803, Feb.	21	11 S.	136 W.	21	9	20	4.2	Asc.	1	Pacific Ocean, Mexico.
1804, Aug.	5	38 S.	66 W.	5	4	6	1.2	Desc.	2	Pacific Ocean, Chile, Argentina.
1806, June	16	42 N.	66 W.	16	4	22	4.6	Desc.	3	New England, Atlantic, Africa.
1807, Nov.	29	11 N.	2 E.	28	23	48	1.4	Asc.	4	Central Africa, Areolia.
1810, April	4	12 N.	154 E.	3	13	41	Ann.	Desc.	5	Pacific Ocean, Borneo.
1811, Mar,	24	39 S.	26 W.	24	2	19	3.4	Desc.	6	South Atlantic to and across South Africa.
1814, July	17	31 N.	84 E.	16	18	33	6.6	Asc.	7	Africa, Central Asia, China.
1815, July	6	88 N.	175 W.	6	11	52	3.2	Asc.	8	Polar Regions, Western Siberia.
1816, Nov.	19	43 N.	30 E.	18	22	9	1.8	Desc.	9	Eastern Europe, Central Asia.
1821, Mar.	4	8 S.	96 E.	3	17	50	4.3	Asc.	1	Indian and Pacific Oceans.
1822, Aug.	16	36 S.	176 W.	16	11	22	1.4	Desc.	2	Australia, Pacific Ocean.
1824, June	26	47 N.	175 W.	26	11	43	4.4	Desc.	3	Pacific Ocean, Japan, China.
1825, Dec.	9	9 N.	127 W.	9	8	27	1.5	Asc.	4	Pacific Ocean, Mexico.
1828, April	14	18 N.	39 E.	13	21	18	0.3	Desc.	5	Northern Africa, India.
1829, April	3	32 S.	149 W.	3	10	24	4.1	Desc.	6	South Pacific Ocean.
1832, July	27	24 N.	28 W.	27	2	2	6.8	Asc.	7	West Indies and across Central Africa.
1833, July	17	78 N.	76 E.	16	19	16	3.5	Asc.	8	North-eastern Asia and Polar Regions.
1834, Nov.	30	40 N.	101 W.	30	6	48	1.9	Desc.	9	Southern and Western United States.
1835, Nov.	20	10 S.	20 E.	19	22	31	4.6	Desc.	10	Central Africa, Madagascar.
1839, Mar.	15	6 S.	31 W.	15	2	14	4.4	Asc.	1	South America, Africa, Egypt.
1840, Aug.	27	34 S.	72 E.	26	18	45	1.6	Desc.	2	Africa, Madagascar, Indian Ocean.
1842, July	8	51 N.	77 E.	7	19	2	4.1	Desc.	3	Spain, France, Russia to China, and Pacific Ocean.
1843, Dec.	21	8 N.	102 E.	20	17	10	1.6	Asc.	4	Indian and North Pacific Oceans and India.
1846, April	25	25 N.	75 W.	25	4	49	0.9	Desc.	5	Mexico, West Indies, Africa.
1847, April	15	24 S.	90 E.	14	18	22	4.7	Desc.	6	Indian Ocean, Australia.
1850, Aug.	7	18 N.	142 W.	7	9	34	6.8	Asc.	7	Pacific Ocean.
1851, July	28	70 N.	34 W.	28	2	41	3.7	Asc.	8	Scandinavia, Russia and North America.
1852, Dec.	11	37 N.	127 E.	10	15	32	2.0	Desc.	9	China, Pacific Ocean.
1857, Mar.	25	4 S.	155 W.	25	10	30	4.5	Asc.	1	Pacific Ocean, Mexico.
1858, Sept.	7	33 S.	41 W.	7	2	16	1.7	Desc.	2	Peru, South Brazil, Uruguay.
1860, July	18	56 N.	31 W.	18	2	21	3.7	Desc.	3	British America, France, Egypt.
1861, Dec.	31	9 N.	29 W.	31	1	55	1.8	Asc.	4	Caribbean Sea to North Africa.
1864, May	6	32 N.	173 E.	5	12	14	1.4	Desc.	5	Pacific Ocean.
1865, April	25	16 S.	30 W.	25	2	13	5.3	Desc.	6	Brazil to Central Africa.
1868, Aug.	18	10 N.	103 E.	17	17	12	6.8	Asc.	7	India to Pacific Ocean.
1869, Aug.	7	61 N.	145 W.	7	10	8	3.8	Asc.	8	United States and Alaska.
1870, Dec.	22	36 N.	5 W.	22	0	19	2.1	Desc.	9	Gibraltar, Northern Africa, Sicily.
1871, Dec.	12	12 S.	118 E.	11	16	2	4.4	Desc.	10	Southern India, Northern Australia.
1875, April	6	2 S.	83 E.	5	18	36	4.7	Asc.	1	Indian Ocean, Siam, Pacific.
1876, Sept.	17	33 S.	156 W.	17	9	54	1.8	Desc.	2	Pacific Ocean.
1878, July	29	60 N.	139 W.	29	9	40	3.2	Desc.	3	United States and Canada.
1880, Jan.	11	10 N.	160 W.	11	10	40	2.1	Asc.	4	Pacific Ocean, California.
1882, May	17	39 N.	63 E.	16	19	34	1.8	Desc.	5	Egypt, Central Asia, China.
1883, May	6	9 S.	147 W.	6	9	58	6.0	Desc.	6	Pacific Ocean, Caroline Islands.
1886, Aug.	29	3 N.	14 W.	29	0	54	6.6	Asc.	7	South America, Central Africa.
1887, Aug.	19	53 N.	102 E.	18	17	39	3.8	Asc.	8	Northern Europe, Siberia, Japan.
1889, Jan.	1	37 N.	138 W.	1	9	8	2.2	Desc.	9	California, Oregon, British America.
1889, Dec.	22	12 S.	13 W.	22	0	52	4.2	Desc.	10	Central Africa and South America.
1893, April	16	1 S.	37 W.	16	2	35	4.8	Asc.	1	Venezuela to West Africa.
1894, Sept.	29	34 S.	86 E.	28	17	43	1.8	Desc.	2	East Africa, Indian Ocean.
1896, Aug.	9	65 N.	112 E.	8	17	2	2.7	Desc.	3	North Europe, Siberia, Japan.
1898, Jan.	22	13 N.	69 E.	21	19	24	2.3	Asc.	4	East Africa, India, China.
1900, May	28	45 N.	45 W.	28	2	50	2.1	Desc.	5	United States, Spain, North Africa.
1901, May	18	2 S.	97 E.	17	17	38	6.5	Desc.	6	Sumatra, Borneo.
1904, Sept.	9	5 S.	133 W.	9	8	43	6.4	Asc.	7	Pacific Ocean.
1905, Aug.	30	45 N.	12 W.	30	1	13	3.8	Asc.	8	Canada, Spain, North Africa.
1907, Jan.	14	39 N.	89 E.	13	17	57	2.3	Desc.	9	Russia, Central Asia.
1908, Jan.	3	12 S.	145 W.	3	9	44	4.2	Desc.	10	Pacific Ocean.
1911, April	28	1 S.	155 W.	28	10	26	5.0	Asc.	1	Australia, Polynesia.
1912, Oct.	10	35 S.	33 W.	10	1	41	1.8	Desc.	2	Colombia, Ecuador, Brazil.
1914, Aug.	21	71 N.	2 E.	21	0	27	2.1	Desc.	3	Scandinavia, Russia, Asia Minor.
1916, Feb.	3	16 N.	62 W.	3	4	6	2.5	Asc.	4	Pacific Ocean, Venezuela, West Indies.
1918, June	8	51 N.	152 W.	8	10	3	2.4	Desc.	5	British Columbia, United States.
1919, May	29	4 N.	18 W.	29	1	12	6.9	Desc.	6	Peru, Brazil, Central Africa.
1922, Sept.	21	12 S.	106 E.	20	16	38	6.1	Asc.	7	East Africa, Australia.
1923, Sept.	10	38 N.	128 W.	10	8	53	3.6	Asc.	8	California, Mexico, Central America.
1925, Jan.	24	42 N.	44 W.	24	2	46	2.4	Desc.	9	United States.
1926, Jan.	14	10 S.	82 E.	13	18	35	4.2	Desc.	10	East Africa, Sumatra, Philippines.
1927, June	29	78 N.	84 E.	28	18	32	0.7	Asc.	11	England, Scotland, Scandinavia.
1929, May	9	1 S.	89 E.	8	18	8	5.1	Asc.	1	Sumatra, Malacca, Philippines.
1930, Oct.	21	36 S.	155 W.	21	9	47	1.9	Desc.	2	Pacific Ocean, Patagonia.
1932, Aug.	31	78 N.	109 W.	31	7	55	1.5	Desc.	3	Canada.
1934, Feb.	14	19 N.	168 E.	13	12	44	2.7	Asc.	4	Borneo, Celebes.
1936, June	19	56 N.	101 E.	18	17	15	2.5	Desc.	5	Greece to Central Asia and Japan.
1937, June	8	10 N.	131 W.	8	8	43	7.1	Desc.	6	Pacific Ocean, Peru.
1940, Oct.	1	19 S.	16 W.	1	0	42	5.7	Asc.	7	Colombia, Brazil, South Africa.
1941, Sept.	21	30 N.	114 E.	20	16	39	3.3	Asc.	8	Central Asia, China, Pacific Ocean.
1943, Feb.	4	47 N.	176 W.	4	11	31	2.5	Desc.	9	China, Alaska.
1947, May	20	2 S.	25 W.	20	1	44	5.2	Asc.	1	Argentina, Paraguay, Central Africa.
1948, Nov.	1	37 S.	82 E.	31	18	3	1.9	Desc.	2	Central Africa, Congo.
1952, Feb.	25	22 N.	39 E.	24	21	17	3.0	Asc.	4	Nubia, Persia, Siberia.
1954, June	30	62 N.	5 W.	30	0	27	2.5	Desc.	5	Canada, Scandinavia, Russia, Persia.
1955, June	20	15 N.	117 E.	19	16	12	7.2	Desc.	6	Ceylon, Siam, Philippines.
1958, Oct.	12	26 S.	139 W.	12	8	52	5.2	Asc.	7	Chile, Argentina.
1959, Oct.	2	23 N.	6 W.	2	0	32	3.0	Asc.	8	Canaries, Central Africa.
1961, Feb.	15	53 N.	53 E.	14	20	11	2.6	Desc.	9	France, Italy, Austria, Siberia.
1962, Feb.	5	4 S.	179 E.	4	12	11	4.1	Desc.	10	New Guinea.
1963, July	20	62 N.	126 W.	20	8	43	1.5	Asc.	11	Alaska, Hudson’s Bay Territory.
1965, May	30	4 S.	137 W.	30	9	14	5.3	Asc.	1	Pacific Ocean.
1966, Nov.	12	38 S.	43 W.	12	2	27	1.9	Desc.	2	Bolivia, Argentina, Brazil.
1972, July	10	67 N.	111 W.	10	7	40	2.7	Desc.	5	North-East Asia, North-East America and Atlantic Ocean.
1973, June	30	19 N.	6 E.	29	23	39	7.2	Desc.	6	South America, Africa and Atlantic Ocean.
1974, June	20	32 S.	107 E.	19	16	56	5.3	Desc.	12	South-West Australia and Indian Ocean.
1976, Oct.	23	31 S.	95 E.	22	17	10	4.9	Asc.	7	Africa, Australia, Indian and Pacific Oceans.
1977, Oct.	12	16 N.	127 W.	12	8	31	2.8	Asc.	8	Venezuela, Pacific Ocean.
1979, Feb.	26	61 N.	77 W.	26	4	47	2.7	Desc.	9	United States, British America, Pacific Ocean, N. Polar Sea.
1980, Feb.	16	1 N.	48 E.	15	20	52	4.3	Desc.	10	Africa, Atlantic and Indian Oceans, and India.
1981, July	31	54 N.	127 E.	30	15	53	2.2	Asc.	11	Pacific Ocean, Asia.
1983, June	11	7 S.	111 E.	10	16	38	5.4	Asc.	1	Java, Atlantic Ocean.
1984, Nov.	22	39 S.	170 W.	22	10	58	2.1	Desc.	2	Pacific Ocean, Patagonia.
1987, Mar.	29	17 S.	6 W.	29	0	45	0.3	Asc.	13	Atlantic, Equatorial Africa.
1988, Mar.	18	28 N.	146 E.	17	14	3	4.0	Asc.	4	Indian and Pacific Oceans, Sumatra.
1990, July	22	72 N.	142 E.	21	14	54	2.6	Desc.	5	Finland, North Atlantic.
1991, July	11	22 N.	105 W.	11	7	6	7.1	Desc.	6	Pacific Ocean, Hawaii, Central America.
1992, June	30	26 S.	5 W.	30	0	19	5.4	Desc.	12	South Atlantic.
1994, Nov.	3	36 S.	31 W.	3	1	36	4.6	Asc.	7	Pacific Ocean, South America.
1995, Oct.	24	10 N.	110 E.	23	16	37	2.4	Asc.	8	Pacific and Indian Oceans.
1997, Mar.	9	71 N.	154 E.	8	13	16	2.8	Desc.	9	North-East Asia, Arctic Sea.
1998, Feb.	26	6 N.	81 W.	26	5	27	4.4	Desc.	10	Pacific and Atlantic Oceans, Central America.
1999, Aug.	11	46 N.	18 E.	10	23	8	2.6	Asc.	11	Central and Southern Europe touching England.

Recurrence of Remarkable Eclipses.

From the property of the Saros it follows that eclipses remarkable for their duration, or other circumstances depending on the relative positions of the sun and moon, occur at intervals of one saros (18 y. 11 d.). Of interest in this connexion is the recurrence of total eclipses remarkable for their duration. The absolute maximum duration of a total eclipse is about 7′ 30″; but no actual eclipse can be expected to reach this duration. Those which will come nearest to the maximum during the next 500 years belong to the series numbered 4 and 6 and in the list which precedes. These occurring in the years 1937, 1955, &c., will ultimately fall little more than 20″ below the maximum. But the series 4, though not now remarkable in this respect, will become so in the future, reaching in the eclipse of June 25, 2150, a duration of about 7′ 15″ and on July 5, 2168, a duration of 7′ 28″, the longest in human history. The first of these will pass over the Pacific Ocean; the second over the southern part of the Indian Ocean near Madras.

All the national annual Ephemerides contain elements of the eclipses of the sun occurring during the year. Those of England, America and France also give maps showing the path of the central line, if any, over the earth’s surface; the lines of eclipse beginning and ending at sunrise, &c., and the outlines of the shadow from hour to hour. By the aid of the latter the time at which an eclipse begins or ends at any point can be determined by inspection or measurement within a few minutes.

V. Methods of computing Eclipses of the Sun.

The complete computation of the circumstances of an eclipse ab initio requires three distinct processes. The geocentric positions of the sun and moon have first to be computed from the tables of the motions of those bodies. The second Elements of eclipses. step is to compute certain elements of the eclipse from these geocentric positions. The third step is from these elements to compute the circumstances of the eclipse for the earth generally or for any given place on its surface. The national Astronomical Ephemerides, or “Nautical Almanacs,” give in full the geocentric positions of the sun and moon from at least the early part of the 19th century to an epoch three years in advance of the date of publication. It is therefore unnecessary to undertake the first part of the computation except for dates outside the limits of the published ephemerides, and for many years to come even this computation will be unnecessary, because tables giving the elements of eclipses from the earliest historic periods up to the 22nd century have been published by T. Ritter von Oppolzer and by Simon Newcomb. We shall therefore confine ourselves to a statement of the eclipse problem and of the principles on which such tables rest.

Two systems of eclipse elements are now adopted in the ephemerides and tables; the one, that of F.W. Bessel, is used in the English, American and French ephemerides, the other—P. A. Hansen’s—in the German and in the eclipse tables of T. Ritter von Oppolzer. The two have in common certain geometric constructions. The fundamental axis of reference in both systems is the line passing through the centres of the sun and moon; this is the common axis of the shadow cones, which envelop simultaneously the sun and moon as shown in figs. 1, 2, 3. The surface of one of these cones, that of the umbra, is tangent to both bodies externally. This cone comes to a point at a distance from the moon nearly equal to that of the earth. Within it the sun is wholly hidden by the moon. Outside the umbral cone is that of the penumbra, within which the sun is partially hidden by the moon. The geometric condition that the two bodies shall appear in contact, or that the eclipse shall begin or end at a certain moment, is that the surface of one of these cones shall pass through the place of the observer at that moment. Let a plane, which we call the fundamental plane, pass through the centre of the earth perpendicular to the shadow axis. On this plane the centre of the earth is taken as an origin of rectangular co-ordinates. The axis of Z is perpendicular to the plane, and therefore parallel to the shadow axis; that of Y and X lie in the plane. In these fundamental constructions the two methods coincide. They differ in the direction of the axis of Y and X in the fundamental plane. In Bessel’s method, which we shall first describe, the intersection of the plane of the earth’s equator with the fundamental plane is taken as the axis of X. The axis of Y is perpendicular to it, the positive direction being towards the north. The Besselian elements of an eclipse are then:—x, y, the co-ordinates of the shadow axis on the fundamental plane; d, the declination of that point in which the shadow axis intersects the celestial sphere; μ, the Greenwich hour angle of this point; l, the radius of the circle, in which the penumbral or outer cone intersects the fundamental plane; and l’, the radius of the circle, in which the inner or umbral cone intersects this plane, taken positively when the vertex of the cone does not reach the plane, so that the axis must be produced, and negatively when the vertex is beyond the plane.

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.

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 is 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 shadow-axis 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 all the elements of the penumbral cone intersect the earth. In this case we shall have both a northern and a 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 central 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.

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