PHILIP HERBERT COWELL

1870-1949

Philip H erbert Cowell was born on 7 August 1870, at Calcutta, the second of the five children of Herbert and Alice Cowell. His father, born at Ipswich in 1837, who was a barrister, first in India and then in England, of the Middle Temple, was related to E. B. Cowell, professor of Sanskrit at Cambridge, and to J. E. Cowell Weldon, successively headmaster of Harrow, bishop of Calcutta and dean of Durham: his mother, born in 1842, was the third daughter of Newson Garrett, merchant, of Aldeburgh, Suffolk. Mr Garrett’s second daughter, Elisabeth (Elisabeth Garrett Anderson, M.D.), was the first woman to qualify in medicine; and his fifth daughter, Millicent (Dame Millicent Garrett Fawcett), became the wife of the Rt Hon. Henry Fawcett, the blind Postmaster- General in the Gladstone ministry of 1880-1885.

From 1881 to 1883 Cowell was at a private school at Stoke Poges, the head­master of which, the Rev. E. St John Parry, was the father of Reginald St John Parry, afterwards a brother Fellow of Cowell’s at Trinity. In 1883 he went to Eton, as a King’s Scholar: it was his good fortune that the Rev. Edmond Warre, who became headmaster in 1884, considerably enlarged the opportunities for boys whose bent was mathematical; and, beginning with Cowell, the school produced several excellent mathematicians in succession: in the year below him was G. H. J. Hurst, second wrangler in 1893 and Fellow of King’s, and in the next year F. W. Lawrence (now Lord Pethick-Lawrence), fourth wrangler in 1894 and Fellow of Trinity. Cowell, Hurst and Lawrence all won the Tomline mathematical prize three years before they left Eton, over the heads of their seniors.

He left Eton in 1889 with an entrance scholarship to Trinity College, Cambridge, and a leaving scholarship from the school. As an undergraduate he worked only three hours a day, but he seemed to remember almost every­thing that he had ever read; and in 1891 he won the Sheepshanks Exhibition in astronomy, graduating in 1892 as Senior Wrangler: the distinction of being ‘above the Senior Wrangler’ had been gained two years earlier by his first cousin Philippa Fawcett. In 1894 he was appointed to the Isaac Newton student­ship in astronomy or physical optics, which had been founded three years previously by the generosity of Mr Frank McClean: Cowell elected to work on celestial mechanics.

It is well known that the positions of the moon and planets can be predicted with great accuracy by means of the law of gravitation: and as the theory created by Newton, Laplace and their successors was already extremely detailed and

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intricate, it might carelessly be inferred that there was not much point in attempt­ing to carry the developments still further. But immense importance attaches to the question whether gravitation accounts , or only ,for the observed motions: unexplained discordances, which can only be very small, may furnish the key to discoveries of the first importance in physics. An example of this possibility was provided by the discrepancy between the Newtonian theory and observation for the motion of the perihelion of the planet Mercury, which was adduced as the first observational confirmation of Einstein’s theory of general relativity.

The celestial body with which Cowell concerned himself was the moon: in order to introduce his researches, something must be said about the state of the lunar theory at that time.

Kepler had discovered that the orbits of the planets about the sun, and of the moon about the earth, are, to a very close approximation, ellipses: and Newton had shown that elliptic motion according to Kepler’s laws was a consequence of the law of gravitation. For nearly two centuries after Newton, the lunar theorists based their investigations on this discovery, regarding all the effects of the sun’s gravitation on the moon as small perturbations of the elliptic orbit that would be described under the influence of the earth alone. This method was soon found to have the disadvantage, that the line of apsides (i.e. the major axis of the ellipse) and the node (i.e. the intersection of the plane of the moon’s orbit with the plane of reference) were being displaced continu­ously, so that the elliptic orbit thus modified no longer satisfied the undisturbed equations of motion. In 1877 a revolutionary proposal was made by an American astronomer, G. W. Hill, namely that the ellipse should be discarded as the ‘undisturbed’ orbit, and should be replaced, as the starting-point of the theory, by another orbit which may be described as follows.

About 1590 Tycho Brahe discovered observationally an inequality which causes the moon’s true place to be in advance of her mean place from new moon to first quarter: from first quarter to full moon it is behind the mean place: from full moon to last quarter it is again in advance, and from last quarter to new moon is again in arrear. This inequality, which is called the ,amounts to about 32', by which the moon’s true place is sometimes before and sometimes behind the mean place: and it was explained by Newton as a conse­quence of the sun’s gravitational action. An orbit which differs from uniform circular motion round the earth only by the variational inequality is a closed curve or periodic orbit, whose shorter axis points towards the sun: the moon under suitable initial conditions could describe this orbit, if the eccentricity of the earth’s orbit round the sun were zero: and Hill now proposed to take it as the fundamental or indisturbed orbit of the moon, in place of Kepler’s ellipse. Hill’s ideas, for the realization of which he invented a new mathematical tool, the infinite determinant, promised to yield expansions far better suited for computation than anything hitherto known.

Cowell, in his capacity as Isaac Newton student, calculated, by Hill’s methods, the terms in the lunar theory which depend on the inclination of the moon’s

376 Obituary Notices

orbit, as far as the first three powers of the inclination, and also that part of the motion of the node which contains the square of the inclination as a factor. In October 1894 he submitted his work (afterwards published as 3)1 as a fellow­ship dissertation, and was elected a Fellow of Trinity.

In the summer of 1894 he made the acquaintance of Professor E. W. Brown, who after graduating as sixth wrangler in 1887 and obtaining a Fellowship at Christ’s, had accepted an appointment as Professor of Applied Mathematics in Haverford College, Pennsylvania, U.S.A., and had resolved to devote him­self to the new developments following on Hill’s work in lunar theory. Brown had come from America to Cambridge on a prolonged visit, and soon formed with Cowell a close friendship: his treatise on Lunar Theory was being printed at the time, and Cowell read the proof sheets.

During 1894 and 1895 he continued to live in his pleasant rooms in the north-west corner of the Great Court of Trinity, carrying on research in celestial mechanics in accordance with the terms of the Isaac Newton studentship. At the end of 1895, however, the Astronomer Royal (Sir William Christie) obtained the Board of Admiralty’s sanction to the appointment of a second Chief Assistant at the Royal Observatory, Greenwich, where F. W. Dyson had been Chief Assistant for a year or more; and the post was offered to Cowell, who accepted it and left Cambridge in April 1896 to take up his new duties. Christie’s intention was that the second Chief Assistant should undertake the superintendence of the astrophysical work to be carried out with the newly erected 26-inch photographic refractor, presented to the Observatory by Sir Henry Thompson, and the 30-inch reflector, the gift of Dr A. A. Common. Thus Cowell was confronted by entirely new types of work, such as general administration, the superintendence and carrying out of routine observations, the adjustment of instruments, etc.—work outside his previous experience, and not congenial to a man of his attainments and temperament, nor did it afford him an opportunity to exercise his natural gifts. He did not always perform it to Christie’s satisfaction, but he introduced improved methods of computation in various departments of the Observatory, and was ultimately enabled to devote himself more and more to those departments of astronomy for which he had a predilection.

Before an account is given of his achievements in these fields, mention must be made of his marriage, which took place on 24 June 1901, at St Mary Abbots, Kensington. The bride was Phyllis, second daughter of Holroyd Chaplin. She was related to the Ayrton family, of which the Rt Hon. Acton Ayrton, Professor William Ayrton, F.R.S., and Mrs Ayrton Zangwill were distinguished members: and she was descended, with no taint of illegitimacy, from Edward I and his second wife, Margaret of France, the grand-daughter of St Louis. Cowell’s devotion to her was one of the great sources of happiness in his life.

He now began the series of investigations on which his scientific reputation rests. At this time, the predictions of the moon’s place in the sky, as given for instance in the Nautical Almanac, were based on the Tables de la Lune of the

1 The numbers in heavy type refer to the list of published papers at the end of this notice.

Philip Herbert Cowell 377

Danish astronomer P. A. Hansen, published at London in 1857. There were, however, some serious outstanding discordances between Hansen’s Tables and the Greenwich Meridian Observations, and it was to an analysis of these that Cowell now applied himself. His object in this research was to correct the coefficients of the periodic terms in the moon’s motion, of which Hansen had taken account, and to detect the existence of other periodic terms which had not been included in the construction of the tables. The observations considered in the investigation extended over the period 1847-1901: for terms of longer period than two months, or for the separation of two terms of nearly coincident period, the observations from 1750 were also included. Cowell introduced some original labour-saving devices which permitted him to extend the analysis to all the inequalities, instead of the comparatively limited number of them examined by previous investigators. One improvement, which greatly facilitated the computations, was to take, as the unit period for the analysis, 400 lunar days instead of (as previous workers had done) one year. His reason for this was that 400 lunar days is 15 periods of the mean anomaly, 14 of the parallactic inequality, and 13 of the evection;2 it is also a round number. He also devised a method of taking account of the unequal distribution of the observations during the lunation—that is to say, the absence of observations near new moon: on this see 13. A full description of the method, together with the analysis for 145 terms in the moon’s longitude, is given in 21, and a coefficient is deduced from the observation for every periodic term.

The author then compares his results with Hansen’s coefficients when the latter are supplemented by certain planetary terms which had been calculated by R. Radau of Paris {Ann. de VObs. de Par, , t. 21) and by Hill’s terms,dependent on the figure of the earth ( Pap. Wash. 3). This comparison shows that, with a few exceptions, the coefficients derived from the gravitational theory agree closely with Cowell’s results derived purely from observation.

The exceptions occur chiefly in connexion with the inequalities into the determination of which the moon’s semidiameter enters. One of these is the variation. Another is the parallactic inequality, an effect which is due to the fact that, since the earth’s distance from the sun is finite, his disturbing force at new moon differs from that at full moon. This inequality naturally involves the value of the solar parallax, and Laplace actually obtained a value of 8" *6 for the parallax by comparing the theoretical formula with lunar observations. The difficulty in determining these inequalities observationally is due to the changes in the apparent semidiameter of the moon, which are produced not only by the state of illumination of the background of the sky, but by the considerable variety in the personal equation of the observers when observing opposite limbs. Cowell assumed that, in the mean of many observers, this personality may be taken to be constant; he therefore introduced the moon’s

2 The evection which is the largest of the inequalities affecting the moon’s place, was dis­covered by Ptolemy, from observation, about a.d . 140. The moon’s longitude is sometimes increased 1° 15', and sometimes diminished as much, by this irregularity, which depends on the position of the perigee with respect to the sun.

378 Obituary Notices

semidiameter as an unknown constant in his analysis of the observations, and finally obtained a determination of the parallactic inequality (16) which corre­sponds to a value of 8"-787 for the solar parallax, with an apparent probable error of d=0,/*007, to which, for the above reasons, he assigned a real uncertainty of i0"-02. The value determined by the present Astronomer Royal in 1941, which is 8"-790, is extremely close to Cowell’s value.

A striking proof of the accuracy and efficiency of Cowell’s work was furnished by a wave with a period of 17^ years from crest to crest in the values of certain coefficients, which had been discovered empirically by Newcomb in 1876, and afterwards correctly attributed by E. Nevill to the action of Jupiter, and named the Jovian evection. Cowell found from his discussion of the observa­tions the value —T'T for its coefficient, whereas Hill and Radau had found theoretically the value —0"*9. Later both Newcomb and Brown found by theory precisely Cowell’s value.

In 25 Cowell dealt with the moon’s latitude in a way similar to that of his investigations on the longitude. His coefficients could now be compared with the theoretical values given by Brown’s Lunar , which had meanwhilebeen published. The agreement was extremely satisfactory.

So far, only the periodic terms in the lunar motion had been considered. There still remained the question of the so-called secular acceleration of the moon’s mean motion. From a study of ancient eclipses recorded by Ptolemy and the Arabian astronomers, Halley concluded in 1693 that the mean motion of the moon had been becoming continually more rapid ever since the epoch of the earliest recorded observations. The theoretical explanation was sought in vain by Newton, Euler and Lagrange, and was not found until 1787, when Laplace showed that the inequality is not truly secular, but periodic, though the period is immensely long, in fact millions of years: and that it is due to a diminution in the solar action on the moon, caused by the diminishing eccen­tricity of the terrestrial orbit, which in its turn is the result of the action of the planets on the earth. Laplace’s first value for the secular acceleration of the mean motion was 11"* 135 per century, which was later reduced to 10"T8. In Hansen’s Tables de la Lune the value adopted was 12"*18. J. C. Adams, however, showed not long afterwards that all these values are much too large, and gave the theoretical value 5 "-64, while Brown in 1897 found 5"-91.

In Cowell’s first paper on the subject (27) he determined the secular accelera­tions of the moon’s longitude and node from the solar eclipses of the years —1062, —762, —647, —430 and +197, and found that the same two supposi­tions satisfied the conditions for all five eclipses; namely that the observed secular acceleration of the moon exceeded the theoretical value of 5"*9 by 5%' and that there was also deducible from observation a secular acceleration of the sun (i.e. of the earth’s orbital motion) of amount 4". These assertions led to a controversy with the eminent American astronomer Simon Newcomb,, who held that ‘there being no observations of times or phases [in the ancient records], the only fact we can take as the base of a conclusion is that a well- identified eclipse was total at a known place. By a curious fatality there is always

Philip Herbert Cowell 379

some weak point in each of the small number of cases in which this condition is presumably satisfied by the narration.’ Newcomb further objected to Cowell’s proposed secular acceleration in the mean longitude of the sun, that ‘no such acceleration is shown by 150 years of modern observations of the sun; and if it exists, it must be due to some cause other than gravitation’.

Cowell put forward in 35 a tentative explanation of the secular acceleration of the sun, which rested on the following principles: (i) a slowing of the earth’s rotation from tidal friction necessarily involves an increase in the moon’s distance, in order that the angular momentum of the earth-moon system may be conserved; (ii) the lengthening of the moon’s period arising from this may be nearly as great as the apparent shortening due to the increase in the length of the day; (iii) in the case of the sun’s period, the shortening due to the latter cause would act unimpeded, so that an apparent acceleration of the sun, com­parable with that of the moon, would result.3

The merit of Cowell’s contributions to lunar theory was recognized in 1906 by his election to Fellowship of the Royal Society. His activities now took a new direction.

At the meeting of the Royal Astronomical Society in December 1906, Cowell’s colleague at Greenwich, A. C. D. Crommelin, read a paper on the approaching return of Halley’s comet. This object, which was observed by Halley in 1682, was by him identified with a comet which had appeared in 1607 and some earlier dates, and he predicted (correctly) that it would be seen in 1758: it was again visible in 1835, and was due to return in 1910. Crommelin now pointed out that computations made by A. J. Angstrom in 1862 led to a value 1913-08 for the time of its perihelion passage, whereas the Comte de Pontecoulant in 1864 had found 1910-37, a discordance of 2-7 years; and he stressed the need for a recalculation. Soon after the meeting he suggested to Cowell that they should undertake it jointly; and to this Cowell agreed.

In the earlier stages, the work had the character of being an extension of Pontecoulant’s investigation, the perturbations being computed for the time from 240 b.c. to a.d . 1531 (Pontecoulant had carried his calculations back to 1531).

Meanwhile, however, their attention was called in the early spring of 1908 to a discovery which proved to be of great theoretical interest. From 1610 to 1892 it had been supposed that the four moons of Jupiter discovered by Galileo were the only members of the Jovian system; but in 1892, 1904 and 1905, three more satellites were discovered at the Lick Observatory. P. J. Melotte, of the Royal Observatory, Greenwich, determined to search for others, and after labours extending over three successive oppositions of Jupiter, he found photographically on 28 February 1908 an object of the 17th magnitude, which he was able to trace back as far as 27 January of the same year on plates taken in the interval. Further photographs were obtained in March and April, but it was as yet uncertain whether the object was a member of Jupiter’s family

3 Subsequent developments regarding Cowell’s suggestion of a secular acceleration of the sun are discussed by E. A* Milne, 1936. Proc. Roy. Soc. A> 156, at pp. 81-85.

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or a minor planet. Crommelin with remarkable acumen pointed out that if it were a satellite with direct motion (as were all motions then known), its period of revolution must be about four years, and that no orbit with such a period could be stable, as was shown by an examination of the periodic orbits com­puted in connexion with Hill’s lunar theory: he therefore made the revolu­tionary suggestion that the motion might be retrograde, as in fact was found to be the case. Since Laplace’s well-known proof of the stability of the solar system involves the assumption that all the motions are direct, the discovery of a violation of this condition presented a difficult problem to cosmologists. It appeared that the orbit was inclined as much as 30° to the plane of Jupiter’s orbit, and that, as the distance from the planet was more than twelve times that of the outermost of the Galilean satellites, the perturbations by the sun were very great, amounting to as much as ten per cent of Jupiter’s attraction. Cowell and Crommelin, who undertook to calculate the motion, decided to abandon any attempt to find a satisfactory analytical solution and resolved to perform the computations directly from the differential equations by a method of mechanical quadratures, which was in great part newly devised by Cowell (53 and 55).

Cowell and Crommelin were so impressed by the advantages of this method that they proceeded to apply it in order to compute the motion of Halley’s comet between 1759 and 1910 (59 and 60, the latter Essay being awarded the Lindemann Prize of the Astronomische Gesellschaft). The solution was built up by calculating the portions of the trajectory described in successive intervals, namely intervals of two days when the comet was near the sun, and intervals of four days, eight days, . . . , 256 days, as the solar distance increased. The perturbations were very considerable, the action of Jupiter alone accelerating the return of 1910 by no less than 800 days.

The time of perihelion predicted by Cowell and Crommelin was about 2-7 days earlier than the time deduced later from actual observations. This discrepancy could not be accounted for by any defect in the calculations, and it would seem therefore that there was some small disturbing cause or causes at work, other than gravitational attraction. The nature of these was not known, but it may be recalled that Backlund had found unexplained accelerations of the mean motion of Encke’s comet at certain points of its trajectory.

The methods introduced in these researches of Cowell and Crommelin represented a great advance in the practice of orbit computation: they have proved extremely convenient for computation with calculating machines, of which, however, Cowell himself never made any use.

His work was recognized in 1910 by the conferment of an Honorary D.Sc. of the University of Oxford, and in 1911 by the award of the Gold Medal of the Royal Astronomical Society.

In 1910 the position of Superintendent of the Nautical Almanac fell vacant: Cowell saw a possibility of making the Nautical Almanac office a centre for research in dynamical astronomy, and applied for the appointment, to which he was elected. The Admiralty, however, gave no support or encouragement

Philip Herbert Cowell 381

towards the fulfilment of his plan, and he settled down with some disgust to perform the traditional work of the office. He was, however, active and successful in devising improvements in computational procedure, and much time and labour were saved.

The one position which would have suited him was the Plumian professor­ship of astronomy at Cambridge, which was purely a research chair in celestial mechanics, not attached to an Observatory and involving no fixed obligations beyond a few lectures to advanced students. In 1912 this chair was vacated by the death of Sir George Darwin, and Cowell was a candidate for it: but A. S. Eddington was chosen. A year later the Lowndean professorship of astronomy and geometry at Cambridge was vacated by the death of Sir Robert Ball: in this case Cowell had actually been giving the lectures connected with the chair during Ball’s last illness; but again he was unsuccessful, H. F. Baker being elected. These disappointments, intensified by the feeling that the Astronomer Royal had not supported his candidatures, helped to complete the alienation from astronomical science and its exponents which had begun soon after his appointment to the Nautical Almanac office: and henceforth he could no longer be regarded as an active research worker in the subject, and was not to be seen at scientific meetings.

Cowell was keenly interested in bridge, and had theories of his own on calling which he developed in a series of articles published by his friend Sir Percy Everett in the magazine edited by him. After his wife’s death, when living alone at Blackheath, he would spend long afternoons and evenings at the Berkeley Club, on partnership days taking with him a player whom he had previously instructed in his system of calling. That the system was sound and his play good was attested by its success. He was also an excellent chess player, and after his retirement played for Suffolk in county matches.

He took to motoring in the 1920’s, regarding it to the end of his life as a sport rather than as a means of getting from place to place. He used to argue (quoting the kinetic theory of gases) that the higher the speed, the less was the chance of a collision: with the result that some of his friends at Greenwich were reluctant to travel with him as passengers.

On attaining the age of sixty in 1930, he arranged that his nephew should be waiting with a car on his birthday, at the precise hour of his birth, outside the Nautical Almanac office: as he emerged for the last time, hatless and with the light on his thick white hair, he said on approaching the car ‘Old Kaspar’s work is done’. He retired to Aldeburgh, which had long been the home of many of his relatives.

His affection for Trinity never failed: he often went up to Commemoration, and the College was remembered in his bequests.

His wife predeceased him on 27 July 1924: there were no children of the marriage.

He died of cardiac asthma on 6 June 1949.I have to acknowledge with thanks help received in the preparation of this

notice from Mrs Hew Stevenson (Cowell’s sister) and her son and daughter,

382 Obituary Notices

and from the Astronomer Royal, the Astronomer Royal for Scotland, the Superintendent of the Nautical Almanac, Lord Pethick-Lawrence, Dr R. d’E. Atkinson, Dr L. J. Comrie, Mr P. J. Melotte and Mr W. F. Sedgwick.

Edmund T. W hittaker

Philip Herbert Cowell 383

(1) 1895.(2) 1896.

(3) 1896.

(4) 1898.

(5) 1899.(6) 1900.(7) 1900.(8) 1901.

(9) 1902.

(10) 1903.

(11) 1903.

(12) 1904.(13) 1904.

(14) 1904.

(15) 1904.

(16) 1904.

(17) 1904.

(18) 1904.(19) 1904.(20) 1904.

(21) 1904.

(22) 1905.(23) 1905.(24) 1905.

(25) 1905.(26) 1905.

(27) 1905.

BIBLIOGRAPHY

Recent developments of lunar theory. Brit. p. 614.Note on the value of the longitude in the lunar theory when the sun’s mass

is put zero. Mon. Not. R. Astr. Soc. 56, 3.On the inclinational terms in the moon’s co-ordinates. Atner. J. Math. 18,

99.Refraction tables arranged for use at the Royal Observatory, Greenwich.

Greenwich Ohs. (Append. I), iv + 11 pp.On the tables in the prayer-book for finding Easter. Observatory, 22, 304. Note on the formulae for star corrections. Mon. Not. R. Astr. Soc. 60, 607. On the form of planetary ephemerides. Observatory, 23, 447.The normal equations that arise in the usual schemes of observation for

division errors and their solutions. Mon. Not. R. Astr. Soc. 61, 527. Reductions of extra-meridian observations of planets. Mon. Not. R. Astr.

Soc. 62, 503.Errors in the moon’s tabular longitude as affecting the comparison of the

Greenwich meridian observations from 1750 with theory. Mon. Not. R. Astr. Soc. 64, 23.

On the semi-diameter, parallactic inequality, and variation of the moon from Greenwich meridian observations, 1847-0 to 1901-5. Mon. Not. R. Astr. Soc. 64, 85.

Transformation of Hansen’s tables. Mon. Not. R. Astr. Soc. 64, 159. Methods of analysis of moon’s errors and some results. Mon. Not. R. Astr.

Soc. 64, 412.Some further analyses of the moon’s errors of longitude. Mon. Not. R. Astr.

Soc. 64, 535.Methods of correcting moon’s tabular longitude. Mon. Not. R. Astr. Soc.

64, 571.Further analysis of moon’s errors with mean elongation as argument,

1847-1901. Mon. Not. R. Astr. Soc. 64, 579.Analyses of errors of moon’s longitude for inequalities of longer periods.

Methods and results. Mon. Not. R. Astr. Soc. 64, 684.The parallactic inequality: a reply. Mon. Not. R. Astr. Soc. 64, 694.New empirical term in the moon’s longitude. Mon. Not. R. Astr. Soc. 64, 838. A discussion of the long-period terms in the moon’s longitude. Mon. Not.

R. Astr. Soc. 65, 34.Analysis of 145 terms in the moon’s longitude, 1750-1901. Mon. Not. R.

Astr. Soc. 65, 108.The longitude of the moon’s perigee. Mon. Not. R. Astr. Soc. 65, 268. Reply to Professor Turner’s further note. Mon. Not. R. Astr. Soc. 65, 562. The coefficient of the principal term in the moon’s latitude. Mon. Not. R.

Astr. Soc. 65, 564.The moon’s observed latitude, 1847—1901. Mon. Not. R. Astr. Soc. 65, 721. On the discordant values of the principal elliptic coefficient in the moon’s

longitude. Mon. Not. R. Astr. Soc. 65, 745.On the secular accelerations of the moon’s longitude and node. Mon. Not.

R. Astr. Soc. 65, 861.

384 Obituary Notices(28) 1905. On the value of ancient solar eclipses. Not. R. Astr. Soc. 65, 867.(29) 1905. On the secular acceleration of the earth’s orbital motion. Mon. Not. R. Astr.

Soc. 66, 3.(30) 1905. On the Ptolemoic eclipses of the moon recorded in the Almagest. Mon.

Not. R. Astr. Soc. 66, 5.(31) 1905. Reply to Professor Newcomb’s note. Mon. Not. R. Astr. Soc. 66, 35.(32) 1905. On the transits of Mercury, 1677—1881. Mon. Not. R. Astr. Soc. 66, 36.(33) 1906. Discussion of Greenwich Observations of the sun, 1864—1900. Mon. Not.

R. Astr. Soc. 66, 302.(34) 1906. Discussion of Greenwich Observations of Venus, 1869—1900. Mon. Not.

R. Astr. Soc. 66, 307.(35) 1906. A tentative explanation of the apparent secular acceleration of the earth’s

orbital motion. Mon. Not. R. Astr. Soc. 66, 352.(36, 37) 1906. On ancient eclipses. Mon. Not. R. Astr. Soc. 66, 473, 523.(38) 1906. The mediaeval eclipses of Celoria. Mon. Not. R. Astr. Soc. 67, 17.(39) 1906. Hansteen’s eclipse at Stiklastad, 31 August 1030. Mon. Not. R. Astr. Soc.

67, 136.(40, 41, 42) 1907. (With A. C. D. Crommelin.) The perturbations of Halley’s comet.

Mon. Not. R. Astr. Soc. 67, 174, 386, 511.(43) 1907. On the Jupiter evection term. Mon. Not. R. Astr. Soc. 67, 356.(44) 1907. Ancieift eclipses. Mon. Not. R. Astr. Soc. 67, 508.(45) 1907. On ancient eclipses. Mon. Not. R. Astr. Soc. 68, 109.(46, 47, 48, 49, 50) 1907-1908. (With A. C. D. Crommelin.) The perturbations of

Halley’s comet in the past. Mon. Not. R. Astr. Soc. 68, 111, 173, 375, 510, 665.

(51) 1908. (With A. C. D. Crommelin.) The perturbations of Halley’s comet, 1759-1910. Mon. Not. R. Astr. Soc. 68, 379.

(52) 1908. (With A. C. D. Crommelin.) Tables giving approximate values of theperturbations of Halley’s comet by Jupiter and Saturn in the first and fourth quadrants of the orbit. Mon. Not. R. Astr. Soc. 68, 458.

(53) 1908. (With A. C. D. Crommelin.) The orbit of Jupiter’s eighth satellite. Mon.Not. R. Astr. Soc. 68, 576.

(54) 1909. Development of the disturbing function in planetary theory, in terms ofthe mean anomalies and constant elliptic elements. Mon. Not. R. Astr.Soc. 69, 170.

(55) 1909. (With A. C. D. Crommelin & C. D avidson.) On the orbit of Jupiter’s eighthsatellite. Mon. Not. R. Astr. Soc. 69, 421.

(56) 1909. Note on Mr Nevill’s paper on the date employed in Oppolzer’s Canon derFinsternisse. Mon. Not. R. Astr. Soc. 69, 434.

(57) 1909. On ancient eclipses. Mon. Not. R. Astr. Soc. 69, 617.(58) 1909. (With A. C. D. Crommelin.) Note on the time of perihelion passage of

Halley’s comet. Mon. Not. R. Astr. Soc. 70, 3.(59) 1910. (With A. C. D. Crommelin.) Investigation of the motion of Halley’s comet

from 1759 to 1910. Appendix to the 1909 volume of Greenwich Observa­tions.

(60) 1910. (With A. C. D. Crommelin.) Essay on the return of Halley’s comet. Publ.Astr. Ges. Lpz. no. 23.

(61) 1919. Numerical differences for the year 1923 between Professor E. W. Brown’stabular places of the moon and the places according to Hansen’s tables (1857) with Newcomb’s corrections (1878). Mon. Not. R. Astr. Soc. 79, 391.

4