XXII. MATHEMATICS AND ASTRONOMY MATHEMATICS E. B. WILSON Collegiate Mathematics. The Mathematical Aeronautics.-During American Mathematical Society has the past few years a new application now about 700 members. It publishes of mathematics has arisen in the theits Bulletin (edited by Prof. F. N. ory of the aeroplane. The aeroplane, Cole, Columbia University, New except for its motive and guiding apYork), which contains reports of the paratus, is essentially a rigid body meetings of the Society, book reviews, which in flight moves through the personal notes, lists of new books, fluid air, and the forces which susand short original articles; the vol- tain the machine and control its moume runs to about 550 pages per tion against the attraction of gravity annum. It publishes also its Trans- are the pressures of the disturbed air actions, a journal devoted to research upon the machine. The problem is, of high caliber, and of some 500 quar- therefore, fundamentally one in hy to pages per annum. The work of dromechanics. Unfortunately the thethe Society is therefore for advanced ory of fluid motion is very complimathematicians; although a large cated and is not yet developed to the number of the members are probably point where the pressures upon the not themselves prosecuting advanced aeroplane, and their points of appliresearches, but are occupied almost cation, can be accurately determined. entirely with teaching and executive Even when, for simplicity, it is aswork, the Society does not pretend to sumed that the wings, tail, fins, etc., do much for elementary collegiate of the machine are plane surfaces, the mathematics and its teachers, except theoretical deductions from hydroindirectly. The American Mathemat- mechanics are not sufficiently near to ical Monthly (edited by Prof. H. E. the results derived from experiment Slaught, University of Chicago) ca- to furnish an entirely safe foundation ters precisely to this large class of for a theory of flight. Nevertheless, persons interested in collegiate Prof. G. H. Bryan (Bangor, Wales), mathematics. The Monthly contains one of the first theorists in the field, book reviews, personal notes, prob- about five years ago obtained imlems proposed for solution and their portant results in his Stability in solutions, discussions, historical arti- Aviation (Macmillans), based on the cles, brief elementary researches, and assumption of flat surfaces. At that runs to some 200 pages per annum. time he treated problems connected During 1915 a proposition was made with steady flight in a horizontal line, to the American Mathematical So- and has during 1915 written an imciety to take over the Monthly and to portant monograph upon the problem widen the Society's activities by tak- of circling flight, using the same asing an active and effective interest in sumptions. the collegiate field. The Council of the Society, which directs its affairs, felt that this step would be inadvisable. There is therefore under way the organization of another national mathematical society whose field will be collegiate mathematics. In the meantime, the National Physical Laboratory at Teddington, England, and other laboratories, such as that at the Massachusetts Institute of Technology, which deal with problems of aeronautical engineering, have proceeded in a different way. The forces, which act upon the machine are determined directly by experiments upon small models placed in a wind tunnel and by calculations which scale up the results from the model to the full-sized machine. With the forces thus determined, the hydromechanical part of the problem has been solved by experiment and the theory of the motion of the machine reduces to the theory of a rigid body moving under these forces. Even as thus reduced the problem remains sufficiently mathematical (cf. the Technical Reports of the Advisory Committee for Aeronautics, Darling & Son, London). Mathematical science has arrived to about the point where the motion of the aeroplane in a straight line or in a curve which is nearly a straight line can be followed in still or in gusty air and the safety of the machine under normal conditions of flight may therefore be determined from the design. Sir George Greenhill has prepared for the Advisory Committee a Report on Gyroscopic Theory (Darling & Son, London), containing the various methods of theoretically explaining the motion of a spinning body, a subject likely in the future to have important utility in its bearing on the steering and stability of a flying machine. ing work by a great English mathematician of the middle of the last century. Dr. H. Bateman (Johns Hopkins) whose work was referred to in the last issue of the YEAR BOOK (p. 583) has published an account of his theories in a book entitled The Mathematical Analysis of Electrical and Optical Wave Motion on the Basis of Maxwell's Equations (Cambridge, England). Prof. Pierpont (Yale) has published (Ginn) an elaborate work on the Theory of Functions (complex variable); he had previously printed a two-volume work on the functions of real variables. During the spring of 1915 Prof. C. J. de la Vallée-Poussin (Louvain, Belgium), an exile from his country, delivered a course of lectures at Harvard setting forth some of his recent important investigations. Necrology. The country has lost a number of mathematicians who had become eminent as teachers or executives through long periods at their respective institutions. J. H. Van Amringe of Columbia, for a long time head of the mathematical department and dean, founder and first president of the American Mathematical Society, died on Sept. 10 at the age of 80. A. W. Phillips, author of a number of texts, formerly head of his department and dean of the GradPersonal Notes.-Prof. H. S. White uate School at Yale, died on Jan. 20, (Vassar) was elected in 1915 to the aged 70. S. W. Shattuck (Illinois), National Academy of Sciences F. A. Sherman (Dartmouth) and J. (Washington), section of mathemat- J. Hardy (Lafayette) are other seics. At the end of 1914, Prof. E. B. nior members of the profession who Van Vleck (Wisconsin) finished his have died. A serious international biennial term as president of the loss was Prof. G. B. Guccia (PalerAmerican Mathematical Society, and mo, Italy), who founded 30 years ago was succeeded by Prof. E. W. Brown the Circolo Matematico di Palermo (Yale). Prof. Brown has been award- and with untiring energy and coned the gold medal of the Royal So- siderable personal financial support ciety (London) for his researches, in developed the Circolo into the foremathematical astronomy. Prof. R. E. most place as an international matheMoritz (University of Washington)|matical society counting among its has published his Memorabilia Ma- members a large number of American thematica or the Philomath's Quota- mathematicians. The official organ tion Book (Macmillans), which con- of the Circolo, the Rendiconti, has tains a large variety of sayings on acquired a position as one of the leadvarious mathematical topics by ing international mathematical pemany eminent students. Prof. D. E. riodicals; it has printed a number of Smith (Columbia) has brought out a researches by Americans, and in 1915 new edition in two large volumes, an- contained (among other American notated, of Augustus De Morgan's papers) a long contribution by Prof. Budget of Paradoxes (Open Court Pub- G. D. Birkhoff (Harvard) on the lishing Co.), a curious and interest- "Problem of Three Bodies." ASTRONOMY R. S. DUGAN Instruments and Observation.—Con- | small island with a large stretch of sistent performance of large reflectors ocean about it would be a good place is to be expected only when tempera- for an observatory, as an almost comture deformation of the mirror is plete absence of convection currents carefully guarded against. Refrac- in the atmosphere, and consequent tors, on the other hand, can ordinari- good seeing, is to be expected in such ly be used over a large range of tem- a location. Pickering reports wonderperature without scruple. As the de- fully fine observing conditions on the sire for accuracy becomes more in- island of Jamaica. sistent, precautions multiply. Schlesinger has found it worth while to install an electric fan to draw out the air from behind the 30-in. objective of the Thaw telescope at Allegheny, thereby getting rid of troublesome changes in spherical aberration. Michelson has finished the ruling of a 10-in. grating having about 622 lines to the millimetre, which gives superb resolution in the eighth-order spectrum. In measuring distances between star images on a photographic plate, a troublesome error enters, depending on the size of the image or brightness of the star. Kapteyn suggests a device by which this source of systematic error may be eliminated. A photograph taken a little inside the focus of the telescope will show enlarged star images of nearly equal size but of various densities. This plate is developed and used as a through which to expose the main plate at the focus, the cone of light from a star going through its intrafocal image before striking the main plate. By a suitable length of exposure the images of all stars, bright or faint, will be of nearly equal size (Ap. J., xli, 77). screen It is Evershed's opinion that a References to periodicals are given under the following abbreviations: A. J., Astronomical Journal, Albany. Ap. J., Astrophysical Journal, Chi- cago. des Sciences, Paris. M. N., Monthly Notices of the Royal Astronomical Society, London. Obs., Observatory, London. The Roman numerals denote volumes; the Arabic numerals pages, except in references to Astronomische Nachrich ten and the Observatory, where they denote individual numbers of the periodical. The Earth.-The discussion of the observations obtained during the carrying out of the elaborate programme for determining by the aid of wireless signals the difference in longitude between Washington and Paris (A. Y. B., 1914, p. 608) gave the value 5h 17 m 36 s.658 ± 0.0029, which is in good agreement with the older value obtained with the aid of the cable. The signals were found to travel at the rate of 175,000 ± 16,000 miles per second. The war, contrary to expectation, did not interfere with latitude observations during 1914 at the six international stations maintained for the study of small variations in latitude caused by the wandering of the terrestrial poles. The amplitude of the motion of the pole, which reached a very small value in 1913, increased throughout 1914 (A. N., 4802). The Cookson floating zenith-telescope at Greenwich has now been in use for three years. The preliminary value found for the constant of aberration-the angle through which the line of sight to a star is displaced by the orbital motion of the earth at right angles to the direction of the star-is 20".4670".006, in com plete agreement with the value found from the solar parallax and the velocity of the earth deduced therefrom. The Moon. The positions of the moon are now being determined with high accuracy photographically. Since January, 1911, photographs of the moon have been taken at Harvard at short intervals. The plate is screened from the moon's light during the relative long time required for the exposure on the neighboring stars, whose distances from the moon are measured. In Harvard Annals (lxxvi, No. 7) Russell and Miss Fowler give the results from the first 204 plates Some years ago Hale discovered that sun-spots are magnetic vortices. A quarter-wave plate and a Nicol prism mounted over the slit of the spectroscope show the direction of rotation of the electrons in the spot measured at Princeton. The results | astronomers found a new and strong are compared with the Greenwich line in the red at about A 6376. The measures, with which they are in good usual green line at X 5303.7 was abagreement. The photographic posi- sent. Cortie finds that certain streamtions are probably more uniformly ers of the solar corona seen at the good, however, as measures are made August eclipse had their apex or raall along the visible limb and are dis- diant area in the region of the large cussed with due regard to the de- sun-spot disturbance which persisted pressions and elevations. Brown has for several months. Four other such published the values of the elements connections between coronal streamof the moon's orbit adopted for use ers and long-continued sun-spot activin the new lunar tables, on which he ity have been previously shown (M. has long been working, with a brief N., lxxv, 496). summary of the reasons for each choice. The value of 1/294 for the earth's ellipticity has been adopted in order to reconcile the theoretical and observed values of the mean motion of perigee and node-a value which also reconciles the best deter- vortex. Later, evidence of the existmination of the lunar parallax with its gravitational value (M. N., lxxv, 508); compare, however, de Sitter (Obs., 490), who considers 1/296 the most reliable value. Brown adopts the theoretical value of the secular acceleration of the mean longitude, inserting a single empirical termthe large one with a period of some two and a half centuries. In the accurate prediction of eclipses, additional empirical terms may in each case be deduced from observations of the few previous years. He will give a few short additional tables for the investigation of ancient eclipses. Brown has been unable to explain satisfactorily certain outstanding ir regularities in the moon's orbital motion. It is conceivable that the cause lies in irregularities in the earth's rotation, which is used as a measure of time. Such a change in the rate of rotation of the earth should become apparent as an error of position of any body moving fast enough to show the effect (M. N., lxxv, 211). An investigation by Glauert of the available observations of Mercury, Venus, and the sun culminates in the conclusion that the errors in longitude of the moon and these three bodies may be accounted for by a rather irregular variation in the rate of rotation of the earth amounting to something like 0s.01 in the third of a century covered by the observations (M. N., lxxv, 489). The Sun.-From eclipse plates taken on Aug. 21, 1914, Spanish and French ence of bipolar spot groups was found. Such pairs of spots lie on a line usually making a small angle with the solar equator. Sometimes one spot is replaced by a group of small spots or only by flocculi, but the eastern and western extremities of the affected area show rotation in opposite directions. The direction of rotation of preceding spots near the equator in the northern hemisphere is counter-clockwise, as it is in the terrestrial tornado, while in the southern hemisphere the rotation is clockwise. In higher latitudes the directions of rotation are reversed. Hale has devised several ingenious laboratory experiments to show that at a certain speed of rotation the lower end of a columnar vortex rises to the surface and a bipolar, semicircular ring vortex is formed, similar to a smoke ring, which has a motion of translation at right angles to its plane, in the direction corresponding with that of the inner edge. This may prove to be an explanation of the observed fact that high-latitude spots in either hemisphere show a slight general tendency to move away from the equator, those in low latitudes to move toward it. Adams and Miss Burwell, in their paper describing their success in photographing without an eclipse the spectrum of the reversing layer and chromosphere of the sun with the 60ft. tower telescope at Mt. Wilson, find that they have been able to photograph the chromosphere to a lower level than that reached by eclipse | evidence has been found of internal plates, and that the accuracy in de- movement, rotatory or otherwise. termining wave-lengths is about two and one-half times as great as that attained by Mitchell in his measurement of eclipse photographs. One of the general conclusions drawn from the data is that the lines of elements of high atomic weight are relatively stronger at the lower levels in the sun's atmosphere (Ap. J., xli, 116). The largest proper motions found are 0".056 and 0".054 per year of two planetary nebulae. Most of the proper motions found may well be ascribed to errors of measurement. These nebulae, none of which are spirals, must be enormously distant. The spectrograph, with which Wolf and Slipher discovered rotatory motion in certain spiral nebulae (Ă. Y. B., 1914, p. 587), measures motion in the line of sight independently of distance. Planets and Satellites.-The mean distance of the ninth satellite of Jupiter, discovered by Nicholson (A. Y. B., 1914, p. 586), is nearly one-fifth Eleven of the 19 gaseous nebulae that of the earth from the sun. No in the Greater Magellanic Cloud are other discovered satellite in the solar found (at the Lick South American system is at so great a distance from station) to be receding at velocities its primary. At the time of discov- of 237 to 287 km. per sec.; also the ery, the orbit was inclined about 157° only known bright-line nebula in the to the earth's orbit, and had an ec- Smaller Cloud is receding at the vecentricity of 0.11. The satellite re- locity of 160 km. per sec. There are quires two years and two months to no other gaseous nebulae in the neighmake one revolution about Jupiter.borhood of the Clouds. The agree These elements are subject to consid- ment of the velocities of the nebulae erable change through the attraction in the Greater Cloud indicate a close of the sun. relationship to each other, and perComets. Comet 1915a was discov-haps to the Cloud (Proc. Nat'l Acad., ered by Mellish on Feb. 10 when it i, 496). was of about the 10th magnitude. It was nearest the earth in June, when it brightened to the third magnitude with a tail six to ten degrees long and was easily visible to the naked eye in the southern hemisphere. Thiele found on May 5 that the nucleus had separated into two parts of unequal brightness. Barnard found two subsidiary nuclei on May 12, one of them conspicuous and at a distance of 28" from the main nucleus, the other faint and at an intermediate point in the same line. Winnecke's periodic comet was discovered by Thiele on its return this year as 1915b. It was very faint, of the 16th magnitude. 1915c was Tempel's periodic comet, discovered on May 16 by Delavan. Mellish discovered another telescopic comet, 1915d, during September. 1915e, a telescopic comet, was discovered Dec. 2 by Taylor at the Cape of Good Hope. Aitken has finished his search, shared for several years by Hussey, for double stars among the stars to the ninth magnitude from the North Pole to 18° south declination. He concludes that at least one star in eighteen is a close double star visible in the 36-inch telescope; that the percentage is greatest in the Milky Way; and that relatively small orbits are in the majority. Most of our knowledge of the proper motions of stars-their change in apparent position-has been accumulated by the comparison of star catalogues of widely different epochs. The comparison of photographic plates is of late giving results in this line. Sometimes a glance at two photographs taken some years apart will show that a star has moved among its fellows. Wolf announces (A. N., 4775) a 13th magnitude star with the large, for so faint a star, annual proper motion of 1".1 in right ascension, and practically none in declination. This motion carried it directly across the line of sight to a 15th magnitude star, causing an unobserved occultation at some time between 1894, when the two stars are shown Stars and Nebulae.-A new series of plates of the nebulae photographed by Keeler and Perrine is being taken with the Crossley reflector at the Lick Observatory. One-fourth of the plates have been measured and thus far no |