XXII. MATHEMATICS AND ASTRONOMY MATHEMATICS E. B. WILSON Path of Falling Bodies.-The theory | gon becomes infinite and each side beof the southerly (or northerly) devia- comes infinitesimal. For ordinary real tion of falling bodies was mentioned curves this means that the limit of in the YEAR BOOK for 1913 (p. 613). the ratio of an arc to its chord is It has been taken up during 1914 by unity as the arc and chord approach Professor Roever (St. Louis) and Pro- zero. For regular curves the value of fessor Moulton (Chicago), from the the arc is expressible as an integral. latter of whom we may quote some- Professor Kasner (Columbia) has diswhat freely. The subject of the devia- covered that there are regular, though tion of freely falling bodies has been imaginary, curves for which the ratio considered in many memoirs from the of the arc, as thus defined by an intime of Laplace and Gauss to the re- tegral, to the chord approaches a limit cent work of Roever and Woodward. which is real and less than unity at All writers have agreed that a body certain points of the curve (Bull. Am. falling from rest near the surface of Math. Soc., July, 1914). This unexthe earth will deviate to the eastward pected result is interesting and imwith respect to a plumb-line hung portant; it shows how careful one from the initial point, but a great va- must be in pure mathematics not to riety of results have been obtained re- assume that his intuition is infallible. garding the deviation measured along It has long been known that by makthe meridian. For example, Laplace ing a curve sufficiently crinkly the found no meridional deviation, Gauss limit of the arc to chord may become a small deviation toward the equator, greater than unity. Roever a deviation toward the equator several times that of Gauss, and Woodward a small deviation away from the equator. The variety of these results is due to various methods of approximation not strictly comparable one with another and not accurately discussed with respect to the quantities neglected in the approximation. Professor Moulton gives a concise and elegant theory which leads to a result essentially concordant with that of Professor Roever, namely, that the deviation of a body freely falling a small distance near the earth's surface is toward the equator for all latitudes, and of the amount determined by Professor Roever. (Annals of Mathematics, June, 1914.) Ratio of Arc to Chord.-The length of a plane curve is defined as the limit of the sum of an inscribed polygon when the number of sides of the poly The Discontinuous in Physics.-Considerable attention in theoretical physics has been paid during the year to the structure of the atom and molecule. The corresponding mathematical work is by no means satisfactorily complete, but a great deal of attention has been given to the mathematical treatment of various discontinuities which modern physicists suppose may exist, and which would not have been considered as possible of existence a few years ago. For instance, H. Bateman (Johns Hopkins) delivered a paper at the annual meeting of the American Mathematical Society, held in New York early in January, in which he presented some possible foundations for a theory of the Faraday tube which, as a physical concept, has been so prominently used by Sir J. J. Thompson. Like the theory of quanta originated by Planck and Einstein, Bateman's theory and Thompson's hypothesis cannot yet be regarded as final; it is safe to say that the question of the discontinuous will receive before very long some more definite position in physics or will again be rejected from the science. (See also XXIV, Physics.) Personal Notes.-American mathematics has suffered more than ordinary losses in the past year. George W. Hill, past president of the American Mathematical Society, renowned all over the world for his original contributions to mathematical astronomy, and in particular for the introduction of infinite determinants, has died. Another serious loss was Alexander MacFarlane of Canada, president of the International Association for the Promotion of Quaternions and allied subjects. He himself had contributed valuable researches, original and bibliographical, to this field. Another death was that of B. O. Pierce (Harvard), president of the American Physical Society and vice-president of the American Mathematical Society, a man whose great skill and wisdom in both mathematics and physics was only equaled by his unusual modesty. Still another loss was Charles S. Pierce, who, over 40 years ago, had contributed original ideas to pure mathematics and had produced researches in mathematical logic, particularly in the theory of relatives, which have not yet been accorded their full value; in this latter field he was evidently 40 to 50 years ahead of his time. More recently, he has been active in philosophy, to which his scientific mind of great insight has brought valuable ideas. Professor Bôcher (Harvard) was in Europe during the spring as exchange professor from Harvard to the Sorbonne. We were to have had in this country during the year as exchange professors Maurice Fréchet (Poitiers, France) at the University of Illinois, and W. Voight (Göttingen, Germany) at Harvard; but the European War has deprived us, so far at any rate, of their presence. Professor Boutroux (Princeton) has had to return to France. Prof. E. W. Brown (Yale) was invited by the British Association to give an address on the "Theory of the Moon," in which he is an unexcelled expert, at the meeting in Australia during the past summer; this international exchange was fortunately not prevented. It is a pleasure to record the recognition of our fellow countryman, E. B. Van Vleck (Wisconsin), contained in the conferring upon him of the degree of Doctor of Mathematics by the University of Groningen at its tercentennial. During the month of October a disastrous fire in University Hall, Columbia, destroyed very valuable records and publications of the American Mathematical Society, a loss which it will be very difficult to make good. ASTRONOMY with the gift of $150,000 by Thomas Cawthron. This will fill the largest gap-between California and Indiain the present chain of solar observatories. Personal. The Paris Academy awarded the Valz prize to Fowler for researches in spectroscopy; Wolf received the Royal Astronomical Society's gold medal for research in celestial photography and spectroscopy; The large telescopes recently inand Backlund the Bruce medal of the stalled or under construction are Astronomical Society of the Pacific. chiefly of the reflector type. The disk The losses to astronomy by death have of the 100-in. reflector for Mt. Wilson been heavy. Sir David Gill, Winslow promises to be fully as satisfactory Upton, E. S. Holden, Scheiner, as the 60-inch now in use. The CanaWeinek, Lehmann-Filhés, Pechûle, dian 72-inch reflector will be located Klein and G. W. Hill have all died near Victoria, B. C., where the diurduring the year. nal range in temperature is peculiarly small and the "seeing" 50 per cent. better than at Ottawa. A 60-in. reflector is being figured at the Cordoba Observatories and Instruments.-A very important new solar observatory is to be established in New Zealand Observatory. The 30-in. mirror, which | a combination of methods, radial ve is to be mounted in Helwan, is completed and pronounced remarkably good. The 30-in. objective for the Allegheny refractor is finally ready. At Mt. Wilson the Koch registering micro-photometer has been applied by its inventor to the study of certain types of laboratory spectra, and is found to show the relative intensities of different lines, under varying conditions, and the distribution of the intensity within the line itself, far better than can be derived from eye estimates (Ap. J.,1 xxxix, 213). From the work of Rosenberg, Guthnick and others, it appears probable that the photo-electric photometer will excel even the selenium photometer in accuracy. This is the latest means devised for eliminating the fallible judgment of the human eye in the measurement of the comparative brightness of two stars. An electrode formed of one of the alkali metals in a well exhausted tube emits electrons when exposed to light at a rate proportional to the intensity of the light. The resulting current can be accurately measured. Ross has devised a mechanism for moving a photographic plate in an east-west direction in the focus of a zenith telescope, and applied it with much success to the investigation of the variation of latitude (A. N., 4713). The disadvantage of the objective prism in competition with the slit spectograph in the detection and measurement of line shifts has lain in the difficulty of placing comparison spectra on the same plate. Schwarzschild (Potsdam Publ. No. 69) finds that by 1 References to periodicals are given under the following abbreviations: A. J., Astronomical Journal, Albany. Ꭺ. N., Astronomische Nachrichten, Kiel. cago. Ap. J., Astrophysical Journal, ChiC. R., Comptes Rendus de l'Académie des Sciences, Paris. locities can be determined with a probable error of 7 km. per second. This error, while not large, hardly repays one for the amount of labor involved in the calculation. Hamy describes (C. R., clviii, 81) a device for utilizing the objective prism in the measurement of radial velocities. The Earth.-The variation of latitude showed a further decrease to an unusually small value at the middle of 1913, followed by a slight increase to the end of the year. The rigidity and viscosity of the earth have been investigated by Michelson, Gale and Moulton with elaborate apparatus installed near the Yerkes Observatory. Two six-inch pipes, each 500 ft. in length, were buried in trenches six feet deep running east and west and north and south, half filled with water, and sealed. The change in level of the water, caused by the change in the attractions of sun and moon, was measured through the glass ends of the pipes every hour from 6 a. m. to midnight and every two hours from midnight to 6 a. m., for a period of two months. The periods of these tides indicate, in a preliminary discussion, that the rigidity and viscosity of the earth are comparable with that of steel (Ap. J., xxxix, 105). (See also XXIV, Physics.) The Moon. Hayn, in his Selenographische Koordinaten (4te Abhandlung), gives drawings of the irregular limb of the moon derived from the measurement of many photographs taken under as many positions of libration as possible. These drawings are to serve in correcting measures of the moon's limb. Brown has published (M. N.) a number of papers in which he discusses the constant of the moon's motion, derived from the Greenwich observations according to the new lunar theory. In the May number of Observatory he states that the outstanding difficulties are not due to neglect of terms or to errors in the gravitation theory, but to the existence of unknown forces. Addressing the British Association for the Advancement ten and the Observatory, where they de- of Science at the Australia meeting, note individual numbers of the periodi- Brown set forth fully the present status of the lunar problem. To sat 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 cal. isfy the observations of the moon it is | faint in the solar spectrum. St. John necessary to take the earth's ellipticity as 1/294 instead of 1/297, the value adopted by the Nautical Almanac directors in 1911 (Science, Sept. 18, 1914). The Sun.-The European War interfered sadly with the observation of the total solar eclipse of Aug. 21. The parties which went to Minsk and Hernösand were successful. The corona was of the square type. Wilsing (Potsdam Publ. No. 66) has determined the effective temperature of the sun by comparison, photographically, with a black body. Measurements by Angström and Kennard of the energy in the green part of the spectrum, and hence independent of aqueous absorption, made during seven days on Mt. Whitney with an Angström pyrheliometer (Ap. J., xxxix, 350), fail to show any variation of the sun during that time in this part of the spectrum exceeding two per cent. Three theories have been advanced to explain the observed displacements toward the red of the lines of the solar spectrum. Jewell, Fabry and Buisson concluded that the phenomenon was due to a pressure of some five to seven atmospheres in the reversing layer. Evershed (Obs., 472) has found, among other difficulties, that the iron lines most sensitive to pressure are shifted on the average least, and by an amount corresponding to a pressure of but one-quarter of an atmosphere. He ascribes the displacements to motion in the line of sight and supposes a descending movement of the cooler vapor all over the sun. Julius (Obs., 475; Ap. J., x1, 1) rejects both these hypotheses as insufficient and interprets this phenomenon, as well as the Evershed distortion of sun-spot lines, recently explained by St. John as due to the horizontal motion of the vapors, on the basis of anomalous dispersion. He finds a mutual influence of the Fraunhofer lines, foreseen by the anomalous dispersion theory. S. A. Mitchell has published an exhaustive study of the wave-lengths of the chromospheric lines found in spectra obtained at the total eclipse of 1905 (Ap. J., xxxviii, 407). By skillxposure he obtained an unusually number of the lines which are finds, in a more recent paper (Ap. J., xl, 45), that Mitchell's observational material indicates a distribution of vapors in the solar atmosphere similar to that based on the former's interpretation of the Evershed distortion of sun-spot lines; that the vapors of the elements ascend in detectable amounts to different heights, and that the lines of any one element originate at depths increasing with decrease of solar intensity. During the present minimum the sun has been more completely free from spots than at any time since 1810. The Planets.-Saturn has been photographed by Tikhoff (Pulk. Mitt., vi, 25) through a variety of filters with the interesting result that the equatorial zone is found to be richest in yellow-green light, the ring in indigo, while the edges of the disk are bluish. This is an indication that very little light penetrates the atmosphere. On July 21 a ninth satellite of Jupiter was discovered at the Lick Observatory by Nicholson. It is about one magnitude fainter than the eighth satellite, has a period of about three years, and a retrograde motion. The observations of several astronomers show that four satellites of Jupiter and five of Saturn vary in their brightness. Guthnick, discussing all available observations (A. N., 4741), concludes that the variation of the inner satellites as a group differs from that of the outer satellites. Many new asteroids have been discovered and many known ones reobserved by an increasingly large number of observers. To prevent duplication of work, the Rechen Institut at Berlin has divided the zones of the sky where asteroids are found, and assigned fields of labor to the astronomers interested. Comets. Comet 1913f, Delavan, discovered in Dec., 1913, has been the most conspicuous comet of 1914. The great length of time between discovery and perihelion passage-ten monthswas very unusual. In September and October the comet was a fine object to the naked eye or opera glass, not far from the Dipper. The head became as bright as the third magnitude. Barnard found one part of the tail photo graphically bright, the other visually | Nicholson finds that there can be but bright. Four new comets were discov- nine of these simple atomic systems; ered during 1914: 1914a, Kritzinger, that ordinary dynamics is valid for and 1914c, Neujmin, were very faint; their free vibrations; and that the 1914b, Zlatinsky, became quite bright masses of the nuclei are proportional in May; 1914e was discovered by sev- to the squares of their charges. The eral observers in the southern hemis- proto-elements 4e and 6e, "nebulium" phere. At the time of discovery 1914e and "archonium," should give, among was a naked-eye object, but became others, the lines A 5007 and λ 3729. rapidly fainter as it came north. These are found in the spectra of neEncke's comet was discovered photo-bulæ. The first evolution products graphically on its return in 1914. show lines in the Wolf-Rayet stars. Stars and Nebulæ.-The ninth series The strength of Nicholson's point of of the very valuable Wolf-Palisa pho- view has been apparently increased by tographic charts has been published. the interferometer measurements by Quite accurate star positions can be Fabry, Buisson and Bourget (C. R., read from them with a scale. Auwers April 6, 1914) of the line X 3729 in the has compared several catalogues, in- spectrum of the Orion nebula. From cluding the A. G. zone catalogues, the width of the line the atomic weight with Küstner's 1900 catalogue, and of nearly three is found. Archonium has determined reduction tables for has a theoretical atomic weight of them, regarding the corrections as de- 2.95. The authors find, however, that pending on magnitude (A. N., 4743). this result leads them to a temperaVolume xi of the Lick Publications ture of 15,000° for the Orion nebula. contains a superb set of reproductions of photographs taken by Barnard with the 6-in. Willard lens. Spectroscopy. Barnes has photographed the spectra of magnesium, calcium and sodium vapors in an electric furnace of his own construction, which gave conditions of density, temperature and excitation producing spectra resembling that of sun spots (Ap. J., xxxix, 370). King finds that there are but few titanium lines for which a low temperature is favorable. Many of the lines are excited in a limited range of temperature (ibid., 139). St. John and Ware (ibid., 5), in continuation of their former work, have examined the international secondary standards of wave-length from A 4282 to ▲ 5506, and have determined the wave-lengths in international units of a series of 198 lines in the arc spectrum of iron between X 4118 and A5506, as tertiary standards. An ordinary lead-pencil is found by Lunt (Annals Cape Obs., x, pt. 4) to give an unexpectedly rich spectrum of fine sharp metallic lines, offering a new and convenient comparison spectrum. J. W. Nicholson presents (M. N., Dec., 1913; Jan., March, April, May, 1914) a complete theoretical scheme for the simple-ring elements and their evolution products, which, he believes, are present in nebulæ, novæ, WolfRayet stars and the solar corona. To avoid this difficulty it is necessary to assume that part of the width of the line is produced by irregular radial motion within the nebula. The atomic weight found is, therefore, a minimum value. The true value may be much larger. Miss Cannon has classified and described the spectra of 150 gaseous nebulæ and 20 novæ as well as those of several groups of stars which have bright lines (Harv. Ann. lxxvi, No. 3). An interesting general development of novæ is brought out by Adams and Pease by a comparison of spectra of four novæ of various ages. A spectrum of Nova Aurige of 1891, now of the 14th magnitude, was obtained with the 60-in. in 16 hours. As the disturbance subsides, the nebulæ lines apparently gradually disappear, the continuous background brightens, and the spectrum settles down essentially to the Wolf-Rayet type. The results achieved with the spectrograph become each year more startling. Following upon the announcement last year of the large radial velocity of the Andromeda nebula, come the astounding statements by Slipher that some of the white nebula are moving in the line of sight at velocities as great as 1000 km. per second; and that opposite ends of the long axis of the Virgo nebula NGC 4594, which is one of those moving through |