memoir by a theoretical consideration of the action of a resisting medium, and concludes that facts agree better on the assumption that the medium acts directly as the velocity, and inversely as the square of the radius vector. The investigations of Oppolzer and Haerdt indicate that there is an acceleration also in the mean motion of Winnecke's comet. These two are the only comets whose motions so far as they are known cannot be fully satisfied by Newton's law. The large eccentricities of the orbits of periodic comets have thus far rendered it impracticable to compute tables of their motions as has been done in the case of planets and their satellites. The general perturbations of Encke's comet have indeed been partially determined, but the results reached, though interesting and valuable, can by no means be considered satisfactory. We have thus glanced briefly at the present condition of our knowledge of the motions of the principal bodies of the solar system. Only four cases have been found in which we cannot fully explain these motions so far as known, by Newton's law of gravity. The unexplained discordances are the motion of the perihelion of Mercury, and the accelerations of the mean motions of the moon and the two periodic comets just named. But in each case, except perhaps the first, there exists a plausible explanation other than a modification of Newton's law. If we go beyond the solar system, we cannot tell whether Newton's law does or does not apply without modification to all parts of the universe. It is principally in the hope of answering this question that double star observations are carried on, and in the case of the many binary systems already detected, Newton's law is satisfied within the errors of observation. Nevertheless, this evidence is purely negative, and its value, it seems to me, not at all commensurate with the labor expended upon it, unless it be in the case of such objects as Sirius, whose observations may assist in the solution of the problem of irregular so-called proper motion. The angles subtended are in general so small that relatively large personal errors are unavoidable, so that, even though their motions be controlled by a law or laws of gravity widely dif ferent from that of Newton, it is not likely that such differences can be proved with any degree of certainty. It is rather to the study of the proper motions of the fixed stars and of the nebulæ, and then only after a lapse of hundreds, and perhaps thousands of years, that we must look for a solution of this question. PAPERS READ. OBLITERATION FROM ILLUMINATION (STELLAR PHOTOMETRY). By HENRY M. PARKHURST, New York, N. Y. [ABSTRACT.] In order to establish the principle that illumination of the field tends to vary the scale of the wedge, I observed Arcturus two hours before sunset with four different apertures differing by half a magnitude. It required about twenty minutes for the approaching twilight to cause the star to become visible with an aperture half a magnitude smaller. It required only four minutes from the first glimpse of the star with each cap for the star to pass entirely across the wedge unextinguished. The effective value of the wedge under that illumination was only about m.1, whereas it had been determined to be with a dark sky about 2 m.3. Daylight illumination reduced the effect of the wedge in extinguishing stars to less than onetenth. The formulæ showing the effect of illumination upon obliteration, applicable to a dark sky, to moonlight, and to twilight, have been deduced by an indirect process, observing the effect of shades. From these formulæ subsidiary tables have been formed, from which it appears that there is a second difference amounting to m.2 or m.3, resulting from the illumination of the dark sky with a large aperture and low power. For stars always observed at the same point of a wedge this becomes a systematic error. Although needing further verification, the substantial accuracy of these tables has been confirmed. The ratio of illumination to obliteration has been approximately ascertained, enabling tables for the application of the second differences to be adjusted to different magnifying powers and different apertures. From observations at different ages of the moon and at different distances from the moon, including observations of Nova Orionis within 2° from the full moon, I have formed tables of the obliteration by the moonlight. From observations in the twilight, I have also formed tables of obliteration from twilight. But there remains to be formed a table from which the combined effect of the two can be ascertained. Whenever the stars to be observed are sufficiently bright, I prefer to observe them with my deflecting apparatus (described in Vol. XVIII of the Annals of the Harvard Observatory, in a paper just issued) which is unaffected by illumination. But logarithmic caps cut off four-fifths of the light, so that the wedge will measure stars 1m.7 fainter. If the necessary corrections are applied, the wedge can be used for stars up to the quadrature of the moon and half an hour before the end of twilight, which could not be seen at all with the logarithmic caps. A METHOD OF REPRESENTING THE IMAGINARY ELEMENTS OF A GEOMETRIC FIGURE AND OF USING THEM IN CONSTRUCTION. BY JAMES MCMAHON, Ithaca, N. Y. [ABSTRACT.] 1. Point-graphs. Call the real points (a ± a', b± b') the graphs of the conjugate imaginary points (a ±ia', b±ib'). Harmonic properties. 2. Graph-locus. If a line meet a conic in two conjugate imaginary points, to find the locus traced by the graphs of these points, as the line moves parallel to itself. To find the real chords joining the imaginary intersections of conics, in certain cases. 3. Line-graphs. Call the lines (l±l') x + ( m ± m') y + (n ±n') = 0 the graphs of the conjugate imaginary lines (lil') x + (m ± im') y + (n± in') = 0. These two line-pairs have the same angle-bisectors, and are cut by a transversal perpendicular to an angle-bisector in an imag. inary point-pair and their real graph-pair. 4. Orthographs and skew-graphs. For transversals in a given direction not perpendicular to an angle-bisector, the graph-locus is easily constructed; it is a pair of lines, which may be called the skew-graphs for the given transverse direction, to distinguish them from the orthographs (for the orthogonal direction). All the graphs for different directions form an involution, of which the imaginary line-pair are the double lines. 5. Given the respective graph-pairs of two imaginary points, not conjugate, to construct (by means of line-graphs) the imaginary line joining them; and reciprocally. 6. Given (by means of graphs) an imaginary line and an imaginary point in it, to construct the imaginary line that passes through the point making a given angle with the first line. Special applications. ON THE VALUE OF THE SOLAR PARALLAX DEDUCIBLE FROM THE AMERICAN PHOTOGRAPHS OF THE LAST TRANSIT OF VENUS. By Prof. WM. HARKNESS, Washington, D. C. [ABSTRACT] In this paper an account was given of the instruments and processes employed by the United States Transit of Venus Commission to determine the solar parallax from photographs of the transit of Venus which occurred in December, 1882. Let be the solar parallax, and dA and ¿D respectively the corrections to the right ascensions and declinations of Venus given by Hill's tables of that planet. Then, upon the assumption that Hansen's tables of the sun are correct, there resulted from measurements of the distances between the centers of the sun and Venus made upon 1475 photographs, taken respectively at Washington, D. C.; Cedar Keys, Fla.; San Antonio, Tex.; Cerro Roblero, N. M.; Wellington, South Africa; Santa Cruz, Patagonia; Santiago, Chili; Auckland, New Zealand; Princeton, N. J.; and the Lick Observatory, Cal.; and the corresponding mean distance from the earth to the sun is 92,385,000 miles, with a probable error of only 125,000 miles. These numbers are doubtless close approximations to the results which will be obtained from the complete discussion of all the photographs, but they cannot be regarded as final for several reasons, chief among which is the fact that the reduction of the position angles of Venus relatively to the sun's center is still unfinished. It is likely that when these angles are combined with the distances the probable error of the parallax will be somewhat reduced. The photographs taken at the Lick Observatory seem to indicate that for altitudes four thousand feet above the sea the values of the refraction given by the tables in general use are somewhat too large. THE DIRECTIONAL THEORY OF SCREWS. By Prof. E. W. HYDE, Station D, Cincinnati, Ohio. DEFLECTIONS OF THE PLUMB-LINE AND VARIATIONS OF GRAVITY IN THE HAWAIIAN ISLANDS. BY ERASMUS D. PRESTON, U. S. C. and G. Sur vey, Washington, D. C. ON A NEW METHOD OF CONSTRUCTION FOR EQUATORIAL DOMES. By Prof. ON A NEW CATALOGUE OF VARIABLE STARS. By SETH C. CHANDLER, Cambridge, Mass. A NEW SHORT-PERIOD VARIABLE IN AUTLIA. By Prof. HENRY M. PAUL, U. S. Naval Observatory, Washington, D. C. LAWS OF FREQUENCY OF ERRORS OF INTERPOLATED LOGARITHMS, ETC., DEPENDENT ON FIRST DIFFERENCES; AND A COMPARISON OF THE THEORETICAL WITH THE ACTUAL DISTRIBUTION OF THE ERRORS OF 1000 INTERPOLATED VALUES. By Prof. R. S. WOODWARD, Geological Survey, Washington, D. C. SOME CONSIDERATIONS ON THE FUNDAMENTAL IDEAS OF QUATERNIONS. By Prof. E. W. HYDE, Station D, Cincinnati, Ohio. |