a (xxx+xxx+xxx) + b (xx y + xxy + xxy)+c (x y + xy + xy) + e (yy + y y + y y) + ƒ (x + x + x) = 3. Prime two-thirds of the x's and of the y's, leaving the equation linear, and we have the equation required: 3 axx+b (2 x'y' + x12 y) + c ( x y' + x' y + x'y') + e (2 yy' + y2) + ƒ (x+2x') = 3. 3. Similarly, taking ax + bx3y + c x2 y2 + h x2 + k x2y +exy+fy2 + gx = 1. Multiplying by 4, priming three-fourths, etc., and we shall have the equation of the tangent: 4 a x13 x + b (3 x12 y' x + x13 y) + c (2 x' y12 x + 2 x'2 y' y) + h (3 x'2 x + x13) +k (2 x'y' x + x12 y + x2 y13) +e(xy' + x' y + 2x' y') +ƒ (2 y y'+2 y12) +g (x+3x') = 4. 4. Generally а ax" + bx"y"=1 in which n is not less than m+p. The equation of the tangent is an x-1x+b(mx!¬1yx+px'yp1y+(n−m—p)x'y')=n 5. Three things will be readily seen: (a) The close analogy to differentiation. (b) That constant terms come in through terms of less than the nth de gree. (c) Hence, from the last remark, it is evident that if the given equation be strictly homogeneous (without an absolute term), the equation to its tangent will have no absolute term. The tangent will, therefore, always pass through the origin, and the original equation represents either a point (the origin); a system of straight lines intersective at the origin; or an imaginary locus. 6. It is easily seen that the methods above given would derive equations of tangent planes from equations of surfaces. A NEW DEMONSTRATION OF CAYLEY'S THEOREM ON THE INTERSECTIONS OF By H. B. FINE, Princeton, N. J. [ABSTRACT.1 CURVES. ATTENTION called to the inadequacy of proofs of algebraic and geometric theorems by method of enumeration of constants: especially as exemplified by this theorem in the consideration and statement of which Cayley overlooked an important class of exceptions. The corrected theorem stated and demonstrated by use of the RiemannRoch theorem, the demonstration differing formally though not essentially from Bacharach's (Math. Ann. XXVI). 0 is cut by another p have any of the values 1, 2, passes through m n — ' of the points of intersection of fn= 0 and &m= :0 must pass through the remainingμ-1.μ-2 points also, UNLESS THESE LIE ON A CURVE OF THE ORDER -3. 2 The theorem is then extended to the case of curves, adjoint and nonadjoint, intersecting fn in singular points. ANY POINT OR PLANE (TANGENTIAL) SINGULARITY IN AN ELEMENT OF A CURVE OF DOUBLE CURVATURE CAN BE COMPLETELY DEFINED BY AID OF THREE SINGULARITY INDICES K1, K2, K3. By H. B. FINE, Princeton, N. J. [ABSTRACT.] THE above is one of the central theorems of a paper of mine which appeared in the American Journal of Mathematics, Vol. VIII, No. 2. I have here given a much simpler proof and one which admits of simple extension to curves of any curvature whatsoever. Indeed, the demonstration given formally contemplates this most general case. The instrument used is the Grassmann Ausdehnungslehre Analysis. PHOTOGRAPHIC DETERMINATIONS OF STELLAR POSITIONS. By DR. B. A. GOULD, Cambridge, Mass. IT has been suggested that a short account of my work upon stellar photographs for the attainment of accurate observations might be acceptable to the Astronomical Section. My intention had been to attend this meeting as a listener and learner only; but I comply with the suggestion the more readily, since by a notable coincidence I spoke upon the same subject in this place just twenty years ago, this week. It is true that my communication then was but an oral one and never reduced to writing; for the successful establishment of the Atlantic cable, of which I had just received notice, called me away suddenly, before the time fixed for the regular presentation; but an elaborate written memoir upon the subject had been presented to the National Academy, ten days previous, at Northampton. The early history of celestial photography is demonstrably and exclusively American; and its use as a method of delicate quantitative research is very markedly so. Without entering upon the historical data, which are of easy access to every investigator, I may mention that No. 77 of the Astronomical Journal contained nineteen photographic impressions, of as many different phases, of the solar eclipse of 1854 May 26,- the moment of each impression being given to the nearest tenth of a second. These were taken at West Point under the direction of Professor Bartlett of the U. S. Military Academy, and form a part of his memoir, in which he also gives the distances between the cusps as measured by himself with the micrometer in the telescope. Ten years later, in 1864, Mr. Rutherfurd constructed the 114-inch photographic object-glass which has acquired so conspicuous a place in astronomical history; and with this, in addition to its other achievements, he obtained sharp photographic stellar images with a definition previously unknown, taking for the first time distinct impressions of stars invisible to the naked eye, in fact to the 8 magnitude for white stars. After constructing a micrometer of great delicacy, for the measurement of these plates, he measured with this the relative distances and positionangles of the stars which they contained; and in the spring of 1866 he kindly placed in my hands the results thus derived from three plates of the Pleiades, each containing two impressions, taken on the evening of March 10. One of these plates contained forty stars. Bessel's memoirs upon the Pleindes, published in 1844, gave the relative positions of fifty-four stars, measured with the Königsberg heliometer during the years 1829 to 1841. Six of these fifty-four do not belong within the limits of the plate (which contains about one square degree) and ten of them are too faint for the photographic record, so that sixteen of Bessel's list are wanting; but on the other hand there are two additional ones, not observed by him. From this fact alone it may be perceived that, ainong the great benefits which astronomy may be justified in expecting from celestial photography, the accurate determination of magnitudes does not find place. The chemical action of the stellar light upon the film is so dependent upon the character of that light, that, in the absence of a correct knowledge of its composition, we are very easily deceived regarding its amount. Thus one of Bessel's stars which was not recorded upon any of Mr. Rutherfurd's plates is estimated by Argelander as of the magnitude 8.0, and by Wolf as 7; while five are distinctly recorded which Argelander calls 84 or less, and eight which Wolf so estimates. The spectroscope would doubtless show a deficiency of the more refrangible rays in the light of the former, and a preponderance of the same in that of the latter. This series of measurements by Mr. Rutherfurd, together with the computations to which the results were submitted, constitutes if I am not mistaken, the first application of the photographic method to exact astronomical determinations. And the investigation necessarily demanded especial care, both for guarding the numerical results against sources of unsuspected error, and for fixing the limits within which known theoretical errors would remain unappreciable. The importance of the successful application of a method so different from all previous ones, and so full of promise, and also the considerable time which would inevitably elapse before the memoir could be printed, led me at the same time to communicate to the Astronomische Nachrichten, at Altona, some of the resultant values. In a comparatively short note, written about the middle of August, 1866, I gave for the ten most conspicuous stars of the Pleiades, after Alcyone, the corrections, derived from one of the photographic plates of March 10, for the values published by Bessel, of the position-angles and distances from Alcyone in 1840; as likewise the average discordance found for a single measure. In the next following year, the Academy had not the means of printing its memoirs; and, as in the meanwhile, Mr. Rutherfurd had measured five more of the plates of the Pleiades previously taken, as well as six additional ones taken in the months of January and February 1867, these were also computed, and the results added to those from the first three plates in the memoir already written. Various circumstances combined to delay the publication, chief among them being what seemed to me a manifest impropriety in printing the results derived from photographs and measurements made by Mr. Rutherfurd and by his own methods, before some account of these methods should have been published by him. His communication on the subject had been made to the National Academy immediately previous to my own, but was not yet in such form as he desired for publication. The result showed a very remarkable accordance with Bessel's determination for 1840, although the total amount of relative proper-motion during the elapsed twenty-six years was comprised in the differences. This memoir still remains in its original form, but unpublished; the results being deduced from twenty-four photographic impressions, upon fourteen plates. In the next year, 1868, I had the gratification of receiving from Mr. Rutherfurd, the results of his measurements of thirty-two stars of the cluster Præsepe, derived from eleven impressions. These were computed in the same way that those of the Pleiades had been, and an analogous memoir upon this cluster was prepared for the National Academy. Before leaving the country, early in 1870, I gave these two memoirs to Mr. Rutherfurd, with the request that he would send them to the printer at the same time with his own paper, already mentioned, but not before. The condition of his health prevented him from attending to the matter for some time; and in the interval he arrived at the unpleasant discovery that the screw of his micrometer had suffered from wear, and to an extent which led him to fear a want of that accuracy of which the method is susceptible, and which he hoped to see demonstrated by its very first applications. Notwithstanding this possible blemish, it seems to me that the results ought to be now made public in their original form, after due mention of the circumstances; and it is among my hopes to be able soon to publish these two memoirs from the original manuscripts of so many years ago. The method was received with manifest distrust and disregard abroad; and, as was but natural for so essential a deviation from former methods, very many grounds of criticism and objection were brought up. One of the principal of these was the possible distortion of the collodion film after receiving the impressions and before the measurements; but Mr. Rutherfurd speedily disposed of this point, at least so far as the albuminized plates are concerned; and moreover the combination of measurements of the same stars, derived from various plates, will at once make manifest the degree of confidence to which the several values and their mean are respectively entitled. A far more serious obstacle to accuracy is presented by the difficulty of obtaining absolutely round images. Irregularity of form in the dots formed by the stellar impressions is almost incompatible with precision of measurement; and, as the time of exposure must often be long, the chief problem was not so much to obtain the images, as to insure uniformity of motion in the telescope during the period of exposure. Not that the photographic processes were not troublesome enough, before the introduction of the dry-plate processes,-for very great care and numerous precautions were often necessary to prevent the plates from drying too fast; but far the greatest difficulty consisted in obtaining sufficient precision in the clockwork and equatorial motion of the telescope. It may easily be imagined how great was my desire, when leaving home for South America, to extend this new method of observation to the southern hemisphere. But the obstacles encountered in the endeavor cannot be easily imagined. Upon these I will not enlarge here, farther than by saying that in Cordoba also the attainment of circular dots for the star-images offered incomparably the greatest of all the difficulties of a practical character. The time of exposure was limited by the maximum size allowable for the large stars; and, previous to 1878, also by the drying of the plate, although exposures for twenty minutes were not unusual. Nevertheless, by dint of specially constructed governors and regulators, and by ceaseless attention, we did succeed in obtaining impressions which to the unaided eye appear absolutely round. This necessity of long-continued and minute uniformity in the motion of the telescope is, of course, largely diminished by the employment of instruments of large aperture, inasmuch as the necessary time of exposure is diminished in the same ratio in which the amount of light is increased. It is yet further and most notably diminished by the manifold greater sensitiveness of the dry gelatine plates. Yet, notwithstanding all this, the attainment of round images, while almost indispensable for giving to stellar photography that increased accuracy, to which it may lay claim as a means of research in practical astronomy, still demands especial care and precaution. The Argentine Government cordially afforded every assistance which I deemed it proper to ask, for these investigations. And, although the chief energies of the Cordoba Observatory were absorbed by those investigations for which the institution was established, I had the satisfaction of obtaining a sufficient number of stellar photographs to occupy not only my own life-time, but many more, in their measurement and proper computation. |