circle to a constant unit which is independent of the subdivisions to be investigated. The success which has attended the use of the electro-magnet clamps in the dividing engine constructed for the writer at the Waltham watch factory, suggested the application of the same principle to the investigation of the errors of a meridian circle, and to the feasibility of the graduation of the circle in situ if this investigation should be successful. In this engine, there is an arm which at one end moves between two stops, of which one is movable, while the other end rests upon the cylindrical shoulder of the screw which is to receive equal increments of revolution. Two magnets are attached to this arm, the cores of which are fitted to the curvature of the index circle of the screw. A third magnet of similar construction is attached to the bed-plate of the machine, and independent of the arm. When the two upper magnets are in circuit, the arm becomes firmly attached to the index without the slightest disturbance of position, and the index is carried forward the required amount by moving the arms between the two stops. During the upward motion of the arm the lower magnet holds the index while the two upper magnets are free, thus allowing the arm to make contact with the upper stop in preparation for the next downward stroke. It has been found from experiment that under similar conditions as many as 5,000 movements of the arm will in repeated trials give the same arc of revolution. It did not, therefore, seem too much to expect that the same method might be successfully applied to the movement of a meridian circle over equal arcs of revolution under exactly the conditions which prevail in actual work with this instrument. It was therefore determined to try the method with the meridian circle of Harvard College Observatory. Professor Pickering kindly authorized the expense of the construction of the necessary apparatus, which was designed by Mr. Geo. B. Clark, of the firm of Alvan Clark & Sons, and which was made under his superintendence. A ring having an outside face of two inches was made in two halves and securely fastened to the axis of the telescope. The magnet arm was made in such a manner that the only connection with the ring was made by the contact of the cores of the magnets at the periphery of the ring. A very heavy bed-plate of iron was securely clamped to the marble pier in such a manner that the edge might be made perpendicular to the axis of rotation and at a distance of about five feet from the centre of the axis. The stops are heavy plates of iron with projecting oval surfaces of tempered steel which move along this table, and which are held in position by heavy clamping screws. They are arranged for a movement of the telescope over arcs varying between 0° and 30°. With the aid of the graduated circle of the telescope it is found easy to set the stops quickly and accurately by tapping the stop-plates with a light hammer. It was found that a bichromate battery of six cells was sufficient to clamp the magnet-arm securely to the ring. It will be at once understood that unless the ring upon which the magnet-arm rests is truly circular, the arm will rise and fall with the revolution of the telescope, thus giving rise to periodic errors proportional in amount to the deviation of the periphery of the ring from a true circle. The test of this circular form was made by means of a microscope attached to the iron bed-plate with which the movement of the arm vertically was observed and measured, a graduated polished metal plate being clamped to the arm for this examination. It was found that during one-half of the revolution of the telescope very little motion of the arm could be detected, but that during the remaining half the maximum rise of the arm amounted to about mm. As was to be expected the chief part of the disturbance occurred at those points at which the magnets passed the junction of the two halves of the ring. It does not seem advisable to encumber this paper with the details of the observations which were made with the ring in its original form. An attempt was made to compare the 30° divisions of the graduated circle by a reference to the fixed distance between the stops, four microscopes being read for each contact, with the expectation that the effect of the error in the form of the ring could be measured by means of a microscope of high power which should measure directly the accumulated error of the arc of revolution at the contact points for each arc of 30°. It will be seen that this expectation was not realized for the summed series of errors of the 30° points of the circle. From seven sets of observations extending from July 3 to July 15, the following relative errors were found, the polar point being taken for the origin. The following are the accumulated errors at contact points from measures with microscope: It will be seen that the general form of the two summed series is the same, but they widely differ in the maximum value. It became at once obvious from this preliminary investigation that it would be necessary to grind the circle, upon which the magnet arm rests, to an exactly circular form. This was very successfully accomplished by Mr. Clark in the following way: A slide rest carrying an emery wheel was firmly mounted nearly opposite the axis of the instrument. The motion to the emery wheel was given by means of an old-fashioned spinning wheel. The operation of grinding was conducted as follows: One assistant turned the spinning-wheel, Mr. Clark managed the slide rest, which governed the movement of the emery wheel, while I, at one end, with the aid of an assistant stationed at the other end, gave to the telescope a nearly uniform motion in revolution. When the operation of grinding was completed, it was found that a complete revolution of the telescope could be made without the slightest trace of disturbance in the position of the magnet-arm under a microscope having twelve times the magnifying power of microscopes attached to the telescope. Before proceeding to describe the new series of observations, it will be necessary to refer to the means employed to neutralize the momentum of the telescope produced by the shock of contact with the stops. It was found that, on account of the disturbance produced by contact made by a hand movement of the arm, there was a liability to a maximum error of about 3". This amount was sensibly reduced by inserting a piece of writing paper between the arm and the stop, and then completing the movement by withdrawing the paper. The next experiment was with an air-buffer attached to the stop-plates, with which the initial contact was made; but it was found that the movement against the air-spring for the remaining distance to the stop was not sufficiently uniform, although there was a decided improvement over results previously obtained. A water-buffer was then employed; the water from a cylinder attached to the stop-plates being forced by a weight-pressure through a cylinder of small diameter. With this buffer as a momentum arrester, it appears from a long series of observations for contact, that the probable error of a single contact is about 0·03′′ or about one-tenth of the ordinary value for a single observation with four microscopes. Eight sets of comparisons of the 30° divisions of the circle have already been made with the improved apparatus, with the following results: July 28, +0.22 -1.02 Aug. 1, -1.38 -1.16 66 о 30-60 60-90 90-100 120-150 150-180 +1.63 +1.43 +0.07 +1.47 +2.39 -0.15 +2.91 +2.49 +3.00 +2.34 -0.79 -0.16 +2.03 +1.35 -0.84 +0.60 -0.19 +0.97 +0.96 -0.94 +0.45 +0.59 +0.61 +0.37 -1.10 +0.69 -0.29 -1.16 +1.54 +1.46 -1.23 +1.02 -0.02 -0.72 +0.74 -0.73 -1.53 -0.85 -0.98 -0.06 -0.08 +0.40 +0.30 -0.13 +0.07 +0.26 Taking the means of the separate determinations of the 30° spaces we have: It is the intention to continue the examination of the subdivisions of the circle throughout the coming year. The subdivision of the 30° spaces will be made by setting the stops for that number of subdivisions, which can be safely made at one time without danger of the introduction of errors depending on the temperature. With two assistants, one to read the 4 microscopes, and one to record, the examination of the 30° divisions can be completed in about one hour and twenty minutes. If it shall be conclusively shown from an investigation of the present graduations of the circle that this method will give greater accuracy than former methods of investigation, it is the purpose to produce a new set of graduations just inside of the original graduations. In conclusion it may be said that an excellent test of the method will be had when the telescope is reversed and the errors of the other circle are determined. The errors of this circle have already been determined for each single degree by comparison with a graduated metal arc of 15° with which each 15° of the circle was compared. In the complete discussion which will follow, care will be taken to eliminate any systematic errors which may appertain to the form of the ring. THE AVERAGE ASTEROID ORBIT AND THE ASTEROID RING. By Prof. M. W. HARRINGTON, University of Michigan, Ann Arbor, Mich. [ABSTRACT.] THIS paper treats the asteroid system on the principles of averages; 230 orbits are employed. If a plane be conceived perpendicular to the ecliptic and passing through the vernal and autumnal equinox, and another through the solstices, they will intersect each |