in which the coördinates and velocities lie severally between the following limits, viz., between

L dx dy, dz, dx, etc. dx dy, dz, dx, etc.

The manner in which the quantity L varies with the time is given by the equation

where t, 1, 1, 21, 22, etc., 21, y1, 21, x2, etc., are the independent variables, and the summation relates to all the coördinates.

The object of the paper is to establish this proposition (which is not claimed as new, but which has hardly received the recognition which it deserves) and to show its applications to astronomy and thermodynamics.

LINEAR FUNCTIONS OF POINTS, LINES AND PLANES.

W. HYDE, Cincinnati, Ohio.

[ABSTRACT.]

By Prof. E.

Ir is shown in this paper that, if p, L and P represent respectively a variable point, line and plane, and if y be the symbol of a linear function; then cp, L, P', have geometrical applications and uses, whose relation to trilinear, quadriplanar and tangential coördinates is analogous to that between Hamilton's geometrical development of gp and ordinary Cartesian coördinates, the elegance and conciseness of the processes being as great in the one case as in the other.

COLORS OF THE STARS. By Prof. E. C. PICKERING, Director of Harvard College Observatory, Cambridge, Mass.

[ABSTRACT]

OBSERVATIONS of the colors of about four hundred of the brightest stars are now being made at the IIarvard College Observatory. The light is analyzed by a direct vision prism, and the intensity of rays of various wave-lengths is measured by a wedge of shade glass.

ON THE COLORS OF VARIABLE STARS. By S. C. CHANDLER, Jr., Harvard College Observatory, Cambridge, Mass.

[ABSTRACT]

In this paper the author gives the results of an extensive series of observations upon the colors of telescopic variable stars, made by him with the 61 inch Clacey equatorial at the Harvard College Observatory in 1883 and 1884, and also the inferences drawn from a discussion of them.

The observations are of two kinds: First, estimates by the decimal scale method practised by Schmidt and others; second, by a new method of the author's, by which the redness of a star's light is measured by the relative diminution of its brilliancy compared with that of a colorless star, as seen through colored glass shades. The discussion shows that the relation pointed out by the author several years ago, and independently by Schmidt, between the depth of color tint and the length of the periods of variable stars is strikingly confirmed by both series of observations here given and he regards the existence of this relation as thus demonstrated. This result may be considered as of the utmost importance in its bearing on the theory of the causes of stellar variability.

ON THE ORIGINAL GRADUATION OF THE HARVARD COLLEGE MERIDIAN CIRCLE in situ. By Prof. W. A. ROGERS, Harvard College Observatory, and GEO. B. CLARK, Cambridge, Mass. In the ordinary method of discussing the errors of graduation in a meridian circle, it is assumed that the graduated arc retains its original form throughout the revolution of the telescope. It is certain that this constancy of form is not maintained in the Harvard College meridian circle and it is probable that in all instruments of this class, with the possible exception of the later instruments of Repsold, more or less irregular flexure of the circles occurs. In so far as the bending is symmetrically distributed, an elimination of the error thus produced is effected, but any change of form at unknown points in the arc of revolution will not only tend to vitiate any system of errors derived from observed measures of given subdivisions, but may actually introduce the very class of errors which it is the aim to eliminate.

It is therefore a matter of supreme importance that some method of investigating the errors of a meridian circle shall be devised which does not depend upon unknown variations in the form of the circle whose errors are desired. This necessity is emphasized by the outstanding systematic differences which, at the present time, exist between the observations in declination made at the principal observatories of the world. The maximum amount of this difference is about five times as great as the probable error of a single observation. That is, assuming, e. g., that the declination of a star at 20° south, as observed at Greenwich, is correct, the observation of the same star at Pulkowa may have an error at least five times as great as a skilful observer ought to make in a single observation, on the supposition that his result is affected by accidental errors of observation only. It is doubtless true that the larger share of these outstanding differences is due to errors in the refraction tables employed for low altitudes, but a certain portion are certainly due to errors of graduation augmented by those errors of flexure and eccentricity of pivots which are not eliminated by the employment of four microscopes. Until each of these errors has been determined it is useless to attempt the determination of the errors of the refraction tables. In the present paper, an attempt has been made to refer the subdivisions of the graduated

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 de tails 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.