(with respect to a configuration in the sky) from the position. for a western elongation. A simple means for obviating any ill effects from this cause, is to mount the object-glass in such a way that it may be revolved in its own plane, around the axis of the telescope tube. On the latter two stops may be placed so that the corresponding positions of the object-glass shall differ by 180°. Throughout the whole series of observations the observer must be careful to reverse the object-glass whenever he reverses the telescope. In this way he will be enabled to present the objective in the same relative position to a configuration, no matter whether in the east or in the west. Optical distortion will then shift all the images of the same star alike, and will not affect the parallax.

The last source of error which will be considered here has its origin in the construction of the measuring machine. It is hardly necessary to point out that all the measures should be differential; thus, the distance between a certain pair of stars on any plate should always be measured with the same parts of the scale and of the micrometer-screw. A simple arrangement of the plan of measuring is possible, by which the parallax is rendered practically independent of most of the instrumental errors.

Other precautions are necessary in parallax work, but as they are not peculiar to photographs they need not be discussed here.

To sum up, the whole process is as follows: put the unexposed plate into a properly constructed template and stain a narrow central strip to a degree previously determined by experiment for plates of the same manufacture. Permit the plate to dry for several weeks at least. Then insert it in the plateholder of the telescope, in such a way that the stained strip shall be parallel with an hour-circle. Make two pictures of the same region, shifting the plate a millimeter in declination between the two exposures. Store the plate in a dark room, undeveloped. Six months later, with the object-glass reversed 180° if the telescope mounting demands it, take two more pictures on the plate. These should be 3 millimeters apart, and each I millimeter from one of the former pictures. Begin a new

plate by making two exposures 1 millimeter apart, and put this by for six months; and so on till a sufficiently long chain of plates is secured to give good values both for the parallax and for relative proper motions, with respect to the comparison stars.

I may remark in conclusion that if we confine ourselves to the measurement of distances only, much simplification is possible in the measuring machine, and some in the reductions. According to a conservative estimate, a single observer, working fifteen or eighteen hours per week at the telescope, and employing the rest of his time in measuring and reducing, could give us in three or four years the parallaxes of 200 stars with an accuracy hitherto attained for only a score.

UKIAH, CAL.

August 29, 1899.

THE DISTRIBUTION OF STARS IN THE CLUSTER

MESSIER 13, IN HERCULES.

By H. K. PALMER.

In all drawings of the great cluster in Hercules which were made prior to the introduction of photographie processes into astronomy, no attempt is made to plat accurately the places of individual stars, the observers having contented themselves with representing the general appearance of the cluster. Accurate charting, and a study of the distribution of the stars in such objects, first became possible with the aid of the photographic plate. Excellent photographs of the Hercules cluster have been made by the Henry brothers, Dr. Roberts, Mr. W. E. Wilson, and others, and with the 36-inch refractor of the Lick Observatory. Some of these have been reproduced in popular journals. Among the researches based on photographic methods is especially to be mentioned the elaborate investigation of Professor Scheiner, in which the positions of 833 stars are determined with all possible precision and arranged in the form of a catalogue.

In the course of the photographic work with the Crossley reflector, which is being carried on by Professor Keeler with my assistance, the Hercules cluster was photographed four times, in June and July of the present year, with exposures ranging from ten minutes to two hours. The large scale of the negatives, the great light-gathering power and fine definition of the mirror, the purity of the sky and steadiness of the images at the time of observation, all combined to make these photographs of unusual excellence. The plate to which the very full exposure of two hours was given is the best. It was very lightly developed; with the result that the images of the brighter stars did not become too large and dense to obscure the multitude of minute stars which was brought out by the long exposure, while even 1 Abh. d. K. Akad. d. W. Berlin, 1892.

the middle of the cluster is resolvable. The number of stars shown on this plate, fairly within the limits of the cluster, is over five thousand.

In view of the large number of stars shown on this photograph, an investigation of their distribution seemed to be desirable, and at Professor Keeler's suggestion I have undertaken this work in the present paper. An accurate determination of the places of all stars, such as was made by Scheiner, would also be very desirable, but it would require more time and labor than can at present be devoted to such a purpose.

To study the distribution of the stars in the cluster, a positive on glass, with an enlargement of 5.6 diameters, was made from the original negative. On the positive the cluster occupies the center of a rectangle which measures 192 mm east and west, and 241 mm north and south. The rings, which are referred to farther below, were traced in red ink on a clear plate of the same size as the enlargement, and the two plates were then bound together with their films in contact.

The focal length of the Crossley reflector is 17 feet 6.1 inches, so that on the original negative 1' 0.061104 inch, or 1.55 mm. On the enlargement 1'= 8.79 mm. The average star disk is 3.5"

in diameter, the brightest being about 4" and the faintest 3". The primary object of this examination was to compare the actual distribution of the stars with the distribution of stars in a uniform globular cluster; by which I mean a globular cluster in which the stars are uniformly distributed. To make this as simple as possible, the sphere was supposed to contain several concentric cylinders, whose volumes within its surface were in the ratio 1, 2, 3 n, and whose axis was parallel to the line of sight. Therefore, when projected on a plane perpendicular to the line of sight, the cluster would appear to be divided into several concentric rings, and if it were uniformly globular, each ring would contain an equal number of stars. The radii of these cylinders were obtained from the formula

π V (R2 —- r2 )3

in which R is the radius of the sphere, r the radius of the cylinder, and ʼn the ratio of the volume of the cylinder to that of the sphere. The limits of the cluster were very indefinite; so the radius of the sphere was assumed to be 80mm on the enlargement, which corresponded to an angular distance of 9.2'. This seemed to include all stars which were so arranged as to come within the limits of a globular structure, and at the same time excluded those which had only a radial arrangement. This circle of 80mm radius was then divided into eight rings, the radius of the different circles being as follows:

Since the stars outside the last circle forming the rays seemed to belong to the cluster, and as these rays extended to the edge of the plate, the dimensions of the plate have been included. To facilitate the counting, and to see if there was any particular radial structure, the circles were divided into twelve sectors of 30° each.

In a general way it may be said that the stars in the Hercules cluster are of two distinct orders of brightness; for although intermediate magnitudes occur, they are less numerous than we should expect them to be in a purely fortuitous assemblage of stars of different sizes.

In counting, the number of faint stars was kept separate from that of the bright stars. The term "bright" was applied to all stars which, in the positive, showed clear glass, while the "faint” stars were those which contained visible silver grains. A comparison of several stars, which were just on the dividing line,