necessities of the arts. In order to represent as closely as possible a curved figure by a series of polygons, Tchebitcheff takes six terms of the integral formula corresponding to six ordinates selected in the following manner: Let the surface be inclosed by the curved line A B C D, and the straight line A D; subdivide A D at E. The one half of A E, multiplied by the measured values of its ordinate, ist then to be set off on either side of E, thus marking the places where new ordinates are to be measured, which are themselves to be multiplied by one half of A E; the products again set off on either side of E, and then a third pair of ordinates measured. In this way three pairs, or six ordinates, are obtained, whose values have a certain relation to each other and to the given curved line. The desired area is found by multiplying one sixth of the sum of these ordinates by the length of the line A E D. Other methods generally give results somewhat less than the truth. The method of Tchebitcheff generally gives larger results than the others. "Mittheilungen" Austrian Hydrogr. Office, 1874, p. 530.

ASTRONOMICAL WORK AT CORDOBA.

In his annual report, as Director of the National Observatory of the Argentine Republic, for the year 1874, Dr. Gould states that the three principal undertakings of that observatory, viz., the uranometry, the zones, and the smaller catalogue of stars, have satisfactorily advanced toward their completion. An inevitable delay having occurred in the publication of the first mentioned of these works, the opportunity was seized to revise some portions of it—a revision which indicates that the accuracy attained is quite commensurable with Gould's original hopes and expectations. Having secured the necessary funds, it is now expected that in the course of the present year the publication of the charts will be completed. These will be thirteen in number, comprising the whole of the southern heavens. The total number of stars whose positions and magnitudes will be given will be not far from 8500. With reference to the zones of stars, he reports that some 12,500 additional observations have been made, bringing the total number up to 82,537. It is not improbable that the number of observations yet to be made will swell the total to more than 100,000; which work he

then (March, 1875) hoped would be completed by the end of July, 1875. The greatest hinderance to the prosecution of this undertaking consists in the difficulty of securing the services of an adequate number of trained astronomical computers.

Of the large number of stars observed in these zones, a small portion have been selected as fit to form a special catalogue of brighter stars. This catalogue includes nearly 5000 stars, and some 12,400 observations upon these were made during the year. Dr. Gould adds that not one hour of unclouded sky between sunset and midnight was lost by his assistants during the whole time of his recent visit to the United States, notwithstanding that other observations were also going on by night, and continual computations by day. The equatorial telescope has been as busily employed as the meridian circle. Coggia's comet was observed from the 27th of July to the 18th of October. Standard Cordoba time has been given regularly from the observatory without a single case of failure; and latterly the exact Buenos Ayres time has been telegraphically transmitted to that city for the convenience of the shipping. Meteorological observations have been conducted and reported regularly to the Meteorological Office. Dr. Gould's corps of assistants has consisted of four persons, with occasional aid from others competent to act as copyists and computers. The assiduity of the labors of all concerned is abundantly testified to by the record of their results.-Annual Report, March, 1875.

PROPERTIES OF THE TETRÆDRON,

In an exhaustive memoir by Dostor on the application of determinants to certain problems in solid geometry, we find the following theorems relating to tetrædrons: The sine of a triedral angle is equal to the product of the sines of two of its faces, multiplied by the sine of the inclosed diedral angle. Again, in every tetrædron each face is equal to the sum of the products which we obtain by multiplying each of the three other faces by the cosine of its inclination to the first face. And, again, in every tetrædron the faces are to each other in the same proportion as the sines of the supplements of the opposite triedral angles. The volume of the tetrædron is equal to one sixth of the product of three

contiguous edges multiplied by the sine of the triedral angles formed by these edges. Its volume may also be expressed as equivalent to multiplying one half of the product of two opposed edges by the sine of the angle comprised between them, and by one third of the shortest distance between these faces. In the regular tetrædron, the radius of the circumscribed sphere is triple the radius of the inscribed sphere.—Grunert's Archiv, LVII., 113.

ORBIT OF THE DOUBLE STAR 42, COME BERENICES. The star 42, Coma Berenices, was discovered to be double in 1826 by the elder Struve, but it appeared single in 1833, since which time it has been observed regularly either by the discoverer or by his son, Otto Struve, as well as by other astronomers. Since 1826 it has four times presented the appearance of a single star, one of the bodies being actually occulted by the other. The very accurate observations of Otto Struve made since 1840, after having been corrected for the personal errors peculiar to his observations, and which have been most carefully investigated by himself, have sufficed to enable him to determine with very considerable accuracy the position and apparent dimensions of the relative orbits of these stars. The plane of their orbits coincides so nearly with the line joining them to the sun, that we can not certainly state that there is any appreciable inclination between the two. We have therefore to adopt 0° as the inclination between the line of sight and the orbit of the stars, and there results 11°, or the mean of all observed directions, as the angle between the ascending node and the declination circle. The remaining elements of the orbit of the stars, viz., the mean annual motion, the eccentricity, the major axis, the time of passage through the periaster and the angle in the orbit between the periaster and the ascending node, must all be deduced from micrometric meas, ures of the relative distances of two stars. Observations of this nature are proverbially so difficult that up to this time astronomers have avoided employing them when position angles could be used instead. The great accuracy of Struve's micrometric observations, however, is fully illustrated by the remarkable agreement between the observed distance and those computed in accordance with the numerical values

found by Struve for the time of revolution and eccentricity, and the other elements of its orbit. Of the thirty-eight positions given from 1827 to 1874, only two cases occur in which the discordances amount to one tenth of a second of arc; and these causes, it is promised, will be, at least in part, explained away in a forthcoming memoir relating to the peculiar systematic errors that attach themselves to the observations made by Otto Struve in 1840-41. Of the remaining thirty-six discordances, eight slightly exceeded one twentieth of a second of arc. The remainder are less than that quantity. The probable error of a single observed distance or the result of a single night's work is 0.046. Observations of these stars made by other astronomers agree satisfactorily with the orbit determined by Otto Struve, although the average of the discordances is somewhat larger in their observations than in his own.-Notices of the Royal Astronomical Society, May, 1875, 372.

METHOD OF CONSTRUCTING CHARTS OF STARS.

In constructing the new charts of the stars in the neighborhood of the ecliptic, the French astronomers, under the general direction of Le Verrier, have adopted some novel and excellent methods. The brothers Paul and Prosper Henry, in that portion of the work which they have performed, have made use of two equatorials, having apertures of about nine inches, and by a duplicate examination of each portion of the heavens have been able to discover many small planets and comets. The great equatorial of the observatory has been furnished with a micrometer of special construction, in which advantage is taken of the precision with which the telescope is made, by means of the regulator of Foucault, to follow the diurnal movements of the stars. This micrometer gives immediately the co-ordinates of any star comprised in the field of view of the telescope, with reference to a given standard point, and that in such shape. that these figures may be entered directly upon the chart. This micrometer is also now being applied to the mapping of the individual stars in some of the clusters. The accuracy of the work done with this instrument is such that the star places given upon the charts are reliable within a second of time and one tenth of a minute of arc: a result some

what surprising when we consider the extent of the work and the rapidity with which it is done. A portion of the zones will cross the Milky Way, and it will be attempted to give the position of every star visible in this region with the help of telescopes of ten inches' aperture.-Bullet. Hebdoma daire, 1875, 335.

ON THE RECTILINEAR RELATIVE MOTION OF THE COMPONENTS OF THE STAR 61 CYGNI.

Mr. Wilson has examined the relative motion of the components of the double star 61 Cygni, with the intention of ascertaining how far recent measures confirm Struve's conclusions that this motion is rectilinear. If these stars were physically connected in a binary system, it would be highly improbable that their apparent motions as seen from the earth would be sensibly straight lines. And yet, during the past century, the observations, which have been numerous, show that their motions really are so. On the other hand, the fact that they both have very large proper motions, being respectively 517 and 509 seconds per century, and in the same directions, leads to the conclusion that in all probability there must be some connection between them. We have thus the remarkable phenomena of two stars close together, animated by an unusually great proper motion, yet whose physical connection is still in doubt. Mr. Wilson's studies upon this subject seem not to contribute any thing toward a solution of our present doubts. He is merely able to confirm the fact that all known observations may be suf ficiently well explained by the assumption that the two stars are moving in straight lines.-Notices of the Royal Astronomical Society, April, 1875, 324.

THE TRIPLE STAR ZETA CANCRI.

The triple star Zeta Cancri has for many years formed an object of study on the part of Otto Struve, who has recently published an excellent memoir on the relative movements of its components. The first observations of this star were by Tobias Mayer in 1756, who recognized it as double, and determined the relative position of the components. Similar observations were made in 1778 by Christian Mayer. Sir William Herschel, in 1781, made the interesting discovery