through the discrepancies of clocks and watches is very considerable, and is directly felt by each individual in the missing of appointments or the needless loss of time in waiting. On very many accounts the country throughout the whole region east of the Rocky Mountains would be benefited by the introduction of some uniform standard of time which should replace the innumerable and often erroneous "local times," and by which not only railroad, telegraph, and stock business might be managed, but which should be adopted also in governmental and in private matters.-Description of the City Hall, Pittsburgh.

PROPERTIES OF PRIME NUMBERS..

As the conclusion of an investigation by Goering into the "Theta" functions of Jacobi, and as an application of his results, the author shows that every prime number of the form 6m+1 is always divisible, although only in one special way, into the sum of a simple and a triple square; and, again, that the product of n prime numbers of the form 6m+1 can always be considered as the sum of a simple and a triple square.- Goering, Inaugural Dissertation, 1874, p. 382.

APPLICATIONS OF PEAUCELLIER CELLS.

Mr. Darwin has given an account of some applications of what are now familiarly known as Peaucellier cells. Among other things he illustrates the fact that it might become possible to construct by means of these a model that shall give an ocular and correct proof of the elliptic motion of the planets about the sun, under the influence of the force varying inversely as the square of the distance in that fixed point. Mr. Sylvester states that he himself had attempted the same problem, but failed.

HAMILTON'S EQUATION OF MOTION.

A decided advance in the principles of theoretical mechanics seems to have been made by Professor Müller, of Zurich, who has developed certain considerations based upon what is known as Sir William Hamilton's general equation of motion. That distinguished mathematician has shown that when a system of material points moves under the influence of forces proceeding from the reciprocal attraction and re

pulsion of the points of the system, all the integral equations of the motions can be represented by the partial differential quotients of a certain function, called the Primary Function, of their co-ordinates in a manner similar to that in which, according to La Grange, the differential equations of the motions can be represented by the partial differential quotients of a function known as La Grange's function of the forces. The primary function of Sir William Hamilton is a complete solution of the partial differential equations of La Grange's function, as was shown by Jacobi. The integration of this differential equation was developed by Jacobi, since whose time the theory has undergone expansion in two respects, by Zipschitz and Schering, to whose researches Müller adds the following propositions: First, the sum of such changes in the primary function and in the expenditure of force as may be produced by the variations of the initial and final co-ordinates alone, is, in the variation of every motion that presup poses a force function, and neither explicitly nor implicitly contains the time, equal to zero. This proposition he designates as "The principle of Energy." Correlated to the preceding is Müller's second proposition, which he calls "The principle of Action," which may be enunciated as follows: That change of the action which is conditioned by the variation of the initial and final co-ordinates alone vanishes with the change of every motion that presupposes a force function, and does not contain the time either explicitly or implicitly. Here, as in the previous proposition, if we imagine the whole series of constantly altered motions to be run through with, they will in general be distinguished by different values of potential and kinetic force and energy; in proportion as by the mere alteration of the co-ordinates the potential diminishes, so does the kinetic increase. These propositions, which are represented by Müller in algebraic language, are exemplified by several applications. Applying the first proposition to a simple case, he by it develops the motion of the or dinary pendulum; but his most interesting results relate to the theory of heat. If according to the mechanical theory heat be considered as molecular motion, the application to this hypothesis of Müller's "Principle of Energy" leads immediately to the well-known first law of thermo-dynamics; while, if we apply to these molecular motions the theorem of

action, we arrive at a well-known equation already demonstrated by Clausius, and equivalent to the so-called second law of the mechanical theory of heat. We are thus able to derive these important laws from the original principle of Sir William Hamilton's theory of motion, and his general equation thus becomes the connecting band for the two propositions of the mechanical theory of heat.-7 A, XLVIII., 274.

ON THE SOLUTION OF NUMERICAL EQUATIONS.

A remarkable theorem relative to the solution of numerical equations whose roots are real is given by La Guerre. He first shows how to draw a certain curve having certain relations to the equation to be solved, and then demonstrates that if from any point whatever of this curve we draw two lines at right angles to each other, the two points where these lines cut the axis correspond to the desired roots.-3 B, XXXV., 457.

THE DENSITY OF THE LUMINIFEROUS ETHER.

In a paper on the heat of bodies, Puschel, of Vienna, attempts to explain this property as consisting mainly in a motion of ether identical with the luminiferous ether; and concludes that we may as the lower limit of the density of this substance consider that it must be more than one twentysixth billionth of the density of water.-12 A, X., 278.

A FINE DOUBLE STAR.

In a recent number of the monthly notices of the Royal Astronomical Society, Mr. Burnham, of Chicago, gives an account of the discovery of the duplicity of Nu Scorpit, which is an interesting illustration of the steady progress made in detecting new double stars. As the case now stands, the star in question is quadruple. It was, however, known to Herschel in the last century simply as a double star, whose components appeared single in his own, his son's, and all other large telescopes, up to the year 1847, in which year Jacob, at Madras, found that the fainter or companion star was itself double. In 1873, with his beautiful six-inch telescope by Alvan Clark, and favored by his own remarkably acute vision, Mr. Burnham writes that he had examined the star several times, and was impressed by an apparent elonga

tion of the principal star in a direction nearly north and south. Professor Young, of Dartmouth College, was requested to examine it with his splendid refractor, and reported that he suspected that it was double, but could not be certain. During the summer of 1874, Mr. Burnham with his sixinch telescope, Mr. Newcomb with his great twenty-six-inch refractor at Washington, and Baron Dembowski, at Florence, with a nine-inch telescope, all nearly simultaneously were able to see that the principal star was double, and to measure the relative positions. We have, therefore, in this case a star which to the naked eye appears of the fourth magnitude, resolved by fine telescopes and sharp eyes into four stars, of the fourth, sixth, seventh, and eighth magnitudes respectively. The last-named and most distinguished observer of double stars says that "this is one of the finest multiple stars known." There are others of the same kind, but none presenting the same striking assemblage of brilliant objects within such narrow bounds.-Burnham on Nu Scorpii.

HERSCHEL'S CATALOGUE OF DOUBLE STARS.

It is well known to astronomers that Sir John Herschel in his later years engaged himself in collecting, arranging, and revising the previous literary and scientific labors of his life. His general catalogue of all nebula discovered up to 1863 was published in the Transactions of the Royal Society of London for the following year. His arrangement of all the double stars observed by his father, Sir William Herschel, was published by the Royal Astronomical Society. The last great work undertaken by him was that of collecting in one catalogue all the trustworthy observations of multiple and double stars which had been recorded up to the date of the undertaking. This catalogue, containing over 10,000 stars, together with a synoptical history of all the known observations of about two fifths of them, was completed at the time of the death of Sir John Herschel. It was bequeathed to the Royal Astronomical Society, at whose expense it has been recently published. This important work will be welcomed heartily by those astronomers and amateurs interested in double-star observations. It unfortunately does not contain. any indication of the magnitudes and distances of the double stars of which it treats, but, by giving the positions in right

ascension and north polar distance of every known double star, it becomes a valuable aid to those who may be searching for new ones, or to those who wish to add to our present knowledge of these interesting subjects of observation.Mem. of Roy. Astr. Soc., XL.

ORBIT OF A DOUBLE STAR.

The double star, 70, Ophiuchi, which consists of a bright yellow star of the 44 magnitude, and a rose-colored star of the sixth magnitude, was first observed by Sir William Herschel in 1779, and has since formed a favorite subject of observation for observers in both hemispheres. Some computations based on these observations have lately been made by Flammarion, in order to determine the apparent orbit and, if possible, the true orbit of this sidereal system. Flammarion's results are practically identical with those of Klinkerfues, as deduced a number of years ago. Flammarion, assuming the parallax as determined by Krüger, concludes the distance of these stars from the earth to be 1,400,000 times that of the sun, and the actual distance of the two stars from each other to be somewhat less than the distance of Neptune from the sun. The relative movement of the stars is, according to Klein, 1.65 that of Neptune and the sun. The two stars have, however, a common movement through space, which is three and a half times as great as their orbital velocities about each other.-19 C, VIII., 46.

THE ORBIT OF THE DOUBLE STAR "MU BOOTIS." Among the theses published by the University of Kasan, in Russia, is an investigation into the orbit of the double star Mu Bootis, by Venogradski. Observations of this star have been made since 1782, when it was first observed by the elder Herschel; and its orbit has been investigated once previously by Wilson, but the computations of Venogradski take precedence, inasmuch as he has had access to very accurate and long-continued observations of Otto Struve and Dembowski. During the past ninety years the smaller star has described nearly one half of its orbit about the larger one; and the mutual distance has diminished from one and a half seconds to less than half a second. According to the present computation, the periodic time of these stars is about