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These tables give the four theta functions which form the numerators and denominators of the three elliptic functions. The calculations relating to these functions have been carried to ten decimal places, and the printed results will occupy about four hundred pages.-12 Å, X., 372. .
NEW FORMULA FOR DETERMINING THE ALTITUDE FROM BARO
METRIC OBSERVATIONS, M. St. Robert, of France, has published the concluding volume of his memoirs, among which we notice a new formula for determining the altitude for barometric observations. This formula embodies the results of Glaisher's balloon observations.
THE REDUCTION OF ELLIPTIC INTEGRALS. From a mathematical paper by Meissel, Professor in Kiel, we take the following thcorem, whose enunciation will be of interest to mathematicians. He states that in a great pumber of cases he has been able to represent the complete elliptic integral of the second order by means of algebraic formulæ, and demonstrates, in general, that the complete integral of the second order can be converted into a complete integral of the first order.---Archiv der Mathematik, LVI., 337.
THE TRISECTION OF AN ANGLE, The problem of the trisection of a circular arc has lately been solved by Dr. Hippauf in a simple manner by means of an auxiliary curve, which may be designated as the conchoid on a circular base. This circular conchoid is the locus of a series of points found by drawing through one extremity of the diameter of a circle a series of lines, and finding, upon each, that point which is at a distance from the circumference of the circle equal to the radius. Having described such a circular conchoid for the circle an arc of which we wish to trisect, we draw the chord belonging to the latter arc, and then through the origin of the conchoid a parallel chord; this latter is equal to the chord of the third part of the arc to be trisected. Three other methods of effecting this trisection are also given by Hippanf by the aid of the same curve; and many other curious properties are found by Professor Sidler, who has shown that this conchoid may also be
described as the locus of the feet of a series of perpendiculars let fall upon all possible tangents to a circle, from a point outside the circle, and at a distance from the centre thereof equal to its diameter. The conchoid is likewise casily described graphically by a point fastened to a given circle which rolls around a fixed circle, provided that the two circles have the same diameter, and that the point be fastened to the rolling circle at a distance from its centre equal to the diameter thereof.-Mitth. der Naturf. Gesell., Berne, 1873, 31.
STANDARD TIME IN PITTSBURGH. The question of the regular distribution throughout the community of standard uniform time has been well tested by Professor Langley, of Pittsburgh, who, during the past five years, has steadily extended the system of telegraphic connections between the astronomical observatory of that city and the railroads that centre therein. The magnificent new City Hall has in its turret a large tower clock, built by the Messrs. Howard of Boston, which by electrical connections is made to beat, second by second, in perfect unison with the standard clock at the observatory. A person at the latter building can, if necessary, even adjust the tower clock by telegraph, and can at any moment ascertain whether its indications are correct or not. The large bell of the tower is struck with the utmost accuracy at noon, and at every third hour throughout the day and night, and the public appreciation of the convenience and utility of the general system of absolutely accurate time is noticed in the universal comparison of watches daily at the stroke of noon. This ordinarily causes a movement so general and simultaneous throughout the city as on the one hand to amuse a stranger, and on the other hand to demonstrate how nervously anxious Americans are to secure the highest attainable accuracy in the timekeepers on which they depend for the regulation of private as well as public business. During nearly two years that the system has been in operation it is stated that there has not been any interruption from the failure of electric mechanism, and the utility of the system certainly more than justifies the expense which the city has been to in establishing this now recognized public necessity, which can not hereafter be dispensed with. In fact, the amount of time wasted 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 cau 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 presupposes 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 ordinary 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 thermodynamics; 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 twenty. sixth 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 Scorpië, 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 bis own remarkably acute vision, Mr. Burnham writes that he had examined the star several times, and was impressed by an apparent elonga